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Heat and moisture transport in unsaturated porous media

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Title:
Heat and moisture transport in unsaturated porous media a coupled model in terms of chemical potential
Creator:
Sullivan, Eric R
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
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English
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Subjects / Keywords:
Darcy's law ( lcsh )
Fourier analysis ( lcsh )
Porous materials -- Mathematical models ( lcsh )
Groundwater flow ( lcsh )
Darcy's law ( fast )
Fourier analysis ( fast )
Groundwater flow ( fast )
Porous materials -- Mathematical models ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Review:
Transport phenomena in porous media are commonplace in our daily lives. Examples and applications include heat and moisture transport in soils, baking and drying of food stuffs, curing of cement, and evaporation of fuels in wild fires. Of particular interest to this study are heat and moisture transport in unsaturated soils. Historically, mathematical models for these processes are derived by coupling classical Darcy's, Fourier's, and Fick's laws with volume averaged conservation of mass and energy and empirically based source and sink terms. Recent experimental and mathematical research has proposed modifications and suggested limitations in these classical equations. The primary goal of this thesis is to derive a thermodynamically consistent system of equations for heat and moisture transport in terms of the chemical potential that addresses some of these limitations. The physical processes of interest are primarily diffusive in nature and, for that reason, we focus on using the macroscale chemical potential to build and simplify the models. The resulting coupled system of nonlinear partial differential equations is solved numerically and validated against the classical equations and against experimental data. It will be shown that under a mixture theoretic framework, the classical Richards' equation for saturation is supplemented with gradients in temperature, relative humidity, and the time rate of change of saturation. Furthermore, it will be shown that restating the water vapor diffusion equation in terms of chemical potential eliminates the necessity for an empirically based fitting parameter.
Thesis:
Thesis (Ph. D.)--University of Colorado Denver. Applied mathematics
Bibliography:
Includes bibliographical references.
General Note:
Department of Mathematical and Statistical Sciences
Statement of Responsibility:
by Eric R. Sullivan.

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|University of Colorado Denver
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|Auraria Library
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862935185 ( OCLC )
ocn862935185

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Full Text
HEAT AND MOISTURE TRANSPORT IN UNSATURATED POROUS MEDIA:
A COUPLED MODEL IN TERMS OF CHEMICAL POTENTIAL
by
Eric R. Sullivan
B.S., Iowa State University, 1998
M.S., University of Colorado at Colorado Springs, 2007
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics
2013


This thesis for the Doctor of Philosophy degree by
Eric R. Sullivan
has been approved
by
Lynn Schreyer-Bennethum, Advisor
Julien Langou, Chair
Jan Mandel
Richard Naff
Kathleen Smits
Date: April 19, 2013
11


Sullivan, Eric R. (Ph.D., Applied Mathematics)
Heat and Moisture Transport in Unsaturated Porous Media:
A Coupled Model in Terms of Chemical Potential
Thesis directed by Associate Professor Lynn Schreyer-Bennethum
ABSTRACT
Transport phenomena in porous media are commonplace in our daily lives. Ex-
amples and applications include heat and moisture transport in soils, baking and
drying of food stuffs, curing of cement, and evaporation of fuels in wild fires. Of
particular interest to this study are heat and moisture transport in unsaturated soils.
Historically, mathematical models for these processes are derived by coupling clas-
sical Darcys, Fouriers, and Ficks laws with volume averaged conservation of mass
and energy and empirically based source and sink terms. Recent experimental and
mathematical research has proposed modifications and suggested limitations in these
classical equations. The primary goal of this thesis is to derive a thermodynami-
cally consistent system of equations for heat and moisture transport in terms of the
chemical potential that addresses some of these limitations. The physical processes
of interest are primarily diffusive in nature and, for that reason, we focus on using
the macroscale chemical potential to build and simplify the models. The resulting
coupled system of nonlinear partial differential equations is solved numerically and
validated against the classical equations and against experimental data. It will be
shown that under a mixture theoretic framework, the classical Richards equation for
saturation is supplemented with gradients in temperature, relative humidity, and the
time rate of change of saturation. Furthermore, it will be shown that restating the
water vapor diffusion equation in terms of chemical potential eliminates the necessity
for an empirically based fitting parameter.
m


The form and content of this abstract are approved. I recommend its publication.
Approved: Lynn Schreyer-Bennethum
IV


DEDICATION
To Johnanna
v


ACKNOWLEDGMENT
There are several people who have either directly or indirectly made this work
possible. First and foremost, I would like to extend many thanks to Lynn Schreyer-
Bennethum. Without her patience, support, amazing talents as a teacher, mathe-
matical and physical insights, and constant questions I would likely not have been
able to get involved so heavily in mathematical modeling. I would like to thank Kate
Smits for agreeing to join my committee so late and for sharing her expertise with
experimentation. I would also like to thank Jan Mandel, Julien Langou, and Rich
Naff for their unending support and patience as my committee members.
There are several people in the UCD community that I would like to thank. To
Keith Wojciechowski, who was one of the primary reasons that I started working
with Lynn. To Cannanut Chamsri, Tom Carson, and Mark Mueller, my research
group comrades, for putting up with my ramblings and never-ending slue of equations
during our weekly meetings. To Mark especially, for his thought provoking questions
and constant push to tie the math and physics together. To Jeff Larson and Henc
Bouwmeester, who have become life-long friends. I appreciated the comic relief, the
deep mathematical conversations, the computer help, and the fun that we had. Of
course, I cannot forget to thank the many other people who helped me to persevere
through graduate school: Jenny Diemunsch, Cathy Erbes, Piezhun Zhu, Brad Lowry,
Tim Morris, Samantha Graffeo, and many other.
Finally, I would like to thank my family and friends. Without my support network
this would have been much tougher. To my parents, Drew and Donna, who have
taught me what real hard work really is. To my siblings, Doug, Emily, and Julie,
for helping me get to this point in my life. To Emily, especially, who went through
grad school along with me. To my friends Fred Hollingworth, Jodie Collins, and
Erik Swanson who reminded me that it was equally important to head up into the
mountains and go climbing as it was to work. To my teacher friends Scott Strain,
Ginger Anderson, and Krista Bruckner who have helped to make me a better teacher
and better person. Lastly, to Johnanna. I couldnt have done this without you.
vi


TABLE OF CONTENTS
Figures ..................................................................... xi
Tables....................................................................... xv
Chapter
1. Introduction............................................................... 1
1.1 Previous Work....................................................... 1
1.2 Hybrid Mixture Theory and Thesis Goals.............................. 3
1.3 Thesis Outline...................................................... 5
2. Ficks Law and Microscale Advection Diffusion Models....................... 9
2.1 Comparison of Ficks Laws........................................... 9
2.2 Transient Diffusion Models......................................... 14
2.3 Conclusion......................................................... 17
3. Hybrid Mixture Theory..................................................... 19
3.1 The Averaging Procedure............................................ 19
3.1.1 The REV and Averaging........................................ 20
3.2 Macroscale Balance Laws............................................ 26
3.2.1 Macroscale Mass Balance...................................... 27
3.2.2 Macroscale Momentum Balance.................................. 30
3.2.3 Macroscale Energy Balance.................................... 31
3.3 The Entropy Inequality............................................. 32
3.3.1 A Brief Derivation of the Entropy Inequality................. 33
4. New Independent Variables and Exploitation of the Entropy Inequality . 36
4.1 A Choice of Independent Variables.................................. 37
4.1.1 The Expanded Entropy Inequality .......................... 41
4.2 Exploiting the Entropy Inequality.................................. 46
4.2.1 Results That Hold For All Time .............................. 46
4.2.1.1 Fluid Lagrange Multipliers............................. 47
vii


4.2.1.2 Solid Phase Identities................................. 47
4.2.2 Equilibrium Results.......................................... 50
4.2.2.1 Fluid Stress Tensor.................................... 50
4.2.2.2 Momentum Transfer Between Phases....................... 51
4.2.2.3 Momentum Transfer Between Species...................... 52
4.2.2.4 Partial Heat Flux...................................... 53
4.2.3 Near Equilibrium Results.................................... 53
4.3 Pressures in Multiphase Porous Media............................ 55
4.4 Chemical Potential in Multiphase Porous Media................... 59
4.5 Derivations Constitutive Equations................................. 62
4.5.1 Darcys Law ................................................. 63
4.5.2 Darcys Law In Terms of Chemical Potential................... 65
4.5.3 Ficks Law .................................................. 68
4.5.4 Fouriers Law................................................ 70
4.6 Conclusion......................................................... 73
5. Coupled Heat and Moisture Transport Model ............................. 75
5.1 Introduction and Historical Work .................................. 75
5.1.1 Richards Equation for Fluid Flow............................ 75
5.1.2 Phillip and De Vries Diffusion Model ....................... 77
5.1.3 De Vries Heat Transport Model .............................. 78
5.2 Assumptions........................................................ 79
5.3 Derivation of Heat and Moisture Transport Model.................... 81
5.3.1 Mass Balance Equations....................................... 81
5.3.2 Energy Balance Equation...................................... 84
5.3.2.1 Energy Transfer in the Total Energy Equation........... 88
5.3.2.2 Stress in the Total Energy Equation.................... 89
5.3.2.3 Total Energy Balance Equation.......................... 91
viii


5.4 Simplifying Assumptions A Closed System.......................... 97
5.4.1 Saturation Equation......................................... 103
5.4.1.1 Capillary Pressure and Dynamic Capillary Pressure . 104
5.4.2 Gas Phase Diffusion Equation................................ Ill
5.4.3 Total Energy Equation....................................... 117
5.4.3.1 Dimensional Analysis.................................. 121
5.4.4 Constitutive Equations...................................... 122
5.5 Conclusion and Summary............................................ 126
6. Existence and Uniqueness Results........................................ 130
6.1 Saturation Equation with r / 0 ................................... 130
6.2 Alt and Luckhaus Existence and Uniqueness Theorems ............... 132
6.2.1 Existence and Uniqueness for Richards Equation............. 135
6.2.2 Vapor Diffusion Equation.................................... 138
6.2.3 Limits of the Alt and Luckhaus Theorem ................. 142
6.3 Heat Transport Equation........................................... 143
6.4 Conclusion........................................................ 146
7. Numerical Analysis and Sensitivity Studies.............................. 148
7.1 Saturation Equation............................................... 150
7.2 Vapor Diffusion Equation.......................................... 158
7.3 Coupled Saturation and Vapor Diffusion............................ 165
7.4 Coupled Heat and Moisture Transport System........................ 168
7.4.1 Experimental Setup, Material Parameters, and IBCs.......... 168
7.4.2 Numerical Simulations....................................... 177
7.5 Conclusion........................................................ 185
8. Conclusions and Future Work............................................. 189
IX


Appendix
A. Microscale Nomenclature................................................. 195
B. Macroscale Appendix..................................................... 197
B.l Nomenclature ..................................................... 197
B.2 Upscaled Definitions.............................................. 202
B. 3 Identities Needed to Obtain Inquality 3.40 205
C. Exploitation of the Entropy Inequality An Abstract Perspective....... 206
C. l Results that Hold For All Time.................................... 207
C.2 Equilibrium Results............................................... 208
C.3 Near Equilibrium Results.......................................... 209
C. 4 Linearization and Entropy ........................................ 210
D. Summary of Entropy Inequality Results................................... 212
D. l Results that Hold For All Time.................................... 212
D.2 Equilibrium Results............................................... 213
D.3 Near Equilibrium Results.......................................... 214
D.4 Constitutive Equations............................................ 215
E. Dimensional Quantities.................................................. 216
References.................................................................. 218
x


FIGURES
Figure
3.1 Illustration of the definition of the REV via a sequence of porosities cor-
responding to a sequence of shrinking volumes. (Image similar to Figures
1.3.1 and 1.3.2 in Bear [5])............................................ 21
3.2 Cartoon of the microscale, REV, and macroscale in a granular soil. The
right-hand plot depicts the mixture of all phases....................... 22
3.3 Local coordinates in and REV.............................................. 22
5.1 Densities as functions of temperature..................................... 99
5.2 van Genuchten relative permeability curves. The red curve shows the
non-wetting phase, nrnw(Se), and the blue curves show the wetting phase,
Krw(Se), each for m = 0.5, 0.67, 0.8, and 1......................... 104
5.3 Contact angle and effective radius in a capillary tube geometry. 6 is the
contact angle, r is the effective radius, and k is the radius of curvature of
the interface............................................................. 105
5.4 Examples of van Genuchten capillary pressure saturation curves for var-
ious parameters........................................................... 108
5.5 Comparison of different diffusion models at constant temperature (T =
295.15iF). The value for the saturated permeability was chosen to match
that of [78] (ks = 1.04 x 10-lom2), where they found a fitting parameter
a = 18.2. The Present Model refers to equation (5.68) (with VT = 0
and no mass transfer) and the Enhancement Model refers to equation
(5.3) along with (5.70), (5.72), and (5.73) for the diffusion coefficient,
enhancement factor, and tortuosity respectively......................... 116
5.6 Johansen thermal conductivity model with Cote-Konrad Ke S relation-
ship (with k = 15) plotted in blue, and the weighted sum of the thermal
conductivities of the individual phases plotted in red.................. 120
xi


5.7 Three proposed functional forms of r = r(S)........................... 124
5.8 Level curves of mass transfer rate functions.......................... 126
6.1 The function b{h) = S(h) 1 for m = 0.8 and various values of a...... 136
6.2 Kirchhoff transformation JC for m = 0.8 and various values of a....... 137
7.1 Cartoon of a 1-dimensional packed column experimental apparatus. . . . 148
7.2 Log of Peclet numbers for various values of saturation. The point at
(a = 5.7,m = 0.94) indicates the values used in Smits et al. [78]. Warmer
colors are associated with higher Peclet number and therefore associated
with an advective solution..................................... 152
7.3 Saturation profiles at various times in a drainage experiment with a =
5.7, n = 17.............................................................. 154
7.4 Convergence test for drainage experiment depicted in Figure 7.3. N is the
number of spatial grid points. In Figure 7.3, t\ = 0.025tc, ^2 = 0.050fc,
f3 = 0.075tc, and 14 = 0.010tc........................................... 156
7.5 Saturation profiles in a imbibition experiment with a = 2.5, n = 5.... 157
7.6 Convergence test for imbibition experiment depicted in Figure 7.5. In
Figure 7.5, t\ = 0.025tc, f2 = 0.050fc, f3 = 0.075tc, and f4 = 0.010fc . 158
7.7 Sample diffusion experiment comparing the enhancement model to the
present model. Here, a = 25, m = 0.9 (n = 10), and k = 10-10 with
Dirichlet boundary conditions and an exponential initial profile....160
7.8 Comparison of diffusion coefficients for various van Genuchten parameters
all taken with ks = 1.04 x 10-10 and e = 0.334 to match the experiment
in [78].................................................................... 161
7.9 Comparison of diffusion coefficients for various van Genuchten parameters
all taken with ks = 4.0822 x 10-11 and e = 0.385 to match the experiment
in [68].................................................................... 162
xii


7.10 The blue and green curves show the left-hand side of equation (7.8) for
different saturated permeabilities, and the red lines show level curves for
right-hand side for various values of a. The blue and green curves can be
used to predict the value of a before experimentation.................. 165
7.11 Comparison of coupled saturation-diffusion models for various weights of
Clv with parameters: ks = 1.04 x 10_lo,e = 0.334, H0 = 10_3,o: = 4, and
m = 0.667.............................................................. 167
7.12 Schematic of the Smits et al. experimental apparatus. Saturation and
temperature sensors numbered 1-11, temperature sensors 12 15, and
relative humidity sensors 1 and 2 [78]. The geometric x coordinate is
shown on the left. (Image recreated with permission from [78]). 169
7.14 Relative humidity and temperature data showing measurement variations
in the first few days of the experiment. (Image recreated with permission
from [78])......................................................... 170
7.13 Broken sensor data. Saturation sensor ^3 shown in blue and relative
humidity sensor #1 shown in red. It is evident from these plots that
these sensors are not working properly as they give non-physical readings.
(Image recreated with permission from [78])........................... 171
7.15 Relative humidity and temperature data at a window beginning roughly
12.5 days into the experiment. This window is chosen since the sensor noise
is qualitatively minimal in this region. (Image recreated with permission
from [78])............................................................... 172
7.16 Approximations to relative humidity and temperature boundary condi-
tions at the surface of the soil......................................... 173
7.17 Approximate initial conditions at the 2000t/l data point (t ~ 13.9 days).
Error bars indicate approximate sensor accuracy......................... 174
7.18 Illustration of how the gas-phase domain might evolve in time......... 175
xiii


7.19 Comparison of relative humidity and saturation for the fully coupled
saturation-diffusion-temperature model as compared to data from [78].
Boundary conditions are taken from a sinusoidal approximation of bound-
ary data. Thermal conductivities are taken as either weighted sum (5.78)
or Cote-Konrad (5.80).................................................. 183
7.20 Comparison of relative humidity and saturation for the fully coupled
saturation-diffusion-temperature model as compared to data from [78].
Boundary conditions are taken from a smoothed square wave approxi-
mation of boundary data. Thermal conductivities are taken as either
weighted sum (5.78) or Cote-Konrad (5.80)............................... 184
7.21 Blowup comparison of relative humidity and saturation for the fully cou-
pled saturation-diffusion-temperature model as compared to data from
[78]. The inset plots give a closer look at the behavior exhibited by these
particular solutions............................................... 185
7.22 Comparison of temperature solutions for the fully coupled saturation-
diffusion-temperature model as compared to data from [78]. Boundary
conditions are taken from a sinusoidal approximation of boundary data. . 186
7.23 Comparison of temperature solutions for the fully coupled saturation-
diffusion-temperature model as compared to data from [78]. Bound-
ary conditions are taken from a smoothed square wave approximation
of boundary data........................................................ 187
xiv


TABLES
Table
2.1 Mass and molar flux forms of Ficks law................................ 10
7.1 Measured and predicted value of the fitting parameter a based on equation
(7.8)................................................................... 165
7.2 Material parameters for experimental setup [78]........................ 170
7.3 Relative errors measured using equation (7.13) for the classical mathemat-
ical model consisting of Richards equation for saturation, the enhanced
diffusion model for vapor diffusion, and the de Vries model for heat trans-
port. These are compared for the two thermal conductivity functions of
interest (weighted sum (5.78) and Cote-Konrad (5.80))................... 180
7.4 Relative errors measured using equation (7.13) for instances within the
parameter space consisting of the thermal conductivity function (weighted
sum (5.78) and Cote-Konrad (5.80)), C^, ClT, and r. These are taken for a
(smoothed) square wave approximation to the boundary conditions. (The
starred rows indicate failure of the numerical method, and the errors from
the classical model are repeated for clarity) ........................ 181
7.5 Percent improvement of the present model over the classical model using
equation (7.13) as the error metric................................... 188
E.l Dimensional quantities................................................... 216
E.2 Typical values of hydraulic conductivity (K) for water and air, and as-
sociated values for permeability (k). Note that K = npg/p where
p9 = 1 kg/rn3, pl = 1000kg/m3 pi = 10_3Pa s, and pg = 10~5Pa s.
Modified from Bear pg. 136 [5] ............................................. 217
xv


1. Introduction
Water flow, water vapor diffusion, and heat transport within variably saturated
soils (above the groundwater and below the soil surface) are important physical pro-
cesses in evaporation studies, contaminant transport, and agriculture. The mathe-
matical models governing these physical processes are typically combinations of clas-
sical empirical law (e.g. Darcys law) and volume averaged conservation laws. The
resulting equations are valid in many situations, but recent experimental and math-
ematical research has suggested modifications and corrections to these models. The
primary goal of this thesis is to build a thermodynamically consistent mathematical
model for heat and moisture transport that takes these recent advancements into con-
sideration. To realize this goal Hybrid Mixture Theory (HMT) and the macroscale
chemical potential are used as the primary modeling tools.
1.1 Previous Work
In 1856, Henri Darcy published his research on the use of sand filters to clean
the water sources for the fountains in Dijon, France. He found that the flux of water
across sand filters was directly proportional to the gradient of the pressure head. This
simple observation has become known as Darcys law and is one of the main modeling
tools in hydrology and soil science [31]. Darcys law is an example of a historical rule
(or law) that has perpetuated to the present day. Darcys law was originally derived
for saturated porous media, but near the turn of the century it was extended for use
in unsaturated soils. In 1931, L.A. Richards coupled Darcys law with liquid mass
balance to derive what is now known as Richards equation. This equation relies on
the assumption that Darcys law is valid for unsaturated media, but it also relies on
an empirically-derived relationship between pressure and saturation.
The pressure-saturation relationship is known to be hysteretic in nature (de-
pends on the direction of wetting), and only recently have researchers been able to
move toward functional relationships that capture this effect [48]. Correction terms
1


in Richards equation have been proposed via HMT that suggest that the rate at
which the capillary pressure is changing may play a role in the overall dynamics of
the saturation [46, 47]. This proposed, third-order, term in Richards equation has
only recently been studied mathematically and experimentally. One possible physical
interpretation of this term is that is accounts for how fast the liquid-gas interfaces
are rearranging at the pore scale.
In the 1950s, Philip and deVries published their works on vapor and heat transport
in porous media [32, 61]. Their approach accounts for water flow in both the liquid
and fluid phases in response to water content and temperature gradients in the soil.
In order to account for the observation that Fickian diffusion inadequately describes
diffusion in porous media, Philip and deVries implemented an enhancement factor, rj,
to adjust the diffusion coefficient. This factor is fitted to the measured diffusion data
for a particular medium. In [24], Cass et al. found that rj increases with saturation
(with rj ~ 1 for dry soils). Philip and deVries proposed that thermal gradients and the
condensation and evaporation through liquid islands are the pore-scale mechanisms
that cause the observed enhancement. Counter intuitively, the governing equation
holds at the macroscale while these mechanisms are inherently pore-scale. The Philip
and de Vreis model has not been validated in a laboratory setting.
More recent works question the validity of the Philip and deVries model. Shokri
et al. [73] suggests that the coupling between the water flow and Fickian diffusion
is the key to estimating the vapor flux. They suggest that under this consideration
there is no need for the enhancement factor. Webb [80], and more recently Shahareeni
et al. [70], showed that enhancement can exist in the absence of thermal gradients.
It was initially thought that enhancement couldnt occurin the absence of thermal
gradients and was therefore ignored. Based on the observtion that enhancement can
occur without the need for thermal gradients, it is clear that the Philip and de Vries
model needs modification. Cass [24] and Campbell [23] give a functional form of the
2


enhancement factor that is commonly used (e.g. [68, 78]), but relies on an empirical
fitting parameter. In the present work we take the view of Shokri et al. that there is
no need for the enhancement fact, and wederive a diffusion equation simply based on
liquid and vapor flow. The novelty of the present approach is the use of the chemical
potential as the driving force for both types of flow.
For energy transport, the 1958 deVries model [32] is still commonly used (e.g.
[68, 78]). Similar to the enhanced diffusion model, deVries built this model so as
to account for the flux of the fluid phases. This is sensible as the fluid phases will
certainly transport heat. More recently, Bennethum et al. [14] and Kleinfelter [52]
used Hybrid Mixture Theory to derive heat transport equations in porous media
(Bennethum et al. studies saturated porous media and Kleinfelter studied multiscale
unsaturated media). They verified many of the findings by deVries but also proposed
several new terms associated with the physical processes of heat transport. In this
work we extend the Bennethum et al. and Kleinfelter approachs to unsaturated
media.
1.2 Hybrid Mixture Theory and Thesis Goals
To build the models in this work we make extensive use of Hybrid Mixture The-
ory (HMT). HMT, statistical upscaling, and homogenization have all been used as
techniques to re-derive, confirm, and extend Darcys, Ficks, and Fouriers laws in
porous media. For a technical summary of some of these methods see [30]. HMT is a
term for the process of using volume averaged pore-scale conservation laws along with
the second law of thermodynamics to give thermodynamically consistent constitutive
equations in porous media. The technique as applied to porous media was developed
by several parties, the most notable being Hassanizadeh and Gray [39, 43] and Cush-
man et al. [11, 12, 29], but the general principles were developed by Coleman and Noll
[27].
3


In the present work we use HMT to derive new extensions to these laws in the
case of unsaturated porous media. These extensions are then used to derive a model
for total moisture transport in unsaturated soils. While this sort of modeling has
been done in the past, rarely have the three principle physical process (movement
of saturation fronts, vapor diffusion, and thermal conduction) been considered from
first principles and put on the same theoretical footing (HMT in this case). No
known work attempts to couple these three different effects together with one physical
measurement: the chemical potential. This is one of the unique features of this work.
In the most general sense, the chemical potential is a measure of the tendency of
a substance (thinking particularly of a fluid or species) to diffuse. This diffusion could
be of a species within a mixture (e.g. water vapor diffusing through air) or it could
be a phase diffusing into another (e.g. water into a Darcy-type sand filter). The fact
that most physical processes in porous media are of this type gives an inspiration for
the potential usefulness of the chemical potential as a modeling tool. That is, from a
broad point of view, it should be possible to restate the physical processes of moving
saturation fronts and vapor diffusion more naturally by the chemical potential. This
approach has not been thoroughly explored in the past since the chemical potential
is not directly measureable and the theoretical footings of upscaling the chemical
potential are relatively new. In the saturated case, extensions to Darcys law have
also been developed via HMT, and the results indicate that the macroscale chemical
potential is a viable modeling tool for diffusive velocity in saturated porous media
[15, 69, 81]. In the present work we give chemical potential forms of Darcys and
Ficks law as well as presenting simplifications to Fouriers law based on the chemical
potential. In the case of a pure liquid phase we will show that the chemical potential
form of Darcys law is no different than the more traditional pressure formulation. In
the gas phase, on the other hand, we will show that the pairings of chemical potential
forms of Ficks and Darcys laws gives a new form of the diffusion coefficient that
4


does not need the enhancement factor indicated in the work by Phillip and DeVries
[61]. The chemical potential will finally be used to derive a novel form of Fouriers
law for heat conduction in multiphase media.
Once we have derived new forms of the classical constitutive equations we pair
these equations with volume averaged conservation laws to give a coupled system of
partial differential equations governing heat and moisture transport. The second law
of thermodynamics is used to suggest additional closure conditions for each of the
equations. Together, the system consists of a nonlinear pseudo-parabolic equation
for saturation, a nonlinear parabolic equation for vapor diffusion, and a nonlinear
parabolic-hyperbolic equation for heat transport.
In summary, this work serves several purposes: (1) it is a step toward better
understanding the role of the chemical potential in multiphase porous media, (2) it
makes strides toward understanding the phenomenon of enhanced vapor diffusion in
porous media, and (3) finally we propose a novel coupled system of equations for heat
and moisture transport.
1.3 Thesis Outline
In Chapter 2 we take a step back from porous media and discuss pore-scale
diffusion models. This is done in an attempt to elucidate the assumptions, derivations,
and models used in various disciplines as there tends to be confusion about where the
miriad of assumptions are valid. In this chapter we give mathematical and physical
reasons for the many commonly used assumptions as well as proposing an alternative
advection-diffusion model as compared to the popular Bird, Stewart, and Lightfoot
model [18].
In Chapter 3 we present the necessary background information in order to un-
derstand volume averaging and the exploitation of the entropy inequality. Much of
this chapter is paraphrased from previous works, such as [11, 12, 39, 43, 81, 86]. In
the beginning of Chapter 4 we use the tools from Chapter 3 to build and exploit a
5


version of the entropy inequality specific for multiphase media where each phase con-
sists of multiple species. The remainder of Chapter 4 is dedicated to the exploitation
of the entropy inequality for a novel choice of independent variables describing these
media. Appendix C serves as a companion to this discussion as it gives the abstract
formulation and logic of the entropy inequality. Throughout Chapter 4, the goal is to
derive new forms of Darcys, Ficks, and Fouriers laws and to propose extensions to
these laws in terms of the macroscale chemical potential. As part of these derivations
we arrive at new expressions for the pressure and wetting potentials in unsaturated
media. All of these derivations are done in a general sense with as few assumptions
as possible. This leaves open the possibilities of future research.
In Chapter 5 we couple the results found from the exploitation of the entropy
inequality (Chapter 4) with the volume averaged conservation laws derived in Chapter
3. In Section 5.1, a more in-depth historical perspective of the classical equations used
for heat and moisture transport is given to orient the reader to the recent research.
Fluid transport equations are presented in Section 5.3.1 along with a discussion of
the relationship between mass transfer and chemical potential. Considerable effort
is put toward deriving a heat transport equation with the final equation presented
in Section 5.3.2.3. In Section 5.4 several simplifying assumptions are presented in
order to close the system of equations. In particular, Sections 5.4.1, 5.4.2, and 5.4.3
give simplifications, assumptions, and dimensional analysis for the liquid, gas, and
heat equations respectively. In Section 5.4.4 we present the remaining constitutive
equations necessary to close the system of equations. Since so many assumptions and
simplifications are made throughout the chapter a summary of all of the results is
presented in Section 5.5.
In Chapter 6 we examine the proper regularity and assumptions needed for exis-
tence and uniqueness of solutions. These results are preliminary and do not constitute
a complete existence and uniqueness study for these equations.
6


In Chapter 7 we perform numerical analysis on the equations derived in Chapter 5.
In Sections 7.1, 7.2, and 7.3 we examine numerical solutions and parameter sensitivity
for the saturation equation, vapor diffusion equation, and the coupled saturation-
vapor diffusion equations respectively. In Section 7.4 we compare numerical solutions
to the fully coupled heat and moisture transport model to the experimental data
collected in [78].
In Chapters 5 7 we work toward building and analyzing the saturation, vapor
diffusion, and heat equations. The flow of thought for these chapters is to apply each
set of new assumptions or simplifications to each of the three equations before moving
to the next set of assumptions. That is, if a set of assumptions are proposed then the
subsequent sections will apply those assumptions to the saturation, vapor diffusion,
and heat equations in turn. Only then will the next set of assumptions be discussed.
This is done so that each set of assumptions are only stated once and since many of
the assumptions create interleaving effects between the equations.
Finally, as an aid to the reader there are several appendices. Appendix A contains
a nomenclature index for the pore-scale diffusion processes considered in Chapter 2.
Appendix B.l contains a nomenclature index for the macroscale results in the re-
maining chapters. There is some overlap between the nomenclature for these distinct
parts, and effort has been made to not create any excessive notational confusions
(even though this work is necessarily notation heavy). Appendix B.2 gives a list (in
alphabetical order) of the upscaled definitions of variables defined in chapters 3 and 4.
As mentioned previously, Appendix C gives an abstract view of the entropy inequal-
ity in an effort to make the exploitation process more clear to the interested reader.
Appendix D gives a summary of the results extracted from the entropy inequality
in Chapter 4. This is done for ease of reference mostly on the authors part, but it
is also done to provide an index of these results for use in future research. Finally,
Appendix E gives several tables of dimensional quantities used throughout.
7


It is suggested that the detail-oriented reader have Appendix A at hand when
reading Chapter 2 and Appendix B.l at hand when reading chapters 3-5. There
are some minor abuses of notation, but effort was made to bring them to the readers
attention whenever possible and to use notation that didnt confuse the immediate
discussion.
8


2. Ficks Law and Microscale Advection Diffusion Models
This chapter consists of a short technical note related to pore-scale diffusion prob-
lems. Vapor diffusion in macroscale porous media is an important phenomenon with
many applications (e.g. evaporation from soils, moisture transport through filters, and
CO2 sequestration). In order to better understand macroscale diffusion it behooves
the researcher to first understand pore-scale mechanics and models. This chapter at-
templts to elucidate the models and assumptions used for diffusion at the pore-scale
so that when we turn our attention to macroscale diffusion we are firmly grounded.
A secondary goal of this chapter is to give a thorough discussion of the diffusion coef-
ficient used in Ficks law. This is necessary since this coefficient is typically wrongly
assumed constant for all choices of dependent variables (mass concentration, molar
concentration, chemical potential, etc.).
To make matters simpler, we focus our pore-scale discussion on the, so called,
Stefan diffusion tube problem. This is a well-studied problem that models the dif-
fusion of a species through an ideal binary gas mixture above a liquid-gas interface
[6, 18, 22, 33, 51, 84, 85]. This is an idealization of the juxtaposition of phases in a
capillary tube geometry, and a capillary tube geometry is an idealization of geometry
of pore-scale porous media. To derive a mathematical model for the time evolution
of the evaporating (or condensing) species, one typically couples Ficks first law with
the mass balance equation.
In Section 2.1 we discuss the various forms of Ficks law and briefly discuss the
relationships between the diffusion coefficients. In Section 2.2 we derive the tran-
sient diffusion equations associated with Ficks law and compare with the associated
equation of Bird, Stewart, and Lightfoot [18] (henceforth referred to as BSL).
2.1 Comparison of Ficks Laws
For a system consisting of an ideal mixture of water vapor, gvi and inert air,
ga, Ficks law can be written in terms of molar concentration, mass concentration,
9


Table 2.1: Mass and molar flux forms of Ficks law
Flux Type Flux Expression Ficks Law
mass flux [ML~2T] J9p = p9jv9j,a Jf = -p9DpVC9> (2.1)
molar flux [(:mol)L~2T] J'i = e'J V9 9 Jf = -c9DcVx9 (2.2)
mass flux [ML~2T] = t'9 V9 9 D^(f,T)vK (2.3)
mole flux {(mol)L~2T] = cfvV = ~D,,c (ffj (2.4)
or the chemical potential. This is potentially confusing since there are inherently
different diffusion coefficients for the different forms of Ficks law. The purpose of
this subsection is to clarify the relationships between these coefficients. In porous
media it is common to use mass flux for Ficks law, but in chemistry (and related
fields) it is more common to use molar flux. As such, we will make most of our
comparisons between mass and molar flux.
According to BSL [18], the mass and molar forms of Ficks law are given by
equations (2.1) and (2.2) (modified from BSL Table 17.8-2). In Table 2.1, v9j,a =
v9j va js the diffusive velocity relative to a mass weighted velocity, vf9 = v9j v9
is the diffusive velocity relative to a mole weighted velocity, p9j is the mass density
of species j, C9j = p9j /p9 is the mass concentration of species j in the mixture,
c9j = mol(gj)/vol(g) is the molar density of species j, and x9j = c9j/c9 is the molar
concentration of species j in the mixture.
The chemical potential forms of Ficks law can be given in terms of two differ-
ent types of chemical potential: mass weighted (equation (2.3)) or mole weighted
(equation (2.4)). In physical chemistry and thermodynamics [21, 56] the chemical
potential is known as the tendency for a species to diffuse, and for this reason it is a
natural candidate for the statement of Ficks law (an exact thermodynamic definition
10


will be presented in subsequent chapters). In equations (2.3) and (2.4), p9^ is the
mole weighted chemical potential [J{mol)~l] and p9p is the mass weighted chemical
potential [JM~1].
The reader should first note that the two fluxes are measured with respect to
different velocities. The mass weighted velocity is p9v9 = a p9jv9j and the mole
weighted velocity is cgvg = ac9jv9j where v9j is the velocity of the species
relative to a fixed coordinate system. This means that that there cannot be a direct
comparison between the two different types of flux without considering them relative
to the same frame of reference. Using these definitions of v9 and v9 we see that the
difference between the two bulk velocities, v9 v9 is
In (2.5), the summation over j indicates that this is an accumulation over the N
species in the gas mixture. In future work we will be interested in the diffusion of
water vapor (j = v) and will consider the gas mixture as binary: j G {v,a}, where
j = a represents the mixture of all species that are not water vapor. Therefore we
can write (2.5) as
v9 v9
E
j=v,a

x9j p9
V
9j
(x9aC9v x9vC9a) v9v + (x9vC9a x9aC9v)v9a.
(2.6)
Converting to a mass weighted velocity we see that v9c39 = v9j,g + (v9 v9), and
therefore the molar flux is J9J = c9jv939 = c9jv9j9 + c9j(v9 v9). Assuming that
c9jx9kC91 < 1 we see that the difference between the frame of reference is potentially
quite small.
Next note that the diffusion coefficients are (initially) assumed to be different for
each choice of independent variable as indicated by the subscripts. To compare Dp
11


and Dc we note that J9pj = m9j J9j and
VCSi
m9jm9k
(xgimgj + xgkmgk)
Vx9>
(2.7)
to conclude that
C9k
QQ9k
Dn = Dr
(2.8)
where m9j is the molar mass of species j and the minuscule, k, represents the other
species. If the molar density form of the diffusion coefficient, Dc, is assumed to be
constant (at constant temperature) we conclude that the mass density version of the
diffusion coefficient is not constant (and visa versa). The fraction, C9k/x9k can be
interpreted as the ratio of the molar mass of species k to the molar mass of the
mixture. If j = v is the water vapor in an air-water mixture then k = a is the inert
air and the scaling factor between the diffusion coefficients is the ratio of molar mass
of the air to the molar mass of the mixture. For sufficiently dilute systems (where
the amount of water vapor is small) the ratio is approximately 1 and the diffusion
coefficients can be considered as approximately equal.
For ideal air-water mixtures, the densities are related through p9 = p9v + p9a and
the water vapor density is related to the relative humidity through p9v = p9svatp>. Here
we are taking p9svat as the saturated vapor density and

standard temperature and pressure we note that p9vat ~ 0.02kg/rri3 and p9 ~ 1 kg/rri3.
This indicates that at standard temperature and pressure we can likely assume that
the mixture is always sufficiently dilute. Therefore, in the systems under consideration
we can assume that the diffusion coefficients are approximately equal.
For the diffusion coefficients associated with the chemical potential forms of Ficks
law we first observe that if we multiply and divide the right-hand side of the molar
form by the molar mass of species j then
J9j
U U,c
( C9>
[~RT
m9j
mgi

Dyx
mgi
p9j
RgiT

(2.9)
12


Here, R9j = R/m9j is the specific gas constant, and we have used p9j = p^ /m9j.
Again noting that J9j = m9j J9jc we conclude that DPtC = D
It remains to compare Dp to DPtP and Dc to DPtC.
fluxes without loss of generality. If the mass fluxes are equal, then in particular
r&V
We focus here on the mass
p9DpVC9v = D
Rearranging, it can be seen that
_P___
9 1 R9.T
V jJL'
9v
DpVC9v
P
Qv
CD^ i RT
(assuming constant temperature). From physical chemistry [56], recall that the chem-
ical potential is related to a reference chemical potential (p9v) and the ratio of partial
pressure, to bulk pressure, p9, via
A
9v
p9P + R^T In ( ^
p9
(2.10)
Therefore,
A
9v
VHnl^
19v
p9
J9v
p_
p9
(2.11)
W"T j V Vp9 } J Vp9*
Using Daltons law for ideal gases, p9 = p9v +p9a, and using the specific gas constants
we note that the partial pressure of species j can be written as p9j = R9jTp9j.
Therefore,
f = R9vTp9v + R9aTp9a
(2.12)
and, after simplifying,
)9v
P
19v
P
P9 P9v + {w^)p9a
(2.13)
From the values found in Appendix E we see that R9a/R9v 0.6 and therefore
equation (2.13) is similar, but not equal to, the mass concentration, C9v = p9v/{p9v +
p9a). Defining C9v = p9v jpp we see that
Dp
D
CJ9v \ WC9v
C9v j V C9v
(2.14)
13


where division is understood component wise (that is, equation (2.14) represents three
equations when the gradient is understood in three spatial dimensions). The right-
hand side of equation (2.14) is not constant at 1 for all densities, but the variation
in the right-hand side depends mostly on the variation in p9a in the gas mixture.
Fortunately, the water vapor density is much smaller than the air-species density,
and hence p9a is approximately constant. In one spatial dimension, the right-hand
side of (2.14) can therefore be approximated by
( x 1 d ( x 1
l( \___Vx-t-l / dx V a;+0.6 /
( x ) A. ( x )
Vic+0.6/ dx Vik+1/
(where x = p9v and p9a ~ 1). It is easy to show that 0.98 < d(x) < 1 for 0 < x < pasvat.
Furthermore, for sufficiently dilute mixtures, d(x) ~ 1, and we therefore conclude
that Dp Dp.
The conclusion from this subsection is that while the diffusion coefficients for the
molar and mass flux forms of Ficks law are not the same, for dilute mixtures they
can be approximated as equal.
2.2 Transient Diffusion Models
In porous media there is a phenomenon known as enhanced uapor diffusion [61].
This phenomenon states that vapor diffusion in porous media occurs faster than as
predicted by Fickian diffusion models. This is merely a statement about the observed
imbalance between Fickian diffusion and experimental measure. Since the ultimate
goal of this work is to develop macroscale advection diffusion models, we seek to
understand the pore-scale diffusion models so that in subsequent chapters we can
tackle the enhanced diffusion problem.
To build a transient model for molecular diffusion we couple Ficks law with the
appropriate form of the mass balance equation. In the previous subsection we showed
that the various forms of Ficks law are approximately equal (for sufficiently dilute
mixtures), so the results stated here will only be in terms of the mass flux form of
Ficks law (equation (2.1)).
14


The mass balance equation for species j in the gas phase can be written as
dp9j
~dt~
+ V (p9>V9>)

(2.15)
where v9j is the velocity of species j within the gas mixture relative to a fixed frame of
reference, and r9j is a mass exchange term accounting for chemical reactions between
species [81, 85]. In the present work we assume that no chemical reactions occur, and
therefore r9j = 0. The combination of the mass balance equation with the mass flux
form of Ficks law (for j = v) gives a transport equation for the mass of water vapor
via advective, p9vv9, and diffusive, J9v, fluxes:
dp9v
~df
+ V- (J9v +p9vv9)
0.
(2.16)
Substituting the mass flux form of Ficks law1 (from equation (2.1)) we get
dp9v
~df
+ V- (p9*v9)
DV (p9VC9v).
(2.17)
Notice here that the diffusion coefficient has been factored out of the divergence
operator. This is only valid in constant temperature environments. If the gas-phase
density were constant in space then we would arrive at the traditional advection
diffusion equation (by dividing equation (2.17) by p9 of by rewriting the diffusion term
as DV Wp9v) and would need an expression for the bulk velocity in terms of density
(or concentration) to close the equation. Unfortunately, if the density of the water
vapor is allowed to vary then the density of the gas varies. Again, for sufficiently
dilute mixtures the variation in gas-phase density is very small and the nonlinear
diffusion on the right-hand side can be approximated by the linear diffusion term
D V Vp9v. It should be noted here that this later case is what is typically thought
of as Ficks law and is what leads to the traditional linear diffusion equation (when
the advection term is neglected) [28].
1The subscripts on the flux and the diffusion coefficient have been dropped since all of the versions
presented in Table 2.1 are approximately equal
15


A different form of equation (2.17), suggested in BSL [18], is derived by consid-
ering the mass weighted bulk velocity. In a binary system,
Solving for p9vv9v
1 O
(2.18)
Using Ficks law for p9vv9v9, and eliminating p9vv9v in (2.15) with (2.18) gives
Whitaker [85] suggested that one can develop convincing arguments in favor of ...
neglecting the air-species flux term. Certainly at steady state we can assume (as is
done in BSL) that v9a is approximately zero at the interface since there is no net
motion of [water vapor] away from the interface [18], but in the transient case this
would constitute a change of frame of reference. This new frame of reference would
be such that the inert air molecules are viewed as stationary with the water vapor
diffusing through them.
In either equation (2.17) or (2.19) one must find appropriate conditions or equa-
tions to either neglect or rewrite the advective term, p9vv9 or p9vv9a respectively.
Typically this term is neglected in a pure diffusion problem. As these are two differ-
ent simplifications of the same equation one must have different reasons for neglecting
the advective term. The easiest fix for this issue is to couple with either the bulk
gas mass balance equation or the air-species mass balance equation and to use the
mass-weighted velocity: p9v9 = ^2^=1 P9jv9j. The point being that one cannot simply
neglect the advection term in the transient case of either equation without proper con-
sideration of the implications: a fixed bulk velocity or a changing frame of reference
respectively.
16


A final comment can be make regarding equations (2.17) and (2.19). The bulk
density term, p9, on the right-hand side of these equations is often factored out of the
divergence operator. This is an error committed by several researchers [18, 22, 51].
The reasoning for assuming that the density is constant (and hence returning to a
linear diffusion model in the absence of advection) is that in an ideal gas, p9 = p9R9T.
Under constant temperature conditions, and if the pressure is assumed constant, then
the density is assumed be constant. There are two possible mistakes here. (1): If the
species densities are allowed to vary then the bulk density must vary. (2): The value
of R9 will vary with the changing composition of the mixture (since the molar mass
of the mixture changes). The effect of this is that, while the pressure may remain
constant, the component parts are not necessarily constant and therefore cannot be
factored from the divergence operator.
2.3 Conclusion
In this chapter we have compared various forms of Ficks law for molecular diffu-
sion. We have shown that, while the diffusion coefficients are indeed different, under
certain common circumstances the diffusion coefficients can be considered as approx-
imately equal. It is common to take the diffusion coefficient as constant (or only a
function of temperature), and in many cases it is safe to assume the same diffusion
coefficient may be used in the common forms of Ficks law.
In the transient case there are two natural formulations for (the mass flux form
of) Ficks second law. In either case, the natural governing equation is a nonlinear
advection diffusion equation that must be closed with the use of another mass balance
equation. When considering the advection term, it is the authors opinion that equa-
tion (2.17) is the more natural choice. The reason for this is that the bulk velocity,
v9, is likely more naturally measured as compared to that of the species velocity. This
chapter concludes our discussion on pore-scale modeling. We now turn our attention
to building macroscale models, but in doing so we keep in mind the diffusion models
17


at the pore scale and use cues from this scale to help make proper assumptions about
the larger scale.
18


3. Hybrid Mixture Theory
In this chapter we use a combination of classical mixture theory and rational
thermodynamics (henceforth called Hybrid Mixture Theory (HMT)) to study novel
extensions to Darcys law, Ficks law, and Fouriers law in variably saturated porous
media. This approach was pioneered by Hassanizadeh and Gray in the 70s and 80s
[39, 43, 44, 45] and later extended by Bennethum, Cushman, Gray, Hassanizadeh,
and many others [29, 30, 41, 81] to model multi-phase, multi-component, and multi-
scale media. HMT involves volume averaging, or upscaling, pore-scale balance laws
to obtain macroscale analogues. The second law of thermodynamics is then used
to derive constitutive restrictions on these macroscale balance laws. Constitutive
relations are particular to the medium being studied, and hence depend on a judicious
choice of independent variables for the energy of each phase in the medium. There are
many excellent resources for the curious reader to gain a more thorough understanding
of HMT (eg [30, 81]). For that reason we will not derive every identity along the
way. Instead partial derivations of the identities necessary to understand the present
application of HMT are presented.
To begin this overview we consider the upscaling of pore-scale balance laws (con-
servation laws) via a mixture theoretic approach. The subsequent sections in this
and the next chapter introduce the entropy inequality and its exploitation to derive
constitutive laws. A judicious choice of independent variables for the energy of each
phase in the medium is chosen and is used to derive novel versions of Darcys, Ficks,
and Fouriers laws. These constitutive equations will be used in subsequent chapters
to develop models for moisture transport in variably saturated porous media.
3.1 The Averaging Procedure
When considering a porous medium one cannot avoid discussing the various scales
involved. This particular work deals with two principal scales: the microscale and
19


the macroscale. At the microscale the phases are separate and distinguishable. Typ-
ical microscale porous media will have pores that measure on the order of microns
to millimeters (depending on the type of solid). At the macroscale the phases are
indistinguishable and the typical measurements range from millimeters to meters.
The macroscale is where most physical measurements are made, and as such, we seek
to derive governing equations that hold at this scale. The microscale structure may
vary dramatically for different media depending on the type of solid phase and the
microscale behavior of the fluid phases. As such, the microscale geometry can have a
dramatic influence on flow and phase interaction.
For any given phase at the pore scale the mass, linear momentum, angular mo-
mentum, and energy balance laws must hold. The problem is that it is difficult to
obtain geometric information everywhere at this scale For this reason we seek to
average (or upscale) the microscale balance laws to the macroscale.
There are many methods for mathematically averaging balance laws. Flere we
choose the simplest method of weighted integration. Before introducing the technical
details of the weighted integration we must first introduce the concept of a Represen-
tative Elementary Volume and local geometry in a porous medium. This elementary
volume will become our basic unit of volume throughout this research. The follow-
ing discussions closely follow and paraphrase those of Bear [5], Bennethum [12, 13],
Hassanizadeh and Gray [39, 43], Weinstein [81], and Wojciechowski [86].
3.1.1 The REV and Averaging
In this work we consider unsaturated porous media. Characteristic to these media
is the juxtaposition of liquid, solid, and gas phases within the pore matrix. We make
the assumption that a representative elementary volume (REV), in the sense of Bear
[5], exists at every point in space. To properly define the REV we first define the
porosity.
20


Consider a sequence of small volumes within a porous medium, (8V)k, each with
centroid x G R3. For each k, let (8VVOid)k be the volume of the void space within
(8V)k- The porosity for the kth volume is given as the ratio
i __ {8Vvoid)k
(5U)
Generate the sequence, {
(5V)-2 > (SV)^ > . As k increases, the porosity will certainly fluctuate due to
heterogeneities in the medium. As (8V)k shrinks there will be a certain value, k = k*,
such that for k > k* the fluctuations in porosity become small and are only due to
fluctuations in the arrangement of the solid matrix. If (8V)k is reduced well beyond
(8V)k* the sequence of volumes will eventually converge to x. The point, x only
lies within one phase, so the limit of the sequence of porosities will either be 0 or 1
(completely in the void space or completely in the solid). This indicates that there
will be some other intermediate volume, (8V)k** < (8V)k*, where the sequence of
porosities begins to fluctuate again as k gets larger. We define the REV, 8V, as any
particular volume (8V)k** < 8V < (8V)k* Without loss of generality we can simply
choose 8V = (8V)k**- Figure 3.1 illustrates two typical sequences of porosities, the volume is decreased (right to left).
Figure 3.1: Illustration of the definition of the REV via a sequence of porosities
corresponding to a sequence of shrinking volumes. (Image similar to Figures 1.3.1
and 1.3.2 in Bear [5])
21


Figure 3.2: Cartoon of the microscale, REV, and macroscale in a granular soil. The
right-hand plot depicts the mixture of all phases.
Consider now a coordinate system superimposed on the porous medium. Let x
be the centroid of the REV, and let r be some other vector inside the REV. Define
the vector, as a vector originating from the centroid of the REV such that
r = x + £. (3.2)
We can now view £ as a local coordinate in the REV as in Figure 3.3.
Figure 3.3: Local coordinates in and REV.
Define the phase indicator function as
7a(r,t)
1, r G Q'-phase
0, r a:-phase
(3.3)
where r is a position vector as indicated in Figure 3.3. The averaging technique
involved multiplying a micro scale quantity (such as density) by 7a and integrate
over the REV. This effectively smears out the phases. A consequence of this is that
22


the averaged value may not accurately represent the actual values being measured at
the pore scale. A further mathematical complication arises since the integrations may
not make sense in the traditional (Riemannian) sense. Therefore, we must understand
all of the following mathematics in the distributional sense (integrals are understood
to be Lebesgue, and derivatives are understood to be generalized derivatives). For
more specifics on these mathematical tools see standard graduate texts on functional
analysis (eg. [59]).
To find the volume of the a phase in the REV we simply integrate 7a over the
REV. Define this volume as \5Va[.
\5Va\= f 7a(r,t)dv= [ 7a(x + £,t) dv(£). (3.4)
Jsv Jsv
The a-phase volume fraction, ea, is defined as 1 *
f(M)=i§f (a5)
Since 0 < \8Va\ < |4V| it is clear that 0 < ea < 1. Furthermore, since the REV is
made up of all of the phases,
= L (3-6)
a
The volume fraction is the first example of a macroscale variable. That is, it is
a variable that describes a pore-scale property but is upscaled to the larger, more
measurable, scale.
It is useful to note that there are two main types of averaging that will be used:
mass averaging and volume averaging [43, 41, 81]. Let be the jth constituent of
some quantity of interest. To volume average we define
(^3)a = [ ^3(r^h(r,t)dv($,), (3.7)
\8Va\ Jsv
1Note: the notation e for the volume fraction is not necessarily standard. Some authors use
0, or V. Furthermore, the superscript notation is sometimes replaced by subscripts.
The present notation is chosen to be consistent with the primary references for Hybrid Mixture
Theory mentioned in the introduction to this chapter.
23


and to mass average ipj we define
(.PT \Wa\ JSV
L
p{r, *)*>(£)
(3.8)
Implicitly in (3.8) we see that density is volume averaged. That is,
(3.9)
(Note: some authors use the mass averaged notation on density even though it is
technically volume averaged. Given the definition of a mass averaged quantity there
is usually little confusion.)
The basic rules of thumb for deciding whether to volume or mass average were
originally proposed by Hassanizadeh and Grey in 1979 [43]. They propose four cri-
teria, listed below, for making this decision. In these criteria it is emphasized that
the microscale quantities correspond to small scale pre-averaged quantities, while
macroscale quantities are defined via the averaging process.
1. When an averaging operation involves integration, the integrand multiplied
by the infinitesimal element of integration must be an additive quantity. For
example, the internal energy density function, E, is not additive, but the total
internal energy, pEdv is additive and an average defined in terms of this quantity
will be physically meaningful.
2. The macroscopic quantities should exactly account for the total correspond-
ing microscopic quantity. For example, total macroscopic momentum fluxes
through a given boundary must be equal to the total microscopic momentum
fluxes through that boundary.
3. The primitive concept of a physical quantity, as first introduced into the clas-
sical continuum mechanics must be preserved by proper definition of the macro-
scopic quantity. For instance, heat is a mode of transfer of energy through a
24


boundary different from work. The definition of macroscopic heat flux must
also be a mode of energy transfer different from macroscopic work.
4. The averaged value of a microscopic quantity must be the same function that is
most widely observed and measured in a field situation or in laboratory practice.
For example, velocities measured in the field are usually mass averaged quan-
tities; therefore, the macroscopic velocity should be a mass averaged quantity.
This ensures applicability of the resulting equations.
In the upscaling procedure to follow we wish to apply a weighted integration to a
pore-scale balance law (a partial differential equation). This will involve terms such
and to ensure that we properly define the macroscale variables as either volume
or mass averaged quantities, we need a theorem that allows for the interchange of
integration and differentiation. This theorem is due to Whitaker and Slattery [75, 82,
83] and a generalization of this theorem is due to Cushman [29].
Theorem 3.1 (Averaging Theorem) Ifwap is the microscopic velocity of the a/3
interface and na is the outward unit normal vector of 5Va indicating that the integrand
should be evaluated in the limit as the afd interface is approached from the a side,
then
as
(3.10b)
(3.10a)
where f is the quantity to be averaged.
25


Keep in mind that / could be a scalar or a vector quantity. In the latter case, the
symbol / is replaced with / and appropriate tensor contractions are inserted.
The averaging procedure is now carried out in the following steps:
1. State the pore-scale balance law for a particular species (or phase).
2. Multiply the equation by ^a.
3. Average each term over the REV.
4. Apply Theorem 3.1 to arrive at terms representing macroscale quantities.
5. Define physically meaningful macroscopic quantities.
We now turn our attention to averaging pore-scale balance laws in the sense listed
above. In the following discussion, v3 is the microscopic velocity of constituent j, wapj
is the velocity of the jth constituent in the a[3 interface, and na is the outward unit
normal vector of 5Va. A full nomenclature index can be found in Appendix B.l.
3.2 Macroscale Balance Laws
As it is the simplest balance law, let us first consider the mass balance equation
for a single constituent:
Dp3
Dt
+ p1V v3 = p3P.
(3.11)
Here, pP is the density of the constituent, v3 is the velocity of the constituent, and
any source of mass from chemical reactions between the constituents is given as f3.
Recall that the material (Lagrangian) derivative is
IV(.) 5(-)
v3 V(-)
(3.12)
Dt dt
This derivative contains the usual Eulerian derivative along with an advective term.
Written in terms of the Eulerian time derivative, (3.11) is
^ + V (p>vj) = ppf3.
at
(3.13)
26


While this is specifically the mass balance equation, it takes the prototypical form of
all balance laws: a time derivative plus a flux is equal to any source.
The constituent momentum balance can be written in a similar manner:
nrl + v (f/v1 s v g)
(3.14)
Here, tj is the Cauchy stress tensor on species j, and the sources on the right-hand
side are gravity, momentum transfer from other constituents, and momentum gained
from chemical reactions respectively. These equations describe the change in mass
and momentum over time and space within a specific constituent. They are sufficient
held equations for modeling systems composed of a single phase gas, liquid, or solid,
but equations (3.13) and (3.14) are insufficient for modeling multiphase and multi-
constituent systems as a mixture because the interactions between the phases and
constituents are not present.
In this work we consider a porous medium consisting of a solid phase and two
fluid phases with multiple constituents within each phase. The phases will be denoted
as a = l,g, and s for liquid, gas, and solid respectively. The constituents will be
enumerated j = 1 : N (using MATLAB-style notation to indicate j = 1,2,..., N).
The following derivations follow similar derivations given by Gray [43], Weinstein [81]
and Wojciechowski [86].
3.2.1 Macroscale Mass Balance
To obtain macroscale equations in multiphase and multi-constituent media we
multiply a constituent balance equations by the phase indicator function, 7a, integrate
over 8V, and divide by |4H|. Applying the averaging theorem (3.1) to the appropriate
terms in equation (3.13) we have
27


(ep) Y tE f £M>s, naia, (3.15a)
V U, liV I J^i
9 ( a
m
\sv\
!&V
V (p-'V') 7adv = V
\5Va
pv^ocdv
E
L|W||5V;| Jw
p3v3 nada
1^1
V eap-

p3v3 nada, and
\5Va
p3f3dv = eapj rj
(3.15b)
(3.15c)
\SV\\8Va\ Jsv1
Substituting equations (3.15)(a)-(c) into equation (3.13), recognizing the volume
fraction terms, and recognizing the averaged mass and velocity gives the upscaled
mass balance equation:
d(ea7a) (
+ V (e p3 v3 )
[ p3 (wafi] v3) na + £apiaria. (3.16)
1v I JSA,
dt
E
p3 (Wafjj V3) na + £apjfj0.
ia/3
Rewriting equation (3.16) in terms of the material time derivative and defining
P 3 = P3 ,
vai = -IP
£
eP ~
\sva
pp (wapj v3) nada, and
'a\ JSAap
raj = £apajWa
(3.17)
(3.18)
(3.19)
(3.20)
respectively to be the averaged mass over 8Va, the mass averaged velocity, the net
rate of mass gained by constituent j in phase a from phase /?, and the rate of mass
gain due to interaction with other species within phase a, we get
Dj ^OtpOtj^
Dt
+ e"p"j V
v
Ed'
j3^a
+ r
(3.21)
28


Next we define the corresponding bulk phase variables so that the macroscale
equations are consistent with experimentally measured terms as much as possible.
Define:
N
pa = ^ paj, and
i=i
ca> =
(3.22)
(3.23)
respectively to be the mass density of the a phase, and the mass concentration of the
jth constituent in the a phase. If equation (3.21) is rewritten as
d (eapaCai)
dt
V {eapaCa>va>) = N3
e + f ^
and then summed over j = 1 : N we obtain a mass balance equation for the a phase:
d (eapa)
dt
V (eapava) = J2e
j3^QL
a
(3.24)
Here we have used the fact that e? = fhe rate of mass transfer to the
a
phase from the f3 phase is the sum of the rates of mass transfer to each individual
constituent in the a phase from the f3 phase.
Now use the definition of the material time derivative to write the mass balance
equation for the a phase as
Da (£apa)
Dt
eapaV -va = J2e'
4
(3.25)
(3^a
where the following restrictions have been applied:
N
raj = 0, Vcu, and
3 = 1
a
0, j = 1 : N.
(3.26)
(3.27)
Restriction (3.26) states that the rate of net gain of mass within species a from
chemical reactions alone must be zero. Equation (3.27) states that the rate of mass
gained by phase a from phase f3 is equal to the rate of mass gained by phase f3 from
phase a.
29


3.2.2 Macroscale Momentum Balance
We now turn our attention to the momentum balance equation, (3.14). We can
apply the same principles to upscale this equation (for full details see [81]). The
macroscopic linear momentum balance equation for constituent j in the a phase is
Dtj v&j
£apaj~5t v' _ £apajgaj
j3y^a
(3.28)
and the macroscopic linear momentum balance equation for the a phase is
£pnr ~ v' (eT) eps
E*
a
(3.29)
where taj and ta are the Cauchy stress tensors for the species and the phase and
Tp and T^ are momentum transfer terms. Most specifically, for the momentum
transfer terms, the former represents momentum transfered to constituent j in the
a phase through mechanical interactions from phase /3, and the latter represents
the momentum transfered to phase a through mechanical interactions from phase /3.
Also notable is the i 3 term. This term represents the rate of momentum gain due
to mechanical interactions with other species within the same phase.
In the processes of deriving these equations the following restrictions were en-
forced:
N
3 + fajvaj,a^J = 0 Vcu, and (3.30)
3 = 1 V
££(i? + JeV) = 0 J = 1:iV- <3-31>
a j3y^a
Restriction (3.30) states that linear momentum can only be lost due to interactions
with other phases (not within the species), and restriction (3.31) states that the
interface can hold no linear momentum. The comma in the superscript of (3.30)
indicates a relative term: vaj,a = vaj va. For a complete list of notation see
Appendix B.l.
30


Lastly, to tie the momentum transfer and stress tensor for the a phase to those
of the species we note two identities that were used in the derivation:
N
ta = Y, (rj pajvaja 0 vaja) (3.32)
3 = 1
N
T = E (t7 + . (3.33)
3 = 1
These identities will be used later and so are presented here for conciseness.
3.2.3 Macroscale Energy Balance
The derivations for the macroscale angular momentum and energy balance laws
are more algebraically complicated. The angular momentum equation will not be
used in this work since we assume that were dealing with granular-type media where
the angular momentum balance results in the solid phase Cauchy stress tensor being
symmetric [15, 43, 81]. The energy balance equation, on the other hand, will allow
us to derive a novel form of the heat equation in porous media. For this reason we
state the full equation here.
Applying the same routine as in the mass and linear momentum equations we
arrive (after significant simplification) at a balance law for the energy in species j:
T)ai (pai 1
eapa>- V (eaqa*) eap : Vva> £apa^h
Qa> + Q
P
(3.34)
(see [7, 81] for details on the derivation). Here, haj is the external supply of energy,
eaj is the energy density, qaj is the partial heat flux vector for the jth component
of the a phase, Qaj is the rate of energy gain due to interaction with other species
within the a phase, and Q^ is the rate of energy transfer from the f3 phase to the a
phase not due to mass or momentum transfer.
Again, following the derivation of [81], the bulk phase energy equation is
Daec
£apa-
Dt
- V (£aqa) £at : VtC £apaha = Y,Q
(3.35)
(3^a
31


where
To arrive at this form of the energy equation we enforced the following restrictions:
Restriction (3.36a) states that energy gained or lost due to species interactions within
the a phase must be gained or lost due to interactions with other phases. Restriction
(3.36b) states that the rate of energy gained or lost by one component in one phase
must go to another component or phase. That is, this second restriction states that
the interface retains no energy.
A system of equations governed by mass, momentum, and energy balance requires
each of the upscaled equations listed. A count of the variables indicates that there
are far more variables than equations. It is at this point where we need a method
for deriving constitutive equations for these remaining variables. The method cho-
sen for this work uses another macroscale balance law based on the second law of
thermodynamics.
3.3 The Entropy Inequality
The development of constitutive laws is central to the modeling process. As
we mentioned previously, this has historically been a process of fitting mathemati-
cal models to empirical evidence. The construct of Hybrid Mixture Theory (HMT)
couples the averaging theorems discussed in the previous section and the second law
of thermodynamics to provide us with restriction on the form of the constitutive
relations; hence narrowing down the experiments required to those that are thermo-
dynamically admissible. It is then up to the experimentalists to verify and refine these
models. Both theoretical and experimental directions of study have their merits, but
Vcc, and (3.36a)
a /3^a L
g ai _|_ ^yaj>a^
0 j = 1 : N. (3.36b)
32


putting the constitutive equations on a firm theoretical footing is ultimately preferred
whether it is before or after the experiments are run. In this section we give a brief
derivation of the upscaled entropy inequality, and we then use this inequality, along
with a judicious choice of variables, to derive constitutive equations for unsaturated
porous media.
3.3.1 A Brief Derivation of the Entropy Inequality
The second law of thermodynamics states that entropy will never decrease as a
system evolves toward equilibrium [4, 21]. The microscale entropy balance equation
that describes this phenomenon is
. I)31/3 -
P
Dt
+ V (p3 fPV = p3 + f3v3 + A,
(3.37)
where p3 is the entropy density of constituent j, (p3 is the entropy flux, b3 is the
external supply of entropy, p3 is entropy gained from other constituents, and A is the
entropy production. Since the second law of thermodynamics must hold we know
that A > 0 for all time.
Applying Theorem 3.1 to equation (3.37) and defining appropriate macroscale
definitions of the variables gives the upscaled entropy balance equation:
'na3
£apai 1
Dt
- V (ea(t>a*) £apa3 ba3 = Y $7 + Vai + A"b
(3.38)
where the terms on the right-hand side of equation (3.38) represent transfer of entropy
through mechanical interaction, entropy gained due to interactions with other species,
and the rate of entropy generation respectively.
Next, assume that the material we are modeling is simple in the sense of Coleman
and Noll [27]. This means that we assume that the entropy flux and external supply
are due to heat fluxes and sources respectively. To remove the dependence on external
heat sources we add (1/T times) the upscaled conservation of energy equation (3.34),
JJCKj ^OLj
£apa3
Dt
V (eaqa>) eaa> : - £ap^ha^ = Qa> +
(3^a
33


At this point we perform a Legendre transformation in order to convert convert
internal energy, eaj, to Helmholtz potential, ripaj (see any thermodynamics text, eg
[21]):
= eaj rjajT. (3.39)
This is done because internal energy has entropy as a natural independent vari-
able, and entropy is difficult to measure experimentally. It should be noted that
the Helmholtz potential is only one choice of thermodynamic potential we could have
made. This is done primarily for historical reasons, but the Gibbs potential and possi-
bly the Grand Canonical potential could have also been viable choices. The appeal of
the Helmholtz potential is that it naturally has independent variables of temperature
and volume (or, in intensive variables, density).
To arrive at a simplified entropy inequality for the total production of entropy
(across all constituents and phases) we now solve for Aaj and then sum over a = l, g, s
and j = 1 : N. This step requires significant algebra so the details of the derivation are
omitted for brevity sake. After much simplification, the entropy inequality becomes
£apa f Dartp
£
~T
T \ Dt
N
Ef
rt
Dst
~Dt
d=i
e"VT
j=i l \/3y
£
N
E(f
3 = 1
vaj,a
p JV
£
-Vaj,a vaj,a
+ + V (£apa*ipa*)
v
a N
%- (tai pa£ajL) :
T
1
T
1
3 = 1
Y{Tap + eapa^T}-v
(3^a
N
2 T
EE'
vaj:a vaj:a
3 = 1
V/3ga
34


(3.40)
where A is the rate of entropy generation.
Several new terms have appeared in (3.40). First, da = (Vva)sym is the rate of
deformation tensor (also known as the strain rate). As before, terms with a comma
in the superscript are relative terms: va,b = va vb.
Several identities were needed to derive (3.40). A complete list of these identities
has been included in Appendix B.3. The next step is to expand the Helmholtz
potential in terms of constitutive independent variables that describe our system.
This allows freedom to make choices about which variables control behavior of the
system. The choice of these variables is generally non-trivial so in the next section
we discuss motivations for the choice of variables.
35


4. New Independent Variables and Exploitation of the Entropy
Inequality
Now that we have an expression for the entropy inequality we must choose a set
of independent variables that describes our system of interest. We seek to describe
a multiphase system where the solid phase may undergo finite deformation, where
the relative saturations of the two fluid phases vary in time and space, and where
phase changes between the fluids possibly occurs throughout the porous medium.
Hassanizadeh and Gray have modeled similar media in the past [41, 44, 45]. These
models include effects from common interfaces, common lines (where three phases
meet), and common points (where four phases meet). These models are very thorough
and follow the same HMT approach. The down sides to their models, in the authors
opinion, are three fold: (1) the complexity of the resulting equations is such that in
order to use these equations a host of simplifying assumptions must be made, (2) the
thermodynamics of the common points and lines make sense physically but are likely
negligible relative to other effects, and (3) constitutive equations must be derived for
transfer rates between interfaces, common lines, common points, and phases. This
final drawback indicates that a detailed knowledge of the pore-scale physics must
be somehow upscaled. Approaches have been taken recently to do just this, but the
proposed theories have not yet gained widespread acceptance. Examples of such work
include those of Gray et al. [42, 40]
In the present approach we choose not to directly model interfaces and instead
strive to eventually write our governing equations in terms of the macroscale chemical
potential. The chemical potential is known from physical chemistry and thermody-
namics as a generalized driving force that is a function of pressure and temperature.
It is well known that mass transfer from liquid to gas states is driven by gradients
of chemical potential [56], so if we can write constitutive equations (such as Darcys,
Ficks, and Fouriers laws) in terms of this potential we can possibly couple the rele-
36


vant effects into much simpler governing equations; for example, equations that track
changes in chemical potential instead of pressure or concentration. The immediate
drawback to the present modeling efforts is that the recent work by Hassanizadeh et
al. seems to indicate that saturation and capillary pressure are linked to the amount
of interfacial area between phases within the medium [20, 48]. In the present work
we will not directly model the fluid-fluid and fluid-solid interfaces. We proceed with
the present modeling effort despite the results proposed by Hassanizadeh et al. We
will discuss this drawback as we run up against it in future sections and chapters.
4.1 A Choice of Independent Variables
In this section we present a choice of independent variables for the Helmholtz free
energy (potential) so as to expand the entropy inequality and to derive the relevant
forms of Darcys law, Ficks law, and Fouriers law. These variables are known as
constitutive independent variables as they represent a postulation of the variables that
control the energy in the system. Deriving physically meaningful results depends on
our ability to relate thermodynamically defined variables to physically interpretable
quantities [81]. To that end, we use our a priori knowledge of thermodynamics to
choose some of the variables. For the remainder of this work we restrict our attention
to a three-phase system consisting of an elastic solid, a viscous liquid phase, and a gas
phase. To begin the modeling process we assume that each of these phases consists
of N constituents (also called species or components), and all interfacial effects are
neglected. Examples of the constituents include dissolved minerals in the liquid,
species evaporated into the gas, or precipitated minerals associated with the solid
phase.
The motivation for choosing some of the variables is relatively trivial. For ex-
ample, to allow for a heat conducting medium, temperature, T, and the gradient of
temperature, VT, are included in the list of independent variables. The pore space is
expected to be variably saturated with the two fluid phases so the volume fractions,
37


el and e9, must be included in the set of variables. The fact that = 1 precludes
us from using all three volume fractions since they are not independent of each other.
In future chapters we will further restrict this assumption since for a rigid solid phase
the sum of the fluid phase volume fractions is equal to the fixed porosity
£l+£g = £. (4.1)
The reason for not making this assumption initially is that it allows us to develop
models for deformable media as well as for media with a rigid solid phase (hence, a
more general model may be derived from these assumptions later if necessary).
Recall from thermodynamics that the change in extensive Helmholtz potential,
A, with respect to volume is minus the pressure: dA/dV = p. In terms of intensive
variables this means that p2drtp/dp = p. To remain consistent with the extensive
definition of the Helmholtz potential, the densities must then be included in the set
of independent variables. Given the fact that there are N constituents in each phase,
this could be done in two different ways: (1) we could include the mass concentrations,
Caj, for j = 1 : N 1 along with the phase density, or (2) we could include all of
the constituent densities, paj for j = 1 : N. Bennethum, Murad, and Cushman
[15], and also Weinstein [81] took the first of these options when using HMT to
derive constitutive relations involving chemical potentials. The trouble with this
approach is that the mass concentration of the Nth constituent is dependent on the
mass concentrations of the previous N 1 constituents (since the concentrations
sum to 1). These results indicate that the behavior of the constituents depends
on how they are labeled instead of simply being independent. Various techniques
were successfully developed in [15] to deal with this complication. To avoid these
complications we choose the second option and include the species densities, paj for
j = 1 : N. Since each constituent is free to move within each phase, the spatial
gradients of the species densities, Vpb and Vpflb are also included.
38


Darcys law and Ficks law are classical empirical expressions for creeping flow
and constitutive diffusion. Darcys law is a statement about the relative velocity of
a fluid phase in a porous medium, and Ficks law is a statement about the relative
diffusive velocity of a species within a phase. Since we seek novel forms of these two
laws we include va,s and vaj,a for a = l, g, s in the list of independent variables. It
should be noted that neither of these variables is objective in the sense that they are
not frame invariant. This poses a problem since any governing equation should not
depend on an observers frame of reference. In [34], Eringen proposed a modification
to Darcys law that creates a frame invariant relative velocity. The new terms needed
for this new relative velocity are second order and are assumed to be negligible in
Darcy flow. A similar argument can be used for Ficks law.
The reasoning given in the previous few paragraphs leads us to the set of inde-
pendent variables for rtpa to include:
T, VT, k
V /'" -V// V/b v
l,s
V
9,s
V
,vgj,g, and Vs h
where j = 1 : N. It is apparent, now, that solid-phase terms corresponding to the
density and gradient of density are missing. The principle of equipresence, from
constitutive theory in continuum mechanics, states that all constitutive variables
are a function of the same set of independent variables [69]. To give symmetry
between the phases we include ps and Vps. The Stokes assumption for the Cauchy
stress tensor in a viscous fluid states that stress is the sum of the fluid pressure and
the strain rate. For this reason we include the strain rate (also known as the rate
of deformation tensor) for the fluid phases: dl and dg. The theory of equipresence
also states that if we include strain rate in the fluid phases then we must include a
comparable term in the solid phase.
A natural choice of variables for the solid phase are the solid phase volume frac-
tion, density, and the (averaged) strain. Weinstein [81] pointed out that these three
variables are not independent, as explained below, and used a modified set of inde-
39


pendent variables for the solid phase. The same modified set will be used here, so
the following simply states Weinsteins results with brief derivations.
Let Js be the Jacobian of the solid phase given by Js =det ((FS)T Fs), where
F_s is the deformation gradient
jps dx'l
= ~ax
(4.2)
K
Xs is the Eulerian coordinate, and Xs is the Lagrangian coordinate. Using standard
identities from Continuum Mechanics, the Jacobian can be rewritten as
Js = det (2Jf + L) (4.3)
Furthermore, through the conservation of mass, the Jacobian is also a scaling factor
for volumetric changes, Js = (eqPq) / (es ps). This clearly shows the dependence of the
three variables. To mitigate this issue, Weinstein [81] adopted ideas from solid me-
chanics and considered a multiplicative decomposition of the deformation gradient,
F_s, and the Greens deformation tensor, Cs, as
Qa = (Js)2/3g*, (4.4)
= (Js)1/3ES, (4.5)
where (J5)1/3/ and (Js)2/3/ represent volumetric deformation, and llF_s and C_s are
the modified deformation gradient and the modified right Cauchy-Green tensor, re-
spectively. With this modification to the solid strain, the solid phase variables we
consider here are JS,C_S,CSk and XCSk where k = 1 : N 1. We note here that
in order to get physically meaningful results for phase change, we include the same
components, so that CSj,Clj, and C9j all refer to the same component. Pairing the
mass concentrations and the Jacobian gives a description of the density of the solid
phase, and the modified Cauchy-Green tensor is used in place of strain.
The principle of equipresence states that all of the constitutive variables must be
a function of the same set of the postulated independent variables. In particular, we
40


postulate that the Helmholtz potential for each phase is a function of the following
set of variables:
[T, VT,
\e9}p\p9^Vp\Vp
fh
v
d\d\vc

VC**}
(4.6)
where a = l, g, s; j = 1 : N and k = 1 : At 1. We postulate that a three phase porous
medium with an elastic solid phase and N constituents per phase can be modeled by
set (4.6).
4.1.1 The Expanded Entropy Inequality
Consider now that the first line of the entropy inequality, (3.40), contains a ma-
terial time derivative of the Helmholtz potential for the a phase. Using the identity
£>() Ds(-)
Dt
Dt
v
v(-).
(4.7)
and applying the chain rule, the entropy inequality can be expanded to include each
of our constitutive independent variables. The central idea to the exploitation of the
second law of thermodynamics is that no term in the entropy inequality can take
values such that entropy generation is negative. A close examination of the expanded
entropy inequality reveals that there are many terms that show up linearly. In these
linear terms we notice some that are neither independent nor constitutive. Examples
of such coefficients are VT, VC*^, Vpb, Vp9j ,d? ,d9 ,vl,s ,v9,s ,vSjyS ,vlj1, v9j,9} T, p"q
and VtUa" (where the dot notation (e.g. T) indicates a material time derivative).
Loosely speaking, we have no control over these variables and they could take values
that violate the second law. For example, take as a thought experiment a process
where all of these variables except T are zero. From Bennethum [10],
Since none of the other terms in the entropy inequality are a function
of T, by varying the value of T we can make the left-hand side of the
entropy inequality as large positive or as large negative as we want hence
violating the entropy inequality. Since the entropy inequality must hold


for all processes (including those for which T is any value), the entropy
inequality can be violated unless the coefficient of T is zero.
In order not to violate the inequality in (4.13), the coefficients of all of these factors
therefore be left out of the expansion of (3.40) for brevity. The time rates of change
of volume fractions are not this type of variable since they are constitutive; that is,
we assume a rule for the time rates of change of volume fractions that depends on
the specific medium of interest.
With this simplification in mind, (3.40) becomes
must be zero. This implies that terms such as {eapa§^are zero and will
42


E
e"VT
N
T
9 (t' "J' PoJ' (if + )
E E { E *7 + ^ + v fpaf)}
a j=1 l \/3^a /
{ N
Y \ ea Y (r pajipaj£)
a l j=1
^EElfEdV^VT
a j= 1 L \fi^a / J
EEM^ + )(^'>2)}>o
a fi^za ^ ^
(4.8)
The next step is to enforce two additional relationships using Lagrange multipli-
ers. In doing so, the Lagrange multipliers become unknowns of the system. We will
see in subsequent sections that the Lagrange multipliers are associated with partial
pressures and chemical potentials of species in the fluid phases. The first relationship
considered is the dependence of the diffusive velocities:
N
Y(Ca>va>a) = 0. (4.9)
3 = 1
One can see this since
N
YJ Cajvaja
3 = 1
N N
(Ja3va3 CajVa = va ~ Va
3 = 1 3 = 1
0.
(4.10)
The implication is that if we know the concentrations and diffusive velocities of the
first N 1 constituents, then we would know the concentration and diffusive velocity
of the Nth constituent. Multiplying by the density, taking the gradient, and using
the product rule gives the following relationship:
(N \ N
Ypa>vaa ] = Y + vaaVpa>) = 0. (4.11)
3 = 1 J 3 = 1
Following Bennethum, Murad, and Cushman [15], we enforce this relationship with
a Lagrange multiplier so as to account for the Nth term dependence.
43


The second relationship to be enforced with Lagrange multipliers is the mass
balance equation for each of the constituents (3.21)
Dj [gapaj)
Dt
£apaj'V Vai = ^
efij + faj
Let AM denote A from equation (4.8), and let Xaj and XaN be the Lagrange
multipliers for the mass balance and Nth term dependencies, (4.11), respectively.
The entropy inequality is rewritten as follows:
N
A,
Aid+Y
Xaj
Dj (epj)
Dt
£apaj V Va> Y
a j=1 L \Ppa
N a
Y Y y=n : v (pj vaj,a)
a j=1
efij + raj
(4.12)
After a significant amount of algebraic simplification (with no additional physical
assumptions), this yields the following form of the entropy inequality:
' dripc
ta = Y{ £apa
dT
+ Va }T
E
P=l,9
N-1
E
3 = 1
EbvgJ
N
E hv
3 = 1

E£>
d%l)c
dC~si
- Xsi£sps
Csi
f N
EE
p=l,9 Kj=l
EI ef>
E^v
dtfj
~dTs
,dil)a
dpPi
1
3 Js
- X13^13
Ph
N
J2tr^s
d=i
r

N
((=')"' (!T~t
2 \ v=
N
E r
T'~t
,3 = 1
E Av
,3 = 1
c
E
P=l,9
iXP
dT
4s/ U^ + p) vr + £ |AVC
3 = 1
44


d¥ 1 £aV' +
de>3 e^p13 ^
3=1
dtp13
dpyj
Vp7A
^ f dip13 Xf3j\ g. ^
+ T,{g^-y) ^P+Y,
dip13 TS dtp13 _ 1 -/3 -/3
+ ^-VJs + : V (C )} + Tc + T,
dJs dC v=
v
/3,s
N
E £T(f
P=i,g l i=i
n ; I) : (V
E
eaVT
T
N
"-EU
3 = 1
%.Oij Oij.Oi
J V
__ pa3'Ua3>a | a ^a3^a
N
E E i E bf1 + *"J + v w"')
a J=1 l \/3^a
-A"aV (e"p"0 e"Aajv Vp"A} vaja
N
+ J2Y1 {£atai + epJ (AJ V^) I + e"p"J Aajv } : V^a
a J=1
N
- ^2 j+vO (Aflj+v,fl)
3 = 1
+i (('-)2 (-)2) + i ((jV')2 (t,-)2) |
N ( 11
- E4 | (A>< + r) (A" + P) 2 Kf + 2
3 = 1 ^
N f 11
- E-5? (A + - W + P) + 2 (a'*)2 + 2 t,S')2 (c'')2
7 = 1 V
> 0
(4.13)
The exploitation of equation (4.13) will be the source of all of the constitutive
relations for the remainder of this work. The next section outlines the details of this
exploitation to form constitutive relations specific to multiphase media governed by
45


our choice of constitutive independent variables, (4.6).
4.2 Exploiting the Entropy Inequality
In this section we exploit the entropy inequality, (4.13), in the sense of Colman
and Noll [27]. The basic principle here is that, according to the second law of ther-
modynamics, entropy is always non-decreasing as time evolves. This fact is used to
extract constitutive relationships from the entropy inequality. Not every result from
this exploitation is relevant to the current study, so we only present the more notable
and useful results in the next subsections. Furthermore, we exploit equation (4.13)
with an eye toward deformable, multiphase, media. The assumption of deformable
media will be removed in the future, but this leaves open the possibility of returning
to these results for future work. For an abstract summary of how the exploitation
of the entropy inequality works, along with subtle but important assumptions, see
Appendix C.
4.2.1 Results That Hold For All Time
As mentioned in Section 4.1.1, several of the terms that appear linearly in the
entropy inequality have factors that are neither independent nor constitutive. We now
use this fact to derive relationships that must hold for all time in order to not violate
the second law of thermodynamics. To illustrate this point consider the coefficient
of T. If this coefficient is set to zero we recover with the thermodynamic constraint
that temperature and entropy are conjugate variables,
di\)a
~w
(4.14)
This is a classical result known from thermodynamics.
4.2.1.1 Fluid Lagrange Multipliers
46


For the gas and liquid phases, the definitions of the Lagrange multipliers stem
from the coefficient of paj and Setting the coefficient of paj to zero gives the
definition of the Lagrange multiplier for the mass balance equations:
eapa dip**
A'% = V
(4.15)
£P Qpfij
Setting the coefficient of 'Vvaj,a to zero, summing over j = 1 : N, and solving for
XaN yields an expression for the other Lagrange multiplier:
N
XaN = J2 \iaj + (pJ AJ) L] + i>aL
pa j=1
(4.16)
4.2.1.2 Solid Phase Identities
Several identities for the solid phase can be derived from the terms associated with
the time derivatives of the solid phase Jacobian, Js, and the modified Cauchy-Green,
.S
C terms. From the Js term we see that
1
3
N
J2tr (D
a
£apa
dtfja
Jl1
Next, consider the identity
N
la = (fai _|_ paivaha 0 vaha
3 = 1
(4.17)
(4.18)
resulting from upscaling the momentum balance equation. Taking the trace of (4.18),
neglecting the diffusive terms, and substituting this into (4.17) gives a definition for
the solid phase pressure:
<4-i9)
a ^ '
This is a generalization of the solid phase pressure found by Weinstein for saturated
porous media in [81].
S
The coefficient of the C term gives a relationship for the stress in the solid phase.
This will give a generalization of the solid phase stress [8, 9] and closely follows the
47


derivations of Bennethum [9] and Weinstein [81]. Setting the coefficient of the C_
term to zero, left multiplying by the modified deformation gradient, Ff, and right
multiplying by the transpose of the deformation gradient gives a relationship that
defines the Lagrange multiplier for the solid phase, XSj:
N N
]Tr' + a sjpsji = -se!
QCS
. (T't
(4.20)
3 = 1 3 = 1
Using identity (4.18) in the stress term of (4.20), neglecting the diffusive velocities,
taking one-third the trace of the result, and using equation (4.19) for the solid-phase
pressure yields a relationship for the solid phase Lagrange multiplier:
N
' V
^ \sj nsj
3 = 1
P- + ^Ehr§):Q.
(4/21)
Substituting (4.21) back into (4.20) gives the following relation for the solid phase
stress:
v = -pL + F
E "V
QnJjCt
r)T-^E(^ )=££
This can be rewritten as
(4.22)
e ey
ts = -psI + t8 + t + ti
e £-.s h £-.s h
(4.23)
where
t- = 2 [ p-r X (Tf XX c-i,,
d£ y=J 3 1 dQ ==!
3F-.X-(F)T-^X:C£
= dc y=J r dc = =
F-X- (F-'f XX : Cl
= dc y=J x dc = =
(4.24a)
(4.24b)
(4.24c)
The stresses above are termed the effective stress, hydrating stress for the liquid
phase, and hydrating stress for the gas phase respectively. Equation (4.22) states
48


that the stress in the solid phase can be decomposed into the solid pressure and
stresses felt due to the presence of the fluid phases. It is here that the modifications
of the deformation gradient and Cauchy-Green tensors become clear. If we take the
trace of the stress tensor then we see that (1/3)tr(ts) = ps\ which is how the
solid phase pressure is measured. Therefore, this thermodynamic definition of ps is
consistent with experimental measure. Furthermore, the effective stress and hydrating
stresses are terms associated with the interaction between the solid and the fluids.
For saturated porous media, Bennethum [8] states the following:
The effective stress tensor is the stress of the solid phase due to the
strain of the porous matrix, and the hydrating stress tensor is the stress
the liquid phase supports due to the strain of the solid matrix (which
would be negligible if the liquid and solid phase were not interactive, but
which becomes significant for swelling porous materials).
One final note on the solid phase stress is that the total stress in the porous
medium is related to the pressures in all three phases. This can be seen by taking
the weighted sum of the stresses in each phase:
i = eT = -£Spsl + £% +£l (ll + i) + £ + i9) (4-25)
a
Taking one-third the trace of the total stress, and recalling that the effective and
hydrating stresses are trace free, gives
Q)tr (I) = ~£Sps + \tr (iO + (ifl) (4-26)
In order to fully understand the stresses in the fluid phases we must continue our
examination of the results coming from the entropy inequality. Equation (4.26) is
similar to the Terzaghi stress principle; suggesting that the fluid phases help to sup-
port the pore space in the medium.
4.2.2 Equilibrium Results
49


There are several more relationships that we can extract from the entropy in-
equality. In particular, we now seek relationships between the Lagrange multipliers
and the fluid-phase pressures. We also seek relationships for the momentum and en-
ergy exchange terms. At equilibrium the production of entropy is minimized. Since
this is a minimum, the gradient of A with respect to the set of independent variables
(4.6) is zero. This indicates that the coefficients of the independent variables that
appear linearly in the entropy inequality are zero at equilibrium. In the case of a three
phase porous medium of this nature, we define equilibrium to be when a subset of
the independent variables are zero. In particular, equilibrium is defined when no heat
conduction occurs, VT = 0, the strain rates in the fluid phases are zero, dP = 0, and
all relative velocities are zero, vaj,a = 0 and va,s = 0. This definition of equilibrium
is particular to this type of media and is chosen as it gives physically relevant and
meaningful results. Another way to look at this is to say that equilibrium is exactly
the state when all of these variables are zero.
4.2.2.1 Fluid Stress Tensor
The first notable equilibrium result comes from the coefficient of the rate of
deformation tensor, dP. Setting the coefficient of dP to zero, eliminating the sum of
the constituent stress tensors using the identity
N
t8 = ^ \tPj + pAyAA 0 yAAj ; (4.27)
3 = 1
and noting that at equilibrium the diffusive velocity is zero, yields
N 1
J] AApA = -P-tr (f) = /. (4.28)
3=1 3
Equation (4.28) links the Lagrange multipliers to the equilibrium pressure of the fluid
phases. This is the classical definition of pressure in a fluid: minus one-third the trace
of the stress tensor. Using equation (4.15) the pressure in the fluid phases can now
50


be written as
eapapl3j dif)a
eP dpPj
(4.29)
With the definition of pressure in equation (4.29) we note that the coefficient of
i13 can now be rewritten as
The second term is the change in energy with respect to volumetric changes, and
is therefore interpreted as the relative affinity for one phase to another. That is,
this term is related to the wetability of the a and solid phases by the f3 fluid phase.
state (that is not yet known), but rewriting the coefficient as in (4.30) hints at the
fact that the equation of state is related to the pressure and the wettability of the
phases. Furthermore, pressure, wettability, and surface tension are related to capillary
pressure; hence indicating that the equation of state for the time rate of change of
volume fraction is related to capillary pressure. It is here that we note the drawback
to the present modeling effort. Recall that in the present expansion of the entropy
inequality we do not include interfacial effects. If we were to include these effects then
a surface tension term would appear here (as shown in Hassanizadeh and Gray [41])
and these terms together would more readily be associated with capillary pressure.
More discussion will be dedicated to the exact equation of state for the time rate of
change of volume fraction after a discussion on cross coupling pressures in Section 4.3
and capillary pressure in Chapters 5 and 7.
4.2.2.2 Momentum Transfer Between Phases
The next notable equilibrium result we can extract from (4.13) comes from the
coefficient of the fluid phase relative velocities, v13^. Setting this coefficient to zero,
recalling that VT = 0 at equilibrium, using the definition of the fluid phase pressure,
(4.30)
The time rate of change of volume fluid phase volume fraction is an equation of
51


(4.29), the definition of the fluid phase Lagrange multipliers, (4.15), and solving for
the momentum transfer terms gives
- [5 - [5
T +T
s 1 7
V- lA + £fipfi
' .1=1
dijjP
dC1^
VC8*
N
E
3 = 1 L
W + fV'S ) V/'
dpPi
+ £Pp!
fiJij^VJS + £fipfi^:V(&
dpPi
]rtpp
dW
3 =
N

3 = 1
(4.31)
where 7 is the other fluid phase not equal to j3. This particular result will be coupled
with the conservation of momentum to yield novel forms of Darcys law in Section
4.5.1.
4.2.2.3 Momentum Transfer Between Species
Another notable equilibrium results comes from the coefficient of the diffusive
velocity, vaj,a. Equation (4.28) indicates that at equilibrium the definition of the
Lagrange multiplier, equation (4.16), simplifies to
Aajv = ipaL (4.32)
This implies that, at equilibrium, the stress tensor for constituent j (from the coeffi-
cient of V/i,a) can be written as
P = -£apa> (A"J ipa*)L £apa^(paL
Consider the diffusive velocity term in the entropy inequality:
- E E ((e *7)+* + v )
a j=1 t \/3y /
- A"J V (eapai) e"Aajv | va*a
Add J2a J2f=i(pajAajv Vea) vaj,a = 0 and simplify to get
- E E ((E *7) + + v (=>' )
a j=1 {_ \/3ga /
(4.33)
52


- A"J V (Eapai) AajV : V (Eapai) | Va*a.
At equilibrium the diffusive velocity is assumed to be zero. By the logic used herein
for the exploiting the entropy inequality, and given the fact that XaN = fjaI_ at
equilibrium, we observe that for each j,
Y Tap + T3 = V {£apaH^) + A"J V (Eapai) + fV (ep0 . (4.34)
This is an expression for the momentum transfer for species j in the a phase. This
result will be coupled with the constituent conservation of momentum equation to
derive a form of Ficks law in Section 4.5.3.
4.2.2.4 Partial Heat Flux
To conclude the equilibrium results we examine the VT term in the entropy
inequality. We have assumed that VT = 0 and vaj,a = 0 at equilibrium, so by the
logic used above we see that the coefficient of VT must be zero and
Ye a
at equilibrium. This is the partial heat flux of the entire porous media and will be
used in Section 4.5.4 to derive a generalized Fouriers law.
4.2.3 Near Equilibrium Results
The next step in exploiting the entropy inequality is to derive near equilibrium
results. These results arise by linearizing the equilibrium results about the equilibrium
state. The linearization process is simply the first-order terms of the Taylor series,
but one must keep in mind that each of the derivatives is a function of all of the
constitutive independent variables that are not zero at equilibrium. For example,
if / = feq at equilibrium, then near equilibrium, / fnear = feq + (0//0(VT))
VT + + (df/dvl,s) vl,s. This full expansion may yield terms that are not readily
53


physically interpretable. For this reason, considerable efforts must be made to relate
the linearization constants to measurable parameters. For a thorough explanation of
the linearization process with the entropy inequality see Appendix C.
For the momentum transfer in the fluid phases, the linearization of equation (4.31)
can be simply written as
fe A) = fe T) (A2St (4.36)
W/3 / near W/3 / eq
The linearization constant, RP, is related to the resistivity of a porous medium; the
inverse of the hydraulic conductivity. It should be noted that we have only expanded
about one of the possible variables: v/3,s. Strictly speaking this is incorrect and
we should expand about all other variables which are zero at equilibrium. A more
thorough expansion is
+ Mf VT + Jf vW +: g + , (4.37)
where // and Jp are second-order tensors and LP is a third-order tensor. The ellipses
at the end of this equation indicates that there are higher order terms that are not
being written explicitly. The left-hand side of (4.37) is the rate of momentum transfer
due to mechanical means. It is reasonable to think that this transfer term might be a
function of fluid velocity, but the effects due to thermal gradients, diffusive velocity,
and velocity gradients are likely small in comparison. To be completely correct we
would have to include these terms in the modeling problems to follow. The trouble
is that each of the coefficients needs to be associated with a physical parameter.
We will see that RP is physically associated with a material parameter of the porous
medium, but it is presently unclear what the physical interpretations are for the other
coefficients. Neglecting these terms simply leaves the door open for future modeling
research.
54


Proceeding in a similar manner, the linearized constituent momentum transfer
from equation (4.34) is
eq
(4.38)
The linearization constant is related to the inverse of the diffusion tensor. The lin-
earized partial heat flux from equation 4.35 is
(recalling that the partial heat flux is zero at equilibrium), and the linearization
constant is related to the thermal conductivity.
In each of these linearization results, the factors of volume fraction and density
are chosen so that the linearization constants better match experimentally measured
coefficients. The signs are chosen so that the entropy inequality is not violated.
Several of the relationships resulting from the entropy inequality rely on proper
definitions of the partial derivatives of the energy with respect to particular indepen-
dent variables. The pressure is one such quantity, but there are several others that
appear in the preceding results. For this reason, we now turn our attention to the
exact definitions of pressure and chemical potential under our choice of independent
variables. This will help to simplify and to attach physical meaning to the terms
appearing in each of the linearized results. In saturated swelling porous material,
Bennethum and Weinstein [16] showed that there are three pressures acting on the
system. These results are extended in the next section to media with multiple fluid
phases.
4.3 Pressures in Multiphase Porous Media
We will see in this subsection that the three pressures defined in [16] can be
extended to broader definitions in multiphase media. These definitions will help to
simplify and attach physical meaning to the terms appearing in each of the linearized
(4.39)
55


results discussed in the previous subsection. We will also define several new pressures
acting as coupling terms between the phases in multiphase media. It will be shown
that we can return to the three pressure relationship of Bennethum and Weinstein if
we simplify these results to a single fluid phase.
Recall from the entropy inequality that the equilibrium pressure in multiphase
media can be written as an accumulation of cross effects as follows:
N
a j=1
£apapl3j 8l{)C
(4.40)
£$ dp^i
The partial derivative is taken while holding ea, pak, e13, and p13 fixed where k = 1 : N
and m = 1 : N, m yt j. Define a cross-coupling pressure as
f eapapf3j dipc
j=i
P
e13 Qpfij
(4.41)
,pak ,£& ,pfin
so that the /3-phase pressure can be simply written as the sum of these cross-coupling
pressures
p
P

for /3g {/,(/} and a e{l,g,s}.
(4.42)
Now we derive an identity that is analogous to the three pressure relationship
derived by Bennethum and Weinstein [16]. To that end, consider the Helmoltz po-
tential as a function of two sets of independent variables where there is a one-to-one
relationship between the two sets.
t[)a = t[)a (ea, eapai, ^, eVJ) and (ea, pa>, e?, p^)
(4.43)
where a £ {l,g} and /3 a, s. The Helmholtz potential is actually a function of
several other variables, but these are suppressed here to make the notation more
readable. Since rtpa and tpa are functions of an equivalent set of variables, the total
differentials must be equal to each other. Setting dipa = dtpa gives
di\)a =
dipc
dec
dec
*pak y£& y£&p@k
N
E
dip0
j' d(eapaj)
£a ,£apam ,£&,£$p@k
d(eapaj)
56


dipa
deP
N
de13 +
dpc
d{ep(fi) (4.44)
xpak}el3pfik j_ 1 P 3) £a }£apak }el3pfin
where in each case we are taking m = 1 : N such that m ^ j, and k = 1 : N. Now
take the partial derivative with respect to e13 while holding ea,pak, and pPk fixed. In
this case, the dea and d(eapaj) terms will be zero. This leaves us with:
dpa dpa deP
deP deP £a ,pak yP^k £a pak y£& y£&p@k
£a,pak,pl3k
N
dpc

d^13 pPi)
la ,£a pak pfin
deP
ypakyp^k
dpc
de13
N
E
pPj Qpa
Xpk,£j,£jp?k ^ ^
(4.45)
la ,£a pak y£& y£@ p&n
Now multiply by eapa to get
a ad^
£ p
deP
a a
= ~ £ p
ypakyp@k
deP
N
£apapPj dpa
t<* Oak .£&.£& pPk 7 £>3
E
x p^k ^£P ^£P p^k
Sa ,£a pak ,£& ,£&p&m
(4.46)
Notice that the third term is pa^ from equation (4.41). Define the following new
terms:

7T
dxbc
_£apa "V
deP
a a d^C
ep
deP
Za,£apak ,£ppPk
>ak .O^k
(4.47)
(4.48)
S^yp^k ,pt
to get the relationship
p(/3) =p( +7r"(. (4.49)
Note that the new definitions only hold if N3 yt o. This can be seen if one returns
back to the Lagrange multiplier equation (at the beginning of this section) for the
pressure. Furthermore, this relationship holds if we had taken the derivative with
respect to ea instead of N3.
57


For completeness sake we define p13 and tt13 so that our definitions are consistent
with [16]:
pP := (4.50)
a
n13 := (4-51)
a
With these definitions we recover the three pressure relationship derived by Ben-
nethum and Weinstein
pP=pl3 + nl3. (4.52)
The physical meaning of p13 is the change in energy with respect to changes in
volume while holding mass fixed. In terms of extensive variables this is the same
definition as pressure encountered in classical thermodynamics for a single phase.
For this reason we call pi3 the thermodynamic pressure. The physical meaning of tt13 is
the change in energy with respect to changes in saturation while holding the densities
fixed. This pressure (or swelling potential as it is called in [16]) relates the deviation
between the classical pressure, pP, and the thermodynamic pressure. It can be seen as
a preferential wetting function that measures the affinity for one phase over another.
With these physical considerations in mind we now return to the coefficient of the e13
terms in the entropy inequality. With the present definitions, the coefficient is
Using (4.52) this is clearly pf3. Since the time rate of change of volume fraction is
taken as a constitutive variable, the linearization result for this term can now be
stated as
f
= f
n.eq.
+ T £,S.
(4.53)
eq.
The coefficient r arose from linearization and is formally defined as
dp13
T
di13
eq.
58


Equation (4.53) does little to make clear the exact meaning of this equation. The
exact meaning will become clear in Chapter 5 under the assumption that the fluid-
phase volume fractions are not independent.
The definitions of the three pressures allow us to attach more physical meaning
(and more convenient notation) to the results found when building constitutive equa-
tions in the next sections. Before building these equations we define the upscaled
chemical potential for a multiphase system, and after this point we will have all of
the tools necessary to derive the new constitutive equations.
4.4 Chemical Potential in Multiphase Porous Media
Chemical potential is defined thermodynamically as the change in energy with
respect to changes in the number of molecules in the system [4, 21]. This classical
definition has the following characteristics [15]: (1) it is a scalar and measures the
energy required to insert a particle into the system, (2) its gradient is the driving
force for diffusive flow (Ficks law), and (3) it is constant for a single constituent in
two phases at equilibrium. In [69], Bennethum proposed a definition for chemical
potential in saturated porous media that satisfies all three of these criteria:


,a | a
r
di{)c
dp
d (patfja) d (eapapa)
S*,p*m 9paj £a>pam d(eapap
(4.54)
£a ,pam
for m = 1 : N and m yt j. In saturated media, if the changes in energy in the solid
phase due to changes in liquid density are assumed to be zero, then the numerator of
the right-hand side of (4.54) can be seen as the total energy in a saturated system.
Under this assumption, the chemical potential can be rewritten as
d'tpT

(4.55)
£a,pam
d (eapai)
This indicates that in a saturated porous medium, the chemical potential of the jth
constituent in the aphase is the change in total energy with respect to changes
in mass of constituent j. We now extend this definition to multiphase unsaturated
systems.
59


Extending this idea to multiphase and multiconstituent media, we define chemical
potential to be the change in total energy with respect to changes in mass in the
constituent. In multiphase media we cannot make the assumption that the energy in
one phase is not effected by changes in other phases. With this in mind, we recall that
the total energy can be given by pT = '^2aeapapa. Therefore, the present definition
of chemical potential is
A
A -
dtfjT
d (ePph)
£0 5pak
E
d (eapapa)
d (e^pA)
(4.56)
Sa )£$ ,p^rn
where again, k = 1 : N and m = 1 : N, mp j. Notice that if dpa/<9pA = 0 for a p ft
then this definition collapses to equation (4.54). Furthermore, recalling the definition
of the Lagrange multiplier, Xaj, from (4.15), equation (4.56) can be rewritten as
£apa dpc

A + E
dpPi
p13 + A A.
(4.57)
ta ,£@ ,pak yp&'
Equation (4.57) only holds for [3 = 1 and (3 = g. A definition of the solid phase
chemical potential is beyond the scope of this work.
As a result of this definition of chemical potential we observe an immediate effect
on the rate of mass transfer terms in the entropy inequality. Using equation (4.57),
the last three terms in the entropy inequality, (4.13), can be rewritten as
N
- ^2 § {(Alj+V'O - +ip9)
3 = 1
60


(4.58)
The square of the relative velocities are likely zero since these models are designed
with creeping flow in mind. With these simplifications, the mass transfer terms from
the entropy inequality are rewritten as
N N N
- Y } Y ^ (,;A' + vo -1'1} Y e0!/,!/ (A'j + voi
j=i i=i i=i
(4.59)
At equilibrium we assume that the mass transfer between phases is zero. Take
note that this is an assumption about how the constitutive variable behaves at equi-
librium and not an assumption about the equilibrium state itself. This fine point
is made since in several works this assumption is made as part of the definition of
equilibrium (for example, [16]). In the authors opinion this is a subtle mistake. The
assumption that elg = 0 at equilibrium implies a final equilibrium relationship; the
mass transfer between the fluid phases is proportional to the difference in chemical
potentials
|n.eq = ^ \eq + [(/V7 ~ p9] ) M~\ (//J )
= [{pl] P9]) M] (pli p9) . (4.60)
This helps to verify our choice of upscaled chemical potential by satisfying the third
criteria set forth at the beginning of this subsection. Furthermore, this suggests a
natural coupling between the liquid and gas phase mass balance equations. The mass
transfer coefficient is chosen to have a factor of the difference in densities so as to
better match experimental measures [78]. Given that the units of the rate of mass
transfer are [ML-3!-1] we see that the units of the linearization constant are
[M]
t
1?
1
wwr
61


A further verification that we have properly defined the multiphase chemical
potential correctly can be seen through the Gibbs-Duhem relationship from thermo-
dynamics [21]. Simply stated, the Gibbs-Duhem relationship states that the Gibbs
potential of the a phase is the weighted sum of the chemical potentials:
N
r* = ^Ca*naK (4.61)
3 = 1
Equation (4.61) specifies the relationship between the Gibbs potential, T", and the
chemical potential. Substituting (4.57) into the right-hand side of (4.61), carrying out
the summation, and applying the definition of pressure, (4.28), gives the equation

(4.62)
which is the standard thermodynamic relationship between the Helmholtz potential
and the Gibbs potential. This clearly demonstrates that the definition of multiphase
chemical potential used here is consistent with the classical thermodynamic definition.
At this point we turn our attention toward using the relationships derived from
the entropy inequality to develop novel expressions for Darcys, Ficks, and Fouriers
laws of flow, diffusion, and heat conduction. For a concise summary of all of the
results derived in this chapter, see Appendix D.
4.5 Derivations Constitutive Equations
In this section we derive general forms of Darcys, Ficks, and Fouriers laws
based on the HMT results in the previous sections. These equations will be coupled
with mass and energy balance equations to form a macroscale model for heat and
moisture transport for unsaturated media. The results derived in this section extend
the classical forms of each of these laws. These extensions suggest terms that, in the
authors knowledge, are previously unreported. Also, we propose new forms of these
laws in terms of the macroscale chemical potential. This suggests that the chemical
potential is a generalized driving force for flow, diffusion, and heat transport.
62


4.5.1 Darcys Law
In 1856, Henri Darcy proposed his empirical law governing flow through saturated
porous media [31]. This was derived through experimentation on sand filters used to
purify the water in the fountains of Dijon, Prance. In its simplest form, Darcys law
states that the averaged fluid flux is proportional to the gradient of hydraulic head
(or fluid pressure)
elv18 = k'Vh. (4.63)
Under the construct of Hybrid Mixture Theory, Darcys law is obtained by coupling
the momentum balance equation for a fluid phase, (3.29), with the linearized constitu-
tive equation for the momentum transfer from other phases. This has been illustrated
by several authors (some examples include [12, 13, 43, 81]), and depending on the
set of independent variables postulated for the Helmholtz potential, the momentum
transfer term can suggest different forms of Darcys law.
In the present case, we recall from equation (4.36) that the linearized momentum
transfer terms can be written as
(t? + T^J = It (Ws) (tt^) /) VU3 7
N r
-VE&^' + E
3 = 1
N
3 = 1 L
) Vpft
dip1
dpU
d pPj
V E *<£), (4-64)
7=1 1
fdt/j13,
~dJs
i dip13 ^ f-^s
Ws
where we recall that Ra is related to the resistivity of the medium and arose from
the linearization process.
Linearization of the stress-pressure relationship for the fluid phases gives an ex-
pression for the stress near equilibrium:
/. = p / + /V : (V.
(4.65)
63


The fourth-order tensor multiplying the rate of deformation tensor can be simplified,
in most cases, to correspond to the viscosity of the medium (see any text on contin-
uum mechanics). Ignoring the acceleration terms in the momentum balance equation
(3.29), and substituting equation (4.64) for momentum transfer and (4.65) for the
stress tensor gives the following generalization of Darcys law:
It (Ws) = -ePVjP TT^Ve13 TT^yVe7 + > N
E
3 = 1 L
£ + v/>
8pPi dp^i J
N
£'
dijjP

3 = 1
e^pP^Vr e?p13^ : V (Q) + V ( yP : g ) (4.66)
dJs
To arrive at this form of Darcys law we have also assume that 'VCSj ~ 0 since it
assumed that concentration gradients in the solid phase do not affect flow. The first
term indicates that flow is primarily due to pressure gradients, as expected. The
eighth and ninth terms were previously reported by Weinstein in [81]. Note that the
extra factor of e13 on the left-hand side of the equation can be moved to the right. If
all but the first term on the right-hand side are then ignored we arrive at the classical
Darcys Law
R? (TV,*) =-Vpfi,
(4.67)
where q = e^v138 is known as the Darcy Flux.
The linearization constant, RP, is related to the resistivity of the porous medium,
the inverse of which is assumed to exist, and we define K3 = (R4) \ The tensor
k[3 is related to the hydraulic conductivity. To determine the exact meaning of the
linearization constant we consider the units of the simplest terms:
(f ~ _kP v/.
The units of the Darcy flux are length per time [L/t\, and the units of the pressure are
mass per length per time squared [M/(L t2)). This indicates that the linearization
64


constant has units [(L3 t)/M], which can be rewritten as [(L2)/(M/(L t))]. The
numerator of this fraction has units of permeability, k, and the denominator has units
of dynamic viscosity, pp. This suggests that
K13
K
h/3
(4.68)
and this relationship is confirmed in equations (11.4) and (11.5) of Pinder et al. [62],
The hydraulic conductivity of a porous medium is defined as
pPgK gK
V/3 Vp
(4.69)
where up is the kinematic viscosity of the fluid. This indicates that K1 can also be
defined as
k
K ' (4.70)
pPg
It is clear from these relationships that K1 is a function of both the type of fluid and
the geometry of the porous medium. This coefficient describes, in some sense, the
ability of the porous medium to transmit fluid [62], In saturated porous media, the
permeability is typically assumed only to be a function of geometry. Under Hybrid
Mixture Theory we must note that the permeability is a function of any variable which
is not necessary zero at equilibrium. Typically it is assumed that the permeability of
an unsaturated medium is a function of the volume fractions [5, 62], The tensorial
notation may be dropped in isotropic media, but for anisotropic media it is assumed
that the permeability may depend on the direction of flow. We will expand upon this
idea in later chapters when building a macroscale mass balance model.
4.5.2 Darcys Law In Terms of Chemical Potential
Equation (4.66) couples all of the physical processes that we wished to model at
the outset; multiphase flow with constituents in each phase and a deformable solid. In
order to build reasonable models based on this constitutive equation, functional forms
65


for the wetting potentials, ttGm and tt13^ and the changes in energy with respect to
density are needed. The solid-phase terms are likely negligible for non-deformable
media, but dealing with the remaining terms represent a significant modeling task.
The goal of this subsection is to greatly simplify this model while maintaining the
physical interpretation. This is done by switching thermodynamic potentials.
Recall from thermodynamics that the Gibbs potential, T", can be written in
terms of the Helmholtz potential, xj)a, as
Taking the gradient of the Gibbs potential, expanding the resulting gradient of
Helmholtz potential in terms of the constitutive independent variables, and multi-
plying by e^pP yields the equation:
Matching the common terms between (4.66) and (4.72), and recognizing the resulting
chemical potential terms yields
,a
(4.71)
- TT^) VG3 TT^) Ve7 Rv/
(4.72)
P
(4.73)
Observe that the summation in eq. (4.73) can be simplified to
N N
[s13 (/J ¥) Vp^-] = R (R3 Vp13 + eV ^2 VC'/3j)
66


by expanding the gradient of pA and using the Gibbs-Duhem equation (4.61). Sub-
stituting this back into (4.73) and simplifying yields the chemical potential form of
Darcys law:
£?B (Ws)
= -eV - GVVVT + V V; + V (v? : g) (4.74)
3 = 1 '
Cancelling the factor of e^ from the left-hand side, rewriting the coefficient of the
VpA term, and multiplying by the inverse of RP gives
GV-5 = K
N
E
-.7=1
VpA) + pPrfVT p^g -^V [y? : g
(4.75)
Equation (4.75) states an amazing fact: the flow of phase f3 is due only to gradients
in chemical potential, temperature, gravity, and viscous forces. The viscous forces
are often neglected in creeping flow. This gives

-K
N

pV VT ppg
.3 = 1
(4.76)
It should be emphasized that no additional assumptions were made to arrive at this
equation. That is, we still assume multiphase flow with a possibly swelling solid
phase. All of the actions, interrelations, and cross coupling effects are tied up within
the chemical potential term. This further indicates that the chemical potential is a
generalized force that, in effect, incorporates several driving forces.
A final simplification is to consider a pure fluid phase where only one constituent
is present. In this case, Darcys law is rewritten as
e/V,s = K [p4vr/3 + p/y VT pPg] (4.77)
where D3 is the macroscale Gibbs potential. The entropy coefficient of the gradient
of temperature poses a significant modeling issue as the entropy is not readily mea-
surable. The fact stated by equation (4.77) is that the Darcy flux of a pure species
67


is truly controlled by gradients in temperature and Gibbs potential. This is a gener-
alization of the classical pressure formulation that captures a wider range of physical
effects.
4.5.3 Ficks Law
We now turn our attention to diffusion and Ficks law. In 1855, Adolf Fick
published the first mathematical treatment of diffusion [37]. The empirically based
equation simply states that the diffusive flux of a species through a mixture is propor-
tional to the gradient in concentration of the species. This has since been generalized
through thermodynamics and physical chemistry [21, 56] to state that the diffusive
flux is proportional to the gradient in chemical potential of the species. In this sub-
section we apply the Hybrid Mixture Theory construct to derive a version of Ficks
law for multiphase porous media. It should be noted here that the classical chemical
potential from the thermodynamic definitions of Ficks law for diffusion in a liquid
not in a porous medium is the not the same chemical potential as that defined for the
porous media. In mixture theory we view the porous medium as a mixture of phases
(and species), but the classical thermodynamic definition considers one phase with a
mixture of species. With this difference in mind, it is not immediately clear that the
multiphase version of Ficks law will be the same.
To derive the present version of Ficks law we first consider a linearization of the
coefficient of the 'Vvaj,a term in the entropy inequality. The gradient of the diffusive
velocity, is taken to be zero at equilibrium so this coefficient is zero (since
entropy generation is minimized at equilibrium). Therefore,
ea¥* + £apa* (A"33 ip) 1 + £apai Aajv = fi for all j.
Using equation (4.32) for the definition of the Lagrange multiplier, XaN at equilibrium
and linearizing the coefficient of Vt)3',a about Vi)3',a gives
£ap = £a§?* ; Vl)3a + (-£apajXaj + £apaHp £apai\j)a) l, (4.78)
68


where Sfj is a fourth-order tensor that arises from linearization. Now consider the
species conservation of momentum equation (3.28). We ignore the inertial terms
since diffusion is assumed to be slow (this is discussed in some detail in Chapter 2).
Now eliminate the momentum transfer terms using the linearized momentum transfer
derived from the entropy inequality, (4.38), use (4.78) for the stress tensor, and using
the fact that paj = Xaj + fa gives a generalized form of Ficks law:
(£a)2pa*B?j va*a =
- £ap^'Vpffi + V (eaga^ : + Eapa*g. (4.79)
The term containing the gradient of diffusive velocity is likely negligible as it is second
order. If not, we would have to relate the fourth-order tensor, S_aj, with some physical
process (similar to viscosity for fluid flow). If we neglect this term then Ficks law
can be written as
£apa^i vai'a = -p^'VpT' + pa*g. (4.80)
Despite the novel choice of variables for this work, this form of Ficks law is identical
to that found by Bennethum and Murad [15] and Weinstein [81].
The linearization coefficient in Ficks law has a similar meaning to that of the
resistivity tensor in Darcys law. In this case, though, we wish to associate the inverse
of this tensor with the diffusivity tensor from classical Ficks law. Assuming that the
inverse exists we have
= . [Vpffi g\. (4.81)
The units of the left-hand side are [M L~2t~l], and the left-hand side term is commonly
known as flux. Therefore, the units of D is simply time [t\. Typically the diffusivity
constant in a gas is measured as [L2t~l], so we correlate D1 to the diffusion coefficient
for that phase, D9, via the relationship
D9, (4.82)
B?>T )=' y J
69


where R9j is the specific gas constant for constituent j. The units of R9jT are [L2t~2]
and the units of D9 are [L2f_1], hence making the units of D' [t]. This is consistent
with the forms of Ficks law from thermodynamics and physical chemistry [21, 56].
Hence, the gas phase form of Ficks law is
epv- = f £-) n [V/, g]. (4.83)
To close this subsection we finally recall from our discussion of pore-scale diffusion
(see Chapter 2) that the diffusive velocities are related via
N
J2pa>va>a = 0. (4.84)
j=i
Multiplying by the volume fraction and recognizing the left-hand side of Ficks law
indicates that
E{(^?)s>-[V41-9]}=0 (4.85)
near equilibrium. Equation (4.85) simply states that the gradients in chemical po-
tential are not independent of each other. This fact will be used in future chapters
as part of a moisture transport model.
4.5.4 Fouriers Law
The final result in this chapter is an extension to Fouriers Law for heat conduc-
tion. Notice that in the chemical potential form of Darcys Law, (4.76), there is a
term that involves the gradient of temperature. That is, the Darcy flux is partially
driven by a gradient in temperature. This means that Darcy flow is naturally driven
by gradients in temperature as well as gradients in chemical potential. To properly
handle this coupling we can either assume that the gradient of temperature is zero
(constant temperature) or consider the energy balance equation and track tempera-
ture as well as chemical potential. To move toward a closed system of equations, we
70


derive a version of Fouriers Law from the entropy inequality so that we have an ex-
pression of heat flux in the energy balance equation. At the outset we first recall that
in the entropy inequality weve assumed only one temperature for the entire porous
medium. This implies that weve assumed that the separate phases are in thermal
equilibrium. For this reason, we will develop an analogue to Fouriers Law that holds
for the entire (bulk) medium.
Following Bennethum and Cushmans work on heat transport in porous media
[14] we observe that if we sum the energy equation (3.35) over a we obtain the bulk
energy balance equation
De
p- t : Vv V q ph = 0, (4.86)
where
P = j2£apa a (4.87a)
x ~ ry ry ry pv = yep v a (4.87b)
rx rx u = v - V (4.87c)
t = J2 [£ pe = [sapaea + eapaua ua} a (4.87e)
=E a £aqa + .u pu + lu<* uc^ (4.87f)
ph = y^^eapaha. (4.87g)
a
Given identities (a) (g), the derivation of (4.86) follows after some significant algebra.
Define the medium velocity, v, as the weighted velocity of the medium, and the
relative velocity, ua = va v, is the aphase velocity relative to the medium. Note
that va,s = va vs vjrv = ua us = ua,s. In the case where the velocity of the solid
phase relative to the medium is zero (us = 0) we immediately see that ua = va,s. This
71


assumption along with equation (4.87f) indicates that there is naturally a coupling
between the relative velocities, va,s, and the total heat flux, q.
Using the near equilibrium result, 'Yha£aqa = K_ VT (equation (4.39)), we can
write the total heat flux as
q = K-^T + YJ
a
If we were to (wrongly) neglect all of the terms in the summation we would arrive
at Fouriers Law for heat conduction. The trouble here is that the terms in the
summation are not negligible, and therefore the total heat flux in a porous medium
must be a function of the gradient of temperature, the relative velocities, the stress
in the fluid phases, and the internal energy.
Since vs,s = Vs Vs = 0 the right-hand side of equation (4.88) is only a function
of the fluid velocities relative to the solid phase. Neglecting viscous terms we recall
that the fluid-phase stress tensors can be rewritten as ta = paI_. Neglecting the
second-order term, va,s va,s, the total heat flux is now written as
ta vas pavc
2V
v-
(4.88)
q = K-VT- [Op" + p"e") vas]. (4.89)
a=l,g
At this point we replace the internal energy term with Gibbs energy in hopes of
deriving an extended Fouriers Law in terms of the chemical potential. Recall from
thermodynamics that the Gibbs potential and internal energy are related through
wa
e = p a+Tr]a_P__ (49Q)
Therefore, the total heat flux can be written in terms of the Gibbs potential as
q = K VT ^ \Pa (F" + TVa) vas] (4.91)
a=l,g
Using the Gibbs-Duhem relationship, (4.61), this can be rewritten in terms of the
chemical potential as
q = K-VT- Y,
a=l,g
(4.92)
72


The trouble with both (4.91) and (4.92) is that they both rely on measurements of
entropy. One way to work around this issue is to assume that the entropy is only a
function of temperature, and then to recall that the specific heat is defined as
= = drf_
p dT dT'
Solving this separable ordinary differential equation (under the assumption that the
variation of specific heat with temperature negligible) gives
1)(r) = crln(K)+>to (4-93)
where T0 is a reference temperature, and t/q is a reference entropy. While this is only
an approximation it does allow us to move forward without direct measurements of
entropy.
The extended Fouriers Law (4.91) presented here frames the equations presented
in [14] in terms of the Gibbs potential. This will allow for easier coupling with
the chemical potential forms of Ficks and Darcys Laws presented in the previous
subsections. The caveat is that the equation for total energy balance, (4.86), is not
particularly useful since we do not have constitutive relations for the total stress and
total energy. For that reason, we will not use equation (4.91) or (4.92) for Fouriers
law in the energy equation. Instead we will use the linearized partial heat flux and
the constitutive relations for the phase stresses and relative velocities to derive a
generalized heat equation.
4.6 Conclusion
In this chapter we have shown that a novel and judicious choice of independent
variables for the Helmholtz Free Energy can be used to derive forms of Darcys,
Ficks, and Fouriers Laws for multiphase porous media. These equations are similar
to those found in [11, 14, 15, 81]. Each equation can be written with an eye toward
the macroscale chemical potential, and in each case the chemical potential form is
73


more mathematically appealing in the sense that there are fewer terms and many
of the physical processes are manifested in the chemical potentials. This illustrates
the usefulness of the chemical potential as a modeling tool. Furthermore, since the
chemical potential appears naturally in each of these equations we have set the stage
for a more natural method of coupling the fluid flow, diffusion, and heat transport. In
Chapters 5 and 7 we will couple these equations with the upscaled mass, momentum,
and energy balance equations to yield a system of equations that will govern total
moisture transport and heat flux in unsaturated porous media.
74


5. Coupled Heat and Moisture Transport Model
To form governing equations for heat and moisture transport in porous media we
pair the constitutive equations derived in Chapter 4 with upscaled mass, momentum,
and energy balance equations derived in Chapter 3. There are several existing mod-
els for each physical process of interest (fluid flow, diffusion, and heat transport) and
recent research indicates a need to understand the fully coupled system of equations
as it relates to moisture transport, evaporation, heat transport, and other physical
phenomena. In this chapter we derive a model for coupled heat and moisture trans-
port using Hybrid Mixture Theory and knowledge of pore-scale effects. To begin this
modeling task we first investigate the classical models used within the past century in
Section 5.1. In Sections 5.2 and 5.3 we pair our constitutive equations from Chapter 4
with upscaled balance laws from Chapter 3, perform a dimensional analysis, and dis-
cuss forms of the linearization coefficients arising from HMT. This is done in an effort
to generate a closed system of governing equations. Several simplifying assumptions
are made to close the system in Section 5.4. The solution(s) to the closed system will
be discussed in Chapter 7.
5.1 Introduction and Historical Work
To give the reader a better understanding of the work from the past century, we
present three classical models here with some discussion on their advantages and dis-
advantages. First we discuss Richards equation for unsaturated fluid flow in Section
5.1.1, second we discuss Phillip and De Vries enhanced diffusion model in Section
5.1.2, and lastly we discuss De Vries heat transport model in Section 5.1.3.
5.1.1 Richards Equation for Fluid Flow
The classical equation for fluid flow in unsaturated media is known as Richards
equation (also called the saturation equation). This equation was first derived in 1931
by L.A. Richards at Cornell University [65]. It takes a postulated form of the mass
75


balance equation (similar to equation (3.25)) and replaces the flux term with Darcys
law. The gradient of pressure is rewritten in terms of pressure head (h = p/(pg)),
and then a constitutive relation is assumed for the pressure head as a function of
saturation (or volume fraction). Another constitutive relation relating the relative
permeability of the medium to saturation is assumed. There are several versions of
the constitutive relations, but one of the more popular in recent research are those
of van Genuchten [79, 62], Another more recently investigated relationship is the
Fayer-Simmons model [36, 68, 78], which is an extension of the van Genuchten model
to cover the case of very low saturations.
The result of the assumption and substitutions in the mass balance equation is a
nonlinear diffusion equation where the primary unknown is the percent saturation of
the medium
fj Q
= v [D(S)VS K(S)z]
(5.1)
where K(S) is the hydraulic conductivity function and D(S) is the product of K(S)
and the derivative of capillary pressure with saturation. Recall that saturation is
defined as
S =
volume of liquid
(5.2)
1 es eg + el £ volume of pore space
and is understood as the volume of liquid per volume of pore space.
This model has been effectively used for several decades, but there are a few dis-
advantages of note. First of all, this equation does not allow for phase change between
the liquid and gas. The original model was proposed for systems with immiscible flu-
ids, where phase changes likely dont occur, but it is also used for unsaturated soils
where phase change is possible and air is always availabe.. A second disadvantage is
that humidity and temperature gradients are not considered. A third disadvantage
is that the pressure head saturation curve is hysteretic (depends on the history of
76


flow). The constitutive laws for pressure head dont account for this hysteretic behav-
ior directly. Instead, it is often assumed that fitting parameters change with changing
direction of flow. This leads to the final disadvantage: the use of the van Genuchten
capillary pressure saturation relation. This is a widely used relationship, but re-
lies heavily on two fitting parameters. The measurement of these fitting parameters
is difficult, and they are typically found by fitting numerical solutions of Richards
equation to experimental data.
Several extensions and modifications to Richards equation have been made re-
cently, the most notable of which is that of Hassanizadeh et al. [49, 47]. In these
papers, they propose a dynamic relationship between capillary pressure and satura-
tion based on Hybrid Mixture Theory with interfaces. They also propose that the
hysteretic effect observed in the capillary pressure saturation curves is due to the
(postulated) fact that the capillary pressure, saturation, and interfacial area density,
elg, form a unique surface. This partially explains hysteretic effects by seeing them
as a projection of this surface onto the capillary pressures saturation plane in the
'pc S elg space. This model is gaining in popularity, but is far from widespread
acceptance. Some of the relevant publications are [48, 49, 50, 46, 47, 58].
In the present chapter we present a modification to the Richards equation that
incorporates the dynamic capillary pressure relationship of Hassanizadeh et al. The
major differences between the present derivations and their work are: (1) modeling
in terms of chemical potential, (2) allowing for phase transition, and (3) allowing for
humidity and temperature gradients. Our present modeling effort will account for all
of these effects, and hence, constitutes a generalization of the existing model.
5.1.2 Phillip and De Vries Diffusion Model
In 1957, Phillip and de Vries published their comprehensive work on diffusion of
water vapor in porous media [61]. In their model they postulate an enhanced Ficks
77


law
q9v = p9Dr)'VC9v, (5.3)
where q9v is the water vapor flux and rj is an enhancement factor that is a function of
the toruosity, volume fraction of air, and a mass-flow factor. The mass-flow factor
is then postulated as a function of pore-scale gradients in saturation and temperature.
This model has successfully been applied to several diffusion and evaporation prob-
lems (e.g. [78]), but the trouble is that the exact form of the enhancement is based on
empirical evidence. Furthermore, this model has come under recent scrutiny due to
the fact that the proposed factors affecting rj are pore-scale effects and are therefore
difficult to accurately measure [25, 71, 72, 70, 73, 74, 78, 80]. Many of these works
use x-ray tomography to attempt to measure these pore-scale effects directly.
In the work by Cass et al. [24], an empirical form of the enhancement factor was
proposed. In this work, a fitting parameter is used in the enhancement factor to
arrive at good agreement with experimental data. This model has been used in more
recent works (e.g. [68, 78]) in conjunction with a mass balance equation for the water
vapor in the gas phase. The resulting model is a nonlinear diffusion equation for
concentration of water vapor that deviates from the more classical de Vries model.
Aside from the empirical fitting parameter, the mass transfer between phases also
relies on a fitting parameter and an empirically-derived functional form.
In the present chapter we build a model for diffusion based on using the chemi-
cal potential as a primary unknown and the Hybrid Mixture Theory construct. The
enhanced diffusion is not incorporated into these models, and the mass transfer is
modeled by the difference in chemical potentials; a more physically natural formula-
tion. A comparison will be made to the model of Cass et al.
5.1.3 De Vries Heat Transport Model
In 1958, de Vries published a second paper coupling heat and moisture transport
in porous media [32], In this research, he proposed an extended heat transport
78


model for porous media that is still used today. Neither his diffusion nor his heat
transport model were thermodynamically derived. Instead, he began each derivation
with a postulation of the forms of diffusive and heat flux. For the heat transport
equation he included terms similar to the classical Fouriers law, but also proposed
that heat transport was due to advective transport in the fluid phases. This model
is still popularly used today to couple heat and mass transport in unsaturated media
[5, 77, 78]. That being said, the effects included in this equations are based solely on
de Vries supposition of the factors affecting heat flow.
In 1999 Bennethum and Cushman published (to the authors knowledge) the first
work using Hybrid Mixture Theory to derive an extended de Vries model for heat
transport in swelling saturated porous media [14]. In the present chapter we take a
similar approach using HMT to derive a thermodynamically consistent model for heat
transport in non-swelling unsaturated media. This is done with an eye toward using
gradients in temperature as the thermal diffusion process and the chemical potential
to describe the secondary processes such as advection.
5.2 Assumptions
In this section we state the baseline assumptions that will be used throughout the
remainder of this work. These assumptions are meant to make minimal limitations
on the applicability of the resulting models, but at the same time they are meant to
keep the mathematics tractable. Possible relaxations to these assumptions (and the
source of possible avenues of future research) will be stated as they are encountered.
The simple set of baseline assumptions are as follows:
Assumption #1: The solid phase is rigid, incompressible, and inert.
Assumption ^2: The liquid and gas phases are each made up of N constituents.
Assumption ^3: No chemical reactions take place in any of the phases.
79


The first assumption is the most restrictive. Mathematically it corresponds to
setting the Lagrangian derivatives of both density and volume fraction for the solid
phase to zero. Assuming that the solid is inert simply means that no mass will
precipitate onto, or dissolve away from, the solid phase. With these assumptions, the
solid phase mass balance equation (from equation (3.25)) becomes
V Vs = 0. (5.4)
If a deformable solid is considered where the solid-phase volume fraction can change,
then this assumption would need to be relaxed. One particular relaxation of this
assumption is to allow for incompressibility and inertness of the solid phase but relax
the rigidity assumption. Under this relaxation, the solid phase mass balance equation
becomes
Ds £s
------esV vs = 0. (5.5)
Dt v '
A consequence of fixing the solid phase volume is that el + e9 = 1 es := e,
where e is known as the porosity of the porous medium. A further consequence is
that the liquid and gas phase volume fractions are no longer independent of each
other. Note that we could have made this assumption up front and exploited the
entropy inequality with this assumption (this is done in [44, 45] for a different set
of independent variables), but proceeding in this order allows us to return to the
present entropy inequality results and consider a deformable solid in the future. Since
the fluid-phase volume fractions are no longer independent we can replace them by
saturation as defined by
S =
£l + £g £ '
This implies that the volume fractions are related via £l = £S and £9 = e(l S).
(5.6)
Assumption ^2 is a byproduct of the principle of equipresence and will be relaxed
later for simplicity. In the most general sense, this assumption states that every
80


species that exists in one fluid phase also exists in the other. In reality this is likely
not true. For example, if a constituent is present in the liquid phase it is possible that
evaporated particles of the constituent are not be present in the gas phase. Another
example would be if we were to extend this model to an oil-water system. The two
fluids in this case are immiscible and it is unlikely that every species in the water
phase is present in the oil phase (and visa versa). We take this into account by
setting the appropriate concentrations to zero after the constitutive equations have
been derived.
Assumption ^3 indicates that the rate of mass exchange due to chemical reac-
tions, raj, is zero for all phases. The consequence of this is that the rate of mass
generation of a constituent in a phase only occurs between two phases. This is true
for some porous media, but chemical reactions can occur in some specific cases such as
remediation problems. Under this assumption these cases are henceforth eliminated
from the discussion.
Other simplifying assumptions exist for many media, but the three presented
herein constitute a set that leads to several mathematical simplifications with as few
physical restrictions as possible.
5.3 Derivation of Heat and Moisture Transport Model
In the remainder of this chapter we focus on using the results from Chapters 3
and 4, along with the assumptions from Section 5.2, to derive a closed system of
equations for heat and mass transport in unsaturated porous media. This will be
done with an eye toward using the chemical potential as the driving force for these
processes. We will show that under certain additional simplifying assumptions that
a closed system can be derived.
5.3.1 Mass Balance Equations
We first build generalized mass balance equations in terms of the chemical po-
tential under assumptions #1 #3. Recall from Chapter 3 that the mass balance
81


equation for the jth constituent in the aphase is (from equation (3.21))
Dj [gapaj)
Dt
£apaV va* = J2ei3J +r

(5.7)
(3y^a
The last term can be dropped under assumption ^3 in Section 5.2. Because of the
form of the constitutive equation, and to adhere to the principle of frame invariance,
it is convenient to rewrite this equation relative to the solid phase. To do so we recall
the identities
Dai(-) Da(-)
Dt Dt
£>(.) £>*(.)
Dt Dt
and expand the Lagrangian time derivatives accordingly to get
Ds (£apai)
+ Va£a V(-) (5.8a)
+ * V(-) (5.8b)
Dt
V
V (e"p"0 + vas V (e"p"0 + eapaj'V va> = ^
(5.9)
Taking the definition of the Lagrangian time derivative,
D = 771 + vs. V(-)
Dt dt + U
adding and subtracting £apai V va, and subtracting £apai V Vs = 0 gives
d (£apai )
dt
+ V (eapa*va*a) + V (£apa>vas) =
(5.10)
Notice the use of Assumption #1 in the last step, and observe that if Assumption
#1 is relaxed then the mass balance equation would involve a time derivative of the
solid-phase volume fraction (at least).
Equation (5.10) is the general mass balance equation for both of the fluid phases.
Notice that we are not replacing the volume fractions with saturation here since we
dont know if a is the liquid or gas phase. Substituting Ficks law for the diffusive
flux and Darcys law for the Darcy flux gives the chemical potential form of the full
mass balance equation for species j in phase a:
d (£apai )
dt
- V {paigi [VpT' g
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(5.11)
paKa
N
Y (p"fc Vp"fc) + papaVT pag
,k= 1
N
= £T
j3^a
It should be noted here that the Eulerian and Lagrangian time derivatives are equal
under the assumption that the solid-phase velocity is zero (Assumption #1). Also
note that if we sum over all constituents then we arrive at the mass balance equation
for the phase (where we have used pajvaj,a = 0)
dt I =
f U=i
The chemical potential form of the mass balance equation is only one form. We
could have used the pressure formulation for Darcys law and arrived at a pressure -
chemical potential form of the mass balance equation.
The rate of mass transfer term on the right-hand side of the mass balance equation
can be rewritten in terms of a linearized result from the entropy inequality. Recall
from equation (4.60) that the mass transfer term can be written as
Y (p"fc Vp"fc) + p"p"VT pag
£
73*
(5.12)
% = [(p> -/>) M] , (5.13)
where the coefficient (paj pA ) is chosen to be consistent with equation (9) of [78].
Also recall that since the interface is assumed to contain no mass we must have that
the rate of mass gained from the f3 phase to the jth species in the a phase must be
equal to the rate of mass lost from the a phase to the jth species of the f3 phase:
If the chemical potential of the liquid phase is larger than the chemical potential of
the water vapor then mass will transfer from liquid to gas and elg < 0. Similarly, if
the chemical potential of the liquid phase is smaller than that of the water vapor then
mass will transfer from gas to liquid and elg > 0. Recall from the discussion adjacent
to equation (4.60) that the units of M are the reciprocal of flux.
83


There are clearly more unknowns than equations in the 2N fluid equations since
we must account for the densities, temperature, volume fractions, and entropies as
well as the chemical potentials. Certain sets of simplifying assumptions can be used
to reduce the number of unknowns (e.g. incompressibility of a fluid phase). These
will be discussed in Section 5.4. Instead of making these assumptions up front we now
turn our attention to deriving a generalized energy balance equation to account for
the temperature. This will give one more equation but will add no more unknowns
to the system of equations.
5.3.2 Energy Balance Equation
As another step toward developing a closed system of equation equations for
heat and moisture transport we next examine the energy balance equation. This will
give an equation in terms of temperature, chemical potentials, saturation (volume
fractions), entropy, and densities; increasing the equation count but not increasing
the variable count. Since we assumed at the outset that all of the phases are in thermal
equilibrium we will only have one equation for energy balance. This will be derived
by considering the sum of each of the phase energy balance equations. Counter-
intuitively, we will not use the form of Fouriers Law (equation (4.87f) or (4.92))
derived for the total heat flux since the energy equation derived in that section is
more cumbersome to work with than the individual phase energy equations. Instead
we will use the partial heat flux for each phase as derived from linearization about
equilibrium (4.39).
From equation (3.35), the volume averaged energy balance equation is
Dae
£apa:
Dt
- Eat d? V = '^2Q$.
(5.14)
(3y^a
Using the identity
Da(-) Ds(-)
Dt
Dt
va,s V(-) and using dot notation for material time
derivatives allows us to rewrite the energy equation as
eapaea + £apavas Ve" £af : £ ~ V (£aqa) + £apaha = Q% (5-15)
84


The trouble with (5.15) is that the first and second terms contain the interal energy
density, e". To tie this equation back to the HMT framework weve used throughout
(and to give the equation a more natural set of dependent variables) we perform a
Legendre transformation to change the energy term into the Helmholtz potential via
the thermodynamic identity ea = + Trja. The energy equation is now written as
NT
j3y^a
a
£ P ~Dt
eapava's Vipa + eapaTi]a + £apaTvas Vrf
+ £apapaT + £apapavas VT £ata : da V (£aqa) + £apaha. (5.16)
Next we seek to remove the Helmholtz potential and entropy terms from the
energy equation. To do this we recall that the Helmholtz potential is a function of
all of the variables listed in (4.6). Under the assumptions listed in Section 5.2 we
drop the solid phase terms from this list. Furthermore, we know that under these
conditions the volume fractions are not independent so we could replace both £l and
£9 by saturation, S. This is not done (yet) as the entropy inequality was exploited
while assuming that they are independent. The switch can be made at any point
later. Therefore, under the present assumptions,
ipa = Ipa (£l, £9, pli, p9', T) forj = 1 : N.
Entropy, rja, is assumed to be a function of the same set of variables (since r)a =
_d%J)a/8T). Using the chain rule to expand all of the derivatives of -0" and rf in
equation (5.16) we arrive at an expanded form of the energy equation:

^ =£~p~
dtf)01
~dT
+ £apa
+ T
+ £apa
dipa
&£l
N
^ I dp
j=i
dt[ja
~dT
r9ya' T
dT
dp
d£l
dripc
+ T
£l +
d r]a
dtba dr]c
r + T- '
d£g d£g
N
+ pa + T
dpd
dr]a
~dT
p' + E La^
.7 = 1
VT
dtba dp'
r +T '
dpgi
p9j
85


Full Text

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HEATANDMOISTURETRANSPORTINUNSATURATEDPOROUSMEDIA: ACOUPLEDMODELINTERMSOFCHEMICALPOTENTIAL by EricR.Sullivan B.S.,IowaStateUniversity,1998 M.S.,UniversityofColoradoatColoradoSprings,2007 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy AppliedMathematics 2013

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ThisthesisfortheDoctorofPhilosophydegreeby EricR.Sullivan hasbeenapproved by LynnSchreyer-Bennethum,Advisor JulienLangou,Chair JanMandel RichardNa KathleenSmits Date:April19,2013 ii

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Sullivan,EricR.Ph.D.,AppliedMathematics HeatandMoistureTransportinUnsaturatedPorousMedia: ACoupledModelinTermsofChemicalPotential ThesisdirectedbyAssociateProfessorLynnSchreyer-Bennethum ABSTRACT Transportphenomenainporousmediaarecommonplaceinourdailylives.Examplesandapplicationsincludeheatandmoisturetransportinsoils,bakingand dryingoffoodstus,curingofcement,andevaporationoffuelsinwildres.Of particularinteresttothisstudyareheatandmoisturetransportinunsaturatedsoils. Historically,mathematicalmodelsfortheseprocessesarederivedbycouplingclassicalDarcy's,Fourier's,andFick'slawswithvolumeaveragedconservationofmass andenergyandempiricallybasedsourceandsinkterms.Recentexperimentaland mathematicalresearchhasproposedmodicationsandsuggestedlimitationsinthese classicalequations.Theprimarygoalofthisthesisistoderiveathermodynamicallyconsistentsystemofequationsforheatandmoisturetransportintermsofthe chemicalpotentialthataddressessomeoftheselimitations.Thephysicalprocesses ofinterestareprimarilydiusiveinnatureand,forthatreason,wefocusonusing themacroscalechemicalpotentialtobuildandsimplifythemodels.Theresulting coupledsystemofnonlinearpartialdierentialequationsissolvednumericallyand validatedagainsttheclassicalequationsandagainstexperimentaldata.Itwillbe shownthatunderamixturetheoreticframework,theclassicalRichards'equationfor saturationissupplementedwithgradientsintemperature,relativehumidity,andthe timerateofchangeofsaturation.Furthermore,itwillbeshownthatrestatingthe watervapordiusionequationintermsofchemicalpotentialeliminatesthenecessity foranempiricallybasedttingparameter. iii

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Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:LynnSchreyer-Bennethum iv

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DEDICATION ToJohnanna v

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ACKNOWLEDGMENT Thereareseveralpeoplewhohaveeitherdirectlyorindirectlymadethiswork possible.Firstandforemost,IwouldliketoextendmanythankstoLynnSchreyerBennethum.Withoutherpatience,support,amazingtalentsasateacher,mathematicalandphysicalinsights,andconstantquestionsIwouldlikelynothavebeen abletogetinvolvedsoheavilyinmathematicalmodeling.IwouldliketothankKate Smitsforagreeingtojoinmycommitteesolateandforsharingherexpertisewith experimentation.IwouldalsoliketothankJanMandel,JulienLangou,andRich Nafortheirunendingsupportandpatienceasmycommitteemembers. ThereareseveralpeopleintheUCDcommunitythatIwouldliketothank.To KeithWojciechowski,whowasoneoftheprimaryreasonsthatIstartedworking withLynn.ToCannanutChamsri,TomCarson,andMarkMueller,myresearch groupcomrades,forputtingupwithmyramblingsandnever-endingslueofequations duringourweeklymeetings.ToMarkespecially,forhisthoughtprovokingquestions andconstantpushtotiethemathandphysicstogether.ToJeLarsonandHenc Bouwmeester,whohavebecomelife-longfriends.Iappreciatedthecomicrelief,the deepmathematicalconversations,thecomputerhelp,andthefunthatwehad.Of course,Icannotforgettothankthemanyotherpeoplewhohelpedmetopersevere throughgraduateschool:JennyDiemunsch,CathyErbes,PiezhunZhu,BradLowry, TimMorris,SamanthaGraeo,andmanyother. Finally,Iwouldliketothankmyfamilyandfriends.Withoutmy supportnetwork thiswouldhavebeenmuchtougher.Tomyparents,DrewandDonna,whohave taughtmewhatrealhardworkreallyis.Tomysiblings,Doug,Emily,andJulie, forhelpingmegettothispointinmylife.ToEmily,especially,whowentthrough gradschoolalongwithme.TomyfriendsFredHollingworth,JodieCollins,and ErikSwansonwhoremindedmethatitwasequallyimportanttoheadupintothe mountainsandgoclimbingasitwastowork.Tomy teacherfriends ScottStrain, GingerAnderson,andKristaBrucknerwhohavehelpedtomakemeabetterteacher andbetterperson.Lastly,toJohnanna.Icouldn'thavedonethiswithoutyou. vi

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TABLEOFCONTENTS Figures.......................................xi Tables........................................xv Chapter 1.Introduction...................................1 1.1PreviousWork..............................1 1.2HybridMixtureTheoryandThesisGoals...............3 1.3ThesisOutline..............................5 2.Fick'sLawandMicroscaleAdvectionDiusionModels...........9 2.1ComparisonofFick'sLaws.......................9 2.2TransientDiusionModels.......................14 2.3Conclusion................................17 3.HybridMixtureTheory.............................19 3.1TheAveragingProcedure........................19 3.1.1TheREVandAveraging.....................20 3.2MacroscaleBalanceLaws........................26 3.2.1MacroscaleMassBalance....................27 3.2.2MacroscaleMomentumBalance.................30 3.2.3MacroscaleEnergyBalance...................31 3.3TheEntropyInequality.........................32 3.3.1ABriefDerivationoftheEntropyInequality..........33 4.NewIndependentVariablesandExploitationoftheEntropyInequality..36 4.1AChoiceofIndependentVariables...................37 4.1.1TheExpandedEntropyInequality...............41 4.2ExploitingtheEntropyInequality...................46 4.2.1ResultsThatHoldForAllTime................46 4.2.1.1FluidLagrangeMultipliers................47 vii

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4.2.1.2SolidPhaseIdentities...................47 4.2.2EquilibriumResults.......................50 4.2.2.1FluidStressTensor....................50 4.2.2.2MomentumTransferBetweenPhases..........51 4.2.2.3MomentumTransferBetweenSpecies..........52 4.2.2.4PartialHeatFlux.....................53 4.2.3NearEquilibriumResults....................53 4.3PressuresinMultiphasePorousMedia.................55 4.4ChemicalPotentialinMultiphasePorousMedia...........59 4.5DerivationsConstitutiveEquations...................62 4.5.1Darcy'sLaw...........................63 4.5.2Darcy'sLawInTermsofChemicalPotential..........65 4.5.3Fick'sLaw............................68 4.5.4Fourier'sLaw...........................70 4.6Conclusion................................73 5.CoupledHeatandMoistureTransportModel................75 5.1IntroductionandHistoricalWork...................75 5.1.1Richards'EquationforFluidFlow...............75 5.1.2PhillipandDeVries'DiusionModel.............77 5.1.3DeVries'HeatTransportModel................78 5.2Assumptions...............................79 5.3DerivationofHeatandMoistureTransportModel..........81 5.3.1MassBalanceEquations.....................81 5.3.2EnergyBalanceEquation....................84 5.3.2.1EnergyTransferintheTotalEnergyEquation.....88 5.3.2.2StressintheTotalEnergyEquation...........89 5.3.2.3TotalEnergyBalanceEquation.............91 viii

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5.4SimplifyingAssumptions{AClosedSystem.............97 5.4.1SaturationEquation.......................103 5.4.1.1CapillaryPressureandDynamicCapillaryPressure..104 5.4.2GasPhaseDiusionEquation..................111 5.4.3TotalEnergyEquation......................117 5.4.3.1DimensionalAnalysis...................121 5.4.4ConstitutiveEquations......................122 5.5ConclusionandSummary........................126 6.ExistenceandUniquenessResults.......................130 6.1SaturationEquationwith 6 =0....................130 6.2AltandLuckhausExistenceandUniquenessTheorems.......132 6.2.1ExistenceandUniquenessforRichards'Equation.......135 6.2.2VaporDiusionEquation....................138 6.2.3LimitsoftheAltandLuckhausTheorem...........142 6.3HeatTransportEquation........................143 6.4Conclusion................................146 7.NumericalAnalysisandSensitivityStudies..................148 7.1SaturationEquation...........................150 7.2VaporDiusionEquation........................158 7.3CoupledSaturationandVaporDiusion................165 7.4CoupledHeatandMoistureTransportSystem............168 7.4.1ExperimentalSetup,MaterialParameters,andIBCs.....168 7.4.2NumericalSimulations......................177 7.5Conclusion................................185 8.ConclusionsandFutureWork.........................189 ix

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Appendix A.MicroscaleNomenclature............................195 B.MacroscaleAppendix..............................197 B.1Nomenclature..............................197 B.2UpscaledDenitions...........................202 B.3IdentitiesNeededtoObtainInquality3.40..............205 C.ExploitationoftheEntropyInequality{AnAbstractPerspective.....206 C.1ResultsthatHoldForAllTime.....................207 C.2EquilibriumResults...........................208 C.3NearEquilibriumResults........................209 C.4LinearizationandEntropy.......................210 D.SummaryofEntropyInequalityResults....................212 D.1ResultsthatHoldForAllTime.....................212 D.2EquilibriumResults...........................213 D.3NearEquilibriumResults........................214 D.4ConstitutiveEquations.........................215 E.DimensionalQuantities.............................216 References ......................................218 x

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FIGURES Figure 3.1IllustrationofthedenitionoftheREVviaasequenceofporositiescorrespondingtoasequenceofshrinkingvolumes.ImagesimilartoFigures 1.3.1and1.3.2inBear[5]..........................21 3.2Cartoonofthemicroscale,REV,andmacroscaleinagranularsoil.The right-handplotdepictsthemixtureofallphases..............22 3.3LocalcoordinatesinandREV........................22 5.1Densitiesasfunctionsoftemperature....................99 5.2vanGenuchtenrelativepermeabilitycurves.Theredcurveshowsthe non-wettingphase, rnw S e ,andthebluecurvesshowthewettingphase, rw S e ,eachfor m =0 : 5 ; 0 : 67 ; 0 : 8 ; and1..................104 5.3Contactangleandeectiveradiusinacapillarytubegeometry. isthe contactangle, r istheeectiveradius,and istheradiusofcurvatureof theinterface..................................105 5.4ExamplesofvanGenuchtencapillarypressure-saturationcurvesforvariousparameters................................108 5.5Comparisonofdierentdiusionmodelsatconstanttemperature T = 295 : 15 K .Thevalueforthesaturatedpermeabilitywaschosentomatch thatof[78] S =1 : 04 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 m 2 ,wheretheyfoundattingparameter a =18 : 2.ThePresentModel"referstoequation.68with r T = 0 andnomasstransferandtheEnhancementModel"referstoequation .3alongwith.70,.72,and.73forthediusioncoecient, enhancementfactor,andtortuosityrespectively...............116 5.6JohansenthermalconductivitymodelwithC^ote-Konrad K e )]TJ/F20 11.9552 Tf 11.369 0 Td [(S relationshipwith =15plottedinblue,andtheweightedsumofthethermal conductivitiesoftheindividualphasesplottedinred............120 xi

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5.7Threeproposedfunctionalformsof = S ................124 5.8Levelcurvesofmasstransferratefunctions.................126 6.1Thefunction b h = S h )]TJ/F15 11.9552 Tf 11.955 0 Td [(1for m =0 : 8andvariousvaluesof .....136 6.2Kirchhotransformation K for m =0 : 8andvariousvaluesof .....137 7.1Cartoonofa1-dimensionalpackedcolumnexperimentalapparatus....148 7.2LogofPecletnumbersforvariousvaluesofsaturation.Thepointat =5 : 7 ;m =0 : 94indicatesthevaluesusedinSmitsetal.[78].Warmer colorsareassociatedwithhigherPecletnumberandthereforeassociated withanadvectivesolution...........................152 7.3Saturationprolesatvarioustimesinadrainageexperimentwith = 5 : 7 ;n =17...................................154 7.4ConvergencetestfordrainageexperimentdepictedinFigure7.3. N isthe numberofspatialgridpoints.InFigure7.3, t 1 =0 : 025 t c t 2 =0 : 050 t c t 3 =0 : 075 t c ,and t 4 =0 : 010 t c ........................156 7.5Saturationprolesinaimbibitionexperimentwith =2 : 5 ;n =5.....157 7.6ConvergencetestforimbibitionexperimentdepictedinFigure7.5.In Figure7.5, t 1 =0 : 025 t c t 2 =0 : 050 t c t 3 =0 : 075 t c ,and t 4 =0 : 010 t c ...158 7.7Samplediusionexperimentcomparingtheenhancementmodeltothe presentmodel.Here, a =25, m =0 : 9 n =10,and =10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(10 with Dirichletboundaryconditionsandanexponentialinitialprole.....160 7.8ComparisonofdiusioncoecientsforvariousvanGenuchtenparameters alltakenwith s =1 : 04 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(10 and =0 : 334tomatchtheexperiment in[78]......................................161 7.9ComparisonofdiusioncoecientsforvariousvanGenuchtenparameters alltakenwith s =4 : 0822 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(11 and =0 : 385tomatchtheexperiment in[68]......................................162 xii

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7.10Theblueandgreencurvesshowtheleft-handsideofequation.8for dierentsaturatedpermeabilities,andtheredlinesshowlevelcurvesfor right-handsideforvariousvaluesof a .Theblueandgreencurvescanbe usedtopredictthevalueof a beforeexperimentation............165 7.11Comparisonofcoupledsaturation-diusionmodelsforvariousweightsof C l withparameters: s =1 : 04 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 ;" =0 : 334, H 0 =10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 ; =4,and m =0 : 667...................................167 7.12SchematicoftheSmitsetal.experimentalapparatus.Saturationand temperaturesensorsnumbered1-11,temperaturesensors12-15,and relativehumiditysensors1and2[78].Thegeometric x coordinateis shownontheleft.Imagerecreatedwithpermissionfrom[78]......169 7.14Relativehumidityandtemperaturedatashowingmeasurementvariations intherstfewdaysoftheexperiment.Imagerecreatedwithpermission from[78]...................................170 7.13Brokensensordata.Saturationsensor#3showninblueandrelative humiditysensor#1showninred.Itisevidentfromtheseplotsthat thesesensorsarenotworkingproperlyastheygivenon-physicalreadings. Imagerecreatedwithpermissionfrom[78]................171 7.15Relativehumidityandtemperaturedataatawindowbeginningroughly 12.5daysintotheexperiment.Thiswindowischosensincethesensornoise isqualitativelyminimalinthisregion.Imagerecreatedwithpermission from[78]...................................172 7.16Approximationstorelativehumidityandtemperatureboundaryconditionsatthesurfaceofthesoil.........................173 7.17Approximateinitialconditionsatthe2000 th datapoint t 13 : 9days. Errorbarsindicateapproximatesensoraccuracy..............174 7.18Illustrationofhowthegas-phasedomainmightevolveintime.......175 xiii

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7.19Comparisonofrelativehumidityandsaturationforthefullycoupled saturation-diusion-temperaturemodelascomparedtodatafrom[78]. Boundaryconditionsaretakenfromasinusoidalapproximationofboundarydata.Thermalconductivitiesaretakenaseitherweightedsum.78 orC^ote-Konrad.80.............................183 7.20Comparisonofrelativehumidityandsaturationforthefullycoupled saturation-diusion-temperaturemodelascomparedtodatafrom[78]. Boundaryconditionsaretakenfromasmoothedsquarewaveapproximationofboundarydata.Thermalconductivitiesaretakenaseither weightedsum.78orC^ote-Konrad.80.................184 7.21Blowupcomparisonofrelativehumidityandsaturationforthefullycoupledsaturation-diusion-temperaturemodelascomparedtodatafrom [78].Theinsetplotsgiveacloserlookatthebehaviorexhibitedbythese particularsolutions..............................185 7.22Comparisonoftemperaturesolutionsforthefullycoupledsaturationdiusion-temperaturemodelascomparedtodatafrom[78].Boundary conditionsaretakenfromasinusoidalapproximationofboundarydata..186 7.23Comparisonoftemperaturesolutionsforthefullycoupledsaturationdiusion-temperaturemodelascomparedtodatafrom[78].Boundaryconditionsaretakenfromasmoothedsquarewaveapproximation ofboundarydata...............................187 xiv

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TABLES Table 2.1MassandmolaruxformsofFick'slaw..................10 7.1Measuredandpredictedvalueofthettingparameter a basedonequation .8......................................165 7.2Materialparametersforexperimentalsetup[78]...............170 7.3Relativeerrorsmeasuredusingequation.13fortheclassicalmathematicalmodelconsistingofRichards'equationforsaturation,theenhanced diusionmodelforvapordiusion,andthedeVriesmodelforheattransport.Thesearecomparedforthetwothermalconductivityfunctionsof interestweightedsum.78andC^ote-Konrad.80...........180 7.4Relativeerrorsmeasuredusingequation.13forinstanceswithinthe parameterspaceconsistingofthethermalconductivityfunctionweighted sum.78andC^ote-Konrad.80, C l ;C l T ; and .Thesearetakenfora smoothedsquarewaveapproximationtotheboundaryconditions.The starredrowsindicatefailureofthenumericalmethod,andtheerrorsfrom theclassicalmodelarerepeatedforclarity................181 7.5Percentimprovementofthepresentmodelovertheclassicalmodelusing equation.13astheerrormetric......................188 E.1Dimensionalquantities............................216 E.2Typicalvaluesofhydraulicconductivity K forwaterandair,andassociatedvaluesforpermeability .Notethat K = g= where g =1 kg=m 3 ; l =1000 kg=m 3 l =10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 Pa s ,and g =10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 Pa s ModiedfromBearpg.136[5].......................217 xv

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1.Introduction Waterow,watervapordiusion,andheattransportwithinvariablysaturated soilsabovethegroundwaterandbelowthesoilsurfaceareimportantphysicalprocessesinevaporationstudies,contaminanttransport,andagriculture.Themathematicalmodelsgoverningthesephysicalprocessesaretypicallycombinationsofclassicalempiricallawe.g.Darcy'slawandvolumeaveragedconservationlaws.The resultingequationsarevalidinmanysituations,butrecentexperimentalandmathematicalresearchhassuggestedmodicationsandcorrectionstothesemodels.The primarygoalofthisthesisistobuildathermodynamicallyconsistentmathematical modelforheatandmoisturetransportthattakestheserecentadvancementsintoconsideration.TorealizethisgoalHybridMixtureTheoryHMTandthemacroscale chemicalpotentialareusedastheprimarymodelingtools. 1.1PreviousWork In1856,HenriDarcypublishedhisresearchontheuseofsandlterstoclean thewatersourcesforthefountainsinDijon,France.Hefoundthattheuxofwater acrosssandlterswasdirectlyproportionaltothegradientofthepressurehead.This simpleobservationhasbecomeknownasDarcy'slawandisoneofthemainmodeling toolsinhydrologyandsoilscience[31].Darcy'slawisanexampleofahistoricalrule or law thathasperpetuatedtothepresentday.Darcy'slawwasoriginallyderived forsaturatedporousmedia,butneartheturnofthecenturyitwasextendedforuse inunsaturatedsoils.In1931,L.A.RichardscoupledDarcy'slawwithliquidmass balancetoderivewhatisnowknownasRichards'equation.Thisequationrelieson theassumptionthatDarcy'slawisvalidforunsaturatedmedia,butitalsorelieson anempirically-derivedrelationshipbetweenpressureandsaturation. Thepressure-saturationrelationshipisknowntobehystereticinnaturedependsonthedirectionofwetting,andonlyrecentlyhaveresearchersbeenableto movetowardfunctionalrelationshipsthatcapturethiseect[48].Correctionterms 1

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inRichards'equationhavebeenproposedviaHMTthatsuggestthattherateat whichthecapillarypressureischangingmayplayaroleintheoveralldynamicsof thesaturation[46,47].Thisproposed,third-order,terminRichards'equationhas onlyrecentlybeenstudiedmathematicallyandexperimentally.Onepossiblephysical interpretationofthistermisthatisaccountsforhowfasttheliquid-gasinterfaces arerearrangingattheporescale. Inthe1950s,PhilipanddeVriespublishedtheirworksonvaporandheattransport inporousmedia[32,61].Theirapproachaccountsforwaterowinboththeliquid anduidphasesinresponsetowatercontentandtemperaturegradientsinthesoil. InordertoaccountfortheobservationthatFickiandiusioninadequatelydescribes diusioninporousmedia,PhilipanddeVriesimplementedan enhancement factor, toadjustthediusioncoecient.Thisfactoristtedtothemeasureddiusiondata foraparticularmedium.In[24],Cassetal.foundthat increaseswithsaturation with 1fordrysoils.PhilipanddeVriesproposedthatthermalgradientsandthe condensationandevaporationthroughliquidislands"arethepore-scalemechanisms thatcausetheobservedenhancement.Counterintuitively,thegoverningequation holdsatthemacroscalewhilethesemechanismsareinherentlypore-scale.ThePhilip anddeVreismodelhasnotbeenvalidatedinalaboratorysetting. MorerecentworksquestionthevalidityofthePhilipanddeVriesmodel.Shokri etal.[73]suggeststhatthecouplingbetweenthewaterowandFickiandiusion isthekeytoestimatingthevaporux.Theysuggestthatunderthisconsideration thereisnoneedfortheenhancementfactor.Webb[80],andmorerecentlyShahareeni etal.[70],showedthatenhancementcanexistintheabsenceofthermalgradients. Itwasinitiallythoughtthatenhancementcouldn'toccurintheabsenceofthermal gradientsandwasthereforeignored.Basedontheobservtionthatenhancementcan occurwithouttheneedforthermalgradients,itisclearthatthePhilipanddeVries modelneedsmodication.Cass[24]andCampbell[23]giveafunctionalformofthe 2

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enhancementfactorthatiscommonlyusede.g.[68,78],butreliesonanempirical ttingparameter.InthepresentworkwetaketheviewofShokrietal.thatthereis noneedfortheenhancementfact,andwederiveadiusionequationsimplybasedon liquidandvaporow.Thenoveltyofthepresentapproachistheuseofthechemical potentialasthedrivingforceforbothtypesofow. Forenergytransport,the1958deVriesmodel[32]isstillcommonlyusede.g. [68,78].Similartotheenhanceddiusionmodel,deVriesbuiltthismodelsoas toaccountfortheuxoftheuidphases.Thisissensibleastheuidphaseswill certainlytransportheat.Morerecently,Bennethumetal.[14]andKleinfelter[52] usedHybridMixtureTheorytoderiveheattransportequationsinporousmedia Bennethumetal.studiessaturatedporousmediaandKleinfelterstudiedmultiscale unsaturatedmedia.TheyveriedmanyofthendingsbydeVriesbutalsoproposed severalnewtermsassociatedwiththephysicalprocessesofheattransport.Inthis workweextendtheBennethumetal.andKleinfelterapproachstounsaturated media. 1.2HybridMixtureTheoryandThesisGoals TobuildthemodelsinthisworkwemakeextensiveuseofHybridMixtureTheoryHMT.HMT,statisticalupscaling,andhomogenizationhaveallbeenusedas techniquestore-derive,conrm,andextendDarcy's,Fick's,andFourier'slawsin porousmedia.Foratechnicalsummaryofsomeofthesemethodssee[30].HMTisa termfortheprocessofusingvolumeaveragedpore-scaleconservationlawsalongwith thesecondlawofthermodynamicstogivethermodynamicallyconsistentconstitutive equationsinporousmedia.Thetechniqueasappliedtoporousmediawasdeveloped byseveralparties,themostnotablebeingHassanizadehandGray[39,43]andCushmanetal.[11,12,29],butthegeneralprinciplesweredevelopedbyColemanandNoll [27]. 3

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InthepresentworkweuseHMTtoderivenewextensionstotheselawsinthe caseofunsaturatedporousmedia.Theseextensionsarethenusedtoderiveamodel fortotalmoisturetransportinunsaturatedsoils.Whilethissortofmodelinghas beendoneinthepast,rarelyhavethethreeprinciplephysicalprocessmovement ofsaturationfronts,vapordiusion,andthermalconductionbeenconsideredfrom rstprinciplesandputonthesametheoreticalfootingHMTinthiscase.No knownworkattemptstocouplethesethreedierenteectstogetherwithonephysical measurement:thechemicalpotential.Thisisoneoftheuniquefeaturesofthiswork. Inthemostgeneralsense,thechemicalpotentialisameasureofthetendencyof asubstancethinkingparticularlyofauidorspeciestodiuse.Thisdiusioncould beofaspecieswithinamixturee.g.watervapordiusingthroughairoritcould beaphasediusingintoanothere.g.waterintoaDarcy-typesandlter.Thefact thatmostphysicalprocessesinporousmediaareofthistypegivesaninspirationfor thepotentialusefulnessofthechemicalpotentialasamodelingtool.Thatis,froma broadpointofview,itshouldbepossibletorestatethephysicalprocessesofmoving saturationfrontsandvapordiusionmorenaturallybythechemicalpotential.This approachhasnotbeenthoroughlyexploredinthepastsincethechemicalpotential isnotdirectlymeasureableandthetheoreticalfootingsofupscalingthechemical potentialarerelativelynew.Inthesaturatedcase,extensionstoDarcy'slawhave alsobeendevelopedviaHMT,andtheresultsindicatethatthemacroscalechemical potentialisaviablemodelingtoolfordiusivevelocityinsaturatedporousmedia [15,69,81].InthepresentworkwegivechemicalpotentialformsofDarcy'sand Fick'slawaswellaspresentingsimplicationstoFourier'slawbasedonthechemical potential.Inthecaseofapureliquidphasewewillshowthatthechemicalpotential formofDarcy'slawisnodierentthanthemoretraditionalpressureformulation.In thegasphase,ontheotherhand,wewillshowthatthepairingsofchemicalpotential formsofFick'sandDarcy'slawsgivesanewformofthediusioncoecientthat 4

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doesnotneedthe enhancementfactor indicatedintheworkbyPhillipandDeVries [61].ThechemicalpotentialwillnallybeusedtoderiveanovelformofFourier's lawforheatconductioninmultiphasemedia. Oncewehavederivednewformsoftheclassicalconstitutiveequationswepair theseequationswithvolumeaveragedconservationlawstogiveacoupledsystemof partialdierentialequationsgoverningheatandmoisturetransport.Thesecondlaw ofthermodynamicsisusedtosuggestadditionalclosureconditionsforeachofthe equations.Together,thesystemconsistsofanonlinearpseudo-parabolicequation forsaturation,anonlinearparabolicequationforvapordiusion,andanonlinear parabolic-hyperbolicequationforheattransport. Insummary,thisworkservesseveralpurposes:itisasteptowardbetter understandingtheroleofthechemicalpotentialinmultiphaseporousmedia,it makesstridestowardunderstandingthephenomenonof enhancedvapordiusion in porousmedia,andnallyweproposeanovelcoupledsystemofequationsforheat andmoisturetransport. 1.3ThesisOutline InChapter2wetakeastepbackfromporousmediaanddiscusspore-scale diusionmodels.Thisisdoneinanattempttoelucidatetheassumptions,derivations, andmodelsusedinvariousdisciplinesastheretendstobeconfusionaboutwherethe miriadofassumptionsarevalid.Inthischapterwegivemathematicalandphysical reasonsforthemanycommonlyusedassumptionsaswellasproposinganalternative advection-diusionmodelascomparedtothepopularBird,Stewart,andLightfoot model[18]. InChapter3wepresentthenecessarybackgroundinformationinordertounderstandvolumeaveragingandtheexploitationoftheentropyinequality.Muchof thischapterisparaphrasedfrompreviousworks,suchas[11,12,39,43,81,86].In thebeginningofChapter4weusethetoolsfromChapter3tobuildandexploita 5

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versionoftheentropyinequalityspecicformultiphasemediawhereeachphaseconsistsofmultiplespecies.TheremainderofChapter4isdedicatedtotheexploitation oftheentropyinequalityforanovelchoiceofindependentvariablesdescribingthese media.AppendixCservesasacompaniontothisdiscussionasitgivestheabstract formulationandlogicoftheentropyinequality.ThroughoutChapter4,thegoalisto derivenewformsofDarcy's,Fick's,andFourier'slawsandtoproposeextensionsto theselawsintermsofthemacroscalechemicalpotential.Aspartofthesederivations wearriveatnewexpressionsforthepressureandwettingpotentialsinunsaturated media.Allofthesederivationsaredoneinageneralsensewithasfewassumptions aspossible.Thisleavesopenthepossibilitiesoffutureresearch. InChapter5wecoupletheresultsfoundfromtheexploitationoftheentropy inequalityChapter4withthevolumeaveragedconservationlawsderivedinChapter 3.InSection5.1,amorein-depthhistoricalperspectiveoftheclassicalequationsused forheatandmoisturetransportisgiventoorientthereadertotherecentresearch. FluidtransportequationsarepresentedinSection5.3.1alongwithadiscussionof therelationshipbetweenmasstransferandchemicalpotential.Considerableeort isputtowardderivingaheattransportequationwiththenalequationpresented inSection5.3.2.3.InSection5.4severalsimplifyingassumptionsarepresentedin ordertoclosethesystemofequations.Inparticular,Sections5.4.1,5.4.2,and5.4.3 givesimplications,assumptions,anddimensionalanalysisfortheliquid,gas,and heatequationsrespectively.InSection5.4.4wepresenttheremainingconstitutive equationsnecessarytoclosethesystemofequations.Sincesomanyassumptionsand simplicationsaremadethroughoutthechapterasummaryofalloftheresultsis presentedinSection5.5. InChapter6weexaminetheproperregularityandassumptionsneededforexistenceanduniquenessofsolutions.Theseresultsarepreliminaryanddonotconstitute acompleteexistenceanduniquenessstudyfortheseequations. 6

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InChapter7weperformnumericalanalysisontheequationsderivedinChapter5. InSections7.1,7.2,and7.3weexaminenumericalsolutionsandparametersensitivity forthesaturationequation,vapordiusionequation,andthecoupledsaturationvapordiusionequationsrespectively.InSection7.4wecomparenumericalsolutions tothefullycoupledheatandmoisturetransportmodeltotheexperimentaldata collectedin[78]. InChapters5-7weworktowardbuildingandanalyzingthesaturation,vapor diusion,andheatequations.Theowofthoughtforthesechaptersistoapplyeach setofnewassumptionsorsimplicationstoeachofthethreeequationsbeforemoving tothenextsetofassumptions.Thatis,ifasetofassumptionsareproposedthenthe subsequentsectionswillapplythoseassumptionstothesaturation,vapordiusion, andheatequationsinturn.Onlythenwillthenextsetofassumptionsbediscussed. Thisisdonesothateachsetofassumptionsareonlystatedonceandsincemanyof theassumptionscreateinterleavingeectsbetweentheequations. Finally,asanaidtothereaderthereareseveralappendices.AppendixAcontains anomenclatureindexforthepore-scalediusionprocessesconsideredinChapter2. AppendixB.1containsanomenclatureindexforthemacroscaleresultsintheremainingchapters.Thereissomeoverlapbetweenthenomenclatureforthesedistinct parts,andeorthasbeenmadetonotcreateanyexcessivenotationalconfusions eventhoughthisworkisnecessarilynotationheavy.AppendixB.2givesalistin alphabeticalorderoftheupscaleddenitionsofvariablesdenedinchapters3and4. Asmentionedpreviously,AppendixCgivesanabstractviewoftheentropyinequalityinaneorttomaketheexploitationprocessmorecleartotheinterestedreader. AppendixDgivesasummaryoftheresultsextractedfromtheentropyinequality inChapter4.Thisisdoneforeaseofreferencemostlyontheauthor'spart,butit isalsodonetoprovideanindexoftheseresultsforuseinfutureresearch.Finally, AppendixEgivesseveraltablesofdimensionalquantitiesusedthroughout. 7

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Itissuggestedthatthedetail-orientedreaderhaveAppendixAathandwhen readingChapter2andAppendixB.1athandwhenreadingchapters3-5.There aresomeminorabusesofnotation,buteortwasmadetobringthemtothereader's attentionwheneverpossibleandtousenotationthatdidn'tconfusetheimmediate discussion. 8

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2.Fick'sLawandMicroscaleAdvectionDiusionModels Thischapterconsistsofashorttechnicalnoterelatedtopore-scalediusionproblems.Vapordiusioninmacroscaleporousmediaisanimportantphenomenonwith manyapplicationse.g.evaporationfromsoils,moisturetransportthroughlters,and CO 2 sequestration.Inordertobetterunderstandmacroscalediusionitbehooves theresearchertorstunderstandpore-scalemechanicsandmodels.Thischapterattempltstoelucidatethemodelsandassumptionsusedfordiusionatthepore-scale sothatwhenweturnourattentiontomacroscalediusionwearermlygrounded. AsecondarygoalofthischapteristogiveathoroughdiscussionofthediusioncoefcientusedinFick'slaw.Thisisnecessarysincethiscoecientistypicallywrongly assumedconstantforallchoicesofdependentvariablesmassconcentration,molar concentration,chemicalpotential,etc.. Tomakematterssimpler,wefocusourpore-scalediscussiononthe,socalled, Stefandiusiontubeproblem.Thisisawell-studiedproblemthatmodelsthediffusionofaspeciesthroughanidealbinarygasmixtureabovealiquid-gasinterface [6,18,22,33,51,84,85].Thisisanidealizationofthejuxtapositionofphasesina capillarytubegeometry,andacapillarytubegeometryisanidealizationofgeometry ofpore-scaleporousmedia.Toderiveamathematicalmodelforthetimeevolution oftheevaporatingorcondensingspecies,onetypicallycouplesFick'srstlawwith themassbalanceequation. InSection2.1wediscussthevariousformsofFick'slawandbrieydiscussthe relationshipsbetweenthediusioncoecients.InSection2.2wederivethetransientdiusionequationsassociatedwithFick'slawandcomparewiththeassociated equationofBird,Stewart,andLightfoot[18]henceforthreferredtoasBSL. 2.1ComparisonofFick'sLaws Forasystemconsistingofanidealmixtureofwatervapor, g v ,andinertair, g a ,Fick'slawcanbewrittenintermsofmolarconcentration,massconcentration, 9

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Table2.1:MassandmolaruxformsofFick'slaw FluxType FluxExpression Fick'sLaw massux[ ML )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 T ] J g j = g j v g j ;g J g j = )]TJ/F20 11.9552 Tf 9.298 0 Td [( g D r C g j .1 molarux[ mol L )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 T ] J g j c = c g j v g j ;g c J g j c = )]TJ/F20 11.9552 Tf 9.298 0 Td [(c g D c r x g j .2 massux[ ML )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 T ] J g j ; = g j v g j ;g J g j ; = )]TJ/F20 11.9552 Tf 9.299 0 Td [(D ; g j R g j T r g j .3 moleux[ mol L )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 T ] J g j ;c = c g j v g j ;g c J g j ;c = )]TJ/F20 11.9552 Tf 9.299 0 Td [(D ;c c g j RT r g j c .4 orthechemicalpotential.Thisispotentiallyconfusingsincethereareinherently dierentdiusioncoecientsforthedierentformsofFick'slaw.Thepurposeof thissubsectionistoclarifytherelationshipsbetweenthesecoecients.Inporous mediaitiscommontousemassuxforFick'slaw,butinchemistryandrelated eldsitismorecommontousemolarux.Assuch,wewillmakemostofour comparisonsbetweenmassandmolarux. AccordingtoBSL[18],themassandmolarformsofFick'slawaregivenby equations.1and.2modiedfromBSLTable17.8-2.InTable2.1, v g j ;g = v g j )]TJ/F40 11.9552 Tf 11.485 0 Td [(v g isthediusivevelocityrelativetoamassweightedvelocity, v g j ;g c = v g j )]TJ/F40 11.9552 Tf 11.485 0 Td [(v g c isthediusivevelocityrelativetoamoleweightedvelocity, g j isthemassdensity ofspecies j C g j = g j = g isthemassconcentrationofspecies j inthemixture, c g j = mol g j =vol g isthemolardensityofspecies j ,and x g j = c g j =c g isthemolar concentrationofspecies j inthemixture. ThechemicalpotentialformsofFick'slawcanbegivenintermsoftwodierenttypesofchemicalpotential:massweightedequation.3ormoleweighted equation.4.Inphysicalchemistryandthermodynamics[21,56]thechemical potentialisknownasthe tendencyforaspeciestodiuse ,andforthisreasonitisa naturalcandidateforthestatementofFick'slawanexactthermodynamicdenition 10

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willbepresentedinsubsequentchapters.Inequations.3and.4, g j c isthe moleweightedchemicalpotential[ J mol )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 ]and g j isthemassweightedchemical potential[ JM )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 ]. Thereadershouldrstnotethatthetwouxesaremeasuredwithrespectto dierentvelocities.Themassweightedvelocityis g v g = P j = v;a g j v g j andthemole weightedvelocityis c g v g c = P j = v;a c g j v g j where v g j isthevelocityofthespecies relativetoaxedcoordinatesystem.Thismeansthatthattherecannotbeadirect comparisonbetweenthetwodierenttypesofuxwithoutconsideringthemrelative tothesameframeofreference.Usingthesedenitionsof v g and v g c weseethatthe dierencebetweenthetwobulkvelocities, v g )]TJ/F40 11.9552 Tf 11.956 0 Td [(v g c is v g )]TJ/F40 11.9552 Tf 11.955 0 Td [(v g c = N X j =1 g j )]TJ/F20 11.9552 Tf 11.955 0 Td [(x g j g g v g j : .5 In.5,thesummationover j indicatesthatthisisanaccumulationoverthe N speciesinthegasmixture.Infutureworkwewillbeinterestedinthediusionof watervapor j = v andwillconsiderthegasmixtureasbinary: j 2f v;a g ,where j = a representsthemixtureofallspeciesthatarenotwatervapor.Thereforewe canwrite.5as v g )]TJ/F40 11.9552 Tf 11.955 0 Td [(v g c = X j = v;a g j )]TJ/F20 11.9552 Tf 11.955 0 Td [(x g j g g v g j = x g a C g v )]TJ/F20 11.9552 Tf 11.955 0 Td [(x g v C g a v g v + x g v C g a )]TJ/F20 11.9552 Tf 11.955 0 Td [(x g a C g v v g a : .6 Convertingtoamassweightedvelocityweseethat v g j ;g c = v g j ;g + v g )]TJ/F40 11.9552 Tf 12.658 0 Td [(v g c ,and thereforethemolaruxis J g j c = c g j v g j ;g c = c g j v g j ;g + c g j v g )]TJ/F40 11.9552 Tf 12.51 0 Td [(v g c .Assumingthat c g j x g k C g l 1weseethatthedierencebetweentheframeofreferenceispotentially quitesmall. Nextnotethatthediusioncoecientsareinitiallyassumedtobedierentfor eachchoiceofindependentvariableasindicatedbythesubscripts.Tocompare D 11

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and D c wenotethat J g j = m g j J g j c and r C g j = m g j m g k x g j m g j + x g k m g k 2 r x g j .7 toconcludethat C g k x g k D = D c ; .8 where m g j isthemolarmassofspecies j andtheminuscule, k ,representsthe other species.Ifthemolardensityformofthediusioncoecient, D c ,isassumedtobe constantatconstanttemperatureweconcludethatthemassdensityversionofthe diusioncoecientisnotconstantandvisaversa.Thefraction, C g k =x g k canbe interpretedastheratioofthemolarmassofspecies k tothemolarmassofthe mixture.If j = v isthewatervaporinanair-watermixturethen k = a istheinert airandthescalingfactorbetweenthediusioncoecientsistheratioofmolarmass oftheairtothemolarmassofthemixture.Forsucientlydilutesystemswhere theamountofwatervaporissmalltheratioisapproximately1andthediusion coecientscanbeconsideredasapproximatelyequal. Foridealair-watermixtures,thedensitiesarerelatedthrough g = g v + g a and thewatervapordensityisrelatedtotherelativehumiditythrough g v = g v sat .Here wearetaking g v sat asthesaturatedvapordensityand astherelativehumidity.At standardtemperatureandpressurewenotethat g v sat 0 : 02 kg=m 3 and g 1 kg=m 3 Thisindicatesthatatstandardtemperatureandpressurewecanlikelyassumethat themixtureisalways sucientlydilute .Therefore,inthesystemsunderconsideration wecanassumethatthediusioncoecientsareapproximatelyequal. ForthediusioncoecientsassociatedwiththechemicalpotentialformsofFick's lawwerstobservethatifwemultiplyanddividetheright-handsideofthemolar formbythemolarmassofspecies j then J g j ;c = )]TJ/F20 11.9552 Tf 9.298 0 Td [(D ;c c g j RT m g j m g j r g j c = )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( D ;c m g j g j R g j T r g j : .9 12

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Here, R g j = R=m g j isthespecicgasconstant,andwehaveused g j = g j c =m g j Againnotingthat J g j ; = m g j J g j ;c weconcludethat D ;c = D ; Itremainstocompare D to D ; and D c to D ;c .Wefocushereonthemass uxeswithoutlossofgenerality.Ifthemassuxesareequal,theninparticular g D r C g v = D g v R g v T r g v : Rearranging,itcanbeseenthat D r C g v = C g v D r g v R g v T assumingconstanttemperature.Fromphysicalchemistry[56],recallthatthechemicalpotentialisrelatedtoareferencechemicalpotential g v andtheratioofpartial pressure, p g v ,tobulkpressure, p g ,via g v = g v + R g v T ln p g v p g : .10 Therefore, r g v R g v T = r ln p g v p g = p g p g v r p g v p g : .11 UsingDalton'slawforidealgases, p g = p g v + p g a ,andusingthespecicgasconstants wenotethatthepartialpressureofspecies j canbewrittenas p g j = R g j T g j Therefore, p g = R g v T g v + R g a T g a ; .12 and,aftersimplifying, p g v p g = g v g v + )]TJ/F21 7.9701 Tf 6.675 -4.977 Td [(R g a R g v g a : .13 FromthevaluesfoundinAppendixEweseethat R g a =R g v 0 : 6andtherefore equation.13issimilar,butnotequalto,themassconcentration, C g v = g v = g v + g a .Dening C g v = p g v =p g weseethat D D = C g v C g v r C g v r C g v ; .14 13

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wheredivisionisunderstoodcomponentwisethatis,equation.14representsthree equationswhenthegradientisunderstoodinthreespatialdimensions.Therighthandsideofequation.14isnotconstantat1foralldensities,butthevariation intheright-handsidedependsmostlyonthevariationin g a inthegasmixture. Fortunately,thewatervapordensityismuchsmallerthantheair-speciesdensity, andhence g a isapproximatelyconstant.Inonespatialdimension,theright-hand sideof.14canthereforebeapproximatedby d x := )]TJ/F21 7.9701 Tf 12.086 -4.976 Td [(x x +1 )]TJ/F21 7.9701 Tf 15.379 -4.976 Td [(x x +0 : 6 d dx )]TJ/F21 7.9701 Tf 15.379 -4.977 Td [(x x +0 : 6 d dx )]TJ/F21 7.9701 Tf 12.086 -4.976 Td [(x x +1 where x = g v and g a 1.Itiseasytoshowthat0 : 98
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Themassbalanceequationforspecies j inthegasphasecanbewrittenas @ g j @t + r g j v g j =^ r g j .15 where v g j isthevelocityofspecies j withinthegasmixturerelativetoaxedframeof reference,and^ r g j isamassexchangetermaccountingforchemicalreactionsbetween species[81,85].Inthepresentworkweassumethatnochemicalreactionsoccur,and therefore^ r g j =0.Thecombinationofthemassbalanceequationwiththemassux formofFick'slawfor j = v givesatransportequationforthemassofwatervapor viaadvective, g v v g ,anddiusive, J g v ,uxes: @ g v @t + r J g v + g v v g =0 : .16 SubstitutingthemassuxformofFick'slaw 1 fromequation.1weget @ g v @t + r g v v g = D r g r C g v : .17 Noticeherethatthediusioncoecienthasbeenfactoredoutofthedivergence operator.Thisisonlyvalidinconstanttemperatureenvironments.Ifthegas-phase densitywereconstantinspacethenwewouldarriveatthetraditionaladvection diusionequationbydividingequation.17by g ofbyrewritingthediusionterm as D r r g v andwouldneedanexpressionforthebulkvelocityintermsofdensity orconcentrationtoclosetheequation.Unfortunately,ifthedensityofthewater vaporisallowedtovarythenthedensityofthegasvaries.Again,for suciently dilute mixturesthevariationingas-phasedensityisverysmallandthenonlinear diusionontheright-handsidecanbeapproximatedbythelineardiusionterm D r r g v .Itshouldbenotedherethatthislatercaseiswhatistypicallythought ofasFick'slaw"andiswhatleadstothetraditionallineardiusionequationwhen theadvectiontermisneglected[28]. 1 Thesubscriptsontheuxandthediusioncoecienthavebeendroppedsincealloftheversions presentedinTable2.1areapproximatelyequal 15

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Adierentformofequation.17,suggestedinBSL[18],isderivedbyconsideringthemassweightedbulkvelocity.Inabinarysystem, g v v g v = g v v g v ;g + g v v g = g v v g v ;g + g v C g v v g v + C g a v g a : Solvingfor g v v g v g v v g v = g v 1 )]TJ/F20 11.9552 Tf 11.956 0 Td [(C g v v g v ;g + g v v g a : .18 UsingFick'slawfor g v v g v ;g ,andeliminating g v v g v in.15with.18gives @ g v @t + r g v v g a = r D g 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(C g v r C g v : .19 Whitaker[85]suggestedthatonecandevelopconvincingargumentsinfavorof..." neglectingtheair-speciesuxterm.Certainlyatsteadystatewecanassumeasis doneinBSLthat v g a isapproximatelyzeroattheinterfacesincethereisnonet motionof[watervapor]awayfromtheinterface"[18],butinthetransientcasethis wouldconstituteachangeofframeofreference.Thisnewframeofreferencewould besuchthattheinertairmoleculesareviewedasstationarywiththewatervapor diusingthroughthem. Ineitherequation.17or.19onemustndappropriateconditionsorequationstoeitherneglectorrewritetheadvectiveterm, g v v g or g v v g a respectively. Typicallythistermisneglectedinapurediusionproblem.Asthesearetwodierentsimplicationsofthesameequationonemusthavedierentreasonsforneglecting theadvectiveterm.Theeasiestxforthisissueistocouplewitheitherthebulk gasmassbalanceequationortheair-speciesmassbalanceequationandtousethe mass-weightedvelocity: g v g = P N j =1 g j v g j .Thepointbeingthatonecannotsimply neglecttheadvectionterminthetransientcaseofeitherequationwithoutproperconsiderationoftheimplications:axedbulkvelocityorachangingframeofreference respectively. 16

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Analcommentcanbemakeregardingequations.17and.19.Thebulk densityterm, g ,ontheright-handsideoftheseequationsisoftenfactoredoutofthe divergenceoperator.Thisisanerrorcommittedbyseveralresearchers[18,22,51]. Thereasoningforassumingthatthedensityisconstantandhencereturningtoa lineardiusionmodelintheabsenceofadvectionisthatinanidealgas, p g = g R g T Underconstanttemperatureconditions,andifthepressureisassumedconstant,then thedensityisassumedbeconstant.Therearetwopossiblemistakeshere.:Ifthe speciesdensitiesareallowedtovarythenthebulkdensitymustvary.:Thevalue of R g willvarywiththechangingcompositionofthemixturesincethemolarmass ofthemixturechanges.Theeectofthisisthat,whilethepressuremayremain constant,thecomponentpartsarenotnecessarilyconstantandthereforecannotbe factoredfromthedivergenceoperator. 2.3Conclusion InthischapterwehavecomparedvariousformsofFick'slawformoleculardiusion.Wehaveshownthat,whilethediusioncoecientsareindeeddierent,under certaincommoncircumstancesthediusioncoecientscanbeconsideredasapproximatelyequal.Itiscommontotakethediusioncoecientasconstantoronlya functionoftemperature,andinmanycasesitissafetoassumethesamediusion coecientmaybeusedinthecommonformsofFick'slaw. Inthetransientcasetherearetwonaturalformulationsforthemassuxform ofFick'ssecondlaw.Ineithercase,thenaturalgoverningequationisanonlinear advectiondiusionequationthatmustbeclosedwiththeuseofanothermassbalance equation.Whenconsideringtheadvectionterm,itistheauthor'sopinionthatequation.17isthemorenaturalchoice.Thereasonforthisisthatthebulkvelocity, v g ,islikelymorenaturallymeasuredascomparedtothatofthespeciesvelocity.This chapterconcludesourdiscussiononpore-scalemodeling.Wenowturnourattention tobuildingmacroscalemodels,butindoingsowekeepinmindthediusionmodels 17

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attheporescaleandusecuesfromthisscaletohelpmakeproperassumptionsabout thelargerscale. 18

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3.HybridMixtureTheory Inthischapterweuseacombinationofclassicalmixturetheoryandrational thermodynamicshenceforthcalledHybridMixtureTheoryHMTtostudynovel extensionstoDarcy'slaw,Fick'slaw,andFourier'slawinvariablysaturatedporous media.ThisapproachwaspioneeredbyHassanizadehandGrayinthe70'sand80's [39,43,44,45]andlaterextendedbyBennethum,Cushman,Gray,Hassanizadeh, andmanyothers[29,30,41,81]tomodelmulti-phase,multi-component,andmultiscalemedia.HMTinvolvesvolumeaveraging,orupscaling,pore-scalebalancelaws toobtainmacroscaleanalogues.Thesecondlawofthermodynamicsisthenused toderiveconstitutiverestrictionsonthesemacroscalebalancelaws.Constitutive relationsareparticulartothemediumbeingstudied,andhencedependonajudicious choiceofindependentvariablesfortheenergyofeachphaseinthemedium.Thereare manyexcellentresourcesforthecuriousreadertogainamorethoroughunderstanding ofHMTeg[30,81].Forthatreasonwewillnotderiveeveryidentityalongthe way.Insteadpartialderivationsoftheidentitiesnecessarytounderstandthepresent applicationofHMTarepresented. Tobeginthisoverviewweconsidertheupscalingofpore-scalebalancelawsconservationlawsviaamixturetheoreticapproach.Thesubsequentsectionsinthis andthenextchapterintroducetheentropyinequalityandit'sexploitationtoderive constitutivelaws.Ajudiciouschoiceofindependentvariablesfortheenergyofeach phaseinthemediumischosenandisusedtoderivenovelversionsofDarcy's,Fick's, andFourier'slaws.Theseconstitutiveequationswillbeusedinsubsequentchapters todevelopmodelsformoisturetransportinvariablysaturatedporousmedia. 3.1TheAveragingProcedure Whenconsideringaporousmediumonecannotavoiddiscussingthevariousscales involved.Thisparticularworkdealswithtwoprincipalscales:themicroscaleand 19

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themacroscale.Atthemicroscalethephasesareseparateanddistinguishable.Typicalmicroscaleporousmediawillhaveporesthatmeasureontheorderofmicrons tomillimetersdependingonthetypeofsolid.Atthemacroscalethephasesare indistinguishableandthetypicalmeasurementsrangefrommillimeterstometers. Themacroscaleiswheremostphysicalmeasurementsaremade,andassuch,weseek toderivegoverningequationsthatholdatthisscale.Themicroscalestructuremay varydramaticallyfordierentmediadependingonthetypeofsolidphaseandthe microscalebehavioroftheuidphases.Assuch,themicroscalegeometrycanhavea dramaticinuenceonowandphaseinteraction. Foranygivenphaseattheporescalethemass,linearmomentum,angularmomentum,andenergybalancelawsmusthold.Theproblemisthatitisdicultto obtaingeometricinformationeverywhereatthisscaleForthisreasonweseekto averageor upscale themicroscalebalancelawstothemacroscale. Therearemanymethodsformathematicallyaveragingbalancelaws.Herewe choosethesimplestmethodofweightedintegration.Beforeintroducingthetechnical detailsoftheweightedintegrationwemustrstintroducetheconceptofaRepresentativeElementaryVolumeandlocalgeometryinaporousmedium.Thiselementary volumewillbecomeourbasicunitofvolumethroughoutthisresearch.ThefollowingdiscussionscloselyfollowandparaphrasethoseofBear[5],Bennethum[12,13], HassanizadehandGray[39,43],Weinstein[81],andWojciechowski[86]. 3.1.1TheREVandAveraging Inthisworkweconsiderunsaturatedporousmedia.Characteristictothesemedia isthejuxtapositionofliquid,solid,andgasphaseswithintheporematrix.Wemake theassumptionthatarepresentativeelementaryvolumeREV,inthesenseofBear [5],existsateverypointinspace.ToproperlydenetheREVwerstdenethe porosity. 20

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Considerasequenceofsmallvolumeswithinaporousmedium, V k ,eachwith centroid x 2 R 3 .Foreach k ,let V void k bethevolumeofthevoidspacewithin V k .Theporosityforthe k th volumeisgivenastheratio k = V void k V k : .1 Generatethesequence, f k g ,bygraduallyshrinking V k about x suchthat V 1 > V 2 > V 3 > .As k increases,theporositywillcertainlyuctuatedueto heterogeneitiesinthemedium.As V k shrinkstherewillbeacertainvalue, k = k suchthatfor k>k theuctuationsinporositybecomesmallandareonlydueto uctuationsinthearrangementofthesolidmatrix.If V k isreducedwellbeyond V k thesequenceofvolumeswilleventuallyconvergeto x .Thepoint, x only lieswithinonephase,sothelimitofthesequenceofporositieswilleitherbe0or1 completelyinthevoidspaceorcompletelyinthesolid.Thisindicatesthatthere willbesomeotherintermediatevolume, V k < V k ,wherethesequenceof porositiesbeginstouctuateagainas k getslarger.WedenetheREV, V ,asany particularvolume V k
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Figure3.2:Cartoonofthemicroscale,REV,andmacroscaleinagranularsoil.The right-handplotdepictsthemixtureofallphases. Considernowacoordinatesystemsuperimposedontheporousmedium.Let x bethecentroidoftheREV,andlet r besomeothervectorinsidetheREV.Dene thevector, ,asavectororiginatingfromthecentroidoftheREVsuchthat r = x + : .2 Wecannowview asalocalcoordinateintheREVasinFigure3.3. V x r Figure3.3:LocalcoordinatesinandREV. Denethe phaseindicatorfunction as r ;t = 8 > < > : 1 ; r 2 -phase 0 ; r 62 -phase ; .3 where r isapositionvectorasindicatedinFigure3.3.Theaveragingtechnique involvedmultiplyingamicroscalequantitysuchasdensityby andintegrate overtheREV.Thiseectivelysmearsoutthephases.Aconsequenceofthisisthat 22

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theaveragedvaluemaynotaccuratelyrepresenttheactualvaluesbeingmeasuredat theporescale.Afurthermathematicalcomplicationarisessincetheintegrationsmay notmakesenseinthetraditionalRiemanniansense.Therefore,wemustunderstand allofthefollowingmathematicsinthedistributionalsenseintegralsareunderstood tobeLebesgue,andderivativesareunderstoodtobegeneralizedderivatives.For morespecicsonthesemathematicaltoolsseestandardgraduatetextsonfunctional analysiseg.[59]. Tondthevolumeofthe phaseintheREVwesimplyintegrate overthe REV.Denethisvolumeas j V j : j V j = Z V r ;t dv = Z V x + ;t dv : .4 The -phasevolumefraction, ,isdenedas 1 x ;t = j V j j V j : .5 Since0 j V jj V j itisclearthat0 1.Furthermore,sincetheREVis madeupofallofthephases, X =1 : .6 Thevolumefractionistherstexampleofamacroscalevariable.Thatis,itis avariablethatdescribesapore-scalepropertybutisupscaledtothelarger,more measurable,scale. Itisusefultonotethattherearetwomaintypesofaveragingthatwillbeused: massaveragingandvolumeaveraging[43,41,81].Let j bethe j th constituentof somequantityofinterest.Tovolumeaverage j wedene j = 1 j V j Z V j r ;t r ;t dv ; .7 1 Note:thenotation "forthevolumefractionisnotnecessarilystandard.Someauthorsuse ", ",or n ".Furthermore,thesuperscriptnotationissometimesreplacedbysubscripts. ThepresentnotationischosentobeconsistentwiththeprimaryreferencesforHybridMixture Theorymentionedintheintroductiontothischapter. 23

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andtomassaverage j wedene j = 1 h j i j V j Z V j r ;t j r ;t r ;t dv : .8 Implicitlyin.8weseethatdensityisvolumeaveraged.Thatis, j = 1 j V j Z V j r ;t r ;t dv : .9 Note:someauthorsusethemassaveragednotationondensityeventhoughitis technicallyvolumeaveraged.Giventhedenitionofamassaveragedquantitythere isusuallylittleconfusion. Thebasicrulesofthumbfordecidingwhethertovolumeormassaveragewere originallyproposedbyHassanizadehandGreyin1979[43].Theyproposefourcriteria,listedbelow,formakingthisdecision.Inthesecriteriaitisemphasizedthat themicroscalequantitiescorrespondtosmallscalepre-averagedquantities,while macroscalequantitiesaredenedviatheaveragingprocess. 1.Whenanaveragingoperationinvolvesintegration,theintegrandmultiplied bytheinnitesimalelementofintegrationmustbeanadditivequantity.For example,theinternalenergydensityfunction, E ,isnotadditive,butthetotal internalenergy, Edv isadditiveandanaveragedenedintermsofthisquantity willbephysicallymeaningful." 2.Themacroscopicquantitiesshouldexactlyaccountforthetotalcorrespondingmicroscopicquantity.Forexample,totalmacroscopicmomentumuxes throughagivenboundarymustbeequaltothetotalmicroscopicmomentum uxesthroughthatboundary." 3.Theprimitiveconceptofaphysicalquantity,asrstintroducedintotheclassicalcontinuummechanicsmustbepreservedbyproperdenitionofthemacroscopicquantity.Forinstance,heatisamodeoftransferofenergythrougha 24

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boundarydierentfromwork.Thedenitionofmacroscopicheatuxmust alsobeamodeofenergytransferdierentfrommacroscopicwork." 4.Theaveragedvalueofamicroscopicquantitymustbethesamefunctionthatis mostwidelyobservedandmeasuredinaeldsituationorinlaboratorypractice. Forexample,velocitiesmeasuredintheeldareusuallymassaveragedquantities;therefore,themacroscopicvelocityshouldbeamassaveragedquantity." Thisensuresapplicabilityoftheresultingequations. Intheupscalingproceduretofollowwewishtoapplyaweightedintegrationtoa pore-scalebalancelawapartialdierentialequation.Thiswillinvolvetermssuch as Z V @ j @t dv; andtoensurethatweproperlydenethemacroscalevariablesaseithervolume ormassaveragedquantities,weneedatheoremthatallowsfortheinterchangeof integrationanddierentiation.ThistheoremisduetoWhitakerandSlattery[75,82, 83]andageneralizationofthistheoremisduetoCushman[29]. Theorem3.1AveragingTheorem If w isthemicroscopicvelocityofthe interfaceand n istheoutwardunitnormalvectorof V indicatingthattheintegrand shouldbeevaluatedinthelimitasthe interfaceisapproachedfromthe side, then 1 j V j Z V @f @t dv = @ @t 1 j V j Z V f dv )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X 6 = 1 j V j Z A f w n da .10a 1 j V j Z V r f dv = r 1 j V j Z V f dv + X 6 = 1 j V j Z A f n da; .10b where f isthequantitytobeaveraged. 25

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Keepinmindthat f couldbeascalaroravectorquantity.Inthelattercase,the symbol f isreplacedwith f andappropriatetensorcontractionsareinserted. Theaveragingprocedureisnowcarriedoutinthefollowingsteps: 1.Statethepore-scalebalancelawforaparticularspeciesorphase. 2.Multiplytheequationby 3.AverageeachtermovertheREV. 4.ApplyTheorem3.1toarriveattermsrepresentingmacroscalequantities. 5.Denephysicallymeaningfulmacroscopicquantities. Wenowturnourattentiontoaveragingpore-scalebalancelawsinthesenselisted above.Inthefollowingdiscussion, v j isthemicroscopicvelocityofconstituent j w j isthevelocityofthe j th constituentinthe interface,and n istheoutwardunit normalvectorof V .AfullnomenclatureindexcanbefoundinAppendixB.1. 3.2MacroscaleBalanceLaws Asitisthesimplestbalancelaw,letusrstconsiderthemassbalanceequation forasingleconstituent: D j Dt + j r v j = j ^ r j : .11 Here, j isthedensityoftheconstituent, v j isthevelocityoftheconstituent,and anysourceofmassfromchemicalreactionsbetweentheconstituentsisgivenas^ r j RecallthatthematerialLagrangianderivativeis D j Dt = @ @t + v j r .12 ThisderivativecontainstheusualEulerianderivativealongwithanadvectiveterm. WrittenintermsoftheEuleriantimederivative,.11is @ j @t + r )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( j v j = j ^ r j : .13 26

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Whilethisisspecicallythemassbalanceequation,ittakestheprototypicalformof allbalancelaws:atimederivativeplusauxisequaltoanysource. Theconstituentmomentumbalancecanbewritteninasimilarmanner: @ j v j @t + r )]TJ/F20 11.9552 Tf 5.48 -9.683 Td [( j v j v j )]TJ/F40 11.9552 Tf 11.955 0 Td [(t j = j g + ^ i j +^ r j v j : .14 Here, t j istheCauchystresstensoronspecies j ,andthesourcesontheright-hand sidearegravity,momentumtransferfromotherconstituents,andmomentumgained fromchemicalreactionsrespectively.Theseequationsdescribethechangeinmass andmomentumovertimeandspacewithinaspecicconstituent.Theyaresucient eldequationsformodelingsystemscomposedofasinglephasegas,liquid,orsolid, butequations.13and.14areinsucientformodelingmultiphaseandmulticonstituentsystemsasamixturebecausetheinteractionsbetweenthephasesand constituentsarenotpresent. Inthisworkweconsideraporousmediumconsistingofasolidphaseandtwo uidphaseswithmultipleconstituentswithineachphase.Thephaseswillbedenoted as = l;g; and s forliquid,gas,andsolidrespectively.Theconstituentswillbe enumerated j =1: N usingMATLAB-stylenotationtoindicate j =1 ; 2 ;:::;N ThefollowingderivationsfollowsimilarderivationsgivenbyGray[43],Weinstein[81] andWojciechowski[86]. 3.2.1MacroscaleMassBalance Toobtainmacroscaleequationsinmultiphaseandmulti-constituentmediawe multiplyaconstituentbalanceequationsbythephaseindicatorfunction, ,integrate over V ,anddivideby j V j .Applyingtheaveragingtheorem.1totheappropriate termsinequation.13wehave 1 j V j Z V @ j @t dv = @ @t j V j j V jj V j Z V j dv )]TJ/F26 11.9552 Tf 11.956 11.358 Td [(X 6 = 1 j V j Z A j w j n da 27

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= @ @t j )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X 6 = 1 j V j Z A j w j n da; .15a 1 j V j Z V r )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( j v j dv = r j V j j V jj V j Z V j v j dv + X 6 = 1 j V j Z A j v j n da = r j v j + X 6 = 1 j V j Z A j v j n da; and .15b j V j j V jj V j Z V j ^ r j dv = j ^ r j : .15c Substitutingequations.15a-cintoequation.13,recognizingthevolume fractionterms,andrecognizingtheaveragedmassandvelocitygivestheupscaled massbalanceequation: @ j @t + r j v j = X 6 = 1 j V j Z A j )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(w j )]TJ/F40 11.9552 Tf 11.955 0 Td [(v j n + j ^ r j : .16 Rewritingequation.16intermsofthematerialtimederivativeanddening j j ; .17 v j v j ; .18 ^ e j j V j Z A j )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(w j )]TJ/F40 11.9552 Tf 11.955 0 Td [(v j n da; and.19 ^ r j j ^ r j .20 respectivelytobetheaveragedmassover V ,themassaveragedvelocity,thenet rateofmassgainedbyconstituent j inphase fromphase ,andtherateofmass gainduetointeractionwithotherspecieswithinphase ,weget D j j Dt + j r v j = X 6 = ^ e j +^ r j : .21 28

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Nextwedenethecorrespondingbulkphasevariablessothatthemacroscale equationsareconsistentwithexperimentallymeasuredtermsasmuchaspossible. Dene: N X j =1 j ; and.22 C j j .23 respectivelytobethemassdensityofthe phase,andthemassconcentrationofthe j th constituentinthe phase.Ifequation.21isrewrittenas @ C j @t + r C j v j = X 6 = ^ e j +^ r j andthensummedover j =1: N weobtainamassbalanceequationforthe phase: @ @t + r v = X 6 = ^ e : .24 Herewehaveusedthefactthat^ e = P N j =1 ^ e j ;therateofmasstransfertothe phasefromthe phaseisthesumoftheratesofmasstransfertoeachindividual constituentinthe phasefromthe phase. Nowusethedenitionofthematerialtimederivativetowritethemassbalance equationforthe phaseas D Dt + r v = X 6 = ^ e ; .25 wherethefollowingrestrictionshavebeenapplied: N X j =1 ^ r j =0 ; 8 ; and.26 X X 6 = ^ e j =0 ;j =1: N: .27 Restriction.26statesthattherateofnetgainofmasswithinspecies from chemicalreactionsalonemustbezero.Equation.27statesthattherateofmass gainedbyphase fromphase isequaltotherateofmassgainedbyphase from phase 29

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3.2.2MacroscaleMomentumBalance Wenowturnourattentiontothemomentumbalanceequation,.14.Wecan applythesameprinciplestoupscalethisequationforfulldetailssee[81].The macroscopiclinearmomentumbalanceequationforconstituent j inthe phaseis j D j v j Dt )]TJ/F43 11.9552 Tf 11.956 0 Td [(r )]TJ/F20 11.9552 Tf 5.479 -9.683 Td [(" t j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" j g j = ^ i j + X 6 = ^ T j ; .28 andthemacroscopiclinearmomentumbalanceequationforthe phaseis D v Dt )]TJ/F43 11.9552 Tf 11.955 0 Td [(r )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(" t )]TJ/F20 11.9552 Tf 11.955 0 Td [(" g = X 6 = ^ T ; .29 where t j and t aretheCauchystresstensorsforthespeciesandthephaseand ^ T j and ^ T aremomentumtransferterms.Mostspecically,forthemomentum transferterms,theformerrepresentsmomentumtransferedtoconstituent j inthe phasethroughmechanicalinteractionsfromphase ,andthelatterrepresents themomentumtransferedtophase throughmechanicalinteractionsfromphase Alsonotableisthe ^ i j term.Thistermrepresentstherateofmomentumgaindue tomechanicalinteractionswithotherspecieswithinthesamephase. Intheprocessesofderivingtheseequationsthefollowingrestrictionswereenforced: N X j =1 ^ i j +^ r j v j ; =0 8 ; and.30 X X 6 = ^ T j + v j ^ e j =0 j =1: N: .31 Restriction.30statesthatlinearmomentumcanonlybelostduetointeractions withotherphasesnotwithinthespecies,andrestriction.31statesthatthe interfacecanholdnolinearmomentum.Thecommainthesuperscriptof.30 indicatesarelativeterm: v j ; = v j )]TJ/F40 11.9552 Tf 13.062 0 Td [(v .Foracompletelistofnotationsee AppendixB.1. 30

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Lastly,totiethemomentumtransferandstresstensorforthe phasetothose ofthespecieswenotetwoidentitiesthatwereusedinthederivation: t = N X j =1 )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(t j )]TJ/F20 11.9552 Tf 11.956 0 Td [( j v j ; v j ; .32 ^ T = N X j =1 ^ T j +^ e j v j ; : .33 Theseidentitieswillbeusedlaterandsoarepresentedhereforconciseness. 3.2.3MacroscaleEnergyBalance Thederivationsforthemacroscaleangularmomentumandenergybalancelaws aremorealgebraicallycomplicated.Theangularmomentumequationwillnotbe usedinthisworksinceweassumethatwe'redealingwithgranular-typemediawhere theangularmomentumbalanceresultsinthesolidphaseCauchystresstensorbeing symmetric[15,43,81].Theenergybalanceequation,ontheotherhand,willallow ustoderiveanovelformoftheheatequationinporousmedia.Forthisreasonwe statethefullequationhere. Applyingthesameroutineasinthemassandlinearmomentumequationswe arriveaftersignicantsimplicationatabalancelawfortheenergyinspecies j : j D j e j Dt )]TJ/F43 11.9552 Tf 11.955 0 Td [(r q j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" t j : r v j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" j h j = ^ Q j + ^ Q j .34 see[7,81]fordetailsonthederivation.Here, h j istheexternalsupplyofenergy, e j istheenergydensity, q j isthepartialheatuxvectorforthe j th component ofthe phase, ^ Q j istherateofenergygainduetointeractionwithotherspecies withinthe phase,and ^ Q j istherateofenergytransferfromthe phasetothe phasenotduetomassormomentumtransfer. Again,followingthederivationof[81],thebulkphaseenergyequationis D e Dt )]TJ/F43 11.9552 Tf 11.956 0 Td [(r q )]TJ/F20 11.9552 Tf 11.955 0 Td [(" t : r v )]TJ/F20 11.9552 Tf 11.955 0 Td [(" h a = X 6 = ^ Q .35 31

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where Toarriveatthisformoftheenergyequationweenforcedthefollowingrestrictions: N X j =1 ^ Q j + ^ i j v j ; +^ r j e j + 1 2 v j ; 2 =0 8 ; and.36a X X 6 = ^ Q j + ^ T j v j +^ e j e j + 1 2 v j ; 2 =0 j =1: N: .36b Restriction.36astatesthatenergygainedorlostduetospeciesinteractionswithin the phasemustbegainedorlostduetointeractionswithotherphases.Restriction .36bstatesthattherateofenergygainedorlostbyonecomponentinonephase mustgotoanothercomponentorphase.Thatis,thissecondrestrictionstatesthat theinterfaceretainsnoenergy. Asystemofequationsgovernedbymass,momentum,andenergybalancerequires eachoftheupscaledequationslisted.Acountofthevariablesindicatesthatthere arefarmorevariablesthanequations.Itisatthispointwhereweneedamethod forderivingconstitutiveequationsfortheseremainingvariables.Themethodchosenforthisworkusesanothermacroscalebalancelawbasedonthesecondlawof thermodynamics. 3.3TheEntropyInequality Thedevelopmentofconstitutivelawsiscentraltothemodelingprocess.As wementionedpreviously,thishashistoricallybeenaprocessofttingmathematicalmodelstoempiricalevidence.TheconstructofHybridMixtureTheoryHMT couplestheaveragingtheoremsdiscussedintheprevioussectionandthesecondlaw ofthermodynamicstoprovideuswithrestrictionontheformoftheconstitutive relations;hencenarrowingdowntheexperimentsrequiredtothosethatarethermodynamicallyadmissible.Itisthenuptotheexperimentaliststoverifyandrenethese models.Boththeoreticalandexperimentaldirectionsofstudyhavetheirmerits,but 32

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puttingtheconstitutiveequationsonarmtheoreticalfootingisultimatelypreferred whetheritisbeforeoraftertheexperimentsarerun.Inthissectionwegiveabrief derivationoftheupscaledentropyinequality,andwethenusethisinequality,along withajudiciouschoiceofvariables,toderiveconstitutiveequationsforunsaturated porousmedia. 3.3.1ABriefDerivationoftheEntropyInequality Thesecondlawofthermodynamicsstatesthatentropywillneverdecreaseasa systemevolvestowardequilibrium[4,21].Themicroscaleentropy balance equation thatdescribesthisphenomenonis j D j j Dt + r j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j b j =^ j +^ r j v j + ^ ; .37 where j istheentropydensityofconstituent j j istheentropyux, b j isthe externalsupplyofentropy,^ j isentropygainedfromotherconstituents,and ^ isthe entropyproduction.Sincethesecondlawofthermodynamicsmustholdweknow that ^ 0foralltime. ApplyingTheorem3.1toequation.37anddeningappropriatemacroscale denitionsofthevariablesgivestheupscaledentropybalanceequation: j D j j Dt )]TJ/F43 11.9552 Tf 11.955 0 Td [(r j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" j b j = X 6 = ^ j +^ j + ^ j ; .38 wherethetermsontheright-handsideofequation.38representtransferofentropy throughmechanicalinteraction,entropygainedduetointeractionswithotherspecies, andtherateofentropygenerationrespectively. Next,assumethatthematerialwearemodelingis simple inthesenseofColeman andNoll[27].Thismeansthatweassumethattheentropyuxandexternalsupply areduetoheatuxesandsourcesrespectively.Toremovethedependenceonexternal heatsourcesweadd =T timestheupscaledconservationofenergyequation.34, j D j e j Dt )]TJ/F43 11.9552 Tf 11.955 0 Td [(r q j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" t j : r v j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" j h j = ^ Q j + X 6 = ^ Q j : 33

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AtthispointweperformaLegendretransformationinordertoconvertconvert internalenergy, e j ,toHelmholtzpotential, j seeanythermodynamicstext,eg [21]: j = e j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j T: .39 Thisisdonebecauseinternalenergyhasentropyasanaturalindependentvariable,andentropyisdiculttomeasureexperimentally.Itshouldbenotedthat theHelmholtzpotentialisonlyonechoiceofthermodynamicpotentialwecouldhave made.Thisisdoneprimarilyforhistoricalreasons,buttheGibbspotentialandpossiblytheGrandCanonicalpotentialcouldhavealsobeenviablechoices.Theappealof theHelmholtzpotentialisthatitnaturallyhasindependentvariablesoftemperature andvolumeor,inintensivevariables,density. Toarriveatasimpliedentropyinequalityforthetotalproductionofentropy acrossallconstituentsandphaseswenowsolvefor ^ j andthensumover = l;g;s and j =1: N .Thissteprequiressignicantalgebrasothedetailsofthederivationare omittedforbrevitysake.Aftermuchsimplication,theentropyinequalitybecomes 0 ^ = X )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(" T D Dt + D s T Dt + T N X j =1 t j : d + r T T 2 q )]TJ/F21 7.9701 Tf 16.804 14.944 Td [(N X j =1 t j v j ; )]TJ/F20 11.9552 Tf 11.955 0 Td [( j v j ; j + 1 2 v j ; v j ; )]TJ/F15 11.9552 Tf 14.467 8.088 Td [(1 T N X j =1 X 6 = ^ T j + ^ i j + r j j v j ; + T N X j =1 )]TJ/F40 11.9552 Tf 5.479 -9.683 Td [(t j )]TJ/F20 11.9552 Tf 11.956 0 Td [( j j I : r v j ; )]TJ/F15 11.9552 Tf 14.467 8.088 Td [(1 T X 6 = n ^ T + r T o v ;s )]TJ/F15 11.9552 Tf 17.394 8.088 Td [(1 2 T N X j =1 X 6 = ^ e j +^ r j v j ; v j ; 34

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)]TJ/F15 11.9552 Tf 11.811 8.088 Td [(1 T X 6 = ^ e + 1 2 v ;s v ;s ; .40 where ^ istherateofentropygeneration. Severalnewtermshaveappearedin.40.First, d = r v sym istherateof deformationtensoralsoknownasthestrainrate.Asbefore,termswithacomma inthesuperscriptarerelativeterms: v a;b = v a )]TJ/F40 11.9552 Tf 11.955 0 Td [(v b Severalidentitieswereneededtoderive.40.Acompletelistoftheseidentities hasbeenincludedinAppendixB.3.ThenextstepistoexpandtheHelmholtz potentialintermsofconstitutiveindependentvariablesthatdescribeoursystem. Thisallowsfreedomtomakechoicesaboutwhichvariablescontrolbehaviorofthe system.Thechoiceofthesevariablesisgenerallynon-trivialsointhenextsection wediscussmotivationsforthechoiceofvariables. 35

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4.NewIndependentVariablesandExploitationoftheEntropy Inequality Nowthatwehaveanexpressionfortheentropyinequalitywemustchooseaset ofindependentvariablesthatdescribesoursystemofinterest.Weseektodescribe amultiphasesystemwherethesolidphasemayundergonitedeformation,where therelativesaturationsofthetwouidphasesvaryintimeandspace,andwhere phasechangesbetweentheuidspossiblyoccursthroughouttheporousmedium. HassanizadehandGrayhavemodeledsimilarmediainthepast[41,44,45].These modelsincludeeectsfromcommoninterfaces,commonlineswherethreephases meet,andcommonpointswherefourphasesmeet.Thesemodelsareverythorough andfollowthesameHMTapproach.Thedownsidestotheirmodels,intheauthor's opinion,arethreefold:thecomplexityoftheresultingequationsissuchthatin ordertousetheseequationsahostofsimplifyingassumptionsmustbemade,the thermodynamicsofthecommonpointsandlinesmakesensephysicallybutarelikely negligiblerelativetoothereects,andconstitutiveequationsmustbederivedfor transferratesbetweeninterfaces,commonlines,commonpoints,andphases.This naldrawbackindicatesthatadetailedknowledgeofthepore-scalephysicsmust besomehowupscaled.Approacheshavebeentakenrecentlytodojustthis,butthe proposedtheorieshavenotyetgainedwidespreadacceptance.Examplesofsuchwork includethoseofGrayetal.[42,40] Inthepresentapproachwechoosenottodirectlymodelinterfacesandinstead strivetoeventuallywriteourgoverningequationsintermsofthemacroscalechemical potential.Thechemicalpotentialisknownfromphysicalchemistryandthermodynamicsasa generalizeddrivingforce thatisafunctionofpressureandtemperature. Itiswellknownthatmasstransferfromliquidtogasstatesisdrivenbygradients ofchemicalpotential[56],soifwecanwriteconstitutiveequationssuchasDarcy's, Fick's,andFourier'slawsintermsofthispotentialwecanpossiblycoupletherele36

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vanteectsintomuchsimplergoverningequations;forexample,equationsthattrack changesinchemicalpotentialinsteadofpressureorconcentration.Theimmediate drawbacktothepresentmodelingeortsisthattherecentworkbyHassanizadehet al.seemstoindicatethatsaturationandcapillarypressurearelinkedtotheamount ofinterfacialareabetweenphaseswithinthemedium[20,48].Inthepresentwork wewillnotdirectlymodeltheuid-uidanduid-solidinterfaces.Weproceedwith thepresentmodelingeortdespitetheresultsproposedbyHassanizadehetal.We willdiscussthisdrawbackaswerunupagainstitinfuturesectionsandchapters. 4.1AChoiceofIndependentVariables InthissectionwepresentachoiceofindependentvariablesfortheHelmholtzfree energypotentialsoastoexpandtheentropyinequalityandtoderivetherelevant formsofDarcy'slaw,Fick'slaw,andFourier'slaw.Thesevariablesareknownas constitutive independentvariablesastheyrepresentapostulationofthevariablesthat controltheenergyinthesystem.Derivingphysicallymeaningfulresultsdependson ourabilitytorelatethermodynamicallydenedvariablestophysicallyinterpretable quantities"[81].Tothatend,weuseouraprioriknowledgeofthermodynamicsto choosesomeofthevariables.Fortheremainderofthisworkwerestrictourattention toathree-phasesystemconsistingofanelasticsolid,aviscousliquidphase,andagas phase.Tobeginthemodelingprocessweassumethateachofthesephasesconsists of N constituentsalsocalledspeciesorcomponents,andallinterfacialeectsare neglected.Examplesoftheconstituentsincludedissolvedmineralsintheliquid, speciesevaporatedintothegas,orprecipitatedmineralsassociatedwiththesolid phase. Themotivationforchoosingsomeofthevariablesisrelativelytrivial.Forexample,toallowforaheatconductingmedium,temperature, T ,andthegradientof temperature, r T ,areincludedinthelistofindependentvariables.Theporespaceis expectedtobevariablysaturatedwiththetwouidphasessothevolumefractions, 37

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" l and g ,mustbeincludedinthesetofvariables.Thefactthat P =1precludes usfromusingallthreevolumefractionssincetheyarenotindependentofeachother. Infuturechapterswewillfurtherrestrictthisassumptionsinceforarigidsolidphase thesumoftheuidphasevolumefractionsisequaltothexedporosity l + g = ": .1 Thereasonfornotmakingthisassumptioninitiallyisthatitallowsustodevelop modelsfordeformablemediaaswellasformediawitharigidsolidphasehence,a moregeneralmodelmaybederivedfromtheseassumptionslaterifnecessary. RecallfromthermodynamicsthatthechangeinextensiveHelmholtzpotential, A ,withrespecttovolumeisminusthepressure: @A=@V = )]TJ/F20 11.9552 Tf 9.298 0 Td [(p .Intermsofintensive variablesthismeansthat 2 @=@ = )]TJ/F20 11.9552 Tf 9.299 0 Td [(p .Toremainconsistentwiththeextensive denitionoftheHelmholtzpotential,thedensitiesmustthenbeincludedintheset ofindependentvariables.Giventhefactthatthereare N constituentsineachphase, thiscouldbedoneintwodierentways:wecouldincludethemassconcentrations, C j ,for j =1: N )]TJ/F15 11.9552 Tf 12.405 0 Td [(1alongwiththephasedensity,orwecouldincludeallof theconstituentdensities, j for j =1: N .Bennethum,Murad,andCushman [15],andalsoWeinstein[81]tooktherstoftheseoptionswhenusingHMTto deriveconstitutiverelationsinvolvingchemicalpotentials.Thetroublewiththis approachisthatthemassconcentrationofthe N th constituentisdependentonthe massconcentrationsoftheprevious N )]TJ/F15 11.9552 Tf 13.117 0 Td [(1constituentssincetheconcentrations sumto1.Theseresultsindicatethatthebehavioroftheconstituentsdepends onhowtheyarelabeledinsteadofsimplybeingindependent.Varioustechniques weresuccessfullydevelopedin[15]todealwiththiscomplication.Toavoidthese complicationswechoosethesecondoptionandincludethespeciesdensities, j for j =1: N .Sinceeachconstituentisfreetomovewithineachphase,thespatial gradientsofthespeciesdensities, r l j and r g j ,arealsoincluded. 38

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Darcy'slawandFick'slawareclassicalempiricalexpressionsforcreepingow andconstitutivediusion.Darcy'slawisastatementabouttherelativevelocityof auidphaseinaporousmedium,andFick'slawisastatementabouttherelative diusivevelocityofaspecieswithinaphase.Sinceweseeknovelformsofthesetwo lawsweinclude v ;s and v j ; for = l;g;s inthelistofindependentvariables.It shouldbenotedthatneitherofthesevariablesis objective inthesensethattheyare notframeinvariant.Thisposesaproblemsinceanygoverningequationshouldnot dependonanobserver'sframeofreference.In[34],Eringenproposedamodication toDarcy'slawthatcreatesaframeinvariantrelativevelocity.Thenewtermsneeded forthisnewrelativevelocityaresecondorderandareassumedtobenegligiblein Darcyow.AsimilarargumentcanbeusedforFick'slaw. Thereasoninggiveninthepreviousfewparagraphsleadsustothesetofindependentvariablesfor toinclude: T; r T;" l ;" g ; l j ; g j ; r l j ; r g j ; v l;s ; v g;s ; v l j ;l ; v g j ;g ; and v s j ;s where j =1: N .Itisapparent,now,thatsolid-phasetermscorrespondingtothe densityandgradientofdensityaremissing.Theprincipleofequipresence,from constitutivetheoryincontinuummechanics,statesthatallconstitutivevariables areafunctionofthesamesetofindependentvariables"[69].Togivesymmetry betweenthephasesweinclude s and r s .TheStokesassumptionfortheCauchy stresstensorinaviscousuidstatesthatstressisthesumoftheuidpressureand thestrainrate.Forthisreasonweincludethestrainratealsoknownastherate ofdeformationtensorfortheuidphases: d l and d g .Thetheoryofequipresence alsostatesthatifweincludestrainrateintheuidphasesthenwemustincludea comparableterminthesolidphase. Anaturalchoiceofvariablesforthesolidphasearethesolidphasevolumefraction,density,andtheaveragedstrain.Weinstein[81]pointedoutthatthesethree variablesarenotindependent,asexplainedbelow,andusedamodiedsetofinde39

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pendentvariablesforthesolidphase.Thesamemodiedsetwillbeusedhere,so thefollowingsimplystatesWeinstein'sresultswithbriefderivations. Let J s betheJacobianofthesolidphasegivenby J s =det )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [( F s T F s ,where F s isthedeformationgradient F s = @x s k @X s K ; .2 x s istheEuleriancoordinate,and X s istheLagrangiancoordinate.Usingstandard identitiesfromContinuumMechanics,theJacobiancanberewrittenas J s =det )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 E s + I : .3 Furthermore,throughtheconservationofmass,theJacobianisalsoascalingfactor forvolumetricchanges, J s = s 0 s 0 = s s .Thisclearlyshowsthedependenceofthe threevariables.Tomitigatethisissue,Weinstein[81]adoptedideasfromsolidmechanicsandconsideredamultiplicativedecomposition"ofthedeformationgradient, F s ,andtheGreen'sdeformationtensor, C s ,as C s = J s 2 = 3 C s ; .4 F s = J s 1 = 3 F s ; .5 where J s 1 = 3 I and J s 2 = 3 I representvolumetricdeformation,and F s and C s are themodieddeformationgradientandthemodiedrightCauchy-Greentensor,respectively."Withthismodicationtothesolidstrain,thesolidphasevariableswe considerhereare J s ; C s ;C s k and r C s k where k =1: N )]TJ/F15 11.9552 Tf 12.638 0 Td [(1.Wenoteherethat inordertogetphysicallymeaningfulresultsforphasechange,weincludethesame components,sothat C s j ;C l j ; and C g j allrefertothesamecomponent.Pairingthe massconcentrationsandtheJacobiangivesadescriptionofthedensityofthesolid phase,andthemodiedCauchy-Greentensorisusedinplaceofstrain. Theprincipleofequipresencestatesthatalloftheconstitutivevariablesmustbe afunctionofthesamesetofthepostulatedindependentvariables.Inparticular,we 40

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postulatethattheHelmholtzpotentialforeachphaseisafunctionofthefollowing setofvariables: T; r T;" l ;" g ; l j ; g j ; r l j ; r g j ; v ;s ; d l ; d g ; v j ; ;J s ; C s ;C s k ; r C s k ; .6 where = l;g;s ; j =1: N and k =1: N )]TJ/F15 11.9552 Tf 10.079 0 Td [(1.Wepostulatethatathreephaseporous mediumwithanelasticsolidphaseand N constituentsperphasecanbemodeledby set.6. 4.1.1TheExpandedEntropyInequality Considernowthattherstlineoftheentropyinequality,.40,containsamaterialtimederivativeoftheHelmholtzpotentialforthe phase.Usingtheidentity D Dt = D s Dt + v ;s r ; .7 andapplyingthechainrule,theentropyinequalitycanbeexpandedtoincludeeach ofourconstitutiveindependentvariables.Thecentralideatotheexploitationofthe secondlawofthermodynamicsisthatnotermintheentropyinequalitycantake valuessuchthatentropygenerationisnegative.Acloseexaminationoftheexpanded entropyinequalityrevealsthattherearemanytermsthatshowuplinearly.Inthese lineartermswenoticesomethatareneitherindependentnorconstitutive.Examples ofsuchcoecientsare r T; r C s j ; r l j ; r g j ; d l ; d g ; v l;s ; v g;s ; v s j ;s ; v l j ;l v g j ;g T ,_ j and r v j ; wherethedotnotatione.g. T indicatesamaterialtimederivative. Looselyspeaking,wehavenocontroloverthesevariablesandtheycouldtakevalues thatviolatethesecondlaw.Forexample,takeasathoughtexperimentaprocess whereallofthesevariablesexcept T arezero.FromBennethum[10], Sincenoneoftheothertermsintheentropyinequalityareafunction of T ,byvaryingthevalueof T wecanmaketheleft-handsideofthe entropyinequalityaslargepositiveoraslargenegativeaswewant-hence violatingtheentropyinequality.Sincetheentropyinequalitymusthold 41

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forallprocessesincludingthoseforwhich T isanyvalue,theentropy inequalitycanbeviolatedunlessthecoecientof T iszero." Inordernottoviolatetheinequalityin.13,thecoecientsofallofthesefactors mustbezero.Thisimpliesthattermssuchas P )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" @ @ r T arezeroandwill thereforebeleftoutoftheexpansionof.40forbrevity.Thetimeratesofchange ofvolumefractionsarenotthistypeofvariablesincetheyareconstitutive;thatis, weassumearuleforthetimeratesofchangeofvolumefractionsthatdependson thespecicmediumofinterest. Withthissimplicationinmind,.40becomes T = X @ @T + T )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X @ @" l l )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X @ @" g g )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X N )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 X j =1 @ @C s j C s j )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X N X j =1 @ @ l j l j )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X N X j =1 @ @ g j g j )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X @ @J s J s )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X @ @ )]TJETq1 0 0 1 343.766 353.43 cm[]0 d 0 J 0.478 w 0 0 m 10.601 0 l SQBT/F40 11.9552 Tf 343.766 343.554 Td [(C s : C s )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( l l @ l @T + l r T + @ l @" l r l + @ l @" g r g + N )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 X j =1 @ l @C s j r C s j + N X j =1 @ l @ l j r l j + N X j =1 @ l @ g j r g j + @ l @J s r J s + @ l @ C s : r )]TJETq1 0 0 1 321.945 247.677 cm[]0 d 0 J 0.478 w 0 0 m 10.601 0 l SQBT/F40 11.9552 Tf 321.945 237.801 Td [(C s + ^ T l s + ^ T l g # v l;s )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( g g @ g @T + g r T + @ g @" l r l + @ g @" g r g + N )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 X j =1 @ g @C s j r C s j + N X j =1 @ g @ l j r l j + N X j =1 @ g @ g j r g j + @ g @J s r J s + @ g @ C s : r )]TJETq1 0 0 1 322.807 132.996 cm[]0 d 0 J 0.478 w 0 0 m 10.601 0 l SQBT/F40 11.9552 Tf 322.807 123.12 Td [(C s + ^ T g s + ^ T g l # v g;s + X " N X j =1 t j : d # 42

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+ X " r T T q )]TJ/F21 7.9701 Tf 16.804 14.944 Td [(N X j =1 t j v j ; )]TJ/F20 11.9552 Tf 11.955 0 Td [( j v j ; j + 1 2 v j ; 2 # )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X N X j =1 X 6 = ^ T j + ^ i j + r j j v j ; + X N X j =1 )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(t j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j j I : r v j ; )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 X N X j =1 X 6 = ^ e j +^ r j v j ; 2 )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X X 6 = ^ e + 1 2 v ;s 2 0.8 ThenextstepistoenforcetwoadditionalrelationshipsusingLagrangemultipliers.Indoingso,theLagrangemultipliersbecomeunknownsofthesystem.Wewill seeinsubsequentsectionsthattheLagrangemultipliersareassociatedwithpartial pressuresandchemicalpotentialsofspeciesintheuidphases.Therstrelationship consideredisthedependenceofthediusivevelocities: N X j =1 C j v j ; = 0 : .9 Onecanseethissince N X j =1 C j v j ; = N X j =1 C j v j )]TJ/F21 7.9701 Tf 16.804 14.944 Td [(N X j =1 C j v = v )]TJ/F40 11.9552 Tf 11.955 0 Td [(v = 0 : .10 Theimplicationisthatifweknowtheconcentrationsanddiusivevelocitiesofthe rst N )]TJ/F15 11.9552 Tf 11.159 0 Td [(1constituents,thenwewouldknowtheconcentrationanddiusivevelocity ofthe N th constituent.Multiplyingbythedensity,takingthegradient,andusing theproductrulegivesthefollowingrelationship: r N X j =1 j v j ; = N X j =1 j r v j ; + v j ; r j = 0 : .11 FollowingBennethum,Murad,andCushman[15],weenforcethisrelationshipwith aLagrangemultipliersoastoaccountforthe N th termdependence. 43

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ThesecondrelationshiptobeenforcedwithLagrangemultipliersisthemass balanceequationforeachoftheconstituents.21: D j j Dt + j r v j = X 6 = ^ e j +^ r j : Let old denotefromequation.8,andlet j and N betheLagrange multipliersforthemassbalanceand N th termdependencies,.11,respectively. Theentropyinequalityisrewrittenasfollows: new = old + X N X j =1 j T D j j Dt + j r v j )]TJ/F26 11.9552 Tf 11.956 20.444 Td [( X 6 = ^ e j +^ r j !# + X N X j =1 T N : r j v j ; : .12 Afterasignicantamountofalgebraicsimplicationwithnoadditionalphysical assumptions,thisyieldsthefollowingformoftheentropyinequality: T = X @ @T + T )]TJ/F26 11.9552 Tf 13.84 11.357 Td [(X = l;g X @ @" )]TJ/F21 7.9701 Tf 16.804 14.944 Td [(N X j =1 )]TJ/F20 11.9552 Tf 5.48 -9.683 Td [( j j # )]TJ/F21 7.9701 Tf 11.955 14.944 Td [(N )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 X j =1 X @ @C s j )]TJ/F20 11.9552 Tf 11.955 0 Td [( s j s s # C s j )]TJ/F26 11.9552 Tf 13.84 11.357 Td [(X = l;g N X j =1 X @ @ j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j # j )]TJ/F26 11.9552 Tf 11.955 20.443 Td [(" X @ @J s )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 s J s N X j =1 tr )]TJ/F40 11.9552 Tf 5.48 -9.684 Td [(t s j !# J s )]TJ/F26 11.9552 Tf 11.955 20.444 Td [(" X @ @ )]TJETq1 0 0 1 234.905 177.77 cm[]0 d 0 J 0.478 w 0 0 m 10.601 0 l SQBT/F40 11.9552 Tf 234.905 167.894 Td [(C s )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(" s 2 )]TJETq1 0 0 1 309.891 187.325 cm[]0 d 0 J 0.478 w 0 0 m 10.145 0 l SQBT/F40 11.9552 Tf 309.891 177.449 Td [(F s )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 N X j =1 t s j )]TJETq1 0 0 1 415.468 187.325 cm[]0 d 0 J 0.478 w 0 0 m 10.145 0 l SQBT/F40 11.9552 Tf 415.468 177.449 Td [(F s )]TJ/F21 7.9701 Tf 6.587 0 Td [(T )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(" s 2 )]TJETq1 0 0 1 208.5 145.172 cm[]0 d 0 J 0.478 w 0 0 m 10.145 0 l SQBT/F40 11.9552 Tf 208.5 135.296 Td [(F s )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 )]TJETq1 0 0 1 253.971 145.172 cm[]0 d 0 J 0.478 w 0 0 m 10.145 0 l SQBT/F40 11.9552 Tf 253.971 135.296 Td [(F s )]TJ/F21 7.9701 Tf 6.586 0 Td [(T N X j =1 s j s j !# : C s )]TJ/F26 11.9552 Tf 13.84 11.357 Td [(X = l;g 6 = " @ @T + r T + N )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 X j =1 @ @C s j r C s j 44

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+ @ @" )]TJ/F15 11.9552 Tf 21.641 8.088 Td [(1 N X j =1 j j r + @ @" r + N X j =1 @ @ j )]TJ/F20 11.9552 Tf 13.15 8.088 Td [( j r j + N X j =1 @ @ j r j + @ @J s r J s + @ @ C s : r )]TJETq1 0 0 1 321.058 624.649 cm[]0 d 0 J 0.478 w 0 0 m 10.601 0 l SQBT/F40 11.9552 Tf 321.058 614.772 Td [(C s + ^ T s + ^ T # v ;s + X = l;g N X j =1 )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(t j + j j I : d + X " r T T q )]TJ/F21 7.9701 Tf 16.804 14.944 Td [(N X j =1 t j v j ; )]TJ/F20 11.9552 Tf 15.542 0 Td [( j v j ; j + 1 2 v j ; v j ; )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X N X j =1 X 6 = ^ T j + ^ i j + r j j )]TJ/F20 11.9552 Tf 9.299 0 Td [( j r j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" N r j v j ; + X N X j =1 t j + j j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j I + j N : r v j ; )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X N X j =1 ^ r j j + 1 2 v j ; 2 )]TJ/F21 7.9701 Tf 16.804 14.944 Td [(N X j =1 ^ e l j g )]TJ/F20 11.9552 Tf 12.453 -9.684 Td [( l j + l )]TJ/F15 11.9552 Tf 11.955 0 Td [( g j + g + 1 2 )]TJ/F40 11.9552 Tf 5.479 -9.683 Td [(v l;s 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( v g;s 2 + 1 2 )]TJ/F40 11.9552 Tf 5.48 -9.683 Td [(v l j ;l 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [( v g j ;g 2 )]TJ/F21 7.9701 Tf 16.804 14.944 Td [(N X j =1 ^ e s j l s j + s )]TJ/F26 11.9552 Tf 11.955 9.684 Td [()]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( l j + l )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 )]TJ/F40 11.9552 Tf 5.48 -9.684 Td [(v l;s 2 + 1 2 v s j ;s 2 )]TJ/F26 11.9552 Tf 11.955 9.684 Td [()]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(v l j ;l 2 )]TJ/F21 7.9701 Tf 16.804 14.944 Td [(N X j =1 ^ e g j s g j + g )]TJ/F15 11.9552 Tf 11.955 0 Td [( s j + s + 1 2 v g;s 2 + 1 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( v g j ;g 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( v s j ;s 2 0 .13 Theexploitationofequation.13willbethesourceofalloftheconstitutive relationsfortheremainderofthiswork.Thenextsectionoutlinesthedetailsofthis exploitationtoformconstitutiverelationsspecictomultiphasemediagovernedby 45

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ourchoiceofconstitutiveindependentvariables,.6. 4.2ExploitingtheEntropyInequality Inthissectionweexploittheentropyinequality,.13,inthesenseofColman andNoll[27].Thebasicprinciplehereisthat,accordingtothesecondlawofthermodynamics,entropyisalwaysnon-decreasingastimeevolves.Thisfactisusedto extractconstitutiverelationshipsfromtheentropyinequality.Noteveryresultfrom thisexploitationisrelevanttothecurrentstudy,soweonlypresentthemorenotable andusefulresultsinthenextsubsections.Furthermore,weexploitequation.13 withaneyetowarddeformable,multiphase,media.Theassumptionofdeformable mediawillberemovedinthefuture,butthisleavesopenthepossibilityofreturning totheseresultsforfuturework.Foranabstractsummaryofhowtheexploitation oftheentropyinequalityworks,alongwithsubtlebutimportantassumptions,see AppendixC. 4.2.1ResultsThatHoldForAllTime AsmentionedinSection4.1.1,severalofthetermsthatappearlinearlyinthe entropyinequalityhavefactorsthatareneitherindependentnorconstitutive.Wenow usethisfacttoderiverelationshipsthatmustholdforalltimeinordertonotviolate thesecondlawofthermodynamics.Toillustratethispointconsiderthecoecient of T .Ifthiscoecientissettozerowerecoverwiththethermodynamicconstraint thattemperatureandentropyareconjugatevariables, @ @T = )]TJ/F20 11.9552 Tf 9.298 0 Td [( : .14 Thisisaclassicalresultknownfromthermodynamics. 4.2.1.1FluidLagrangeMultipliers 46

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Forthegasandliquidphases,thedenitionsoftheLagrangemultipliersstem fromthecoecientof_ j and r v j ; .Settingthecoecientof_ j tozerogivesthe denitionoftheLagrangemultiplierforthemassbalanceequations: j = X " @ @ j : .15 Settingthecoecientof r v j ; tozero,summingover j =1: N ,andsolvingfor N yieldsanexpressionfortheotherLagrangemultiplier: N = )]TJ/F15 11.9552 Tf 13.555 8.088 Td [(1 N X j =1 t j + j j I + I : .16 4.2.1.2SolidPhaseIdentities Severalidentitiesforthesolidphasecanbederivedfromthetermsassociatedwith thetimederivativesofthesolidphaseJacobian, J s ,andthemodiedCauchy-Green, C s terms.Fromthe J s termweseethat 1 3 N X j =1 tr )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(t s j = J s s X @ @J s : .17 Next,considertheidentity t = N X j =1 )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(t j + j v j ; v j ; .18 resultingfromupscalingthemomentumbalanceequation.Takingthetraceof.18, neglectingthediusiveterms,andsubstitutingthisinto.17givesadenitionfor thesolidphasepressure: p s := )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 3 tr )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(t s = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(J s s X @ @J s : .19 ThisisageneralizationofthesolidphasepressurefoundbyWeinsteinforsaturated porousmediain[81]. Thecoecientofthe C s termgivesarelationshipforthestressinthesolidphase. Thiswillgiveageneralizationofthesolidphasestress[8,9]andcloselyfollowsthe 47

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derivationsofBennethum[9]andWeinstein[81].Settingthecoecientofthe C s termtozero,leftmultiplyingbythemodieddeformationgradient, F s ,andright multiplyingbythetransposeofthedeformationgradientgivesarelationshipthat denestheLagrangemultiplierforthesolidphase, s j : N X j =1 t s j + N X j =1 s j s j I = 2 s F s X @ @ C s !# )]TJETq1 0 0 1 441.987 608.637 cm[]0 d 0 J 0.478 w 0 0 m 10.145 0 l SQBT/F40 11.9552 Tf 441.987 598.761 Td [(F s T : .20 Usingidentity.18inthestresstermof.20,neglectingthediusivevelocities, takingone-thirdthetraceoftheresult,andusingequation.19forthesolid-phase pressureyieldsarelationshipforthesolidphaseLagrangemultiplier: N X j =1 s j s j = p s + 2 3 s X @ @ C s : C s : .21 Substituting.21backinto.20givesthefollowingrelationforthesolidphase stress: t s = )]TJ/F20 11.9552 Tf 9.298 0 Td [(p s I + 2 s F s X @ @ C s !# )]TJETq1 0 0 1 340.563 389.003 cm[]0 d 0 J 0.478 w 0 0 m 10.145 0 l SQBT/F40 11.9552 Tf 340.563 379.127 Td [(F s T )]TJ/F15 11.9552 Tf 18.097 8.087 Td [(2 3 s X @ @ C s : C s I : .22 Thiscanberewrittenas t s = )]TJ/F20 11.9552 Tf 9.299 0 Td [(p s I + t s e + l s t l h + g s t g h .23 where t s e =2 s F s @ s @ C s )]TJETq1 0 0 1 315.875 221.199 cm[]0 d 0 J 0.478 w 0 0 m 10.145 0 l SQBT/F40 11.9552 Tf 315.875 211.323 Td [(F s T )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 s @ s @ C s : C s I ; .24a t l h =2 l F s @ l @ C s )]TJETq1 0 0 1 314.581 181.348 cm[]0 d 0 J 0.478 w 0 0 m 10.145 0 l SQBT/F40 11.9552 Tf 314.581 171.472 Td [(F s T )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 l @ l @ C s : C s I ; .24b t g h =2 g F s @ g @ C s )]TJETq1 0 0 1 316.275 141.498 cm[]0 d 0 J 0.478 w 0 0 m 10.145 0 l SQBT/F40 11.9552 Tf 316.275 131.621 Td [(F s T )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 g @ g @ C s : C s I : .24c Thestressesabovearetermedtheeectivestress,hydratingstressfortheliquid phase,andhydratingstressforthegasphaserespectively.Equation.22states 48

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thatthestressinthesolidphasecanbedecomposedintothesolidpressureand stressesfeltduetothepresenceoftheuidphases.Itisherethatthemodications ofthedeformationgradientandCauchy-Greentensorsbecomeclear.Ifwetakethe traceofthestresstensorthenweseethat = 3 tr t s = )]TJ/F20 11.9552 Tf 9.298 0 Td [(p s ;whichishowthe solidphasepressureismeasured.Therefore,thisthermodynamicdenitionof p s is consistentwithexperimentalmeasure.Furthermore,theeectivestressandhydrating stressesaretermsassociatedwiththeinteractionbetweenthesolidandtheuids. Forsaturatedporousmedia,Bennethum[8]statesthefollowing: Theeectivestresstensoristhestressofthesolidphaseduetothe strainoftheporousmatrix,andthehydratingstresstensoristhestress theliquidphasesupportsduetothestrainofthesolidmatrixwhich wouldbenegligibleiftheliquidandsolidphasewerenotinteractive,but whichbecomessignicantforswellingporousmaterials." Onenalnoteonthesolidphasestressisthatthetotalstressintheporous mediumisrelatedtothepressuresinallthreephases.Thiscanbeseenbytaking theweightedsumofthestressesineachphase: t = X t = )]TJ/F20 11.9552 Tf 9.299 0 Td [(" s p s I + s t s e + l t l h + t l + g t g h + t g : .25 Takingone-thirdthetraceofthetotalstress,andrecallingthattheeectiveand hydratingstressesaretracefree,gives 1 3 tr )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(t = )]TJ/F20 11.9552 Tf 9.298 0 Td [(" s p s + l 3 tr )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(t l + g 3 tr )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(t g : .26 Inordertofullyunderstandthestressesintheuidphaseswemustcontinueour examinationoftheresultscomingfromtheentropyinequality.Equation.26is similartotheTerzaghistressprinciple;suggestingthattheuidphaseshelptosupporttheporespaceinthemedium. 4.2.2EquilibriumResults 49

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Thereareseveralmorerelationshipsthatwecanextractfromtheentropyinequality.Inparticular,wenowseekrelationshipsbetweentheLagrangemultipliers andtheuid-phasepressures.Wealsoseekrelationshipsforthemomentumandenergyexchangeterms.Atequilibriumtheproductionofentropyisminimized.Since thisisaminimum,thegradientof ^ withrespecttothesetofindependentvariables .6iszero.Thisindicatesthatthecoecientsoftheindependentvariablesthat appearlinearlyintheentropyinequalityarezeroatequilibrium.Inthecaseofathree phaseporousmediumofthisnature,wedeneequilibriumtobewhenasubsetof theindependentvariablesarezero.Inparticular,equilibriumisdenedwhennoheat conductionoccurs, r T = 0 ,thestrainratesintheuidphasesarezero, d = 0 ,and allrelativevelocitiesarezero, v j ; = 0 and v ;s = 0 .This denition ofequilibrium isparticulartothistypeofmediaandischosenasitgivesphysicallyrelevantand meaningfulresults.Anotherwaytolookatthisistosaythatequilibriumisexactly thestatewhenallofthesevariablesarezero. 4.2.2.1FluidStressTensor Therstnotableequilibriumresultcomesfromthecoecientoftherateof deformationtensor, d .Settingthecoecientof d tozero,eliminatingthesumof theconstituentstresstensorsusingtheidentity t = N X j =1 t j + j v j ; v j ; ; .27 andnotingthatatequilibriumthediusivevelocityiszero,yields N X j =1 j j = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 3 tr )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(t = p : .28 Equation.28linkstheLagrangemultiplierstotheequilibriumpressureoftheuid phases.Thisistheclassicaldenitionofpressureinauid:minusone-thirdthetrace ofthestresstensor.Usingequation.15thepressureintheuidphasescannow 50

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bewrittenas p = )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 3 tr )]TJ/F40 11.9552 Tf 5.48 -9.683 Td [(t = N X j =1 X j @ @ j : .29 Withthedenitionofpressureinequation.29wenotethatthecoecientof cannowberewrittenas p )]TJ/F26 11.9552 Tf 11.956 11.358 Td [(X @ @" : .30 Thesecondtermisthechangeinenergywithrespecttovolumetricchanges,and isthereforeinterpretedastherelativeanityforonephasetoanother.Thatis, thistermisrelatedtothewetabilityofthe andsolidphasesbythe uidphase. Thetimerateofchangeofvolumeuidphasevolumefractionisanequationof statethatisnotyetknown,butrewritingthecoecientasin.30hintsatthe factthattheequationofstateisrelatedtothepressureandthewettabilityofthe phases.Furthermore,pressure,wettability,andsurfacetensionarerelatedtocapillary pressure;henceindicatingthattheequationofstateforthetimerateofchangeof volumefractionisrelatedtocapillarypressure.Itisherethatwenotethedrawback tothepresentmodelingeort.Recallthatinthepresentexpansionoftheentropy inequalitywedonotincludeinterfacialeects.Ifweweretoincludetheseeectsthen asurfacetensiontermwouldappearhereasshowninHassanizadehandGray[41] andthesetermstogetherwouldmorereadilybeassociatedwithcapillarypressure. Morediscussionwillbededicatedtotheexactequationofstateforthetimerateof changeofvolumefractionafteradiscussiononcrosscouplingpressuresinSection4.3 andcapillarypressureinChapters5and7. 4.2.2.2MomentumTransferBetweenPhases Thenextnotableequilibriumresultwecanextractfrom.13comesfromthe coecientoftheuidphaserelativevelocities, v ;s .Settingthiscoecienttozero, recallingthat r T = 0 atequilibrium,usingthedenitionoftheuidphasepressure, 51

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.29,thedenitionoftheuidphaseLagrangemultipliers,.15,andsolvingfor themomentumtransfertermsgives )]TJ/F26 11.9552 Tf 11.291 13.27 Td [( ^ T s + ^ T = @ @" )]TJ/F20 11.9552 Tf 11.955 0 Td [(p r + @ @" r + N )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 X j =1 @ @C s j r C s j )]TJ/F21 7.9701 Tf 16.804 14.944 Td [(N X j =1 @ @ j + s s @ s @ j r j + N X j =1 @ @ j r j + @ @J s r J s + @ @ C s : r )]TJETq1 0 0 1 380.972 580.468 cm[]0 d 0 J 0.478 w 0 0 m 10.601 0 l SQBT/F40 11.9552 Tf 380.972 570.592 Td [(C s ; .31 where isthe other uidphasenotequalto .Thisparticularresultwillbecoupled withtheconservationofmomentumtoyieldnovelformsofDarcy'slawinSection 4.5.1. 4.2.2.3MomentumTransferBetweenSpecies Anothernotableequilibriumresultscomesfromthecoecientofthediusive velocity, v j ; .Equation.28indicatesthatatequilibriumthedenitionofthe Lagrangemultiplier,equation.16,simpliesto N = I : .32 Thisimpliesthat,atequilibrium,thestresstensorforconstituent j fromthecoecientof r v j ; canbewrittenas t j = )]TJ/F20 11.9552 Tf 9.299 0 Td [(" j j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j I )]TJ/F20 11.9552 Tf 11.955 0 Td [(" j I : .33 Considerthediusivevelocitytermintheentropyinequality: )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X N X j =1 X 6 = ^ T j + ^ i j + r j j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j r j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" N r j v j ; : Add )]TJ/F26 11.9552 Tf 11.291 8.966 Td [(P P N j =1 j N r v j ; =0andsimplifytoget )]TJ/F26 11.9552 Tf 11.956 11.358 Td [(X N X j =1 X 6 = ^ T j + ^ i j + r j j 52

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)]TJ/F20 11.9552 Tf 11.955 0 Td [( j r j )]TJ/F40 11.9552 Tf 11.955 0 Td [( N : r j v j ; : Atequilibriumthediusivevelocityisassumedtobezero.Bythelogicusedherein fortheexploitingtheentropyinequality,andgiventhefactthat N = I at equilibrium,weobservethatforeach j X 6 = ^ T j + ^ i j = )]TJ/F43 11.9552 Tf 9.299 0 Td [(r j j + j r j + r j : .34 Thisisanexpressionforthemomentumtransferforspecies j inthe phase.This resultwillbecoupledwiththeconstituentconservationofmomentumequationto deriveaformofFick'slawinSection4.5.3. 4.2.2.4PartialHeatFlux Toconcludetheequilibriumresultsweexaminethe r T termintheentropy inequality.Wehaveassumedthat r T = 0 and v j ; = 0 atequilibrium,sobythe logicusedaboveweseethatthecoecientof r T mustbezeroand X q = 0 .35 atequilibrium.Thisisthepartialheatuxoftheentireporousmediaandwillbe usedinSection4.5.4toderiveageneralizedFourier'slaw. 4.2.3NearEquilibriumResults Thenextstepinexploitingtheentropyinequalityistoderive nearequilibrium results.Theseresultsarisebylinearizingtheequilibriumresultsabouttheequilibrium state.Thelinearizationprocessissimplytherst-ordertermsoftheTaylorseries, butonemustkeepinmindthateachofthederivativesisafunctionofallofthe constitutiveindependentvariablesthatarenotzeroatequilibrium.Forexample, if f = f eq atequilibrium,thennearequilibrium, f f near = f eq + @f=@ r T r T + + @f=@ v l;s v l;s .Thisfullexpansionmayyieldtermsthatarenotreadily 53

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physicallyinterpretable.Forthisreason,considerableeortsmustbemadetorelate thelinearizationconstantstomeasurableparameters.Forathoroughexplanationof thelinearizationprocesswiththeentropyinequalityseeAppendixC. Forthemomentumtransferintheuidphases,thelinearizationofequation.31 canbesimplywrittenas X 6 = ^ T near = X 6 = ^ T eq )]TJ/F26 11.9552 Tf 11.955 9.684 Td [()]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" 2 R v ;s : .36 Thelinearizationconstant, R ,isrelatedtotheresistivityofaporousmedium;the inverseofthehydraulicconductivity.Itshouldbenotedthatwehaveonlyexpanded aboutoneofthepossiblevariables: v ;s .Strictlyspeakingthisisincorrectand weshouldexpandaboutallothervariableswhicharezeroatequilibrium.Amore thoroughexpansionis X 6 = ^ T n:eq = X 6 = ^ T eq )]TJ/F26 11.9552 Tf 11.956 9.683 Td [()]TJ/F20 11.9552 Tf 5.479 -9.683 Td [(" 2 R v ;s + H r T + J v j ; + L : d + ; .37 where H and J aresecond-ordertensorsand L isathird-ordertensor.Theellipses attheendofthisequationindicatesthattherearehigherordertermsthatarenot beingwrittenexplicitly.Theleft-handsideof.37istherateofmomentumtransfer duetomechanicalmeans.Itisreasonabletothinkthatthistransfertermmightbea functionofuidvelocity,buttheeectsduetothermalgradients,diusivevelocity, andvelocitygradientsarelikelysmallincomparison.Tobecompletelycorrectwe wouldhavetoincludethesetermsinthemodelingproblemstofollow.Thetrouble isthateachofthecoecientsneedstobeassociatedwithaphysicalparameter. Wewillseethat R isphysicallyassociatedwithamaterialparameteroftheporous medium,butitispresentlyunclearwhatthephysicalinterpretationsarefortheother coecients.Neglectingthesetermssimplyleavesthedooropenforfuturemodeling research. 54

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Proceedinginasimilarmanner,thelinearizedconstituentmomentumtransfer fromequation4.34is X 6 = ^ T j + ^ i j near = X 6 = ^ T j + ^ i j eq )]TJ/F20 11.9552 Tf 11.955 0 Td [(" j R j v j ; : .38 Thelinearizationconstantisrelatedtotheinverseofthediusiontensor.Thelinearizedpartialheatuxfromequation4.35is X q = K r T .39 recallingthatthepartialheatuxiszeroatequilibrium,andthelinearization constantisrelatedtothethermalconductivity. Ineachoftheselinearizationresults,thefactorsofvolumefractionanddensity arechosensothatthelinearizationconstantsbettermatchexperimentallymeasured coecients.Thesignsarechosensothattheentropyinequalityisnotviolated. Severaloftherelationshipsresultingfromtheentropyinequalityrelyonproper denitionsofthepartialderivativesoftheenergywithrespecttoparticularindependentvariables.Thepressureisonesuchquantity,butthereareseveralothersthat appearintheprecedingresults.Forthisreason,wenowturnourattentiontothe exactdenitionsofpressureandchemicalpotentialunderourchoiceofindependent variables.Thiswillhelptosimplifyandtoattachphysicalmeaningtotheterms appearingineachofthelinearizedresults.Insaturatedswellingporousmaterial, BennethumandWeinstein[16]showedthattherearethreepressuresactingonthe system.Theseresultsareextendedinthenextsectiontomediawithmultipleuid phases. 4.3PressuresinMultiphasePorousMedia Wewillseeinthissubsectionthatthethreepressuresdenedin[16]canbe extendedtobroaderdenitionsinmultiphasemedia.Thesedenitionswillhelpto simplifyandattachphysicalmeaningtothetermsappearingineachofthelinearized 55

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resultsdiscussedintheprevioussubsection.Wewillalsodeneseveralnew pressures actingascouplingtermsbetweenthephasesinmultiphasemedia.Itwillbeshown thatwecanreturntothethreepressurerelationshipofBennethumandWeinsteinif wesimplifytheseresultstoasingleuidphase. Recallfromtheentropyinequalitythattheequilibriumpressureinmultiphase mediacanbewrittenasanaccumulationofcrosseectsasfollows: p = X N X j =1 j @ @ j : .40 Thepartialderivativeistakenwhileholding ; k ;" ; and m xedwhere k =1: N and m =1: N;m 6 = j .Deneacross-couplingpressureas p := N X j =1 j @ @ j ; k ;" ; m .41 sothatthe -phasepressurecanbesimplywrittenasthesumofthesecross-coupling pressures p = X p for 2f l;g g and 2f l;g;s g : .42 Nowwederiveanidentitythatisanalogoustothethreepressurerelationship derivedbyBennethumandWeinstein[16].Tothatend,considertheHelmoltzpotentialasafunctionoftwosetsofindependentvariableswherethereisaone-to-one relationshipbetweenthetwosets. = )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(" ;" j ;" ;" j and ^ = ^ )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" ; j ;" ; j ; .43 where 2f l;g g and 6 = ;s .TheHelmholtzpotentialisactuallyafunctionof severalothervariables,butthesearesuppressedheretomakethenotationmore readable.Since and ^ arefunctionsofanequivalentsetofvariables,thetotal dierentialsmustbeequaltoeachother.Setting d ^ = d gives d ^ = @ @" k ;" ;" k d" + N X j =1 @ @ j ;" m ;" ;" k d j 56

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+ @ @" ;" k ;" k d" + N X j =1 @ @ j ;" k ;" ;" m d j .44 whereineachcasewearetaking m =1: N suchthat m 6 = j ,and k =1: N .Now takethepartialderivativewithrespectto whileholding ; k ; and k xed.In thiscase,the d" and d j termswillbezero.Thisleavesuswith: @ ^ @" ; k ; k = @ @" k ;" ;" k @" @" ; k ; k + N X j =1 @ @ j ;" k ;" ;" m @ j @" ; k ; k = @ @" k ;" ;" k + N X j =1 j @ @ j ;" k ;" ;" m : .45 Nowmultiplyby )]TJ/F20 11.9552 Tf 9.298 0 Td [(" toget )]TJ/F20 11.9552 Tf 12.487 0 Td [(" @ ^ @" ; k ; k = )]TJ/F20 11.9552 Tf 12.487 0 Td [(" @ @" k ;" ;" k )]TJ/F21 7.9701 Tf 16.804 14.944 Td [(N X j =1 j @ @ j ;" k ;" ;" m : .46 Noticethatthethirdtermis p fromequation.41.Denethefollowingnew terms: p := )]TJ/F20 11.9552 Tf 9.299 0 Td [(" @ @" ;" k ;" k .47 := @ @" ; k ; k .48 togettherelationship p = p + : .49 Notethatthenewdenitionsonlyholdif 6 =0.Thiscanbeseenifonereturns backtotheLagrangemultiplierequationatthebeginningofthissectionforthe pressure.Furthermore,thisrelationshipholdsifwehadtakenthederivativewith respectto insteadof 57

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Forcompletenesssakewedene p and sothatourdenitionsareconsistent with[16]: p := X p .50 := X : .51 WiththesedenitionswerecoverthethreepressurerelationshipderivedbyBennethumandWeinstein p = p + : .52 Thephysicalmeaningof p isthechangeinenergywithrespecttochangesin volumewhileholdingmassxed.Intermsofextensivevariablesthisisthesame denitionaspressureencounteredinclassicalthermodynamicsforasinglephase. Forthisreasonwecall p the thermodynamic pressure.Thephysicalmeaningof is thechangeinenergywithrespecttochangesinsaturationwhileholdingthedensities xed.This pressure orswellingpotentialasitiscalledin[16]relatesthedeviation betweenthe classicalpressure p ,andthethermodynamicpressure.Itcanbeseenas apreferentialwettingfunctionthatmeasurestheanityforonephaseoveranother. Withthesephysicalconsiderationsinmindwenowreturntothecoecientofthe_ termsintheentropyinequality.Withthepresentdenitions,thecoecientis p )]TJ/F20 11.9552 Tf 11.955 0 Td [( : Using.52thisisclearly p .Sincethetimerateofchangeofvolumefractionis takenasaconstitutivevariable,thelinearizationresultforthistermcannowbe statedas p n:eq: = p eq: + : .53 Thecoecient arosefromlinearizationandisformallydenedas = @ p @ eq: 58

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Equation.53doeslittletomakecleartheexactmeaningofthisequation.The exactmeaningwillbecomeclearinChapter5undertheassumptionthattheuidphasevolumefractionsarenotindependent. Thedenitionsofthethree pressures allowustoattachmorephysicalmeaning andmoreconvenientnotationtotheresultsfoundwhenbuildingconstitutiveequationsinthenextsections.Beforebuildingtheseequationswedenetheupscaled chemicalpotentialforamultiphasesystem,andafterthispointwewillhaveallof thetoolsnecessarytoderivethenewconstitutiveequations. 4.4ChemicalPotentialinMultiphasePorousMedia Chemicalpotentialisdenedthermodynamicallyasthechangeinenergywith respecttochangesinthenumberofmoleculesinthesystem[4,21].Thisclassical denitionhasthefollowingcharacteristics[15]:itisascalarandmeasuresthe energyrequiredtoinsertaparticleintothesystem,itsgradientisthedriving forcefordiusiveowFick'slaw,anditisconstantforasingleconstituentin twophasesatequilibrium.In[69],Bennethumproposedadenitionforchemical potentialinsaturatedporousmediathatsatisesallthreeofthesecriteria: j = + @ @ j ; m = @ @ j ; m = @ @ j ; m ; .54 for m =1: N and m 6 = j: Insaturatedmedia,ifthechangesinenergyinthesolid phaseduetochangesinliquiddensityareassumedtobezero,thenthenumeratorof theright-handsideof.54canbeseenasthetotalenergyinasaturatedsystem. Underthisassumption,thechemicalpotentialcanberewrittenas j = @ T @ j ; m .55 Thisindicatesthatinasaturatedporousmedium,thechemicalpotentialofthe j th constituentinthe )]TJ/F15 11.9552 Tf 9.298 0 Td [(phaseisthechangeintotalenergywithrespecttochanges inmassofconstituent j .Wenowextendthisdenitiontomultiphaseunsaturated systems. 59

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Extendingthisideatomultiphaseandmulticonstituentmedia,wedenechemical potentialtobethechangeintotalenergywithrespecttochangesinmassinthe constituent.Inmultiphasemediawecannotmaketheassumptionthattheenergyin onephaseisnoteectedbychangesinotherphases.Withthisinmind,werecallthat thetotalenergycanbegivenby T = P .Therefore,thepresentdenition ofchemicalpotentialis j = @ T @ j ;" ; k ; m = X @ @ j ;" ; k ; m ; .56 whereagain, k =1: N and m =1: N;m 6 = j .Noticethatif @ =@ j =0for 6 = thenthisdenitioncollapsestoequation.54.Furthermore,recallingthedenition oftheLagrangemultiplier, j ,from.15,equation.56canberewrittenas j = + X " @ @ j ;" ; k ; m = + j : .57 Equation.57onlyholdsfor = l and = g .Adenitionofthesolidphase chemicalpotentialisbeyondthescopeofthiswork. Asaresultofthisdenitionofchemicalpotentialweobserveanimmediateeect ontherateofmasstransfertermsintheentropyinequality.Usingequation.57, thelastthreetermsintheentropyinequality,.13,canberewrittenas )]TJ/F21 7.9701 Tf 16.804 14.944 Td [(N X j =1 ^ e l j g )]TJ/F20 11.9552 Tf 12.453 -9.684 Td [( l j + l )]TJ/F15 11.9552 Tf 11.955 0 Td [( g j + g + 1 2 )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(v l;s 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [( v g;s 2 + 1 2 )]TJ/F40 11.9552 Tf 5.48 -9.684 Td [(v l j ;l 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( v g j ;g 2 )]TJ/F21 7.9701 Tf 16.805 14.945 Td [(N X j =1 ^ e s j l s j + s )]TJ/F26 11.9552 Tf 11.955 9.683 Td [()]TJ/F20 11.9552 Tf 5.479 -9.683 Td [( l j + l )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(1 2 )]TJ/F40 11.9552 Tf 5.48 -9.683 Td [(v l;s 2 + 1 2 v s j ;s 2 )]TJ/F26 11.9552 Tf 11.955 9.683 Td [()]TJ/F40 11.9552 Tf 5.48 -9.683 Td [(v l j ;l 2 )]TJ/F21 7.9701 Tf 16.805 14.944 Td [(N X j =1 ^ e g j s g j + g )]TJ/F15 11.9552 Tf 11.955 0 Td [( s j + s + 1 2 v g;s 2 + 1 2 )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [( v g j ;g 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( v s j ;s 2 = )]TJ/F21 7.9701 Tf 16.805 14.944 Td [(N X j =1 ^ e l j g l j )]TJ/F20 11.9552 Tf 11.956 0 Td [( g j + 1 2 )]TJ/F40 11.9552 Tf 5.479 -9.683 Td [(v l;s 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( v g;s 2 + 1 2 )]TJ/F40 11.9552 Tf 5.479 -9.683 Td [(v l j ;l 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( v g j ;g 2 )]TJ/F21 7.9701 Tf 16.805 14.944 Td [(N X j =1 ^ e s j l s j + s )]TJ/F20 11.9552 Tf 11.955 0 Td [( l j )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 )]TJ/F40 11.9552 Tf 5.479 -9.683 Td [(v l;s 2 + 1 2 v s j ;s 2 )]TJ/F26 11.9552 Tf 11.955 9.683 Td [()]TJ/F40 11.9552 Tf 5.48 -9.683 Td [(v l j ;l 2 60

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)]TJ/F21 7.9701 Tf 16.805 14.944 Td [(N X j =1 ^ e g j s g j )]TJ/F15 11.9552 Tf 11.955 0 Td [( s j + s + 1 2 v g;s 2 + 1 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( v g j ;g 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( v s j ;s 2 : .58 Thesquareoftherelativevelocitiesarelikelyzerosincethesemodelsaredesigned withcreepingowinmind.Withthesesimplications,themasstransfertermsfrom theentropyinequalityarerewrittenas )]TJ/F21 7.9701 Tf 16.14 14.944 Td [(N X j =1 ^ e l j g l j )]TJ/F20 11.9552 Tf 11.955 0 Td [( g j )]TJ/F21 7.9701 Tf 16.805 14.944 Td [(N X j =1 ^ e s j l s j + s )]TJ/F20 11.9552 Tf 11.955 0 Td [( l j )]TJ/F21 7.9701 Tf 16.805 14.944 Td [(N X j =1 ^ e g j s f g j )]TJ/F15 11.9552 Tf 11.955 0 Td [( s j + s g : .59 Atequilibriumweassumethatthemasstransferbetweenphasesiszero.Take notethatthisisanassumptionabouthowtheconstitutivevariablebehavesatequilibriumandnotanassumptionabouttheequilibriumstateitself.Thisnepoint ismadesinceinseveralworksthisassumptionismadeaspartofthedenitionof equilibriumforexample,[16].Intheauthor'sopinionthisisasubtlemistake.The assumptionthat^ e l j g =0atequilibriumimpliesanalequilibriumrelationship;the masstransferbetweentheuidphasesisproportionaltothedierenceinchemical potentials ^ e l j g j n:eq =^ e l j g j eq + )]TJ/F20 11.9552 Tf 10.461 -9.683 Td [( l j )]TJ/F20 11.9552 Tf 11.955 0 Td [( g j M )]TJ/F20 11.9552 Tf 12.453 -9.683 Td [( l j )]TJ/F20 11.9552 Tf 11.956 0 Td [( g j = )]TJ/F20 11.9552 Tf 10.461 -9.684 Td [( l j )]TJ/F20 11.9552 Tf 11.955 0 Td [( g j M )]TJ/F20 11.9552 Tf 12.453 -9.684 Td [( l j )]TJ/F20 11.9552 Tf 11.956 0 Td [( g j : .60 Thishelpstoverifyourchoiceofupscaledchemicalpotentialbysatisfyingthethird criteriasetforthatthebeginningofthissubsection.Furthermore,thissuggestsa naturalcouplingbetweentheliquidandgasphasemassbalanceequations.Themass transfercoecientischosentohaveafactorofthedierenceindensitiessoasto bettermatchexperimentalmeasures[78].Giventhattheunitsoftherateofmass transferare[ ML )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 t )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 ]weseethattheunitsofthelinearizationconstantare [ M ]= t L 2 = 1 L 2 =t : 61

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Afurthervericationthatwehaveproperlydenedthemultiphasechemical potentialcorrectlycanbeseenthroughtheGibbs-Duhemrelationshipfromthermodynamics[21].Simplystated,theGibbs-DuhemrelationshipstatesthattheGibbs potentialofthe phaseistheweightedsumofthechemicalpotentials: )]TJ/F21 7.9701 Tf 7.314 4.936 Td [( = N X j =1 C j j : .61 Equation4.61speciestherelationshipbetweentheGibbspotential,)]TJ/F21 7.9701 Tf 378.537 4.338 Td [( ,andthe chemicalpotential.Substituting.57intotheright-handsideof.61,carryingout thesummation,andapplyingthedenitionofpressure,.28,givestheequation )]TJ/F21 7.9701 Tf 7.315 4.936 Td [( = + p ; .62 whichisthestandardthermodynamicrelationshipbetweentheHelmholtzpotential andtheGibbspotential.Thisclearlydemonstratesthatthedenitionofmultiphase chemicalpotentialusedhereisconsistentwiththeclassicalthermodynamicdenition. Atthispointweturnourattentiontowardusingtherelationshipsderivedfrom theentropyinequalitytodevelopnovelexpressionsforDarcy's,Fick's,andFourier's lawsofow,diusion,andheatconduction.Foraconcisesummaryofallofthe resultsderivedinthischapter,seeAppendixD. 4.5DerivationsConstitutiveEquations InthissectionwederivegeneralformsofDarcy's,Fick's,andFourier'slaws basedontheHMTresultsintheprevioussections.Theseequationswillbecoupled withmassandenergybalanceequationstoformamacroscalemodelforheatand moisturetransportforunsaturatedmedia.Theresultsderivedinthissectionextend theclassicalformsofeachoftheselaws.Theseextensionssuggesttermsthat,inthe author'sknowledge,arepreviouslyunreported.Also,weproposenewformsofthese lawsintermsofthemacroscalechemicalpotential.Thissuggeststhatthechemical potentialisageneralizeddrivingforceforow,diusion,andheattransport. 62

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4.5.1Darcy'sLaw In1856,HenriDarcyproposedhisempiricallawgoverningowthroughsaturated porousmedia[31].Thiswasderivedthroughexperimentationonsandltersusedto purifythewaterinthefountainsofDijon,France.Initssimplestform,Darcy'slaw statesthattheaverageduiduxisproportionaltothegradientofhydraulichead oruidpressure l v l;s = )]TJ/F20 11.9552 Tf 9.298 0 Td [(k r h: .63 UndertheconstructofHybridMixtureTheory,Darcy'slawisobtainedbycoupling themomentumbalanceequationforauidphase,.29,withthelinearizedconstitutiveequationforthemomentumtransferfromotherphases.Thishasbeenillustrated byseveralauthorssomeexamplesinclude[12,13,43,81],anddependingonthe setofindependentvariablespostulatedfortheHelmholtzpotential,themomentum transfertermcansuggestdierentformsofDarcy'slaw. Inthepresentcase,werecallfromequation.36thatthelinearizedmomentum transfertermscanbewrittenas ^ T s + ^ T = )]TJ/F20 11.9552 Tf 9.298 0 Td [(" R )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" v ;s )]TJ/F26 11.9552 Tf 11.955 9.684 Td [()]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( )]TJ/F20 11.9552 Tf 11.955 0 Td [(p r )]TJ/F20 11.9552 Tf 11.955 0 Td [( r )]TJ/F20 11.9552 Tf 11.955 0 Td [(" N )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 X j =1 @ @C s j r C s j + N X j =1 @ @ j + s s @ s @ j r j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" N X j =1 @ @ j r j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" @ @J s r J s )]TJ/F20 11.9552 Tf 11.955 0 Td [(" @ @ C s : r )]TJETq1 0 0 1 473.203 245.613 cm[]0 d 0 J 0.478 w 0 0 m 10.601 0 l SQBT/F40 11.9552 Tf 473.203 235.737 Td [(C s ; .64 wherewerecallthat R isrelatedtotheresistivityofthemediumandarosefrom thelinearizationprocess. Linearizationofthestress-pressurerelationshipfortheuidphasesgivesanexpressionforthestressnearequilibrium: t = )]TJ/F20 11.9552 Tf 9.298 0 Td [(p I + : d : .65 63

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Thefourth-ordertensormultiplyingtherateofdeformationtensorcanbesimplied, inmostcases,tocorrespondtotheviscosityofthemediumseeanytextoncontinuummechanics.Ignoringtheaccelerationtermsinthemomentumbalanceequation .29,andsubstitutingequation.64formomentumtransferand4.65forthe stresstensorgivesthefollowinggeneralizationofDarcy'slaw: R )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(" v ;s = )]TJ/F20 11.9552 Tf 9.298 0 Td [(" r p )]TJ/F20 11.9552 Tf 11.955 0 Td [( r )]TJ/F20 11.9552 Tf 11.955 0 Td [( r + g + N X j =1 @ @ j + s s @ s @ j r j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" N X j =1 @ @ j r j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" @ @J s r J s )]TJ/F20 11.9552 Tf 11.955 0 Td [(" @ @ C s : r )]TJETq1 0 0 1 383.595 516.532 cm[]0 d 0 J 0.478 w 0 0 m 10.601 0 l SQBT/F40 11.9552 Tf 383.595 506.656 Td [(C s + r : d : .66 ToarriveatthisformofDarcy'slawwehavealsoassumethat r C s j 0 sinceit assumedthatconcentrationgradientsinthesolidphasedonotaectow.Therst termindicatesthatowisprimarilyduetopressuregradients,asexpected.The eighthandninthtermswerepreviouslyreportedbyWeinsteinin[81].Notethatthe extra factorof ontheleft-handsideoftheequationcanbemovedtotheright.If allbutthersttermontheright-handsidearethenignoredwearriveattheclassical Darcy'sLaw R )]TJ/F20 11.9552 Tf 5.479 -9.683 Td [(" v ;s = )]TJ/F43 11.9552 Tf 9.299 0 Td [(r p ; .67 where q = v ;s isknownasthe DarcyFlux Thelinearizationconstant, R ,isrelatedtotheresistivityoftheporousmedium, theinverseofwhichisassumedtoexist,andwedene K = )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(R )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 .Thetensor K isrelatedtothehydraulicconductivity.Todeterminetheexactmeaningofthe linearizationconstantweconsidertheunitsofthesimplestterms: q )]TJ/F40 11.9552 Tf 21.918 0 Td [(K r p : TheunitsoftheDarcyuxarelengthpertime[ L=t ],andtheunitsofthepressureare massperlengthpertimesquared[ M= L t 2 ].Thisindicatesthatthelinearization 64

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constanthasunits[ L 3 t =M ],whichcanberewrittenas[ L 2 = M= L t ].The numeratorofthisfractionhasunitsofpermeability, ,andthedenominatorhasunits ofdynamicviscosity, .Thissuggeststhat K = ; .68 andthisrelationshipisconrmedinequations.4and.5ofPinderetal.[62]. Thehydraulicconductivityofaporousmediumisdenedas k c = g = g ; .69 where isthekinematicviscosityoftheuid.Thisindicatesthat K canalsobe denedas K = k c g : .70 Itisclearfromtheserelationshipsthat K isafunctionofboththetypeofuidand thegeometryoftheporousmedium.Thiscoecientdescribes,insomesense,the abilityoftheporousmediumtotransmituid"[62].Insaturatedporousmedia,the permeabilityistypicallyassumedonlytobeafunctionofgeometry.UnderHybrid MixtureTheorywemustnotethatthepermeabilityisafunctionofanyvariablewhich isnotnecessaryzeroatequilibrium.Typicallyitisassumedthatthepermeabilityof anunsaturatedmediumisafunctionofthevolumefractions[5,62].Thetensorial notationmaybedroppedinisotropicmedia,butforanisotropicmediaitisassumed thatthepermeabilitymaydependonthedirectionofow.Wewillexpanduponthis ideainlaterchapterswhenbuildingamacroscalemassbalancemodel. 4.5.2Darcy'sLawInTermsofChemicalPotential Equation.66couplesallofthephysicalprocessesthatwewishedtomodelat theoutset;multiphaseowwithconstituentsineachphaseandadeformablesolid.In ordertobuildreasonablemodelsbasedonthisconstitutiveequation,functionalforms 65

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forthewettingpotentials, and ,andthechangesinenergywithrespectto densityareneeded.Thesolid-phasetermsarelikelynegligiblefornon-deformable media,butdealingwiththeremainingtermsrepresentasignicantmodelingtask. Thegoalofthissubsectionistogreatlysimplifythismodelwhilemaintainingthe physicalinterpretation.Thisisdonebyswitchingthermodynamicpotentials. RecallfromthermodynamicsthattheGibbspotential,)]TJ/F21 7.9701 Tf 303.069 4.339 Td [( ,canbewrittenin termsoftheHelmholtzpotential, ,as )]TJ/F21 7.9701 Tf 7.314 4.936 Td [( = + p : .71 TakingthegradientoftheGibbspotential,expandingtheresultinggradientof Helmholtzpotentialintermsoftheconstitutiveindependentvariables,andmultiplyingby )]TJ/F20 11.9552 Tf 9.299 0 Td [(" yieldstheequation: )]TJ/F20 11.9552 Tf 11.956 0 Td [( r )]TJ/F20 11.9552 Tf 11.955 0 Td [( r )]TJ/F20 11.9552 Tf 11.955 0 Td [(" r p = N X j =1 @ @ j r j + N X j =1 @ @ j r j )]TJ/F20 11.9552 Tf 11.956 0 Td [(" r )]TJ/F21 7.9701 Tf 7.314 4.936 Td [( )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(" p r )]TJ/F20 11.9552 Tf 11.955 0 Td [(" r T: .72 Matchingthecommontermsbetween.66and.72,andrecognizingtheresulting chemicalpotentialtermsyields R )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" v ;s = )]TJ/F20 11.9552 Tf 9.299 0 Td [(" r )]TJ/F21 7.9701 Tf 7.314 4.936 Td [( )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(" p r )]TJ/F20 11.9552 Tf 11.955 0 Td [(" r T + g + N X j =1 )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( j )]TJ/F20 11.9552 Tf 11.955 0 Td [( r j + r : d : .73 Observethatthesummationineq..73canbesimpliedto N X j =1 )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( j )]TJ/F20 11.9552 Tf 11.955 0 Td [( r j = )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [()]TJ/F21 7.9701 Tf 7.314 4.936 Td [( )]TJ/F20 11.9552 Tf 11.955 0 Td [( r + N X j =1 )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( j r C j 66

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byexpandingthegradientof j andusingtheGibbs-Duhemequation.61.Substitutingthisbackinto.73andsimplifyingyieldsthechemicalpotentialformof Darcy'slaw: R )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(" v ;s = )]TJ/F20 11.9552 Tf 9.299 0 Td [(" N X j =1 )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(C j r j )]TJ/F20 11.9552 Tf 11.956 0 Td [(" r T + g + r : d .74 Cancellingthefactorof fromtheleft-handside,rewritingthecoecientofthe r j term,andmultiplyingbytheinverseof R gives v ;s = )]TJ/F40 11.9552 Tf 9.299 0 Td [(K N X j =1 )]TJ/F20 11.9552 Tf 5.48 -9.683 Td [( j r j + r T )]TJ/F20 11.9552 Tf 11.955 0 Td [( g )]TJ/F15 11.9552 Tf 15.793 8.087 Td [(1 r : d # : .75 Equation.75statesanamazingfact:theowofphase isdueonlytogradients inchemicalpotential,temperature,gravity,andviscousforces.Theviscousforces areoftenneglectedincreepingow.Thisgives v ;s = )]TJ/F40 11.9552 Tf 9.298 0 Td [(K N X j =1 )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( j r j + r T )]TJ/F20 11.9552 Tf 11.955 0 Td [( g # : .76 Itshouldbeemphasizedthatnoadditionalassumptionsweremadetoarriveatthis equation.Thatis,westillassumemultiphaseowwithapossiblyswellingsolid phase.Alloftheactions,interrelations,andcrosscouplingeectsaretiedupwithin thechemicalpotentialterm.Thisfurtherindicatesthatthechemicalpotentialisa generalizedforce that,ineect,incorporatesseveraldrivingforces. Analsimplicationistoconsiderapureuidphasewhereonlyoneconstituent ispresent.Inthiscase,Darcy'slawisrewrittenas v ;s = )]TJ/F40 11.9552 Tf 9.299 0 Td [(K r )]TJ/F21 7.9701 Tf 7.314 4.937 Td [( + r T )]TJ/F20 11.9552 Tf 11.955 0 Td [( g .77 where)]TJ/F21 7.9701 Tf 41.522 4.339 Td [( isthemacroscaleGibbspotential.Theentropycoecientofthegradient oftemperatureposesasignicantmodelingissueastheentropyisnotreadilymeasurable.Thefactstatedbyequation.77isthattheDarcyuxofapurespecies 67

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istrulycontrolledbygradientsintemperatureandGibbspotential.Thisisageneralizationoftheclassicalpressureformulationthatcapturesawiderrangeofphysical eects. 4.5.3Fick'sLaw WenowturnourattentiontodiusionandFick'slaw.In1855,AdolfFick publishedtherstmathematicaltreatmentofdiusion[37].Theempiricallybased equationsimplystatesthatthediusiveuxofaspeciesthroughamixtureisproportionaltothegradientinconcentrationofthespecies.Thishassincebeengeneralized throughthermodynamicsandphysicalchemistry[21,56]tostatethatthediusive uxisproportionaltothegradientinchemicalpotentialofthespecies.InthissubsectionweapplytheHybridMixtureTheoryconstructtoderiveaversionofFick's lawformultiphaseporousmedia.Itshouldbenotedherethattheclassicalchemical potentialfromthethermodynamicdenitionsofFick'slawfordiusioninaliquid notinaporousmediumisthenotthe same chemicalpotentialasthatdenedforthe porousmedia.Inmixturetheoryweviewtheporousmediumasamixtureofphases andspecies,buttheclassicalthermodynamicdenitionconsidersonephasewitha mixtureofspecies.Withthisdierenceinmind,itisnotimmediatelyclearthatthe multiphaseversionofFick'slawwillbethesame. ToderivethepresentversionofFick'slawwerstconsideralinearizationofthe coecientofthe r v j ; termintheentropyinequality.Thegradientofthediusive velocity, r v j ; ,istakentobezeroatequilibriumsothiscoecientiszerosince entropygenerationisminimizedatequilibrium.Therefore, t j + j j )]TJ/F20 11.9552 Tf 11.956 0 Td [( j I + j N = 0 forall j: Usingequation.32forthedenitionoftheLagrangemultiplier, N atequilibrium andlinearizingthecoecientof r v j ; about r v j ; gives t j = S j : r v j ; + )]TJ/F20 11.9552 Tf 9.299 0 Td [(" j j + j j )]TJ/F20 11.9552 Tf 11.956 0 Td [(" j I ; .78 68

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where S j isafourth-ordertensorthatarisesfromlinearization.Nowconsiderthe speciesconservationofmomentumequation.28.Weignoretheinertialterms sincediusionisassumedtobeslowthisisdiscussedinsomedetailinChapter2. Noweliminatethemomentumtransfertermsusingthelinearizedmomentumtransfer derivedfromtheentropyinequality,.38,use.78forthestresstensor,andusing thefactthat j = j + givesageneralizedformofFick'slaw: 2 j R j v j ; = )]TJ/F20 11.9552 Tf 11.955 0 Td [(" j r j + r S j : r v j ; + j g : .79 Thetermcontainingthegradientofdiusivevelocityislikelynegligibleasitissecond order.Ifnot,wewouldhavetorelatethefourth-ordertensor, S j ,withsomephysical processsimilartoviscosityforuidow.IfweneglectthistermthenFick'slaw canbewrittenas j R j v j ; = )]TJ/F20 11.9552 Tf 9.299 0 Td [( j r j + j g : .80 Despitethenovelchoiceofvariablesforthiswork,thisformofFick'slawisidentical tothatfoundbyBennethumandMurad[15]andWeinstein[81]. ThelinearizationcoecientinFick'slawhasasimilarmeaningtothatofthe resistivitytensorinDarcy'slaw.Inthiscase,though,wewishtoassociatetheinverse ofthistensorwiththediusivitytensorfromclassicalFick'slaw.Assumingthatthe inverseexistswehave j v j ; = )]TJ/F20 11.9552 Tf 9.298 0 Td [( j D j [ r j )]TJ/F40 11.9552 Tf 11.956 0 Td [(g ] : .81 Theunitsoftheleft-handsideare[ ML )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 t )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 ],andtheleft-handsidetermiscommonly knownas ux .Therefore,theunitsof D j issimplytime[ t ].Typicallythediusivity constantinagasismeasuredas[ L 2 t )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 ],sowecorrelate D g j tothediusioncoecient forthatphase, D g ,viatherelationship D g j = 1 R g j T D g ; .82 69

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where R g j isthespecicgasconstantforconstituent j .Theunitsof R g j T are[ L 2 t )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 ] andtheunitsof D g are[ L 2 t )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 ],hencemakingtheunitsof D g j [ t ].Thisisconsistent withtheformsofFick'slawfromthermodynamicsandphysicalchemistry[21,56]. Hence,thegasphaseformofFick'slawis g g j v g j ;g = )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( g j R g j T D g [ r g j )]TJ/F20 11.9552 Tf 11.955 0 Td [(g ] : .83 Toclosethissubsectionwenallyrecallfromourdiscussionofpore-scalediusion seeChapter2thatthediusivevelocitiesarerelatedvia N X j =1 j v j ; = 0 : .84 Multiplyingbythevolumefractionandrecognizingtheleft-handsideofFick'slaw indicatesthat N X j =1 j R g j T D [ r j )]TJ/F40 11.9552 Tf 11.955 0 Td [(g ] = 0 .85 nearequilibrium.Equation.85simplystatesthatthegradientsinchemicalpotentialarenotindependentofeachother.Thisfactwillbeusedinfuturechapters aspartofamoisturetransportmodel. 4.5.4Fourier'sLaw ThenalresultinthischapterisanextensiontoFourier'sLawforheatconduction.NoticethatinthechemicalpotentialformofDarcy'sLaw,.76,thereisa termthatinvolvesthegradientoftemperature.Thatis,theDarcyuxispartially drivenbyagradientintemperature.ThismeansthatDarcyowisnaturallydriven bygradientsintemperatureaswellasgradientsinchemicalpotential.Toproperly handlethiscouplingwecaneitherassumethatthegradientoftemperatureiszero constanttemperatureorconsidertheenergybalanceequationandtracktemperatureaswellaschemicalpotential.Tomovetowardaclosedsystemofequations,we 70

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deriveaversionofFourier'sLawfromtheentropyinequalitysothatwehaveanexpressionofheatuxintheenergybalanceequation.Attheoutsetwerstrecallthat intheentropyinequalitywe'veassumedonlyonetemperaturefortheentireporous medium.Thisimpliesthatwe'veassumedthattheseparatephasesareinthermal equilibrium.Forthisreason,wewilldevelopananaloguetoFourier'sLawthatholds fortheentirebulkmedium. FollowingBennethumandCushman'sworkonheattransportinporousmedia [14]weobservethatifwesumtheenergyequation.35over weobtainthebulk energybalanceequation De Dt )]TJ/F40 11.9552 Tf 11.955 0 Td [(t : r v )]TJ/F43 11.9552 Tf 11.955 0 Td [(r q )]TJ/F20 11.9552 Tf 11.955 0 Td [(h =0 ; .86 where = X .87a v = X v .87b u = v )]TJ/F40 11.9552 Tf 11.956 0 Td [(v .87c t = X t )]TJ/F20 11.9552 Tf 11.956 0 Td [(" u u .87d e = X [ e + u u ].87e q = X q + t u )]TJ/F20 11.9552 Tf 11.955 0 Td [( u e + 1 2 u u .87f h = X h a : .87g Givenidentitiesa-g,thederivationof.86followsaftersomesignicantalgebra. Denethemediumvelocity, v ,astheweightedvelocityofthemedium,andthe relativevelocity, u = v )]TJ/F40 11.9552 Tf 11.651 0 Td [(v ,isthe )]TJ/F15 11.9552 Tf 9.298 0 Td [(phasevelocityrelativetothemedium.Note that v ;s = v )]TJ/F40 11.9552 Tf 9.753 0 Td [(v s )]TJ/F40 11.9552 Tf 9.752 0 Td [(v + v = u )]TJ/F40 11.9552 Tf 9.753 0 Td [(u s = u ;s .Inthecasewherethevelocityofthesolid phaserelativetothemediumiszero u s = 0 weimmediatelyseethat u = v ;s .This 71

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assumptionalongwithequation.87findicatesthatthereisnaturallyacoupling betweentherelativevelocities, v ;s ,andthetotalheatux, q Usingthenearequilibriumresult, P q = K r T equation.39,wecan writethetotalheatuxas q = K r T + X t v ;s )]TJ/F20 11.9552 Tf 11.955 0 Td [( v ;s e + 1 2 v ;s v ;s : .88 Ifweweretowronglyneglectallofthetermsinthesummationwewouldarrive atFourier'sLawforheatconduction.Thetroublehereisthatthetermsinthe summationarenotnegligible,andthereforethetotalheatuxinaporousmedium mustbeafunctionofthegradientoftemperature,therelativevelocities,thestress intheuidphases,andtheinternalenergy. Since v s;s = v s )]TJ/F40 11.9552 Tf 11.637 0 Td [(v s = 0 theright-handsideofequation.88isonlyafunction oftheuidvelocitiesrelativetothesolidphase.Neglectingviscoustermswerecall thattheuid-phasestresstensorscanberewrittenas t = )]TJ/F20 11.9552 Tf 9.298 0 Td [(p I .Neglectingthe second-orderterm, v ;s v ;s ,thetotalheatuxisnowwrittenas q = K r T )]TJ/F26 11.9552 Tf 13.979 11.357 Td [(X = l;g [ p + e v ;s ] : .89 AtthispointwereplacetheinternalenergytermwithGibbsenergyinhopesof derivinganextendedFourier'sLawintermsofthechemicalpotential.Recallfrom thermodynamicsthattheGibbspotentialandinternalenergyarerelatedthrough e =)]TJ/F21 7.9701 Tf 19.74 4.936 Td [( + T )]TJ/F20 11.9552 Tf 13.232 8.088 Td [(p : .90 Therefore,thetotalheatuxcanbewrittenintermsoftheGibbspotentialas q = K r T )]TJ/F26 11.9552 Tf 13.979 11.358 Td [(X = l;g [ )]TJ/F21 7.9701 Tf 11.866 4.937 Td [( + T v ;s ] : .91 UsingtheGibbs-Duhemrelationship,.61,thiscanberewrittenintermsofthe chemicalpotentialas q = K r T )]TJ/F26 11.9552 Tf 13.978 11.358 Td [(X = l;g N X j =1 j j + T v ;s # : .92 72

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Thetroublewithboth.91and.92isthattheybothrelyonmeasurementsof entropy.Onewaytoworkaroundthisissueistoassumethattheentropyisonlya functionoftemperature,andthentorecallthatthespecicheatisdenedas c p = T @ @T = T d dT : Solvingthisseparableordinarydierentialequationundertheassumptionthatthe variationofspecicheatwithtemperaturenegligiblegives T = c p ln T T 0 + 0 .93 where T 0 isareferencetemperature,and 0 isareferenceentropy.Whilethisisonly anapproximationitdoesallowustomoveforwardwithoutdirectmeasurementsof entropy. The extended Fourier'sLaw.91presentedhereframestheequationspresented in[14]intermsoftheGibbspotential.Thiswillallowforeasiercouplingwith thechemicalpotentialformsofFick'sandDarcy'sLawspresentedintheprevious subsections.Thecaveatisthattheequationfortotalenergybalance,.86,isnot particularlyusefulsincewedonothaveconstitutiverelationsforthetotalstressand totalenergy.Forthatreason,wewillnotuseequation.91or.92forFourier's lawintheenergyequation.Insteadwewillusethelinearizedpartialheatuxand theconstitutiverelationsforthephasestressesandrelativevelocitiestoderivea generalizedheatequation. 4.6Conclusion Inthischapterwehaveshownthatanovelandjudiciouschoiceofindependent variablesfortheHelmholtzFreeEnergycanbeusedtoderiveformsofDarcy's, Fick's,andFourier'sLawsformultiphaseporousmedia.Theseequationsaresimilar tothosefoundin[11,14,15,81].Eachequationcanbewrittenwithaneyetoward themacroscalechemicalpotential,andineachcasethechemicalpotentialformis 73

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more mathematicallyappealing inthesensethattherearefewertermsandmany ofthephysicalprocessesaremanifestedinthechemicalpotentials.Thisillustrates theusefulnessofthechemicalpotentialasamodelingtool.Furthermore,sincethe chemicalpotentialappearsnaturallyineachoftheseequationswehavesetthestage foramorenaturalmethodofcouplingtheuidow,diusion,andheattransport.In Chapters5and7wewillcoupletheseequationswiththeupscaledmass,momentum, andenergybalanceequationstoyieldasystemofequationsthatwillgoverntotal moisturetransportandheatuxinunsaturatedporousmedia. 74

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5.CoupledHeatandMoistureTransportModel Toformgoverningequationsforheatandmoisturetransportinporousmediawe pairtheconstitutiveequationsderivedinChapter4withupscaledmass,momentum, andenergybalanceequationsderivedinChapter3.Thereareseveralexistingmodelsforeachphysicalprocessofinterestuidow,diusion,andheattransportand recentresearchindicatesaneedtounderstandthefullycoupledsystemofequations asitrelatestomoisturetransport,evaporation,heattransport,andotherphysical phenomena.InthischapterwederiveamodelforcoupledheatandmoisturetransportusingHybridMixtureTheoryandknowledgeofpore-scaleeects.Tobeginthis modelingtaskwerstinvestigatetheclassicalmodelsusedwithinthepastcenturyin Section5.1.InSections5.2and5.3wepairourconstitutiveequationsfromChapter4 withupscaledbalancelawsfromChapter3,performadimensionalanalysis,anddiscussformsofthelinearizationcoecientsarisingfromHMT.Thisisdoneinaneort togenerateaclosedsystemofgoverningequations.Severalsimplifyingassumptions aremadetoclosethesysteminSection5.4.Thesolutionstotheclosedsystemwill bediscussedinChapter7. 5.1IntroductionandHistoricalWork Togivethereaderabetterunderstandingoftheworkfromthepastcentury,we presentthreeclassicalmodelsherewithsomediscussionontheiradvantagesanddisadvantages.FirstwediscussRichards'equationforunsaturateduidowinSection 5.1.1,secondwediscussPhillipandDeVriesenhanceddiusionmodelinSection 5.1.2,andlastlywediscussDeVries'heattransportmodelinSection5.1.3. 5.1.1Richards'EquationforFluidFlow TheclassicalequationforuidowinunsaturatedmediaisknownasRichards' equationalsocalledthesaturationequation.Thisequationwasrstderivedin1931 byL.A.RichardsatCornellUniversity[65].Ittakesapostulatedformofthemass 75

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balanceequationsimilartoequation.25andreplacestheuxtermwithDarcy's law.Thegradientofpressureisrewrittenintermsofpressurehead h = p= g andthenaconstitutiverelationisassumedforthepressureheadasafunctionof saturationorvolumefraction.Anotherconstitutiverelationrelatingtherelative permeabilityofthemediumtosaturationisassumed.Thereareseveralversionsof theconstitutiverelations,butoneofthemore popular inrecentresearcharethose ofvanGenuchten[79,62].Anothermorerecentlyinvestigatedrelationshipisthe Fayer-Simmonsmodel[36,68,78],whichisanextensionofthevanGenuchtenmodel tocoverthecaseofverylowsaturations. Theresultoftheassumptionandsubstitutionsinthemassbalanceequationisa nonlineardiusionequationwheretheprimaryunknownisthepercentsaturationof themedium @S @t = r [ D S r S )]TJ/F20 11.9552 Tf 11.955 0 Td [(K S z ] ; .1 where K S isthehydraulicconductivityfunctionand D S istheproductof K S andthederivativeofcapillarypressurewithsaturation.Recallthatsaturationis denedas S = l 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(" s = l g + l = l = volumeofliquid volumeofporespace .2 andisunderstoodasthevolumeofliquidpervolumeofporespace. Thismodelhasbeeneectivelyusedforseveraldecades,butthereareafewdisadvantagesofnote.Firstofall,thisequationdoesnotallowforphasechangebetween theliquidandgas.Theoriginalmodelwasproposedforsystemswithimmiscibleuids,wherephasechangeslikelydon'toccur,butitisalsousedforunsaturatedsoils wherephasechangeispossibleandairisalwaysavailabe..Aseconddisadvantageis thathumidityandtemperaturegradientsarenotconsidered.Athirddisadvantage isthatthepressurehead-saturationcurveishystereticdependsonthehistoryof 76

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ow.Theconstitutivelawsforpressureheaddon'taccountforthishystereticbehaviordirectly.Instead,itisoftenassumedthatttingparameterschangewithchanging directionofow.Thisleadstothenaldisadvantage:theuseofthevanGenuchten capillarypressure-saturationrelation.Thisisawidelyusedrelationship,butreliesheavilyontwottingparameters.Themeasurementofthesettingparameters isdicult,andtheyaretypicallyfoundbyttingnumericalsolutionsofRichards' equationtoexperimentaldata. SeveralextensionsandmodicationstoRichards'equationhavebeenmaderecently,themostnotableofwhichisthatofHassanizadehetal.[49,47].Inthese papers,theyproposeadynamicrelationshipbetweencapillarypressureandsaturationbasedonHybridMixtureTheorywithinterfaces.Theyalsoproposethatthe hystereticeectobservedinthecapillarypressure-saturationcurvesisduetothe postulatedfactthatthecapillarypressure,saturation,andinterfacialareadensity, lg ,formauniquesurface.Thispartiallyexplainshystereticeectsbyseeingthem asaprojectionofthissurfaceontothecapillarypressures-saturationplaneinthe p c )]TJ/F20 11.9552 Tf 12.344 0 Td [(S )]TJ/F20 11.9552 Tf 12.344 0 Td [(" lg space.Thismodelisgaininginpopularity,butisfarfromwidespread acceptance.Someoftherelevantpublicationsare[48,49,50,46,47,58]. InthepresentchapterwepresentamodicationtotheRichards'equationthat incorporatesthedynamiccapillarypressurerelationshipofHassanizadehetal.The majordierencesbetweenthepresentderivationsandtheirworkare:modeling intermsofchemicalpotential,allowingforphasetransition,andallowingfor humidityandtemperaturegradients.Ourpresentmodelingeortwillaccountforall oftheseeects,andhence,constitutesageneralizationoftheexistingmodel. 5.1.2PhillipandDeVries'DiusionModel In1957,PhillipanddeVriespublishedtheircomprehensiveworkondiusionof watervaporinporousmedia[61].Intheirmodeltheypostulatean enhanced Fick's 77

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law, q g v = )]TJ/F20 11.9552 Tf 9.299 0 Td [( g D r C g v ; .3 where q g v isthewatervaporuxand isan enhancement factorthatisafunctionof thetoruosity,volumefractionofair,andamass-owfactor".Themass-owfactor isthenpostulatedasafunctionofpore-scalegradientsinsaturationandtemperature. Thismodelhassuccessfullybeenappliedtoseveraldiusionandevaporationproblemse.g.[78],butthetroubleisthattheexactformoftheenhancementisbasedon empiricalevidence.Furthermore,thismodelhascomeunderrecentscrutinydueto thefactthattheproposedfactorsaecting arepore-scaleeectsandaretherefore diculttoaccuratelymeasure[25,71,72,70,73,74,78,80].Manyoftheseworks usex-raytomographytoattempttomeasurethesepore-scaleeectsdirectly. IntheworkbyCassetal.[24],anempiricalformoftheenhancementfactorwas proposed.Inthiswork,attingparameterisusedintheenhancementfactorto arriveatgoodagreementwithexperimentaldata.Thismodelhasbeenusedinmore recentworkse.g.[68,78]inconjunctionwithamassbalanceequationforthewater vaporinthegasphase.Theresultingmodelisanonlineardiusionequationfor concentrationofwatervaporthatdeviatesfromthemoreclassicaldeVriesmodel. Asidefromtheempiricalttingparameter,themasstransferbetweenphasesalso reliesonattingparameterandanempirically-derivedfunctionalform. InthepresentchapterwebuildamodelfordiusionbasedonusingthechemicalpotentialasaprimaryunknownandtheHybridMixtureTheoryconstruct.The enhanceddiusionisnotincorporatedintothesemodels,andthemasstransferis modeledbythedierenceinchemicalpotentials;amorephysicallynaturalformulation.AcomparisonwillbemadetothemodelofCassetal. 5.1.3DeVries'HeatTransportModel In1958,deVriespublishedasecondpapercouplingheatandmoisturetransport inporousmedia[32].Inthisresearch,heproposedanextendedheattransport 78

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modelforporousmediathatisstillusedtoday.Neitherhisdiusionnorhisheat transportmodelwerethermodynamicallyderived.Instead,hebeganeachderivation withapostulationoftheformsofdiusiveandheatux.Fortheheattransport equationheincludedtermssimilartotheclassicalFourier'slaw,butalsoproposed thatheattransportwasduetoadvectivetransportintheuidphases.Thismodel isstillpopularlyusedtodaytocoupleheatandmasstransportinunsaturatedmedia [5,77,78].Thatbeingsaid,theeectsincludedinthisequationsarebasedsolelyon deVries'suppositionofthefactorsaectingheatow. In1999BennethumandCushmanpublishedtotheauthor'sknowledgetherst workusingHybridMixtureTheorytoderiveanextendeddeVriesmodelforheat transportinswellingsaturatedporousmedia[14].Inthepresentchapterwetakea similarapproachusingHMTtoderiveathermodynamicallyconsistentmodelforheat transportinnon-swellingunsaturatedmedia.Thisisdonewithaneyetowardusing gradientsintemperatureasthethermaldiusionprocessandthechemicalpotential todescribethesecondaryprocessessuchasadvection. 5.2Assumptions Inthissectionwestatethebaselineassumptionsthatwillbeusedthroughoutthe remainderofthiswork.Theseassumptionsaremeanttomakeminimallimitations ontheapplicabilityoftheresultingmodels,butatthesametimetheyaremeantto keepthemathematicstractable.Possiblerelaxationstotheseassumptionsandthe sourceofpossibleavenuesoffutureresearchwillbestatedastheyareencountered. Thesimplesetofbaselineassumptionsareasfollows: Assumption#1: Thesolidphaseisrigid,incompressible,andinert. Assumption#2: Theliquidandgasphasesareeachmadeupof N constituents. Assumption#3: Nochemicalreactionstakeplaceinanyofthephases. 79

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Therstassumptionisthemostrestrictive.Mathematicallyitcorrespondsto settingtheLagrangianderivativesofbothdensityandvolumefractionforthesolid phasetozero.Assumingthatthesolidisinertsimplymeansthatnomasswill precipitateonto,ordissolveawayfrom,thesolidphase.Withtheseassumptions,the solidphasemassbalanceequationfromequation.25becomes r v s =0 : .4 Ifadeformablesolidisconsideredwherethesolid-phasevolumefractioncanchange, thenthisassumptionwouldneedtoberelaxed.Oneparticularrelaxationofthis assumptionistoallowforincompressibilityandinertnessofthesolidphasebutrelax therigidityassumption.Underthisrelaxation,thesolidphasemassbalanceequation becomes D s s Dt )]TJ/F20 11.9552 Tf 11.955 0 Td [(" s r v s =0 : .5 Aconsequenceofxingthesolidphasevolumeisthat l + g =1 )]TJ/F20 11.9552 Tf 12.737 0 Td [(" s := where isknownastheporosityoftheporousmedium.Afurtherconsequenceis thattheliquidandgasphasevolumefractionsarenolongerindependentofeach other.Notethatwecouldhavemadethisassumptionupfrontandexploitedthe entropyinequalitywiththisassumptionthisisdonein[44,45]foradierentset ofindependentvariables,butproceedinginthisorderallowsustoreturntothe presententropyinequalityresultsandconsideradeformablesolidinthefuture.Since theuid-phasevolumefractionsarenolongerindependentwecanreplacethemby saturationasdenedby S = l l + g = l : .6 Thisimpliesthatthevolumefractionsarerelatedvia l = "S and g = )]TJ/F20 11.9552 Tf 11.955 0 Td [(S Assumption#2isabyproductoftheprincipleofequipresenceandwillberelaxed laterforsimplicity.Inthemostgeneralsense,thisassumptionstatesthatevery 80

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speciesthatexistsinoneuidphasealsoexistsintheother.Inrealitythisislikely nottrue.Forexample,ifaconstituentispresentintheliquidphaseitispossiblethat evaporatedparticlesoftheconstituentarenotbepresentinthegasphase.Another examplewouldbeifweweretoextendthismodeltoanoil-watersystem.Thetwo uidsinthiscaseareimmiscibleanditisunlikelythateveryspeciesinthewater phaseispresentintheoilphaseandvisaversa.Wetakethisintoaccountby settingtheappropriateconcentrationstozeroaftertheconstitutiveequationshave beenderived. Assumption#3indicatesthattherateofmassexchangeduetochemicalreactions,^ r j ,iszeroforallphases.Theconsequenceofthisisthattherateofmass generationofaconstituentinaphaseonlyoccursbetweentwophases.Thisistrue forsomeporousmedia,butchemicalreactionscanoccurinsomespeciccasessuchas remediationproblems.Underthisassumptionthesecasesarehencefortheliminated fromthediscussion. Othersimplifyingassumptionsexistformanymedia,butthethreepresented hereinconstituteasetthatleadstoseveralmathematicalsimplicationswithasfew physicalrestrictionsaspossible. 5.3DerivationofHeatandMoistureTransportModel IntheremainderofthischapterwefocusonusingtheresultsfromChapters3 and4,alongwiththeassumptionsfromSection5.2,toderiveaclosedsystemof equationsforheatandmasstransportinunsaturatedporousmedia.Thiswillbe donewithaneyetowardusingthechemicalpotentialasthedrivingforceforthese processes.Wewillshowthatundercertainadditionalsimplifyingassumptionsthat aclosedsystemcanbederived. 5.3.1MassBalanceEquations Werstbuildgeneralizedmassbalanceequationsintermsofthechemicalpotentialunderassumptions#1-#3.RecallfromChapter3thatthemassbalance 81

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equationforthe j th constituentinthe )]TJ/F15 11.9552 Tf 9.299 0 Td [(phaseisfromequation.21 D j j Dt + j r v j = X 6 = ^ e j +^ r j : .7 Thelasttermcanbedroppedunderassumption#3inSection5.2.Becauseofthe formoftheconstitutiveequation,andtoadheretotheprincipleofframeinvariance, itisconvenienttorewritethisequationrelativetothesolidphase.Todosowerecall theidentities D j Dt = D Dt + v j ; r .8a D Dt = D s Dt + v ;s r .8b andexpandtheLagrangiantimederivativesaccordinglytoget D s j Dt + v j ; r j + v ;s r j + j r v j = X 6 = ^ e j : .9 TakingthedenitionoftheLagrangiantimederivative, D s Dt = @ @t + v s r ; addingandsubtracting j r v ,andsubtracting j r v s = 0 gives @ j @t + r j v j ; + r j v ;s = X 6 = ^ e j : .10 NoticetheuseofAssumption#1inthelaststep,andobservethatifAssumption #1isrelaxedthenthemassbalanceequationwouldinvolveatimederivativeofthe solid-phasevolumefractionatleast. Equation.10isthegeneralmassbalanceequationforbothoftheuidphases. Noticethatwearenotreplacingthevolumefractionswithsaturationheresincewe don'tknowif istheliquidorgasphase.SubstitutingFick'slawforthediusive uxandDarcy'slawfortheDarcyuxgivesthechemicalpotentialformofthefull massbalanceequationforspecies j inphase : @ j @t )]TJ/F43 11.9552 Tf 11.955 0 Td [(r j D j [ r j )]TJ/F40 11.9552 Tf 11.955 0 Td [(g ] 82

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)]TJ/F43 11.9552 Tf 11.955 0 Td [(r j K N X k =1 k r k + r T )]TJ/F20 11.9552 Tf 11.956 0 Td [( g # = X 6 = ^ e j : .11 ItshouldbenotedherethattheEulerianandLagrangiantimederivativesareequal undertheassumptionthatthesolid-phasevelocityiszeroAssumption#1.Also notethatifwesumoverallconstituentsthenwearriveatthemassbalanceequation forthephasewherewehaveused P N j =1 j v j ; = 0 @ @t )]TJ/F43 11.9552 Tf 11.955 0 Td [(r K N X k =1 k r k + r T )]TJ/F20 11.9552 Tf 11.955 0 Td [( g # = X 6 = ^ e : .12 Thechemicalpotentialformofthemassbalanceequationisonlyoneform.We couldhaveusedthepressureformulationforDarcy'slawandarrivedatapressurechemicalpotentialformofthemassbalanceequation. Therateofmasstransfertermontheright-handsideofthemassbalanceequation canberewrittenintermsofalinearizedresultfromtheentropyinequality.Recall fromequation4.60thatthemasstransfertermcanbewrittenas ^ e j = )]TJ/F20 11.9552 Tf 10.461 -9.683 Td [( j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j M )]TJ/F20 11.9552 Tf 12.454 -9.683 Td [( j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j ; .13 wherethecoecient j )]TJ/F20 11.9552 Tf 12.083 0 Td [( j ischosentobeconsistentwithequationof[78]. Alsorecallthatsincetheinterfaceisassumedtocontainnomasswemusthavethat therateofmassgainedfromthe phasetothe j th speciesinthe phasemustbe equaltotherateofmasslostfromthe phasetothe j th speciesofthe phase: ^ e j = )]TJ/F15 11.9552 Tf 9.735 0 Td [(^ e j : Ifthechemicalpotentialoftheliquidphaseislargerthanthechemicalpotentialof thewatervaporthenmasswilltransferfromliquidtogasand^ e l g < 0.Similarly,if thechemicalpotentialoftheliquidphaseissmallerthanthatofthewatervaporthen masswilltransferfromgastoliquidand^ e l g > 0.Recallfromthediscussionadjacent toequation.60thattheunitsof M arethereciprocalofux. 83

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Thereareclearlymoreunknownsthanequationsinthe2 N uidequationssince wemustaccountforthedensities,temperature,volumefractions,andentropiesas wellasthechemicalpotentials.Certainsetsofsimplifyingassumptionscanbeused toreducethenumberofunknownse.g.incompressibilityofauidphase.These willbediscussedinSection5.4.Insteadofmakingtheseassumptionsupfrontwenow turnourattentiontoderivingageneralizedenergybalanceequationtoaccountfor thetemperature.Thiswillgiveonemoreequationbutwilladdnomoreunknowns tothesystemofequations. 5.3.2EnergyBalanceEquation Asanothersteptowarddevelopingaclosedsystemofequationequationsfor heatandmoisturetransportwenextexaminetheenergybalanceequation.Thiswill giveanequationintermsoftemperature,chemicalpotentials,saturationvolume fractions,entropy,anddensities;increasingtheequationcountbutnotincreasing thevariablecount.Sinceweassumedattheoutsetthatallofthephasesareinthermal equilibriumwewillonlyhaveoneequationforenergybalance.Thiswillbederived byconsideringthesumofeachofthephaseenergybalanceequations.Counterintuitively,wewillnotusetheformofFourier'sLawequation.87for.92 derivedforthetotalheatuxsincetheenergyequationderivedinthatsectionis morecumbersometoworkwiththantheindividualphaseenergyequations.Instead wewillusethepartialheatuxforeachphaseasderivedfromlinearizationabout equilibrium.39. Fromequation.35,thevolumeaveragedenergybalanceequationis D e Dt )]TJ/F20 11.9552 Tf 11.955 0 Td [(" t : d )]TJ/F43 11.9552 Tf 11.955 0 Td [(r q + h = X 6 = ^ Q : .14 Usingtheidentity D Dt = D s Dt + v ;s r andusing dot notationformaterialtime derivativesallowsustorewritetheenergyequationas e + v ;s r e )]TJ/F20 11.9552 Tf 11.955 0 Td [(" t : d )]TJ/F43 11.9552 Tf 11.955 0 Td [(r q + h = X 6 = ^ Q : .15 84

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Thetroublewith.15isthattherstandsecondtermscontaintheinteralenergy density, e .TotiethisequationbacktotheHMTframeworkwe'veusedthroughout andtogivetheequationamorenaturalsetofdependentvariablesweperforma LegendretransformationtochangetheenergytermintotheHelmholtzpotentialvia thethermodynamicidentity e = + T .Theenergyequationisnowwrittenas X 6 = ^ Q = D s Dt + v ;s r + T + T v ;s r + T + v ;s r T )]TJ/F20 11.9552 Tf 11.955 0 Td [(" t : d )]TJ/F43 11.9552 Tf 11.955 0 Td [(r q + h : .16 NextweseektoremovetheHelmholtzpotentialandentropytermsfromthe energyequation.TodothiswerecallthattheHelmholtzpotentialisafunctionof allofthevariableslistedin.6.UndertheassumptionslistedinSection5.2we dropthesolidphasetermsfromthislist.Furthermore,weknowthatunderthese conditionsthevolumefractionsarenotindependentsowecouldreplaceboth l and g bysaturation, S .Thisisnotdoneyetastheentropyinequalitywasexploited whileassumingthattheyareindependent.Theswitchcanbemadeatanypoint later.Therefore,underthepresentassumptions, = l ;" g ; l j ; g j ;T for j =1: N: Entropy, ,isassumedtobeafunctionofthesamesetofvariablessince = )]TJ/F20 11.9552 Tf 9.298 0 Td [(@ =@T .Usingthechainruletoexpandallofthederivativesof and in equation.16wearriveatanexpandedformoftheenergyequation: X 6 = ^ Q = @ @T + + T @ @T T + @ @" l + T @ @" l l + @ @" g + T @ @" g g + N X j =1 @ @ l j + T @ @ l j l j + N X j =1 @ @ g j + T @ @ g j g j + @ @T + + T @ @T r T 85

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+ @ @" l + T @ @" l r l + @ @" g + T @ @" g r g + N X j =1 @ @ l j + T @ @ l j r l j + N X j =1 @ @ g j + T @ @ g j r g j v ;s )]TJ/F20 11.9552 Tf 11.955 0 Td [(" t : d )]TJ/F43 11.9552 Tf 11.955 0 Td [(r q + h : .17 Fromtheentropyinequalityweknowthatthetemperatureandentropyareconjugatevariables.Forthisreasonwecancancelthesetermsfromthe T and r T coecients. Equation.17isanexpressionofenergybalanceforphase ,butsinceweare workingundertheassumptionthatthephasesareinthermalequilibriumwenow sumoverallofthephasestoformoneenergybalanceequationfortheentireporous medium.Thesumis: X X 6 = ^ Q = X T @ @T T .18 + X @ @" l + T @ @" l l + X @ @" g + T @ @" g g + X N X j =1 @ @ l j + T @ @ l j l j + X N X j =1 @ @ g j + T @ @ g j g j + X = l;g T @ @T r T + @ @" l + T @ @" l r l + @ @" g + T @ @" g r g + N X j =1 @ @ l j + T @ @ l j r l j + N X j =1 @ @ g j + T @ @ g j r g j v ;s )]TJ/F26 11.9552 Tf 11.955 11.358 Td [(X t : d )]TJ/F43 11.9552 Tf 11.955 0 Td [(r X f q g + X f h g .19 86

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The T termcanberewrittenas P T @ @T T = c p T ,andindoingsowe implicitlydenethevolumetricheatcapacityoftheentiremedium: c p = X T @ @T : Nextwerecallfromequation.39thatthepartialheatuxcanbewrittenas P f q g = K r T morewillbesaidaboutthefunctionalformof K infuture sections.Theheatsourcetermcanberewrittenas P f h g = h ,where h is anyinternalsourceorsinkofheatontheentiremediumi.e.heatsourcesthatare notboundaryconditions. Noticethatseveralofthegradienttermsarethesameasthoseinthelinearized constitutiveequationforthemomentumtransfer,.31and.36.Replacingthese termswiththeremainderofthemomentumbalancetermsandsimplifyinggives X X 6 = ^ Q = c p T )]TJ/F43 11.9552 Tf 11.955 0 Td [(r )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(K r T + h )]TJ/F26 11.9552 Tf 11.955 11.357 Td [(X t : d + X @ @" l + T @ @" l l + X @ @" g + T @ @" g g + X N X j =1 @ @ l j + T @ @ l j l j + X N X j =1 @ @ g j + T @ @ g j g j + X = l;g )]TJ/F26 11.9552 Tf 11.291 11.357 Td [(X 6 = ^ T + p r + N X j =1 X @ @ j + T @ @ j r j # + T @ @T r T + T @ @" l r l + T @ @" g r g + N X j =1 T @ @ j r j # v ;s : .20 Equation.20expressestheenergybalanceforthebulkporousmedium.Several ofthetermscanbesimpliedatthispoint.Towardthisgoal,wewill 87

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1.derivearelationfortheenergytransferterms: P P 6 = ^ Q 2.rewritethestressterm, P t : d ,usingconstitutiverelationshipsfor t 3.rewritethemomentumtransferterms, ^ T ,usingthelinearizedmomentum transferfromtheentropyinequality,.36 4.rewritetheadvectiveterms, v ;s ,usingDarcy'slaw,and 5.relatethechangesinentropy, @ @ ,tomaterialcoecients. Thersttwoofthesearediscussedinthefollowingtwosubsections.Thethirdand fourthcomeasaconsequenceofthersttwo,andthefthwillbediscussedunder propersimplicationsinfuturesections. 5.3.2.1EnergyTransferintheTotalEnergyEquation Considertheenergytransferandstressterms: ^ Q ; ^ Q j ,and t .Fromequations .36aand.36bwerecallthattherestrictionsontheinterfaceare N X j =1 ^ Q j + ^ i j v j ; +^ r j e j + 1 2 v j ; v j ; =0 8 ; .21a X X 6 = ^ Q j + ^ T j v j +^ e j e j + 1 2 v j v j =0 j =1: N: .21b Wealsonotetheidentity ^ Q = N X j =1 ^ Q j + ^ T j v j ; +^ e j e j ; + 1 2 v j ; v j ; .22 seeAppendixA.2of[81].Withthesethreeidentities,thesumoftheenergytransfer termscanbewrittenas X X 6 = ^ Q = X X 6 = N X j =1 ^ Q j + ^ T j v j ; +^ e j e j ; + 1 2 v j ; v j ; = N X j =1 X X 6 = h ^ Q j i 88

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+ X X 6 = ^ T j v j ; +^ e j e j ; + 1 2 v j ; v j ; = N X j =1 )]TJ/F26 11.9552 Tf 11.291 11.358 Td [(X X 6 = ^ T j v j +^ e j e j + 1 2 v j v j + X X 6 = ^ T j v j ; +^ e j e j ; + 1 2 v j ; v j ; = )]TJ/F21 7.9701 Tf 16.14 14.944 Td [(N X j =1 X X 6 = ^ T j v +^ e j e )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 ^ e j v j ; v j ; )]TJ/F40 11.9552 Tf 11.955 0 Td [(v j v j : .23 Nextweexaminethemomentumtransfertermappearinginequation.23. Recallfromequation.33that ^ T = N X j =1 h ^ T j +^ e j v j ; i : .24 Rearrangingthisidentityandmultiplyingbythe )]TJ/F15 11.9552 Tf 9.298 0 Td [(phasevelocityweseethat N X j =1 ^ T j v = ^ T v )]TJ/F21 7.9701 Tf 16.804 14.944 Td [(N X j =1 ^ e j v j ; v : .25 Substituting.25into.23,simplifying,andneglectingthesecond-ordertermsin velocityweseethat X X 6 = n ^ Q o = )]TJ/F26 11.9552 Tf 11.291 11.357 Td [(X 6 = l n ^ T l v l;s o )]TJ/F26 11.9552 Tf 11.956 11.357 Td [(X 6 = g n ^ T g v g;s o )]TJ/F15 11.9552 Tf 12.392 0 Td [(^ e l g )]TJ/F20 11.9552 Tf 5.48 -9.683 Td [(e l )]TJ/F20 11.9552 Tf 11.955 0 Td [(e g : .26 Noticefromthissimpliedversionthatwehaveeliminatedtheenergytransferin favorofthemassandmomentumtransfertermsaftersummingover andneglecting second-ordereects. 5.3.2.2StressintheTotalEnergyEquation Wenextderivetheproperformofthestressterminequation.20.The )]TJ/F15 11.9552 Tf 9.298 0 Td [(phasestressnearequilibriumisgivenby t = )]TJ/F20 11.9552 Tf 9.298 0 Td [(p I + : d fromthelinearizationoftheuidphasestresstensorsaboutequilibrium.Forthesolidphase 89

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stresstensor,ontheotherhand,wewillnotuseconstitutiverelationsfor t s butkeep inmindthatitisthesumofeectiveandhydratingstressesseeequation.23. Therefore, X t : d = X = l;g )]TJ/F20 11.9552 Tf 9.298 0 Td [(p I + : d : d + s t s : d s = )]TJ/F26 11.9552 Tf 13.315 11.358 Td [(X = l;g p I : d + X = l;g : d : d + s t s : d s : .27 Thesecondtermislikelynegligibleastheviscoustermstypicallyplaylittlerolein creepingow.Thismeansthat P t : d canbeapproximatedby X t : d = )]TJ/F26 11.9552 Tf 13.315 11.357 Td [(X = l;g p I : d + s t s : d s : .28 Usingindicialnotationwenotethatfortheuidphases, I : d = I : r v sym = ij v j;i = v i;i = r v ; andthereforethestresstensortermscanbesimpliedto X t : d = )]TJ/F26 11.9552 Tf 13.314 11.357 Td [(X = l;g f p r v g + s t s : d s : .29 Thesolidphaserate-of-deformationtensorisrelatedtothestrainrateofthesolid phase.Assumingthatthestrainrateiszeroforarigidandincompressiblesolid, wecanneglectthisterm.Thisimpliesthatthestresstensortermin.20canbe approximatedby X t : d = )]TJ/F26 11.9552 Tf 13.315 11.357 Td [(X = l;g f p r v g : UsingAssumption#1fromSection5.2forthedivergenceofthesolid-phasevelocity r v s =0,wenallyconcludethatthestresstermin.20canbesimpliedto X t : d = )]TJ/F20 11.9552 Tf 9.298 0 Td [(" l p l r v l;s )]TJ/F20 11.9552 Tf 11.956 0 Td [(" g p g r v g;s : .30 Notsurprisingly,thisstatesthatthestressisrelatedtotheuidpressures. 5.3.2.3TotalEnergyBalanceEquation 90

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Inthissubsectionweuseequations.26and.30tosimplifytheenergybalanceequation,.20.Substitutingtheseinto.20andcancelingthemomentum transfertermsgives 0= c p T )]TJ/F43 11.9552 Tf 11.955 0 Td [(r )]TJ/F40 11.9552 Tf 5.48 -9.684 Td [(K r T + h + l p l r v l;s + g p g r v g;s +^ e l g )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(e l )]TJ/F20 11.9552 Tf 11.955 0 Td [(e g + X @ @" l + T @ @" l l + X @ @" g + T @ @" g g + X N X j =1 @ @ l j + T @ @ l j l j + X N X j =1 @ @ g j + T @ @ g j g j + X = l;g p r + N X j =1 X @ @ j + T @ @ j r j # + T @ @T r T + T @ @" l r l + T @ @" g r g + N X j =1 T @ @ j r j # v ;s : .31 Nextwediscussthe p r v ;s and p r v ;s terms.Usingtheproduct ruleitisclearthatthesumofthesetwotermsgives p r v ;s .Achoiceismade heretoremovethesetermsinlieuofmasstransferterms.Todoso,werecallfrom themassbalanceequationthat D s Dt + r v ;s = X 6 = ^ e ; andsolvefor r v ;s : r v ;s = )]TJ/F20 11.9552 Tf 9.299 0 Td [( )]TJ/F20 11.9552 Tf 11.955 0 Td [(" )]TJ/F20 11.9552 Tf 11.955 0 Td [(" v ;s r +^ e : Wehavedroppedthesummationonthemasstransfertermsinceweareassuming thatthesolidphaseisinertandthatthereareonlytwouidphases.Multiplyingby p = givesanexpressionfor p r v ;s : p r v ;s = )]TJ/F20 11.9552 Tf 9.298 0 Td [(p )]TJ/F26 11.9552 Tf 11.956 16.857 Td [( p )]TJ/F26 11.9552 Tf 11.956 16.857 Td [( p v ;s r + p ^ e : .32 91

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Substitutingthisintotheenergyequationgives 0= c p T )]TJ/F43 11.9552 Tf 11.955 0 Td [(r )]TJ/F40 11.9552 Tf 5.48 -9.683 Td [(K r T + h + p l l + e l )]TJ/F26 11.9552 Tf 11.955 16.856 Td [( p g g + e g ^ e l g + )]TJ/F20 11.9552 Tf 9.299 0 Td [(p l + X @ @" l + T @ @" l l + )]TJ/F20 11.9552 Tf 9.299 0 Td [(p g + X @ @" g + T @ @" g g + N X j =1 )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( l p l l + X @ @ l j + T @ @ l j # l j + N X j =1 )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( g p g g + X @ @ g j + T @ @ g j # g j + X = l;g N X j =1 )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( p + X @ @ j + T @ @ j # r j + T @ @T r T + T @ @" l r l + T @ @" g r g + N X j =1 T @ @ j r j # v ;s : .33 Thereareseveralmoresimplicationsthatcanbemade.Tohelpwiththese simplicationsrecallthefollowingdenitionsforenthalpy,pressure,wettingpotential, chemicalpotential,andentropyrespectively: H = p + e .34a p = X N X j =1 j @ @ j .34b = X @ @" .34c j = + X " @ @ j .34d = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(@ @T : .34e Withtheseidentitiesinmindwemakethefollowingfoursimplications: 92

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1.coecientofthemasstransferterm: p l l + e l )]TJ/F26 11.9552 Tf 11.956 16.856 Td [( p g g + e g ^ e l g = )]TJ/F20 11.9552 Tf 5.479 -9.683 Td [(H l )]TJ/F20 11.9552 Tf 11.955 0 Td [(H g ^ e l g := L ^ e l g Recallingthat^ e l g istherateofmasstransferbetweentheuidphases, L is understoodasthelatentheatofevaporationsincethisrepresentstheheatlost orgainedduetophaseexchangedbetweentheuids.Thisisconsistentwith thechemist'sdenitionoflatentheatasthechangeinenthalpy. 2.coecientofthetimeratesofchangeofvolumefractions: )]TJ/F20 11.9552 Tf 11.955 0 Td [(p + X @ @" + T @ @" = )]TJ/F20 11.9552 Tf 9.298 0 Td [(p + + T @ @T = )]TJETq1 0 0 1 283.771 434.14 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 283.771 427.319 Td [(p )]TJ/F20 11.9552 Tf 11.955 0 Td [(T @ @T ; wherewerecallthat p isthermodynamicpressureasdenedinChapter4 p = p + : Atthispointwecanexchangethetimeratesofchangeofvolumefractionsfor timeratesofchangeofsaturation.Thatis,recall_ l = S and_ g = )]TJ/F20 11.9552 Tf 9.298 0 Td [(" S .The sumofthetwoassociatedtermsis )]TJETq1 0 0 1 254.958 242.439 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 254.958 235.618 Td [(p l )]TJ/F20 11.9552 Tf 11.955 0 Td [(T @ l @T S )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( )]TJETq1 0 0 1 363.34 242.439 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 363.34 235.618 Td [(p g )]TJ/F20 11.9552 Tf 11.955 0 Td [(T @ g @T S = )]TJETq1 0 0 1 274.109 209.135 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 274.109 202.314 Td [(p g )]TJETq1 0 0 1 299.409 209.135 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 299.409 202.314 Td [(p l + T @ g @T )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(@ l @T S: Fromthenearequilibriumresultsfromtheentropyinequalitywenowrecall fromequation.53that p n:eq: = p eq: )]TJ/F20 11.9552 Tf 11.955 0 Td [( .35 93

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Therefore,the S termbecomes p g eq: )]TJETq1 0 0 1 272.321 678.931 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 272.321 672.11 Td [(p l eq: +2 S + T @ @T )]TJ/F20 11.9552 Tf 5.48 -9.683 Td [( g )]TJ/F20 11.9552 Tf 11.955 0 Td [( l S: .36 Therstsetofparenthesisin.36approximatelyrepresentsthecapillary pressureasmeasuredatequilibrium, p g eq: )]TJETq1 0 0 1 332.266 583.08 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 332.266 576.259 Td [(p l eq: = p c : ThiswillbediscussedinmoredetailinSection5.4.1.1.Themiddletermin .36isaneectofthedynamicpressure-saturationrelationshipequation .53.Thetemperaturederivativecanbeinterpretedastheeectoftemperatureontherelativewettingpotential.Thatis,howmuchdoestemperature aecttherelativeanityforonephaseovertheother.Itislikelythataconstitutiveequationisneededforthisrelationship. 3.coecientoftimeratesofchangeofdensities: Wewishtorewritethesecoecientsintermsofenthalpyandchemicalpotential sinceitprovidesamathematicallysimplerexpression. N X j =1 )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( p + X @ @ j + T @ @ j # j = N X j =1 )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( p + )]TJ/F20 11.9552 Tf 5.479 -9.683 Td [( j )]TJ/F20 11.9552 Tf 11.955 0 Td [( )]TJ/F20 11.9552 Tf 11.955 0 Td [(" T @ @T )]TJ/F20 11.9552 Tf 5.48 -9.683 Td [( j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j = )]TJ/F20 11.9552 Tf 9.299 0 Td [(" p + + T + N X j =1 j )]TJ/F20 11.9552 Tf 11.955 0 Td [(T @ j @T j = )]TJ/F20 11.9552 Tf 9.299 0 Td [(" H + N X j =1 j )]TJ/F20 11.9552 Tf 11.955 0 Td [(T @ j @T j wherewerecallthat H istheenthalpyofphase 4.coecientofrelativevelocity: Forthiscoecientweagainusethedenitionsofpressure,chemicalpotential, 94

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andentropy.WealsorelyontheGibbs-Duhemrelationship.61. N X j =1 )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( p + X @ @ j + T @ @ j # r j + T @ @T r T + T @ @" l r l + T @ @" g r g + N X j =1 T @ @ j r j = )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( p r + N X j =1 )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( j )]TJ/F20 11.9552 Tf 11.955 0 Td [( r j + c p r T + T @ @T @ @" l r l + @ @" g r g + N X j =1 @ @ j r j + N X j =1 @ @ j r j = )]TJ/F20 11.9552 Tf 9.298 0 Td [(" )]TJ/F21 7.9701 Tf 7.315 4.936 Td [( r + N X j =1 j r j + c p r T + T @ @T )]TJ/F26 11.9552 Tf 11.291 13.27 Td [( ^ T s + ^ T + p r + X N X j =1 @ @ j r j = )]TJ/F20 11.9552 Tf 9.298 0 Td [(" )]TJ/F21 7.9701 Tf 7.315 4.936 Td [( r + N X j =1 j r j + c p r T + T @ @T )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" 2 R v ;s + p r + N X j =1 )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( j )]TJ/F20 11.9552 Tf 11.955 0 Td [( r j = )]TJ/F20 11.9552 Tf 9.298 0 Td [(" )]TJ/F21 7.9701 Tf 7.315 4.936 Td [( r + N X j =1 j r j + c p r T + T @ @T )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" 2 R v ;s + p r )]TJ/F20 11.9552 Tf 11.956 0 Td [(" r + N X j =1 j r j = )]TJ/F20 11.9552 Tf 9.298 0 Td [(" )]TJ/F21 7.9701 Tf 7.315 4.936 Td [( r + N X j =1 j r j + c p r T + T @ @T )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" 2 R v ;s )]TJ/F20 11.9552 Tf 11.955 0 Td [(" )]TJ/F21 7.9701 Tf 7.314 4.936 Td [( r + p r )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" + N X j =1 j r j Sincethiscoecientiscontractedwiththerelativevelocity, v ;s ,wecanlikely neglecttherelativevelocityterminthetemperaturederivativeasitwillresult 95

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insecond-ordereects.Thissimpliesthecoecientoftherelativevelocityto )]TJ/F20 11.9552 Tf 11.956 0 Td [(" )]TJ/F21 7.9701 Tf 7.314 4.936 Td [( r + N X j =1 j r j + c p r T + T @ @T )]TJ/F20 11.9552 Tf 9.299 0 Td [(" )]TJ/F21 7.9701 Tf 7.314 4.936 Td [( r + N X j =1 j r j + p r )]TJ/F20 11.9552 Tf 5.479 -9.683 Td [(" : .37 Afterthesefoursimplicationsandrearrangements,equation.33isnowrewrittenas 0= c p T )]TJ/F43 11.9552 Tf 11.955 0 Td [(r )]TJ/F40 11.9552 Tf 5.48 -9.684 Td [(K r T + h + L ^ e l g + p g eq: )]TJETq1 0 0 1 198.981 504.867 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 198.981 498.046 Td [(p l eq: +2 S + T @ @T )]TJ/F20 11.9552 Tf 5.479 -9.683 Td [( g )]TJ/F20 11.9552 Tf 11.955 0 Td [( l S )]TJ/F20 11.9552 Tf 11.955 0 Td [("SH l l + "S N X j =1 l j )]TJ/F20 11.9552 Tf 11.956 0 Td [(T @ l j @T l j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" )]TJ/F20 11.9552 Tf 11.955 0 Td [(S H g g + )]TJ/F20 11.9552 Tf 11.955 0 Td [(S N X j =1 g j )]TJ/F20 11.9552 Tf 11.955 0 Td [(T @ g j @T g j + X = l;g )]TJ/F20 11.9552 Tf 9.299 0 Td [(" )]TJ/F21 7.9701 Tf 7.315 4.936 Td [( r + N X j =1 j r j + c p r T + T @ @T )]TJ/F20 11.9552 Tf 9.299 0 Td [(" )]TJ/F21 7.9701 Tf 7.314 4.937 Td [( r + N X j =1 j r j + p r )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" # v ;s : .38 Equation.38dependsontemperature,wettingpotentials,enthalpies,chemical potentials,Gibbspotentials,saturation,densities,pressures,andrelativevelocities. SincetheGibbspotentialsarefunctionsofdensitiesandchemicalpotentialsthis doesnotaddmoreunknownstothesystemofequations.Thepressuresandrelative velocitiescanbepairedwithformsofDarcy'slaw,andconstitutiveequationsare neededfortheenthalpiesandwettingpotentials.Wenowturnourattentiontothe couplingoftheuid-phasemassbalanceequationsandthepresentenergyequation. 5.4SimplifyingAssumptions{AClosedSystem 96

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Ahostofsimplifyingassumptionscanbemadeonthesystemconsistingofequations.11for = l;g and.38.Thesearemadetoreducethenumberof unknownsandequationstoacountthatismoreeasilyhandledbynumericalsolvers. Thisisalsodonetoavoidhavingtomodelanysecondarypossiblysecond-order physicalprocessesexamplesofwhichincludeveryslowprocessessuchasthoseon theorderof v l;s 2 or v j ; 2 .TheseassumptionsareinadditiontoAssumptions #1-#3madeinSection5.2. Assumption#4: Assumethattheliquidphaseiscomposedofapureuidwith noadditionalspecies.Strictlyspeakingthisisnotrealisticsincethewaterin eldmeasurementscontainscontaminants,dissolvedsolids,chargedionssuch assodium,andotherimpurities.Theconsequenceofthisassumptionisthat thediusivetermswithintheliquidmassbalanceequationarezero v l j ;l = v l;l = 0 : Assumption#5: Theliquidphaseisassumedtobeincompressible.Thisassumptionisvalidundermoderatepressuresandallowsustoremovetheliquidphase materialtimederivativeofdensityfromtheliquidmassbalanceequation D l l Dt =0 : Inisothermalconditionsthedensityoftheliquidphasecanbeassumedconstant inspaceandtime.Inthepresenceofthermalgradients,ontheotherhand,we presumethatthedensityoftheliquidphaseisafunctiononlyoftemperature givenbytheempiricalmodel l T =10 3 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(7 : 37 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(6 T )]TJ/F15 11.9552 Tf 11.955 0 Td [(277 : 15 2 +3 : 79 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(8 T )]TJ/F15 11.9552 Tf 11.955 0 Td [(277 : 15 3 .39 measuredin kg=m 3 andwhere[ T ]= K .SeeFigure5.1a. 97

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Assumption#6: Thegasphaseisassumedtobeanidealbinarymixtureofwater vaporandinert air .Therearemostcertainlymorethantwospeciesinmost practicalgasmixtures,buthereweareconcernedwithwiththediusion,evaporation,andcondensationofwatervaporwithinthegasmixture.Theother speciesareassumedtobenon-reactiveandarethereforeallgroupedtogether intothe air species.Wechoosethemixturetobeidealsothatwecantake advantageoftheidealgaslaw.Thisisvalidsinceathegaspressuresunder mostexperimentalconsiderationsareclosetoatmospheric,bunderRichards' assumption[62,65],thebulkgaspressuredoesn'tvarymuchundermostexperimentalconsiderations,andcthetemperaturesunderconsiderationaren't far fromstandardroomtemperature.Theuseofanidealgasmixturewill breakdownunderhigherpressures,highertemperatures,andpossiblyunder highvariationsintemperature. Assumption#7: Thegas-phasechemicalpotentialsanddensitiesareonlyfunctionsoftherelativehumidityandtemperature g j = g j ';T g j = g j ';T : .40 Wemakethisassumptionbasedonthefactthatattheporescalewecaneasily convertbetweenthechemicalpotential,thedensity,andtherelativehumidity. Furthermore,thisallowsforustotiethegas-phasemassbalanceequationto experimentallymeasurablequantitiessuchastherelativehumidity. Justasattheporescale,wedenethemacroscalerelativehumidity, ,viathe saturatedvapordensity, sat ,andthedensityofthewatervaporinthemixture: g v = sat '; .41 where sat = sat T canbeexpressedthroughtheempiricalequation sat = exp : 37 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6014 : 79 =T )]TJ/F15 11.9552 Tf 11.955 0 Td [(7 : 92 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 T T 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 : .42 98

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seeFigure5.1b. Thechemicalpotentialofthewatervaporisdenedthroughtheidealgaslaw as g v = g v + R g v T ln .43 where = p sat =p isafunctionoftemperaturefrom.42and p isatmospheric pressure. Thereasonwearecallingthisanassumption"isthat,strictlyspeaking,these relationshipsholdforthepore-scalechemicalpotentialsandpressures.Weare dealingwithaveragedupscaledquantitiessowemaketheassumptionthat thesequantitiesfollowthesamefunctionalforms.Itisknownthattheupscaled pressure,density,andchemicalpotentialarenotthesameasthepore-scale pressure,soineectwearedeningtheupscaledrelativehumiditythrough theserelationships. aLiquiddensityvs.temperature bSaturatedvapordensityvs.temperature Figure5.1:Densitiesasfunctionsoftemperature Underassumptions4and5ontheliquidphasewereectnowonthechoiceof theformofDarcy'slawfortheliquidphase.Intheabsenceofspeciesitmaynotbe 99

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reasonabletousethechemicalpotentialformandinsteadreverttothepressureform. Recallfromequation.77thattheDarcyuxforauidwithonespeciesisdriven bygradientsinGibbspotentialandtemperature.Recallalsothatthecoecientof thetemperaturegradientisthemacroscaleentropy.Tosidestepthenecessityof modelingtheliquidphaseentropyandGibbspotentialdirectlyweusethepressure formoftheDarcyux:equation.66.Givenoneliquidspecies,arigidsolidphase, twogasspeciesseeassumption#6,andtheassumptionthatthegasdensitiesare functionsoftemperatureandrelativehumidityseeassumption#7,theDarcyux fortheliquidphasecanbewrittenas l R l )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(" l v l;s = )]TJ/F20 11.9552 Tf 9.298 0 Td [(" l r p l )]TJ/F20 11.9552 Tf 11.956 0 Td [( l l r l )]TJ/F20 11.9552 Tf 11.956 0 Td [( l g r g + l l g + g g @ g @ l + s s @ s @ l r l )]TJ/F20 11.9552 Tf 11.955 0 Td [(" l l X j = v;a @ l @ g j r g j = )]TJ/F20 11.9552 Tf 9.298 0 Td [(" l r p l )]TJ/F20 11.9552 Tf 11.956 0 Td [(" )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( l l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l g r S + l l g + g g @ g @ l + s s @ s @ l @ l @T r T )]TJ/F20 11.9552 Tf 11.955 0 Td [(" l l X j = v;a @ l @ g j @ g j @T r T + @ g j @' r = )]TJ/F20 11.9552 Tf 9.298 0 Td [(" l r p l )]TJ/F20 11.9552 Tf 11.956 0 Td [(" )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( l l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l g r S + l l g )]TJ/F20 11.9552 Tf 11.955 0 Td [(" l C l T r T )]TJ/F20 11.9552 Tf 11.955 0 Td [(" l C l r ': .44 Thefunctions C l T and C l areimplicitlydenedbyequations.44andmaybe functionsofanyvariablesfromthesetofindependentvariablesfortheHelmholtz Potential.Thecoecientofthesaturationgradientcanberewrittenas )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( l l )]TJ/F20 11.9552 Tf 11.956 0 Td [( l g = "" l l @ l @" l )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(@ l @" g = "" l l @ l @" l )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(@ l @" g =2 l l @ l @S := l C l S : .45 Thiscoecientfunctionmeasuresthechangesinliquidenergyduetochangesin saturationwhileholdingdensityxed.Thenotationchosenforthesecoecients 100

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ismeanttobedescriptive;thesubscriptindicatestheassociatedgradientandthe superscriptindicatesthephase. Dividingbothsidesof.44by l givesthesimpliedpressure,saturation,temperature,andrelativehumidityformulationoftheliquidDarcyux R l )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" l v l;s = )]TJ/F43 11.9552 Tf 9.299 0 Td [(r p l + l g )]TJ/F20 11.9552 Tf 11.955 0 Td [(C l S r S )]TJ/F20 11.9552 Tf 11.955 0 Td [(C l T r T )]TJ/F20 11.9552 Tf 11.955 0 Td [(C l r ': .46 Thersttwotermsontheright-handsidearetheclassicalDarcyterms,andthe functions C l S ;C l T ; and C l are,asofyet,unknown.Allofthesenewfunctionsmeasure crosscouplingeectsduetothepresenceofotherphases.Thoughtexperimentsused tomakesenseofthesenewtermswillbepresentedinSection5.4.1.1afteradeeper discussionofcapillarypressure. Underassumptions#1-#7,theheatandmasstransportsystemcannowbe writtenas: @S @t )]TJ/F43 11.9552 Tf 11.955 0 Td [(r K l r p l + C l S r S + C l T r T + C l r )]TJ/F20 11.9552 Tf 11.955 0 Td [( l g = M )]TJ/F20 11.9552 Tf 5.48 -9.683 Td [( l )]TJ/F20 11.9552 Tf 11.956 0 Td [( g v l )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v .47a )]TJ/F20 11.9552 Tf 11.955 0 Td [(S @ g v @t )]TJ/F20 11.9552 Tf 11.956 0 Td [(" g v @S @t )]TJ/F43 11.9552 Tf 11.956 0 Td [(r g v D g v [ r g v )]TJ/F40 11.9552 Tf 11.955 0 Td [(g ] )]TJ/F43 11.9552 Tf 11.956 0 Td [(r g v K g [ g v r g v + g a r g a + g g r T )]TJ/F20 11.9552 Tf 11.955 0 Td [( g g ] = )]TJ/F20 11.9552 Tf 9.299 0 Td [(M )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v )]TJ/F20 11.9552 Tf 12.952 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v .47b 0= c p T )]TJ/F43 11.9552 Tf 11.955 0 Td [(r )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(K r T + h + L ^ e l g + p g eq: )]TJETq1 0 0 1 216.077 196.896 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 216.077 190.075 Td [(p l eq: +2 S + T @ @T )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( g )]TJ/F20 11.9552 Tf 11.956 0 Td [( l S )]TJ/F20 11.9552 Tf 11.956 0 Td [(" )]TJ/F20 11.9552 Tf 11.955 0 Td [(S H g g + )]TJ/F20 11.9552 Tf 11.955 0 Td [(S X j = v;a g j )]TJ/F20 11.9552 Tf 11.955 0 Td [(T @ g j @T g j + l c l p + e l d l dT r T + T l @p l @T r l )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(" l v l;s + )]TJ/F15 11.9552 Tf 9.298 0 Td [()]TJ/F21 7.9701 Tf 7.314 4.936 Td [(g r g + X j = v;a [ g j r g j ]+ g c g p r T 101

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+ T @ @T )]TJ/F15 11.9552 Tf 9.298 0 Td [()]TJ/F21 7.9701 Tf 7.315 4.936 Td [(g r g + X j = v;a [ g j r g j ]+ p g g g r g g # g v g;s : .47c Thissystemofequationsoriginatedfrommass,momentum,andenergyconservation andwassupplementedwithconstitutiveformsoftheratesofmass,momentum,and energytransfer.Weusedtheincompressibilityoftheliquidphasetoarriveatthe fourthlineoftheenergyequation.Inthegasphase,thechangeinpressurewith temperatureisgivenviatheidealgaslaw: p g = g R M g T; .48 @p g @T = g R M g + TR M g @ g @T .49 where M g isthemolarmassofthegasmixtureand R istheuniversalgasconstant. Intheliquidphase,thechangeinpressurewithtemperatureistheratioofisobaric andisothermalcompressibilitiesofliquidwater @p l @T = )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( 1 V l @V l @T )]TJ/F15 11.9552 Tf 13.841 8.088 Td [(1 V l @V l @p l = l l : Recallthat l = l T g v = g v ';T g j = g j ';T l = "S g = )]TJ/F20 11.9552 Tf 12.145 0 Td [(S and = T .Furthermore, v ;s istheDarcyuxassociatedwiththe )]TJ/F15 11.9552 Tf 9.298 0 Td [(phase seeequation.76andthelatentheat, L ,isanempiricallybasedfunctionoftemperature.Therefore,assumingthattheenthalpy,internalenergy,andthelinearization coecientsareknownfunctionsofthesesamevariables,equations.47a-.47c canbeseenasaclosedsystemofequationsinsaturation S ,relativehumidity andtemperature T .Itremainstondrelationshipsforthelinearizationcoecients, thecrosscouplingDarcyterms,thegas-phaseentropy,theenthalpy,andthechemical potentials.Inthenextsubsectionswediscussdimensionalanalysis,functionalforms ofthecoecients,andfurthersimplicationsforeachequationoneatatime. 5.4.1SaturationEquation 102

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Intheliquidphase,thelinearizationconstant, K l ,isafunctionoftheeasein whichuidowsthroughthemedium.Thisisknownasthehydraulicconductivity ofthemedium.Thehydraulicconductivityisalsoknowntobeafunctionofthe permeabilityofthemedium.Insaturatedrigidmediathisisconsideredconstant oratleastatensor,butinunsaturatedmediatheyaretypicallytakenasfunctions ofsaturation.Inthepresentcase,acarefulinspectionoftheunitsindicatethat K l = l = k c l g ; .50 where isthepermeabilitytensorofthemedium, k c isthehydraulicconductivity tensor,and l isthedynamicviscosity[5,62].Notationally "withasuperscript willdenotechemicalpotential,and "withasubscriptwilldenotedynamic viscosity. Thepermeability, ,istypicallyseparatedintoasaturatedpermeability, s ,and arelativepermeability, r .Therelativepermeabilityisassumedtobeafunctionof saturationanddependsonwhether isthewettingornon-wettingphase[62].There areseveralfunctionalformsof r ,butoneofthemorecommonlyusedisthatofvan Genuchten[79], rl = rw = S e 1 = 2 n 1 )]TJ/F26 11.9552 Tf 11.955 13.27 Td [(h 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [( S e 1 =m i m o 2 .51a rg = rnw =[1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( S e ] 1 = 3 h 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( S e 1 =m i 2 m ; .51b where m isattingparameter,and S e istheeectivesaturationdenedby S e = S )]TJ/F20 11.9552 Tf 11.956 0 Td [(S min S max )]TJ/F20 11.9552 Tf 11.955 0 Td [(S min S e 2 [0 ; 1] : .52 Typicalvaluesof m arelessthan1where m =2 = 3iscommonlyusedasastarting pointforttingnumericalmodelstoexperimentaldata.Typicalrelativepermeability curvesareshowninFigure5.2.Thereaderistokeepinmindthatthereareseveral suchmodelsintheliterature[5,62].ThevanGenuchtenmodelsimplyconstitutes awidelyusedrelativepermeabilitymodel.Notethatthereisnotasymmetryin 103

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k rnw and k rw inthesensethat k rnw S e 6 = k rw )]TJ/F20 11.9552 Tf 12.031 0 Td [(S e aswouldnaivelybeassumed. Thisisamanifestationofthefactthatunsaturatedmedia behave dierentlyduring imbibitionanddrainage.Thevalueof s ischosenbasedonthetypeofmedium.If themediumisisotropicthenthetensorialnotationcanbedroppedandvaluesfrom TableE.2canbeused. Figure5.2:vanGenuchtenrelativepermeabilitycurves.Theredcurveshowsthe non-wettingphase, rnw S e ,andthebluecurvesshowthewettingphase, rw S e eachfor m =0 : 5 ; 0 : 67 ; 0 : 8 ; and1. 5.4.1.1CapillaryPressureandDynamicCapillaryPressure Thecapillarypressure, p c ,istypicallydenedasthedierencebetweenthenonwettinggasandwettingliquidphasepressureswhenmeasuredinatubeatequilibrium p c = p non )]TJ/F21 7.9701 Tf 6.587 0 Td [(wetting )]TJ/F20 11.9552 Tf 11.955 0 Td [(p wetting : .53 104

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Atthemicrosale,thedierenceisrelatedtothesurfacetensionoftheuid,thecontact angle,andtheeectiveradiusthroughtheYoung-LaplaceequationseeFigure5.3 p c = 2 cos r : .54 Thequestioniswhich pressure thermodynamic,classical,orwettingseeSection 1 = r Figure5.3:Contactangleandeectiveradiusinacapillarytubegeometry. isthe contactangle, r istheeectiveradius,and istheradiusofcurvatureoftheinterface. 4.3representsthenon-wettingandwettingpressuresinequation.53.Thecapillarypressureismeasuredwithaforcetransducerinthesamemannerthatthe classical pressureismeasured.Forthisreasonwedenethecapillarypressureas p c = p g )]TJ/F20 11.9552 Tf 11.956 0 Td [(p l : .55 Nowthatweunderstandwhich pressure isassociatedwiththecapillarypressure weturntotheentropyinequalitytoderiveaconstitutiveequationequationforthe timerateofchangeofsaturation.InRichards'equationitisstandardpracticeas mentionedinSection5.1.1totakethecapillarypressureasafunctionofsaturation. Theserelationsarereasonableforanequilibriumrelationships.Inthepresentmodelingeortwelooktowardtheentropyinequalitytodetermineanappropriateformof p c awayfromequilibrium.Intheentropyinequalityequation.13therearetwo termsassociatedwiththetimerateofchangeofsaturation: )]TJETq1 0 0 1 279.294 78.821 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 279.294 72 Td [(p l l and )]TJETq1 0 0 1 353.771 78.821 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 353.771 72 Td [(p g g : 105

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Since_ g = )]TJ/F15 11.9552 Tf 11.387 0 Td [(_ l thesetermscanbecombinedtogive p g )]TJETq1 0 0 1 430.099 714.866 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 430.099 708.045 Td [(p l l .Thetimerate ofchangeofvolumefractionisaconstitutivevariablesotheassociatedlinearized equationis )]TJETq1 0 0 1 241.695 630.97 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 241.695 624.149 Td [(p g )]TJETq1 0 0 1 266.996 630.97 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 266.996 624.149 Td [(p l n:eq = )]TJETq1 0 0 1 324.658 630.97 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 324.658 624.149 Td [(p g )]TJETq1 0 0 1 349.958 630.97 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 349.958 624.149 Td [(p l eq )]TJ/F20 11.9552 Tf 11.955 0 Td [( l ; .56 wheretheequilibriumstateisnotnecessarilyzeroandtheminussignischosentobe consistentwiththeentropyinequality.Fromthethreepressuresrelationship,.52, theclassicalpressureisgivenas p = p + where p isthe thermodynamic pressureand isawettingpotential.Thedierence inthermodynamicpressuresisthereforerewrittenas p c := p g )]TJETq1 0 0 1 216.347 415.289 cm[]0 d 0 J 0.478 w 0 0 m 5.875 0 l SQBT/F20 11.9552 Tf 216.347 408.468 Td [(p l = p g )]TJ/F20 11.9552 Tf 11.955 0 Td [( g )]TJ/F26 11.9552 Tf 11.955 9.684 Td [()]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(p l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l = p c )]TJ/F26 11.9552 Tf 11.955 9.684 Td [()]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( g )]TJ/F20 11.9552 Tf 11.956 0 Td [( l = p c )]TJ/F20 11.9552 Tf 11.955 0 Td [( c andequation.56becomes p c )]TJ/F20 11.9552 Tf 11.955 0 Td [( c n:eq = p c )]TJ/F20 11.9552 Tf 11.955 0 Td [( c eq )]TJ/F20 11.9552 Tf 11.956 0 Td [( l : .57 Rewritingweget p c n:eq = p c eq + c n:eq )]TJ/F20 11.9552 Tf 11.956 0 Td [( c eq )]TJ/F20 11.9552 Tf 11.955 0 Td [( l .58 Weassumethattheeectofthesolidphaseonthecapillarypressureiscompletely capturedbythepreferentialwetting, c .Withoutthesolidphase,thenormalpressuresoftheliquidandgasphasesarezerothisisthecasewithaatinterface. Withthisassumptionthethermodynamicpressuresareequalacrossthephasesat equilibrium.Therefore, p c j eq =0.Thisimpliesthat p c j eq = p c j eq + c j eq = c j eq Thereforethecapillarypressureatequilibriumisinterpretedasthedierenceinwettingpotentialandwearriveatanexpressionthatissimilartothatfoundin[47]. 106

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Toavoidpossibleconfusionwewillcontinuetousethesymbols p c j eq inplaceof c j eq eventhoughtheyareunderstoodtobethesame. Wenallyarriveatanexpressionrelatingtheclassicalliquid-phasepressurethat appearsinDarcy'slaw, p l j n:eq ,andthecapillarypressure, p c j eq : )]TJ/F20 11.9552 Tf 9.298 0 Td [(p l n:eq = p c eq + c n:eq )]TJ/F20 11.9552 Tf 11.955 0 Td [( c eq )]TJ/F20 11.9552 Tf 11.955 0 Td [(p g n:eq )]TJ/F20 11.9552 Tf 11.955 0 Td [( l : .59 Ifthedeviationinthewettingpotentialfromequilibriumisassumedtobesmall relativetothepressureandthedynamiceectswecanapproximatetheliquidpressure as )]TJ/F20 11.9552 Tf 9.298 0 Td [(p l n:eq p c eq )]TJ/F20 11.9552 Tf 11.955 0 Td [(p g n:eq )]TJ/F20 11.9552 Tf 11.955 0 Td [( l ; .60 whereitispossiblethat p g 0aswellinfact,thisisacommonassumption.To seewhythedeviationinwettingpotentialmightbesmall,considerthatinequation .58if p c j n:eq p c j eq thenthesaturationdynamicsisdrivenbythedeviationin wettingpotential.Thedeviationinwettingpotentialmeasureshowmuchtheshape ofthecurvedliquid-gasinterfaceisawayfromequilibrium.Inslowowsitisunlikely thatthisdeviationissignicant. AsmentionedinSection5.1.1,theequilibriumcapillarypressurecanberelated totheeectivesaturationthroughthevanGenuchten p c )]TJ/F20 11.9552 Tf 13.192 0 Td [(S relationship.This relationshipdependsonseveralttingparametersandisgivenas p c S e = 1 )]TJ/F20 11.9552 Tf 5.479 -9.683 Td [(S )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 =m e )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 )]TJ/F21 7.9701 Tf 6.586 0 Td [(m ; .61 where hasunitsofreciprocalpressureand m isthesamettingparameterasinthe relativepermeabilities.51a[5,62].SeeFigure5.4forseveralexamplesofcapillary pressure-saturationcurvesforvarioussetsofparameters.Generallyspeaking, m increasestoward1asthesoilbecomesmoredenselypacked. 107

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Figure5.4:ExamplesofvanGenuchtencapillarypressure-saturationcurvesfor variousparameters. SubstitutingthecapillarypressureandvanGenuchtenrelationshipsintoDarcy's law,.46,theuiduxbecomes R )]TJ/F20 11.9552 Tf 5.479 -9.683 Td [(" l v l;s = dp c dS )]TJ/F20 11.9552 Tf 11.955 0 Td [(C l S r S )]TJ/F20 11.9552 Tf 11.955 0 Td [(" r S + l g )]TJ/F43 11.9552 Tf 11.956 0 Td [(r p g )]TJ/F20 11.9552 Tf 11.956 0 Td [(C l r )]TJ/F20 11.9552 Tf 11.955 0 Td [(C l T r T + r c j n:eq )]TJ/F20 11.9552 Tf 11.955 0 Td [( c j eq | {z } 0 : .62 Tounderstandthenewlytermsproposedhere,wemakethefollowingthreecomments: 1.Firstconsiderthegaspressureandrelativehumidityterms.Intheabsence ofgravity,ifthesaturation,temperature,andthechangeincapillarywetting potentialareheldxedthen.62statesthatowisdrivenbygradientsinrelativehumidityandgas-phasepressure.Thegas-phasepressureandtherelative 108

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humidityareproportionaltoeachotherwheretheconstantofproportionality isafunctionoftemperatureandthespeciesdensities.Withthisinmind,these twotermstogethercanberewrittenasagradientingaspressure.Whileagradientingaspressurecancertainlycauseow,itiscommonlyassumedthat p g isapproximatelyconstantknownasRichards'assumption[62]andtherefore thesetermsaretypicallyneglected.Ifthesetermsarenotneglectedthenthey arebestwrittenasasinglegradientofrelativehumidityforeasycouplingwith thegas-phasediusionequation C l r r p g + C l r ': 2.Nextconsiderthegradientoftemperatureterm.Intheabsenceofgravity, ifsaturationandrelativehumidityareheldxedthen.62statesthatow isdrivenbyagradientintemperature.Saitoetal.[67]indicatedthatthe thermallyinducedowwasnegligibleascomparedtoisothermalowalso discussedin[78,80].Thisindicatesthatthe r T termin.62islikelyquite small. 3.Finallywediscusstheroleof C l S =2 l @ l @S .Thisfunctionorconstantrelates thechangesinenergywithrespecttosaturation.Thetermisalreadyassociated withthegradientinsaturationasseeninequation.62.Fromthe r S term inthisequationwecansee C l S asan enhancement ofthecapillarypressuresaturationrelationshipthatdirectlymodelstheanityfortheliquidphaseto theotherphases.Itisentirelylikelythatthistermissocloselylinkedwiththe capillarypressurethatinexperimentalsettingsitisimpossibletodiscernthis eectfromothers. Thesaturationequationcannallybewrittenas @S @t )]TJ/F43 11.9552 Tf 11.955 0 Td [(r K S )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(dp c dS + C l S r S e + r S e + C l T r T + C l r )]TJ/F20 11.9552 Tf 11.956 0 Td [( l g 109

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= M l )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v )]TJ/F20 11.9552 Tf 12.952 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v ; .63 wherewehaveassumedthat c j n:eq )]TJ/F20 11.9552 Tf 11.248 0 Td [( c j eq 0and,abusingnotationslightly,the C l termhasberedenedtoincorporatechangesinthegaspressure. Toaccountfortheresidualminimumsaturation,thesaturationisscaledto the eectivesaturation accordingto S e = S )]TJ/F20 11.9552 Tf 12.791 0 Td [(S min = S max )]TJ/F20 11.9552 Tf 12.79 0 Td [(S min .Dening S astheproductofporosityandthedierenceinmaximalandminimalsaturation, S := S max )]TJ/F20 11.9552 Tf 12.144 0 Td [(S min ,andletting S notationallystandfor S e allowsustowritethe saturationequationas @S @t )]TJ/F43 11.9552 Tf 11.955 0 Td [(r )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 S K S )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(dp c dS + C l S r S + S r S + C l T r T + C l r )]TJ/F20 11.9552 Tf 11.955 0 Td [( l g = M l S l )]TJ/F20 11.9552 Tf 5.48 -9.683 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v )]TJ/F20 11.9552 Tf 12.951 -9.683 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v .64 Ataquickglance,thesignofthe r S termlookssuspiciousasitseemstoindicate abackwardheatequation.Observethat p 0 c S < 0forallvaluesof S .Taking onlytherstlinewith C l S =0returnsRichards'equationsexactly.The r S term henceforthreferredtoasthe dynamicsaturationterm wasoriginallyproposedby Hassanizadehetal.inseveralpublicationsexamplesinclude[47,49]andisgaining morewidespreadacceptanceintheporousmediacommunity.Takingalloftheterms ontherstlineagainwith C l S =0alongwiththedynamicsaturationtermgives aclosedpseudo-parabolicequationinsaturation.The C l S ;C l T ;C l termsalongwith theformoftheright-handsideareallnoveltothiswork.Thetemperatureand relativehumiditycouplingtermscancertainlybetakentobezeroincertainphysical instances,butgenerallytherelativeweightandfunctionalformsofthesetermsis,as ofyet,unknown. Wenowturnoutattentiontothegasphasediusionequation.Analysisand numericalsolutionstothesaturationequationwillbeconsideredinChapter7. 5.4.2GasPhaseDiusionEquation 110

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Inthissubsectionwemakecertainsimplicationstothegas-phasediusionequationsoastotiethechemicalpotentialformulationtothemoreclassicalenhanced diusionmodel.Asarststeptowardthissimplicationweconsiderthefactthat thegasphasechemicalpotentialsarerelatedtoeachotherthroughequation.85; theexpressionfortherelativemotionofdiusingspeciesinabinarysystem: N X j =1 j R g j T D [ r j )]TJ/F40 11.9552 Tf 11.955 0 Td [(g ] = 0 : Withthis,thegradientofchemicalpotentialoftheinertairin.47bcanberewritten asafunctionofthewatervaporchemicalpotential g a r g a = )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( R g a g v R g v r g v )]TJ/F40 11.9552 Tf 11.956 0 Td [(g + g a g : Thismeansthatthegas-phasemassbalanceequationcanberewrittenas @ @t g v sat )]TJ/F20 11.9552 Tf 11.955 0 Td [(S )]TJ/F43 11.9552 Tf 11.955 0 Td [(r g v D g v + g v 1 )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(R g a R g v K g [ r g v )]TJ/F40 11.9552 Tf 11.955 0 Td [(g ] )]TJ/F43 11.9552 Tf 11.955 0 Td [(r g g v g K g r T = )]TJ/F20 11.9552 Tf 9.299 0 Td [(M )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v )]TJ/F20 11.9552 Tf 12.952 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v : .65 Typically,onewouldchooseafunctionalformof D g v tomatchtheenhancementmodeldiscussedinSection5.1.2andthefunctionalformof K g fromthevan GenuchtenmodeldiscussedinSection5.4.1.Inthepresentcasewearguetousedifferentfunctionalformsof D g v and K g .Thisisdonebyconsideringtheconversions betweenthepore-scaledensityandchemicalpotentialtotherelativehumidity.For simplicitythetensorialnotationisdroppedandweassumethatthediusionand conductivitytensorsareallscalarmultiplesoftheidentitymatrix. Webeginwithsomelogicalconsiderationsforthegas-phasediusioncoecient. Ifthegas-phasevolumefractionweretodroptozerothentherewouldbenogasin theporespaceortheirwouldbenoporespaceandthediusioncoecientshould droptozero.Similarly,ifthegas-phasevolumefractionweretoincreaseto1% 111

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gaswithnosolidorliquid,thenthediusioncoecientshouldreturntotheFickian diusioncoecient D g .Withthesetwolimitingcasesinmindwerstproposethat D g v = C" g D g where C isascalingparameter. AsseeninChapter2,thediusioncoecientismodiedforFick'slawbasedon thedependentvariableofinterest.Inequations.1and.3weseeascalarfactor of1 = R g v T betweenthemassandchemicalpotentialformsofFick'slaw.Making thesamemodicationherealongwiththefactorof g suggestedaboveweget g v D g v r g v g v g R g v T D g r g v = sat )]TJ/F20 11.9552 Tf 11.955 0 Td [(S R g v T D g r g v .66 where D g isthesamepore-scalediusioncoecientasfoundinChapter2.One simplewaytolookatthisconversionisthatitscalesouttheunitsandmagnitudeof thechemicalpotentialwhenconvertingtorelativehumidity.Thatis, D g v r g v and D g = R g v T r havethesameunitsandmagnitude.Afurtherjusticationofthisis foundbyrecallingthepore-scaledenitionofthechemicalpotential: g v = g v + R g v T ln p g v p g = g v + R g v T ln ; .67 where = p g v sat =p g and p g v sat isthepartialpressureofthewatervaporundersaturated conditions.Takingthegradientof.67andneglectingthetemperaturevariation gives r g v R g v T r ': HenceweseetheexactconversionusedinFick'slaw. Nextweturnourattentiontothehydraulicconductivitytermthatarosefrom Darcy'slaw: g v K g r g v .SimilartothatofFick'slaw,weneedtoscaletheconductivitytoaccountforthefactthatwe'reusingthechemicalpotentialasthedependent variable.UnliketheFickiandiusioncoecient,thistermalreadyhastheproper unitssincetheunitsof g v r g v arethesameasthegradientofpressure.Therefore 112

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weseekascalingthatisunitlessbutscalesthemagnitudeofthechemicalpotential downtothatofpressure.Thatis,weneedaconstant, c ,suchthat c g v K g r g v and K g r p g haveapproximatelythesamemagnitude. Takingthegradientofbothsidesoftherstlineofequation.67wearriveat r g v = R g v Tp g p g v r p g v p g = R g v Tp g p g v 1 p g r p g v )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( p g v p g 2 r p g = R g v T p g v r p g v )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( R g v T p g r p g : Thecoecientofthegradientofgas-phasepressurecanberewrittenas R g v T p g = sat R g v T sat p g = p g v sat sat p g = sat : Sincethechemicalpotentialformalreadyhasafactorof g v = g v sat wescale K g by toaccountforthedierenceinmagnitudebetweenthechemicalpotentialandthe pressure.Hence,theDarcyterminequation.65isrewrittenas g v 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(R g a R g v K g r g v g v 1 )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(R g a R g v K g r g v : Keepinmindthatthisisascalingofthehydraulicconductivity;justasthefactorof 1 = R g v T isascalingofthediusioncoecientinFick'slaw. Onepointofinterestforthischoiceofscalingfactoristhatitisinvisiblewhen weconsidera pure gasphase.Thatis, =1whennospeciesareconsideredsincethe saturatedpartialpressurewillsimplybethebulkpressure.Thisindicatesthatwe havenotactuallychangedDarcy'slaw.Insteadwehavesimplymadeaconversionto accountfortheuseofadierentdependentvariable. Nextwefocusonwritingthegas-phasediusionequation.65intermsof relativehumidity,saturation,andtemperature.Todothiswereplacethechemical potentialwithrelativehumidityandtemperatureviaequation.67.Takingthe gradientofthechemicalpotentialinequation.67weget r g v = R g v T r + R g v T d dT + R g v ln r T: 113

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WiththeFickianandDarcytermswrittenintermsoftherelativehumidity,along withthefactthatthesaturatedvapordensityisafunctionoftemperature,thevapor diusionequationscanbewrittenas @ @t sat )]TJ/F20 11.9552 Tf 11.955 0 Td [(S )]TJ/F43 11.9552 Tf 11.955 0 Td [(r sat )]TJ/F20 11.9552 Tf 11.956 0 Td [(S R g v T D g + sat 1 )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(R g a R g v K g R g v T r + R g v T d dT + R g v ln r T )]TJ/F43 11.9552 Tf 11.955 0 Td [(r f g sat g K g r T g = )]TJ/F20 11.9552 Tf 9.299 0 Td [(M )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( sat )]TJ/F20 11.9552 Tf 12.952 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v : Combiningliketerms,dividingbytheporosity,replacingthehydraulicconductivity bythesaturatedandrelativepermeabilities,andsimplifyinggives @ @t sat )]TJ/F20 11.9552 Tf 11.955 0 Td [(S )]TJ/F43 11.9552 Tf 11.955 0 Td [(r n sat D ';S;T h r )]TJ/F40 11.9552 Tf 19.073 8.087 Td [(g R g v T io )]TJ/F43 11.9552 Tf 11.955 0 Td [(r f sat N g ';S;T r T g = )]TJ/F20 11.9552 Tf 10.494 8.847 Td [(M l )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v ; .68 wherethefunctions D and N g are D ';S;T := )]TJ/F20 11.9552 Tf 11.955 0 Td [(S D g + sat 'R g v T 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(R g a R g v s g rg S and.69a N g ';S;T := D ';S;T 1 d dT + R g v ln T + g sat g S g rg S .69b TheenhancementmodelsuggestedbydeVries,andsubsequentlyusedbyseveralauthors[24,67,78,68,80],isamultiplicativecombinationofthepureFickian diusioncoecient, D g ,thetortuosity, = g ,andanenhancementfactor, : D = D g : .70 Intheseworks,thefunctionalformoftheenhancementfactoristakentobeofthe formsuggestedbyCassetal.[24] a = a +3 l )]TJ/F15 11.9552 Tf 11.955 0 Td [( a )]TJ/F15 11.9552 Tf 11.955 0 Td [(1exp )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( 1+ 2 : 6 p f c l 3 : .71 114

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Here, f c isthemassfractionofclayinthesoil.Intheabsenceofclaytheenhancement factoristakenas a = a +3 l s .72 foranexamplewhere f c 6 =0seeSaitoetal.[67].Thetortuosityistakentobea functionofthevolumetricgascontent, = = 3 g : .73 Usingequations.72and.73inthemultiplicativeexpansionofthediusion coecient,.70givesadiusioncoecientof D = a +3 l 2 3 g D g : .74 Thetortuosityandtheporositycommunicatetothediusioncoecientthetypeof geometryunderconsideration.Thepresentmodelequation.68communicates thisinformationviatheporosity,therelativepermeability,andthesaturatedpermeability.Thediusionmodelusingequation.74reliesonattingparameter,while thepresentmodelavoidsthistrouble.Intheauthor'sopinion,thishighlightsthe mainadvantagetousingthechemicalpotentialasamodelingtool. ComparingtheenhancementmodelofCassetal.usingthematerialparameters fromtheexperimentbySmitsetal.[78]tothepresentmodel,wesee,inFigure 5.5,thattherelativehumiditylevelcurvesofthepresentmodelunderestimatethe enhancedmodelformanyvaluesofthettingparameter, a .Thatbeingsaid,these curvesdosuggestanenhancementoverregularFickandiusionand,dependingonthe parametersofinterst,give similar levelsofenhancementasthemodelusedin[78].We simplystateherethatthepresentmodeloersamodiedviewoftheenhancement model.Thereareseveralparametersthatplayrolesinthismodel,buttheadvantage tothepresentapproachisthatalloftheparametersarereadilymeasuredforagiven 115

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Figure5.5:Comparisonofdierentdiusionmodelsatconstanttemperature T = 295 : 15 K .Thevalueforthesaturatedpermeabilitywaschosentomatchthatof [78] S =1 : 04 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 m 2 ,wheretheyfoundattingparameter a =18 : 2.The PresentModel"referstoequation.68with r T = 0 andnomasstransferand theEnhancementModel"referstoequation.3alongwith.70,.72,and .73forthediusioncoecient,enhancementfactor,andtortuosityrespectively. mediumatleastinlaboratoryexperiments.Thereisno tting parameter,sothe typeofmaterialshoulddictatethelevelofenhancement. Anotherwaytolookatthepresentmodelistoconsiderthatinmostclassical situationsthegas-phasepressureisconsideredconstant.Theeectofthisassumption isthattheDarcytermsinthegas-phasemassbalanceequationareneglected.This assumptionisvalidinmanycases,butinthepresentcasetheDarcytermisbroken intocomponentpartsairandwatervaporviathechemicalpotentials.Thechemical potentialformulationdrawsinuencefromtheDarcy-typemovement,alongwith theFickiandiusion,oftheindividualconstituentstodenethegeneraldiusion coecient. 116

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ItisemphasizedherethatthetraditionaldeVries-typeviewofdiusionin porousmediaisnottakenhere.Shokri[73]suggestedthatthemechanismofenhanced diusionisdrivenbythecouplingofDarcyandFickiandiusion.Thenoveltyhere isthattheadvectionanddiusionaremodeledintermsofthesamedependent variable;thechemicalpotential.Thissuggeststhattheenhanceddiusionproblem canbemodeledbycouplingDarcy-typeowalongwithFickiandiusioninthegas phase.Therelationshipbetweentheenhancementmodelandthepresentmodelwill bediscussedwhenweconsidernumericalsolutionsinChapter7. 5.4.3TotalEnergyEquation Continuingwiththeequation-by-equationderivationofthetotalheatandmoisturetransportmodel,wenowturnoutattentiontothetotalenergyequation.This picksupfromequation.38andweapplythesimplifyingassumptionspresentedin thebeginningofSection5.4. Ifweassumethatthevaporandairdensitiesarefunctionsofrelativehumidity andtemperatureonly,thetotalenergyequation.38canbewrittenas 0= c p T )]TJ/F43 11.9552 Tf 11.956 0 Td [(r )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(K r T + h + L ^ e l g + p c +2 S + T @ @T )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( g )]TJ/F20 11.9552 Tf 11.955 0 Td [( l S + )]TJ/F20 11.9552 Tf 11.955 0 Td [(S X j = v;a g j )]TJ/F20 11.9552 Tf 11.956 0 Td [(T @ g j @T @ g j @T )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F21 7.9701 Tf 11.866 4.936 Td [(g + T g @ g @T # T + )]TJ/F20 11.9552 Tf 11.955 0 Td [(S X j = v;a g j )]TJ/F20 11.9552 Tf 11.956 0 Td [(T @ g j @T @ g j @' )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F21 7.9701 Tf 11.866 4.936 Td [(g + T g @ g @' # + l c l p + e l d l dT r T + T S @p l @T r S )]TJ/F20 11.9552 Tf 5.48 -9.683 Td [(" l v l;s + g c g p )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F21 7.9701 Tf 7.314 4.936 Td [(g @ g @T + X j = v;a g j @ g j @T + T @ @T X j = v;a g j @ g j @T )]TJ/F20 11.9552 Tf 11.955 0 Td [( g @ g @T !! r T + )]TJ/F15 11.9552 Tf 9.299 0 Td [()]TJ/F21 7.9701 Tf 7.314 4.937 Td [(g @ g @' + X j = v;a g j @ g j @' + T @ @T X j = v;a g j @ g j @' )]TJ/F20 11.9552 Tf 11.956 0 Td [( g @ g @' !! r )]TJ/F20 11.9552 Tf 24.983 8.088 Td [(T )]TJ/F20 11.9552 Tf 11.955 0 Td [(S @p g @T r S g v g;s : .75 117

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Recallthat = ';S;T p c = p c S e g j = g j ';T g j = g j ';T ,)]TJ/F21 7.9701 Tf 21.173 4.338 Td [(g = )]TJ/F21 7.9701 Tf 7.314 4.338 Td [(g ';T g = g T l = l T .Alsorecallthat v ;s representstheDarcyux forthe phase: l v l;s = )]TJ/F20 11.9552 Tf 9.299 0 Td [(K l h )]TJ/F20 11.9552 Tf 9.298 0 Td [(p 0 c S e + C l S r S e + r S e + C l T r T + C l r )]TJ/F20 11.9552 Tf 11.956 0 Td [( l g i .76a g v g;s = )]TJ/F20 11.9552 Tf 9.299 0 Td [(K g g v 1 )]TJ/F20 11.9552 Tf 13.151 8.088 Td [(R g a R g v @ g v @T r T + @ g v @' r + g g r T )]TJ/F20 11.9552 Tf 11.955 0 Td [( g g : .76b Itisclearthatthereareseveralphysicalprocessesandcouplingsthatoccurfor energybalancetobeachieved.Equation.77belowshowstheclassical1958model ofdeVries[32]whichissimilartothatofBear[5]andisalsopresentedin[14]for thesaturatedcase. c p @T @t )]TJ/F20 11.9552 Tf 11.955 0 Td [(" )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l W l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g W g @S @t = r )]TJ/F40 11.9552 Tf 5.48 -9.684 Td [(K r T )]TJ/F20 11.9552 Tf 11.955 0 Td [(L ^ e l g )]TJ/F26 11.9552 Tf 11.955 20.443 Td [( X = l;g c p " v ;s r T: .77 Inthisformoftheenergyequation, W isa dierentialheatofwetting [14],and theothervariablesarewritteninthepresentnotationforconvenience.Atrst observation,the T S K ,^ e l g ,and r T termsinequation.75aresimilartoterms foundinthedeVriesmodel.Thatis,wecapturethestandardeectsofspecic heatalongwithdierentialheatofwetting,thermalconductivity,masstransfer,and convectiveheating.Implicitinthe S termin.75isthatwerelatethepartial derivativeofthedierenceinwettingpotentials, T@ g )]TJ/F20 11.9552 Tf 12.611 0 Td [( l =@T ,asadierential heatofwetting.Thepresentmodelalsocapturestheeectsofchangingrelative humidity,nonlineareectssuchas r S r S and r r ,andcrosseectssuch as r S r .Itremainstodeterminewhichifanyoftheseeectsarenegligible ascomparedtotheothers.Tomakethisdeterminationweperformadimensional analysisinthenextsubsection.Letusrstfocusonthethermalconductivityterm, r )]TJ/F40 11.9552 Tf 5.48 -9.684 Td [(K r T : 118

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Thefunctionalformofthethermalconductivity, K ,canbeapproximatedin severalways.Arstapproximationistotakethethermalconductivityasaweighted sumoftheconductivitiesoftheindividualphases K = X K T : .78 Comparingtoresultsin[77],wenotethatthisseemstooverestimatethemeasured thermalconductivityaswellasfailtocapturetheexperimentallymeasuredcurvature ofthethermalconductivity-saturationrelationship.Since K isalinearization constantthatarosefromtheentropyinequality,itcandependonanyvariablewhich isnonzeroatequilibrium.Inparticular, K isafunctionofsaturation.Smitset al.[77]useacombinationoftheC^ote-KonradandJohansenmodelstoestimatethe thermalconductivityinthescalarcase: K S = K e S K sat )]TJ/F20 11.9552 Tf 11.955 0 Td [(K dry + K dry ; .79 where K sat istheconductivityofthesaturatedmedium, K dry istheconductivity ofthedrymedium,and K e S isanormalizedthermalconductivityknownasthe Kerstennumber."C^oteandKonradproposedafunctionalformof K e as K e S = S 1+ )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 S : .80 Theparameter, ,isattingparameterthatispresumedtobedierentforeachtype ofsoil.In[77], wasestimatedforseveraltypesofsandsandseveraltypesofsoil packs.Figure5.6showsathermalconductivitycurvefor.79withtightlypacked 30/40sandthathasaporosityof0 : 334.Forcomparison,equation.78isshownin redforthesameexperiment. Wemakesomecommentsheregivingsomepossiblereasonsforthediscrepancy betweentheweightedsummodelequation.78andthemodelthatmoreclosely matcheswhatisexperimentallyobservedequation.79.First,thethermalconductivityofairisneglectedascomparedtothethermalconductivityofwateror 119

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Figure5.6:JohansenthermalconductivitymodelwithC^ote-Konrad K e )]TJ/F20 11.9552 Tf 11.054 0 Td [(S relationshipwith =15plottedinblue,andtheweightedsumofthethermalconductivities oftheindividualphasesplottedinred. solid.Also,thethermalconductivityofliquidismuchsmallerthanthatofthesolid, K l
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5.4.3.1DimensionalAnalysis Todeterminewhich,ifany,termscanbeneglectedfromtheenergytransport equationweperformadimensionalanalysis.Beginbynotingthat K = c p hasunits ofareapertime.Thissuggestsanaturalchoiceoftimescaleforthethermalproblem of t = c p x 2 c K t 0 ; where t 0 isdimensionlesstime.Dividingby c p measuredatareferencestate, introducing x c asacharacteristiclengthe.g.theheightofacolumnexperiment,and multiplyingby t c = c p x 2 c =K givesthedimensionlessformoftheenergyequation thestatementofwhichissuppressedforthesakeofbrevity. Recallthatthevolumetricheatcapacity, c p ,islinearlyrelatedtothespecic heatsoftheindividualphases c p = X )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" c p = "S l c l p + )]TJ/F20 11.9552 Tf 11.955 0 Td [(S g c g p + )]TJ/F20 11.9552 Tf 11.956 0 Td [(" s c s p : Taking S =1asareferencestateorequivalently, S =0givesacharacteristicvalue of c p .UsingvaluesfromAppendixEweseethat c p O 6 .Hence,severalof thequantitiesin.75canbeneglected: c p X j = v;a g j )]TJ/F20 11.9552 Tf 11.955 0 Td [(T @ g j @T @ g j @T )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F21 7.9701 Tf 11.866 4.937 Td [(g + T g @ g @T # O )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 .81a c p X j = v;a g j )]TJ/F20 11.9552 Tf 11.955 0 Td [(T @ g j @T @ g j @' )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F21 7.9701 Tf 11.866 4.937 Td [(g + T g @ g @' # O )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 .81b t c x 2 c c p )]TJ/F15 11.9552 Tf 9.298 0 Td [()]TJ/F21 7.9701 Tf 7.314 4.936 Td [(g @ g @T + X j = v;a g j @ g j @T + T @ @T X j = v;a g j @ g j @T )]TJ/F20 11.9552 Tf 11.955 0 Td [( g @ g @T !# O )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 .81c t c x 2 c c p )]TJ/F15 11.9552 Tf 9.298 0 Td [()]TJ/F21 7.9701 Tf 7.314 4.937 Td [(g @ g @' + X j = v;a g j @ g j @' + T @ @T X j = v;a g j @ g j @' )]TJ/F20 11.9552 Tf 11.955 0 Td [( g @ g @' !# O )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 .81d t c x 2 c c p e l @ l @T O )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 .81e 121

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InordertomaketheseapproximationsitisassumedthatGibbspotentialsaregiven bytheGibbs-Duhemrelationship,.61,andthattheHelmholtzpotentialandinternalenergyareapproximatelythesameorderofmagnitudeastheGibbspotential. Withtheseconsiderationswecanrewritethepresentversionoftheenergyequationas 0= c p @T @t )]TJ/F43 11.9552 Tf 11.955 0 Td [(r )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(K r T + h + L ^ e l g + p c +2 S + T @ @T )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( g )]TJ/F20 11.9552 Tf 11.956 0 Td [( l @S @t + l c l p r T + T S @p l @T r S )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" l v l;s + g c g p r T )]TJ/F20 11.9552 Tf 27.639 8.088 Td [(T )]TJ/F20 11.9552 Tf 11.955 0 Td [(S @p g @T r S g v g;s : .82 Unfortunatelythisanalysisleadsustotheconclusionthatthisnewversionofthe heattransportequationisonly slightly dierentthanthoseproposedinpastworks [14,32].Themajordierencesarethe r S termsassociatedwiththeDarcyuxes,the capillarypressureadjustmenttothedierentialheatofwettingterm,andtheDarcy uxesthemselves.RecallingtheformsoftheDarcyuxesfromequations.76,the energyequationcanberewritteninamorecompactnotationas 0= c p @T @t )]TJ/F43 11.9552 Tf 11.956 0 Td [(r )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(K r T + h + L ^ e l g + W @S @t + 1 r S + 2 r T + 3 r r T + 4 r S + 5 r r + 6 r S r S .83 where W andeach j areimplicitlydenedviaequations.82and.76.Itremains todeterminethefunctionalformsoftheseveralconstitutivevariablesin.83. 5.4.4ConstitutiveEquations Hiddenwithinthecoecientsof.83,.64,and.65areafewnalrelationshipsnecessaryforclosure.Inparticular,weneedconstitutiveequationsfor = @p c @ l .84a 122

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^ e g v l = M )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v )]TJ/F20 11.9552 Tf 12.952 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v .84b W = p c S +2 S + T @ @T )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( g )]TJ/F20 11.9552 Tf 11.955 0 Td [( l = p c S +2 S + W .84c C l S =2 l @ l @S = )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l g .84d C l T = l N X j =1 @ l @ g j @ g j @T )]TJ/F26 11.9552 Tf 14.626 11.357 Td [(X = g;s " l @ @ l @ l @T .84e C l = l X j @ l @ g j @ g j @' : .84f Thesimplestpossibleassumptionwouldbethat ;C l S ;C l T ;C l ; and W areconstants. Thiswouldallowfortheeasiestsensitivityanalysisbutislikelycontrarytophysical reality.Thefollowingparagraphsdiscusseachofthesetermsandproposefunctional formsintermsofsaturation,relativehumidity,andtemperature.Thesensitivityof thenumericalsolutiontoseveraloftheseparametersisdiscussesinChapter7. Itisgenerallyassumedthat inequation.84aisconstant[47,60],butaccordingtothelinearizationprocessinHMT, canbeafunctionofanyvariablethat isnotzeroatequilibrium.Inparticular,itispossiblethat isafunctionof S ; but which function?In[17],theauthorssuggestseveralfunctionalformsconstant, linear,quadratic,Gaussian,anderrorandcomparetoexperimentalndings.Their ndingssuggestthat...anerrorfunctionorGaussianrelationshipforthedamping coecient providesreasonableagreementbetweendataandsimulations."Thuswe considerthefollowingforms: = max .85a = max 2 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(erf S )]TJ/F20 11.9552 Tf 11.956 0 Td [( .85b = max exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [( S )]TJ/F20 11.9552 Tf 11.955 0 Td [( 2 2 2 : .85c Plotsofequations.85areshowninFigure5.7withtypicalmeanandstandard deviationparameters.Totheauthor'sknowledge,nootherexperimentshavebeen conductedtomakeabetterdeterminationastothefunctionalformof .This 123

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beingsaid,since isameasureoftherateatwhichthepore-scalesaturationprole rearrangesinadynamicsituation,itisreasonabletoassumethatas S 1the eectofthistermshouldbeminimizedandas S 0theeectshouldbemaximized. Hence,intheauthor'sopinionanerrorfunctionismoresensible.Itremains,ofcourse, todeterminethevaluesofthemaximum,meanandstandarddeviationparameters whicharelikelythemselvesfunctionsofmaterialproperties. Figure5.7:Threeproposedfunctionalformsof = S Theevaporationrateterm,^ e g v l ,giveninequation.84biswrittenasafunction ofthedierencebetweentheliquidandvaporchemicalpotentials.Thechemical potentialinthewatervaporisafunctionoftemperatureandrelativehumidity[21], g v = g v + R g v T ln : Theliquid-phasechemicalpotential,ontheotherhand,doesnothavesuchanatural description.Atequilibrium, l = g v .Awayfromequilibriumweonlyknowthat l =)]TJ/F21 7.9701 Tf 19.74 4.936 Td [(l = l + p l l ; 124

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andthereforeisafunctionofeveryvariablethat l depends.Inthemostsimplistic formwecanassumethattheliquidchemicalpotentialis l = l + p l )]TJ/F20 11.9552 Tf 11.529 0 Td [(p l = l .This assumptionistakenfromclassicalthermodynamicssee[21]forexample.Furthermore, l g v ifwetakethereferencestatetobeequilibrium.Therefore, ^ e g v l M' l )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v p l )]TJ/F20 11.9552 Tf 11.955 0 Td [(p l 0 l )]TJ/F20 11.9552 Tf 11.955 0 Td [(R g v T ln M' l )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v )]TJ/F20 11.9552 Tf 9.299 0 Td [(p c + S )]TJ/F20 11.9552 Tf 11.955 0 Td [(p l 0 l )]TJ/F20 11.9552 Tf 11.955 0 Td [(R g v T ln ; .86 where M isattingparameter.Thefactorofrelativehumidityisincludedtoachieve abettermatchwithexistingempiricalmodelsdiscussedinthenextparagraph. Thereareseveralempiricalrulesforevaporationinporousmedia.Onesuchrule, givenbyBixler[19]andrepeatedinSmitsetal.[78],is ^ e g v l = b l )]TJ/F20 11.9552 Tf 11.955 0 Td [(" l r R g v T sat )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v ; .87 where b isattingparameterand l r istheresidualvolumetricwatercontent.Equations.86and.87arequitedierent,butunderproperscalingtheyare close as seeninFigures5.8.Fromtheseplotsitisalsoclearthatthereisalargedicrepancy betweenthesemodelatverylowsaturations.Theseplotsaregeneratedatstandardtemperaturewith S =0.Thedynamicsaturationtermwillchangetheshape ofthesecurves,butastheBixlermodel,.87,isnotdynamicwecompareonly withthesteadystateformof.86.Furthermore,thepresentmodeldependson thevanGenuchtenparametersforcapillarypressure.InFigures5.8theparameters m =0 : 944and =5 : 7areusedalongwith b 2 : 1 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 tomatchthevaluesused in[78]. Thedierentialheatofwetting, W ,inequation.84crepresentstheheatgained orlostduetochangesinsaturationandadsorption.Thepresentgeneralization suggeststhatthedierentialheatofwettingbesupplementedbythecapillarypressure andtimerateofchangeofsaturation.Accordingto[64],thetypicalvalueofthe 125

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aComparisonofmasstransferratevs.effectivesaturationshownwithlevelcurvesin relativehumidity. bComparisonofmasstransferratevs.relativehumidityshownwithlevelcurvesinsaturation. Figure5.8:Levelcurvesofmasstransferratefunctions. dierentialheatofwettingisontheorderto10 3 J=kg dependingonthetypeofsoil. Thisvaluewillbetakenasconstantthroughout,butinrealityvalueshouldbea functionofsaturation. Finally,thevaluesof C l S ;C l T ; and C l inequations.84d-.84farenewand hencethereisnoexistingliteratureforwhichtomakeestimatesorcomparisons. Forthisreasonwemaketheinitialassumptionthatthesetermsareconstant.This allowsforrelativelysimplesensitivityanalysiswithoutintroducinganyunnecessary mathematicaldiculties.AsdiscussedinSection5.4.1.1,thevalueof C l T islikely quitesmallsincesomeresearchhasbeendonetodeterminetheaectofthermal gradientsonDarcyow[67]. 5.5ConclusionandSummary Inthischapterwehavederivedseveralnewequationsandtermsforheatand moisturetransportinunsaturatedporousmedia.Forthesakeofreadability,we summarizetheresults,assumptions,andequationsderivedherewithinChapter5. Themainassumptionsare: Assumption#1 Thesolidphaseisrigid,incompressible,andinert. 126

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Assumption#2 Theliquidandgasphasesarecomposedof N constituents.this waslaterrelaxedtolet N =2inthegasphaseand N =1intheliquidphase. Assumption#3 Nochemicalreactionstakeplaceinanyofthesephase. Assumption#4 Diusionwiththeliquidphaseisnegligiblecomparedtotheadvectionoftheliquidphase. Assumption#5 Theliquidphaseisincompressible. Assumption#6 Thegasphaseisanidealbinarygasmixtureofwatervaporand inert air Assumption#7 Thegas-phasechemicalpotentialsanddensitiesarefunctionsof relativehumidityandtemperature. Thesecondaryassumptionsuseduptothispointareinorderofappearance: themediumofinterestisgranularsoangularmomentumconservationyieldsa symmetricstresstensor, thematerialis simple inthesenseofColemanandNoll[27], thephaseinterfacesareassumedtocontainnomass,momentum,orenergy, second-ordereectsinvelocityarenegligiblee.g. v j ; v j ; v thespeciesinthesolidphasedonotdiuse, inertialtermsinthemomentumbalanceequationarenegligible, thecapillarypressure-saturationrelationshipisgivenbythevanGenuchten function, thedeviationinwettingpotentialisapproximatelyzero c j n:eq )]TJ/F20 11.9552 Tf 11.955 0 Td [( c j eq 0, 127

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thecoecientofthedynamicsaturationterm, ,isconstant, Consideringassumptions#1-#7alongwithallofthesecondaryassumptions, thenalsystemofequationsproposedtomodelheatandmoisturetransportinunsaturatedporousmediais: @S @t )]TJ/F43 11.9552 Tf 11.955 0 Td [(r h )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 S K l p 0 c + C l S r S + S r S + C l T r T + C l r )]TJ/F20 11.9552 Tf 11.955 0 Td [( l g i = ^ e g v l l .88a @ @t sat )]TJ/F20 11.9552 Tf 11.955 0 Td [(S )]TJ/F43 11.9552 Tf 11.955 0 Td [(r [ sat D r + N g r T ]= )]TJ/F15 11.9552 Tf 9.735 0 Td [(^ e g v l .88b 0= c p @T @t + W @S @t )]TJ/F43 11.9552 Tf 11.955 0 Td [(r )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(K r T + h + L ^ e l g + 1 r S + 2 r T + 3 r r T + 4 r S + 5 r r + 6 r S r S; .88c wheretherelevantempirical,constitutive,andderivedrelationsare K l S = s l rl = s l p S 1 )]TJ/F26 11.9552 Tf 11.956 9.683 Td [( 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S 1 =m m 2 .89a K g S = s g rg = s g )]TJ/F20 11.9552 Tf 11.955 0 Td [(S 1 = 3 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S 1 =m 2 m .89b p c S = 1 )]TJ/F20 11.9552 Tf 5.479 -9.683 Td [(S )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 =m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 )]TJ/F21 7.9701 Tf 6.587 0 Td [(m .89c = @p c @ l seeequations.85.89d ^ e g v l = M' )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v )]TJ/F20 11.9552 Tf 9.299 0 Td [(p c + S )]TJ/F20 11.9552 Tf 11.955 0 Td [(p l 0 l )]TJ/F20 11.9552 Tf 11.955 0 Td [(R g v T ln .89e D ';S;T := )]TJ/F20 11.9552 Tf 11.955 0 Td [(S D g + sat 'R g v T 1 )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(R g a R g v s g rg S .89f N g ';S;T = D ';S;T 1 d dT + R g v ln T + g sat g S g rg S .89g D g T =2 : 12 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 T 273 : 15 2 .89h c p = X c p .89i 128

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h = X h .89j W = p c +2 S + W .89k K = X K T ; or K = S K sat )]TJ/F20 11.9552 Tf 11.955 0 Td [(K dry 1+ )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 S + K dry .89l l T =3 : 79 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 T )]TJ/F15 11.9552 Tf 11.956 0 Td [(277 : 15 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(7 : 37 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 T )]TJ/F15 11.9552 Tf 11.955 0 Td [(277 : 15 2 +10 3 .89m sat T = 1 T exp 31 : 37 )]TJ/F15 11.9552 Tf 11.955 0 Td [(7 : 92 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 T )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(6014 : 79 T 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 .89n l T = )]TJ/F23 11.9552 Tf 5.48 -9.683 Td [()]TJ/F15 11.9552 Tf 9.298 0 Td [(2 : 56109 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(6 T )]TJ/F15 11.9552 Tf 11.956 0 Td [(273 : 15 3 +0 : 00057672 T )]TJ/F15 11.9552 Tf 11.955 0 Td [(273 : 15 2 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 0469527 T )]TJ/F15 11.9552 Tf 11.955 0 Td [(273 : 15+1 : 7520210 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 .89o g T = 1 : 02312 T 3 10 9 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(3 : 62788 T 2 10 6 +0 : 00665915 T +0 : 11767 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 .89p L T =2 : 501 10 6 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2369 : 2 T )]TJ/F15 11.9552 Tf 11.956 0 Td [(273 : 15.89q g T =6 : 1771 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(4 T )]TJ/F15 11.9552 Tf 11.956 0 Td [(273 : 15 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(7 : 3971 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 T )]TJ/F15 11.9552 Tf 11.955 0 Td [(273 : 15 3 +3 : 1324 T )]TJ/F15 11.9552 Tf 11.955 0 Td [(273 : 15 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(34 : 4817 T )]TJ/F15 11.9552 Tf 11.955 0 Td [(273 : 15+191 : 208 : .89r Equations.88coupledwithequations.89giveseveraladjustmentstothe classicalmodelsforsaturationRichards',vapordiusionPhillipanddeVries,and heattransportdeVriespresentedinSection5.1.Inorderforthepresentmodels tobeacceptedinthehydrologycommunitywemustshowthattheproposedterms arenon-negligibleandinsomewayputsomeoftheempiricalrelationsonarmer theoreticalfooting.Theproposedvapordiusionequation.88bisaprimeexample ofthisastherearenoempiricalttingparameterswithinthediusioncoecient henceremovingtheneedforanempirical enhancementfactor InChapter6wediscussthemathematicalquestionsofexistenceanduniquenessof solutionstotheindividualequations.InChapter7wediscussnumericalsimulations ofthemodels. 129

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6.ExistenceandUniquenessResults Inthischapterwediscussthenecessaryregularityandassumptionsforexistence anduniquenessofsolutionsforthethreeequations.Asthemainthrustofthiswork isnottoproveexistenceanduniquenessforgeneralclassesofsystemsofpartialdifferentialequations,weapproachtheseproblemsbystatingrelevantexistingtheorems fromtheliteratureandsatisfyingthehypothesesofthesetheorems.Thesaturation andgasdiusionequationsarebothofparabolictypeandcanbetreatedsimilarly. Theheattransportequationisanadvection-reaction-diusionequationthat,inprinciple,shouldbeparabolicinnature.Theadvectiontermsforceadierentapproach tothisequation.InSection6.1,anexistenceanduniquenessresultforthesaturation equationwiththethird-ordertermduetoMikelic[57]isoutlined.Thetheoremsof AltandLuckhaus[1,2]areoutlinedinSection6.2andthenusedinSections6.2.1 and6.2.2toproveexistenceanduniquenessresultsforRichards'equationandthe vapordiusionequationrespectively.Finally,anexistenceanduniquenessresultfor aspecialcaseoftheheattransportequationispresentedinSection6.3. 6.1SaturationEquationwith 6 =0 ThesaturationequationhasbeenwellstudiedsinceRichards'rstintroducedit inthe1930's.Recentmodelingeorts,includingthoseofHassanizadehetal.,have introducedanewtermintotheclassicalRichards'equationandthishascauseda resurgenceintheanalyticalstudyofthesaturationequation.The2010paperby AndroMikelic[57]givesthenecessaryconditionsforexistenceanduniquenessofa weaksolutiontothefollowingequation: @S @t = r K S )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(dp c dS r S + r @S @t + e 3 in T = ;T .1a S = S D on)]TJ/F21 7.9701 Tf 23.573 -1.793 Td [(D = @ D ;T .1b K S )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(dp c dS r S + r @S @t + e 3 = R on)]TJ/F21 7.9701 Tf 23.573 -1.793 Td [(N = @ N ;T .1c S x;t =0= S i x on.1d 130

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Here, e 3 isaunitvectorpointingthe z directiontoaccountforgravitationaleects, isanoutwardpointingnormal,andthesubscripts D and N representDirichletand Neumannconditionsrepsectively.Noticethat.1isasimplicationofthepresent saturationequationasitcontainsnoevarporationsourcetermandnocouplingwith relativehumidityortemperature. Mikelic'stheoremisstatedhereforcompleteness. Theorem6.1Mikelic2010[57],Theorems3&4 Considerthefollowinghypotheses: H1: thereareconstants > 0 ;S K > 0 andanonnegativefunction f 2 C 1 0 R such that K isgivenby K z = S k z 1+ S K z f z ;z 2 [0 ; 1] H2: thereexists > 0 ;S p > 0 ;M p > 0 andanarbitraryfunction g 2 C 1 0 R such that )]TJ/F20 11.9552 Tf 9.298 0 Td [(p 0 c iswrittenas )]TJ/F20 11.9552 Tf 9.298 0 Td [(p 0 c z = S p z )]TJ/F21 7.9701 Tf 6.586 0 Td [( 1+ M p z g z ;z 2 [0 ; 1] H3: theproductofthefunctions K and p 0 c isboundedon [0 ; 1] H4: theinitialDirichletdataissmooth: S D 2 C 1 [0 ;T ]; H 1 ,andisbounded awayfromzero 0 0 ; and z z 0 for z< 0 H6: Initialmoisturecontentsatisesaniteentropy"condition: R S 0 x 2 )]TJ/F21 7.9701 Tf 6.587 0 Td [( dx< + 1 where > 2 Underthesehypothesesthereisaweaksolutionfor .1 where S 2 H 1 T suchthat 0 S x;t a:e: on T r @ t S 2 L 2 T and S )]TJ/F20 11.9552 Tf 11.955 0 Td [(S D 2 L 2 ;T ; V for V = H 1 ; 1 131

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Theproofofthistheoremisbeyondthescopeofthiswork,butitindicates thatunderconstantrelativehumidityandtemperatureconditions,wherenomass transferisexpected,thereexistsaweaksolututiontothesaturationequation.The sixthhypothesisrestrictstheshapeoftheinitialcondition.Simplyput,theinitial conditioncannotdroptozeroinsuchawayastomake R S 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [( 0 dx gotoinnity.This avoidsthenaturaldegeneratenatureoftheproblem.Theregularityexpectedfor thesolution H 1 isaniceresultgiventhatthisisactuallyathird-orderdierential equation. 6.2AltandLuckhausExistenceandUniquenessTheorems WenowturnourattentiontodemonstratingthenecessaryconditionsforexistenceanduniquenessofRichards'equationsaturationwith =0andthevapor diusionequationinthespecialcaseswheretheotherdependentvariablesareheld xedpossiblyevenconstant.Thetwoequationsaretreatedtogetherinthissection sincetheybothfallundertheclassofquasi-linearparabolicequations.Assuch,they canbeanalyzedusingsimilartheory.Forthepurposesofdemonstratingexistenceand uniquenessweapplygeneraltheoremsbyAltandLuckhaus[1,2]totheseequations. ThefollowingparagraphsareparaphrasedfromAltandLuckhaus[2]andare presentedheretointroducethereadertothenotationusedthereinandforfuture reference. ConsiderthegeneralinitialboundaryvalueproblemIBVPforasystemofquasilinearelliptic-parabolicdierentialequations @ t b j u )]TJ/F43 11.9552 Tf 11.955 0 Td [(r a j b u ; r u = f j b u in ;T ;j =1: m .2a b u = b 0 on f 0 g .2b u = u D on ;T )-9721(.2c a j b u ; r u =0on ;T @ n \051 ;j =1: m .2d 132

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Inequations.2, u 2 R m b : R m R m ,and a : R m R m N R m where N is thespatialdimensionoftheproblemand m isthenumberofequations. Wecall u intheanespace u D + L r ;T ; V aweaksolutionof.2ifthe followingtwopropertiesarefullled: 1. b u 2 L 1 ;T ; L 1 and @ t b u 2 L r ;T ; V withinitialvalues b 0 ,thatis Z T 0 h @ t b u ; i + Z T 0 Z )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(b u )]TJ/F20 11.9552 Tf 11.955 0 Td [(b 0 @ t =0 foreverytestfunction 2 L r ;T ; V W 1 ; 1 ;T ; L 1 with T =0. 2. a b u ; r u ;f b u 2 L r ;T and u satisesthedierentialequation, thatis, Z T 0 h @ t b u ; i + Z T 0 Z a b u ; r u r = Z T 0 Z f b u forevery 2 L r ;T ; V RecallfromFunctionalAnalysisthat V isthedualspaceofthevectorspace V ,and W k;p = f u 2 L p : D u 2 L p 8j j k g withtheweakderivative D u .The readershouldalsorecallthecommonsimpliednotation W k; 2 = H k Considerthefollowinghypotheses: H1: R n isopen,bounded,andconnectedwithLipschitzboundary,)]TJ/F23 11.9552 Tf 319.307 0 Td [( @ is measurablewith H n )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 \051 > 0and0
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H4: Thefollowinggrowthconditionissatised: j a b z ; p j + j f b z j c )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ B z r )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 =r + j p j r )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 : H5: Weassumethat u D isin L r ;T ; W 1 ;r andin L 1 ;T andwedene V := f v 2 W 1 ;r : v =0on)]TJ/F23 11.9552 Tf 45.753 0 Td [(g H6: Assumeeitherthat b 0 mapsintotherangeof b andthereforethereisameasurablefunction u 0 with b 0 = b u 0 orthat @ t u D 2 L 1 ;T ; L 1 : TheexistenceanduniquenesstheoremsofAltandLuckhaus[2]arestatedhere forconvenienceandreference. Theorem6.2AltandLuckhaus[2],Theorem1.7 Supposethedatasatisfy H1-H6,andassumethat @ t u D 2 L 1 ;T ; L 1 .Thenthereisaweaksolutionto .2 Theorem6.3AltandLuckhaus[2],Theorem2.4 SupposethatthedatasatisfyH1-H6with r =2 and a t;x;b z ; p = A t;x p + e b z where A t;x isasymmetricmatrixandmeasurablein t and x suchthatfor > 0 A )]TJ/F20 11.9552 Tf 11.955 0 Td [(I and A + @ t A arepositivedenite.Moreoverassumethat j e b z 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(e b z 1 j 2 + j f b z 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(f b z 1 j 2 C b z 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(b z 1 z 2 )]TJ/F40 11.9552 Tf 11.955 0 Td [(z 1 : Thenthereisatmostoneweaksolution. 134

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6.2.1ExistenceandUniquenessforRichards'Equation TheexistenceanduniquenessofweaksolutionstoRichardsequationisknown, andageneraltoolforhandlingthisproblemistheAlt-Luckhaustheoremstated above.Inthissubsectionwesetupandstatethetheorems.Wewillshowthatthe hypothesesofTheorems6.2and6.3aresatisedunderrestrictionsontheDirichlet boundaryconditionsandappropriateboundednessassumptions.Theequation @S @t = @ @x rw S l g @p l @x )]TJ/F20 11.9552 Tf 11.956 0 Td [( rw S = @ @x rw S @h @x )]TJ/F20 11.9552 Tf 11.955 0 Td [( rw S .3 isRichards'equationindimensionlesstimeandonespatialdimension.Inthisformulationwetake h asthehydraulichead: h = p l = l g .Recallfrompreviousdiscussions thatthepressureorheadisafunctionofsaturation.Thisrelationshipisinvertible soherewenotethatsaturationcanbewrittenasafunctionofpressureorhead.As suggestedin[2,63],saturationmaybelessregularthanpressure,thereforeweexpect toachievebetter[regularity]resultsbyapplyingaKirchhotransform".AKirchho transformationgivesasmoothedrelationshipbetweenheadandanewunknown;a generalizedpressurehead K : R R as K h = Z h rw S q dq := u: Thepressureheadistakentobenegativebyconventionoppositesignofcapillary pressure.Thevariable u nowbecomestheprimaryunknownof.3sincethespatial derivativecanbewrittenas du dx = d K dh dh dx = rw dh dx ; andif b u isdenedas b u = S K )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 u )]TJ/F15 11.9552 Tf 11.955 0 Td [(1then @ @t b u = @ @x @u @x )]TJ/F20 11.9552 Tf 11.955 0 Td [( rw b u +1 : .4 135

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Noticethatthedenitionof b dependsontheinvertibilityof K .Also,since K )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 u = h b canbeseenasafunctionof h : b h = S h )]TJ/F15 11.9552 Tf 13.914 0 Td [(1seeFigure6.1.GiventhevanGenuchtencapillarypressure-saturationrelationship, S h = h 1 = )]TJ/F21 7.9701 Tf 6.586 0 Td [(m +1 )]TJ/F21 7.9701 Tf 6.586 0 Td [(m ; andthevanGenuchtenrelativepermeabilityfunction, rw S = p S )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F26 11.9552 Tf 11.956 9.684 Td [()]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S 1 =m m 2 ; Figure6.2showsseveralplotsof K forvarious parametervalues.Thereisahorizontalasymptoteas h anditisevidentfrom theplotthat K isone-to-oneandontoforallvaluesof h 2 ; 0,butas h gets large inabsolutevaluetheinversebecomesunstable. Figure6.1:Thefunction b h = S h )]TJ/F15 11.9552 Tf 11.955 0 Td [(1for m =0 : 8andvariousvaluesof 136

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Figure6.2:Kirchhotransformation K for m =0 : 8andvariousvaluesof Matchingtoequation.2awenotethat j =1,dene a b u ; r u as a b u ; @u @x = @u @x + rw b u +1 ; andnoticethat f =0.GivenconstantheadDirichletboundaryconditions,wenally rewriteRichards'equationas @ @t b u = @ @x @u @x )]TJ/F20 11.9552 Tf 11.955 0 Td [( rw b u in ;T .5a u = u D on ;T f 0 ; 1 g .5b u = u 0 on0 .5c WiththisformofRichards'equationweproposethefollowingexistenceanduniquenessresult. Theorem6.4 Supposethatthefollowingconditionsholdforthegeneralizedhead, u inequation .5 137

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1. = ; 1 and t 2 ;T ; 1 2. u D 2 L 1 ;T 3. @ t u D 2 L 1 ;T ; L 1 Thenthereexistsauniqueweak u 2 u D + L 2 ;T ; H 1 0 solutionto .5 TheproofofTheorem6.4hasbeendiscussedinseveralarticles.Inparticular,the transformationofRichardsequationtotheformseeninequations.5arediscussed asamodelproblemfortheAltandLuckhaustheorems[2].Furthermore,thisproof ispresentedin[63]aspartoftheirnumericalformulationofRichards'equation.The fundamentalreasonforpresentingthisresulthereisthatthevalueof inthenew saturationequationsisnotyetwellknowninexperimentalstudies.Presentingthis casesimplycoversallofthepossiblebases. Inthecaseswhere r r T ,ormasstransfertermsarenon-zero,theseterms becomesourcetermsthatdependon x .Thismeansthat f = f x;b u 6 =0.Accordingtosection1.10in[2]itmakesnodierenceif a and f dependon x ."Thisis mademoreclearintheirsubsequentwork[1]wherethetheoremisexplicitlystated toallowfor x and t dependence. Forcomparisonsakeweobservethedierencebetweentheregularityrequiredfor RichardsequationTheorem6.4andfortheextendedsaturationequationwiththe third-ordertermTheorem6.1.Fortheequationwiththethird-orderterm r r S theweaksolutionisin H 1 whileinthesecond-orderequationtheweaksolutionisin L 2 .Thisextrarequiredregularityisexpected. 6.2.2VaporDiusionEquation Toproveexistenceforthegasdiusionproblemweproceedusingthetheorem ofAltandLuckhausaswiththesaturationequation.Recallthatunderconstant temperatureandxedsaturationconditions )]TJ/F20 11.9552 Tf 11.955 0 Td [(S x @' @t )]TJ/F20 11.9552 Tf 16.477 8.088 Td [(@ @x D ';S x @' @x =0.6 138

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where D ';S x =1 )]TJ/F20 11.9552 Tf 11.956 0 Td [(S x D g + sat 'R g v T 1 )]TJ/F20 11.9552 Tf 13.151 8.087 Td [(R g a R g v K S x : .7 Allowing S tobeafunctionof x constitutesadeparturefromtheexactformof theparabolic-ellipticsystemfoundinAltandLuckhausseeequations.2asthis isnowanon-autonomousdierentialequation.In[1]thisproofwasgeneralized toallowfor b = b x;u andfor a = a x;u; r u seesection11of[1].Theonly additionalassumptionsfortheexistencetheoremsarethat b : R R and a : R R N R N aremeasurableintherstargumentandcontinuousinthe others.WiththisadditiontoTheorem6.2weproceedwiththeexistencetheoremfor thegas-phaseequation. Assumethattheinitial-boundaryconditionsare ;t = D; 0 2 ; 1 ; 8 t 2 ;T .8a ;t = D; 1 t 2 ; 1 ; 8 t 2 ;T .8b x; 0= 0 x ;x 2 : .8c NoteherethattheDirichletboundaryconditiononthe right-hand sideofistime dependentandtheoneontheleftisindependentoftime.Theproblemcouldalso berestatedwheretheright-handboundaryisofNeumantype.Theconditionsare chosentobettermatchtheexperimentaldatathatwillbeconsideredinSection7.4. Theorem6.5ExistenceofWeakSolutiontoDiusionEquation Supposethat thefollowingconditionsholdforequations .6 .8 1. = ; 1 and t 2 ;T ; 1 2. 2 ; 1 )]TJ/F20 11.9552 Tf 11.956 0 Td [( ] forall x 2 andforall t 2 ;T where 0 < 1 3. S 2 [ ; 1 )]TJ/F20 11.9552 Tf 11.956 0 Td [( ] and S x 2 C 1 independentoftimewhere 0 < 1 139

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4. D; 1 2 L 2 ;T ; H 1 and L 1 ;T Thenthereexistsaweaksolutionto .6 .8 inthesensethat 2 D + L 2 ;T ; H 1 0 Matchingequation.6totheformofAltandLuckhausequation.2awe have b x;z = )]TJ/F20 11.9552 Tf 11.955 0 Td [(S x z;a x;z;p = D x;z p;f =0 ;m =1 : .9 IntheconditionsforTheorem6.5weusetheparameter todenetwodierentsets. Thisisasmallabuseofnotationsince and S neednotbelongtoexactlythesame set.Wearesimplystatingthatbothofthesefunctionsmustbeboundedawayfrom 0and1. Proof: WeproceedbyverifyinghypothesesH1-H6ofTheorem6.2notingthe extensionproposedin[1]tonon-autonomousfunctions. H1:In1spatialdimensionitisclearthatisanopen,bounded,andconnected domainwithLipschitzboundary.)-278(= f 0 ; 1 g ,and H 0 \051 > 0and0 0since S x 2 ; 1.Since S 2 C 1 assumption #3inthestatementofthetheoremitisclearthat b ismeasurableinthe rstcomponent.Furthermore, b isacontinuousgradientofaconvexfunction inthesecondcomponent.Dene B x;z = b x;z z )]TJ/F15 11.9552 Tf 13.249 0 Td [( x;z + x; 0= )]TJ/F20 11.9552 Tf 11.955 0 Td [(S x z 2 = 2. H3:Since a isalinearfunctionof p itiseasytoseethat )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(a x;z;p )]TJ/F20 11.9552 Tf 11.955 0 Td [(a x;z;p )]TJ/F20 11.9552 Tf 12.951 -9.684 Td [(p )]TJ/F20 11.9552 Tf 11.955 0 Td [(p = D x;z )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(p )]TJ/F20 11.9552 Tf 11.955 0 Td [(p 2 C )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(p )]TJ/F20 11.9552 Tf 11.955 0 Td [(p 2 .10 140

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where D x;z C forall x;z this -dependencereectsthechoiceofthe saturationfunction, S x .Giventhefunctionalformof D itisobviousthat a iscontinuousandboundedon z 2 ; 1, p 2 R ,andismeasurablein x .Hence a satisestheellipticitycondition.Simplystated,theellipticityofthediusion coecientmeansthattheoperatorinquestionisboundedawayfromzeroand isthereforeinvertible. H4:Let z 2 [ ; 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [( ]and p 2 R .Fromthedenitionof a and f j a x;z;p j + j f z j = j a x;z;p j = jD x;z p j c D ; j p j c 1+ p B z + j p j .11 forall z ,where c D ; istheupperboundon D x;z over z .Therefore a and f satisfythegrowthcondition. H5:TheleftDirichletboundaryconditionisxedintime, ;t = D; 0 .Itis assumedthat D; 0 2 L 2 ;T ; H 1 and L 1 ;T .TherightDirichlet boundaryconditionisallowedtovaryintime.Assumption#4inthestatement ofthistheoremguaranteesthathypothesisH5issatisedforthisboundary condition. H6:Since b x;z = )]TJ/F20 11.9552 Tf 12.365 0 Td [(S x z itisclearthat b issurjectivesolongas S x 6 =1 andthat b 0 = 0 = )]TJ/F20 11.9552 Tf 12.568 0 Td [(S x .Thatis,thereexistsafunction 0 = )]TJ/F20 11.9552 Tf 12.568 0 Td [(S x suchthat b 0 = b x;' 0 Giventhenalassumptioninthestatementofthistheoremwehave,inparticular, D 2 L 1 ;T ; L 1 since L 1 L 2 L 1 forsetsofnitemeasure [38].Therefore,fromTheorem6.2thereexistsaweaksolution, ,intheanespace D + L 2 ;T ; V where V = f v 2 H 1 : v =0on f 0 ; 1 gg 141

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Theuniquenessoftheweaksolutionto.6-.8,unfortunately,doesn'tt Theorem6.3becausethediusionoperatorcannotbedecomposedinthemanner required.Thisdoesnotmeanthattheweaksolutionisnotunique,itsimplymeans thatthisisnotthetooltoproveuniqueness.Thissmallproblemisleftforfuture research. 6.2.3LimitsoftheAltandLuckhausTheorem ThetheoremofAltandLuckhausdoesnotapplytotheheattransportequation sincethereareadvection-typetermspresentinthatequationthatcannotsatisfy theassumedformofTheorem6.2.Thenextlogicaldirectionistoseeifthistool canbeusedtoproveexistenceofthecoupledsaturation-humiditysystematconstant temperature.Theforcingtermontheright-handsideofeachequationisnownonzero.Theequationsare @S @t )]TJ/F20 11.9552 Tf 16.477 8.088 Td [(@ @x )]TJ/F20 11.9552 Tf 9.298 0 Td [(D S + C l S @S @x + C l @' @x )]TJ/F20 11.9552 Tf 11.956 0 Td [(K S l g =^ e g v l ';S .12a )]TJ/F20 11.9552 Tf 11.956 0 Td [(S @' @t )]TJ/F20 11.9552 Tf 11.955 0 Td [(' @S @t )]TJ/F20 11.9552 Tf 16.476 8.087 Td [(@ @x D ';S @' @x = )]TJ/F15 11.9552 Tf 9.735 0 Td [(^ e g v l ';S : .12b Ifweweretodene b z : R 2 R 2 as b z = 0 B @ 10 )]TJ/F20 11.9552 Tf 9.298 0 Td [(z 2 )]TJ/F20 11.9552 Tf 11.955 0 Td [(z 1 1 C A 0 B @ z 1 z 2 1 C A onecanshowthattheredoesnotexistafunction: R 2 R suchthat b = r .For thisreasonwerestatetheequationswithaconsolidatedformofthetimederivatives inthesecondequation @S @t )]TJ/F20 11.9552 Tf 16.477 8.088 Td [(@ @x )]TJ/F20 11.9552 Tf 9.298 0 Td [(D S + C l S @S @x + C l @' @x )]TJ/F20 11.9552 Tf 11.955 0 Td [(K S l g =^ e g v l ';S .13a @u @t )]TJ/F20 11.9552 Tf 16.477 8.088 Td [(@ @x D ';S @' @x = )]TJ/F15 11.9552 Tf 9.735 0 Td [(^ e g v l ';S .13b u = )]TJ/F20 11.9552 Tf 11.955 0 Td [(S .13c 142

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Solvingfortherelativehumidityinequationcandsubstitutingintoequationsa andbgives @S @t )]TJ/F20 11.9552 Tf 16.476 8.088 Td [(@ @x )]TJ/F20 11.9552 Tf 9.299 0 Td [(D S + C l S @S @x + C @ @x u 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S )]TJ/F20 11.9552 Tf 11.955 0 Td [(K S l g =^ e g v l u 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S ;S .14a @u @t )]TJ/F20 11.9552 Tf 16.477 8.088 Td [(@ @x D u 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S ;S @ @x u 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S = )]TJ/F15 11.9552 Tf 9.735 0 Td [(^ e g v l u 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S ;S : .14b Itcanbeseenfromthisformthatthecouplinginthetimederivativeshasbeen movedtoastrongercouplingwiththediusionterms.Itcanbeshownthattheassociated a ; functionisnotellipticinthesenserequiredinTheorem6.2.Therefore wehavedeterminedthattheAltandLuckhaustheoremsdon'tapplytothecoupled systeminthisform. Equations.14posesthesysteminaformofstrongcouplingknownasa triangularsystem .Atriangularparabolicsystemhastwoequations;oneparabolicequation withacontributiontodiusionfrombothdependentvariablesandtheotherwitha contributiontodiusionfromonlyonevariable[53].Futureresearchintotheexistenceanduniquenessresultswilllikelystarthereasthetheoryoftriangularsystems isfairlywelldevelopedandmayprovideaspringboardtoresultsforthisproblem. 6.3HeatTransportEquation Inthissectionweconsiderthequestionofexistenceanduniquenessfortheheat transportequation.Thisisdoneundertheassumptionsthattherelativehumidity andsaturationprolesarexedinspaceandtime. Ifthesaturationandtherelativehumidityareconsideredxedandconstantthen thethermaltransportequation.88ccollapsesto c p @T @t )]TJ/F43 11.9552 Tf 11.955 0 Td [(r K r T + h + 2 r T r T =0 ; .15 143

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where 2 isgivenas 2 = l c l p C l T K S + g c g p N S;T : Intheabsenceofheatsourcesandif 2 isneglectedwearriveatthestandardheat equation;theexistenceanduniquenessresultsofwhicharewellknownseeany standardtextonPDEs.Itislikelythat C l T 0since,inSaito[67],theauthors indicatedthatthethermalliquiduxisnegligibleascomparedtoisothermalliquid ux.Theentropytermappearingin N ,ontheotherhand,islikelynon-negligible andthereforemustbeconsidered.Inthecasewhere S and arexedbutnonconstant,thetermsinequation.88cassociatedwith r S and r arecombined asasourcetermwhichdependson x;t; and T .Therefore,weonlyneedtoconsider thermalequationsintheformof.15.If h =0andDirichletboundaryconditions areconsideredthenthisistheexactformoftheequationconsideredbyRinconet al.in[66]: @u @t )]TJ/F43 11.9552 Tf 11.956 0 Td [(r a u r u + b u j r u j 2 =0in ;T .16a u =0on @ ;T .16b u x; 0= u 0 x in : .16c wherewehavedened u suchthat u T )]TJ/F20 11.9552 Tf 12.317 0 Td [(T ref withreferencetemperature T ref Taking h =0meansthatwemustassumethatboth S and areconstantinspace andxedintime.Thisisnotentirelyphysical,butitisasteptowardageneral existenceuniquenesstheoryforthepresentequations.Inthisproblem, a u isthe diusioncoecient, a u = K u ,denedeitherbytheweightedsumofthethermal conductivitiesequation.78orbytheJohansenthermalconductivityfunction equation.79.The b functionisdenedas 2 asabove. FortheRinconexistenceanduniquenesstheoremweconsiderthefollowinghypotheses: H1: a u and b u belongto C 1 R andtherearepositiveconstants a 0 ;a 1 suchthat 144

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a 0 a u a 1 and b u u 0. H2: Thereisapositiveconstant M> 0suchthat max s da du s ; db du s M: H3: u 0 2 H 1 0 H 2 suchthat k u 0 k L 2 < forsomeconstant > 0. Theorem6.6Rinconetal.[66],Theorem2.1 UnderhypothesesH1-H3there existsapositiveconstant 0 suchthatif 0 << 0 thentheproblem .16 admitsa uniquesolutionsatisfying i. u 2 L 2 ;T ; H 2 0 H 2 and @ t u 2 L 2 ;T ; H 1 0 ii. @u @t )]TJ/F43 11.9552 Tf 11.955 0 Td [(r a u r u + b u j r u j 2 =0 in L 2 ;T iii. u = u 0 Theorem6.7 Thereexistsauniquesolutiontoequation .15 underthefollowing conditions: 1. u ;t = u ;t 2. h =0 3. u x; 0= u 0 2 H 1 0 H 2 andthereexists > 0 suchthat k u 0 k L 2 < Hereweareusing u = T )]TJ/F20 11.9552 Tf 13.103 0 Td [(T ref forascaledtemperaturesothatcapital T will representanitetimeasinTheorem6.6. Proof: WewillproceedbyverifyingthehypothesesofTheorem6.6 H1:Fromthederivationoftheheattransportequation, a u isaweightedsumof thermalconductivitiesfromtheindividualphases.Theparticularformofthe weightedsumcomesfromeitherequation.78or.79,butinthisscenario, 145

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thesaturationispresumedtobeconstant.Therefore,inthiscase a u isconstantandistrivially C 1 .Thefunctionalformof b dependsonthefunctional formoftheentropyandthesaturation.Solongasthesaturationisxedaway fromzerothen b isin C 1 .Furthermore, b u ispositiveso b u u isalso positiveforall u H2:Since a isaconstant, da=du =0forall u .Thefunctionalformof b ,ontheother hand,isnotconstantsothishypothesissimplystatesthat 2 needstohavea boundedrstderivative.TakingtheentropytermfromtheDarcyuxas g = c g p ln T T ref + ref ; fromthedenitionofthespecicheatanddening 2 accordinglyweseethat b willhaveaboundedrstderivativesolongas u + T ref = T remainsbounded awayfrom0.Thisis,ofcourse,alwaystruesince T istheabsolutetemperature. H3:Thethirdassumptionofthetheoremsatisesthishypothesis. Therefore,thereexistsauniquesolutionto.15withnosourcesandequalDirichlet boundaryconditions. 6.4Conclusion Atthispointweturnourattentiontotheanalysisandcomparisonsofnumerical solutionsoftheequationsbothindividuallyandcoupled.Theexistenceanduniquenesstheorypresentedhereisbynomeanscomplete.Inparticular,wearemissing auniquenessresultforthevapordiusionequation,thetheoremusedfortheheat transportequationisverylimitingwithrespecttoboundaryconditionsandsources, andwehavenotmentionedresultsforanyofthecoupledsystems.Manynumerical solverswilliteratecoupledsystemsacrosstheequations,soanexistenceanduniquenesstheoryforeachequationisessentialtogivehopethatthenumericalmethod convergesto the solution.Theseresultsareleftforfutureworkastheultimatecrux 146

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ofthisthesisistojustifythemodelingtechniqueagainstphysicalexperimentation andclassicalmodels. 147

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7.NumericalAnalysisandSensitivityStudies Inthischapterwebuildandanalyzethesolutionstotheheatandmoisture transportmodelsummarizedinequations.88a-.88cwithconstitutiveequations summarizedinequations.89a-.89r.Tosimplifymatterswehenceforthassume a1-dimensionalgeometrymodelingacolumnexperimentcommontosoilscience. Figure7.1givesacartoondrawingofatypicalcolumnexperimentwithadenition ofthegeometricvariable x .Thegrainsrepresentapackedporousmedium.Flow, diusion,andheattransportareassumedtotravelsolelyinthe x directionupor down. x =0 x =1 g Figure7.1:Cartoonofa1-dimensionalpackedcolumnexperimentalapparatus. Inthischapterweareinterestedinthebehaviorofequations.88inseveral situationsrelatedtotheapparatusdepictedinFigure7.1;somephysicalandsome merelyhypothetical. 1.Ina drainageexperiment thecolumnissaturatedwiththewettingphase andthenallowedtodrainundertheinuenceofgravity. Possiblesimplifyingassumptionsinclude:constantrelativehumidityandtemperature. 148

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2.Inan imbibitionexperiment thecolumnstartspartiallysaturatedordry andthewettingphaseisintroducedeitherat x =1or x =0.Ifthewetting phaseisintroducedat x =1thentheprimaryforcedrivingtheliquidowwill begravity,andifitisintroducedat x =0thenthepressureheadfromthe reservoirdrivestheow. Possiblesimplifyingassumptionsinclude:constantrelativehumidityandtemperature. 3.In evaporationstudies ,agradientinrelativehumidityisintroducedbetween x =0and x =1andrelativehumidityistrackedthroughoutthecolumn. Possiblesimplifyingassumptionsinclude:constanttemperatureand/orxed saturationprole. 4.In Coupledsaturationandevaporationexperiments thesaturationand relativehumidityaretrackedthroughoutthecolumnunderboundaryconditionsthatdriveboth. Possiblesimplifyingassumptionsinclude:constantoratleastxedtemperature. 5.In fullycoupled systemsweconsideraheatsourcetypicallylocatedat x =1 andboundaryconditionsthatdriveallthreeequations. InChapter6wediscussedthequestionsofexistenceanduniquenessofsolutions toequations.88.Wenowturntonumericalanalysis.InSections7.1,7.2,and7.3 wediscussvariousnumericalsolutionsassociatedwiththesituationsoutlinedabove. Forexample,inSection7.1,weexaminenumericalsolutionsassociatedwithdrainage andimbibitionexperimentstypes1and2above.InSection7.4wecomparewith a1-dimensionalcolumnexperimentoutlinedinSmitsetal.[78].Notwo-orthreedimensionalexperimentsareperformedinthiswork. 7.1SaturationEquation 149

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Inthissubsectionweconsiderthesaturationequation.88awithxedandconstantrelativehumidityandtemperatureandnomasstransfer.Thatis,weconsider @S @t = @ @x )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 S K S )]TJ/F20 11.9552 Tf 9.299 0 Td [(p 0 c S + C l S @S @x + S @ 2 S @x@t )]TJ/F20 11.9552 Tf 11.955 0 Td [( l g : .1 Theseassumptionsarenaturalinanoil-watersystemorsimplyunsaturatedsystems wheretherelativehumidityisconsideredxedexperimentally.Wewouldliketo determinequalitativebehaviorofsolutionstothisequationundercertainboundary conditions,experimentalsetups,vanGenuchtenparameters,andvaluesorfunctional formsof and C l S .Asarststeptowardthisanalysisletusconsiderdimensionless spatialandtemporalscalings.Noticethatthespatialdimensioncanalreadybe viewedasdimensionlessasseeninFigure7.1.Acharacteristictimeforthisequation is t c = x c =k c =1 =k c where k c = l g s = l isthehydraulicconductivity.Multiplying by t c andhenceforthunderstanding t and x asdimensionlessweget @S @t = @ @x t c )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 S K S )]TJ/F20 11.9552 Tf 9.299 0 Td [(p 0 c S + C l S @S @x + @ @x K S @ 2 S @x@t )]TJ/F20 11.9552 Tf 16.477 8.088 Td [(@ @x )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(t c K S )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 S l g : .2 Inthecasewhere =0,thequalitativebehaviorcanbeanalyzedviathePeclet number;theratiooftheadvectivetodiusivecoecients Pe = l g )]TJ/F20 11.9552 Tf 9.299 0 Td [(p 0 c S + C l S = l g l g )]TJ/F21 7.9701 Tf 6.586 0 Td [(m m S )]TJ/F18 7.9701 Tf 6.587 0 Td [(+1 =m S )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 =m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 7.9701 Tf 6.587 0 Td [(m + C l S : .3 Sincethediusivecoecientdependsonthedependentvariableitisimmediately clearthatthePecletnumberwillchangeintimeandspaceinthestudyoflinearPDEs thePecletnumberisaxedratiothatdoesnotdependonthedependentvariable.If Pe< 1thentheproblemis diusiondominated whereasif Pe> 1thentheproblemis advectiondominated .Inadiusiondominatedproblemweexpectasmoothsolution thatspreadsspatiallyintime,andinanadvectiondominatedproblemweexpectmore advectiontransportthansmoothing.Inquasilinearadvectiondiusionequations 150

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seeastandardPDEtextdiscussingthemethodofcharacteristicse.g.[88,35],ifthe diusiontermisnotsignicantlyweightedthentheadvectivetermmayyieldshocktypesolutions.Forexample,ifthematerialparametersforaparticularexperiment arelocatedinthetoprightofFigure7.2thenthediusivetermisweightedvery smallascomparedtotheadvectionandashockismorelikelytodevelop.Thatbeing said,ashock-typesolutionisnon-physicalsoitisnotexpectedintheseexperiments. Thisgivesanindicationthatifashockdoesoccurthentheparametersmustbe non-physicalorthenumericalmethodisnotaccuratelycapturingthediusion. FromthedenitionofthePecletnumberitisclearthattheactionof C l S isto increasethedampingofthediusionterm.GiventheformofthePecletnumber, itstandstoreasonthatdampingsimilartothatof C l S canbeachievedbychoosing dierentvanGenuchtenparameters.Forthisreasonwepresumefortheremainder ofthisworkthattheeectsof C l S areinseparablytiedupwiththeeectsofthe p c )]TJ/F20 11.9552 Tf 10.984 0 Td [(S relationship.Hencewecanassumethat C l S 0.Recallthat C l S isdenedsee equation.45as C l S = l l )]TJ/F20 11.9552 Tf 11.955 0 Td [( l g =2 @ l @S andisinterpretedasawettingpotential. Withtheassumptionthat C l S 0orisatleastinseparableexperimentallyfrom p 0 c S ,thePecletnumberbecomes Pe = m 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(m S +1 =m S )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 =m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 m : ThevanGenuchtenparameters, and m ,areindependentinthisformofthePeclet number.Furthermore,thevanGenuchtencapillarypressure-saturationfunctionis onlyoneofseveralchoicesforthisrelationship.OthercommonformsaretheBrooksCoreyandFayer-Simmonsmodels;eachofwhichwillhavetheirownassociatedPeclet number.Figure7.2showsthenatureofthePecletnumberasafunctionofthese parametersaswellasthesaturation. 151

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aLogofPecletnumberfor S =0 : 1 bLogofPecletnumberfor S =0 : 4 cLogofPecletnumberfor S =0 : 7 dLogofPecletnumberfor S =0 : 99 Figure7.2:LogofPecletnumbersforvariousvaluesofsaturation.Thepointat =5 : 7 ;m =0 : 94indicatesthevaluesusedinSmitsetal.[78].Warmercolors areassociatedwithhigherPecletnumberandthereforeassociatedwithanadvective solution. InFigure7.2itappearsthatthesolutionstothesaturationequationwith =0 becomemorediusiondominatedforsmallervaluesofvanGenuchtenparameters. As S 1thediusiontermgainsmoretractionandhencedampenstheadvection. Ofcourse,onecannotsimplychooseasetofvanGenuchtenparameters.Instead,the parametersaredictatedbythematerialpropertiesofthesoil.InthestudybySmitset al.[78], =5 : 7and m 0 : 94indicatedbythepointinFigure7.2.Inthisinstance, weexpectanadvectiondominatedsolutionwithverylittlediusivedamping.This posesadangernumericallyasitisclosetotheregimewhereshock-typesolutions 152

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couldarise. Thethird-ordertermcanbeanalyzedinasimilarmanner.Totheauthor's knowledgethereisno name fortheratioofthecoecientsofthethird-ordertermto thediusiveterm H := S )]TJ/F20 11.9552 Tf 9.299 0 Td [(t c p 0 c S = S l g s )]TJ/F20 11.9552 Tf 9.299 0 Td [(p 0 c S l = S S l m 1 )]TJ/F20 11.9552 Tf 11.956 0 Td [(m S +1 =m S )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 =m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 m = S S l Pe := H 0 Pe .4 ThustheplotsinFigure7.2aresimplyscaledversionsof H .Thequestionthat remainsiswhateectthethird-ordertermhasonthesolution.Toanswerthisquestionsweexamineafewsolutionplots.Thesesolutionsarefoundusing Mathematica 's NDSolve function.Thisbuild-incommandisageneraldierentialequationsolver handlingordinaryandpartialdierentialequations,systemsofequations,vector equations,andstisystems.ForpartialdierentialequationsitusesanitedierenceapproachtodiscretizethespatialvariableandaversionofGear'smethodfor implicitstitimesteppingfollowingamethod-of-linesapproach[87]. Figure7.3showsadrainageexperimentforvariousvaluesof H 0 = S S = l Theinitialconditionisgiveninblack.ADirichletboundarycondition S = S 0 is givenat x =1andahomogeneousNeumanncondition @S=@x j x =0 =0isimposed at x =0.Gravitypointsinthenegative x direction,sothattheliquidpresentin thecolumnisexpectedtodrainovertime.Figure7.3ashowsthatatearliertimes alargervalueof H 0 givesasteeperfrontwithplausiblyphysicalsaturationproles. Non-physical,non-monotonic,resultsareobservedfor H 0 =10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 asseennear x =0 : 8 inFigures7.3b-7.3d.Forvaluesof H 0 smallerthan10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 wecontinuetoobserve asharperfrontascomparedtosolutionsfor =0showninblue. 153

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aSaturationprolesat t = t 1 bSaturationprolesat t = t 2 cSaturationprolesat t = t 3 dSaturationprolesat t = t 4 Figure7.3:Saturationprolesatvarioustimesinadrainageexperimentwith = 5 : 7 ;n =17. Tobesurethatthenon-physicalresultsobservedfor H 0 =10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 arenotdueto numericalnoisewecompleteanumericalconvergencetestonthisparticularsetof initialboundaryconditions.Atypicalconvergencetestofanumericalmethodwould compareagainstaknownanalyticsolution,butinthiscasethereisnoknownanalytic solution.Forthisreason,weallow Mathematica tosolvetheproblemusingthedefault spatialandtemporaltolerancesandthencomparesolutionswithxedgridsconsisting offewermeshpointstothissolution. Mathematica 'sdierentialequationsolveruses anitedierenceapproachforspatialdiscretization.Thedefaultsforthisscheme arefourth-ordercentraldierenceswherespatialpointsareonastaticgridandthe 154

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numberofgridpointsischosenautomaticallybasedontheinitialcondition.Forthe testsshowninFigure7.3therewere103gridpointsselectedautomatically.Tocheck thissolution,weexaminetherelative L 2 ; 1errorasafunctionoftime, E N t = k S k )]TJ/F20 11.9552 Tf 11.955 0 Td [(S k L 2 k S k L 2 ; where N isthenumberofspatialpoints.InFigure7.4wemeasure E N t for N rangingfrom20spatialpointsto100spatialpoints.Noticethatforanyxed small timetherelativeerrordecreaseswithincreasinggridsize;henceindicatingnumerical convergenceatthatxedtime.Fordimensionlesstimegreaterthanapproximately 0 : 05,ontheotherhand,theerrordecreasesataslowerrateandthereisevidence thatthenumericalmethodmaynotbeconverging.Inallcasestherelativeerror growsintimeuntilapproximately0 : 25.Whilethe bump thatappearsinFigure7.3 iscertainlynon-physical,Figure7.4seemstoindicatethatthenumericalmethodis failinginthiscaseandtheresultsmaynotbetrust-worthyforthissetofparameters andinitialboundaryconditions. 155

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Figure7.4:ConvergencetestfordrainageexperimentdepictedinFigure7.3. N isthe numberofspatialgridpoints.InFigure7.3, t 1 =0 : 025 t c t 2 =0 : 050 t c t 3 =0 : 075 t c and t 4 =0 : 010 t c Figure7.5showsanimbibitionexperimentforvariousvalueof H 0 .Asbefore,the initialconditionisshowninblack.Forthisexperiment,Dirichletboundaryconditions areenforcedatboth x =0and x =1.Gravitypointsinthenegative x direction,and theboundaryconditionat x =1indicatesthatwettinguidisbeingaddedovertime. For H 0 =10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(4 and H 0 =10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 weseeplausiblyphysicalresultsandweseesharper wettingfrontsasinthedrainageexperiment.For H 0 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 wealmostimmediately seeanon-physicalnon-monotonicityappearatthetopedgeofthewettingfront. SimilarbehaviorwasobservedbyPeszynskaandYi[60]fortheirnumericalscheme, andtheystated ...wecannotspeculatewhethertheapparentnonmonotonicityofproles ...relatestoanumericalinstability,ortoaphysicalphenomenon." Itisreasonabletoaskwhetherthisisassociatedwithnumericalnoise,andFigure7.6 showsaconvergencetestsimilartothatshownwiththedrainageexperiment.From 156

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Figure7.6itappearsasifthenumericalmethodisconvergingundermeshrenement fordimensionlesstimeapproximatelylessthan0 : 1.Thenon-monotonicityappears intheregionwherethemethodshouldbestablesowetentativelyconcludethat thiseectisnotanumericalartifact.Finally,weobservethatfor H 0 =10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 the advectiontermhasbeenoverwhelmedbythediusionandthethird-ordertermand thenumericalresultsarecompletelynon-physical. aSaturationprolesat t = t 1 bSaturationprolesat t = t 2 cSaturationprolesat t = t 3 dSaturationprolesat t = t 4 Figure7.5:Saturationprolesinaimbibitionexperimentwith =2 : 5 ;n =5. 157

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Figure7.6:ConvergencetestforimbibitionexperimentdepictedinFigure7.5.In Figure7.5, t 1 =0 : 025 t c t 2 =0 : 050 t c t 3 =0 : 075 t c ,and t 4 =0 : 010 t c Clearlythereareinnitelymanychoicesofinitialboundaryconditions,andthe resultspresentedhereininherentlydependontheconditionschosen.Similartypesof non-physicalbehaviorcanbeobservedforotherfamiliesofvanGenuchtenparameters, buttheassociatedplotsareexcludedhereforbrevity.Anempiricalconclusionisthat for H 0 = S S = l greaterthan10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 possiblyleadstonon-physicalbehaviorinthe numericalsolution. Totheauthor'sknowledge,ananalysisofparametersofthistypehasnotbeen completedintheliterature.Wehaveshowninthissubsectionthatforreasonably smallvaluesof wepredictsharperfrontsthanwiththetraditionalRichards'equation. 7.2VaporDiusionEquation Nextletusconsiderthevapordiusionequationunderassumptionsofxedconstanttemperatureandaxedsaturationprole.Thisparticularstudyisabitpeculiar 158

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sinceitisunlikelythatasaturationprolewillremainxedduringanevaporation orcondensationstudy.Ofcourse,wecouldconsider S 0everywhereandstudy onlyevaporationindryporousmedia,butthisisalsonotrealisticasenhancement modelsdependpartlyonthepresenceofaliquidphase.Inthissectionwecompare thepresentmodelproposedinSection5.4.2totheclassicalenhancementmodeland toFickiandiusion. )]TJ/F20 11.9552 Tf 11.955 0 Td [(S @' @t = @ @x D ';S @' @x presentmodel.5 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S @' @t = @ @x a S S D g @' @x enhancementmodel.6 )]TJ/F20 11.9552 Tf 11.955 0 Td [(S @' @t = D g @ 2 @x 2 Fickiandiusionmodel : .7 Recallthat a S istheempirical enhancementfactor traditionallyused, S isthe tortuosity,and D g istheconstantFickianvapordiusioncoecientseeequations .71,.73,andobviously.7.Thereadershouldnotethatweareslightly abusingnotationgiventhat previouslystoodforintensiveentropyand isthe labelfortherelaxationterminthesaturationequation.Thisabuseofnotationis containedtothissectionandshouldnotcauseconfusion. Qualitatively,the shape ofthediusioncurveinthe x )]TJ/F20 11.9552 Tf 12.065 0 Td [(' planeforthepresent modelisratherdierentthanthoseoftheenhancementandFickianmodels.Figure7.7givesseveralsnapshotsofasamplediusionexperimentwithenhancement parameter a =25,vanGenuchtenparameter m =0 : 9,andsaturatedpermeability S =1 : 04 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 m 2 .Observefurtherthatthesteadystatesolutionsaredierentfor thetwomodels.Thisisnosurprisesincethenonlinearitiesinthediusioncoecient havedierentfunctionalforms. 159

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aRelativehumidityprolesat t = t 1 bRelativehumidityprolesat t = t 2 cRelativehumidityprolesat t = t 3 dRelativehumidityprolesat t = t 4 Figure7.7:Samplediusionexperimentcomparingtheenhancementmodeltothe presentmodel.Here, a =25, m =0 : 9 n =10,and =10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 withDirichlet boundaryconditionsandanexponentialinitialprole. InFigure5.5wesawthatthereispotentiallyamarkeddierencebetweenthe diusioncoecientinthepresentmodelandtheenhancementmodel.Figures7.8 and7.9showacomparisonofthediusioncoecientsforseveralvaluesofthevan Genuchten m parameterandtwodierentsaturatedpermeabilities.Thefunctional dependenceofthediusioncoecientinthepresentmodelonthevanGenuchten parametercanbereadilyseenbetweenFigures7.8aand7.8dsimilarly,7.9a and7.9d,andthefunctionaldependenceonthesaturatedpermeabilitycanbe seenbetweenthetwosetsofgures. 160

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aComparisonfor m =0 : 4 n =1 : 67 bComparisonfor m =0 : 6 n =2 : 5 cComparisonfor m =0 : 8 n =5 dComparisonfor m =0 : 95 n =20 Figure7.8:ComparisonofdiusioncoecientsforvariousvanGenuchtenparameters alltakenwith s =1 : 04 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 and =0 : 334tomatchtheexperimentin[78]. 161

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aComparisonfor m =0 : 4 n =1 : 67 bComparisonfor m =0 : 6 n =2 : 5 cComparisonfor m =0 : 8 n =5 dComparisonfor m =0 : 95 n =20 Figure7.9:ComparisonofdiusioncoecientsforvariousvanGenuchtenparameters alltakenwith s =4 : 0822 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(11 and =0 : 385tomatchtheexperimentin[68]. Inthisthesisweproposethatthereisarelationshipbetweenthetted a value andthematerialproperties. Proposition7.1 GiventhevanGenuchten m orequivalently, n parameterand thesaturatedpermeability, S ,ofthesoilthereisana-prioriestimateofthetting parameter a TheimmediateconsequenceofPropostion7.1isthatifthettingparametercanbe predictedwiththeuseofexperimentationthenitis,indeed,unnecessary. 162

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TotestProposition7.1weuseasimpleheuristicapproachtomatchthematerial coecients, m and S ,tothecalculatedttingparameter, a .Thisisdoneonthe experimentsbySmitsetal.[78]andSakaietal.[68].InSmitsetal., S =1 : 04 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(10 [m 2 ]and m =0 : 944withastatisticallytuned a -valueof18 : 2.InSakaietal., S = 4 : 082 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(11 [m 2 ]and m 0 : 799with a -valuesof5 ; 8,and15considered.Sakai etal.indicatedthebestagreementwith a =8whilesimultaneouslyconsideringa modiedvanGenuchtenFayer-Simmonsmodelforthesoilwaterretentioncurve. WedonotconsidertheFayer-Simmonsmodelhere,butastheFayer-Simmonsmodel isdesignedtogivebetteragreementofthecapillarypressure-saturationrelationship withverylowsaturationswedon'tbelievethisnegatesourapproach. TheheuristictestsofProposition7.1isasfollows.Thesteady-statemassuxes predictedbythepropsednewmodelandthetranditionalenhanceddiusionmodel are sat D m; S ';S r and sat a S S D g r ': Assumingthatthemassuxesareequalgivestheequation D m; S ';S r = a S S D g r ': Makingthefurtherassumptionthatthegradientsinrelativehumidityarethesame atsteadystatethenthediusioncoecientsmustbeequal.Ifthismassuxistaken ataliquid-gasinterfacewecanassumethat =1.Hence,theleft-handsideofthis equationisafunctionof S m ,and s whiletheright-handsideisafunctionof S and a D m; S ;S = a S S D g : Atthispointwecouldproceedbysimplychoosingavaluefor S andmakingcomparisonsorwecouldconsidertheintegraloverallof S toremovethedependenceon thesaturation.Wechoosethelatterasitgivesacumulativeeectofthediusion coecientovertheentirerangeofsaturations. 163

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Therefore,foreach m and S andforxed =1,thereisavalueof a suchthat Z 1 0 D m; S ;S dS = Z 1 0 a S S D g dS: .8 Theleft-handsideisafunctionofmaterialparametersandtheright-handsideis afunctionof a .Theright-handsideofequation.8integrateseasilytoalinear functionof a Z 1 0 a S S D g dS = D g 3 a + 1 1 )]TJ/F20 11.9552 Tf 11.955 0 Td [(" : Theleft-handsideofequation.8,ontheotherhand,isn'treadilyintegrabledue tothenonlinearnatureofthevanGenuchtenrelativepermeabilityfunction.Forthis reasonweseekanapproximatesolutiontoequation.8. Figure7.10showstheleft-andright-handsidesofequation.8.Theintersectionsindicatethetriple m; S ;a wheretheequationistrue,andhenceindicates wherethetwomodelshavethesamecumulativediusiveeectover S 2 [0 ; 1].For example,inFigure7.10,if m 0 : 5and s 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(10 thenwepredictattingparameter of a 30. InFigure7.10,theblueandgreencurvesaretheright-handsidesofequation .8fordierentsaturatedpermeabilities.Thebluecurveisincludedtoshowthe agreementwithSmitsetal.Thegreencurveisincludedtoshowtheagreementwith Sakaietal.Observethattheexperimentalvaluesare close tothevaluesthatmake equation.8truetheintersectionsindicatedinthegure.Thisistosaythat given m and S ,equation.8couldhavebeenusedasana-prioriestimateofthe valueof a inthesetwoexperiments.Table7.1givesamoreconcisesummaryofthe resultsfoundinFigure7.10. 164

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Figure7.10:Theblueandgreencurvesshowtheleft-handsideofequation.8for dierentsaturatedpermeabilities,andtheredlinesshowlevelcurvesforright-hand sideforvariousvaluesof a .Theblueandgreencurvescanbeusedtopredictthe valueof a beforeexperimentation. Table7.1:Measuredandpredictedvalueofthettingparameter a basedonequation .8. a measured a predictedfrom.8 Smitsetal. 18.2 18 Sakaietal. 8 9 ComparisonswiththeexperimentsofSmitsetal.andSakaietal.indicatethat, whilenotperfect,thepresentmodelgivesadiusionequationthatmatchesexperimentalndingsreasonablywellwithoutthenecessityofana-posterioritting parameter. 7.3CoupledSaturationandVaporDiusion Inthissectionwecouplethesaturationandvapordiusionequationsunder reasonableboundaryconditions.Thisisdonewhileholdingthetemperaturexed. 165

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Thepurposeofthisshortstudyistodeterminetherolesofthemasstransfer,^ e g v l ,the C l @'=@x termappearinginthesaturationequation,andthetimerateofchangeof saturationthatappearsinthevapordiusionequation.Theequationsarerestated hereforreference. @S @t )]TJ/F20 11.9552 Tf 16.477 8.088 Td [(@ @x )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 S K S )]TJ/F20 11.9552 Tf 9.299 0 Td [(p 0 c S @S @x + S @ 2 S @x@t + C l @' @x )]TJ/F20 11.9552 Tf 11.955 0 Td [( l g = ^ e g v l S l .9a )]TJ/F20 11.9552 Tf 11.955 0 Td [(S @' @t )]TJ/F20 11.9552 Tf 11.955 0 Td [(' @S @t )]TJ/F20 11.9552 Tf 16.477 8.088 Td [(@ @x )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 S D ';S @' @x = )]TJ/F15 11.9552 Tf 9.735 0 Td [(^ e g v l S sat .9b where ^ e g v l = M' )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( sat )]TJ/F20 11.9552 Tf 9.299 0 Td [(p c + S )]TJ/F20 11.9552 Tf 11.955 0 Td [(p l 0 l )]TJ/F20 11.9552 Tf 11.956 0 Td [(R g v T ln : Thisisasystemofadvection-diusion-reactionequationswithapseudo-parabolic dampingterminsaturationfor 6 =0andnoadvectiontermintherelativehumidity equation.Tojudgetherelativeaectof C l comparetherateofmovementofthewater throughtheliquidphasewiththerateofmovementofwaterinthegasphase.As such,weconsidertheratioofthecoecientofthistermtothesaturationdiusion C l )]TJ/F20 11.9552 Tf 9.298 0 Td [(p 0 c = C l l g Pe = C l m l g )]TJ/F20 11.9552 Tf 11.955 0 Td [(m S 1+1 =m S )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 =m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 m : .10 Intheunlikelycasethatratio.10isapproximately1thenthediusioninrelative humidityhasequaleectasthediusioninsaturationincontrollingthetransientnatureofthesaturation.Thisdoesnottwithourphysicalexperiencesoweconjecture thattheratioismuchsmaller.Figures7.11showtimesnapshotsofanimbibition experimentwithsimultaneousvapordiusionandtemperatureindependentevaporation.BothsaturationandrelativehumidityarecontrolledwithxedDirichlet boundaryconditionsandinitialprolesconsistentiwithimbibitionintoalowsaturationcolumn. 166

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aComparisonat t = t 1 bComparisonat t = t 2 cComparisonat t = t 3 dComparisonat t = t 4 Figure7.11:Comparisonofcoupledsaturation-diusionmodelsforvariousweightsof C l withparameters: s =1 : 04 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 ;" =0 : 334, H 0 =10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 ; =4,and m =0 : 667. FromFigures7.11,ifratio.10isgreaterthanorequalto10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 thennonphysicalresultsareobservedforthisparticularsetofinitialboundaryconditions.As thereareinntielymanysetsofinitialboundaryconditionsweonlypresentthisone particularcaseasaproofofconcept.Ingeneralweobservethatthisratiomustbe keptbelow10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 .Witharatiothissmallwearesimplysayingthattheenhancement insaturationseenduetoincreasedlevelsofrelativehumidityhaveasmallaectas comparedtogradientsincapillarypressureinthecaseofxedtemperature. 7.4CoupledHeatandMoistureTransportSystem 167

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Inthisnalsectionweexaminethefullycoupledsystemofsaturation,vapor diusion,andheattransfer.Theculminatinggoalofthissectionistocompare thenumericalsolutionofthepresentmodelwiththeexperimentalresultsassociated with[78].Dr.Smitswasgenerousenoughtosharetheexperimentalresultsforthis comparison.InSection7.4.1,thephysicalapparatusisdiscussedaswellasmaterial parametersandinitialboundaryconditions.InSection7.4.2thefullsystemissolved numericallyandcomparedtotheexperimentaldata. 7.4.1ExperimentalSetup,MaterialParameters,andIBCs Theexperimentofinterestistotracktemperature,relativehumidity,andsaturationinacolumnofpackedsand.Soilmoisture,relativehumidity,andtemperature sensorswereplacedthroughouta111cmcolumnofpackedsand.Aheatsourcewas turnedonandoabovethesurfaceofthesoiltosimulatenaturaltemperaturecycles.ThegoalofSmitsetal.wastodeterminewhethertheequilibriumassumption betweenphaseswasvalidinporousmediaevaporationstudies.Fortheourpurposes weusethisdatasimplyasavalidationofthepresentmodelingeort. AschematicoftheexperimentalapparatususedinSmitsetal.[78]isshownin Figure7.12recreatedfromDr.Smits'notes.Saturationandtemperaturesensors #1-#10,areplacedevery10cmfromthebottom.Saturationandtemperaturesensor #11is1cmunderthesurfaceofthesand.Sensor#12is10cmabovethesurface. Sensor#13isonthesurfaceingoodcontact".Temperaturesensors#14and#15 areplacedwithintheinsulationsurroundingtheapparatustomeasurethelateral heatlossseethetopviewinFigure7.12.Relativehumiditysensor#1is1cm underthesurface,andsensor#2isonthesurface.ThegrayshadedareainFigure 7.12representsthelocationofthesoilpack.Theinitialwaterlevelisthesurfaceof thesoilpack.Thespatialvariabletobeusednumericallyis x 2 [0 ; 1]where x =0 representsthecoolendoftheapparatusandwhere x =1representsthesurfaceof thesoil111cmabovethecoolend.Thematerialpropertiesusedinthisexperiment 168

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areshowninTable7.2. water 10 cm 111 cm 10 : 16 cm 1 2 3 4 5 6 7 8 9 10 11 12 2 1 13 14 15 2 : 5 cm 4 : 5 cm insulation Top heatsource x =0 x =1 Figure7.12:SchematicoftheSmitsetal.experimentalapparatus.Saturationand temperaturesensorsnumbered1-11,temperaturesensors12-15,andrelativehumiditysensors1and2[78].Thegeometric x coordinateisshownontheleft.Image recreatedwithpermissionfrom[78] Theexperimentwasrunfor32days,atwhichpointtherewasapoweroutageand theexperimentwasstopped.Inthemidstoftheexperimentthereweretwosensors thatfailed:saturationsensor#3afterthe1847 th measurement t> 12 : 8days,and relativehumiditysensor#1afterthe2155 th measurement t> 14 : 9daysseeFigure 7.13.Thesaturationsensorsareaccuratetowithin 2%soilmoisturecontentafter soilcalibrationperformedbySmitsetal..Therelativehumiditysensoraccuracy rangesbetween 2%formid-rangetemperaturesandhumiditiesand 12%for extremetemperaturesandhumidities.Thetemperaturesensorsareaccurateto within0 : 5 Cforthetemperaturerangesofinterest www.decagon.com 169

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Table7.2:Materialparametersforexperimentalsetup[78]. Parameter Value Units SandNumber 30/40 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(] DryBulkDensity 1.77 [gcm )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 ] Porosity 0.318 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(] ResidualWaterContent 0.028 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(] SaturatedHydraulicConductivity 0.104 [cms )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 ] vanGenuchten 5.7 [m )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 ] vanGenuchten n m =1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 =n 17.8.9438 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(] Theinitialandboundaryconditionsfortheforthcomingnumericalexperiments canbetakenfromanypointwithinthedataset.Thelogicalinitialpointforthe numericalexperimentisthebeginningofthephysicalexperiment.Thisparticular pointisofinteresttotheexperimentalistassomeoftheinterestingtransientbehavior occursduringthisperiod.Thatbeingsaid,thereisasignicantamountofsensor noiseintheinitialphasesoftheexperimentseeFigure7.14a,andifasimple proof ofconcept isallthatisneededforthepurposesofthiswork,thenalatertimeis preferredsoastoavoidcomplicationsrelatedtothisnoise. aInitialsensordataforrelativehumidity bInitialsensordatafortemperature Figure7.14:Relativehumidityandtemperaturedatashowingmeasurementvariationsintherstfewdaysoftheexperiment.Imagerecreatedwithpermissionfrom [78] 170

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Figure7.13:Brokensensordata.Saturationsensor#3showninblueandrelative humiditysensor#1showninred.Itisevidentfromtheseplotsthatthesesensors arenotworkingproperlyastheygivenon-physicalreadings.Imagerecreatedwith permissionfrom[78] Thesectionofdatawherewewillinitiallyfocusisbetweentimemeasurements 1800.5daysand2150.9days.Thissectionofdataischosensince,qualitatively,itshowstheleastamountofsensornoiseinbothrelativehumidityand temperature.Saturationsensor#3isfaultyinthistimeregion,buttheadjacentsensorsindicatethatthereislittletonodeviationfromfullsaturationforthesetimes. TherelativehumidityandtemperaturedataforthistimeregionareshowninFigure 7.15. 171

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aSensordataforrelativehumidity bSensordatafortemperature Figure7.15:Relativehumidityandtemperaturedataatawindowbeginningroughly 12.5daysintotheexperiment.Thiswindowischosensincethesensornoiseisqualitativelyminimalinthisregion.Imagerecreatedwithpermissionfrom[78] Thepeaksandvalleysofthetemperatureandrelativehumiditydataassociated withtheon-ocycleoftheheatlamphavesmallvariationsthatarelikelydueto sensornoise.Toavoidmodelingthisnoisedirectlywecanapproximatethedata witheitherasimplesinusoidalfunctionorasquarewaveapproximationfoundby applyingthe sign functiontothesinusoidalapproximation.Thedatasuggestsa squarewaveapproximation,butthejumpsindatamaycausenumericaldiculties asthederivativesatthepointsofdiscontinuityaretechnicallydeltafunctionals.A graphicoftheseapproximationsisshowninFigure7.16. 172

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arelativehumidity btemperature Figure7.16:Approximationstorelativehumidityandtemperatureboundaryconditionsatthesurfaceofthesoil. Anystartingpointcanbetakenwithinthiswindowoftime.Wechoosethe 2000 th timestepastheinitialconditionsomewhatarbitrarilyandtfunctionsto thecoarsespatialdataforsaturation,relativehumidity,andtemperature.Forthe relativehumidityandsaturationproleswechoosehyperbolictangentfunctionssince theyexhibittheprimaryfeaturesobservedinthedataseeFigures7.17aand7.17b respectively.Forthetemperatureinitialconditionwechooseanexponentialfunction seeFigure7.17c. Tosummarize,thusfarwehaveboundaryconditionsforrelativehumidityand temperatureat x =1andwehaveinitialconditionsforallofthevariables.The boundaryconditionsat x =0aremuchsimpler.Forsaturationandrelativehumidity wecantake S x =0 ;t =1and x =0 ;t =1basedonthefactthatthesaturationis xedmechanicallyat100%atthebottomendoftheapparatus.Forthetemperature wecaneithertake T x =0 ;t = T 0 or @T=@x x =0 ;t =0.TheDirichletcondition simplystatesthatthetemperatureisxed,andtheNeumannconditionstatesthat thebottomoftheapparatusisinsulatedsothatnoheatislost.Throughoutthe courseofthisexperiment,thethermaleectsarenotappreciablytranslatedtothe 173

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arelativehumidity bsaturation ctemperature Figure7.17:Approximateinitialconditionsatthe2000 th datapoint t 13 : 9days. Errorbarsindicateapproximatesensoraccuracy. bottomoftheapparatussoeitherboundaryconditionwouldbesucient.Finally,the onlyboundaryconditionremainingisthesaturationconditionatthesurfaceofthe soil.Asimpleconditionistostatethattheuxofliquidiszeroacrossthisboundary. Mathematically,thistranslatestotheNeumanncondition @S=@x x =1 ;t =0. Anepointneedstobestatedregardingtherelativehumidityequation.Thesaturationinitialconditionstatesthatmuchoftheexperimentalapparatusiscompletely saturatedwithliquidwaterbelowsensor#9approximately.Theissueisthatthere isnogasphasepresentwhen S =1.MathematicallythistranslatestoaStefan-type 174

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problemwherethelowerboundaryforthegasphaseisactuallymovingspatiallyas theliquidwaterevaporates.Forthesakeofillustrationletussimplyassumefora momentthatthesaturationequationisahyperboliclinearadvectionequationwhere thefrontsimply advects intime.Figure7.18illustrateshowthegas-phasedomain mightevolveintimeinthissimpliedexample.Ofcourse,thesaturationequation isnotsuchasimpleequationbuttheessenceofthemovingdomainisthesame regardless. Figure7.18:Illustrationofhowthegas-phasedomainmightevolveintime. OnewaytomodelthisStefanproblemistoassumethatwhen S =1then =1. Underthisassumptionwecandenethegas-phaseequationasapiecewise-dened dierentialequation: @' @t = 8 > < > : 0 ;S =1 )]TJ/F20 11.9552 Tf 9.299 0 Td [(' @ @t )]TJ/F20 11.9552 Tf 11.955 0 Td [(S sat + r sat D r +^ e g v l )]TJ/F20 11.9552 Tf 11.955 0 Td [(S sat ;S 2 ; 1 : .11 175

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Thisisasomewhatarticialsetup,butitallowsustoassumethatarelativehumidity existsfortheentiredomainevenwhenagasphasedoesn'tstrictlyspeakingexist. Simplystated,equation.11indicatesthatifthesaturationis100%thenthere isnochangeinrelativehumidityeventhoughtechnicallythereisnogasphase. Thenifwetaketheinitialconditionas x; 0=1when S =1andtheleft-hand boundaryconditionas ;t =1wearticiallycreatearelationshipfortherelative humiditythatholdsovertheentirespatialdomain.Thereareseveralconcernswith thisapproach,nottheleastofwhichisthattheexistenceanduniquenesstheory discussedpreviouslydoesnotcoverthissortofcase.Moreover,numericallyrequiring thatthetransitionpointisexactly1isnotreasonableandsomearticialcuto, S < 1,shouldbeusedtoloosenthisconditioninnumericalsimulations. Asimplerwaytomodeltherelativehumidityinthisexperimentistoprescribean initialsaturationthatallowsfor some gasphasetoexistthroughouttheexperiment foralltime.Thisisachievedbysettingtheinitialsaturationlessthan1.From anumericalstandpointthismakestheequationseasiertosolveastheboundaries areallstationaryintime.Ontheotherhand,thischoiceofinitialconditiondoes notmatchtheexperimentalsetupandisthereforelessdesirableforthepurposes ofmodelvalidation.Theforthcomingnumericalexperimentsareperformedusing acombinationofthesetwoapproaches;if S
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Inthissubsectionweperformnumericalsimulationsofthecoupledsystembased ontheinitialandboundaryconditionsinSection7.4.1.Thesenumericalsimulations arearststeptowardvalidationofthenewlyproposedmathematicalmodel.The mainthrustofthisworkwasnottocreaterobustnumericalsolversforcoupledsystems ofPDEs.Assuch,werelyonthe NDSolve packagebuiltinto Mathematica asthe primarynumericalsolver.Theplottingandpostprocessingareperformedonamixof Mathematica and MATLAB .Thepurposeofthesenumericalexperimentsistoprovide avalidationfortheproposedequations.Assuch,wechosetodirectlymodelthe Stefanproblemwiththerelativehumidityequationasshownin.11. The NDSolve packageisbasedonamethodoflinesapproachtonumericaltime integrationwithanitedierencespatialdiscretization.Intimeweuseafourth-order Gear'sschemeandinspaceweuseacentraldierencingscheme.Onedisadvantage tousingthistypeofspatialschemeinthisproblemisthatthesaturationandheat equationshaveadvectivecomponents.Itiswellknown[54,55]thatupwindschemes aretypicallybetteratcapturingthephysicsofadvectiveowproblemsandthecentral dierencingschemeswillusuallyintroducearticialdiusionintothesolution.In anevaporation-typeexperimentsuchasthisone,theadvectiveowisexpectedto belessdominantthanindrainageorimbibitionexperiments.Hence,thearticial diusionintroducedwithacentraldierenceschemeisexpectedtohavelittleimpact onthesolutionquality.Todate,theuseofnon-centralschemesandadaptivemesh renementarenotsupportedby Mathematica 'sdierentialequationsolvingpackage. Theparametersthatcanbevariedinthisexperimentarethecoecientofthe dynamicsaturationterm, ,theweightoftheevaporationcoecient, M ,andthe weightsofthe r and r T termsinthesaturationequation, C l and C l T .Thereare noparametersthatcanbevariedintherelativehumidityequationduetothenewly proposedmodelfordiusion.Thisisincontrasttothestandardenhancementmodel wheretheempiricalttingparameterfordiusionisusedtomatchtheexperimental 177

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data.Thefactthatwehavethreedierentttingparametersinthesaturation equationsimplyallowsustonetunethe shape ofthesaturationsolutionbeyond whatispredictedbythetraditionalRichards'equation.RecallfromSection7.1that largervaluesofthedynamicsaturationtermaectsthesharpnessofthemovingfront inthesaturationequation. InSection5.4.4itwasdemonstratedthatforcertainchoicesof M intheevaporationrule,thepresentevaporationmodelapproximatedthatofBixler[19] ^ e g v l = b )]TJ/F20 11.9552 Tf 5.48 -9.683 Td [(" l )]TJ/F20 11.9552 Tf 11.955 0 Td [(" l r R g v T sat )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v : In[78],thettingparameterforBixler'smodelwas b =2 : 1 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 .Thecorresponding ttedparameterinthepresentmodelis M 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(15 where^ e g v l isgivenas ^ e g v l = M' )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v )]TJ/F20 11.9552 Tf 12.952 -9.684 Td [( l )]TJ/F20 11.9552 Tf 11.955 0 Td [( g v : Thisisonlyanorderofmagnitudeapproximationandnetuningcanbemadeto bettertthedata. Severalexperimentalestimatesof werepresentedinTable2of[47].Fromthis data, couldpossiblyspanseveralordersofmagnitude:10 4 << 10 8 .Unfortunately,thesoiltypeswereonlylistedassand"ordunesand"andtherelevant permeabilitiesandvanGenuchtenparameterswereabsentfromthissummary.These valuesatleastgivea ballpark estimateforexperimentationwith .Thecoecientsof r and r T inthesaturationequation,ontheotherhand,arenewtothisstudyand appropriatevalueshaveyettobedetermined.Assuch,westudydierentordersof magnitudeforthesevaluestoestimatetheeectofthetermstotheoverallnumerical solution.ThematerialparametersareallchosentomatchthoseinTable7.2. Theinitialsimulationswillberunwithsinusoidalboundaryconditionsasshown inFigures7.16.Thisistogiveaqualitativeestimationofthebehaviorofthesolutionswithoutthetroubleofthejumpdiscontinuitiesassociatedwiththesquarewave approximationordatainterpolation.Asmoothedsquarewaveapproximationalso 178

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showninFigures7.16isthenusedtogivea closer matchtotheexperimentaldata. Thesmoothingisachievedbytakingpiecewisedenedhyperbolictangentfunctions toapproximatethesteps.Tomeasuretheerrorbetweenthedataandthenumerical solutionweuseasumofthesquaresoftheresidualvaluesmeasuredateachsensor location: e u t = P 2 data u data ;t )]TJ/F20 11.9552 Tf 11.955 0 Td [(u num ;t 2 P 2 data u data ;t 2 ; .12 where u representsanyofthethreedependentvariablesofinterest S ,or T and thesubscriptindicateswherethevalueistakenfrom.Statingthat 2 data "simply meansthat spansthesensorlocationsrelevantforthegiven u i.e. 2f 110 = 111 ; 1 g for u = .Obviously e u t isafunctionoftimesotogetasinglemeasurethat describestheerrorwetakethemaximumof e u t overthelengthofanexperimental day E u =max t e u t : .13 Asingleexperimentaldaywaschosenduetonumericaldicultiesandduetolossof relativehumiditysensorinformation.Equation.13givesasinglenumericalvalue measuringthetofthenumericalsolutiontothedata.Intherelativehumiditythis isaverysimplisticexerciseasthereisonly1datapointtocompareagainst;the sensorlocated1cmbelowthesurfaceofthesoil.Forthesaturationandtemperature, ontheotherhand,thisgivesabetterquantitativemeasure. Table7.3giveserrorsmeasuredwithequation.13againstthe classical system ofequationsRichards',EnhancedDiusion,anddeVries.Table7.4showstheerror asmeasuredwithequation.13forvariousvaluesof ,fordierentfunctionalforms ofthethermalconductivityseeSection5.4.3,andforvariousvaluesof C l and C l T Inordertomakecomparisonswith ;C l ; and C l T weusetheratioofthiscoecientas comparedtothediusiveterminthesaturationequation.ThiswasdoneinSections 179

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Table7.3:Relativeerrorsmeasuredusingequation.13fortheclassicalmathematicalmodelconsistingofRichards'equationforsaturation,theenhanceddiusion modelforvapordiusion,andthedeVriesmodelforheattransport.Thesearecomparedforthetwothermalconductivityfunctionsofinterestweightedsum.78and C^ote-Konrad.80. RelativeErrors Conductivity BoundaryCond. Saturation Rel.Humidity Temperature WeightedSum SmoothedSquare 0.00356 1.54048 0.000515 C^ote-Konrad SmoothedSquare 0.00357 1.27818 0.000631 7.1and7.3,andtheratiosofinterestarerepeatedhereforclarity: S )]TJ/F20 11.9552 Tf 9.299 0 Td [(t c p 0 c S = S S l Pe S = H 0 Pe S .14a )]TJ/F26 11.9552 Tf 11.291 20.443 Td [( C l p 0 c S = C l l g Pe S = R 0 Pe S .14b )]TJ/F26 11.9552 Tf 11.291 16.856 Td [( C l T p 0 c S = C l T l g Pe S = 0 Pe S : .14c SinceeachratioisrelativetothePecletnumberwhichisafunctionof S we focusonlyontheratiosontheright-handsidesofequations.14.Duetothe factthatthisisalargeparameterspace,onlysomeofthenotablerelativeerrors arepresented.Meshrenementwasusedinthecomparisonsinseveralinstancesto minimizenumericalartifacts.Spatially,themeshesrangedbetween100and1024 points.Onlyauniformmeshwasconsidered. ItisapparentinTable7.4thatthebesterrorapproximationsforsaturation,relativehumidity,andtemperaturearefoundwithsmallervaluesof H 0 orequivalently, .Thisobservationisparticulartoadrainage-typeexperiment.Iftheexperiment wereanimbibition-typethenitisconjecturedbasedontheresultsinSection7.1 thatthevalueof wouldplayalargerrole.AlsoapparentinTable7.4,weseethat thevaluesof C l and C l T playlittleroleintheoveralldynamicsoftheproblem. Keepinmindthattherelativehumidityerrorsarereallyjustthedierencebetween1singlesensorandthenumericalsolutionatthatphysicallocation.Inthe author'sopinionitisunreasonabletojudgetheeectivenessofthenumericalsolu180

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Table7.4:Relativeerrorsmeasuredusingequation.13forinstanceswithinthe parameterspaceconsistingofthethermalconductivityfunctionweightedsum.78 andC^ote-Konrad.80, C l ;C l T ; and .Thesearetakenforasmoothedsquare waveapproximationtotheboundaryconditions.Thestarredrowsindicatefailure ofthenumericalmethod,andtheerrorsfromtheclassicalmodelarerepeatedfor clarity Parameters&Functions RelativeErrors Conductivity R 0 0 H 0 Saturation Rel.Humidity Temperature WeightedSum ClassicalModel 0.00356 1.54048 0.000515 WeightedSum 0 0 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 : 5 0.011966 1.206005 0.000463 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 : 0 0.009020 1.201793 0.000463 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 : 5 0.006508 1.198274 0.000463 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 : 0 0.004904 1.199174 0.000463 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 : 5 0.004076 1.195180 0.000463 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 : 0 0.003712 1.196750 0.000463 0 0.003536 1.199584 0.000463 WeightedSum 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 0.003712 1.196751 0.000463 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(4 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 0.003712 1.196756 0.000463 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 0.003710 1.196807 0.000463 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 0.003692 1.197041 0.000463 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 0.003509 1.202644 0.000462 ? 1 1 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 ? ? ? C^ote-Konrad ClassicalModel 0.00357 1.27818 0.000631 C^ote-Konrad 0 0 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 : 5 0.011964 0.946441 0.000516 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 : 0 0.009022 0.951573 0.000515 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 : 5 0.006513 0.950024 0.000515 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 : 0 0.004910 0.948023 0.000515 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 : 5 0.004084 0.944724 0.000516 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 : 0 0.003719 0.946599 0.000516 0 0.003545 0.942403 0.000516 C^ote-Konrad 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 0.003719 0.948032 0.000516 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(4 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 0.003720 0.941801 0.000516 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 0.003717 0.945905 0.000515 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 0.003698 0.947884 0.000515 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 0.003510 0.966405 0.000513 ? 1 1 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 ? ? ? tionbasedsolelyononepoint.Onecomplicationthatarosewithinthissolutionis thattherelativehumidityexhibitssmallperiodsofnon-physicalbehaviorforcertain parametersandboundaryconditions.Meshrenementremovessomeofthiseect, 181

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butevenwithfurthermeshrenementnotallofthenon-physicalregionswereremoved.Possiblesourcesofthisproblemare:thefactthat Mathematica usescubic interpolationpolynomialstodeliverthesolutionstonumericaldierentialequations cubicinterpolationcanovershootsharptransitionsindata,andtheStefannatureoftheproblemcausesnumerical stiness atthepointoftransition.Further studiesareneededtodeterminetheexactcauseofthisnon-physicalbehavior. Figures7.19showindividualtimestepsofseveralsolutionsforvariousparameters withasinusoidalapproximationtotherelativehumidityandtemperatureboundary conditions.Figures7.20showthesameplotswithsmoothedsquarewaveboundary conditions.Thesquarewaveboundaryconditionsobviouslygivecloserapproximationtotheboundarydata,andatthesametimetheuseofthesquarewaveboundary conditionsremovesthenon-physicalbehaviorinthiscase.Theplotsassociatedwith thesinusoidalapproximationtotheboundaryconditionsarepresentedhereforcomparisonbetweenverysmoothandslightlysharpertransitionsinboundarydata.A closeupoftheregionsofnon-physicalbehaviorisshowninFigures7.21aand7.21b. Observeintheseguresthatthediusionequationsolvedwithsmallervaluesof H 0 andlargerdiusionfromtheweightedsumequationgivethemostplausiblesolutions.Figures7.21cand7.21dgiveanindicationofthedierenceintherelative humidityequationsgivendierentthermalconductivityfunctions. ThecomparisonsofthetemperaturesolutionsareshowninFigures7.22and7.23. Thereisverylittledierencebetweenthemodelsforvariousvaluesof orequivalently, H 0 ,soonlythecurvesassociatedwith H 0 =10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 areshown.Observethatthe thermalequationdoesapoorjobcapturingtheextentofthediusionnearthetopof theexperimentalapparatus,butitdoeswellotherwise.Possiblesourcesofthiserror comefrom:thetermsneglectedinthesimplicationofthethermalmodel, theinitialcondition,thethermalconductivityfunctionsorparameters,and/or theaccuracyofthesensorinformation. 182

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aComparisonat t =1 600sec bComparisonat t =50 600sec cComparisonat t =100 600sec dComparisonat t =150 600sec Figure7.19:Comparisonofrelativehumidityandsaturationforthefullycoupled saturation-diusion-temperaturemodelascomparedtodatafrom[78].Boundary conditionsaretakenfromasinusoidalapproximationofboundarydata.Thermal conductivitiesaretakenaseitherweightedsum.78orC^ote-Konrad.80. ComparisonoftheerrorestimatesbetweenthetwomodelsTable7.3compared toTable7.4,weseethatthepresentmodelgivesslightlybetterapproximationsas measuredwiththismetric.Table7.5givesthepercentimprovementofthepresent modelovertheclassiclmodel.Thesevaluesarechosenfromthetablespresented herein,andassuchthisisalowerboundonthepercentimprovement.Thefactthat therewasanimprovementinerrorislessimportant,intheauthor'sopinion,thanthe factthatthemodelspredictnearlythesameerrorwhileremovingthenecessity fortheenhanceddiusionparameter,andputtingtheentiresystemofequations 183

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aComparisonat t =1 600sec bComparisonat t =50 600sec cComparisonat t =100 600sec dComparisonat t =150 600sec Figure7.20:Comparisonofrelativehumidityandsaturationforthefullycoupled saturation-diusion-temperaturemodelascomparedtodatafrom[78].Boundary conditionsaretakenfromasmoothedsquarewaveapproximationofboundarydata. Thermalconductivitiesaretakenaseitherweightedsum.78orC^ote-Konrad.80. onarmthermodynamicfooting. Inthisproblemthereareseveralparametersofinterest,butthepresentstudy suggeststhatvariationsintheseparametersplaylittleroleintheoveralldynamicsof theproblem.Thisnarrowsusdowntoonly1ttingparameterforthisproblem:the coecient, M ,intherateofevaporationterm.Thiswastakentobestmatchwith thettedevaporationratein[78]soitisexpectedthatthisvaluecanbeconsidered roughlyconstant.TheclassicalmodelconsistingofRichards'equation,Philipand deVriesdiusionequationwithenhancementttingfactors,andthedeVriesheat 184

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aBlowupcomparisonat t =100 600sec. sinusoidalapproximatedboundary bBlowupcomparisonat t =100 600sec. squarewaveapproximatedboundary cBlowupcomparisonat t =100 600sec. sinusoidalapproximatedboundary dBlowupcomparisonat t =100 600sec. squarewaveapproximatedboundary Figure7.21:Blowupcomparisonofrelativehumidityandsaturationforthefully coupledsaturation-diusion-temperaturemodelascomparedtodatafrom[78].The insetplotsgiveacloserlookatthebehaviorexhibitedbytheseparticularsolutions. transportequationcontainsatleasttwottingparametersthatarecalculatedusing aleastsquaresstatisticalapproach. 7.5Conclusion InSections7.1-7.3,severalnumericalresultswerepresentedindicatingtheconsistencyofthepresentmodelswiththeclassicalmathematicalmodelsforsaturation andrelativehumidity.Ofparticularimportanceistheanalysisoftheenhanceddiusionproblem.TheargumentspresentedinSection7.2indicatethatmodelingvapor 185

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aComparisonat t =1 600sec bComparisonat t =50 600sec cComparisonat t =100 600sec dComparisonat t =150 600sec Figure7.22:Comparisonoftemperaturesolutionsforthefullycoupledsaturationdiusion-temperaturemodelascomparedtodatafrom[78].Boundaryconditionsare takenfromasinusoidalapproximationofboundarydata. diusioninunsaturatedmediawiththechemicalpotentialcaneliminatethenecessityforattedenhancementfactor.Alsoshownwithinthissectionisasensitivity analysisofthe parameterforthedynamiccapillarypressuretermaswellasthe coecientof r thatappearsinthesaturationequation. InSection7.4itwasshownthatthefullycoupledsystemmatchesquantitatively andqualitativelytoexperimentaldataforheatandmoisturetransport.Thereare severalproblemswiththematchingtothisexperimentaldata.Firstofall,thespatial dataisverycoarsesogettinganexacttfortheinitialconditionsisdicult.Secondly, 186

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aComparisonat t =1 600sec bComparisonat t =50 600sec cComparisonat t =100 600sec dComparisonat t =150 600sec Figure7.23:Comparisonoftemperaturesolutionsforthefullycoupledsaturationdiusion-temperaturemodelascomparedtodatafrom[78].Boundaryconditionsare takenfromasmoothedsquarewaveapproximationofboundarydata. thedataisnoisysogettingareasonabletfortheboundaryconditionsespeciallyin theinitialexperimentaltimes,isdicult.Lastly,theStefannatureofthisproblem causesnumericaldiculties. Thenumericalsimulationspresentedhereinindicatethattheproposedmodels matchbothphysicalandexperimentalexpectationsforaheatandmoisturetransport model.Themodelissensitivetothechoiceofthermalconductivityfunctionand furtherinvestigationisneededtodeterminewhichfunctionsareappropriate.The slightnon-physicalnatureoftheresultsforcertainboundaryconditionsneedstobe 187

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Table7.5:Percentimprovementofthepresentmodelovertheclassicalmodelusing equation.13astheerrormetric. %Improvement BoundaryCond. SaturationRel.HumidityTemperature Sinusoidal 1.71%22.85%16.33% Sq.Wave 1.71%5.91%26.78% investigated.Furtherstudiesbothnumericalandexperimentalneedtobeperformed andprovideabaselineforfutureresearchendeavors.Thecouplingofthesethree processes,allofwhichwerederivedfromathermodynamicfoundation,openstothe doortofutureresearchendeavorsoncoupledprocessesinunsaturatedsoils. 188

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8.ConclusionsandFutureWork ThroughoutthisthesisithasbeendemonstratedthatHMTandthemacroscale chemcialpotentialarepowerfulmodelingtoolsthatcanbeusedtoderivemathematicalmodelsforrathercomplexphenomenainporousmedia.Chapter-by-chapter,the resultsareasfollows. InChapter2,ashortdiscussionofdierentFickiandiusioncoecientswas presented.Whilethisworkisnot new inthesenseofthecreationofnewtheories orequations,itservesasanaidtounderstandtheassumptionscommonlyusedin diusion-relatedresearchandexperimentation.Weshowedthatundercertainassumptionsthatallofthepore-scaleFickiandiusioncoecientscanbeassumed constant.InChapter3,thefocusturnedbacktoporousmediaandthefundamental frameworkforvolumeaveragingandHybridMixtureTheorywerepresented.This chapterservesasareferenceforthesetopicsandnonewresultswerepresented. InChapter4wederivedandgeneralizedtheprimaryconstitutiveequationsfor unsaturatedmedia.Inparticular,wegeneralizedtheformsofDarcy'sandFick's lawsproposedbyBennethum[12],Weinstein[81],andothers.TheformofFourier's lawderivedissimilartothatofBennethum[14],butcontainstermsparticularto multiphasemedia.Totheauthor'sknowledge,thechemicalpotentialformofFourier's lawisnewtothiswork.TheexactgeneralizationsofDarcy'sandFick'slawspresented herearenoveltothiswork,butsimilartermshavebeenproposedinotherworks [12,13,81].Whatisnoveltothisworkistheextensionofthesetermstomultiphase owandtheuseofthemacroscalechemicalpotentialasadependentvariableto obtainadditionalinsight. InChapter5,acoupledsystemofequationsforheatandmoisturetransport wasderived.OfparticularimportancearethegeneralizationsofRichardsequation andthePhilipanddeVriesvapordiusionequation.Itwasdemonstratedthatthe enhanceddiusionmodelofPhilipanddeVriescanbere-framedintermsofthe 189

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macroscalechemicalpotential.Thisre-framingremovesthenecessityoftheenhancementfactorproposedbyPhilipanddeVries.Thecouplingofthevapordiusion andsaturationtermsisachievedthroughthechemicalpotential.Therelationship betweenrelativehumidityandchemicalpotentialiswellknowninchemistryand thermodynamics,buttotheauthor'sknowledgeithasnotbeenpreviouslyusedin theporousmedialiterature.AlsoinChapter5,ageneralizationoftheheattransport equationgivenbyBennethum[14]tomultiphasemediawasderivedviaHMT.This newmodelcollapsestoBennethum'smodelinthecaseofasaturatedporousmedium andalsosuggestscorrectionstotheclassicaldeVriesmodelproposedin1958. InChapter6,thequestionsofexistenceanduniquenesswerestudiedforeach individualequationwhileholdingtheotherdependentvariablesxed.Theseresultswereknownforthesaturationequation,buttheresultspresentedforthevapor diusionandthermalequationareuniquetothiswork.Thisis,ofcourse,because theseequationsarenoveltothiswork.Moreworkneedstobedoneonthisfront. Inparticular,theuniquenessresultforvapordiusionequationisabsent.Also,the resultsforthethermalequationdependonstrictboundaryconditionswhicharenot necessarilymetphysically.Itisemphasizedthattheexistenceanduniquenessresults presentedhereinareonlypreliminary. ThenumericalresultsinChapter7areperformedforvalidationpurposes.Wheneverpresentingnewequationsitisnecessarytocompareagainstexistingmodelsand, whenpossible,experimentallyobtaineddata.Tothatend,inSections7.1,7.2and 7.3,numericalsolutionstoeachequationwerepresentedalongwithsensitivityanalysesandcomparisonstoclassicalequations.InSection7.4,numericalsolutionstothe coupledsystemwerepresentedandcomparedtoexperimentaldata.Itwasshown thatundercertainparametervaluesthenewlyproposedmodelagreeswiththeexperimentaldata. 190

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Therearemanyavenuesforfutureresearchleftuncompletedinthiswork.RelaxationoftheassumptionsoutlinedatthebeginningofChapter5Sections5.2and 5.4providetheinitialavenuesforfurtherresearch. 1.Itiswellknownthatclaysoilsswellwhenwetted.Relaxingtherigidityassumptiononthesolidphasewouldallowforthisphenomenon.Mathematically, though,thiscreatesfurthercomplicationsintheliquid-andgas-phasemass conservationequationssincethedivergenceofthesolid-phasevelocitywillno longerbezeroduetotheremainingtermsinthesolid-phasemassbalanceequation: s + s r v s = X 6 = s ^ e s : Theright-handsideofthisequationmaybezeroinmostcasesnodissolutionor precipitationofsolidparticles,butaconstitutiveequationwouldbeneededfor s or_ s .Thishasfurthercomplicationsinthatsaturation, S = l = )]TJ/F20 11.9552 Tf 10.837 0 Td [(" s ,now varieswithsolid-phasevolumefractionaswellasliquidphasevolumefraction. ThestressinthesolidphasewasdiscussedinChapter4asaconsequenceof theentropyinequality.Uponfurtherinvestigationthiswillgiveageneralization totheTerzhagiStressPrinciplewhichstatesthattheuidphasescansupport somestressonthesolidmatrix.Thisprinciplewillbenecessarytoexpressthe stressesonthedeformingsolid. 2.Changingthesoilparametersandstudyingthesensitivitytoempiricaland measuredrelationshipscanbeanotheravenueoffutureresearch.Inthepresent studythesoilisassumedtobeisotropic.Forthisreasonitispossibleto takethepermeabilityasascalarfunction.Thisisnotnecessarilyareasonable assumptioninrealphysicalproblems,andonepossibleavenueofresearchisto relaxthisassumptionandtakeafulltensorrepresentationofthepermeability. Clearlya2-or3-dimensionalsimulationwouldhavetobeconsideredinthis 191

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research. Anotheradjustmenttothesoilpropertiesistotherelativepermeabilityand capillarypressure-saturationfunctions.ThevanGenuchtenrelationshipswere usedforthiswork,butthesearenottheonlyfunctionalformsavailable.The Fayer-Simmonsmodel[36],forexample,isanextensionofthevanGenuchten capillarypressure-saturationrelationshipthataccountsforverysmallsaturationcontent.Asimpleavenueforfutureresearchistoimplementthismodel andotherslikeitandthestudytothesensitivityofthemodelstothesesmall changes. 3.Thepresentmodelcanbeextendedtomultipleuidphases.Inthismodelwe haveonlyconsideredoneliquidphaseandonegasphase.Itisreasonableto assumethatthegasphaseisabinaryidealgas,butitisnotalwaysreasonable toassumethattheonlyuidpresentisapureliquid. Onepossibleavenueforfutureresearchistoassumethattheliquidphasehas adissolvedcontaminant.Inthatcaseitisnotreasonabletoassumethatthe densityoftheliquidisonlyafunctionoftemperature.Furthermore,thediffusivetermintheliquidphaseequationmaynotbeabsentdependingonthe typeofcontaminant. Anotherpossiblecaseiswhereseveralliquidphasesarepresent.Inthiscaseit wouldbenecessarytoincludeamassconservationequationandrelatedDarcytypeconstitutivelawforthisotheruidphase.Thisisacommoncasewhen water,oil,andgasarepresentwithintheporousmatrix.Thistypeofsystem hasseenrecentmediaattentionduetothepracticeoffrackingdrivinghigh pressurewaterandchemicalsintoporousrocktoreleaseoilandnaturalgas. Asmentionedpreviouslyinthischapter,anavenueforfutureresearchistoshift thefocusfrommodelingtoanalysis.Theexistenceanduniquenessresultspresented 192

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inChapter7areincomplete,andmoreresearchneedstobecompletedtogiveafull analyticdescriptionofthebehavioroftheseequations.Ofparticularinterestisthe studyofthefullycoupledsystem.Thereareseveralpapersdiscussing stronglycoupled system ofreactiondiusionequationse.g.[3,53].Thesepapersserveasastarting pointtounderstandingtheanalysisforthecoupledsystem. Athirdavenueforfutureresearchistofocusonthenumericalmethodforsolving thecoupledsystem.ThenumericalsolutionspresentedinChapter7werefoundusing Mathematica 's NDSolve package.Thisisageneralpurposenitedierencesolver designedtosolvewideclassesofordinaryandpartialdierentialequations.Evenso, thereareproblemswiththemethodsusedwithinthatpackage.Theforemostissueis theuseofcentraldierencingschemes.Whensolvingadvectiveorhyperbolic-type equationsitisoftenadvantageoustouseupwind-typenumericalschemes.Thisis notpossiblewiththe NDSolve packageandcausessomeissueswiththenumerical solutionsinadvection-dominatedsimulations. Futureresearchforthenumericalmethodcanbeapproachedinseveralways: 1.Theequationsofinterestinthisworkformasystemofconservationlaws,and assuchanitevolumemethodislikelythebestchoice.PeszynskaandYi [60]derivedacellcenterednitedierencemethodandalocallyconservative Euler-Lagrangemethodbasedonthenitedierencemethodforthesaturation equationwiththethird-orderdynamicalcapillarypressureterm.Theseare bothsimilartonitevolumemethods,andthemethodsderivedthereincan possiblyserveasabasisforextensiontothecoupledsystem.Thispaperis ofparticularinterestasthebulkofthenumericaldicultiesinthesaturation equationariseasaresultoftheweightofthethird-orderterm. 2.Thereareotherpackagesavailabletosolvegeneralclassesofpartialdierential equations. COMSOL ComputationalMultiPhysicsisonesuchpackagethatis commonamongstengineeringgroups.Thisisaparticularlynon-mathematical 193

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approachtofutureresearch,but COMSOL andotherpackagescanbeusedtotest manycasesofparametersandtermssuggestedbytheentropyinequality. 3.Thenumericalsimulationsusedtocomparetotheexperimentaldataweresolved inonespatialdimension.Whilethisassumptionisapproximatelyvalidforthe experimentofinterest,theseequationsneedtobecomparedagainstmultidimensionaldata.Oneavenueoffutureresearchistoexplorethenumerical solutiontothesystemandcompareittoatwo-dimensionalexperiment.One suchexperimentcanbefoundin[76].Thisexperimentisofinterestsincemany oftheparametersarethesameasthoseusedinthecolumnexperiment. FinalWords Theinitialpurposeofthisworkwastoexploretheuseofthechemicalpotentialasa modelingtoolinporousmedia.Itwasdemonstratedthatthechemicalpotentialisa powerfulmodelingtoolwhentheunderlyingphysicalprocessesarediusiveinnature. Thedownsidestousingthechemicalpotentialarethatitisindirectlymeasuredand notwidelyunderstood.Withinthisworkthemainadvantagetousingthechemical potentialwastorewritethegas-phasediusionequation.Thisallowedfortheremoval oftheenhancementfactorfromtheclassicaldiusionequation. Inthisworkwederivedthreenewequationsthat,whencoupled,formasetof governingequationsforheatandmoisturetransportinporousmediathatexplains previouslyunexplainedphenomena.Theideasandquestionsproposedwithinthis chaptersetuparesearchagendaforfutureyearsofscholarlywork. 194

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APPENDIXA.MicroscaleNomenclature ThisappendixcontainsnomenclatureforPartI:Pore-Scalemodeling.While thereissomeoverlapinnotationbetweenthetwoparts,thisappendixallowsthenotationinChapter2tostandalone.Theequationreferencesindicatetheapproximate rstinstanceofthesymbolthesearehyperlinkedinthedigitalversionforeaseof use.Thesupportingtextfortheseequationsusuallygivescontextandmoredetail. Superscripts,Subscripts,andOtherNotations j : j th componentof )]TJ/F15 11.9552 Tf 9.299 0 Td [(phase : )]TJ/F15 11.9552 Tf 9.298 0 Td [(phase a;b :dierenceoftwoquantities, a;b = a )]TJ/F15 11.9552 Tf 11.955 0 Td [( b a :boldsymbolvectorquantity :areferencequantityoraquantityevaluatedatareferencestate LatinSymbols c g j :molarconcentrationofthe j th constituentinthegasphase[mol/length 3 ] .2 C j :Massfractionof j th compontnent C j = j = [-].1 D :diusioncoecient[length 2 /time].1-.4 D :diusioncoecientassociatedwiththe -formofFick'slaw[length 2 /time] .1-.4 J g j :uxofspecies j inthegasphase[mass/length 2 -time].1-.4 195

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m j :molarmassofthe j th constituentinphase [mass/mol].7 p j :partialpressureofspecies j inphase [force/length 2 ].10 p :pressureof phase[force/length 2 ].10 R :universalgasconstant[energy/mass-temperature].4 R g j :specicgasconstantforspecies j inthegasphase[energy/masstemperature].3 T :absolutetemperature.3 t :time.15 v j :velocityofspecies j inphase [length/time].1 v :velocityofphase [length/time].1 x g j :molarconcentrationof j th constituentinthegasmixture[-].2 GreekSymbols g v :chemicalpotentialofwatervaporingasphase[energy/mass].3-.4 g v :chemicalpotentialofwatervaporatstandardtemperatureandpressure [energy/mass].7 j :massdensityofspecies j inphase [mass/length 3 ].1 :massdensityofphase [mass/length 3 ].1 sat :massdensityofwatervaporundersaturatedconditions[mass/length 3 ] paragraphbefore.9 196

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APPENDIXB.MacroscaleAppendix B.1Nomenclature ThisappendixcontainsnomenclatureforPartII:Macroscalemodeling.While thereissomeoverlapinnotationbetweenthetwoparts,thisappendixallowsPartII tostandalone.Thisappendixalsoclariesanynotationaldiscrepanciesbetweenthe pore-scaleandmacroscalemodels.Theequationreferencesindicatetheapproximate rstinstanceofthesymbolthesearehyperlinkedinthedigitalversionforeaseof use.Thesupportingtextfortheseequationsusuallygivescontextandmoredetail. Superscripts,Subscripts,andOtherNotations j : j th componentof )]TJ/F15 11.9552 Tf 9.299 0 Td [(phaseonmacroscale : )]TJ/F15 11.9552 Tf 9.298 0 Td [(phaseonmacroscale ^ :denotesexchangefromotherinterface,phase,orcomponent a;b :dierenceoftwoquantities, a;b = a )]TJ/F15 11.9552 Tf 11.955 0 Td [( b j :porescalepropertyofcomponent j a :boldsymbolvectorquantity A :secondordertensormatrix 0 or :referencestate LatinSymbols a :ttingparameterforenhanceddiusionmodel[-].71 b j ;b :Externalentropysource[energy/mass-time-temperature].37 197

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C j :Massfractionof j th compontnent[-].23 C l u :Coecientsof r u termsingeneralizedDarcy'slaw u = S;T;' .44 C s :RightCauchy-Greentensorofthesolidphase= F s T F s [-].4 C s :ModiedrightCauchy-Greentensor= F s T F s [-].4 D :diusivitytensor[length 2 /time].81 D :generalizeddiusivityfunction[length 2 /time].69a d :Rateofdeformationtensor= r v sym [1/time].40 e j ;e :energydensity[energy/mass].34 ^ e j :rateofmasstransferfromphase tocomponent j inphase perunit massdensity[1/time].21 ^ e :rateofmasstransferfromphase tophase perunitmassdensity [1/time].24 F s :Deformationgradientofthesolidphase[-].5 F s :Modieddeformationgradientofthesolidphase[-].5 g g j ; g :gravity[length/time 2 ].14 h j ;h :externalsupplyofenergy[energy/mass-time].34 ^ i j :rateofmomentumgainduetointeractionwithotherspecieswithinthe samephaseperunitmassdensity[force/mass].28 J s :Jacobianofthesolidphase[-].3 K :hydraulicconductivitytensorfor phase[length/time].36 198

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K :thermalconductivitytensor[energy/mass-time-temperature].39 m :vanGenuchtenparameter m =1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 =n [-].51a n :vanGenuchten pore-sizedistribution parameter n =1 = )]TJ/F20 11.9552 Tf 10.527 0 Td [(m [-].51a p :classicalpressureinthe phase[force/length 2 ].29 p :crosscouplingclassicalpressure[force/length 2 ].41 p c = p g )]TJ/F20 11.9552 Tf 11.955 0 Td [(p ` :capillarypressure[force/length 2 ].53 p :thermodynamicpressure[force/length 2 ].50 p :crosscouplingthermodynamicpressure[force/length 2 ].47 q :heatuxfor phase[energy/length 2 -time].35 q :totalheatux[energy/length 2 -time].87f q :Darcyuxfor phase[length/time].67 ^ Q j :rateofenergygainduetointeractionwithotherspecieswithinthe samephaseperunitmassdensitynotduetomassormomentumtransfer [energy/mass-time].34 ^ Q j :energytransferfromphase toconstituent j inphase perunitmass densitynotduetomassormomentumtransfer[energy/mass-time].34 r :microscalespatialvariable[length].2 ^ r j :rateofmassgainduetointeractionwithotherspecieswithinthesame phaseperunitmassdensity[1/time].21 R :gasconstant[energy/mol-temperature].48 199

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R g j :specicgasconstantfor j th constituentingasphase[energy/masstemperature].82 R :resistivitytensor[time/length].36 S = l = :liquidsaturation[-].2 t :time T :absolutetemperature.39 t j :partialstresstensorforthe j th constituentinthe phase[force/length 2 ] .28 t :totalstresstensorfor phase[force/length 2 ].29 ^ T j :rateofmomentumtransferthroughmechanicalinteractionsfromphase tothe j th constituentofphase [force/length 3 ].28 ^ T :rateofmomentumtransferthroughmechanicalinteractionsfromphase tophase [force/length 3 ].29 v :velocityofthe phaserelativetoaxedcoordinatesystem[length/time] .24 v j ; = v j )]TJ/F40 11.9552 Tf 11.955 0 Td [(v :diusivevelocity[length/time].40 v ;s = v )]TJ/F40 11.9552 Tf 11.955 0 Td [(v s :velocityrelativetosolidphasevelocity[length/time].40 w j :velocityofconstituent j atinterfacebetweenphases and [length/time] .10 GreekSymbols :phase 200

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:vanGenuchtenparameter[-].61 :phase t :Diracdeltafunction[-] A :Portionofthe )]TJ/F15 11.9552 Tf 13.2 0 Td [(interfaceinREV.10 :volumetriccontentof phasepervolumeofREV[-].15 = ` + g :porosity[-].1 :specicentropyof phase[energy/mass-temperature].38 ^ j :entropygainduetointeractionwithotherspecieswithinthesamephase perunitmassdensity[energy/mass-time-temperature].38 :enhancementfactorfordiusion[-].70 )]TJ/F21 7.9701 Tf 7.314 4.339 Td [( :macroscaleGibbspotential[energy/mass].61 :indicatorfunctionwhichis1ifinphase andzerootherwise.10 :permeability[length 2 ].68 ^ :rateofentropyproductionfor phase[entropy/time].38 j :Lagrangemultiplierforthecontinuityequationofphase .12 N :Lagrangemultiplierforthe N th termdependencerelationofthecomponentsinphase .12 = p g v =p g :ratioofpartialpressuretobulkpressureingasphase[-].43 :dynamicviscosityforphase [force-time].68 j :macroscalechemicalpotentialof j th speciesin phase[energy/mass] .56 201

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:kinematicviscosityforphase .69 ^ j :Entropytransferthroughmechanicalinteractionsfromphase tophase perunitmass[energy/mass-time-temperature].38 :wettingpotentialof phase[force/length 2 ].51 :crosscouplingwettingpotential[force/length 2 ].48 :relativehumidity[-].40 :specicHelmholtzpotentialofthe phase[energy/mass].39 :massdensityof phasemass pervolumeof [mass/length 3 ].22 j :massdensityof j th constituentin phasemass j pervolume [mass /length 3 ].22 sat :saturatedvapordensity[force/length 2 ].41 :tortuosity[-].70 :scalingcoecientfordynamicsaturationterm[-].53 B.2UpscaledDenitions Denitionsofbulkphase,species,andaveragedvariablesresultingfromupscaling.Recallthatanoverbarindicatesamassaveragedquantityandangularbrackets indicateavolumeaveragedquantity. h i = 1 j V j Z V dv and = 1 h i j V j Z V dv b j = b j B.1 b = N X j =1 C j b j B.2 202

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C j = j B.3 e j = e j + 1 2 v j v j )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 2 v j v j ; B.4 e = N X j =1 C j e j + 1 2 v j ; v j ; B.5 ^ e j = j V j Z A j )]TJ/F40 11.9552 Tf 5.48 -9.684 Td [(w j )]TJ/F40 11.9552 Tf 11.955 0 Td [(v j n da B.6 ^ e = N X j =1 ^ e j B.7 g j = g j B.8 g = N X j =1 C j g j B.9 h j = h j + g j v j )]TJ/F40 11.9552 Tf 11.956 0 Td [(g j v j ; B.10 h = N X j =1 C j h j + g j v j ; B.11 ^ i j = j ^ i j + ^ r j v j )]TJ/F40 11.9552 Tf 11.955 0 Td [(v j ^ r j B.12 q j = q j + t j v j )]TJ/F40 11.9552 Tf 11.956 0 Td [(t j v j + j v j e j + 1 2 v j v j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j v j e j + 1 2 v j v j ; B.13 q = N X j =1 q j + t j v j ; )]TJ/F20 11.9552 Tf 11.955 0 Td [( j e j + 1 2 v j ; v j ; v j ; B.14 ^ Q j = j ^ Q j + i j v j )]TJ/F26 11.9552 Tf 11.955 16.856 Td [( ^ i j + ^ r j v j )]TJ/F40 11.9552 Tf 11.955 0 Td [(v j ^ r j v j + ^ r j e j + 1 2 v j v j )]TJETq1 0 0 1 291.85 215 cm[]0 d 0 J 0.478 w 0 0 m 9.983 0 l SQBT/F15 11.9552 Tf 292.374 204.604 Td [(^ r j e j + 1 2 v j v j # ; B.15 ^ Q j = j V j Z A q j + t j v j + j e j + 1 2 v j v j )]TJ/F40 11.9552 Tf 5.48 -9.684 Td [(w j )]TJ/F40 11.9552 Tf 11.955 0 Td [(v j n da: )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( e j )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 v j v j Z A j )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(w j )]TJ/F40 11.9552 Tf 11.955 0 Td [(v j n da )]TJ/F40 11.9552 Tf 9.299 0 Td [(v j Z A t j + j v j )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(w j )]TJ/F40 11.9552 Tf 11.955 0 Td [(v j n da B.16 203

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^ Q = N X j =1 ^ Q j + ^ T j v j ; +^ e j e j + 1 2 v j ; v j ; B.17 ^ r j = j ^ r j B.18 t j = t j + j v j v j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j v j v j B.19 t = N X j =1 )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(t j )]TJ/F20 11.9552 Tf 11.955 0 Td [( j v j ; v j ; B.20 ^ T j = j V j Z A t j + j v j )]TJ/F40 11.9552 Tf 5.479 -9.684 Td [(w j )]TJ/F40 11.9552 Tf 11.955 0 Td [(v j n da )]TJ/F40 11.9552 Tf 9.299 0 Td [(v j Z A j )]TJ/F40 11.9552 Tf 5.48 -9.684 Td [(w j )]TJ/F40 11.9552 Tf 11.956 0 Td [(v j n da # B.21 ^ T = N X j =1 ^ T j +^ e j v j ; B.22 v j = v j B.23 v = C j v j B.24 j = j B.25 = N X j =1 C j j B.26 ^ j = j ^ j + ^ r j j )]TJETq1 0 0 1 277.655 328.459 cm[]0 d 0 J 0.478 w 0 0 m 9.983 0 l SQBT/F15 11.9552 Tf 278.179 318.063 Td [(^ r j j B.27 ^ j = ^ j B.28 ^ = N X j =1 ^ j B.29 j = j B.30 = N X j =1 C j j B.31 j = j B.32 = N X j =1 j B.33 B.3IdentitiesNeededtoObtainInquality3.40 204

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N X j =1 j T D j j Dt = j T D j Dt + T ^ e + N X j =1 1 T v j ; r j j )]TJ/F20 11.9552 Tf 13.151 8.087 Td [( j T ^ e j )]TJ/F20 11.9552 Tf 10.494 8.088 Td [( j T ^ r j )]TJ/F20 11.9552 Tf 13.15 8.088 Td [(" j T j r v j ; B.34 N X j =1 j T j D j T Dt = T D T Dt + N X j =1 j T j v j ; r T B.35 N X j =1 T t j : r v j = N X j =1 T t j : r v j ; + T t j : r v j B.36 N X j =1 X 6 = ^ j = )]TJ/F21 7.9701 Tf 16.14 14.944 Td [(N X j =1 X 6 = ^ e j j B.37 N X j =1 ^ Q j = )]TJ/F26 11.9552 Tf 11.291 11.357 Td [(X 6 = ^ i j v j ; +^ r j j + T j 1 2 v j ; 2 B.38 N X j =1 X 6 = ^ Q j = )]TJ/F26 11.9552 Tf 11.291 11.358 Td [(X X 6 = ^ T v ;s + 1 2 ^ e v ;s 2 + N X j =1 ^ T j v j ; + 1 2 ^ e j v j ; 2 + X 6 = N X j =1 ^ e j j + T j B.39 205

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APPENDIXC.ExploitationoftheEntropyInequality{AnAbstract Perspective Thisshortappendixismeanttogiveabriefandabstractdescriptionofhowthe exploitationoftheentropyinqualityworks.Byabstract"wemeanthatwewillnot assignanyphysicalmeaningtothevariables.Insteadwewillsimplystatehowthe variablesrelatetoeachotherandhowtheyrelatetothefullsetofchosenindependent variables.Thesecondarypurposeofthisappendixistomakeclearafewassumptions relatedtoconstitutiveequationsthatarenecessaryinorderfortheexploitationof theentropyinequalitytobesuccessfull.Weconcludewithaninequalitythatdictates howthelinearizationoftheconstitutiverelationsmustbehaveinordernottoviolate thesecondlawofthermodynamics.Thisissimilartothe1968NobelPrizewinning analysisbyOnsager,whoshowedthe reciprocalrelations thatmustholdatequilibrium forirreversibleprocesses. Let S bethesetofallindependentvariablesfortheHelmholtzpotential.This setdenesthephysicalsystemofinterest.Denethefollowingsets: f x j g :=thesetofallvariablesthatareneitherconstitutivenorindependent. Examplestypicallyinclude T; r T; r l j ; f y k g :=thesetofallconstitutivevariableswhicharezeroatequilibrium.Examplestypicallyinclude_ ,^ e j ,and^ r j f ~ y g :=thesetofallconstitutivevariableswhicharenotzeroatequilibrium. Examplestypicallyinclude ^ T j ,and ^ i j f z l g :=thesetofallvariablesthatarezeroatequilibrium.Examplestypically include r T v ;s v j ; ,and d 206

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Sinceeach z l isanindependentvariableitisclearthat f z l g S .Furthermorewe observethat f x j g S = ; : Theconstitutivevariables,ontheotherhand,areknown tobefunctionsofvariablesin S andassuchthestatementthat f y k g[f ~ y g S = ; iseasilymisinterpreted.Itisatruestatementthat f y k g[f ~ y g S = ; ,anditis correctnottochooseconstitutivevariablesasindependentvariables.Theconfusion isinthefactthat y k = y k S forall k andforall Therateofentropygeneration, ^ can bewrittenasalinearcombinationofthe variablesfrom f x j g f y k g ,and f z l g ,wherethecoecientsarefunctionsofvariables from S .Thatis, 0 X ^ = ^ = X j x j X j + X k y k Y k + X l z l Z l 0C.1 where X j = X j S; ~ Y S ;Y k = Y k S; ~ Y S ; and Z l = Z l S; ~ Y S : C.2 Thisisnottheonlywaytoalgebraicallyrearrange ^ ,butthisiswhatiscommonly doneduringtheexploitationprocess. WenowuseinequalityC.1toderiveequationsthatholdforalltime,atequilibrium,andnearequilibrium. C.1ResultsthatHoldForAllTime Wehave nocontrol overthevariables x j sincetheyareneitherconstitutivenor independent.Thismeansthattheycouldbepositiveornegative,largeorsmall.In orderforthesecondlawofthermodynamicstoholdforalltime,thecoecients X j mustthereforebezeroforalltime.Thisimpliesthat ^ = X k y k Y k + X l z l Z l 0 : C.3 C.2EquilibriumResults 207

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Thedenitionofequilibriumis thestateatwhichallofthevariables f y k g are zero .Thisdenitionisbasedonphysicalintuitionandwillvarydependingonthe systemofinterest.Fromthermodynamics,therateofentropygenerationmustbe minimizedatequilibrium.Thisimpliesthatthegradientoftheentropygeneration functionmustbethezerovectorasunderstoodwith S astheindependentvariables forthegradient. 0= @ ^ @s i eq forall i: C.4 Takingthispartialderivativeoftheright-handsideofC.3weseethat 0= @ ^ @s i eq = X k y k @Y k @s i + X k Y k @y k @s i + X l z l @Z l @s i + X l Z l @z l @s i # eq : C.5 Since z l j eq =0= y k j eq forall l;k and @z l @s i = il since z l 2 S 8 l weget 0= @ ^ @s i eq = X k Y k @y k @s i + Z i il # eq : C.6 Atthispointwemakeanassumptionthatgreatlyaectstheconstitutivevariables. Assumption: Atequilibriumwemusthave @y k @z l eq =0forall l;k C.7 Underthisassumptionitisclearthat Z l =0atequilibriumforall l .Noticethat thissaysnothingaboutwhen s i 62f z l g .Fromthisargument,eachequation Z l =0 givesaconstraintonsomeofthevariablesin S .Theassumptionmadecanbeviewed asafurtherrestrictionontheconstitutivevariables,butitisnotclearwhetherthis assumptionisphysical. C.3NearEquilibriumResults 208

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Forthenearequilibriumresultsweconsidertwotypesofvariables:variablesthat arezeroatequilibriumandconstitutivevariables.Ineachcasewelinearizeabout theequilibriumstate.Atypicallinearizationresultforvariableswhicharezeroat equilibriumis Z l j n:eq = Z l j eq | {z } =0 + X m @Z l @z m eq z m + X n @Z l @y n eq y n + = X m @Z l @z m eq z m + X n @Z l @y n eq y n + : C.8 Thevalueof Z l j eq iszerobytheabovearguments,andthepartialderivativesare nowfunctionsofalloftheothervariableswhicharenotzeroatequilibrium: C lm := @Z l @z m eq = @Z l @z m eq 1 ; 2 ;::: ; C.9 D ln := @Z l @y n eq = @Z l @y n eq 1 ; 2 ;::: C.10 for n 2 S nf z l g : Writtenmoresimply Z l j n:eq = C lm z m + D ln y n C.11 wherethesummationsareimplicitoverrepeatedindices. Fortheconstitutivevariableswedoasimilarlinearization,butnotethatthe equilibriumstateisnotnecessarilyzero.Therefore, Y k j n:eq = Y K j eq + X p @Y k @y p eq y p + X q @Y k @z q eq z q + : C.12 Makingsimilardenitionsasbefore, E kp := @Y k @y p eq C.13 F kq := @Y k @z q eq ; C.14 thelinearizationresultfortheconstitutiveequationsis Y k j n:eq = Y k j eq + E kp y p + F kq z q C.15 209

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wherethesummationsareimplicitoverrepeatedindices. Thetroublehereisthatwemusthavesomeinformationabouttheequilibrium stateoftheconstitutivevariable.Thepresumptionthatthisiszeromaybenonphysical.Anexampleofthisisthecapillarypressurerelationshipderivedinmultiphaseunsaturatedmedia. p c = p c j eq + l ; C.16 where p c isthecapillarypressure, = @p c =@ l ,andtheequilibriumcapillarypressure isgivenasafunctionofsaturation p c j eq = p c S viathevanGenuchtenapproximation. Thisequilibriumconstitutiveequationisknownnottobezero.Inothersystems theissuemaybemoresubtle,butinanycaseoneneedstohavesomeinformation whetherfromexperimentsorfromothertheorytodenetheequilibriumstateof theconstitutivevariable. C.4LinearizationandEntropy ConsideragainequationC.3,butnowsubstitutethelinearizedresultsinto Y k and Z l 0 ^ = y k Y k + z l Z l = y k Y k j eq + E kp y p + F kq z q + z l C lm z m + D ln y n = y k Y k j eq + y k E kp y p + y k F kq z q + z l C lm z m + z l D ln y n C.17 summationsareagaintakenoverrepeatedindices.Recognizingthequadraticterms asmatrixproductsandrewritinginblockmatrixformgives 0 y k Y k j eq + 0 B @ y z 1 C A T 0 B @ E F D C 1 C A 0 B @ y z 1 C A = y k Y k j eq + T A C.18 NoticethatintheabsenceofconstitutivevariablesC.18simpliesto 0 z T C z : C.19 210

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Simplystatedthismeansthat C mustbepositivesemideniteinorderforthesecond lawofthermodynamicstohold.ThisisOnsager'sNobelPrizewinningresult.Recall that Z l j n:eq = C lm z m .Ifwetake,forexample, z m = r T andobservethat Z l j n:eq isminustheheatuxnearequilibriumthecoecientintheentropyinequality associatedwith r T isminustheheatux,thenthe l )]TJ/F20 11.9552 Tf 12.366 0 Td [(m entryin C istheheat uxtensor.Onsager'sresultdictatesthepositivityoftheheatuxtensorandthe linearizedresultgiveFourier'slaw: )]TJ/F40 11.9552 Tf 9.298 0 Td [(q = K r T .Inotherwords,thereisnoaccident thatmanyphysicallaws"takethesameformasFourier'slaw;theyarearesultof thenon-negativityof C andtheentropyinequality.Thisisasimpleexample,but itshouldhelptoelucidatetheproblemthatariseswhenconstitutiveequationsare introduced. ReturningtoequationC.18weseethatitisnotimmediatelyobviousthat A needstobepositivesemidenite.Infact,theonlywaythatwecanguaranteethat A hasthispropertyisif y k Y k j eq 0.Thatis,theremustbeaphysicalrestrictionon y k Y k j eq that,whenviolated,oneperceivesnonphysicalresults. Togiveaphysicalexampleofthiswereturntothecapillarypressureexample. Inthiscasewehave y k =_ l and Y k j eq = p c S whereweareignoringallother constitutiveequationsorwearetaking Y r j eq =0forall r 6 = k .Therestriction derivedhereinstatesthat p c S l 0foralltime.Thetimederivativecanclearly takeeithersign,butwhatthisseemstobeindicatingisthatindrainagewhen_ l < 0 theequilibriumcapillarypressuremustbepositive,andinimbibitionwhen_ l > 0 theequilibriumcapillarypressuremustbenegative.Thisisabitcontradictorysince drainage"andimbibition"arenon-equilibriumphenomena,andassuchitisnot possibletomeasurethe equilibrium capillarypressureatthesestates.Thisleavesus withaconundrum:Isthereafundamentalmisinterpretationofthecapillarypressure inthisexample,oristhereisadeep-seatedawintheexploitationoftheentropy inequality. 211

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APPENDIXD.SummaryofEntropyInequalityResults Thefollowingisaconcisecollectionoftheresultsderivedfromtheentropyinequality.Thisappendixistobeusedforreferencewhenbuildingthemacroscale models. D.1ResultsthatHoldForAllTime Helmholtzpotentialandentropyareconjugatevariablesequation.14 @ @T = )]TJ/F20 11.9552 Tf 9.299 0 Td [( : D.1 Lagrangemultiplierforuidphaseequation.15 j = X " @ @ j : D.2 Lagrangemultiplierforthedependenceofthediusivevelocitiesonthe N th speciesequation.16 N = )]TJ/F15 11.9552 Tf 13.555 8.088 Td [(1 N X j =1 t j + j j I + I : D.3 Solidphasepressureequation.19 p s = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 3 tr )]TJ/F40 11.9552 Tf 5.48 -9.684 Td [(t s = )]TJ/F20 11.9552 Tf 10.494 8.088 Td [(J s s X @ @J s : D.4 Solidphasestressequation.23 t s = )]TJ/F20 11.9552 Tf 9.298 0 Td [(p s I + t s e + l s t l h + g s t g h D.5 where t s e =2 s F s @ s @ C s )]TJETq1 0 0 1 330.507 90.843 cm[]0 d 0 J 0.478 w 0 0 m 10.145 0 l SQBT/F40 11.9552 Tf 330.507 80.967 Td [(F s T )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 3 s @ s @ C s : C s I ; D.6a 212

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t l h =2 l F s @ l @ C s )]TJETq1 0 0 1 329.214 708.955 cm[]0 d 0 J 0.478 w 0 0 m 10.145 0 l SQBT/F40 11.9552 Tf 329.214 699.078 Td [(F s T )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 l @ l @ C s : C s I ; D.6b t g h =2 g F s @ g @ C s )]TJETq1 0 0 1 330.907 669.104 cm[]0 d 0 J 0.478 w 0 0 m 10.145 0 l SQBT/F40 11.9552 Tf 330.907 659.227 Td [(F s T )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 g @ g @ C s : C s I : D.6c D.2EquilibriumResults Fluidpressuresequations.29,.41,.42,and.47-.52 p = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 3 tr )]TJ/F40 11.9552 Tf 5.48 -9.683 Td [(t = N X j =1 X j @ @ j D.7 p = N X j =1 j @ @ j ; k ;" ; m D.8 p = X p D.9 p := )]TJ/F20 11.9552 Tf 9.299 0 Td [(" @ @" ;" k ;" k D.10 := @ @" ; k ; k D.11 p = p + D.12 p := X p D.13 := X D.14 p = p + : D.15 Momemtumtransferbetweenphasesequation.31 )]TJ/F26 11.9552 Tf 11.291 13.27 Td [( ^ T s + ^ T = @ @" )]TJ/F20 11.9552 Tf 11.955 0 Td [(p r + @ @" r + N )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 X j =1 @ @C s j r C s j + @ @ l r + @ @ )]TJ/F21 7.9701 Tf 16.805 14.944 Td [(N X j =1 @ @ j + s s @ s @ j r j + N X j =1 @ @ j r j 213

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+ @ @J s r J s + @ @ C s : r )]TJETq1 0 0 1 399.952 711.915 cm[]0 d 0 J 0.478 w 0 0 m 10.601 0 l SQBT/F40 11.9552 Tf 399.952 702.039 Td [(C s ; D.16 Momentumtransferbetweenconstiuentsequation.34 X 6 = ^ T j + ^ i j = )]TJ/F43 11.9552 Tf 9.298 0 Td [(r j j + j r j + r j : D.17 Partialheatuxequation.35 X q = 0 D.18 Chemicalpotentialdenitionequations.56and.57 j = @ T @ j ;" ; k ; m = X @ @ j ;" ; k ; m D.19 = + j D.20 Masstransferequation.60 l j eq = g j eq D.21 D.3NearEquilibriumResults Momentumtransferbetweenphasesequation.36 X 6 = ^ T near = X 6 = ^ T eq )]TJ/F26 11.9552 Tf 11.955 9.684 Td [()]TJ/F20 11.9552 Tf 5.48 -9.684 Td [(" 2 R v ;s : D.22 Momentumtransferbetweenconstituentsequation4.38 X 6 = ^ T j + ^ i j near = X 6 = ^ T j + ^ i j eq )]TJ/F20 11.9552 Tf 11.955 0 Td [(" j R j v j ; : D.23 Partialheatuxequation.39 X q = )]TJ/F40 11.9552 Tf 9.299 0 Td [(K r T D.24 214

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D.4ConstitutiveEquations Darcy'slaw { Pressureformulationequation.66 R )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" v ;s = )]TJ/F20 11.9552 Tf 9.299 0 Td [(" r p )]TJ/F20 11.9552 Tf 11.956 0 Td [( r )]TJ/F20 11.9552 Tf 11.955 0 Td [( r + g + N X j =1 @ @ j + s s @ s @ j r j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" N X j =1 @ @ j r j )]TJ/F20 11.9552 Tf 11.955 0 Td [(" @ @J s r J s )]TJ/F20 11.9552 Tf 11.955 0 Td [(" @ @ C s : r )]TJETq1 0 0 1 370.985 485.288 cm[]0 d 0 J 0.478 w 0 0 m 10.601 0 l SQBT/F40 11.9552 Tf 370.985 475.411 Td [(C s + r : d : D.25 { Chemicalpotentialformulationequation.76 R )]TJ/F20 11.9552 Tf 5.479 -9.684 Td [(" v ;s = )]TJ/F21 7.9701 Tf 16.14 14.944 Td [(N X j =1 )]TJ/F20 11.9552 Tf 5.48 -9.684 Td [( j r j )]TJ/F20 11.9552 Tf 11.955 0 Td [( r T + g + 1 r : d D.26 Fick'slawequation.80 j R j v j ; = )]TJ/F20 11.9552 Tf 9.298 0 Td [( j r j + j g : D.27 Totalheatuxequation.92 q = )]TJ/F40 11.9552 Tf 9.299 0 Td [(K r T )]TJ/F26 11.9552 Tf 13.979 11.357 Td [(X = l;g N X j =1 j j + T v ;s # : D.28 215

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APPENDIXE.DimensionalQuantities Thisappendixcontainstypicalvaluesforthequantitiesfoundinthemacroscale heatandmoisturetransportmodel.Notethatsincemanyofthetablesarelargeso someareturnedsidewaysandsomearebumpedtodierentpagesbydefault. TableE.1:Dimensionalquantities Symbol Quantity Dimensions referencevalue liquidwater gasair volumefraction )]TJETq1 0 0 1 364.076 493.441 cm[]0 d 0 J 0.398 w 0 0 m 0 14.446 l SQBT/F23 11.9552 Tf 402.298 497.775 Td [()]TJETq1 0 0 1 449.818 493.441 cm[]0 d 0 J 0.398 w 0 0 m 0 14.446 l SQBT/F23 11.9552 Tf 488.04 497.775 Td [()]TJETq1 0 0 1 535.56 493.441 cm[]0 d 0 J 0.398 w 0 0 m 0 14.446 l SQq1 0 0 1 108.538 478.316 cm[]0 d 0 J 0.398 w 0 0 m 0 15.124 l SQBT/F20 11.9552 Tf 127.75 482.65 Td [( density ML )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 1000 kg m 3 1 kg m 3 g v vapordensity ML )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 )]TJETq1 0 0 1 449.818 463.192 cm[]0 d 0 J 0.398 w 0 0 m 0 15.124 l SQBT/F15 11.9552 Tf 463.928 467.526 Td [(2 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 kg m 3 g a dryairdensity ML )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 )]TJETq1 0 0 1 449.818 448.068 cm[]0 d 0 J 0.398 w 0 0 m 0 15.124 l SQBT/F15 11.9552 Tf 478.194 452.401 Td [(1 : 2 kg m 3 T temperature K 298 : 15 K 298 : 15 K R g a gasconstantair L 2 t )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 K )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 )]TJETq1 0 0 1 449.818 417.796 cm[]0 d 0 J 0.398 w 0 0 m 0 15.826 l SQBT/F15 11.9552 Tf 468.664 423.468 Td [(286 : 9 J kg K R g v gasconstantvapor L 2 t )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 K )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 )]TJETq1 0 0 1 449.818 401.97 cm[]0 d 0 J 0.398 w 0 0 m 0 15.826 l SQBT/F15 11.9552 Tf 468.664 407.643 Td [(461 : 5 J kg K D diusioncoecient L 2 t )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 2 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(9 m 2 s 2 : 5 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 m 2 s g v chem.potentialvapor L 2 t )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 )]TJETq1 0 0 1 449.818 369.991 cm[]0 d 0 J 0.398 w 0 0 m 0 15.826 l SQBT/F23 11.9552 Tf 456.445 375.663 Td [()]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 27 10 7 J kg g a chem.potentialair L 2 t )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 )]TJETq1 0 0 1 449.818 354.165 cm[]0 d 0 J 0.398 w 0 0 m 0 15.826 l SQBT/F15 11.9552 Tf 461.094 359.837 Td [(1 : 47 10 7 J kg g gravity Lt )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 9 : 81 m s 2 9 : 81 m s 2 permeability L 2 seeTab.E.2 seeTab.E.2 dynamicviscosity ML )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 t )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 Pa s 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 Pa s specicentropy L 2 t )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 K )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 3886 J kg K 6519 J kg K M evaporationcoecient tL )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 216

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TableE.2:Typicalvaluesofhydraulicconductivity K forwaterandair,andassociatedvaluesforpermeability .Note that K = g= where g =1 kg=m 3 ; l =1000 kg=m 3 l =10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 Pa s ,and g =10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 Pa s .ModiedfromBearpg.136[5] K [ m=s ]water 10 0 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(3 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(6 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(7 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(8 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(9 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(10 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(11 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(12 K [ m=s ]air 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(4 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(6 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(7 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(8 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(9 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(11 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(12 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(13 [ m 2 ] 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(7 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(8 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(9 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(11 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(12 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(13 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(14 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(15 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(16 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(17 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(18 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(19 Permeability pervious semipervious impervious Aquifer good poor none SandandGravel cleangravel cleansand nesand ClayandOrganic peat stratiedclay unweatheredclay Rocks oilrocks sandstone limestone granite K [ cm=s ]water 10 2 10 1 10 0 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(1 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(2 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(4 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(6 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(7 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(8 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(9 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 K [ cm=s ]air 10 1 10 0 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(1 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(2 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(5 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(6 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(7 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(8 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(9 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(10 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(11 [ cm 2 ] 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(3 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(4 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(5 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(6 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(7 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(8 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(9 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(10 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(11 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(12 10 )]TJ/F18 7.9701 Tf 6.587 0 Td [(13 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(14 10 )]TJ/F18 7.9701 Tf 6.586 0 Td [(15 217

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