Citation

## Material Information

Title:
An analysis of the stochastic approaches to the problems of flow and transport in porous media
Creator:
Dean, David W
Place of Publication:
Denver, Colo.
Publisher:
Publication Date:
Language:
English
Physical Description:
vi, 227 leaves : illustrations ; 29 cm

## Thesis/Dissertation Information

Degree:
Doctorate ( Doctor of Philosophy)
Degree Grantor:
Degree Divisions:
Department of Mathematical and Statistical Sciences, CU Denver
Degree Disciplines:
Applied Mathematics
Committee Chair:
Russell, Thomas F.
Committee Members:
Franca, Leopoldo P.
Manteuffel, Thomas A.
McCormick, Steven F.

## Subjects

Subjects / Keywords:
Fluid dynamics ( lcsh )
Porosity -- Mathematical models ( lcsh )
Stochastic analysis ( lcsh )
Fluid dynamics ( fast )
Porosity -- Mathematical models ( fast )
Stochastic analysis ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

## Notes

Bibliography:
Includes bibliographical references (leaves 220-227).
General Note:
Submitted in partial fulfillment of the requirements for the degree, Doctor of Philosophy, Applied Mathematics.
Statement of Responsibility:
by David W. Dean.

## Record Information

Source Institution:
Holding Location:
Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
36474144 ( OCLC )
ocm36474144
Classification:
LD1190.L622 1996d .D43 ( lcc )

Full Text
AN ANALYSIS OF THE STOCHASTIC
APPROACHES TO THE PROBLEMS OF FLOW
AND TRANSPORT IN POROUS MEDIA
by
DAVID W. DEAN
B. S., Illinois State University, 1967
M. S., University Of Illinois, 1969
A thesis submitted to the
University of Colorado at Denver in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics
1997

This thesis for the Doctor of Philosophy
degree by
David W. Dean
has been approved
by
Thomas F. Russell
Leopoldo P. Franca
Thomas A. Manteuffel
Steven F. McCormick
Date

Dean, David W. (Ph.D., Applied Mathematics)
An Analysis Of The Stochastic Approaches To The Problems
Of Flow And Transport In Porous Media
Thesis directed by Professor Thomas F. Russell
ABSTRACT
One need in the current theory of subsurface transport in porous
media is an improved understanding of the basic transport physics in highly
heterogeneous subsurface environments using models that are valid at multiple
scales. The thesis addresses this problem by first developing a theoretical back-
ground for the spectral representation of stochastic processes which are then
used to illustrate the more common aspects of the theoretical descriptions of
dispersion. The analysis shows how the dispersion tensor in the homogeneous
case must be modified in order to include mildly heterogeneous permeability
fields and provides a transformation law for the conversion of the spectrum of
velocity perturbations to the spectrum of log hydraulic conductivities. This
theoretical connection is important because in Chapter II a Lagrangian ap-
proach is used to develop a description of dispersion in terms of the covariance
of the hydraulic conductivities using a particle tracking algorithm. Chapter III
describes the numerical methods used to implement the algorithm. Chapter IV
treats the transport equation using stochastic calculus, specifically Itos lemma,
from which weak formulations of the mean and covariance equations can be de-
rived. Chapter V considers the application of the theory of stochastic evolution
equations to the problem of transport. By allowing both the dispersion and
velocity to have random components, the evolution equation can be split into
deterministic and stochastic parts. Using semigroup methods, the solution is
given in terms of a Neumann expansion. Finally, Chapter VI uses the operator
splitting method of Chapter V to illustrate a stochastic finite element method
for solving the transport equation that uses the Karhunen-Loeve expansion,
the Galerkin method and the Homogeneous Chaos spaces of Wiener.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication.
Signed.
m
Thomas F. Russell

CONTENTS
Chapter
1. Introduction................................................ 1
1.1 Overview.................................................... 1
1.2 Stochastic Processes....................................... 13
1.3 Stochastic Measures........................................ 16
1.4 Process With Orthogonal Increments......................... 21
1.5 Spectral Representation.................................... 24
1.6 Space Correlations And Space Spectra....................... 27
1.7 Ergodicity ................................................ 29
1.8 Stochastic Solute Transport And Dispersion................. 31
1.10 Velocity/Permeability Covariance Relationship............. 39
1.11 Summary................................................... 43
2. Dispersivity Coefficients Time And Distance Forms ........ 45
2.1 Time Dependent Dispersivity Coefficients................... 45
2.1.1 Introduction........................................ 45
2.1.2 Transport Equation.................................. 46
2.1.3 Spatial Moments Of The Solute Concentration . 46
2.2 Stochastic Differential Equations.......................... 53
2.2.1 Introduction........................................ 53
2.2.2 Integral Of A Stochastic Process.................... 53
2.2.3 Wiener Process (Brownian Motion) ................... 54
2.2.4 Stochastic Integration.............................. 55
2.2.5 Types Of Stochastic Integrals....................... 56
2.3 The Concentration Equation................................. 56
2.3.1 The Lagrangian Approach ............................ 57
2.3.2 Basic Form Of Transport Equation.................... 61
2.3.3 Solution Of Basic Transport Equation................ 64
2.3.4 Dispersion As Velocity Covariances.................. 69
2.3.5 Dagans Approach.................................... 73
2.4 Distance Dependent Dispersivity Coefficients............... 74
2.4.1 Local Grid Block Dispersivity Coefficients.......... 75
2.4.2 Global Dispersivity Coefficients.................... 81
2.4.3 Molecular Diffusion................................. 86
2.5 Random Variable Generation................................. 86
2.5.1 Independent Random Variables........................ 86
IV

90
93
95
95
95
99
102
104
104
104
106
108
110
116
119
120
122
124
128
131
132
138
138
138
139
142
147
150
151
155
172
174
174
185
187
188
203
214
215
215
2.5.2 Correlated Random Variables..........
2.6 Summary....................................
Methods Of Solution.............................
3.1 General....................................
3.1.1 Simultaneous Solution................
3.1.2 The Mixed Model......................
3.1.3 The Spaces Qh And 14.................
3.2 2D Finite Element Solution.................
3.2.1 General..............................
3.2.2 Finite Elements .....................
3.2.3 Rectangular Elements.................
3.2.4 Numerical Integration................
3.2.5 Groundwater Flow Equation............
3.2.6 Matrix Assembly......................
3.2.7 Velocity Field ......................
3.2.8 The Transport Equation ..............
3.2.9 Imposition Of The Boundary Condition
3.2.10 The Boulder Experiments.............
3.3 Summary....................................
Moment Equations................................
4.1 Moments Derived From Distributed Parameters
4.2 An Ito Calculus Approach...................
4.2.1 System Definition....................
4.2.2 Types Of Problems ...................
4.2.3 Existence Theory ....................
4.2.4 Stochastic Integration...............
4.2.5 Itos Lemma In Hilbert Space.........
4.2.6 Small o Notation.....................
4.2.7 Hilbert Space Structures ............
4.2.8 Moment Equation Derivation...........
4.3 Summary....................................
Stochastic Evolution Equations .................
5.1 General Theoretical Foundations............
5.2 Application To Transport And Scale-Up ....
5.3 Stochastic Parameters......................
5.4 Formal Solution............................
5.5 Convergence................................
5.6 Summary....................................
Future Research.................................
6.1 General....................................
v

6.2 Homogeneous Chaos....................................... 217
6.3 Stochastic Finite Elements ............................. 221
6.4 The Covariance Function................................. 224
References........................................................ 227
vi

1. Introduction
1.1 Overview
This study is concerned with the basic forms of the equations of flow
and transport of solutes or contaminants through porous media. Whenever two
miscible fluids come together to form one phase, there is potential for either the
density of the phase or the viscosity of the phase to change. This change can be
brought about by changes in the concentration and/or changes in the pressure.
In the case that the density is dependent on the concentration of the pollutant,
a coupled system of PDEs is obtained that must be solved simultaneously,
see Equation[ 3.3], page 99. Using Darcys law and the Hubbert potential
for a compressible fluid, the velocity can be related to the pressure via the
permeability of the medium, see Equation[ 3.1], page 97. This equation can
be solved for the velocity, pressure pair by the mixed finite element method.
In the case of constant density, the situation is less complicated. This
is referred to as the tracer case. Here, the system can be written as an uncou-
pled system consisting of the flow equation, which can be solved for the piezo-
metric head, and the transport equation. Once a velocity field has been derived
from the piezometric head distribution, the transport equation is solved for the
concentration plume that evolves over time. Section 3.2 discusses a 2D finite
element implementation of the tracer case. This model is subsequently used to
simulate horizontal tank experiments conducted in Boulder at the University
of Colorados Civil Engineering Department under the direction of Professor
Tissa Illangesekare. The Boulder experiments are discussed in Section 3.2.10.
Briefly, the tank environment is like that of a confined aquifer into which a
tracer (pollutant) is injected. The tank has 45 port locations where tracer in-
jections can be made and samples can be taken. Constant head conditions are
assumed on the ends of the tank, and no flow boundary conditions are assumed
on the sides of the tank.
Both homogeneous and heterogeneous experiments are run in the
tank. In the homogeneous case, the tank is packed with a uniform sand, i.e.,
of uniform hydraulic conductivity. Spectral methods can be used to study the
homogeneous packing. As would be expected, Fourier series can be used to
solve the ID flow equation resulting in a solution of the form
1

where
4>*(x,t) = if)(x,t) + *(Q,t) + y ((l>*(L,t) 0(0, t))
-( ejl\2 JLf /nn\
ip(x, t) = ^2 Cne L s^ sin ( j x
n=1 \ L J
By letting t > oo, the steady state solution is then linear. In the two dimen-
sional case, the constant head contours are nearly parallel, yielding a uniform
velocity field. Both the model results and the experimental results show these
characteristics.
The velocity field is determined from the flow equation or the ve-
locity/pressure equation described earlier. However, prior to solving for the
concentration distribution using the transport, advection-dispersion, equation,
the dispersion parameter must be provided. The first two chapters of this study
focus on the determination of the dispersion parameter. For homogeneous me-
dia, spectral methods can again be used to develop a theoretical formulation
for the treatment of dispersion. These spectral methods can be extended to
include mildly heterogeneous media by using stochastic process techniques. A
stochastic process can be given a spectral representation defined in terms of
special stochastic processes that have orthogonal increments. The first step of
the procedure used to construct an integral spectral representation is to define
orthogonal stochastic measures, Section 1.3. Once this is done, it can be shown
that there is an isomorphism between processes with orthogonal increments,
continuous from the right, and the orthogonal stochastic measures. It is this
association that makes the integral spectral representation work, Sections 1.4
and 1.5.
Based on a review of the literature, it is our view that the theoretical
descriptions of dispersion in porous media found in Gelhar and Axness[45], Neu-
man and Zhang[70] and Dagan[29, 30, 31, 32, 33, 34] are the most prominent.
All of these theories link dispersion to the hydraulic conductivity properties of
the media. However, there are fundamental differences which have generated a
lot of debate in the literature. Section 1.8 seeks to select the common aspects
of the Gelhar and Axness[45] and Neuman and Zhang[70] approaches which
are based on the integral spectral representations of stochastic processes. The
analysis shows how the dispersion tensor associated with the homogeneous case
must be modified to include mildly heterogeneous cases, Equation[ 1.13], page
2

37. The modification is in terms of the integral of the spectrum of the velocity
perturbations. Section 1.10 provides a transformation that allows the conver-
sion of the spectrum of the velocity perturbations to the spectrum of the log
hydraulic conductivity perturbations, Equation[ 1.21], page 43. This theoreti-
cal connection is important because in Section 2.4 a description of dispersion in
terms of the covariance of the hydraulic conductivities using a particle tracking
algorithm will be developed.
Chapter 2 discusses the time and distance forms of the dispersivity
tensor. In this analysis, the solute body spatial moments are given by
M= ncdx
Jn
R
/ nxcdx
M Jn
% = fnn(xi Ri)(xj Rj)c(x,t)dx i, j = 1,2,3
Here M is the mass, R is the centroid coordinate, S^- is the second spatial
moment which characterizes the spread around the centroid, n is the porosity
and c is the contaminant concentration. Starting from the transport equation
with V representing fluid velocity,
Be ->
+ V Vc = V (DVc)
at
and using integration by parts, it can be shown that
Zero Moment => Conservation Of Mass
First Moment => Centroid Of The Mass Concentration Moves
With Velocity V.
Second Moment => D^- =
Although the result that dispersion is related to the time rate of
change of the second moment makes intuitive sense, in order to implement
such a definition requires knowledge of the plume that is probably not avail-
able. For this reason, dispersion is dealt with as a stochastic entity. Dealing
3

with the problem from a stochastic point of view allows a certain level of
uncertainty to be accounted for by the model. This means that instead of
deterministic ordinary and partial differential equations, the parameters and
variables in the differential equations will be allowed to have stochastic com-
ponents. This switch to stochastic differential equations brings with it many
difficulties that require a stochastic calculus to handle. For example, in dis-
cussing the movement of a fluid particle through a porous medium, stochastic
differential equations of the form
will arise where X(t) is the trajectory of the particle through the medium and
W(t) is a special type of stochastic process called a Wiener process. This
equation has the interpretation on the interval [0, t] of
The integrals in this equation cannot be interpreted in the usual sense.
In the case of the first integral on the right hand side, the integrand, a(s, X(s))
is a random function, i.e., for a given s, a(s,X(s)) is a random variable. The
second integral is even more difficult to deal with since the measure part of the
integral is a stochastic process which can be shown to have infinite variation.
Hence, the usual Stieltjes interpretation is not applicable. How these integrals
are dealt with is discussed in Section 2.2 and again in Section 4.2.4. As will
be seen in those sections, the stochastic differential equation and stochastic
integral have more than one interpretation.
Section 2.3 is the first section in which the Lagrangian approach is
used to give the basic form of the transport equation and the basic form of the
dispersion tensor that can be derived from it. In the Lagrangian framework,
transport is characterized in terms of indivisible solute particles i.e., ensembles
of molecules in a small volume, which are transported by the fluid. The total
displacement of the fluid particle can be decomposed into a component due
to convection and a component due to diffusion, Equation[ 2.12], page 58.
The diffusion component is represented by a Brownian motion. The convective
component is described by the fundamental kinematic equation
dX(t) = a(t, X(t j)dt + b(t,X(t))dW(t)
where V(XT) is the Lagrangian velocity field associated with the fluid particle.
Assuming the first order relationship between the displacement of X, X', and
the displacement of V, V is given by
4

X'(t-,x0,t0) = f V'(E[V]t')dt'
Jo
it follows that the displacement covariance tensor is given by the time integra-
tion of the velocity covariances, Equation[ 2.14], page 59. This characterization
of the displacement covariance will play a central role in the development of
the dispersion tensor.
As will be seen in Section 3.1.1, the transport equation is usually
derived from conservation of mass considerations. However, in Section 2.3.2
it will be demonstrated that the basic form of the transport equation can be
derived by treating the fluid particles as obeying the following Ito stochastic
differential equation
dXT = E [V]dt + dXd
Then, the transport equation follows from the Fokker-Planck or Kolmogorov
forward equation. This derivation is restricted to the case where dispersion is
created only by the Brownian motion, Xd.
In Section 2.3.3, the solution of the transport equation is shown to be
a multivariate Gaussian density function, and the dispersion tensor is shown
to be one-half the time rate of change of the covariances of the trajectory dis-
placements. Section 2.3.4 expands on this result to show that the components
of the dispersion tensor should be related to the time rate of changes in the
covariances of the fluid particle or the time integral of the velocity covariances
which approximates the path integral of the velocity covariances along the fluid
particles trajectory. The integrand of this integral is a lagged covariance over
the interval of integration. A numerical formulation of this integral is given
which depends on a particle tracking algorithm which is suitable for implemen-
tation in a computer code. Section 2.3.5 is an extension of these results due to
Kitanidis[58] and Dagan[32],
One of the objectives of this study is to be able to specify the disper-
sion coefficients on a numerical grid block basis so that they can be used in a
finite element model. Section 2.4 contains the details of a proposed method of
doing this. By dividing the domain of the problem into numerical grid blocks on
which it can be assumed that gradients and hydraulic conductivity covariances
are constant, the local displacement covariance tensor, which is represented in
the literature as a bold-faced X, can be expressed as
5

X = E
(KE[K]_1E[f]) (KEfK]-^^)1
where K = K E[K] and K is hydraulic conductivity and the dagger, f,
represents the vector transpose. Since r in this expression is the position vector
of the fluid particle with reference to some injection point, this is really a
distance dependent formula. One of the conclusions of experimental studies is
that dispersivity varies with the distance from the input zone. The manner
in which it varies depends on the degree and location of heterogeneities in
the porous medium domain. The presence of heterogeneities in the porous
medium will cause the velocity field to be non-uniform. In order to get an
adequate representation of exactly where in the domain different magnitudes
of dispersion are to be expected, ensembles of particles must be tracked. Figure
4 in Section 2.4 illustrates the tracking of 5 particles from each of two adjacent
grid blocks located near the center of the domain. These particle paths are
used to identify the numerical grid blocks that are most likely to be reached by
a tracer plume that emanates from the grid blocks containing the origin of the
plume. Once the grid blocks most likely to be reached have been identified,
the previous formulation can be applied on a grid block by grid block basis to
estimate how dispersion will develop over time. In this case, the displacement
covariance matrix is given by
X
n1n1
E EE
i=0 j=0
(Kb+1)E[K(i+1)]-1E[f where the superscripts represent individual block designations. Dispersivity
estimates are derived by differentiating this expression with respect to time.
Since it may be difficult to identify exactly the numerical grid blocks
where a local source is originating, or if more than one point source is involved,
an entire column of grid blocks can be used to determine the particle paths.
Figure 5, Section 2.4, shows the result of tracking a particle from each of the
grid blocks in a column in the center of the domain. Of course, in a simulation,
more than one particle from each grid block would be tracked.
Section 2.5 discusses some methods of random variable generation
that are either currently being used or are planned to be used in the models
of flow and transport being created. Since all of the methodology is based on
exploiting randomness or uncertainty, it is necessary to have methods of simu-
lating this randomness. In Section 2.5.1, it is shown that by using a Box-Muller
6

transformation, two independent uniform random variables can be converted
into two independent Gaussian random variables. It is also shown, Section
2.5.2, how correlated random variables can be simply generated. However,
for more sophisticated simulations in 3 dimensions, efficient computer codes
are available that are capable of cogenerating pairs of 3 dimensional, cross-
correlated random fields with different correlation scales, Robin, et a/[80].
Chapter 3 discusses the numerical method being used to study the
groundwater flow/transport problem. The particular form of the equations
that are to be solved depends on the relationship that is assumed between
the density, viscosity and concentration. For example, in an Enhanced Oil
Recovery problem, a change in the viscosity of the single phase is brought
about by the mixing of a surfactant with the oil in the reservoir. In problems
that involve a pollutant entering an aquifer, many times it is the density that
changes with the concentration of the pollutant. In these cases, it can be argued
using standard definitions of the physical properties of a compressible fluid that
the seepage velocity is related to hydraulic conductivity, pressure and density;
or to the permeability, pressure, density and viscosity, Equation[ 3.1], page 97.
The transport equation is usually derived by applying the Divergence theorem
to the Conservation of Mass law. These equations result in a coupled system of
equations, Equation[ 3.2], page 98. For this coupled system, the mixed finite
element method can be used to solve for the pressure/velocity pair, followed by
a solution of the transport equation by some method. Currently, the code that
is available to solve the coupled system is the SEGMIX code. This code uses
mixed finite elements to solve for the pressure/velocity pair and the modified
method of characteristics (MMOC) to solve the transport equation. SEGMIX
assumes a rectangular domain with no flow boundary conditions on the sides.
In order to use this code to simulate the Boulder tank experiments, it would
have to be modified to accept constant head conditions at the ends of the tank
and no flow boundary conditions on the sides of the tank.
The types of experiments that are being conducted in the Boulder
horizontal tank are tracer experiments. This means that a solute such as
benzene or sodium chloride is injected at a selected port and samples are taken
from a port downstream of the injection port. In this case, the effect on density
is probably minimal, and the uncoupled flow/transport system of equations is
adequate to study the experiments numerically.
The numerical method being used to solve the uncoupled system is
finite elements. Because Gaussian integration formulas are used to evaluate the
7

integrals that arise in the system of equations resulting from the variational
formulation, a reference element which is convenient for integration purposes
is defined which is affinely equivalent to the elements in the domain. Affine
equivalence can be defined in terms of special mappings called pull-backs and
push-forwards. These concepts are explained in Sections 3.2.2 to 3.2.4.
Section 3.2.5 contains a derivation of the local system of equations
that arise from the variational formulation of the flow equation, Equation[ 3.10],
page 116. Once the systems of equations on the local rectangular elements have
been established, they must be assembled into a global system of equations for
the whole domain. This process is described in Section 3.2.6. The derivations
of the velocity field from the piezometric head estimates is given in Section
3.2.7.
Section 3.2.8 shows the derivation of the system of equations that
follow from the variational formulation of the transport equation. Since there
are constant boundary conditions at the ends of the tank, and a pulsed-input
is allowed to take place at an injection point, it is necessary to allow constant
concentration conditions to exist at some grid points. The modification of the
global system of equations to allow certain grid points to maintain a constant
level of concentration is explained in Section 3.2.9. Section 3.2.10 describes
in more detail the horizontal test tank used in the Boulder experiments. Two
types of experiments are conducted in the tank. The homogeneous experiments
are those in which the tank is packed with a single type of sand as rated by its
hydraulic conductivity. In the heterogeneous experiments, the tank is packed
in a block arrangement with 5 different types of sand. The hydraulic conduc-
tivities of the sands range from 3.618 m/day for Sand #1 to 1036.8 m/day
for Sand #5. With this wide span of hydraulic conductivities, a significant
amount of heterogeneity is represented in the tank. The block arrangement of
the sands in the tank is represented graphically in Figures 13 and 14 in Section
3.2.10. Figure 15 provides a flowchart of the basic program components used
and how they interact. Comparisons of computer simulation results shown in
Figures 16 and 17 to actual tank measurements show very good agreement.
Figure 18 illustrates a computed tracer plume.
Chapter 4 actually starts the second part of the thesis. The previous
sections have investigated the components of the equations and the forms of
the equations. However, only the expected or mean value of the concentration
is predicted. Because of the uncertainties involved in specifying the physical
characteristics of the porous medium, the concentration of a solute at a given

point in time is a random variable, and over a period of time it is a stochastic
process. Consequently, in order to more accurately characterize the distribu-
tion of the solute concentration, higher order statistical moments such as the
variance need to be estimated also. In theory, the more moments that can be
predicted, the better this characterization will be. But, in practice, it is usually
a difficult problem just to obtain information on the variance or covariance of
variables in the system.
A much referenced paper in this area is the Graham and McGlaughlin[48]
paper which specifies a set of three equations that are to be solved for the mean
concentration, the velocity-concentration covariance and the concentration co-
variance. Section 4.1 derives and discusses these equations because they will be
used as a basis of comparison for an approach to developing moment equations
based on the Ito calculus.
Randomness can enter the boundary value problem in many different
ways. Equation[ 4.12], page 138, is a statement of the stochastic boundary
value problem, and the discussion following that equation specifies the various
ways in which randomness can enter the picture. Existence theory for the
stochastic boundary value problem is not unlike the nonstochastic case. A
summary is included in Section 4.2.3.
Stochastic integration is again addressed in Section 4.2.4, this time
from the more general perspective of a martingale. The Ito integral then follows
from this more general definition as a special case. The use of the Ito integral
requires that the rules of calculus have to be modified. The reason for this
can be illustrated as follows: Suppose that W(t) is a one dimensional Wiener
process, then it is well known that it can be represented as the limiting form
of a random walk. And, as part of this limiting process the step size of the
random walk goes to zero as the square root of the time interval
Gardiner[42] shows that for the Ito integral this property means that
and that dt dW(t) = 0. The most important new rule is that of Itos lemma.
It is a change of variable formula. The reason the change of variable formula
AIT = C>(AG)
N = 0
N > 0
9

has to be modified is due to the above differential relationships. For example,
if / is a smooth function and X (t) satisfies the Ito differential equation
dX (t) = a dt + b dW (t)
Then, on expanding df[X(t)\
df[X(t)] = f[X(t) + dX(t)]-f[X(t)]
= f[x(t)]dx(t) + r[x(t)] (dX(t))2 + ---
= f'[X(t)] (adt + b dW(t)) + f (adt + b dW(t)f +
2
And, using the differential relationships, it follows that
df[X(t)] = f'[X(t)} (adt + b dW(t)) +
which is different from the ordinary calculus rule. The Ito formula is a stochas-
tic calculus chain-rule. It can also be extended to martingale type processes,
Karatzas[57]. Curtain and Falb[26] have extended Itos lemma to infinite di-
mensional Hilbert spaces. It is this form that is used to derive weak forms of
the moment equations in Sections 4.2.5 to 4.2.8. For the purpose of illustrating
this theory, the key equation is Equation[ 4.17], page 158, which is applied to
two examples. The first example uses this theory to derive mean and covari-
ance equations that in the weak form are identical to those used by Graham
and McLaughlin[48]. The second example is cast in terms of accounting for
the effects of measurement error that is assumed to enter the experiment as a
random perturbation that takes the form of a Wiener process.
Chapter 5 considers the application of the theory of stochastic evolu-
tion equations to the problems of flow and transport. In the deterministic case,
it is well known that the boundary value problem can be recast as an abstract
evolution equation or Cauchy problem. Such a problem takes the form of
fill
- + A(t)u = f(t) 0 u(0) = u o
10

The term abstract is attached to indicate the fact that the functions involved are
mapping a time interval, [0,T] C Sft1, into a Banach or Hilbert space. Hence,
for a Hilbert space H, the function u(t) is an 77-valued function. It follows,
then, that there must be an association between the abstract functions u(t) and
the real-valued functions u(x, t) of the boundary value problem. And, there
must be an association between the differential operator of the boundary value
problem and the operator A(t) of the abstract evolution equation. Section
5.1 explores these connections and the forms of the solution to the abstract
evolution equation for both the autonomous case, A independent of t, and the
nonautonomous, Aft) is dependent on t, cases. The solution, in general, is
given as an integral equation.
Curtain and Falb[27] are able to extend these results to the case
where the forcing term of the abstract evolution equation contains an H-valued
Wiener process, as defined in Section 4.2.4. The solution in this case is given
in terms of an evolution operator U(t, s) generated by Aft) as
u(t) = U(t,0)u0+ f U(t,s)$(s)dW(s) Jo where the integral is now a stochastic integral. Working from the Curtain and Falb solution, weak forms of the mean and covariance equations for the solution u(t) are found that agree with the forms of these equations for the finite dimensional case as given in Astrom[8]. In Section 5.2, the time stochastic process nature of the dispersion tensor is again considered. The fact that the dispersion tensor could be con- sidered as a time dependent quantity was first shown in Section 2.1. But, since then, it has been treated in the models developed up to this section as an effective parameter by some type of averaging process. In the following sections, the time dependent dispersivity coefficient will be incorporated into the basic partial differential equation model. Both the dispersion coefficients and the velocity will be allowed to have random components. Section 5.3 gives the general form of the stochastic PDE that results in terms of the sum of deterministic and stochastic operator components. Section 5.4 represents the solution of the stochastic PDE as an integral equation which has the form uft) = Uft, 0)uo + [ Uft, s)g(s)ds [ U(t, s)Rs xu(s)ds Jo Jo 11 where g(s) is the forcing function which may be stochastic, RSjX is the stochastic operator component and U(t, s) is the evolution operator that is the evolution operator that is derived from treating the deterministic part of the stochastic PDE as an abstract differential equation. As an example of this procedure, the following 1-D transport equation is solved |-(E[D1 + D',(,w,)g + (E[V1+V'M)|=0 Since it is assumed that the E[D] is a constant in this example, the evolution operator U(t, s) can be expressed as a strongly continuous semigroup generated by the operator Using a change of variables to a moving coordinate system, the semi- group is calculated and its properties verified. The solution is then given as an integral equation that has the form of a Volterra type two equation. Section 5.5 applies the classical integral equation methods to characterize the convergence property of the series solution to the integral equation, Hochstadt[53]. In order to provide some insight into the speed of convergence of this series, a test problem is constructed and three scenarios tested. The first case allows the dispersivity to have a random component, but not the velocity. The second case allows the velocity to have a random component, but not the dispersivity. Finally, the last case allows both the dispersivity and the velocity to have random components. The results are contained in the tables at the end of Section 5.5. Monte Carlo methods can be used to generate several sample paths for the random components, and from these, concentration means and variances can be calculated. The Neumann expansion procedure discussed in Sections 5.4 and 5.5 can be extended to a stochastic finite element method by combining the Karhunen-Loeve expansion, a Fourier type expansion of a stochastic process, with the Galerkin method. This approach is, however, subject to the same con- vergence criterion discussed in Section 5.5. Instead of pursuing the Neumann expansion any further, Chapter 6 on Future Research describes an alternative method which is based on the following three components: (1) The Karhunen-Loeve Expansion (2) The Galerkin Method (3) Wieners Homogeneous Chaos Spaces 12 As described in Section 5.4, the formal solution of the equation 9 is developed by splitting the operator CtjX into a deterministic part and a zero mean stochastic part, be., Â£t,x = Lt,x -\- Rt,x and using the Karhunen-Loeve expansion to express the random coefficients in the stochastic component, RtiXl and the Homogeneous Chaoses of Wiener to represent the solution. As described in Section 6.1, the Karhunen-Loeve expansion requires a knowledge of the covariance function of the stochastic process. Specifically, the expansion requires that the eigenvalues and eigenfunctions associated with the covariance function of the process being represented be known. Although certain assumptions can be made regarding the covariance function associ- ated with the stochastic coefficients of the stochastic operator, the covariance function of the solution process is not known. What is required is a way of representing the solution process that does not require a knowledge of its co- variance function. The principal result in this approach is the Homogeneous Chaos decomposition of the L2-space of a Gaussian process. The fundamentals of this type of decomposition are provided in Section 6.2. The stochastic finite element method using Homogeneous Chaoses is outlined in Section 6.3. Since the success of this finite element procedure depends on obtaining the eigenval- ues and eigenfunctions of the covariance function, some details of this problem are covered in Section 6.4. 1.2 Stochastic Processes Some of the more famous and fundamental results on contaminant transport through porous media involve the characterization of dispersion in terms of the spectral representation of stochastic processes, Gelhar and Axness[45], Neuman, et a/[69]. This introduction will trace the development of these spec- tral representations from their beginnings in classical functional analysis to their use in deriving fundamental theoretical formulations of the dispersion tensor. In doing so, the key role that the dispersion tensor plays in the process of scaling-up from small homogeneous laboratory experiments to larger het- erogeneous field problems will become clear. The following definitions can be found in many references, for example Burrill[17], Doob[36] and Todorovic[96]: 13 Definition: Let (Id, B, V) be a probability space. A real (complex) valued measurable function with domain Q is a real (complex) random variable. Definition: A real (complex) stochastic process is a family of real (complex) random variables {X(t) : t E T} defined on a common probability space (n,B,V) with T CK1. Definition: A stochastic process is said to be strictly stationary if its dis- tributions do not change with time, i.e., if for any ti,t2, ,tn E T and for any h E T, the multivariate distribution function of the random variable (Xtl+h, , Xtn+h) does not depend on h. Ex: /(Ali, X2, , Xn\ ti, , tn) f (Xi,X2, , Xn; t\ + h, , tn + h) Definition: A stochastic process {Xt : t E T} is wide sense stationary if E[|Ah|2] < oo B[Xt+r,Xt] =B[Xr,X0] Vt E T where E represents the expected value. Doob[36], shows that for wide sense stationary processes, the following holds: Proposition If {X(t)] t E 3?1} is a process which is wide sense stationary, then there is a group of unitary transformations {Up, t E 3?1} such that for each t, X(t) = Utx( 0) Us+t = Usut U0 = Identity Element U-t = Inverse Of Ut Using the notation (, ) to represent the Hilbert space inner product, the fol- lowing spectral representation theorem exists for unitary operators, Riesz and Sz.-Nagy[79], Section 137, 14 Stones Theorem Every one-parameter group {Ut;t e 3?1} of unitary trans- formations for which (Utf, g) is a continuous function of t, for all elements / and g, admits the spectral representation Ut L oc where {Ex} is a spectral family. Furthermore, E\ is uniquely determined. E\ is an orthogonal projection with the properties (1) Ex (2) E\+o = E\ (3) E\ > 0 forA > oo (4) Ex > I forA > oo A second theorem related to this is, Riesz and Sz.-Nagy[79], Section 138, Bochners Theorem In order for the function p(t) (oo < t < oo) to admit the representation with a nondecreasing and bounded real function V(A), it is necessary and sufficient that p(t) be continuous and nonnegative definite in the sense that whatever the positive integer m, the real numbers ti, t2, , tm and the complex numbers pi, p2,---, pm- The reason that these two theorems are connected is that the function is nonnegative definite, and from this it follows that, Riesz and Sz.-Nagy[79], Section 138, /OC elXtdV{ A) -OC rn Y, ptf? - >0 pit) = (utf, f) (i.i) where Ex is a spectral family. 15 space From the polarization identity, the following holds in a complex Hilbert {E\f,g) =$l(E\f,g) + iS(Exf,g)
= K(Elf,g)+iZ(Elf,g)
= (En,ExS) + ,S(Exf,Exg)
In particular, if g = /, then
(ExfJ) = \\Exf\\2
and Equation [ 1.1] then gives
/OO
eiXtd\\Exf\\2 (1-2)
-00
From the properties of the spectral family,
o < II-Ea/II2 = (EJ,ExJ) = (Elf,J)
= (EJJ)
and, if A > p then Ex > E^ => (Exf, /) > {E^f, /), so that \\Exf\\2 is real
and nondecreasing. Furthermore,
ii-ea/h2 < ii-Eoc/ii2=n/ir
so that ||i?^/||2 is also bounded. Equation[ 1.2] will be useful in the discussion
of the Bochner-Khinchin theorem in Section 1.4.
1.3 Stochastic Measures
In the preceding section, it was shown that a stochastic process X (t)
that is wide sense stationary has the representation
16

f 00
X(t) = UtX( 0) = / elXtdExX( 0)
.7 00
The process {Â£\Ai(0) : A G 9?1} turns out to be a special type of stochastic
process. Also, in order to be able to interpret this integral in the usual sense,
this process must be associated in some way with a type of measure. In this
section a stochastic measure and a stochastic integral are defined that allow
the representation as a stochastic process in terms of a unique orthogonal
stochastic measure which corresponds to a stochastic process with orthogonal
increments, Section 1.4. The details of this section on Stochastic Measures are
found in Todorovic[96].
Definition: A complex-valued random variable Z on (O, B, V) is called second
order if
E[\Z\2] < oo
The family of all such random variables is denoted by
L2{Q,B,V}
Todorovic[96] shows that given the definition of equality that
Zi = Z2 iff Zi = Z2 (a.s.)
ie, they differ on at most a set of measure zero, so that the space L2{Q, B,V}
consists of equivalence classes, and using the inner product definition
where the symbol represents complex conjugation, then L2(Q,B,V) is an
inner product space. The norm for this space is defined as
Using this norm to define distance, L2(VL,B,T) is a metric space.
The Cauchy-Schwarz inequality holds, Todorovic[96], and the inner product is
uniformly continuous, for if Zq, Zi, Z2 G L2{Q, B, V}
Z || = +(Z,Z)
* = E IZIT
$$Z1,Zo)-(Z2,Z0)\ < \(Z1-Z2,Z0)\ < \\Z0 17 so that for jjZ1 Z2\\ < p^y, \(Zt, Z0) (Z2, Z0)\ < e. The mode of convergence on this metric space is that of mean squared convergence and defined as (m2) lim Zn = Z iff IIZn Z\\ 0 as n > oo ny oc And, this definition of convergence leads to the following form of the Riesz-Fisher Theorem If {Zn}^=1 is a Cauchy sequence, then there exists a Z E L2{Cl,B, P} such that (m2) lim Zn = Z Let {S, 5} be an arbitrary measurable space, and let 50 be the algebra of sets that generates the a-algebra S. Definition: A mapping r) : So > L2{Q, B,V} such that 7/(0) = 0 rj(AuB) = 7](A) + r](B) (a.s.) for any disjoint sets A, B E S0 is called an elementary random or stochastic measure. Definition: Let m(A) = ||?7(A)||2 < oo for any A E S0 where || ||2 = E[| |2] Definition: An elementary random measure is said to be orthogonal if (77(A), i](B)) = 0 V disjoint A, B E S0 It is clear from the orthogonality property of the random measure rj(-) that the set function m(-) is finitely additive. And, if it is assumed that m(-) is 18 countably subadditive, then it can be extended to a measure on to Proposition 12.3.9 of Royden[81], all that is required is a semialgebra of sets C that generates S0 and that on C, m(0) = 0, m(-) is finitely additive on C and countable subadditive on C. Furthermore, the measure m(-) so defined on can then be extended to the measurable space {S', 5}. Definition: m(-) is called the measure associated with r](-). Let L2{m) = L2{S, 5, m} the Hilbert space of complex-valued functions on S' which are square integrable with respect to m. Definition: For step functions n Hs) = J2ckIBk(s) disjoint Bk e S0 k=1 define h(s)rj(ds) = Y, ckV(Bk) k=1 It is clear that the mapping ip takes step functions in L2(m) and maps them into random variables in L2(Q,B, V). Furthermore, given two step functions h(s) and f(s) (ip(h), ip(f))L2(n,B,T) n n Yxiv(A),YvMBj) i=1 j=1 L2(n,B,V) n n L2{n,B,T) i=1j=1 and, since the sets Aj and Bj can be written in terms of disjoint sets as At = (At \ Bj) U (Ai n Bj) Bj = (Bj\Ai)U{AinBj) 19 it follows from the orthogonality of g(-) that the inner product (rj(Ai), g(Bj))L^u B ^ can be written as (v(-A-i), rl(Bj))L^n i3 V^ (r)(Ai fl Bj)1 g(Ai fl Bj))L^QBV^ = m(Ai fl Bj) consequently, the inner product {^(h),i/j(f))L2^QBV^ can be written as n n (^(h)^(f))L.2{a,B,T) = EE Xiyjm(Ai n Bj) i=i j=1 = J h(s) f (s)m(ds) = (K /)i2(m) Therefore, the mapping ip preserves the inner product of the step functions from L2(Q,B,V) to L2(m). If g e L2(m), then let {hn}^=1 be a sequence of step functions such that Wd hn\\ ^ 0 as n > oo then from the orthogonality of g(-) ||'0(h) p){hm)\\2 = \\hn hm\\2 > 0 as m, n -> oo So, {ipihn)}^^ is a Cauchy sequence, and by the Riesz-Fisher Theorem, 3 ip(g) e I/2{Q, B, P} such that W'ip(g) ip{hn)\\ 0 as oo Definition: The random variable 'tp(g) is called the stochastic integral of g e L2(m) with respect to the elementary orthogonal random measure rj. It is denoted by ip(g) = [ g(s)r)(ds) (1.3) J S The elementary stochastic measure g which is defined on Sq can be extended to the a-algebra S, Todorovic[96]. Hence, Equation[ 1.3] is taken to be the integral of a g E L2{m) over the set S with respect to the orthogonal stochastic measure V- 20 1.4 Process With Orthogonal Increments Let {S, S,m} = m} where 5R1 = Real Line, and 1Z = c>- algebra of Borel sets of 3?1. Let {Z(t)]t G 3?1} C Z/2{f2,P, P} with E[Z(t)] = 0, Vt G Sft1. Then, using Stones theorem, these stochastic processes can be written as /OO elXtdExZ{ 0) -OO From the properties of the spectral family that the E\ are symmetric and for Ai < A2, EXiE\2 EXl it follows that, for A0 < Ai < A2 E[(Ba,Z(0) EXl>Z(0)) (EX2Z(0) EXlZ(0))] = ((EXl Em)Z{0), (EXi EXl)Z{0)) = (Z(0), (EXl EXa)(Ex, EXl)Z{0)) The operator product in the last term can be expanded as (Pai EXo)(Ex.2 EXl) EXl EX2 EXl EXl EXo EX2 + EXo EXl = pAi pAi Pa0 + Pa0 = 0 Consequently, the stochastic processes ExZ(0), oo < A < oo that appear in the Stone representation have the special property E [(PAlZ(0) EXoZ(0j) (Pa2P(0) EXlZ(0))*] = 0 Todorovic[96] makes the following Definition: The stochastic process {Z(t)]t G 5R1} is said to have orthogonal increments if, for any t0 < E < t2, E[Z(t1)-Z(tQ)][Z(t2)-Z(t1)]* = 0 21 where the symbol represents complex conjugation. Definition: The process Z(t) is right continuous in the mean squared sense if Vt E 9?1, t fixed, \\Z(t) Z(tk)\\ 0 as tkit Todorovic[96] shows that there exists an isomorphism between processes {Z(t) : t E 3?1} C L2{Q,B, P} with orthogonal increments continuous from the right in the mean squared sense and the orthogonal stochastic measures rj with m(-) the measure associated with it. This correspondence is given by Z(t) = 1]) From this, for t > s, Z(t) = r/((-oo,t]) = r/((-oo,s]) U (s,i]) = ri{{-oo,s])+ri{{s,t]) = Z(s)+T)((s,t]) => v((s,t]) = Z(t) Z (s) Ash[6] characterizes the Borel sets in various ways. In particular, they can be defined as the smallest a-algebra of subsets of Sft1 that contains the intervals (00,t], t E 9?1. Furthermore, from this definition the function F(t) can be defined as F(t) = m((-oo,i]) = ||)j((-oo,t])||2 = E[|Z()|2]=E[Z(i)Z(i)*] where represents the complex conjugate. Since m((00, t]) = m((00, s] U (s, t]), then 22 F(t) = F(s) + m((s,t$$
So that,
m((s,t]) = F{t) F{s)
It can be shown, Todorovic[96], Ash[6], that the function
F{t) = E [Z(t)Zity\
is bounded, right continuous and non-decreasing.
Letting T/idt) = dZ(t), then
dF(f) = m(df) = ||??(df)||2
= imm2
= mzm
= E [dZit)dZ{t)*\
Hence, the following holds
dF(t) = E [dZ(t)dZ(t)*]
And, if the process Z(i) has orthogonal increments, then
ndz(t)dz(t'y] = { dp{t)
Definition: The stochastic integral
/+oo
h{t)dZ(t)
-OC
(1.4)
is to be interpreted as
f*+oo
hit)rjidt)
23

From the definition of the stochastic integral, Equation[ 1.3], page 20, h(t) is
a complex-valued function. If
h(t) = elXt
then
/OO
elXtdExZ{ 0)
-OC
/OO
elXtdr](d\)
-OO
where r] is the orthogonal stochastic measure associated with the stochastic
process with orthogonal increments, E\Z(0).
1.5 Spectral Representation
Let {Z(t);t E 9?1} C P} be a wide sense stationary process
with
E [Z(t)\ = 0 C(t) = E [Z(s + t)Z* (s)]
The following proposition follows from the non-negative definiteness of the
covariance function, Todorovic[96].
Proposition The covariance function C(t) is continuous on 5R1 if it is contin-
uous at zero.
From the definition of the inner product,
C(t) = (Z(s + t),Z(s))
Let / = Z(0), then from the discussion in Section 1.2, the covariance can be
expressed in terms of the unitary operators as
C(t) = (Ua+tf,Uaf)
= (UtUJ,UJ)
Letting g = Usf = Z(s) and using Equation[ 1.2], page 16, it follows that
24

/oo
eatd\\Exg\\2
-OC
Furthermore, since Hi-fi'll2 is real, bounded and nondecreasing and E\ is
uniquely determined, then
F(A) = \\Exg\\2 = E [ExZ(s) (ExZ{s))*] = E [Z(\)Z(\)*] (1.5)
and obtain
/OO
elXtdF( A)
-00
In particular, letting 1 = 0 and using the spectral family properties, the covari-
ance can be written as
/OO
d\\Exgf = WE^gf
-OO
= iiair
= E[Z(s)Z*(s)]
The forgoing discussion illustrates to the famous
Bochner-Khinchin Theorem A complex-valued function C(t) defined on
R continuous at zero is the covariance function of a wide sense stationary
stochastic process iff it can be written in the form
/+oo
eltxdF( A)
-OO
where F(-) is a real nondecreasing bounded function on Sft1 called the spectral
distribution of the process Â£(l).
If the spectral distribution function F(-) is absolutely continuous,
then the spectral density of the process is given by
1 r+oo
}(\) = F'(\) = / e~MC(t)dt
Z7T J oo
25

So, the covariance function and the spectral density form a Fourier Transform
pair.
Comparing this with Equation[ 1.4], page 23, and Equation[ 1.5], page 25, the
spectral density can be written as
/(A) is called the continuous spectrum of C(t) and the integral expresses C(t)
as a continuous bundle of waves having amplitudes /(A).
Similarly, the cross-spectrum Sxy{&) of two processes X(t) and Y(t)
is the Fourier transform of their cross-correlation Rxy(t)
Let (Â£(t); t e 9?1} C L2{fl, B, P} be a wide sense stationary process such that
E[Â£(f)] = 0 and C(t)=E[^t + s)C(s)]
Then from Todorovic[96] there is the following existence
Proposition Let the covariance function C(t) be continuous at zero. Then,
there exists a unique orthogonal stochastic measure 7] with values in Z/2{C2, B, V}
such that
f(X)d\ = dF{ A) = E[dZ(X)dZ{X)*}
So that the covariance function takes the form
and
and
26

where m(-) is the Lebesque-Stieltjes measure associated with g and generated
by the spectral distribution F.
Hence, if {Z^(A); A e 9?1} is a stochastic process with orthogonal
increments corresponding to the orthogonal stochastic measure rj(-), the process
{Â£(Â£);Â£ e Sft1} has the spectral representation
/+oo f+OO
etXtrj(d\) = / elMdZ/:{ A)
-oo J oo
1.6 Space Correlations And Space Spectra
The interpretation of the independent variable as a space variable
instead of a time variable does not change things, Lumley and Panofsky[64].
Homogeniety is the property that for the space variable corresponds
to stationarity for the time variable.
C(x)= f+ eiXxdZt(\)
J OO
In 3-dimensions, the corresponding spectral representation is given as
Â£(x) = eit3dZ<:{k)
where k is the wave number and Z^(k) is a stochastic process with orthogonal
increments.
In the case that time is a contributing factor, the spectral represen-
tation can be written as
mt) = J^e*-sdZt(k,t)
As in the one-dimensional case, if Z{k) is a stochastic process with
orthogonal increments, then
EMZMT] = {JCK] (1.6)
where dF(k) = f(k) dk and f(k) is the spectral density of the process.
The cross-spectrum of two processes X and Y is given by
27

Sxyik) ( RXy{x)z l^xdx
(27t)3 Jsr3
and, its Fourier inverse is the cross-correlation of the processes X and Y, given
by
Rxy(x) = [ SXY(k)e^xdk
Jvt 3
For a homogeneous process, the following is true
Rxy(-Q) = E [X(x-e)Y\x)\ = R^x0)
In particular, if X and Y are real stochastic processes,
Rxvi-e) = RYX(e)
and, if X = Y,
Rxx(9) Rxx0)
so that the covariance function of a real stochastic process is an even function.
Also,
Rxy(-O) + Ryx(-O) = RYX0) + Rxy(6)
So, clearly, RXy(9) + Ryx(9) is an even function. Furthermore, if Rxx is an
even function, then using the mapping T(x) = x = r and the change of
variable formula it follows that
Sxx(-k)
(27t)3
1
(27t)3
1
(27t)3
I Rxx(r)e-<-^df
f Rxx{x)e-^{-x)dr
s?3
f RXX{x)eXltx(-l)dx
k3
Sxxik)
The last equality follows from the change of variable formula and the fact that
1 is the Jacobian of the transformation T. Hence, Sxx(k) is an even function.
Similarly, SXY(k) + SYx(k) is an even function.
28

1.7 Ergodicity
For many properties that are to be measured, it is easier to obtain
one observation at each point in time of a random sequence over a long period
of time than to obtain several observations at the same point in time.
The former method is called averaging along the process (time aver-
ages) and the latter method is called averaging across the process (ensemble
averages).
A statement that these two averages are the same is called an ergodic theorem.
Burrill[17] states the following:
Definition: Let M be a 1-1 mapping of onto fi. M is a measure-preserving
transformation whenever the following condition holds: E is a measurable
set iff M(E) is and P(E) = P(M(Ej) for each measurable set E. For every
integer t, the function Mt defined inductively by Mt = M^_1) oM is a measure
preserving transformation.
Weak Ergodic Theorem: If M is a measuring-preserving transformation
and if X G I/2, then
1 n~ i
(m2) lim -----------Y MlX = X
' nmoo 77 777, ,^
1,0 t=m
exists.
The important corollary is
Corollary:
E[X] = E[X]
The transformation M is called ergodic if X is a.s. constant. If this
is true, then from the corollary
(m2
lim -----
nm>oo fi
m
n1
E m = e[.y]
t=rn
where X(t) = MtX. So, it is clear from this that the mean-squared limit
taken over time is equal to the expected value of the random variable X. The
29

term ergodic can have several interpretations. If a stochastic process has the
property that the mean taken along the process is equal to the mean taken
across the process, then the process is said to be mean-ergodic. In general,
a stochastic process is ergodic if the ensemble averages are equal to the time
averages.
In the Hilbert space context, F. Riesz proved the following form of
the
Mean Ergodic Theorem Let H be a Hilbert space and Ut : H > H be a
strongly continuous one parameter unitary group. Let the closed subspace Hq
be defined by
H0 = {x G H : Utx = x V t e Sft1}
and let P be the orthogonal projection onto H0. Then for any x G H
1 rT
lim / Usxds = Px
t-> c T Jo
The statement and proof are found in Abraham, et al[2]. Using the Hilbert
space of second order random variables, V) and the Proposition from
Section 1.2, there is a group of unitary operators {Ut;t e 9?1} such that
X(t) = UtX( 0)
The theorem then says that the limit of the time averages is a random variable
that is invariant under the group of operators, {Ut : t 6 1ft1}.
If X(t,u) is a sample path of a stochastic process X(t,u), then the
integral
f Jo X^^dt
represents a time average of X(t,6j) over the interval [0,T]. From this, the
random variable
Yt(u)) = ^[ X(t, u)dt
1 Jo
can be formed. Clearly, if X(t, to) is a stationary process, then E[JA] =
E[X(t)] V t and
30

EiYT\ = fÂ£nx(t)\dt = nx\
The variance of Yt(uj) is given by
4t = E [{Yt E[Ft])2] = E [{Yt E[X])2]
Clearly, if (m2) lim^oo Yt(uj) = E[Af], then GyT > 0 as T > oo. Similarly, if
Oyt > 0 as T > oo, then (m2) lim^oo Yt(co) = E[X].
Assuming homogeniety, if the independent variable is a spatial vari-
able instead of a time variable, the interpretation is that as more spatial points
are included in the average, the variance of the spatial average tends to zero.
1.8 Stochastic Solute Transport And Dispersion
A key component in the scale-up problem is the correct formulation of
the dispersion tensor used in the transport equation. Using arguments similar
to those found in Gelhar and Axness[45] and Neuman[69], the reason for the
focus on dispersion can be illustrated as follows:
Starting from the transport equation,
= -V n[cV DVc]
dt 1 1
and letting the porosity be constant, it follows that
^ = -V [cV DVc]
dt L J
where c = concentration, V = seepage velocity, and D = molecular diffusion.
From the conservation of mass equation
d(nP) ,
= -v. (pH)
with constant porosity and density, it follows that
V V = 0 and V E[U] = 0
31

Writing the transport equation as
_ + V (cV) = V (DVc) (1.7)
and letting the concentration and velocity be stochastic processes, which are
distributed as follows:
c = E[c] + d E[c'] = 0
V = E[E] + V' E[V'] = 0
the substitution of these distributed parameters into Equation[ 1.7] yields
9(E[1+ C/) + V [(E[c] + d)(E[V] + V')] = V (DV(E[c] + d))
at
Expanding,
9(E[c]
dt
+ V {E[c]E[E] + E[c\V' + c'B[V] + dV'} =
V [DV(E[c] + c')\
(1.8)
Taking expectations and using E[c'] = E[W/] = 0,
<9E[c]
dt
V (E[c]E[E]) + V E[c'W] = V [DV(E[c])]
(1.9)
Subtracting Equation[ 1.9] from Equation[ 1.8],
3d -* -*
+ V [E\c]V' + c'E[E] + dV' E\dV']] = V [DV(c')]
at
Important assumption: Assuming that the perturbations from their means
in c and V are small, the second order perturbation term
dV' E[dV'\
is eliminated because for small perturbations this difference would be close to
zero leaving the first order approximation
32

^ + V [E[c\V' + c'E[V}} V [DV(c')]
Expanding and using the zero divergence of V and E[V],
dc'
+ VE[c] V + Vc' E[V] = V (DVc') (1.10)
(J L
Assuming that d depends on both x and t and that there is spatial homogeneity,
the following spectral representations can be made:
d(x,t) = J e^dZj&t)
g(f) =
v' = xxxy
where k = (ki,k2,k3) is the wave number and x = (xi,X2,x3) is the position
vector.
Defining
dZ^i (k) = (dZyi (k), dZyi (k), dZyt (k^
and making the change of variables
Vi = Xi E[Vi]t, = 1,2,3
and letting
c(x,t) = v(y,t)
=> E[c(x,t)]=E[u(y,t)] => d(x,t) = u'(y,t)
so that v'(y,t) is d(x,t) in the moving coordinate system. In other words,
d and v! are the same stochastic process represented in different coordinate
systems. This allows us to write in terms of the spectral representations,
33

j^dZsik, t)
v'(y,t) = d(x,t) = J e^'xdZci(k,t)
J jHtt+nn)dZcf(ki *)
[ eik-yeik-E[V]tdz^
./5ft 3
The stochastic process z/(y, t) has a unique orthogonal stochastic mea-
sure associated with it, and from the above equality and the isomorphism be-
tween the orthogonal stochastic measures and the processes with orthogonal
increments, continuous from the right, it follows that
eifc-E[v]tdZrJ^,t) = dZv, (k, t)
ik E[V]dZc,(k, t) + ^~ (dZc,(k, t))
(J L
Jk-Emt
d_
dt
(dZvi(k, t)j (1.11)
Substituting u'(y,t) into Equation[ 1.10], page 33, and simplifying yields
dv'{y,t)
dt
Vyu(y,t) E[V] + VyE[u(y,t)]-V' + Vyi/(y,t)- E[V]
= Vj,-[DVj,i/'(y,t)]
Then letting V = Vy, it follows that
+ VE[v{y,t)\ V' = V [DVis'(y,t)\
Substituting the spectral representations for z/ and V in this equation gives
8_
dt
dk = V-
34

Hence, using Equation[ 1.11]
ik-E[V]dZc,(k, t) +
d_
di
('dZd(k, f))
+
I e*-*VE[v(y,t)]-dZf,(k,t)
5R3
= V-
DV
f el^dZc,(k,
SR3
Taking derivatives and simplifying yields
X3 e^Wt (dZc + Xs ^ ^ E W + ' *] ^ ft A)
= X e^VE^y, A)] ft t)
And, using the uniqueness of the spectral representation, the following first
order ordinary differential equation is obtained

[ik-E[V] + Dk-k\ dZd(k, t)
= -VE [v(y,t)]-dZtf,(k,t)
Letting a = \ik E[V] + T)k kj, this equation has the solution, assuming that
dZd(k, 0) = 0,
dZd{k, t) = e~at J* eTVE[i/] dZy,(k)dr
Furthermore, if it is assumed that the gradient VE[^] is constant, then VE[z/] =
VE[c], and it follows by evaluating the integral that
a dZci(k,t) = 6(Â£)VE[c] dZ^,(k)
where b(t) = 1 e~at. The assumption that the gradient of the average con-
centration is constant means that the concentration is spatially linear. This is
another assumption that requires only mild heterogeneities. Multiplying both
sides of this by a = ft E[V] + DA: A:], then gives
- VE [c] dZy, ft \ik E[V] + Bk-k] b(t)
-----------=;---=f-------=;-=;---------- = dZ{k)
(k E[H])2 + (DA: k)2
35

Multiplying both sides by dZyi(k)*, the complex conjugate transpose, taking
expectations and using the fact that
E [(VE[c] dZv, (k)) dZy, (Â£)*] = E [dZy, (k) dZy, (Â£)*]f VE[c]
yields
6M[i^E[V]-D^*](4v(*)VE[c])t
(k E[V])2 + (Dk k)2 c'^'
where Sp/p/ (k) is the cross-spectrum matrix and Sc,^/ (k) is the cross-spectrum
vector.
Now, taking the inverse Fourier Transform
**(*) =
and letting x = 0, it follows that
E[c'V-'] = Bc,f.,(0) = Sd{(k)dk
Jdt6
(1.12)
bit) \-ik-E[V] +Dfc.*l St,^,(k)VE[c]
-dk
/SR3
(k-E[V])2 + (Dk-k)2
Take the kjth term of the matrix S^,^,(k) which, as shown in Section 1.6, has
the property that Sv'v>(k) + Sv'v'(k) is an even function, and assume that
Syiy'(k) = Sy'y'(k) and that D is positive definite, then using the mapping
Vk Vj Vj Vk
T(k) = k = uj and the change of variable formula, it follows that
b(t) \-ik E[V] + Dk k] Sy'y (k)
L i Vk: J
/SR3
(k E[E])2 + (Dk k)2
dk =
b(t) \ik E[V] + Dk-k] Sy'y> (k)
L J Vk_j
/SR3
(k E[V})2 + (Dk kf
dk
36

r b(t) ' E[E] + D(^) (w)] sv!v' ()
+ / -----------------------------^-----dw
Jw (w-E[V})2 + (D(w) (w))2
b(t) \ik E[V] + DA: k] Sv>v> (k) _
1________________J k 3 dk
(k-E[V])2 + (Dk-k)2
, b(t) \ik E[E] + Dk k] Sv _l_ / 1_________________J k j dk
Jw (k-E[V])2 + (Dk-k)2
, b(t)(Dk k)Sv>v> (k) _
= 2 ______________________dk
JiR3 (k E[V})2 + (Dk k)2
Without the symmetry assumption, this result would contain the term Sv>v> (k) +
Vk Vj
Sv'v'(k). Returning to the transport equation
V j Vk
= -V [cV DVc]
dt 1 1
we know that by writing the concentration and velocity in terms of distributed
parameters,
c = E[c] + J E[c'] = 0
V = E[V] + V' E[V'] = 0
we obtain Equation[ 1.9], page 32, namely,
+ V (E[c]E[V]) V (DVE[c]) + V E[c'V'} = 0
The term E[c'v ] is an additional dispersive flux term that has already been
expressed in terms of VE[c] via Equation[ 1.12], page 36. So, the transport
equation for E[c] can be stated in terms of a new dispersion tensor, D, as
^ + V (E[c]E[E]) V (DVE[c]) = 0
where
D
D+ r b(t) (Dk-k) 'S'y'y' (k) ^
J3 (k-E[V])2 + (Dk-k)2
(1.13)
37

It is clear from this formulation that in general, D is a time dependent quantity.
If k E 5R3 and if DA; k is positive definite, then since a = [ik E[V] + DA; k\t,
lim bit) = lim 1 e~at
tyoc tyoc
= 1 lim e-[Dk-k+ik-E[v]\t
t)-oo
= 1 since DA; k > 0
And, assuming that the interchange of limit and integral makes sense, the
asymptotic or steady state limit as t > oo for D is
D
D +
r (d k-k)SrrW ]Â£
t (k-E[V])2 + (Dk-k)2
The assumption of ergodicity is implicit in the study, i.e., the solute
transport in an ensemble of aquifers approximates the real field situation. This
means that if the independent variable is a spatial variable, then the variance
around the average must go to zero as the size of the domain gets arbitrarily
large. Hence, the scale of the system must be large in comparison to the
correlation scale, the length scale over which variables remain correlated. So,
the estimates of macroscopic dispersivity and effective hydraulic conductivity
are meaningful only if the scale of the problem is large in comparison to the
correlation scale. Consequently, Equation[ 1.12], page 36, is valid only after
a large displacement distance has been reached, perhaps tens or hundreds of
meters.
The adequacy of the first order approximation of the solute transport
equation
^ + V [E[c]V' + cTE[V]\ = V [DV(c')]
depends on small perturbations in c and V and is not certain for large variance
of hydraulic conductivity. In this same vein, the assumption that the gradient
of the expected concentration is constant also depends on mild heterogeneities.
38

Existing linear theories predict that transverse dispersivities tend
asymptotically to zero as Fickian conditions are reached. The assumption
of mildly fluctuating hydraulic conductivities has been used to justify elimi-
nating nonlinear terms in establishing the linear theories. However, Rubin[82]
found that higher order terms may cause some reduction in longitudinal mix-
ing and a significant enhancement in the transverse spread. Hence, the im-
portance of non-linearities should not be disregarded in studying dispersion in
geologic media. Although it is the point of view of this study that significant
heterogeneities should be modelled using numerical models, attempts at in-
corporating heterogeneity and therefore nonlinearities in the analytical models
have been made. Neuman and Zhang[70] recover part of the nonlinearity due
to the deviation of the plume particles from their mean trajectories using what
has become known as Corrsins Conjecture, which is a statement relating the
Lagrangian covariance of the velocity to the Eulerian covariance of the velocity
through the probability density of the particles position. From this, nonlinear
analytical expressions for the time dependent dispersivity have been developed.
The relationship between the Lagrangian and Eulerian velocity fields is key to
the developments of dispersion that are to follow.
1.10 Velocity/Permeability Covariance Relationship
Spectral arguments similar to those used in the preceding subsection
can also be used to develop a fundamental relationship between the Fourier
Transforms of the velocity covariances and the log-hydraulic conductivity co-
variances, Equation[ 1.21], page 43. This relationship is important because it
shows the connection between velocity covariances and log-hydraulic conduc-
tivity covariances, and hence permeability covariances. This means that esti-
mates of dispersion can be based on either velocity covariances or permeability
covariances. In essence, the characterization of dispersion as a phenomenon
created by permeability says that dispersion is a result of deviations in local
permeabilities from a global average permeability.
By Darcys law
q = -KV(f)
where K is the hydraulic conductivity, in general for 3?2 a second rank symmet-
ric tensor, and V(/> is the hydraulic gradient. By assuming that the medium
is isotropic, the second rank tensor can be replaced by a scalar hydraulic con-
ductivity, K.
Let Y = In(K) and suppose that
39

Y = E[Y] + Y'\ fi = E[0] + fi'- E['] = 0; E[T'] = 0
Then,
q = -eyV(E [] + And, since K = eY = eE^eY>, it follows by expanding eY' that
q = -eEW(l + Y' + 23^ + -)(VE[0] + Vfi')
Letting J = VE[c/>], then
q = e
Efri
(/- v') + y'(j v') +
Take expectations and dropping terms higher than first order, yields the first
order approximation
E[g] eE[Y] J
From Darcys law and the incompressibility condition, respectively,
q = KVfi, V q = 0
it follows that since K = eY, then
V (eYV) = 0
Expanding this it follows that
erVY V + eYV2(/> = 0
which yields
VT Vfi + V2 = 0
Then using the distributed parameters
(114)
J' = E[y] + Y' = E[0] + 4>
(1.15)
40

with
E [(/>'] = 0; E [Y'\ = 0
and substituting Equation[ 1.15] into Equation[ 1.14], it follows that
V(E[F] + F') V(E[0] + 4f) + V2(E[0] + (/)') = 0
Expanding and treating E[F] as a constant, gives
vf' ve[] + vf' v] + vy = o
And, by retaining only first order perturbation terms, i.e. dropping the VF'
V VF' VE[(/>] + V2E[(/>] + V2' 0
Taking expectations and using E[F'] = 0 and E[(//] = 0, results in
0 = -V2E[] VF' VE[] + V2(f)' (1.16)
Substituting in Equation[ 1.16] the spectral representations
Y' = [ eilsdZY,(k) '= ( eilsdZ^(k)
it follows that
ik VE[ty^dZyik) - ||Â£||VK^(Â£) 0
And, using the uniqueness of the spectral representation theorem,
ik-VE[]dZY>(k) WkfdZ^k) (1.17)
Starting from the expansion
q = -eE[y](l + F' + ^1- + -)(VE[] + V and disregarding terms involving the products of perturbed quantities yields
qn -eE[y](VE[] + V]) (1.18)
41

And, taking expectations and using E[F'] = E[(/>'] = 0,
E[q] -eE[y]VE[]
Then since q = E[ q< -eE[y](V0' + T'VE[0])
Using the spectral representations
q'= [ eilsdZfik)-, Y' = [ ellsdZY,(k); '=/ etlsdZ^(k)
JK 3 ./SR3
it follows that
J ellÂ£dZj(h) -eE[y] (V e^dZ^ik) + ^ e^dZy,(Â£)VE [])
By the uniqueness of the spectral representation theorem
dZj(k) -eE[Y]ikdZ^(k) eE[Y]VE[cl)]dZY'(k)
Using the first order approximation, E[g] eET]j = eE[Y]VE[^5
E[g]dZy/(fc) ieE[Y]kdZÂ¥(k) (1.19)
Then from Equation[ 1.17]
dZ^fik) i||Â£||-2,fc VE[]dZY'(k) (1-20)
Equations [ 1.19] and [ 1.20] along with the first order approximation, E[g] &
eE[Y]j^ anj fa(q ^at E[^]) = k$E[q\ yield dZ-fik) (i II^H-2^) E[q']dZY>(k) where I is the identity matrix. Multiplying this expression by its conjugate transpose yields E[dZj{k)dZj{k)*\ (i \\k\\-2W) E[g]E[gf (i \\k\\-2W) E[dZY>{k)dZY>{k)*] And, using the formulas, Equation[ 1.6], page 27, 42 it follows that E [dZe(k)dZe(k)'] E[dZY,(k)dZY,(k)*] dF(k) = Sjj(k) dk dF(k) = SY>Y'(k) dk Sff(k) (i \\k\rm) E[?-]E[?f (i \\k\\-2kP) SY.Y'(k) (1.21) This result agrees with Dagan[31, Equation 4.10]. 1.11 Summary This section introduced some important concepts of stochastic pro- cesses that will be used and expanded on in subsequent sections to study the general problem of scale-up. Some basic conclusions can be drawn from the presentation in this section. First, from Equation[ 1.13], page 37, it is clear that in the absence of variations in the velocity field no adjustment to the local dispersion tensor D is necessary. Secondly, from Equation[ 1.21], page 43, it is clear that the variations in the velocities are due to the variations in the hydraulic conductivities. Consequently, the conclusion is that the dispersion tensor requires scaling-up in the presence of variations in the hydraulic conduc- tivities. Another way of saying this is that the effects of heterogenieties in the porous medium cannot be adequately modeled using a local dispersion tensor only. This means that although a local dispersion tensor may be adequate for describing plume development in small homogeneous laboratory experiments, it must be modified in a way that takes into consideration either variations in the velocities or variations in the hydraulic conductivities if it is to be used to adequately describe plume development in highly heterogeneous field problems. The linear theories that have been characterized in this section depend on the assumption of only mild heterogeneities being present in the porous medium. However, researchers have found that this restriction may lead to erroneous conclusions regarding both the longitudinal and transverse spread of the plume. For this reason, attempts have been made to modify the linear theories. To this end, Corrsins conjecture has been borrowed from the field of plasma diffusion to aid in the modification of the linear theories. Corrsins conjecture relates the Lagrangian velocity covariance to the Eulerian velocity covariance through the probability density of the particless position. E[C(0)H(f)t] = [ E[V(6,0)V(y,t)']p(y,t)dy Jn 43 As will be seen in the next section, the idea of Lagrangian velocity covariance is key to the characterization of dispersion in a highly heterogeneous environment. Field studies made by various researchers to determine dispersivity values have concluded that: Field dispersivity values are larger than laboratory dispersivity values by a few orders of magnitude. Dispersivity varies with the distance from the solute input zone. Consequently, the upscaled dispersion tensor can be represented either as a time dependent quantity, as in the case of Equation[ 1.13], page 37, or as a distance dependent quantity. Chapter 2 of this study will investigate further the origins of time and distance characterizations of the dispersion tensor. In addition, the next chapter will provide information on some methods that will be used to study the scaling up of dispersion from a stochastic point of view. 44 2. Dispersivity Coefficients Time And Distance Forms 2.1 Time Dependent Dispersivity Coefficients 2.1.1 Introduction In the previous section, it was shown that when heterogenieties are introduced into the porous medium, the component of the transport equation that is significantly affected is the dispersion tensor. And, it was concluded that the dispersion tensor could be characterized as either a time dependent or a distance dependent quantity. In this section, these characterizations will be developed further, al- beit with some simplifying assumptions. The intent is to provide motivation for the formulations of dispersion that will be used in subsequent sections. In Section 2.1.3, a characterization of the dispersion tensor is derived from the transport equation and shown to be equal to half the time rate of change of the second spatial moment tensor. This makes physical sense in that the second spatial moment represents the spread around the centroid of the mass plume. Dispersion, characterized in this manner, then represents how the spread of the plume with respect to the centroid is changing over time. It is also shown that the centroid of the plume moves with the velocity of the flow. So, by using the centroid of the plume as a reference point, this characterization eliminates any changes in the plume due to convective influences. The following is required: Let Q be a bounded, open connected domain in 3?" with a Lipshitz- continuous boundary, dQ. Then the Fundamental Greens Formula, integration by parts, is given by (2-1) and follows from the Divergence Theorem by letting a = pq and using the expansion V (pq) = Vp q + p V q (2.2) 45 2.1.2 Transport Equation Let c(x, t) represent the concentration distribution of a solute, then the mass transport equation of this solute is given by where K is the hydraulic conductivity, n is the porosity and 0 is the hydraulic head. Initially, a divergence free velocity held is assumed, 2.1.3 Spatial Moments Of The Solute Concentration In this section expressions are developed for the first three spatial moments of the solute concentration. In these calculations the porosity, n, is assumed to be constant and included in the definition of c(x,t). Zero Moment (j( * + V Vc = V (DVc) (2.3) Here V is the seepage velocity V V = 0 Taking the time derivative of this integral, gives And, from the mass transport equation and the Divergence theorem d dt M0 a Ja V Vcdtt + / V (DVc)dfi 46 So, the time rate of change of Mq depends on the boundary conditions. And, if c = 0 and Vc = 0 on <912, then = 0 implies M0 is a constant. Since M0 is the total mass, this is a statement about the conservation of mass of the solute. First Moment Mi = [ xc(x, t)dVt Jn Taking the time derivative, d d [ _ [ Be r-Mi = [ xc(x, t)dn = [ x^dn dt dtJn K J Jn dt And, from the Transport Equation[ 2.3], page 46, 4Mi = / x(V Vc)dQ + / f(V (DVc)) dt Jn Jn From Equation[ 2.2], page 45, V Vc = V (cV) cV U By letting {e^}, i = 1, , n be a standard basis of it follows that d n r _Ml = -Ee* JnXiV (cV)dtt + f cfV Vdtt + y et [ xAV (DVc))dQ Jn Jn Using the assumption of a divergence free velocity held, the second integral on the right hand side of the previous equation vanishes so that d .n f -* _n r -M1 = -Eu / x?V-(cU)df2 + Eu / Xi(y (DVc))df2 (2.4) dt J n J o i=1 Let V be independent of x, then from Greens formula[ 2.1] the two integrals on the right are evaluated as [ XjV (cV)dQ = [ (cV) Vx,idQ + [ x,i(cV) Adj Jn Jn Jon = I cVidQ + / ay(cU) Adj ion -Vi / cdtt Jn (2.5) 47 if c = 0 on dQ. / Xi(V (DVc))cA2 = D'Vc-'VxidQ + / Xi(DVc) VdQ In Jn Jon = / (DVc)idfl Jn if Vc = 0 on dQ. Combining these results it follows that d Mi = V V& cdQ-Tei (DVc)idVl = V cdQ - DVcdQ dt Jn Jn Jn Jn Now, if D is dependent on only t, and letting ' A ' A Vc D = A =* DVc = A Vc . A _ Dn-Vc _ it follows that / D'VcdQ = In Jn Di Vc D2 Vc Dn Vc dQ But, by integrating by parts, each component integral is equal to zero, [ Di VcdQ = [ cV DidQ + [ cDi Pdj = 0 (2.6) Jfl JQ J dVL since Di is independent of x and c = 0 on dQ. Then, from Equations [ 2.4], [ 2.5] and [ 2.6] it follows that 4 Mi = V [ cdQ dt Jn And, since M0 = fn cdQ is a constant, this yields 48 (2.7) d /Mi di \M0 V But since is the centroid of the concentration mass, Equation[ 2.7] says that the centroid of the concentration mass moves with velocity V. Second Moment The first moment, Ml5 is a vector quantity and the zero moment is a constant. The quantity R(t) M1 Mo is the centroid (center of mass) and is also a vector quantity. The second mo- ment about the centroid is given by M2 = / (x R(t))(x R(t)yc(x,t)dCl Jn M2 is a matrix since the quantity S = (x R(t))(x R(t)y is a matrix. So, M2 can be written as M2 = / Sc(x, t)dQ, Jn In order to evaluate the time rate of change of M2, first consider the Diagonal Element: i = j First, from the definition of the centroid and Equation[ 2.7], page 49, d dt [ (xi Ri)2c(x, t)dÂ£l = 217 [ (xi RAcdfl + / (xi Ri)2dVt Jn Jn Jn at = -2Vi I (Xi Ri)cdQ + / (Xi- Ri)2[-V Vc + V (DVc)]dfi Jn Jn = -2 Vi J^(xi R,()cdn { J^cV -(xi- RifVdQ, \- f c{Xi Ri)2V udj] + I (xi- RifV (D Vc)dVt Jan J Jn 49 Now, / cV (xi Ri)v dQ, = 2Vi / (xi Ri)cdVl Jn Jn so that the first two terms of the last equality cancel, and using the boundary condition c = 0 on BO,, the boundary integral vanishes so that 4- f (Xi Ri)2c(x,t)dQ = f (xi Ri)2V (DVc)dn dt Jn Jn Integrating by parts and using the boundary condition that Vc = 0 on BQ, [ (xi Ri)2'V (DVc)df2 = f DVc V(xi Ri)2dQ + [ (ay RifDVc Pdj = -2 an n Be n Be r\ ) ) / j tit j=l dxJ j. dxi 0 {x% R{) 0 dn f -J1 ^ dc -2 / fa Ri)dtt in 'Bxj 2 / Vc adVt a = Jn Dji(*C Ri D^2 (j^i Ri = -2 Ri) / cV ddVl + / cd vd'y . Jn Jan = 2 [ cDudQ c(x, t) = 0 x e BVt Jn Consequently, a diagonal element can be written as D, 1 d 2 dt Jn(xj Rj)2c(x, t)dQ fnc(x, t)dQ Off-Diagonal Element: i ^ j In this case, take the time derivative of the ijth component of M2, f (xi Ri){xj Rj)c(x,t)dVt = f |) (xj Rj)c{x,t)dVt Cli~C i/ / \ ttC f 50 + & KMZ'tW f dc 1 J (^Z ^ti ) jjI " = Vf f (xj Rj)cdVt Vjf (xi Ri)cdQ j ^2 J Â£2 + f (xi Ri)(xj ~ Rj)[V Vc + V (DVc)]dn J = Vi [ (xa Rj)cdVt Vjf (xi Ri)cdQ (2.8) Jn Jn + f (xi Ri)(xj Rj)V VcdVt J + f (xi Ri)(xj Rj)V (DVc)dfi J Integrating by parts the integral Rj)V VcdVt cV (xi Ri)(xj Rj)Vdtt Ri)(xj Rj)V udj = J c[(xi Ri)Vj + (xj Rj)Vi\d^2.9) And, integrating by parts the last integral in Equation[ 2.8], it follows from Equation[ 2.1], page 45, xi Ri)(x j Rj)V (DVc)dfl = J DVc V(xi R,t + / (xi-R^ixj- Jan f 0 1 )(xj Rj)dQ - f?j)DVc Vd^f DVc adQ a = (xj Rj) (xi Ri) < ith element < jth element 0 51 0 52 ^ ^nk~Bx~k k=1 L/*x/c L/c=l / / . - tft- LS dxk Consider the integral (xj Rj) (Xi Ri) 0 n Be 11 Be 52 ^ikTRT^xi ~ Rj) 52 ^ik~Q^~(Xi ~ ^) dQ (2.10) k=1 dn n Qq r n Â£)q 52 R*ik (xj ~ Rj)d& = 52 R*ik(xj ~ Rj)d& /ei Bxfc In^ Bxk = / Wc-[3dVt 13 = Jn idn = J cT>ijdQ = D ij J cdQ Dji {xj -Rj) 1 D i2(xj -Ri) . Djra (Xj -Ri). vd^y t) = 0 x G <90 So, Equation[ 2.10], page 52, and the fact that = Dji? yield I (Xi Ri){xj i?j)V (DVc)dfl = 2D ij I cdQ J J And, from Equations [ 2.8], [ 2.9] and [ 2.10] it follows that fn(xi Ri){xj Rj)c(x, t)dQ, ^ . Id D ij(t) = JW 2 dt In c(x, t)dQ Since the mass of the solute is given by M = f c(x, t)dQ Jn this result can be written as 52 which is in agreement with the definition given by Dagan for the Actual Dis- persion Coefficients. 2.2 Stochastic Differential Equations 2.2.1 Introduction In order to proceed further with the investigation of the dispersion tensor in terms of a stochastic analysis, it is necessary to define the type of integration that will be required to analyze certain stochastic differential equa- tions that arise naturally. This will be the subject of Subsections 2.2.2 to 2.2.5. Although the present discussion is very superficial, it will be adequate for the immediate need. However, this material will be expanded later when more details are required. 2.2.2 Integral Of A Stochastic Process According to Jazwinski[54] a stochastic process X(t) is mean square Rie- mann integrable over [a, b] if for a = t0 < ti < < tn = b and P max(tj+i b), t{ f t { b+i % the following mean squared limit exists n-1 fb (m2) lim Y X(t'i) (ti+1 -ti)= X(t)dt Or, lim E p>o n-l ,.h Yx(t'i)(ti+1 ti) / X(t)dt 4n **a 0 As an existence theorem for the mean square Riemann integral, the following can be shown to be true, Jazwinski[54], Theorem: X (t) is mean square Riemann integrable over [a, b] if and only if E[X(f)X(r)] is Riemann integrable over [a, b] x [a, b\. This notion of integrating a stochastic process is different from stochastic in- tegration which will be defined next. 2.2.3 Wiener Process (Brownian Motion) In Section 1.2, the concepts of a random variable and a stochastic process were defined. In short, a stochastic process can be thought of as a mathematical model that describes the occurrence of a random phenomenon at each point in time subsequent to some initial time. Karatzas and Shreve[57] make the following Definition: If (12, B, V) is the probability space on which the stochastic process X(t,u) is defined, then X(t,u) is measurable if the mapping X(t, co) : ([0, oo) x Q, B{[0, oo)) B) (V, B($td))
is measurable. Here Â£>([0, oo)) and B(1R.d) represent the Borel sets of [0, oo) and
respectively.
Definition: Given a probability space (fi},B,V), a nondecreasing family {Bt :
t > 0} of sub-cr-algebras of B, Bs C Bt C B for 0 < s < t < oo is called a
filtration.
Definition: The stochastic process X(t, to) is adapted to the filtration {Bt} if
for each t > 0, X(t,u) is a Ht-measurable random variable.
With these definitions, the one-dimensional Brownian motion can be defined
as follows:
Definition: A one-dimensional Brownian Motion is a continuous, adapted
process W = {Wt, Bt] 0 with the properties:
(1) Wo = 0 (a.s.)
(2) For 0 < s < t, the increment Wt Ws is independent of Bs and is
normally distributed with mean 0 and variance t s.
Definition:
A process Â£(Â£) is a white noise process if its values Â£(Â£,) and C(tj) are uncorre-
lated for every and tj such that tj tj. For a white noise process with zero
mean,
54

EfCWCM] = q(t)S(t-T),
q(t) > 0
The Wiener process can be defined as the limit of a random walk, or as the
integral of a Gaussian white noise process with zero mean.
W(t)= ftC(s)ds
Jo
The following is a block diagram representation of this equation:
c(t)
f
wit)
2.2.4 Stochastic Integration
Because the solutions of the stochastic evolution equations come from
spaces whose members are random functions, the solution process will require
integrating these functions. For reasons given below, this will require a new
type of integration called stochastic integration.
Consider an integral of the form
f B(s)dW(s) (2.11)
Jo
where W(s) is a Brownian motion process.
To illustrate the difficulty involved with interpreting this type of integral in
the usual Riemann-Stieltjes sense, it is necessary to define a sample path of a
Brownian motion process.
Definition: The mapping t ) W{t, lu) (lu fixed) is called a sample path.
The following facts regarding sample paths of a Brownian motion process are
proved in Friedman [40].
Almost all sample paths of a Brownian motion are nowhere differen-
tiable
55

Almost all sample paths of a Brownian motion have infinite variation
on any finite interval
A non-rigorous reason for the first bullet is that since
AW = O (At*)
then
AIR 1
t ^ f y oo as At ^ 0
At At*
Consequently, given the preceding discussion of the interpretation of W(t) as
the integral of a Gaussian white noise, the derivative ^ must be interpreted
as a d-correlated Gaussian white noise which is a purely mathematical ideal-
ization.
So, the integral [ 2.11] cannot be defined as a Stieltjes integral in the usual
sense, for in order to do so, the sample paths would have to have bounded
variation.
2.2.5 Types Of Stochastic Integrals
Stochastic integrals have been defined in different ways. Two of these
methods of defining the stochastic integral are:
Ito Integral
* Applicable to a larger class of functions
* Does not follow the formal rules of calculus
Stratonovich Integral
* Applicable to a restricted class of functions
* Follows the formal rules of calculus
Stochastic integrals are defined in the sense of convergence in measure or mean
squared convergence. Although the definition of stochastic integrals will be
expanded in more detail below, the present discussion is adequate for the im-
mediate purpose of exploring the dispersion tensor.
2.3 The Concentration Equation
Traditionally, the concentration or solute transport equation is de-
veloped from the consideration of conservation of mass in terms of variables
averaged over a Representative Elementary Volume (REV), Bear[12], Gray[49].
In this section, the solute transport equation will be derived using stochastic
considerations. In addition, the all important dispersion tensor will be further
scrutinized with the objective in mind of finding expressions for this tensor
that can be used in out computational work.
56

2.3.1 The Lagrangian Approach
By the Dupuit-Forcheimer equation, Bear[12], the filtration velocity,
V is given in terms of the specific discharge, q, and the porosity, n, as
V=*
n
In a heterogeneous porous medium, the properties of the medium cannot be
precisely known. Hence they are considered to be composed of an average
value plus some type of random component that describes the uncertainty
associated with the medium. Because of this, both q and n are random and so
is the velocity V. The uncertainty associated with the velocity field of a fluid
particle can be illustrated by the situation depicted in Figure 1. The black dot
in this figure represents a fluid particle about to begin its journey through a
porous medium represented by the open circles. As shown, there are several
paths that the particle can ultimately take. And, there is no way of knowing
which path will be the actual path.
A
B
C
D
Figure 1
Figure 1 depicts the mechanical mixing component of dispersion. If
the velocity field could be perfectly described, i.e., if at each juncture it was
known which way the particle was going to go, then there would be no mechan-
ical dispersion to account for, only molecular diffusion. As shown in Chapter
1 and as will be discussed further below, the mechanical dispersion can be
accounted for through the velocity covariances, or equivalently, through the
permeability covariances.
The velocity covariances are defined as
57

V' = V-E[V]
Pjk{x,y) = E\V'j(x)V'k{y)\
In the Lagrangian framework, transport is developed in terms of indivisible
solute particles which are transported by the fluid. As discussed in Bear and
Verruijt[ll], the solute particles can be thought of as ensembles of molecules
in a small volume. If the vector XT represents the total displacement of the
particle which started its motion at x = xq, t = to, then the vector Xt can be
decomposed into
Xr(t',xo,to) X(t] xo, to) + Xd(t; to) (2-12)
s v ^ s v y s v /
Total Displacement Convection Diffusion
In this expression, the first term on the right hand side is due to a mechanical
mixing of the fluid and requires movement of the fluid in order to exist. The
second term on the right hand side is due to molecular diffusion which can take
place without motion of the fluid. Hence, it does not contribute to the velocity
held of the fluid particle. It does, however, contribute to the total displacement
of the fluid particle. The vector Xd represents a Normalized Brownian motion
or Brownian motion with zero drift type of displacement. Hence, Xd can be
defined as the integral of a white noise process
Xd(t) = [ u(s)ds E[z/] = 0
Jo
where the autocovariance of the white noise, assuming a constant spectral
density /(A) = K and using the integral representation for the delta function,
5(r) = ^ Jjji elTXd\, is given by
E[zy(t + T)u(t)*] = J KeirXd\ = 2vrKS(t) = aS(r)
Even though S(r) is not a function in the mathematical sense, the idea is that
if r ^ 0, then the autocovariance is zero.
The vector X comes from convective transport and is related to the
velocity held by the kinematic relationship
X(P,x0,t0) = fv{XT)dt'
Jo
58

(213)
= /W7i+v'(xT)w
Jo
= E[V]t+ [ V'(XT)dt'
Jo
= E[V]f + X'{t; Xo, to)
By taking expectations, it follows that E[XT] = E[X] = E[E]t, and if XT
in the integrand of the above integral is replaced by its average, a first order
approximation of XT in terms of V' is obtained, Dagan[33]
X'(t-x0,tQ) = /V'(E[E]f,)dt'
Jo
Replacing the vector XT{t') with the vector E[E]t' in essence assumes that the
displacements of the velocity about the expected velocity path are not unlike
those about the actual trajectory. This type of assumption is similar to that
used by Taylor who assumed that since the expected velocity was much greater
than the velocity fluctuations, any disturbances or eddies in a wind tunnel were
transported with the expected velocity without significant distortion, Monin
and Yaglom[68].
The displacement covariance tensor, is then expressed in terms
of the velocity covariances as
Xjk(t;x0,t0) = E[X'j(t; x0,t0)X'k(t; x0fi0)\
Pjfc(E[V]*,,E[V]*,/)eM,/
(2.14)
The mass of the indivisible particle is denoted mfp, and the solute is assumed
to be inert, be., it does not react with the fluid that transports it nor with
the solid matrix. A solute particle follows a path through the porous medium
according to Equation[ 2.12]. The velocity of the particle is given by
dXT
dt
V
t > 0
The concentration field associated with the particle is given by
(2-15)
C'fpiXt t) xq, to)
- XT(t;x0,t0))
(2.16)
59

where n is the effective porosity. The concentration is defined as mass per unit
volume, but, only a portion of the unit volume is available for flow, and that
portion is given by the porosity, ffence, if the concentration is multiplied by
the porosity and integrated over !ftn, the result is the mass of the fluid parti-
cle. From Equation[ 2.16], this is equal to rrifp at each point of the particles
trajectory. This illustrates the indivisible nature of the fluid particle.
The concentration held Cfp is a stochastic function, i.e., at each time
t, the function Cfp(x,t) is a random variable of the spatial variable, x. This
follows from the fact that the trajectory XT is a stochastic function. Hence, if
p(Xx,t) represents the probability density function of Xt, then
E [Cfp(x,t]x0,to)\ = / Cfpp(XT-,t,x0,t0)dXT
= f T^e-8{x XT)p(XT-,t,x0,t0)dXT
J 5R n
n
It is clear from this equation that the probability density function of the con-
centration held is the same as that of the trajectory Xt- So, the concentration
held is random because the trajectory is random. As mentioned before, the
velocity held is also a stochastic function. From Equation[ 2.12], page 58, the
following stochastic differential equation can be formed
dXT dX dXd
dt dt dt
And, since Xd is a Wiener process, the second term on the right hand side
is formally a Gaussian white noise. Furthermore, Equation[ 2.13], page 59,
yields,
dXT
dt
E[V] + V'(XT) +
dXd
dt
The resulting stochastic differential equation then takes the form
dXT = (e[H] + V'{XT)) dt + dXA (2.17)
which is to be interpreted in terms of the stochastic integral equation as
60

XT(t) = XT(0) + J* (E[V] + V'(XTj) ds + J* dXd(s)
The uncertainty of the position of the fluid particle, Xt, can be demonstrated
by solving the hnite-difference form of this stochastic differential equation, viz.,
for t0 < ti < < tn,
XT(ti+1) = XT(ti) + (E[E] + V'(XT(ti))) AU + AXd(i)
where
Atj tj+l ti
AXd(t) = Xd(ti+1) Xd(U)
Figure 2 shows various realizations of the path of a fluid particle. The average
velocity vector is along the line x = y. What is immediately clear from this
figure is that the final position of a fluid particle can vary greatly depending on
the exact path taken through the porous medium. And, this is the essence of
dispersion. If we knew exactly which path each fluid particle takes, there would
be no mechanical dispersion. Since this is impossible to know, the transport
equation must include a term to compensate for this uncertainty. This was
the case in Chapter 1 where the dispersion tensor was modified to include a
component in part described by the spectrum of the velocity covariances.
2.3.2 Basic Form Of Transport Equation
It can be shown, Jazwinski[54], that under certain circumstances the
solution process of an Ito stochastic differential equation (SDE) is a Markov
process and that the probability density or transition probability density func-
tion of the solution process solves the Fokker-Planck or Kolmogorov for-
ward equation, i.e., given the Ito SDE
dx = a(x, t)dt + B(x, t)dXd
where x and a are n-vectors, Bisannxm matrix and Xd is an m-vector Wiener
process with Q(t)dt = E \dXd dXj], then the density function, p, satifies
c)t) 1
+ V (pa) = -V V(pBQBt)
It is important to note that the coefficient of the dXd term in Equation[ 2.17],
page 60, which in this case is the identity, is not allowed to be spatially
61

Figure 2 Sample Particle Paths
0 0.02 0.04 0.06 0.08 0.1
62

varying. If this term had a spatially varying coefficient, then the form of
the Fokker-Planck equation would depend on whether the Ito theory or the
Stratonovich theory is applied. For example, Gardiner[42], if the stochastic
differential equation (SDE)
dx = a(x, t)dt + B(T, t)dXd
is an Ito SDE, then the equivalent Stratonovich SDE is given by
dx = as(x, t)dt + Bs'(x, t)dXd
where
af = di
=
2 y B j. /A; Ei ,;
^ j,k
And, the form of the Fokker-Planck equation depends on which approach is
used.
If, for simplicity, it is assumed that V'(Xt) = 0 in Equation[ 2.17],
page 60, then the resulting stochastic differential equation is
dXT = E [V]dt + dXd
Given that the solution process XT is a Markov process, then the probability
density function, p, satisfies
% + V (pÂ£[r]) = V (Iqmvp) (2.18)
So, if we let D = |Q(t), then the tensor D is either a constant or a time
dependent quantity. To be accurate, Equation[ 2.18] represents a diffusion
process with drift coefficient E[D] and a diffusion coefficient D.
Obviously, this equation has the form of the transport equation. And,
the expected value of the concentration field solves the same equation, e.g.,
+ V (E[C]E[E]) = V (DVF[F])
63

Hence, the basic form of the concentration equation follows from the funda-
mental displacement Equation[ 2.12], page 58 and the associated stochastic
differential equation.
Clearly, this equation accomodates only the dispersion created by the
Brownian motion term, Xj. Whereas, according to Equation[ 2.12], page 58,
the total dispersion is going to come from that associated with the convective
transport and that associated with the Brownian motion.
2.3.3 Solution Of Basic Transport Equation
In general terms, the multivariate Gaussian probability density func-
tion (pdf) for n dependent random variables is given by
f(zi,z2,---,zn) = |^y|-exp|-i(i'-yn)t(V-1)(i'-yn)| (2.19)
where
z = (zi, z2, , zny
fl = (E[zi],E[z2], ' ,E[^])f
012 ' ' ' Cpn ^
o\ (Jn2 ' ' ' /
so that V is the variance-covariance matrix. And, the multivariate character-
istic function for the Gaussian probability density is given by
<2, , Cn) = exp (^C exp (< f/l) (2.20)
In theory, the characteristic function of a random variable is given by
m = f(x)e^dx
Clearly, / is the Fourier Transform of the function f(x) so that
V =
f
021
V ^
n 1
64

1
/(<3) =
(27t)" Jw
f(cu)e-ig-sdLU
Given that the trajectory XT is given by Equation[ 2.12], page 58, and assum-
ing the the probability density function is Gaussian and has the form
|V|-2
p(XT,t) = exp {-^(Xr E[TG]t)+(y-1)- E[V]*)}
then this pdf solves a non-divergence form of Equation[ 2.18], page 63, e.g.
^ + E[E] Vp = V DVp
(J L
To see this, first from Ortega[73]:
If D is a real symmetric matrix, then there is an orthogonal matrix
P whose columns are the eigenvectors of D, such that
PfDP = diag(Ai, A2, , An) = D
where Ai, A2, , Xn are the eigenvalues of D. The change of variables
x = P V
has the effect of aligning the principle axes of the matrix D with the coordinate
axes and D becomes a diagonal matrix.
In this case, the non-divergence form of the transport equation in the
original y system, viz.,
^ + E[E] Vp = V DVp
(J L
becomes in the rotated x system
% + ET ]-Vp-f E|?l = P'Ep?]
And, by making the change of variables to a moving coordinate system,
X1 = x1- E[Vi ]t; X2 = x2 E [V2\t] ; Xn = xn E[Vn]t; T = t
65

and,
then it follows that
is(X,T) =p(x,t)
dp dv dv
dt dt ~ dT
V*f> = VaSV = ^xv
d2p 32 V
dx2 dX2
xl
So that the problem becomes
dv n d2v
w = ^DwmrDA" **"'T>0
"(WO) = f(X)
This equation is transformed by forming
iuj'X
0= / edw'
Jirtn
dv ~
DA^
dT
dX
Assuming that v(X,T) and Vv(X,T) 0 as |X| > oo and integrating by
parts, it follows that
/5ft"
vXiXieiX^dX = -u2 elX-av(X,T)dX
/5R"
So, letting vT =
0 = vt{^-, T) + w^Dwi)(w, T)
And, by transforming the initial condition, the following ordinary differential
equation results
z>t(u;,T) = uj^T>cov(co,T)
v(uj, 0) = f(u)
66

which has the solution
v{uj,T) = f{uj)e
u(X,T) can be retrieved by writing
-lu^'DluT
u(X,T) =
(27t)" Jw
v(uj,T)e 1 'udCo

(27t)" 7k
f(u)duj
And, using the transform
it follows that
v(X,T) =
m = / eiaY f (Y)dY
D-iw-{X-Y)-Yt)QT
(27r)n
Next, let [3 = X Y and write the inside integral as
[ e-^e-^^du
J\Rn
and let
dujf(Y)dY
g(uj) = e
-uj^iycuT
then
90) =
e-J.Â£e-ritD tiTdQ
(2n)n Jw
= I e-^-v^XduJr I e-lWn^-t> 27r i-oo 27r J-oo
Then, using the integral from Guenther and Lee[51, p. 167]
e^P-D^Tdu) =
7T \ 2 /32
---- I p 4DT
DTj
it follows that
(2.22)
67

1
9(0) =
n
7T 2 Z-/i = l
e
(27r)"(Du...Dnn)iTf
EE
re
(47tT) 2(Dh D
Consequently, Equation[ 2.22] can be written as
v(X, T)= f K(X -Y, Du-~DnnT)f(Y)dY
where
Hence, if /(H) = 5(H), then
-EC
(47rT)f(Dn---Dr
1 l 4DzzT
v{X,T) = / K(X-Y,t)n---t)nnT)5(Y)dY
wn
= K(X,Bn---i>nnT)
En
i = 1
(47tT) t (DX1 ! e Dnn) 2
1
(27r) f (2DnT 2DnnT)iC
1
(27r) t (2Dnt 2Dnnt) 2
1
(2^)f(2DnC- 2Dnt) 2
1
(2^)f(2DnC- 2Dnt) 2
1 l 4D,, T
|[i(Â£-E[V]t)tD-1(Â£-E[V]t)]
|[(pt(^-E[y]b)t/b-1(pt(y-E[y]b
-|[(y-E[y]qtpJ_D-ipt(y-_E[y]t)]
68

Since, P^D 1 and since det(ptp) = det(I) = 1, then
pWtt) =
(27r) 2
For given y, t, and E[V], the probability p(y,t) is then linked to the size of
the components of the D tensor; which indicates spreading due to dispersion.
Comparing this equation with Equation[ 2.19], page 64, it is seen that p(x, t) is
a Gaussian density function with mean E[V]f and covariance matrix V = 2tD,
from which it follows immediately that
Dp' 2^*7

1 dX.Tij
2 dt
(2.23)
This method does not work if the matrix D is allowed to be a function of t,
for then E[V] would depend on t also by virtue of the dependence of pt on
t. However, in the case that D does depend on t, if it is assumed that p is a
Gaussian density function, then it can be shown by direct differentiation that in
order for p to satisfy the transport equation it is necessary that Equation[ 2.23]
hold.
2.3.4 Dispersion As Velocity Covariances
If X and Xd are not correlated, then it follows from Equation[ 2.14],
page 59, and Equation[ 2.23] that
D
ldX-
Tij
1 dXjj 1 dXdij
13
2 dt
2 dt 2 dt
Id rt rt
Letting
ldX
!4 2 dt

dij
then Xy = /o f(t",t)dt" and by the Liebnitz formula
dX

= f(t,t) +^f(f,t)dt
dt
69

/(M) + f Pij{E[V]t, E[V]t)dt"
J 0
= f Pij(E[V]t', E[V]t)d,H + J* pij(E[V]t, E[V]t")dt"
Furthermore, it is assumed that a type of stationarity or homogeneity (Section
1.2 or Section 1.6) in the sense that E[y ] is constant and
p(E[l?]t',E[V-]t) = E[C',(E[C]t') V^E^i)]
= E[V'i(E[?](t + (t'-t))0j(E[?]i)]
= p.,(E[C]('-))
This means that pij depends only on the separation vector E[E](f' t). Also,
since the stochastic processes considered are all real,
PlJ(E[V}t',E[V}t) = E[0i(E[V]O V';(E[V]*)]
= Pij(E[V]t, E[V]t!)
which leads to the conclusion that
And, if the mapping a is dehned as
a : [0, t] W g
a(t') = E[V]t'
then it follows from the definition of a path integral that
fpijds = f Piy (<^(^), E[T?]t) 11^'
J (T JO
= ||E[y]|| f Pl](E[V}t\E[V}t)dt'
Jo
70

Hence, /g Pij(E[V]t', E[V]t)dt' is a path integral. And, consequently,
n 1
1 dXi:j
= [ N(v[v]t\nv]wt' = ^^,LNds
JO Eh Jv
2 dt Jo L J L J " ||E[V]||
If the true Lagrangian path integral is given by
Pijds
<7t
then the path a is an approximation to the path ot so that for the true La-
grangian path
[ p^ds = [ pij (aT(t'),aT(t)) aT(t')
J dt'
' Pij(E[V]t',E[V]t) \\E[V]\\dt'
so that
Dp-
1 dXij
2 dt
'' PlJ(E[V}t',E[V}t))dt'
(2.24)
This result says that the components of the dispersion tensor should be related
to time rate of changes in the covariances of the position of a fluid particle or
the time integral of the velocity covariances. Furthermore, it also says that
the calculation of the components of the dispersion tensor should take into
consideration the orientation of the path of the fluid particle.
In the pure Lagrangian sense, it is not assumed that the expected
velocity is constant along the fluid particles path. If x and y represent two
points on the particle path, then
pjt(x,y) = E [(g(f) E[g(f)]) (X4 E[U])]
Using the kinematic relationship
X(P,x0,t0) = f V{XT{t'))dt'
Jo
it follows that E[X(t; x0, f0)] = /0*E\V(XT(t'))\dt'. The convective displace-
ment is given by
71

X'(t;x0,t0) = X(t-,x0,t0) E[X(t;x0,t0)\
= J* (v(XT(t')) E[V(XT(t'))]) dt1
And, the displacement covariance is given by
Xjfe(t) = Jo JQ Pjk (XT(t'),XT(t")^ dt'dt"
And, by differentiation,
Dj7c = 2dtXj/c = L Pjk dt'
The integrand represents a lagged covariance. If s = t t', then t' = t s and
Pjk {XT{t'),XT(t)} = pjk (xT(t ~ s),Xt(Â£))
= E [(g(.fT( s)) E[g(VT(i s))])
x (ll(XT(t s)) E[Vt(.YT(i s))])]
In order to implement this type of dispersion estimate, the paths that the fluid
particles take have to be identified. This can be done by generating velocity
fields and computing the paths using a particle tracking algorithm such as
XT((k + 1) At) = XT(kAt) + V(XT(kAt))At k = 0,1, , n 1
Then by choosing an appropriate At so that t = nAt where n is an integer and
letting s = mAt, it follows that t' = t s = (n m)At. The formula for Djk
can be approximated by
Id n
= ~ EE[(lAW((rc-m)At))-E[g(.YT((n-m)At))])
Z al m=0
x (vk(XT(nAt)) -E[Vk(XT(nAt))])\ At
The velocities in this representation are elements of the Lagrangian velocity
field. A distinction must be made between an Eulerian velocity field and the
72

Lagrangian velocity field. It is true that at any given point the fluid particle will
move with the velocity of the fluid at that point. But, since the fluid particle
is following a changing path through the porous medium, it will encounter
only a subset of velocities that comprise the entire velocity field of the domain.
This subset is the Lagrangian velocity field. The velocity field of the entire
domain is the Eulerian velocity field. In Section 1.11 the relationship between
the Lagrangian velocity covariance and the Eulerian velocity covariance was
described in terms of the probability density function of the fluid particless
position. Consequently, the statistical properties of the Lagrangian velocity
field may well be different from those of the Eulerian velocity field.
2.3.5 Dagans Approach
Formulations of the dispersion tensor in terms of the velocity covari-
ances appear quite often in the literature. Section 1.7 describes a version based
on arguments in Gelhar and Axness[45] and Neuman[69]. Dagan[32] offers an
approach that allows the specification of the dispersion tensor on a numerical
grid block by numerical grid block basis.
As discussed in Section 2.1.3, the second spatial moment, Sij, which
characterizes the spread of a mass around its centroid, is given by
Sij = Jf Jnn(X* ~ R*)(Xj ~ Rj)c(X,t)dX i,j = 1,2,3
where M is the mass, R is the centroid coordinate, c(X, t) is the concentration
and n is the porosity.
Since concentration is mass per unit volume, the second spatial moment of the
plume with respect to the centroid can be written as
Sij(t) = ^ ~ Ri(t)][Xj(t, a) Rj(t)]da
The Actual Dispersion Coefficients are defined as half the rate of change of
the plumes second spatial moment with respect to the centroid in the given
realization
1 dSij
2 dt
Since S^- is a random variable, Dagan[30, 32] defines effective dispersion coef-
ficients as:
73

1 dE[Stj]
13 2 dt
The key to Dagans final result is the fundamental relationship, Kitanidis[58],
Dagan[29],
E[Sl3(t)] = Sij(O) + Xl3(t, 0) ^{t)
From which it follows that
^ 1 dE[Sjj] 1 dX.ij 1 dRij
13 ~ 2 dT~ ~ 2 ~dT ~ 2 dt
If the rectangle V0 is of dimension li in the direction of the mean flow, x1} and
l2 be transverse to it, then the dispersion tensor components Dy(f,w) where
uj is the l\ x /2 rectangle are given by Dagan[30, Equation 15]
Dv(t,lnh) = 1^Jo l Jo(h-b1)(l2-b2)
x[pt3(E[V]t',0) l-p,l3(E[V]t'+ bub2)
~^pl3(E[V]t' -b^b^dt'dhdh
So, by assuming a particular autocovariance function (exponential or Gaussian)
the Dij(t,l2) can be solved for, and applied using the following steps:
Determine the log-transmissivity variogram
Calculate D,j for the assumed transverse dimension l2 of the numerical
blocks
Attach the resulting to rows of blocks at a distance x = || V||t from
the input zone
2.4 Distance Dependent Dispersivity Coefficients
In Gelhar[44] it is demonstrated that for a stratified aquifer to which
an hydraulic gradient which varies only in the z direction is applied parallel to
the layers, the variance of the displacement of a fluid particle is given in terms
of the hydraulic conductivity, K, by
a
2
x
a
K
E[/V]
:Ebl
2
74

And, using the definition of dispersivity this means that
D
1 dal 1 da^ dE[x]
2 dt 2 dE[x] dt
a
K
E [K]2
E[rr]E[t>]
This result shows the dependence of the dispersivity on the mean distance. The
following seeks to extend this result to higher dimensions. In order to determine
the effect of one variable on a second variable, it is desirable to have all other
variables held constant during the experiment. In economics this is referred
to as ceteris paribus. If the object is to determine the effect of permeability
or hydraulic conductivity on velocity, the experimenter would take a material
of known permeability, put it in a test tank, establish a hydraulic gradient in
the tank and measure the velocity. Next, a second material with a different
permeability would be placed in the tank, a hydraulic gradient of the same
magnitude as before would be established and the resulting velocity measured.
By maintaining a constant hydraulic gradient in both measurements, the effect
of permeability on velocity can be determined. Because we want to know only
the effect of permeability on velocity, the hydraulic gradient and conductivity
covariance will be assumed to be constant in the following calculations.
2.4.1 Local Grid Block Dispersivity Coefficients
The seepage velocity is given by
V = V(J)
n
And, the mean seepage velocity is given by
E[C]
E[K]
n
V
Let f be the position vector of a fluid particle at time t. This vector is given
by the kinematic relationship
rt , ft k ,
r= / V(t)dt = / V(/)dt
Jo Jo n
The deviation of the position vector f from its expected value is given by
f E[r\, and the matrix of covariances of these deviations is given by
75

X = E
= E
= E
(r E[r])(r E[r])f
( f\K-E[K])dt") ( f\K-E[K])dt'
n
10
n
m (K-m
t
/* r I K' ] ( K'-^ I dt"dt'
lo Jo \ n J \ n J
t rt
E
o Jo
V' V1
dt"dt'

But, since (E\Y'{x)V {y)\) = pij(x,y), this is basically the same form as
Equation[ 2.14], page 59. In order to generalize the one dimensional result,
we can argue as follows:
X =
E[(r E[f])(r E[f])'
= E
ft K
/ V(j)dt E
/ o n
f* Vdi
Jo n
ft K
/ V (f>dt E
lo n
r rt K ,1 v
< -e- R-
Jo n J
= E
(K-E[K])^j j(K-E[K])^J
tv \
n j
Letting K = K E[K], and noting that
E[f] = m[V] = E[K] V0
which implies that
n
-V(j> = E[K]_1E[fl
n
So
X
E
(KEtKj-^r]) (KEfK]-^^)1
Let A be an r x s matrix and B be a t x u matrix, then the Kronecker product
is the rt x su matrix
76

" nB ai2B aisB '
a2iB a22B 2SB
ari B ar2B * CLy* g ^3
In particular, if y G 9tr, 2; G 3?s, then y is r x 1 and Â£ is s x 1 so that y Is is
rs x s and given by
y is
" Vih '
V2h
. Vr^-s _
and Ir z is rs x s and given by
Â£ 0 0 0
0 z 0 0
0 0 0 z
Given that
X
then using the differentiation formula from Marlow[66] that if y G 5Rr, z G
16S", then
where
and
dz
[0) = {y Is) ^ + (Ir
dx
dx
dx
dy_
dx
' diyi d2yi
diy2 d2y2
. di yT d2yr
dny 1 '
dny 2
^nj/r .
diVj
01
dxi
77

In our case, we let r = s = n and
y = z = KE[K]-1E[f] = KE[f]
' E"=i K.jE^] '
Â£=i KE[r,]
. E=i K,yE[ri] _
which is an n x 1 vector. It has been shown that the dispersivity tensor contains
the term
Id 1 dX rfE[r] 1 dX -
2 dt 2 dEi[r\ dt 2 dE[r\
According to Marlow[66], the matrix ^ can be written as
r Mu. i
dt
dX12
dt
dX.
dt
dX\n
dt
dX 7J,1
dt
dXn2
dt
dXn
dt
which is n2 x 1. Also, is an n2 x n matrix and E[V] is an n x 1 vector so
that ^^E[E] is also n2 x 1.
Now,
E
E
(y Ir
dz
dE[r\
+ (1^
dy
dE[r\
Since y 0 In has dimension n2 x n and I n z has dimension n2 x n and
and have dimension n x n, the matrix has dimension n2 x n.
aE[r] 5 aE[r]
The components of this equation are evaluated as follows:
78

And,
y In
E=! KyEfr^In '
. E=1 KjE[rj]In .
z 0 0
0 z 0
0 0 Â£
where
' E"=i KiiE[r,] - ' 0 0
z = . E"=i KniE[r,] . and 0 = . 0 .
Furthermore,
dz
dE[r\
So, we can write
(y In)
Kn to
K2i k22
Knl Kn2
Â£"=i KijE[rj]In '
Ej=i KnjE[rj]I .
k13 K1-
K23 k2
K3
Kn k12 k13
to K22 k23
Ki Kn2 K3
K
nn
which is an n2 x n matrix. And,
(In Z)
dy
dE[r\
79
K> E*>

E?=i Ki;E[r;] 0 0
E=1 KiE[r:] 0 0
0 E"=i KyE[rj] 0 ' Ki! Kl2 Kl3 ' K
0 E"=i KnjE[rj] 0 K2i k22 K23 K
. Kni ~^-n2 K3 K
0 0 >:;, k /I'i/;
0 0 E"=1 k^-e^] _
which is also an n2 x n matrix. The result will be demonstrated using a two
dimensional example:
Local Grid Block
Example: n = 2
In this example we can write
iLx =
2 dt
1 dXn
2 dt
1 dX12
2 dt
1 dXai
2 dt
1 dX22
2 dt
1 dX
2 dE[r]
E[V] = E
(y h
dz
dE[r\
(h
dy
dE[r\
Expanding this yields
E[C]
80

1 d
2 dt
X =
ELi KuKyE[rj
-2E<
E]=i K21KliE[ri
Ei=i KnK2jE[r.-
_ Z2 K21K2jE[;
E'j=i KisKyE^]
Ej=i K22KliE[ri]
E-=i K12K2jE[rj]
EU K22K2jE[rj]
' EU KiiKi.EE-]
E?=i KnK2iEE]
EU K21Ki.EE-]
. E2=i K2iK2.-EE]
Ei=1 K12K..-EE-]
EU Ki2K2iEE]
Ei=1 K22K..-EE-]
EU K22K2iEE]
X
E[Ei]
E[U2]
Using the first order approximation of the expected value of the fluid particles
position, viz.,
E[f] = E [V]t
and assuming that Ky = 0 for E j, it follows that
E^EfE]H -
E[KnK22]E[Ui]E [V2\t
E[KuK22]E[Vi]E [V2]t
E[K22]E[U2]2f .
The presence of the expected velocity vector in this expression is key since it
can be changing from numerical grid block to numerical grid block. Figure
3 shows the effect of a non-constant expected velocity on the path of a fluid
particle. This figure shows the result of solving Equation[ 2.17] first with
E[U] = (0, 0.1), the dotted line, and then allowing E[U] = (0, 0.1) for the
first 25 time steps, E[U] = (0,0.0) for the next 25 time steps, E[U] = (0,0.1)
for the next 25 time steps, and finally, E[U] = (0, 0.1) for the last 25 time
steps, the solid line in Figure 3.
-X =
2 dt
81

0 0.02 0.04 0.06 0.08 0.1
82

2.4.2 Global Dispersivity Coefficients
The presence of heterogeneities in the porous medium will cause the
velocity field to be non-uniform. To maintain dispersive symmetry, the disper-
sion tensor should be recalculated on a grid block by grid block basis, taking
into consideration the expected velocity on the grid block. The preceding for-
mulation can be extended to this case in the following manner: Let Xt be the
trajectory of a fluid particle. Then, the position vector of the particle is given
It is assumed that V is steady state so that it does not depend on if. Fur-
thermore, it is assumed that the porous medium is locally homogeneous, i.e.,
in the sense of Section 1.1 and Section 1.5, and that this local homogeniety
applies to numerical grid blocks.
grid block 2, and from tn_i to tn on grid block n. Then since V0 does not
depend on time and the statistics of K do not depend on time on individual
grid blocks, we can write
by
Suppose that XT spends from t0 to t\ on grid block 1, from ti to t2 on
f =
then,
Using the formulas
(a + b) (c + d) = (a c) + (a d) + (b c) + (b d)
it follows that
83

X
= E[(f-E[rl)(r-E[rl)t]
{-E ,ti+1 ^ e[k(j+1)]^ dfl
. [ i=0 71 J
= E
[ 3+1 (K(J+1) E[K(J+1)]) dt
= E
j=o
H+1
v^'+1> }1
n J
- E 17' k^+i)dt' 777J j_Â£ [*i+1 K^+Vdf
i=0 Jt*
3=0 ^
n J
n1n1
= E EE
i=0 j=0
n1n1
= E EE
z=0 j0
n1n1
= EEe
i=0 j=0
V(/^+1) 11
n j
j fU+1 K^dt' ZEE 1 0 j fh+1 Kti+Vdf
[Jti n J
( y K(W y K*-
[ Jtj n J [ Jtj n J
2 \7^*+1) 1 f JL r*j+i
Y,iu"Kirdt'^
1=1
n
E
^m=l 0
*SJ>**
Define the vectors a and b such that for u, v = 1, 2
.=e r ks+i)*'
/=!*
and
K = t P' K^11*"
m=l 'h n
then
n1n1
X= ^ ^E[a(i+1)(6(2+1))t]
j=0 j=0
where
(<-+>(J<;+.>)t)o = Â£Â£
t=l m=l
rH1K^K^daf^^
u n n
84

If it is assumed that the components E[K^+1^K^+1)] are constant over blocks,
then
n1n1
(X) = E E E
i=0 j=0
2 2
E E At
1_1 _1 ll
(i+1)
Â£=1 m=l
And, using the relationship that
v^+1>
TO A -t
i+1---------AAIj+1
n
At-
E^] = Af*E[y] = E[K(l)]V0w
n
^Ati = E[K]_1E[r]
The displacement covariance matrix becomes
(X)u H iW iW II EK^1 (E[Ki*+1)]-1E[++1']) , E KS (E[KU>]-1E[++>])m
i=0 j=0 .1=1 TO = 1
or
X
n1n1
E EE
i=0 j=0
^Kd+bEfK^1)]-^^1)]) (K(j+1)E[K(j+1)]-1E[r To get an adequate representation of exactly where in the domain different
magnitudes of dispersion are to be expected, ensembles of particles must be
tracked. Such a representation is given in Figure 4. In this example, 5 particles
are tracked from two adjacent numerical grid blocks located in the center of
the domain.
These particle paths are used to identify the numerical grid blocks
that are most likely to be reached by a tracer plume that emanates from the
grid blocks containing the origin of the plume. Once the grid blocks most likely
to be reached have been identified, the previous formulation can be applied on
a grid block by grid block basis to estimate how dispersion will develop over
time. From Equation[ 2.24], page 71,
Dp-
f
o
Pij(E[V]t',-E[V]t)d1? =
J*E[V'i(E[V]t') V%(E[1?}t)]dH
||E[y]|| Jot H J
85

As time t' increases from 0 to t, the particle moves from its original position
at time t = 0 to its position at time t, in block (m). As it moves, it traverses
different grid blocks on its way to block (m). Using the arguments given
previously,
rnl
Dm e e
i=0
m 1
EE
i=0
m 1
i+i
(k(*+1) -E [k(i+1)])

n
x | (k*1 E [K*1]) Vl^|m) | dt'
A<.+1.ZAAAi>+d AwZAt'
n
n
(k(<+1)E [k(*+1)1 ^[r^+blj (K(m)E[K(m)] 1E [u(m)l
i = Q l \ Ju V
m 1
= ^ E [(K(m)E [f C+i)]) (KWE[yWj)
i=0
Or, in terms of the full tensor
i -i
rnl
D=^E (K(i+1)E [r(*+1)]) (K^E [U(m)])
i=0
Since it may be difficult to identify exactly the numerical grid blocks
where a local source is originating, or if more than one point source is involved,
an entire column of grid blocks can be used to determine the particle paths.
Figure 5 shows the result of tracking a particle from each of the grid blocks in
a column in the center of the domain. Of course, in a simulation, more than
one particle from each grid block would be tracked.
2.4.3 Molecular Diffusion
ft is reasonable to have some level below which these dispersivity coef-
ficients cannot fall. This level would, in effect, represent the level of molecular
diffusion. In order to specify such a level, an example from Batchelor[9] is
used. For a solute of NaCl in water, the coefficient of diffusion is found to
be 1.1 xlCU5 cm2/sec at 15 C and for any concentration. For molecules
such as potassium permanganate, KMnCU, which are much larger than wa-
ter molecules, the coefficient of diffusion is found to vary with the level of
concentration. Since this is not a problem with sodium chloride, the level of
86

ST

10 5 cm2/sec, or the equivalent level of 6 x 10 4 cm2/min, will be used in the
examples.
2.5 Random Variable Generation
2.5.1 Independent Random Variables
In order to generate the sample path of the stochastic process, it is as-
sumed that the stochastic process is Gaussian and stationary, he., at each time
t, the random variable X(t, u) has the same mean and variance and that these
two moments are sufficient to describe the random variabless distribution. A
sample path can then be generated by sampling from a Gaussian distribution
with the specified mean and variance. The following analysis shows that by
generating two independent uniform random variables on the interval [0,1]
and using the Box-Muller transformation, Equation[ 2.25], two independent
Gaussian random variables can be produced.
Let Ui and U2 be two independent uniform random variables with the
same density function on the interval [0,1]. Define the random variables
Vx = (-21n(Gi))^cos(2^y2)
X2 = (2 ln([/i)) 2 sin(27rLr2)
The inverse relations are given by
U\ = exp r-w + xi)
2
u2 1 /V9\
= arctan
2n \XJ
Then, by taking derivatives, it follows that
dX\ cos(27r[/2)
dUi ~ (2 ln(Ui))^Ui
^77- = (2 ln(C/i))^ (27T sin(27rf/2))
oU2
88
(2.25)
(2.26)

And, that
dX2
dUx
dX2
dU2
sin(27r U2)
(-21n((71))^1
= (2 ln([/i)) 2 (27t cos{2'kU2))
The Jacobian of the transformation is then
J =
S3; (-2">(^))J-2-n(2^))
ASSk (-21W))h2,cos(2 MJ)
2ir cos2(2kU2) 27t sin2(27rf/2)
t/i
?7i
-2tt
t/i
From Equation[ 2.26],
LJ = exp
Hence,
\J\ =
exp
{Xt + Xfi
2n
j A?+^2)j
If g(Ui, U2) is the joint density function of Ui and U2, then the joint density
function of AJ and X2 is given by
f(XuX2) =
From, Equation[ 2.26], it follows that
gjUuU*)
\J\
f(x 1,X2) =
g (exp [-M+Ail] 7 _L arctan ({*))
2tr
exp -
(xf+xh
89

And, we have that
and,
v xl5 x2,
exp
(^i2+*ir
2
e (0,1)
V xl5 x2,
iarctan(|5) 6 (0,1)
Since g is the joint density of two independent uniform random variables on
(0,1), it follows that
v xux2
9 exp
(^i2 + xl)
2
1 (X 2
arctan
2tt \Xi
1
Hence,
f(X i,X2)
(*i2 + *22)'
2
1 \ X*] 1 \ X'1
r- exp V2n 2 ' r- eXP V2n 2
which shows that X1 and X2 are independent and each has the standard normal
distribution. Hence, the two U(0,1) independent random variables are used to
produce two independent X(0,1) random variables.
Once an N(0,1) distribution has been produced, an N(jn, a2) distri-
bution can be produced by the transformation
V = ju + aX X~N(0,1)
Figure 6 illustrates this method by showing the results of generating 20000
samples from a Gaussian distribution with mean 3.0 and standard deviation
0.3. Plots of the two normal samples are shown along with their associated
lognormal plots. The sample lognormal points are found from the formula
lgn[z] = exp(n[*])
where n[z], i = 1, , m are the sample points from the normal distribution.
90

Figure 6 Sample Distributions
Normal
0 20 40 60 80 100
Normal
91

2.5.2 Correlated Random Variables
Once two independent normal random variables have been produced,
it is possible via a linear transformation to produce two correlated random
variables. Correlated random variables will be required in order to model an
anisotropic porous medium.
Given two independent standard normal random variables, X1 ~ 7V(0,1) and
X2 ~ N(0,1), two new random variables can be defined by letting
^3 CL -h bX\ + cX<2 a,b,c e 3?1
^4 = d + eX1 + fX2 d, e, f G 3?
then E[V3] = a, E[V4] = d and
B[X3X3] = (a + b)2 + c2
E[V4V4] = (d + e)2 + f2
E[V3V4] = ad + be + cf
Example:
Suppose we want to define two random variables V3 and V4 such that
E[x3] = 0 E[V4] = 0
E[V3V3] = 1 E[V4V4] = i E[.Y3.Y] 1 2
then a = d = 0 and
b2 + c2 = 1
e2 + /2 1 3
be + cf 1 9
92

Letting c=l=^& = 0=^/ = |=^e = so that
E[X3X3] = (a + b)2 + c2 = 1
E[X4X4] = (d + e)2 + f2 = -
e[x3x4] = ad + be + cf = J o
as required. Chapter 7 of Law and Kelton[61] gives many approaches to gen-
erating random variables, both correlated and uncorrelated. If a multivariate
normal distribution is to be used, then a particularly simple algorithm exists
for generating a multivariate normal vector
X = (X1,X2,---,Xn)l
The reason for this is that in the joint density function, Equation[ 2.19], page
64, the covariance matrix, V, is symmetric and positive definite. Hence, it can
be factored as
V = ccf
where C is lower triangular. The algorithm consists of the following two steps:
(1) Generate Zi, Z2, , Zn as independent identically distributed 7V(0,1)
random variables following the procedure in Section 2.5.1.
(2) For i = 1, 2, , n, set ^ + Â£}=i CijZj.
It then follows that
X = fl+CZ
In the study of porous media systems, the measurement of the physical
properties of the system at each point of the domain is a practical impossibil-
ity. In the case of these systems, it is common to assume that the physical
properties of the system, i.e., hydraulic conductivity, etc., are realizations of a
underlying random field. Since the development of our dispersion estimates is
based on assumptions about the covariance function describing the hydraulic
conductivities in the spatial domain, a method of generating random fields
that takes into consideration the degree of variance, correlation lengths, cross-
correlations, anisotropies, etc. of the hydraulic conductivity is necessary. Such
93

a method is provided by the Spectral Turning Bands method. In this method,
simulations are performed along several lines using a unidimensional covariance
function, C'i(-), that corresponds to the 2 or 3-dimensional covariance function
given for the spatial domain. Given two spatial points xi, X2 in the domain,
this correspondence is given by
C(xi, x2) = C(h) = [ Ci(h-u)f(u) du
J s
where E represents the unit circle or unit sphere, f(u) is the probability density
function of u, and h = xi x^. The value assigned to a point in the domain
is given by an average of the values generated for the projection of the point
onto the various lines used in the simulation.
Finally, the algorithm described in Robin, et a/[80] is capable of co-
generating pairs of three dimensional cross correlated random variables.
2.6 Summary
In this section, the all important dispersion tensor component of the
transport equation was investigated under some simplifying assumptions that
allow a better understanding of the concept. Section 2.1 looked at the disper-
sion concept from an Eulerian point of view which lead to its characterization as
half the time rate of change of the second spatial moment around the centroid.
Section 2.3 introduced the Lagrangian approach. Here the motion of a fluid
particle was described by a stochastic differential equation, Equation[ 2.17],
page 60, and dispersion was characterized as describing the uncertainty sur-
rounding the path of a fluid particle as it proceeds through the porous medium.
Under the assumption that the trajectory of the fluid particle has a
Gaussian probability density function, it was shown that the dispersion tensor
is equal to half the time rate of change of the covariances of the displacements in
the fluid particles position, which was then related to the velocity covariances
of the particle. Special consideration was given to a method due to Dagan that
allows the specification of the dispersion on a grid block by grid block basis.
In Section 2.4, a one dimensional result that describes the dispersiv-
ity as a distance dependent entity was extended to higher dimensions. This
method shows that the symmetry of the dispersion tensor is with respect to
the average velocity vector. If the dispersion tensor is changed on a grid block
by grid block basis, then the dispersion tensor must be recomputed to take into
consideration changes in the expected velocity on the block. Section 2.4.2 ap-
plies the results of Section 2.4.1, which are applicable to locally homogeneous
94

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AN ANALYSIS OF THE STOCHASTIC APPROACHES TO THE PROBLEMS OF FLOW AND TRANSPORT IN POROUS MEDIA by DAVID W. DEAN B. S., Illinois State University, 1967 M. S., University Of Illinois, 1969 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Doctor of Philosophy Applied Mathematics 1997

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This thesis for the Doctor of Philosophy degree by David W. Dean has been approved by Thomas F. Russell Leopolda P. Franca Karen Kafadar Thomas A. Manteuffel Steven F. McCormick Date ____________________

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Dean, David W. (Ph.D., Applied Mathematics) An Analysis Of The Stochastic Approaches To The Problems Of Flow And Transport In Porous Media Thesis directed by Professor Thomas F. Russell ABSTRACT One need in the current theory of subsurface transport in porous media is an improved understanding of the basic transport physics in highly heterogeneous subsurface environments using models that are valid at multiple scales. The thesis addresses this problem by first developing a theoretical back ground for the spectral representation of stochastic processes which are then used to illustrate the more common aspects of the theoretical descriptions of dispersion. The analysis shows how the dispersion tensor in the homogeneous case must be modified in order to include mildly heterogeneous permeability fields and provides a transformation law for the conversion of the spectrum of velocity perturbations to the spectrum of log hydraulic conductivities. This theoretical connection is important because in Chapter II a Lagrangian ap proach is used to develop a description of dispersion in terms of the covariance of the hydraulic conductivities using a particle tracking algorithm. Chapter III describes the numerical methods used to implement the algorithm. Chapter IV treats the transport equation using stochastic calculus, specifically Ito's lemma, from which weak formulations of the mean and covariance equations can be de rived. Chapter V considers the application of the theory of stochastic evolution equations to the problem of transport. By allowing both the dispersion and velocity to have random components, the evolution equation can be split into deterministic and stochastic parts. Using semigroup methods, the solution is given in terms of a Neumann expansion. Finally, Chapter VI uses the operator splitting method of Chapter V to illustrate a stochastic finite element method for solving the transport equation that uses the Karhunen-Loeve expansion, the Galerkin method and the Homogeneous Chaos spaces of Wiener. This abstract accurately represents the content of the candidate's thesis. I recommend its publication. Signed ____________________ __ Thomas F. Russell lll

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CONTENTS Chapter 1. Introduction 0 0 1 1.1 Overview 0 0 1 1.2 Stochastic Processes 13 1.3 Stochastic Measures 16 1.4 Process With Orthogonal Increments 0 21 1.5 Spectral Representation 0 0 0 24 1.6 Space Correlations And Space Spectra 27 1.7 Ergodicity 0 0 0 0 29 1.8 Stochastic Solute Transport And Dispersion 0 31 1.9 Comments And Limitations 0 0 0 38 1.10 Velocity /Permeability Covariance Relationship 39 1.11 Summary 0 0 0 0 0 43 20 Dispersivity Coefficients Time And Distance Forms 45 201 Time Dependent Dispersivity Coefficients 45 201.1 Introduction 0 0 45 201.2 Transport Equation 0 0 46 201.3 Spatial Moments Of The Solute Concentration 46 202 Stochastic Differential Equations 53 20201 Introduction 0 0 53 20202 Integral Of A Stochastic Process 53 20203 Wiener Process (Brownian Motion) 54 202.4 Stochastic Integration 0 0 55 20205 Types Of Stochastic Integrals 0 56 203 The Concentration Equation 0 0 56 20301 The Lagrangian Approach 57 20302 Basic Form Of Transport Equation 0 61 20303 Solution Of Basic Transport Equation 64 203.4 Dispersion As Velocity Covariances 0 69 20305 Dagan's Approach 0 0 73 2.4 Distance Dependent Dispersivity Coefficients 74 2.401 Local Grid Block Dispersivity Coefficients 75 2.402 Global Dispersivity Coefficients 0 81 2.403 Molecular Diffusion 0 0 86 205 Random Variable Generation 0 0 86 20501 Independent Random Variables 0 86 IV

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2.5.2 Correlated Random Variables 2.6 Summary . 3. Methods Of Solution . . 3.1 General ............ 3.1.1 Simultaneous Solution 3.1.2 The Mixed Model ... 3.1.3 The Spaces Qh And Vh 3.2 2D Finite Element Solution 3.2.1 General ....... 3.2.2 Finite Elements 3.2.3 Rectangular Elements 3.2.4 Numerical Integration 3.2.5 Groundwater Flow Equation 3.2.6 Matrix Assembly . 3.2. 7 Velocity Field . . 3.2.8 The Transport Equation 3.2.9 Imposition Of The Boundary Condition 3.2.10 The Boulder Experiments 3.3 Summary . . . . . 4. Moment Equations . . . . 4.1 Moments Derived From Distributed Parameters 4.2 An Ito Calculus Approach 4.2.1 System Definition 4.2.2 Types Of Problems 4.2.3 Existence Theory 4.2.4 Stochastic Integration 4.2.5 Ito's Lemma In Hilbert Space 4.2.6 Small o Notation ...... 4.2. 7 Hilbert Space Structures 4.2.8 Moment Equation Derivation 4.3 Summary . . . 5. Stochastic Evolution Equations . 5.1 General Theoretical Foundations .. 5.2 Application To Transport And Scale-Up 5.3 Stochastic Parameters 5.4 Formal Solution 5.5 Convergence 5.6 Summary 6. Future Research 6.1 General .. v 90 93 95 95 95 99 102 104 104 104 106 108 110 116 119 120 122 124 128 131 132 138 138 138 139 142 147 150 151 155 172 174 174 185 187 188 203 214 215 215

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602 Homogeneous Chaos 0 217 603 Stochastic Finite Elements 221 6.4 The Covariance Function 224 References 0 227 Vl

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1. Introduction 1.1 Overview This study is concerned with the basic forms of the equations of flow and transport of solutes or contaminants through porous media. Whenever two miscible fluids come together to form one phase, there is potential for either the density of the phase or the viscosity of the phase to change. This change can be brought about by changes in the concentration and/ or changes in the pressure. In the case that the density is dependent on the concentration of the pollutant, a coupled system of PDE's is obtained that must be solved simultaneously, see Equation[ 3.3], page 99. Using Darcy's law and the Hubbert potential for a compressible fluid, the velocity can be related to the pressure via the permeability of the medium, see Equation[ 3.1], page 97. This equation can be solved for the velocity, pressure pair by the mixed finite element method. In the case of constant density, the situation is less complicated. This is referred to as the tracer case. Here, the system can be written as an uncou pled system consisting of the flow equation, which can be solved for the piezo metric head, and the transport equation. Once a velocity field has been derived from the piezometric head distribution, the transport equation is solved for the concentration plume that evolves over time. Section 3.2 discusses a 2D finite element implementation of the tracer case. This model is subsequently used to simulate horizontal tank experiments conducted in Boulder at the University of Colorado's Civil Engineering Department under the direction of Professor Tissa Illangesekare. The Boulder experiments are discussed in Section 3.2.10. Briefly, the tank environment is like that of a confined aquifer into which a tracer (pollutant) is injected. The tank has 45 port locations where tracer in jections can be made and samples can be taken. Constant head conditions are assumed on the ends of the tank, and no flow boundary conditions are assumed on the sides of the tank. Both homogeneous and heterogeneous experiments are run in the tank. In the homogeneous case, the tank is packed with a uniform sand, i.e., of uniform hydraulic conductivity. Spectral methods can be used to study the homogeneous packing. As would be expected, Fourier series can be used to solve the 1D flow equation resulting in a solution of the form 1

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X cjJ*(x, t) = t) + *(0, t) + L (cjJ*(L, t)(0, t)) where By letting t ----+ oo, the steady state solution is then linear. In the two dimen sional case, the constant head contours are nearly parallel, yielding a uniform velocity field. Both the model results and the experimental results show these characteristics. The velocity field is determined from the flow equation or the ve locity /pressure equation described earlier. However, prior to solving for the concentration distribution using the transport, advection-dispersion, equation, the dispersion parameter must be provided. The first two chapters of this study focus on the determination of the dispersion parameter. For homogeneous me dia, spectral methods can again be used to develop a theoretical formulation for the treatment of dispersion. These spectral methods can be extended to include mildly heterogeneous media by using stochastic process techniques. A stochastic process can be given a spectral representation defined in terms of special stochastic processes that have orthogonal increments. The first step of the procedure used to construct an integral spectral representation is to define orthogonal stochastic measures, Section 1.3. Once this is done, it can be shown that there is an isomorphism between processes with orthogonal increments, continuous from the right, and the orthogonal stochastic measures. It is this association that makes the integral spectral representation work, Sections 1.4 and 1.5. Based on a review of the literature, it is our view that the theoretical descriptions of dispersion in porous media found in Gelhar and Axness[45], Neuman and Zhang[70] and Dagan[29, 30, 31, 32, 33, 34] are the most prominent. All of these theories link dispersion to the hydraulic conductivity properties of the media. However, there are fundamental differences which have generated a lot of debate in the literature. Section 1.8 seeks to select the common aspects of the Gelhar and Axness[45] and Neuman and Zhang[70] approaches which are based on the integral spectral representations of stochastic processes. The analysis shows how the dispersion tensor associated with the homogeneous case must be modified to include mildly heterogeneous cases, Equation[ 1.13], page 2

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37. The modification is in terms of the integral of the spectrum of the velocity perturbations. Section 1.10 provides a transformation that allows the conver sion of the spectrum of the velocity perturbations to the spectrum of the log hydraulic conductivity perturbations, Equation[ 1.21], page 43. This theoreti cal connection is important because in Section 2.4 a description of dispersion in terms of the covariance of the hydraulic conductivities using a particle tracking algorithm will be developed. Chapter 2 discusses the time and distance forms of the dispersivity tensor. In this analysis, the solute body spatial moments are given by M = k ncdx .... 1 { .... d"""' R = M ln nxc x S = 2_ f n(x-R)(x-R)c(x t)dx zJ M ln z z J J i,j = 1, 2, 3 Here M is the mass, R is the centroid coordinate, Sij is the second spatial moment which characterizes the spread around the centroid, n is the porosity and c is the contaminant concentration. Starting from the transport equation with V representing fluid velocity, oc .... -+ V. V'c = V'. (DV'c) at and using integration by parts, it can be shown that Zero Moment First Moment Second Moment Conservation Of Mass Centroid Of The Mass Concentration Moves With Velocity V. D-.!_dS;j ZJ -2 dt Although the result that dispersion is related to the time rate of change of the second moment makes intuitive sense, in order to implement such a definition requires knowledge of the plume that is probably not avail able. For this reason, dispersion is dealt with as a stochastic entity. Dealing 3

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with the problem from a stochastic point of view allows a certain level of uncertainty to be accounted for by the model. This means that instead of deterministic ordinary and partial differential equations, the parameters and variables in the differential equations will be allowed to have stochastic com ponents. This switch to stochastic differential equations brings with it many difficulties that require a stochastic calculus to handle. For example, in dis cussing the movement of a fluid particle through a porous medium, stochastic differential equations of the form dX(t) = a(t, X(t))dt + b(t, X(t))dW(t) will arise where X(t) is the trajectory of the particle through the medium and W(t) is a special type of stochastic process called a Wiener process. This equation has the interpretation on the interval [0, t] of X(t) = X(O) +lot a(s, X(s))ds +lot b(s, X(s))dW(s) The integrals in this equation cannot be interpreted in the usual sense. In the case of the first integral on the right hand side, the integrand, a( s, X ( s)) is a random function, i.e., for a givens, a(s, X(s)) is a random variable. The second integral is even more difficult to deal with since the measure part of the integral is a stochastic process which can be shown to have infinite variation. Hence, the usual Stieltjes interpretation is not applicable. How these integrals are dealt with is discussed in Section 2.2 and again in Section 4.2.4. As will be seen in those sections, the stochastic differential equation and stochastic integral have more than one interpretation. Section 2.3 is the first section in which the Lagrangian approach is used to give the basic form of the transport equation and the basic form of the dispersion tensor that can be derived from it. In the Lagrangian framework, transport is characterized in terms of indivisible solute particles i.e., ensembles of molecules in a small volume, which are transported by the fluid. The total displacement of the fluid particle can be decomposed into a component due to convection and a component due to diffusion, Equation[ 2.12], page 58. The diffusion component is represented by a Brownian motion. The convective component is described by the fundamental kinematic equation X(t) =lot V(Xr(t'))dt' where V (Xr) is the Lagrangian velocity field associated with the fluid particle. Assuming the first order relationship between the displacement of X' x'' and the displacement of v' v'' is given by 4

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it follows that the displacement covariance tensor is given by the time integra tion of the velocity covariances, Equation[ 2.14], page 59. This characterization of the displacement covariance will play a central role in the development of the dispersion tensor. As will be seen in Section 3.1.1, the transport equation is usually derived from conservation of mass considerations. However, in Section 2.3.2 it will be demonstrated that the basic form of the transport equation can be derived by treating the fluid particles as obeying the following Ito stochastic differential equation Then, the transport equation follows from the Fokker-Planck or Kolmogorov forward equation. This derivation is restricted to the case where dispersion is created only by the Brownian motion, Xd. In Section 2.3.3, the solution of the transport equation is shown to be a multivariate Gaussian density function, and the dispersion tensor is shown to be one-half the time rate of change of the covariances of the trajectory dis placements. Section 2.3.4 expands on this result to show that the components of the dispersion tensor should be related to the time rate of changes in the covariances of the fluid particle or the time integral of the velocity covariances which approximates the path integral of the velocity covariances along the fluid particle's trajectory. The integrand of this integral is a lagged covariance over the interval of integration. A numerical formulation of this integral is given which depends on a particle tracking algorithm which is suitable for implementation in a computer code. Section 2.3.5 is an extension of these results due to Kitanidis[58] and Dagan[32]. One of the objectives of this study is to be able to specify the disper sion coefficients on a numerical grid block basis so that they can be used in a finite element model. Section 2.4 contains the details of a proposed method of doing this. By dividing the domain of the problem into numerical grid blocks on which it can be assumed that gradients and hydraulic conductivity covariances are constant, the local displacement covariance tensor, which is represented in the literature as a bold-faced X, can be expressed as 5

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where K K -E[K] and K is hydraulic conductivity and the dagger, t, represents the vector transpose. Since fin this expression is the position vector of the fluid particle with reference to some injection point, this is really a distance dependent formula. One of the conclusions of experimental studies is that dispersivity varies with the distance from the input zone. The manner in which it varies depends on the degree and location of heterogeneities in the porous medium domain. The presence of heterogeneities in the porous medium will cause the velocity field to be non-uniform. In order to get an adequate representation of exactly where in the domain different magnitudes of dispersion are to be expected, ensembles of particles must be tracked. Figure 4 in Section 2.4 illustrates the tracking of 5 particles from each of two adjacent grid blocks located near the center of the domain. These particle paths are used to identify the numerical grid blocks that are most likely to be reached by a tracer plume that emanates from the grid blocks containing the origin of the plume. Once the grid blocks most likely to be reached have been identified, the previous formulation can be applied on a grid block by grid block basis to estimate how dispersion will develop over time. In this case, the displacement covariance matrix is given by E [ (:k(i+l)E[K(i+l)tlE[r\i+l)J) (:K(j+l)E[K(J+l)tlE[r\Hl)J) t] i=O j=O where the superscripts represent individual block designations. Dispersivity estimates are derived by differentiating this expression with respect to time. Since it may be difficult to identify exactly the numerical grid blocks where a local source is originating, or if more than one point source is involved, an entire column of grid blocks can be used to determine the particle paths. Figure 5, Section 2.4, shows the result of tracking a particle from each of the grid blocks in a column in the center of the domain. Of course, in a simulation, more than one particle from each grid block would be tracked. Section 2.5 discusses some methods of random variable generation that are either currently being used or are planned to be used in the models of flow and transport being created. Since all of the methodology is based on exploiting randomness or uncertainty, it is necessary to have methods of simu lating this randomness. In Section 2.5.1, it is shown that by using a Box-Muller 6

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transformation, two independent uniform random variables can be converted into two independent Gaussian random variables. It is also shown, Section 2.5.2, how correlated random variables can be simply generated. However, for more sophisticated simulations in 3 dimensions, efficient computer codes are available that are capable of cogenerating pairs of 3 dimensional, cross correlated random fields with different correlation scales, Robin, et al[80]. Chapter 3 discusses the numerical method being used to study the groundwater flow /transport problem. The particular form of the equations that are to be solved depends on the relationship that is assumed between the density, viscosity and concentration. For example, in an Enhanced Oil Recovery problem, a change in the viscosity of the single phase is brought about by the mixing of a surfactant with the oil in the reservoir. In problems that involve a pollutant entering an aquifer, many times it is the density that changes with the concentration of the pollutant. In these cases, it can be argued using standard definitions of the physical properties of a compressible fluid that the seepage velocity is related to hydraulic conductivity, pressure and density; or to the permeability, pressure, density and viscosity, Equation[ 3.1], page 97. The transport equation is usually derived by applying the Divergence theorem to the Conservation of Mass law. These equations result in a coupled system of equations, Equation[ 3.2], page 98. For this coupled system, the mixed finite element method can be used to solve for the pressure/velocity pair, followed by a solution of the transport equation by some method. Currently, the code that is available to solve the coupled system is the SEGMIX code. This code uses mixed finite elements to solve for the pressure/velocity pair and the modified method of characteristics (MMOC) to solve the transport equation. SEGMIX assumes a rectangular domain with no flow boundary conditions on the sides. In order to use this code to simulate the Boulder tank experiments, it would have to be modified to accept constant head conditions at the ends of the tank and no flow boundary conditions on the sides of the tank. The types of experiments that are being conducted in the Boulder horizontal tank are tracer experiments. This means that a solute such as benzene or sodium chloride is injected at a selected port and samples are taken from a port downstream of the injection port. In this case, the effect on density is probably minimal, and the uncoupled flow /transport system of equations is adequate to study the experiments numerically. The numerical method being used to solve the uncoupled system is finite elements. Because Gaussian integration formulas are used to evaluate the 7

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integrals that arise in the system of equations resulting from the variational formulation, a reference element which is convenient for integration purposes is defined which is affinely equivalent to the elements in the domain. Affine equivalence can be defined in terms of special mappings called pull-backs and push-forwards. These concepts are explained in Sections 3.2.2 to 3.2.4. Section 3.2.5 contains a derivation of the local system of equations that arise from the variational formulation of the flow equation, Equation[ 3.10], page 116. Once the systems of equations on the local rectangular elements have been established, they must be assembled into a global system of equations for the whole domain. This process is described in Section 3.2.6. The derivations of the velocity field from the piezometric head estimates is given in Section 3.2.7. Section 3.2.8 shows the derivation of the system of equations that follow from the variational formulation of the transport equation. Since there are constant boundary conditions at the ends of the tank, and a pulsed-input is allowed to take place at an injection point, it is necessary to allow constant concentration conditions to exist at some grid points. The modification of the global system of equations to allow certain grid points to maintain a constant level of concentration is explained in Section 3.2.9. Section 3.2.10 describes in more detail the horizontal test tank used in the Boulder experiments. Two types of experiments are conducted in the tank. The homogeneous experiments are those in which the tank is packed with a single type of sand as rated by its hydraulic conductivity. In the heterogeneous experiments, the tank is packed in a block arrangement with 5 different types of sand. The hydraulic conduc tivities of the sands range from 3.618 m/day for Sand #1 to 1036.8 m/day for Sand #5. With this wide span of hydraulic conductivities, a significant amount of heterogeneity is represented in the tank. The block arrangement of the sands in the tank is represented graphically in Figures 13 and 14 in Section 3.2.10. Figure 15 provides a flowchart of the basic program components used and how they interact. Comparisons of computer simulation results shown in Figures 16 and 17 to actual tank measurements show very good agreement. Figure 18 illustrates a computed tracer plume. Chapter 4 actually starts the second part of the thesis. The previous sections have investigated the components of the equations and the forms of the equations. However, only the expected or mean value of the concentration is predicted. Because of the uncertainties involved in specifying the physical characteristics of the porous medium, the concentration of a solute at a given 8

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point in time is a random variable, and over a period of time it is a stochastic process. Consequently, in order to more accurately characterize the distribu tion of the solute concentration, higher order statistical moments such as the variance need to be estimated also. In theory, the more moments that can be predicted, the better this characterization will be. But, in practice, it is usually a difficult problem just to obtain information on the variance or covariance of variables in the system. A much referenced paper in this area is the Graham and McGlaughlin[48] paper which specifies a set of three equations that are to be solved for the mean concentration, the velocity-concentration covariance and the concentration co variance. Section 4.1 derives and discusses these equations because they will be used as a basis of comparison for an approach to developing moment equations based on the Ito calculus. Randomness can enter the boundary value problem in many different ways. Equation[ 4.12], page 138, is a statement of the stochastic boundary value problem, and the discussion following that equation specifies the various ways in which randomness can enter the picture. Existence theory for the stochastic boundary value problem is not unlike the nonstochastic case. A summary is included in Section 4.2.3. Stochastic integration is again addressed in Section 4.2.4, this time from the more general perspective of a martingale. The Ito integral then follows from this more general definition as a special case. The use of the Ito integral requires that the rules of calculus have to be modified. The reason for this can be illustrated as follows: Suppose that W(t) is a one dimensional Wiener process, then it is well known that it can be represented as the limiting form of a random walk. And, as part of this limiting process the step size of the random walk goes to zero as the square root of the time interval 1 = Gardiner[42] shows that for the Ito integral this property means that [dW(t)]2+N = { dt N = 0 0 1f N > 0 and that dt dW(t) = 0. The most important new rule is that of Ito's lemma. It is a change of variable formula. The reason the change of variable formula 9

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has to be modified is due to the above differential relationships. For example, iff is a smooth function and X(t) satisfies the Ito differential equation dX(t) =a dt + b dW(t) Then, on expanding df[X(t)] df[X(t)] = f[X(t) + dX(t)]-f[X(t)] f'[X(t)]dX(t) + f"[X(t)] (dX(t))2 + ... 2 f'[X(t)] (a dt + b dW(t)) + (a dt + b dW(t))2 + And, using the differential relationships, it follows that df[X(t)] = f'[X(t)] (a dt + b dW(t)) + b2dt which is different from the ordinary calculus rule. The Ito formula is a stochas tic calculus chain-rule. It can also be extended to martingale type processes, Karatzas[57]. Curtain and Falb[26] have extended Ito's lemma to infinite di mensional Hilbert spaces. It is this form that is used to derive weak forms of the moment equations in Sections 4.2.5 to 4.2.8. For the purpose of illustrating this theory, the key equation is Equation[ 4.17], page 158, which is applied to two examples. The first example uses this theory to derive mean and covari ance equations that in the weak form are identical to those used by Graham and McLaughlin[48]. The second example is cast in terms of accounting for the effects of measurement error that is assumed to enter the experiment as a random perturbation that takes the form of a Wiener process. Chapter 5 considers the application of the theory of stochastic evolu tion equations to the problems of flow and transport. In the deterministic case, it is well known that the boundary value problem can be recast as an abstract evolution equation or Cauchy problem. Such a problem takes the form of du dt + A(t)u = f(t) u(O) uo 10

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The term abstract is attached to indicate the fact that the functions involved are mapping a time interval, [0, T] c into a Banach or Hilbert space. Hence, for a Hilbert space H, the function u(t) is an H-valued function. It follows, then, that there must be an association between the abstract functions u(t) and the real-valued functions u(x, t) of the boundary value problem. And, there must be an association between the differential operator of the boundary value problem and the operator A(t) of the abstract evolution equation. Section 5.1 explores these connections and the forms of the solution to the abstract evolution equation for both the autonomous case, A independent oft, and the nonautonomous, A(t) is dependent on t, cases. The solution, in general, is given as an integral equation. Curtain and Falb[27] are able to extend these results to the case where the forcing term of the abstract evolution equation contains an H-valued Wiener process, as defined in Section 4.2.4. The solution in this case is given in terms of an evolution operator U(t, s) generated by -A(t) as u(t) = U(t, O)u0 + ht U(t, s)(s)dW(s) where the integral is now a stochastic integral. Working from the Curtain and Falb solution, weak forms of the mean and covariance equations for the solution u(t) are found that agree with the forms of these equations for the finite dimensional case as given in Astrom[8]. In Section 5.2, the time stochastic process nature of the dispersion tensor is again considered. The fact that the dispersion tensor could be con sidered as a time dependent quantity was first shown in Section 2.1. But, since then, it has been treated in the models developed up to this section as an effective parameter by some type of averaging process. In the following sections, the time dependent dispersivity coefficient will be incorporated into the basic partial differential equation model. Both the dispersion coefficients and the velocity will be allowed to have random components. Section 5.3 gives the general form of the stochastic PDE that results in terms of the sum of deterministic and stochastic operator components. Section 5.4 represents the solution of the stochastic PDE as an integral equation which has the form u(t) = U(t, O)u0 + {t U(t, s)g(s)ds-{t U(t, s)Rs xu(s)ds lo lo 11

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where g( s) is the forcing function which may be stochastic, Rs,x is the stochastic operator component and U(t, s) is the evolution operator that is the evolution operator that is derived from treating the deterministic part of the stochastic PDE as an abstract differential equation. As an example of this procedure, the following 1-D transport equation is solved au ( I ) a2 U ( I ) au at-E[D] +D (t,w) ax2 + E[V] + V (w) ax= 0 Since it is assumed that the E[D] is a constant in this example, the evolution operator U ( t, s) can be expressed as a strongly continuous semigroup generated by the operator a a2 A -E[V]+ E[D]ax ax2 Using a change of variables to a moving coordinate system, the semi group is calculated and its properties verified. The solution is then given as an integral equation that has the form of a Volterra type two equation. Section 5.5 applies the classical integral equation methods to characterize the convergence property of the series solution to the integral equation, Hochstadt[53]. In order to provide some insight into the speed of convergence of this series, a test problem is constructed and three scenarios tested. The first case allows the dispersivity to have a random component, but not the velocity. The second case allows the velocity to have a random component, but not the dispersivity. Finally, the last case allows both the dispersivity and the velocity to have random components. The results are contained in the tables at the end of Section 5.5. Monte Carlo methods can be used to generate several sample paths for the random components, and from these, concentration means and variances can be calculated. The Neumann expansion procedure discussed in Sections 5.4 and 5.5 can be extended to a stochastic finite element method by combining the Karhunen-Loeve expansion, a Fourier type expansion of a stochastic process, with the Galerkin method. This approach is, however, subject to the same con vergence criterion discussed in Section 5.5. Instead of pursuing the Neumann expansion any further, Chapter 6 on Future Research describes an alternative method which is based on the following three components: (1) The Karhunen-Loeve Expansion (2) The Galerkin Method (3) Wiener's Homogeneous Chaos Spaces 12

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As described in Section 5.4, the formal solution of the equation is developed by splitting the operator .Ct,x into a deterministic part and a zero mean stochastic part, i.e., and using the Karhunen-Loeve expansion to express the random coefficients in the stochastic component, Rt,x, and the Homogeneous Chaoses of Wiener to represent the solution. As described in Section 6.1, the Karhunen-Loeve expansion requires a knowledge of the covariance function of the stochastic process. Specifically, the expansion requires that the eigenvalues and eigenfunctions associated with the covariance function of the process being represented be known. Although certain assumptions can be made regarding the covariance function associated with the stochastic coefficients of the stochastic operator, the covariance function of the solution process is not known. What is required is a way of representing the solution process that does not require a knowledge of its co variance function. The principal result in this approach is the Homogeneous Chaos decomposition of the L2-space of a Gaussian process. The fundamentals of this type of decomposition are provided in Section 6.2. The stochastic finite element method using Homogeneous Chaoses is outlined in Section 6.3. Since the success of this finite element procedure depends on obtaining the eigenval ues and eigenfunctions of the covariance function, some details of this problem are covered in Section 6.4. 1.2 Stochastic Processes Some of the more famous and fundamental results on contaminant transport through porous media involve the characterization of dispersion in terms of the spectral representation of stochastic processes, Gelhar and Axness[45], Neuman, et al[69]. This introduction will trace the development of these spectral representations from their beginnings in classical functional analysis to their use in deriving fundamental theoretical formulations of the dispersion tensor. In doing so, the key role that the dispersion tensor plays in the process of scaling-up from small homogeneous laboratory experiments to larger het erogeneous field problems will become clear. The following definitions can be found in many references, for example Burrill[17], Doob[36] and Todorovic[96]: 13

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Definition: Let (D, B, P) be a probability space. A real (complex) valued measurable function with domain D is a real (complex) random variable. Definition: A real (complex) stochastic process is a family of real (complex) random variables {X(t) : t E T} defined on a common probability space (D, B, P) with T c lR1 Definition: A stochastic process is said to be strictly stationary if its dis tributions do not change with time, i.e., if for any t1 t2 tn E T and for any h E T, the multivariate distribution function of the random variable (Xt1 +h, Xtn+h) does not depend on h. Ex: Definition: A stochastic process { Xt : t E T} is wide sense stationary if E[IXtl2 ] < oo E[Xt+n Xt] = E[Xn Xo] Vt E T where E represents the expected value. Doob[36], shows that for wide sense stationary processes, the following holds: Proposition If {X(t); t E lR1 } is a process which is wide sense stationary, then there is a group of unitary transformations {Ut; t E lR1 } such that for each t, U0 Identity Element U -t Inverse Of Ut Using the notation (, ) to represent the Hilbert space inner product, the following spectral representation theorem exists for unitary operators, Riesz and Sz.-Nagy[79], Section 137, 14

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Stone's Theorem Every one-parameter group {Ut; t E lR1 } of unitary trans formations for which (Utf, g) is a continuous function oft, for all elements f and g, admits the spectral representation where { E.x} is a spectral family. Furthermore, E.x is uniquely determined. E.x is an orthogonal projection with the properties (1) E.x :::; EM for .\ < f.t (2) EA+o = E.x (3) E.x ---+ 0 for.\---+ -oo ( 4) E .x ---+ I for.\ ---+ oo A second theorem related to this is, Riesz and Sz.-Nagy[79], Section 138, Bochner's Theorem In order for the function p(t) ( -oo < t < oo) to admit the representation with a nondecreasing and bounded real function V(.\), it is necessary and sufficient that p(t) be continuous and nonnegative definite in the sense that m L p(tp,-tv)Pp,Pv 0 p,,v=l whatever the positive integer m, the real numbers t1 t2 tm and the complex numbers PI, P2, Pm The reason that these two theorems are connected is that the function p(t) = (Utf, f) is nonnegative definite, and from this it follows that, Riesz and Sz.-Nagy[79], Section 138, 100 '.At (Utf,g) = -oo d(E.xf,g) (1.1) where E.x is a spectral family. 15

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From the polarization identity, the following holds in a complex Hilbert space (E;.J,g) ( g) + ( g) In particular, if g = f, then and Equation[ 1.1] then gives (1.2) From the properties of the spectral family, and, if.\ 2: JL then E;.. 2: E11 =* (E;..j, f) 2: (E11j, f), so that IIE;..JII2 is real and nondecreasing. Furthermore, so that IIE;..JII2 is also bounded. Equation[ 1.2] will be useful in the discussion of the Bochner-Khinchin theorem in Section 1.4. 1.3 Stochastic Measures In the preceding section, it was shown that a stochastic process X(t) that is wide sense stationary has the representation 16

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The process { E>..X (0) : A E lR1 } turns out to be a special type of stochastic process. Also, in order to be able to interpret this integral in the usual sense, this process must be associated in some way with a type of measure. In this section a stochastic measure and a stochastic integral are defined that allow the representation as a stochastic process in terms of a unique orthogonal stochastic measure which corresponds to a stochastic process with orthogonal increments, Section 1.4. The details of this section on Stochastic Measures are found in Todorovic[96]. Definition: A complex-valued random variable Z on (0, B, P) is called second order if The family of all such random variables is denoted by L2{0,B, P} Todorovic[96] shows that given the definition of equality that ie, they differ on at most a set of measure zero, so that the space L2{0, B, P} consists of equivalence classes, and using the inner product definition where the symbol represents complex conjugation, then L2(0, B, P) 1s an inner product space. The norm for this space is defined as 1 1 IIZII = +(Z, Z)2 = E [IZI2 ] 2 Using this norm to define distance, L2(0, B, P) is a metric space. The Cauchy-Schwarz inequality holds, Todorovic[96], and the inner product is uniformly continuous, for if Z0 Z1 Z2 E L2{0, B, P} 17

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The mode of convergence on this metric space is that of mean squared convergence and defined as (m2 ) lim Zn = Z iff IIZn-Zll ---+ 0 as n---+ oo n--+oo And, this definition of convergence leads to the following form of the Riesz-Fisher Theorem If { is a Cauchy sequence, then there exists a Z E L2{0, B, P} such that Let { S, S} be an arbitrary measurable space, and let S0 be the algebra of sets that generates the a-algebraS. Definition: A mapping such that TJ : S0 ---+ L2{0, B, P} TJ(0) TJ(A u B) 0 TJ(A) + TJ(B) (a.s.) for any disjoint sets A, B E S0 is called an elementary random or stochastic measure. Definition: Let m(A) = IITJ(A) 112 < oo for any A E So where II 112 = E[l 12 ] Definition: An elementary random measure is said to be orthogonal if (TJ(A), TJ(B)) = 0 V disjoint A, BE S0 It is clear from the orthogonality property of the random measure TJ( ) that the set function m( ) is finitely additive. And, if it is assumed that m( ) is 18

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countably subadditive, then it can be extended to a measure on S0 According to Proposition 12.3.9 of Royden[81], all that is required is a semialgebra of sets C that generates S0 and that on C, m(0) = 0, m() is finitely additive on C and countable subadditive on C. Furthermore, the measure m() so defined on S0 can then be extended to the measurable space {S, S}. Definition: m() is called the measure associated with TJ(). Let the Hilbert space of complex-valued functions on S which are square integrable with respect to m. Definition: For step functions n h(s)='Lck!Bk(s) disjoint BkES0 k=l define It is clear that the mapping 'ljJ takes step functions in L2 ( m) and maps them into random variables in L2(0, B, P). Furthermore, given two step functions h(s) and f(s) n n L L Xi]/j (TJ(Ai), TJ(Bj)h2(0,B,P) i=lj=l and, since the sets Ai and Bj can be written in terms of disjoint sets as 19

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it follows from the orthogonality of ry() that the inner product (ry(Ai), ry(Bj)) L2(n,B,P) can be written as = m(Ai n Bj) consequently, the inner product ('1/J(h), '1jJ(f)h2(n,B,P) can be written as n n ('1/J(h), 'I/JU)h2(n,B,P) = L L xiyjm(Ai n Bj) i=lj=l is h(s )f(s )m(ds) = (h, Jh2(m) Therefore, the mapping 'ljJ preserves the inner product of the step functions from L2(D, B, P) to L2(m). If g E L2(m), then let be a sequence of step functions such that llg-hnll --+ 0 as n--+ oo then from the orthogonality of ry() 11'1/J(hn) '1/J(hm) 112 = llhn-hmll2 --+ 0 as m, n--+ 00 So, { is a Cauchy sequence, and by the Riesz-Fisher Theorem, :3 '1/J(g) E L2{D, B, P} such that 11'1/J(g) '1/J(hn) II --+ 0 as n --+ oo Definition: The random variable '1/J(g) is called the stochastic integral of g E L2(m) with respect to the elementary orthogonal random measure 'T/ It is denoted by '1/J(g) = fsg(s)ry(ds) (1.3) The elementary stochastic measure 'T/ which is defined on S0 can be extended to the a-algebraS, Todorovic[96]. Hence, Equation[ 1.3] is taken to be the integral of a g E L2 ( m) over the set S with respect to the orthogonal stochastic measure 'Tl 20

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1.4 Process With Orthogonal Increments Let {S, S, m} = {R\ n, m} where R1 Real Line, and n aalgebra of Borel sets of R1 Let { Z(t); t E R1 } C L2{0, B, P} with E[Z(t)] = 0, 't:/t E R1 Then, using Stone's theorem, these stochastic processes can be written as From the properties of the spectral family that the E;... are symmetric and for .\1 ::; .\2, it follows that, for .\0 ::; .\1 ::; .\2 The operator product in the last term can be expanded as = 0 Consequently, the stochastic processes E;...Z(O), -oo < ). < oo that appear in the Stone representation have the special property Todorovic[96] makes the following Definition: The stochastic process { Z(t); t E R1 } is said to have orthogonal increments if, for any t0 < t1 < t2 21

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where the symbol represents complex conjugation. Definition: The process Z(t) is right continuous in the mean squared sense if Vt E lR1 t fixed, IIZ(t) -Z(tk) II ---+ 0 as tk 4-t Todorovic[96] shows that there exists an isomorphism between processes {Z(t) : t E lR1 } C L2{D, B, P} with orthogonal increments continuous from the right in the mean squared sense and the orthogonal stochastic measures TJ with m( ) the measure associated with it. This correspondence is given by Z(t) = TJ(( -oo, t]) From this, for t > s, Z(t) TJ(( -oo, t]) = TJ(( -oo, s]) U (s, t]) TJ(( -oo, s]) + TJ((s, t]) Z(s) +TJ((s,t]) =? TJ((s, t]) = Z(t)-Z(s) Ash[6] characterizes the Borel sets in various ways. In particular, they can be defined as the smallest a-algebra of subsets of lR1 that contains the intervals ( -oo, t], t E lR1 Furthermore, from this definition the function F(t) can be defined as F(t) m((-oo,t]) = IITJ((-oo,t])ll2 = E[IZ(t) 12 ] = E[Z(t)Z(t)*] where represents the complex conjugate. Since m(( -oo, t]) = m(( -oo, s] U (s, t]), then 22

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F(t) = F(s) +m((s,t]) So that, m((s, t]) = F(t)-F(s) It can be shown, Todorovic[96], Ash[6], that the function F(t) = E[Z(t)Z(t)*] is bounded, right continuous and non-decreasing. Letting TJ(dt) = dZ(t), then dF(t) m(dt) = IITJ(dt) 112 lldZ(t) 112 E[dZ(t)dZ(t)*] Hence, the following holds dF(t) = E[dZ(t)dZ(t)*] And, if the process Z(t) has orthogonal increments, then { 0 if t =J t' E[dZ(t)dZ(t')*] = dF(t) if t = t' (1.4) Definition: The stochastic integral /_:oo h(t)dZ(t) is to be interpreted as 23

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From the definition of the stochastic integral, Equation[ 1.3], page 20, h(t) is a complex-valued function. If then h(t) = ei>.t j_: ei>.tdE>.Z(O) j_: ei>.tdry(d>.) where T7 is the orthogonal stochastic measure associated with the stochastic process with orthogonal increments, E>.Z(O). 1.5 Spectral Representation Let {Z(t);t E lR1 } c L2{0,B,P} be a wide sense stationary process with E[Z(t)] = 0 C(t) = E[Z(s + t)Z*(s)] The following proposition follows from the non-negative definiteness of the covariance function, Todorovic[96]. Proposition The covariance function C(t) is continuous on lR1 if it is contin uous at zero. From the definition of the inner product, C(t) = (Z(s + t), Z(s)) Let f = Z(O), then from the discussion in Section 1.2, the covariance can be expressed in terms of the unitary operators as Letting g = Usf = Z(s) and using Equation[ 1.2], page 16, it follows that 24

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C(t) = (Utg, g) = /_: ei>.tdiiE>-gll2 Furthermore, since IIE>.gll2 is real, bounded and nondecreasing and E>. 1s uniquely determined, then F(>.) = IIE>.gll2 = E [E>.Z(s) (E>.Z(s))*] = E [Z(>.)Z(>.)*] (1.5) and obtain In particular, letting t = 0 and using the spectral family properties, the covari ance can be written as C(O) /_: diiE>.gll2 = IIEoogll2 119112 = E [Z(s) Z*(s)] The forgoing discussion illustrates to the famous Bochner-Khinchin Theorem A complex-valued function C(t) defined on R continuous at zero is the covariance function of a wide sense stationary stochastic process iff it can be written in the form J+oo C(t) = -oo eit>.dF(>.) where F ( ) is a real nondecreasing bounded function on called the spectral distribution of the process ( t). If the spectral distribution function F() is absolutely continuous, then the spectral density of the process is given by j(>.) = F'(>.) = j+oo e-i>.tc(t)dt 27r -()() 25

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So, the covariance function and the spectral density form a Fourier Transform pair. Comparing this with Equation[ 1.4], page 23, and Equation[ 1.5], page 25, the spectral density can be written as j()..)d).. = dF()..) = E[dZ()..)dZ()..)*] So that the covariance function takes the form j()..) is called the continuous spectrum of C(t) and the integral expresses C(t) as a continuous bundle of waves having amplitudes j()..). Similarly, the cross-spectrum Sxy(w) of two processes X(t) and Y(t) is the Fourier transform of their cross-correlation Rxy ( T) J+oo E[X(t + T)Y*(t)] = Rxy(T) = -oo Sxy(w)eiwT dw and Let { t E lR1 } c L2{D, B, P} be a wide sense stationary process such that = 0 and C(t) = + s)C(s)] Then from Todorovic[96] there is the following existence Proposition Let the covariance function C(t) be continuous at zero. Then, there exists a unique orthogonal stochastic measure 'T/ with values in L2{D, B, P} such that J+oo = -oo ei>.try(d)..) (a.s.) and II'Tl(A)II2 = m(A) = L dF \lA En 26

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where m( ) is the Lebesque-Stieltjes measure associated with 'T/ and generated by the spectral distribution F. Hence, if { .\ E lR1 } is a stochastic process with orthogonal increments corresponding to the orthogonal stochastic measure ry( ), the process t E lR1 } has the spectral representation 1.6 Space Correlations And Space Spectra The interpretation of the independent variable as a space variable instead of a time variable does not change things, Lumley and Panofsky[64]. Homogeniety is the property that for the space variable corresponds to stationarity for the time variable. In 3-dimensions, the corresponding spectral representation is given as where k is the wave number and is a stochastic process with orthogonal increments. In the case that time is a contributing factor, the spectral representation can be written as As in the one-dimensional case, if Z(k) is a stochastic process with orthogonal increments, then __, __, { 0 if k =I k' E[dZ(k)dZ(k')*] = dF(k) if k = k' (1.6) where dF(k) = f(k) dk and f(k) is the spectral density of the process. The cross-spectrum of two processes X and Y is given by 27

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Sxy(k) = k3 Rxy(x)e-ikxdi! and, its Fourier inverse is the cross-correlation of the processes X and Y, given by For a homogeneous process, the following is true Rxy( -if)= E[X(x-if)Y*(i!)] = R'Yx(if) In particular, if X and Y are real stochastic processes, and, if X= Y, Rxx( -if) = Rxx(if) so that the covariance function of a real stochastic process is an even function. Also, So, clearly, Rxy(if) + Ryx(if) is an even function. Furthermore, if Rxx is an even function, then using the mapping T(x) = -if = f and the change of variable formula it follows that Sxx(-k) k3 Rxx(f)e-i(-k)i' df k3 Rxx(if)e-ik(-i')df k3 Rxx(if)e-ikx( -l)di! Sxx(k) The last equality follows from the change of variable formula and the fact that -1 is the Jacobian of the transformation T. Hence, Sxx(k) is an even function. Similarly, S xY ( k) + Sy x ( k) is an even function. 28

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1. 7 Ergodicity For many properties that are to be measured, it is easier to obtain one observation at each point in time of a random sequence over a long period of time than to obtain several observations at the same point in time. The former method is called averaging along the process (time aver ages) and the latter method is called averaging across the process (ensemble averages). A statement that these two averages are the same is called an ergodic theorem. Burrill[17] states the following: Definition: Let M be a 1-1 mapping of D onto D. M is a measure-preserving transformation whenever the following condition holds: E is a measurable set iff M(E) is and P(E) = P(M(E)) for each measurable set E. For every integer t, the function Mt defined inductively by Mt = M(t-1 ) oM is a measure preserving transformation. Weak Ergodic Theorem: If M is a measuring-preserving transformation and if X E L2 then 1 n-1 (m2 ) lim L Mtx =X n-m-+oo n m t=m exists. The important corollary is Corollary: E[X] =E[X] The transformation M is called ergodic if X is a.s. constant. If this is true, then from the corollary 1 n-1 (m2 ) lim L X(t) = E[X] n-m-+oo n m t=m where X(t) = MtX. So, it is clear from this that the mean-squared limit taken over time is equal to the expected value of the random variable X. The 29

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term ergodic can have several interpretations. If a stochastic process has the property that the mean taken along the process is equal to the mean taken across the process, then the process is said to be mean-ergodic. In general, a stochastic process is ergodic if the ensemble averages are equal to the time averages. In the Hilbert space context, F. Riesz proved the following form of the Mean Ergodic Theorem Let H be a Hilbert space and Ut : H --+ H be a strongly continuous one parameter unitary group. Let the closed subspace H0 be defined by Ho = { x E H : Utx = x V t E and let P be the orthogonal projection onto H0 Then for any x E H lim T 1 {T U8xds = Px D t--+oo Jo The statement and proof are found in Abraham, et al[2]. Using the Hilbert space of second order random variables, L2(0, B, P) and the Proposition from Section 1.2, there is a group of unitary operators {Ut; t E such that X(t) = UtX(O) The theorem then says that the limit of the time averages is a random variable that is invariant under the group of operators, {Ut : t E }. If X ( t, w) is a sample path of a stochastic process X ( t, w), then the integral 1 {T T lo X(t, w)dt represents a time average of X ( t, w) over the interval [0, T]. From this, the random variable 1 {T Yr(w) = T lo X(t, w)dt can be formed. Clearly, if X ( t, w) is a stationary process, then E[X] E[X(t)] V t and 30

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1 {T E[Yr] = T Jo E[X(t)]dt = E[X] The variance of Yr(w) is given by Clearly, if (m2 ) limr--+oo Yr(w) = E[X], then ---+ 0 as T---+ oo. Similarly, if ---+ 0 as T---+ oo, then (m2 ) limr--+oo Yr(w) = E[X]. Assuming homogeniety, if the independent variable is a spatial vari able instead of a time variable, the interpretation is that as more spatial points are included in the average, the variance of the spatial average tends to zero. 1.8 Stochastic Solute Transport And Dispersion A key component in the scale-up problem is the correct formulation of the dispersion tensor used in the transport equation. Using arguments similar to those found in Gelhar and Axness[45] and Neuman[69], the reason for the focus on dispersion can be illustrated as follows: Starting from the transport equation, anc ..... -= -\7 n[cV-D\7c] at and letting the porosity be constant, it follows that ac ..... -= -\7 [cV-D\7c] at where c concentration, V seepage velocity, and D molecular diffusion. From the conservation of mass equation with constant porosity and density, it follows that \7 V = 0 and \7 E[V] = 0 31

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Writing the transport equation as ac ..... at + V (cV) = V (DVc) (1.7) and letting the concentration and velocity be stochastic processes, which are distributed as follows: c E[c] +c' E[c'] = 0 E[Y]+V' E[V'] =0 the substitution of these distributed parameters into Equation[ 1. 7] yields c') + v. [(E[c] + c')(E[V] + V')] = v. (DV(E[c] + c')) Expanding, c') + v. {E[c]E[V] + E[c]V' + c'E[V] + c'V'} = (1.8) V [DV(E[c] + c')] Taking expectations and using E[c'] = E[V'] = 0, + V. (E[c]E[V]) + V. E[c'V'] = V [DV(E[c])] (1.9) Subtracting Equation[ 1.9] from Equation[ 1.8], + V [E[c]V' + c'E[V] + c'V'-E[c'V']] = V [DV(c')] Important assumption: Assuming that the perturbations from their means in c and V are small, the second order perturbation term is eliminated because for small perturbations this difference would be close to zero leaving the first order approximation 32

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ad at+ V' [E[c]V1 + c1E[V]] V' [DV'(c1)] Expanding and using the zero divergence of V and E[V], acl I I at+ Y'E[c] V + V'c E[V] = V' (DV'c) (1.10) Assuming that c1 depends on both x and t and that there is spatial homogeneity, the following spectral representations can be made: v' = (V,' v;' v')t 1' 2' 3 where k = (k1 k2 k3 ) is the wave number and x = (x1 x2 x3 ) is the position vector. Defining dZ\7' (k) = (dZv' (k), dZv;' (k), dZv;' (k)t 1 2 3 and making the change of variables Yi =XiE[1/i]t, i = 1, 2, 3 and letting c(x, t) = v(fl, t) E[c(x, t)] = E[v(fl, t)] c1(x, t) = v1(y, t) so that v1(fl, t) is c1(x, t) in the moving coordinate system. In other words, c1 and v1 are the same stochastic process represented in different coordinate systems. This allows us to write in terms of the spectral representations, 33

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The stochastic process v'(iJ, t) has a unique orthogonal stochastic mea sure associated with it, and from the above equality and the isomorphism be tween the orthogonal stochastic measures and the processes with orthogonal increments, continuous from the right, it follows that Substituting v'(i/, t) into Equation[ 1.10], page 33, and simplifying yields Then letting \7 = \7 Y' it follows that t) + \7E[v(iJ, t)] V' = \7. [D\7v'(i/, t)] Substituting the spectral representations for v' and V' in this equation gives :t (l3 eik(Y'+E[VJt)dZc'(k, t)) + \7E[v(iJ, t)]k3 eikxdzv,(k, t) \7 ( D\7 k3 eikxdzc' (k, t)) 34

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Hence, using Equation[ 1.11] k3 eikx [ik E[V]dZc' (k, t) + :t ( dZc' (k, t))] + k3 eikxvE[v(y, t)] dZv, (k, t) V ( DV k3 eikxdzc'(k, t)) Taking derivatives and simplifying yields k3 eikx :t ( dZc' (k, t)) + k3 eikx [ik E[V] + Dk. fJ dZc' (k, t) f eikxvE[v(y, t)] dZv-,(k, t) }'!R3 And, using the uniqueness of the spectral representation, the following first order ordinary differential equation is obtained :t ( dZc' (k, t)) + [ik E[V] + Dk fJ dZc' (k, t) -VE[v(y, t)] dZv-,(k, t) Letting a= [ik E[V] + Dk f], this equation has the solution, assuming that dZc'(k,O) = 0, Furthermore, if it is assumed that the gradient VE[v] is constant, then VE[v] = VE[c], and it follows by evaluating the integral that a dZc'(k, t) = -b(t)VE[c] dZv-,(k) where b(t) = 1-e-at. The assumption that the gradient of the average con centration is constant means that the concentration is spatially linear. This is another assumption that requires only mild heterogeneities. Multiplying both sides of this by a= [ik E[V] + Dk f], then gives -VE[c] dZv'(k) [-ik E[V] + Dk fJ b(t) _. (k. E[V])2 + (Dk. k)2 = dZc'(k) 35

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Multiplying both sides by dZv' (k)*, the complex conjugate transpose, taking expectations and using the fact that E [ (vE[c] dZ-v' (k)) dZ-v' (k)*] yields b(t)[ik E[V] Dk fJ ( (k)VE[cJ) t t _. (k E[V])2 + (Dk k)2 = Sc'v' (k) where Sv'v' (k) is the cross-spectrum matrix and Sc'v' (k) is the cross-spectrum vector. Now, taking the inverse Fourier Transform and letting x = 0, it follows that (1.12) h b(t) [ -ik E[V] + Dk fJ (k)VE[c] _, _. _. _. _. dk lR3 (k. E[V])2 + (Dk. k)2 Take the klh term of the matrix (k) which, as shown in Section 1.6, has the property that (k) + (k) is an even function, and assume that (k) = (k) and that D is positive definite, then using the mapping T(k) = -k = w and the change of variable formula, it follows that h b(t) [ -ik E[V] + Dk f] (k) _. 2 _. _, _. _. dk lR3 (k. E[V])2 + (Dk. k)2 h b(t) [ -ik E[V] + Dk f] (k) _. _. _. _. _. dk lR3 (k. E[V])2 + (Dk. k)2 36

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h b(t) [ -i( w) E[V] + D( w) ( w) J SV,'v' ( w) + ..... k J dw (w. E[VJ)2 + (D(w). (w))2 h b(t) [ -ik E[V] + Dk fJ (k) ..... = --+ -+ --+ --+ J dk (k E[V])2 + (Dk k)2 h b(t) [ik E[V] + Dk f] (k) ..... + ..... ..... ..... ..... dk (k E[V])2 + (Dk k)2 { b(t)(Dk (k) ..... = 2 (k. E[V])2 + (Df. k)2 dk Without the symmetry assumption, this result would contain the term (k)+ (k). Returning to the transport equation ac ..... at= -V [cV-DVc] we know that by writing the concentration and velocity in terms of distributed parameters, c E[c] +c' E[c'] = 0 E[V]+V' E[V'] =0 we obtain Equation[ 1.9], page 32, namely, 8E[c] ..... ....., ---at+ V (E[c]E[V]) V (DVE[c]) + V E[c'V] = 0 The term E[c'V'] is an additional dispersive flux term that has already been expressed in terms of VE[c] via Equation[ 1.12], page 36. So, the transport equation for E[c] can be stated in terms of a new dispersion tensor, D, as where + V. (E[c]E[V]) V (DVE[c]) = 0 h b(t) (nf f) Sv'v' (k) ..... D -D + ..... ..... ..... ..... dk (k E[V])2 + (Dk k)2 37 (1.13)

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It is clear from this formulation that in general, D is a time dependent quantity. If k E R3 and if Dk k is positive definite, then since a= [ik E[V] + Dk k]t, lim b(t) t---+oo lim 1-e-at t---+oo 1 lim e-[of.f+ikE[VJ]t t---+oo = 1 smce D k k ?:_ 0 And, assuming that the interchange of limit integral makes sense, the asymptotic or steady state limit as t--+ oo for D is h (nf f) Sv'v' (k) _. D = D + _. _. _. _. dk (k E[V])2 + (Dk k)2 1.9 Comments And Limitations The assumption of ergodicity is implicit in the study, i.e., the solute transport in an ensemble of aquifers approximates the real field situation. This means that if the independent variable is a spatial variable, then the variance around the average must go to zero as the size of the domain gets arbitrarily large. Hence, the scale of the system must be large in comparison to the correlation scale, the length scale over which variables remain correlated. So, the estimates of macroscopic dispersivity and effective hydraulic conductivity are meaningful only if the scale of the problem is large in comparison to the correlation scale. Consequently, Equation[ 1.12], page 36, is valid only after a large displacement distance has been reached, perhaps tens or hundreds of meters. The adequacy of the first order approximation of the solute transport equation + \7 [E[c]V' + c'E[V]] = \7 [D\7(c')] depends on small perturbations in c and V and is not certain for large variance of hydraulic conductivity. In this same vein, the assumption that the gradient of the expected concentration is constant also depends on mild heterogeneities. 38

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Existing linear theories predict that transverse dispersivities tend asymptotically to zero as Fickian conditions are reached. The assumption of mildly fluctuating hydraulic conductivities has been used to justify eliminating nonlinear terms in establishing the linear theories. However, Rubin[82] found that higher order terms may cause some reduction in longitudinal mix ing and a significant enhancement in the transverse spread. Hence, the im portance of non-linearities should not be disregarded in studying dispersion in geologic media. Although it is the point of view of this study that significant heterogeneities should be modelled using numerical models, attempts at in corporating heterogeneity and therefore nonlinearities in the analytical models have been made. Neuman and Zhang[70] recover part of the nonlinearity due to the deviation of the plume particles from their mean trajectories using what has become known as Corrsin's Conjecture, which is a statement relating the Lagrangian covariance of the velocity to the Eulerian covariance of the velocity through the probability density of the particle's position. From this, nonlinear analytical expressions for the time dependent dispersivity have been developed. The relationship between the Lagrangian and Eulerian velocity fields is key to the developments of dispersion that are to follow. 1.10 Velocity /Permeability Covariance Relationship Spectral arguments similar to those used in the preceding subsection can also be used to develop a fundamental relationship between the Fourier Transforms of the velocity covariances and the log-hydraulic conductivity co variances, Equation[ 1.21], page 43. This relationship is important because it shows the connection between velocity covariances and log-hydraulic conduc tivity covariances, and hence permeability covariances. This means that esti mates of dispersion can be based on either velocity covariances or permeability covariances. In essence, the characterization of dispersion as a phenomenon created by permeability says that dispersion is a result of deviations in local permeabilities from a global average permeability. By Darcy's law if= -K\7 where K is the hydraulic conductivity, in general for R2 a second rank symmet ric tensor, and -\7 is the hydraulic gradient. By assuming that the medium is isotropic, the second rank tensor can be replaced by a scalar hydraulic con ductivity, K. Let Y = ln(K) and suppose that 39

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Y = E[Y] + Y'; = E[] + '; E['] = 0; E[Y'] = 0 Then, And, since K = ey = eE[YleY', it follows by expanding eY' that if= -eE[Yl(1 + Y' + (Y')2 + )(\7E[] + \7') 2 Letting 1 = \7E [ ], then if= eE[YJ [(1\7') + Y'(1\7') + (1\7') + ] Take expectations and dropping terms higher than first order, yields the first order approximation From Darcy's law and the incompressibility condition, respectively, if= -K\7, \7if=O it follows that since K = ey, then Expanding this it follows that which yields (1.14) Then using the distributed parameters Y=E[Y]+Y' = E[] +' (1.15) 40

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with E[/] = 0; E[Y'] = 0 and substituting Equation[ 1.15] into Equation[ 1.14], it follows that \7(E[Y] + Y') \7(E[] + ') + \72(E[] + ') = 0 Expanding and treating E[Y] as a constant, gives And, by retaining only first order perturbation terms, i.e. dropping the \7Y' \7 term, it follows that Taking expectations and using E[Y'] = 0 and E['] = 0, results in Substituting in Equation[ 1.16] the spectral representations it follows that And, using the uniqueness of the spectral representation theorem, Starting from the expansion if= -eE[Yl(1 + Y' + (Y')2 + )(\7E[] + \7') 2 (1.16) (1.17) and disregarding terms involving the products of perturbed quantities yields -eE[Yl(\7E[] + \7' + Y'\7E[]) (1.18) 41

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And, taking expectations and using E[Y'] = E['] = 0, Then since if= E[if] +;},it follows from Equation[ 1.18] that Using the spectral representations it follows that By the uniqueness of the spectral representation theorem Using the first order approximation, E[qj eE[YJj = eE[YlVE[], dZq-;(k) E[qjdZy,(k)-ieE[Y]kdZ1>'(k) Then from Equation[ 1.17] (1.19) (1.20) Equations [ 1.19] and [ 1.20] along with the first order approximation, E[qj eE[YlJ, and the fact that k(k E[qj) = fftE[qj yield where I is the identity matrix. Multiplying this expression by its conjugate transpose yields And, using the formulas, Equation[ 1.6], page 27, 42

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dF(k) = SY'Y'(k) dk it follows that Sq-;q-; (k) (Illkll-2kft) E[t}1E[qjt (Illkll-2kft) SY'Y' (k) This result agrees with Dagan[31, Equation 4.10]. 1.11 Summary (1.21) This section introduced some important concepts of stochastic pro cesses that will be used and expanded on in subsequent sections to study the general problem of scale-up. Some basic conclusions can be drawn from the presentation in this section. First, from Equation[ 1.13], page 37, it is clear that in the absence of variations in the velocity field no adjustment to the local dispersion tensor D is necessary. Secondly, from Equation[ 1.21], page 43, it is clear that the variations in the velocities are due to the variations in the hydraulic conductivities. Consequently, the conclusion is that the dispersion tensor requires scaling-up in the presence of variations in the hydraulic conduc tivities. Another way of saying this is that the effects of heterogenieties in the porous medium cannot be adequately modeled using a local dispersion tensor only. This means that although a local dispersion tensor may be adequate for describing plume development in small homogeneous laboratory experiments, it must be modified in a way that takes into consideration either variations in the velocities or variations in the hydraulic conductivities if it is to be used to adequately describe plume development in highly heterogeneous field problems. The linear theories that have been characterized in this section depend on the assumption of only mild heterogeneities being present in the porous medium. However, researchers have found that this restriction may lead to erroneous conclusions regarding both the longitudinal and transverse spread of the plume. For this reason, attempts have been made to modify the linear theories. To this end, Corrsin's conjecture has been borrowed from the field of plasma diffusion to aid in the modification of the linear theories. Corrsin's conjecture relates the Lagrangian velocity covariance to the Eulerian velocity covariance through the probability density of the particles's position. 43

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As will be seen in the next section, the idea of Lagrangian velocity covariance is key to the characterization of dispersion in a highly heterogeneous environment. Field studies made by various researchers to determine dispersivity values have concluded that: Field dispersivity values are larger than laboratory dispersivity values by a few orders of magnitude. Dispersivity varies with the distance from the solute input zone. Consequently, the upscaled dispersion tensor can be represented either as a time dependent quantity, as in the case of Equation[ 1.13], page 37, or as a distance dependent quantity. Chapter 2 of this study will investigate further the origins of time and distance characterizations of the dispersion tensor. In addition, the next chapter will provide information on some methods that will be used to study the scaling up of dispersion from a stochastic point of view. 44

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2. Dispersivity Coefficients -Time And Distance Forms 2.1 Time Dependent Dispersivity Coefficients 2.1.1 Introduction In the previous section, it was shown that when heterogenieties are introduced into the porous medium, the component of the transport equation that is significantly affected is the dispersion tensor. And, it was concluded that the dispersion tensor could be characterized as either a time dependent or a distance dependent quantity. In this section, these characterizations will be developed further, al beit with some simplifying assumptions. The intent is to provide motivation for the formulations of dispersion that will be used in subsequent sections. In Section 2.1.3, a characterization of the dispersion tensor is derived from the transport equation and shown to be equal to half the time rate of change of the second spatial moment tensor. This makes physical sense in that the second spatial moment represents the spread around the centroid of the mass plume. Dispersion, characterized in this manner, then represents how the spread of the plume with respect to the centroid is changing over time. It is also shown that the centroid of the plume moves with the velocity of the flow. So, by using the centroid of the plume as a reference point, this characterization eliminates any changes in the plume due to convective influences. The following is required: Let 0 be a bounded, open connected domain in lRn with a Lipshitz continuous boundary, 80. Then the Fundamental Green's Formula, integration by parts, is given by { if \1pd0 + { p\1 ifdO = { pif iJdr ln ln len (2.1) and follows from the Divergence Theorem f v ado = f a vd1 ln len by letting a = pif and using the expansion \1 (pi/)= \1p. if+ p\1 if (2.2) 45

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2.1.2 Transport Equation Let c(x, t) represent the concentration distribution of a solute, then the mass transport equation of this solute is given by oc --+ -+ V \7 c = \7 (D \7 c) at Here V is the seepage velocity (2.3) where K is the hydraulic conductivity, n is the porosity and is the hydraulic head. Initially, a divergence free velocity field is assumed, 2.1.3 Spatial Moments Of The Solute Concentration In this section expressions are developed for the first three spatial moments of the solute concentration. In these calculations the porosity, n, is assumed to be constant and included in the definition of c(x, t). Zero Moment Mo = k c(x, t)dD Taking the time derivative of this integral, gives And, from the mass transport equation and the Divergence theorem :tMo k V \7cdD + k \7 (D\7c)dD -[L c\7. VdD + kn cV. vdr] + k \7 (D\7c)dD f cV. vdr + f (D\7c) vdr lan lan 46

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So, the time rate of change of M0 depends on the boundary conditions. And, if c = 0 and V'c = 0 on 80., then ftM0 = 0 implies M0 is a constant. Since M0 is the total mass, this is a statement about the conservation of mass of the solute. First Moment M1 = k xc(x, t)dO. Taking the time derivative, M1 = :t k xc(x, t)dO. = k dO And, from the Transport Equation[ 2.3], page 46, !!_M1 =-f xCV. V'c)dO. + f x(V'. (DV'c))dO. dt ln ln From Equation[ 2.2], page 45, V. V'c = V'. (cV)-cV'. V By letting { ei}, i = 1, n be a standard basis of lRn, it follows that :tM1 k xiV' (cV)dO. + k cxV' 0 v ei k Xi(V' 0 (DV'c))dO. Using the assumption of a divergence free velocity field, the second integral on the right hand side of the previous equation vanishes so that d n --+ n -Ml =L:ei r XiV' 0 (cV)dO.+ L:ei r Xi(V' 0 (DV'c))dO. (2.4) dt i=l ln i=l ln Let V be independent of x, then from Green's formula[ 2.1] the two integrals on the right are evaluated as (2.5) 47

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if c = 0 on an. r D\7c. \7xid0. + r Xi(D\7c). iJdO. ln lao -fo (D\7c)id0. if \7c = 0 on an. Combining these results it follows that d n n --+ -Ml = 2:Viei r cdO.-Lei r (D\7c)id0. = v r cdO.r D\7cd0. dt i=I ln i=l ln ln ln Now, if D is dependent on only t, and letting jjl D1 \7c --+ --+ D= D2 D\7c= D2 \7c ===} --+ --+ Dn Dn\7c it follows that dO But, by integrating by parts, each component integral is equal to zero, (2.6) since Di is independent of x and c = 0 on an. Then, from Equations [ 2.4], [ 2.5] and [ 2.6] it follows that And, since M0 = fn cdO. is a constant, this yields 48

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(2.7) But since is the centroid of the concentration mass, Equation[ 2. 7] says that the centroid of the concentration mass moves with velocity V. Second Moment The first moment, M1 is a vector quantity and the zero moment is a constant. The quantity R(t) = is the centroid( center of mass) and is also a vector quantity. The second mo ment about the centroid is given by M2 = k (x-R(t))(x-R(t))tc(x, t)dO M2 is a matrix since the quantity s = (x-R(t))(x-ii(t))t is a matrix. So, M2 can be written as M2 = k Sc(x, t)dO In order to evaluate the time rate of change of M2 first consider the Diagonal Element: i = j First, from the definition of the centroid and Equation[ 2. 7], page 49, k (xi-Ri)2c(x, t)dO = -21/i k (xiRi)cdO + k (xidO -21/i k (xiRi)cdO + k (xi-Ri)2[V \7c + \7 (D\7c)]d0 -21/i k (xiRi)cdO{k c\7 (xiRi)2V dO + kn c(xiRi)2V iJ&y} + k (xiRi)2\7 (D\7c)d0 49

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Now, so that the first two terms of the last equality cancel, and using the boundary condition c = 0 on an, the boundary integral vanishes so that Integrating by parts and using the boundary condition that V' c = 0 on an, k (xiRi)2Y' (DV'c)dD = k DV'c V'(xi-Ri)2drl +lao (xiRi)2DY'c iJdr 0 -2 LDlj-,,L:Dni-1 [ n ac n ac l o j=l axj j=l axj -2 k V'c adD Dil(xi-Ri) Di2(xi-Ri) D (x-R) -2 [k cV' adD+ lao ca iJdr] 2 k cDiidD c(x, t) = 0 x E an Consequently, a diagonal element can be written as Off-Diagonal Element: i =I j In this case, take the time derivative of the ilh component of M2 50 0

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+ k ((xi-Ri)c(x, t)dD 1 oc + n (xi-Ri)(xj-Rj) ot dD -Vi k (xjRj)cdD-Vj k (xiRi)cdD + L(xiRi)(xjRj)[-V Vc+ V (DVc)]dD -Vi k (xj Rj)cdD-Vj k (xiRi)cdD (2.8) + k (xi-Ri)(xjRj)V VcdD + k (xi-Ri)(xjRj)V (DVc)dD Integrating by parts the integral k (xi-Ri)(xjRj)V VcdD k cV (xiRi)(xjRj)V dD + f c(xi-Ri)(xjRj)V iJd[ lan k c[(xiRi)Vj + (xjAnd, integrating by parts the last integral in Equation[ 2.8], it follows from Equation[ 2.1], page 45, k (xi-Ri)(xjRj)V (DVc)dD =k DVc V(xiRi)(xjRj)dD + { (xi-Ri)(xjRj)DVc iJdr lan 0 fnnvcadD 0 51

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Consider the integral 0 0 iJ= dO Dil(xj-Rj) Di2(xj-Rj) Din(Xj-Rj) r c\1. i]dn + r ciJ. iJd[ ln lan (2.10) k cDijdO. c(x, t) = 0 x E 80. -D { cdO. ln So, Equation[ 2.10], page 52, and the fact that Dij = Dji, yield k (xi-Ri)(xjRj)\1 (D\lc)dO. = 2Dij k cdO. And, from Equations [ 2.8], [ 2.9] and [ 2.10] it follows that Since the mass of the solute is given by M = k c(x, t)dO. this result can be written as 52

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which is in agreement with the definition given by Dagan for the Actual Dis persion Coefficients. 2.2 Stochastic Differential Equations 2.2.1 Introduction In order to proceed further with the investigation of the dispersion tensor in terms of a stochastic analysis, it is necessary to define the type of integration that will be required to analyze certain stochastic differential equa tions that arise naturally. This will be the subject of Subsections 2.2.2 to 2.2.5. Although the present discussion is very superficial, it will be adequate for the immediate need. However, this material will be expanded later when more details are required. 2.2.2 Integral Of A Stochastic Process According to Jazwinski[54] a stochastic process X(t) is mean square Riemann integrable over [a, b] if for a = to < t1 < < tn = b and t < t' < t+l z -z z the following mean squared limit exists Or, lim E L X ( t' i) ( ti+l -ti) -r X ( t) dt = 0 [ln-1 b 12] p---+0 i=O Ja As an existence theorem for the mean square Riemann integral, the following can be shown to be true, Jazwinski[54]. Theorem: X(t) is mean square Riemann integrable over [a, b] if and only if E[X(t)X(T)] is Riemann integrable over [a, b] x [a, b]. 53

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This notion of integrating a stochastic process is different from stochastic in tegration which will be defined next. 2.2.3 Wiener Process (Brownian Motion) In Section 1.2, the concepts of a random variable and a stochastic process were defined. In short, a stochastic process can be thought of as a mathematical model that describes the occurrence of a random phenomenon at each point in time subsequent to some initial time. Karatzas and Shreve[57] make the following Definition: If (D, B, P) is the probability space on which the stochastic process X ( t, w) is defined, then X ( t, w) is measurable if the mapping is measurable. Here B([O, oo)) and represent the Borel sets of [0, oo) and respectively. Definition: Given a probability space (D, B, P), a nondecreasing family {Bt: t 0} of sub-a-algebras of B, Bs C Bt C B for 0 :::; s < t < oo is called a filtration. Definition: The stochastic process X ( t, w) is adapted to the filtration { Bt} if for each t 0, X ( t, w) is a Br measurable random variable. With these definitions, the one-dimensional Brownian motion can be defined as follows: Definition: A one-dimensional Brownian Motion is a continuous, adapted process W = {Wt, Bt; 0 :::; t < oo} defined on the probability space (D, B, P), with the properties: (1) W0 = 0 (a.s.) (2) For 0 :::; s < t, the increment Wt W8 is independent of Bs and is normally distributed with mean 0 and variance t-s. Definition: A process ((t) is a white noise process if its values ((ti) and ((tj) are uncorre lated for every ti and tj such that ti #tj. For a white noise process with zero mean, 54

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E[((t)((T)j = q(t)J(tT), q(t) 2: 0 The Wiener process can be defined as the limit of a random walk, or as the integral of a Gaussian white noise process with zero mean. W ( t) = lot ( ( s) ds The following is a block diagram representation of this equation: ( ( t) -------+1 0f---------+- w ( t) 2.2.4 Stochastic Integration Because the solutions of the stochastic evolution equations come from spaces whose members are random functions, the solution process will require integrating these functions. For reasons given below, this will require a new type of integration called stochastic integration. Consider an integral of the form lot B(s)dW(s) (2.11) where W ( s) is a Brownian motion process. To illustrate the difficulty involved with interpreting this type of integral in the usual Riemann-Stieltjes sense, it is necessary to define a sample path of a Brownian motion process. Definition: The mapping t----+ W(t,w) (w fixed) is called a sample path. The following facts regarding sample paths of a Brownian motion process are proved in Friedman[40]. Almost all sample paths of a Brownian motion are nowhere differen tiable 55

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Almost all sample paths of a Brownian motion have infinite variation on any finite interval A non-rigorous reason for the first bullet is that since then 1 ------+ oo as ----+ 0 Consequently, given the preceding discussion of the interpretation of W(t) as the integral of a Gaussian white noise, the derivative d; must be interpreted as a 6-correlated Gaussian white noise which is a purely mathematical ideal ization. So, the integral [ 2.11] cannot be defined as a Stieltjes integral in the usual sense, for in order to do so, the sample paths would have to have bounded variation. 2.2.5 Types Of Stochastic Integrals Stochastic integrals have been defined in different ways. Two of these methods of defining the stochastic integral are: Ito Integral Applicable to a larger class of functions Does not follow the formal rules of calculus Stratonovich Integral Applicable to a restricted class of functions Follows the formal rules of calculus Stochastic integrals are defined in the sense of convergence in measure or mean squared convergence. Although the definition of stochastic integrals will be expanded in more detail below, the present discussion is adequate for the im mediate purpose of exploring the dispersion tensor. 2.3 The Concentration Equation Traditionally, the concentration or solute transport equation is de veloped from the consideration of conservation of mass in terms of variables averaged over a Representative Elementary Volume (REV), Bear[12], Gray[49]. In this section, the solute transport equation will be derived using stochastic considerations. In addition, the all important dispersion tensor will be further scrutinized with the objective in mind of finding expressions for this tensor that can be used in out computational work. 56

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2.3.1 The Lagrangian Approach By the Dupuit-Forcheimer equation, Bear[12], the filtration velocity, V is given in terms of the specific discharge, if, and the porosity, n, as __, if V= n In a heterogeneous porous medium, the properties of the medium cannot be precisely known. Hence they are considered to be composed of an average value plus some type of random component that describes the uncertainty associated with the medium. Because of this, both if and n are random and so is the velocity V. The uncertainty associated with the velocity field of a fluid particle can be illustrated by the situation depicted in Figure 1. The black dot in this figure represents a fluid particle about to begin its journey through a porous medium represented by the open circles. As shown, there are several paths that the particle can ultimately take. And, there is no way of knowing which path will be the actual path 0 D Figure 1 Figure 1 depicts the mechanical mixing component of dispersion. If the velocity field could be perfectly described, i.e., if at each juncture it was known which way the particle was going to go, then there would be no mechan ical dispersion to account for, only molecular diffusion. As shown in Chapter 1 and as will be discussed further below, the mechanical dispersion can be accounted for through the velocity covariances, or equivalently, through the permeability covariances. The velocity covariances are defined as 57

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V-E[V] In the Lagrangian framework, transport is developed in terms of indivisible solute particles which are transported by the fluid. As discussed in Bear and Verruijt[ll], the solute particles can be thought of as ensembles of molecules in a small volume. If the vector x""'r represents the total displacement of the particle which started its motion at x = x0 t = t0 then the vector x""'r can be decomposed into x""'r(t; xa, to) = X(t; xa, to)+ xd(t; to) (2.12) ..._.,_.._.., '-v-" '-v--' Total Displacement Convection Diffusion In this expression, the first term on the right hand side is due to a mechanical mixing of the fluid and requires movement of the fluid in order to exist. The second term on the right hand side is due to molecular diffusion which can take place without motion of the fluid. Hence, it does not contribute to the velocity field of the fluid particle. It does, however, contribute to the total displacement of the fluid particle. The vector Xd represents a Normalized Brownian motion or Brownian motion with zero drift type of displacement. Hence, Xd can be defined as the integral of a white noise process ___, rt Xd(t) = lo v(s)ds E[v] =0 where the autocovariance of the white noise, assuming a constant spectral density j()..) = K and using the integral representation for the delta function, b(T) = fliF ei7>.d).., is given by Even though 5 ( T) is not a function in the mathematical sense, the idea is that if T =!= 0, then the autocovariance is zero. The vector X comes from convective transport and is related to the velocity field by the kinematic relationship 58

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lot [E [V] + V' ( x--;,) l dt' E[V]t +lot V'(X--;, )dt' E[V]t + X'(t; x---o, to) (2.13) By taking expectations, it follows that E[X--;,] = E[X] = E[V]t, and if x--;, in the integrand of the above integral is replaced by its average, a first order approximation of x--;, in terms of V' is obtained, Dagan[33] Replacing the vector Xr(t') with the vector E[V]t' in essence assumes that the displacements of the velocity about the expected velocity path are not unlike those about the actual trajectory. This type of assumption is similar to that used by Taylor who assumed that since the expected velocity was much greater than the velocity fluctuations, any disturbances or eddies in a wind tunnel were transported with the expected velocity without significant distortion, Monin and Yaglom[68]. The displacement covariance tensor, Xjk, is then expressed in terms of the velocity covariances as xjk(t; x---o, to) = E[X'j(t; x---o, to)X' k(t; x---o, to)] (2.14) The mass of the indivisible particle is denoted m1p, and the solute is assumed to be inert, i.e., it does not react with the fluid that transports it nor with the solid matrix. A solute particle follows a path through the porous medium according to Equation[ 2.12]. The velocity of the particle is given by t>O (2.15) The concentration field associated with the particle is given by (2.16) 59

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where n is the effective porosity. The concentration is defined as mass per unit volume, but, only a portion of the unit volume is available for flow, and that portion is given by the porosity. Hence, if the concentration is multiplied by the porosity and integrated over the result is the mass of the fluid parti cle. From Equation[ 2.16], this is equal to mfp at each point of the particle's trajectory. This illustrates the indivisible nature of the fluid particle. The concentration field C fp is a stochastic function, i.e., at each time t, the function C1p(x, t) is a random variable of the spatial variable, if. This follows from the fact that the trajectory Xr is a stochastic function. Hence, if p(Xr, t) represents the probability density function of Xr, then h mfp __. __. __. __. ..... -b(x-Xr )p(Xr; t, x0 t0)dXr n mfp ( __. t __. t ) --p x; ,xo, o n It is clear from this equation that the probability density function of the con centration field is the same as that of the trajectory Xr. So, the concentration field is random because the trajectory is random. As mentioned before, the velocity field is also a stochastic function. From Equation[ 2.12], page 58, the following stochastic differential equation can be formed And, since Xd is a Wiener process, the second term on the right hand side is formally a Gaussian white noise. Furthermore, Equation[ 2.13], page 59, yields, The resulting stochastic differential equation then takes the form (2.17) which is to be interpreted in terms of the stochastic integral equation as 60

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The uncertainty of the position of the fluid particle, Xr, can be demonstrated by solving the finite-difference form of this stochastic differential equation, viz., for to < t1 < < tn, where ti+l-ti xd(ti+l) xd(ti) Figure 2 shows various realizations of the path of a fluid particle. The average velocity vector is along the line x = y. What is immediately clear from this figure is that the final position of a fluid particle can vary greatly depending on the exact path taken through the porous medium. And, this is the essence of dispersion. If we knew exactly which path each fluid particle takes, there would be no mechanical dispersion. Since this is impossible to know, the transport equation must include a term to compensate for this uncertainty. This was the case in Chapter 1 where the dispersion tensor was modified to include a component in part described by the spectrum of the velocity covariances. 2.3.2 Basic Form Of Transport Equation It can be shown, Jazwinski[54], that under certain circumstances the solution process of an Ito stochastic differential equation (SDE) is a Markov process and that the probability density or transition probability density func tion of the solution process solves the Fokker-Planck or Kolmogorov forward equation, i.e., given the Ito SDE dx = a(x, t)dt + B(x, t)dxd where if and a are n-vectors, B is an nxm matrix and Xd is an m-vector Wiener process with Q(t)dt = E [dXd dXJ], then the density function, p, satifies op + v. (pa) = !v. V(pBQBt) at 2 It is important to note that the coefficient of the dXd term in Equation[ 2.17], page 60, which in this case is the identity, is not allowed to be spatially 61

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Figure 2 Sample Particle Paths X 0 0.02 0.04 0.06 0.08 0.1 62

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varying. If this term had a spatially varying coefficient, then the form of the Fokker-Planck equation would depend on whether the Ito theory or the Stratonovich theory is applied. For example, Gardiner[42], if the stochastic differential equation (SDE) dx = a(x, t)dt + B(x, t)dXd is an Ito SDE, then the equivalent Stratonovich SDE is given by where dx = ii8 (x, t)dt + B8 (x, t)dXd --.s a. 2 L.....J Ju j,k And, the form of the Fokker-Planck equation depends on which approach is used. If, for simplicity, it is assumed that V'(Xr) = 0 in Equation[ 2.17], page 60, then the resulting stochastic differential equation is Given that the solution process Xr is a Markov process, then the probability density function, p, satisfies (2.18) So, if we let D = then the tensor D is either a constant or a time dependent quantity. To be accurate, Equation[ 2.18] represents a diffusion process with drift coefficient E[V] and a diffusion coefficient D. Obviously, this equation has the form of the transport equation. And, the expected value of the concentration field solves the same equation, e.g., + \7 (E[C]E[VJ) = \7 (D\7 E[C]) 63

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Hence, the basic form of the concentration equation follows from the funda mental displacement Equation[ 2.12], page 58 and the associated stochastic differential equation. Clearly, this equation accomodates only the dispersion created by the Brownian motion term, Xd. Whereas, according to Equation[ 2.12], page 58, the total dispersion is going to come from that associated with the convective transport and that associated with the Brownian motion. 2.3.3 Solution Of Basic Transport Equation In general terms, the multivariate Gaussian probability density func tion (pdf) for n dependent random variables is given by f( ) { 1(..... .....)t( -1)(..... .....)} Z1, Z2, Zn = (21r)% exp -2 Zf.t V Zf.t (2.19) where V= ( CJI CJ12 (Jln l <721 D":n CJnl CJn2 (JN so that V is the variance-covariance matrix. And, the multivariate character istic function for the Gaussian probability density is given by (2.20) In theory, the characteristic function of a random variable is given by Clearly, j is the Fourier Transform of the function f(x) so that 64

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Given that the trajectory Xr is given by Equation[ 2.12], page 58, and assum ing the the probability density function is Gaussian and has the form then this pdf solves a non-divergence form of Equation[ 2.18], page 63, e.g., ap _, +E[V] \lp = V D\lp at To see this, first from Ortega[73]: If D is a real symmetric matrix, then there is an orthogonal matrix P whose columns are the eigenvectors of D, such that where .\1 .\2 ,An are the eigenvalues of D. The change of variables x=Ptg has the effect of aligning the principle axes of the matrix D with the coordinate axes and D becomes a diagonal matrix. In this case, the non-divergence form of the transport equation in the original y system, viz., ap _, +E[V] \lp = V D\lp at becomes in the rotated x system ap ---.... n a2p +E[V] \lp = "'D at zz z=l z And, by making the change of variables to a moving coordinate system, 65

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and, then it follows that op at So that the problem becomes ov fJT v(x,o) f(X) v(X, T) = p(x, t) ov ov ------:::; -= -E[V] V -v at ar x This equation is transformed by forming Assuming that v(X, T) and Vv(X, T) ---+ 0 as lXI ---+ oo and integrating by parts, it follows that f vx x-eiXwdX = f eiXwv(X, T)dX }fRn J J J }fRn So, letting vr = o = Dr(w, T) + wt:fiwD(w, T) And, by transforming the initial condition, the following ordinary differential equation results { Dr(w, T) D(w, o) -wt:Dwv(w, T) }(w) 66 (2.21)

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which has the solution A (--+ T) fA(--+) -wt:Dwr v w, = we v(X, T) can be retrieved by writing v(X, T) And, using the transform it follows that Next, let t-f =X Y and write the inside integral as and let then h -iwiJ -wt:Dwrd--+ e e w Rn A(--+) -wt:Dwr g w = e 1 h -iw.iJ -wt:Dwrd--+ --e e w (27r)n Rn Then, using the integral from Guenther and Lee[51, p. 167] T dw = -e-4DT 100 2 ( 7r ) (32 -oo DT it follows that 67 (2.22)

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g(iJ) Consequently, Equation[ 2.22] can be written as where n ( (32 ) ..... 1 Li=1 4:6' T K(f3,DuDnnT)= n-1e (47rT)2(Du Dnn)2 Hence, if f ("Y) = fJ ("Y), then 1 ( 4:;:T) n 1 e ( 47rT) 2 (Du Dnn) 2 1 _l(...l...xtf>-1 x) n 1 e 2 2T (21r)2 (2DnT 2DnnT)2 1 n 1 e (27r)2 (2Dnt 2Dnnt)2 1 [ (Pt(y-E[V]t)) t (Pt(y-E[V]t))] n 1 e (27r)2 (2Dnt 2Dnnt)2 1 n 1 e (27r)2 (2Dnt 2Dnnt)2 68

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Since, = and since det(PtP) = det(I) = 1, then P(y_, t) = e-HW'-E[VJt)t ft-n-1(17-E[VJt)] For given i/, t, and E[V], the probability p(fl, t) is then linked to the size of the components of the D tensor; which indicates spreading due to dispersion. Comparing this equation with Equation[ 2.19], page 64, it is seen that p(x, t) is a Gaussian density function with mean E[V]t and covariance matrix V = 2tD, from which it follows immediately that (2.23) This method does not work if the matrix D is allowed to be a function oft, for then E[V] would depend on t also by virtue of the dependence of pt on t. However, in the case that D does depend on t, if it is assumed that p is a Gaussian density function, then it can be shown by direct differentiation that in order for p to satisfy the transport equation it is necessary that Equation[ 2.23] hold. 2.3.4 Dispersion As Velocity Covariances If j(' and Xd are not correlated, then it follows from Equation[ 2.14], page 59, and Equation[ 2.23] that D _! dXrij ZJ2 dt Letting 1 dXii 1 dXdii 2 dt + 2 ----;It :t lot lot Pii (E[V]t', E[V]t")dt' dt" + f(t", t) =lot Pij(E[V]t', E[V]t")dt' then Xij = JJ f(t", t)dt" and by the Liebnitz formula = f(t, t) +lot :tf(t", t)dt" 69

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f(t, t) +fat Pij(E[V]t, E[V]t")dt" fat Pij (E[V]t'' E[V]t)dt' + fat Pij (E[V]t, E[V]t")dt" Furthermore, it is assumed that a type of stationarity or homogeneity (Section 1.2 or Section 1.6) in the sense that E[V] is constant and Pii(E[V]t',E[V]t) = E [v\(E[V]t') i\(E[V]t)] E [v\(E[V](t + (t'-t)) i\(E[V]t)] Pij(E[V](t't)) This means that Pij depends only on the separation vector E[V](t't). Also, since the stochastic processes considered are all real, Pij(E[V]t',E[V]t) = E [i\(E[V]t') V\(E[V]t)] Pii (E[V]t, E[V]t') which leads to the conclusion that ldxij rt ( [ ..... J(' )) 2.dt = Jo Pii E V t t dt And, if the mapping a is defined as a : [0, t] --+ a(t') E[V]t' then it follows from the definition of a path integral that k Piids = fat Pii(a(t'), E[V]t) lla'(t') lldt' IIE[VJII fat Pij(E[V]t', E[V]t)dt' 70

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Hence, JJ Pii(E[V]t1 E[V]t)dt1 is a path integral. And, consequently, 1 dXij lot ( [-+]I [ --+] )) I 1 1 --d-= Pii E V t E V t dt = --+ Piids 2 t 0 IIE[VJII if If the true Lagrangian path integral is given by Piids la-T then the path a is an approximation to the path ar so that for the true La grangian path so that Piids la-r lot Pii (iir(t1),iir(t)) dt1 lot Pij (E[V]t1 E[VJt) IIE[VJI1dt1 1 dXij lot ( [-+]I [ --+] )) I 1 1 Dij = --d-= Pii E V t, E V t dt --+ Piids 2 t 0 IIE[VJII ifr (2.24) This result says that the components of the dispersion tensor should be related to time rate of changes in the covariances of the position of a fluid particle or the time integral of the velocity covariances. Furthermore, it also says that the calculation of the components of the dispersion tensor should take into consideration the orientation of the path of the fluid particle. In the pure Lagrangian sense, it is not assumed that the expected velocity is constant along the fluid particle's path. If x and iJ represent two points on the particle path, then Using the kinematic relationship X(t; x0 t0 ) =lot V(Xr(t1))dt1 it follows that E[X(t; x0 t0)] = JJ E[V(Xr(t1))]dt1 The convective displace ment is given by 71

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x' (t; xo, to) X(t; x0 t0 ) -E[X(t; x0 t0)] fat (v(Xr(t1))-E[V(Xr(t1))J) dt1 And, the displacement covariance is given by And, by differentiation, 1 d {t ( __, 1 __, ) I Djk = 2 dt Xjk = Jo Pik Xr(t ), Xr(t) dt The integrand represents a lagged covariance. If s = t t1 then t1 = t s and Pik (Xr(t1),Xr(t)) = Pik (Xr(t-s),Xr(t)) E s))s))J) x (vk(Xr(ts)) E[Vk(Xr(ts))J) J In order to implement this type of dispersion estimate, the paths that the fluid particles take have to be identified. This can be done by generating velocity fields and computing the paths using a particle tracking algorithm such as Xr((k + = + k = 0, 1, n-1 Then by choosing an appropriate so that t = where n is an integer and letting s = it follows that t1 = t-s = (nThe formula for Djk can be approximated by n L E m=O The velocities in this representation are elements of the Lagrangian velocity field. A distinction must be made between an Eulerian velocity field and the 72

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Lagrangian velocity field. It is true that at any given point the fluid particle will move with the velocity of the fluid at that point. But, since the fluid particle is following a changing path through the porous medium, it will encounter only a subset of velocities that comprise the entire velocity field of the domain. This subset is the Lagrangian velocity field. The velocity field of the entire domain is the Eulerian velocity field. In Section 1.11 the relationship between the Lagrangian velocity covariance and the Eulerian velocity covariance was described in terms of the probability density function of the fluid particles's position. Consequently, the statistical properties of the Lagrangian velocity field may well be different from those of the Eulerian velocity field. 2.3.5 Dagan's Approach Formulations of the dispersion tensor in terms of the velocity covari ances appear quite often in the literature. Section 1. 7 describes a version based on arguments in Gelhar and Axness[45] and Neuman[69]. Dagan[32] offers an approach that allows the specification of the dispersion tensor on a numerical grid block by numerical grid block basis. As discussed in Section 2.1.3, the second spatial moment, Sij, which characterizes the spread of a mass around its centroid, is given by 1 1 _, _, Sij =-n(Xi-Ri)(Xj-Rj)c(X, t)dX i,j = 1, 2, 3 M n where M is the mass, R is the centroid coordinate, c(X, t) is the concentration and n is the porosity. Since concentration is mass per unit volume, the second spatial moment of the plume with respect to the centroid can be written as The Actual Dispersion Coefficients are defined as half the rate of change of the plume's second spatial moment with respect to the centroid in the given realization 1 dSij 2 dt Since Sij is a random variable, Dagan[30, 32] defines effective dispersion coef ficients as: 73

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D. dE[Sij] ZJ2 dt The key to Dagan's final result is the fundamental relationship, Kitanidis[58], Dagan[29], From which it follows that D _! dE[Sij] _! dXii _! dRij ZJ -2 dt -2 dt 2 dt If the rectangle Vo is of dimension it in the direction of the mean flow, x1 and l2 be transverse to it, then the dispersion tensor components Dij ( t, w) where w is the it x l2 rectangle are given by Dagan[30, Equation 15] So, by assuming a particular autocovariance function (exponential or Gaussian) the Dij(t, l2 ) can be solved for, and applied using the following steps: Determine the log-transmissivity variogram Calculate Dij for the assumed transverse dimension l2 of the numerical blocks Attach the resulting Dij to rows of blocks at a distance x = II VI It from the input zone 2.4 Distance Dependent Dispersivity Coefficients In Gelhar[44] it is demonstrated that for a stratified aquifer to which an hydraulic gradient which varies only in the z direction is applied parallel to the layers, the variance of the displacement of a fluid particle is given in terms of the hydraulic conductivity, K, by 74

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And, using the definition of dispersivity this means that D 1 d0"2 X 2 dt 1 dO"; dE[x] -----2 dE[x] dt This result shows the dependence of the dispersivity on the mean distance. The following seeks to extend this result to higher dimensions. In order to determine the effect of one variable on a second variable, it is desirable to have all other variables held constant during the experiment. In economics this is referred to as ceteris paribus. If the object is to determine the effect of permeability or hydraulic conductivity on velocity, the experimenter would take a material of known permeability, put it in a test tank, establish a hydraulic gradient in the tank and measure the velocity. Next, a second material with a different permeability would be placed in the tank, a hydraulic gradient of the same magnitude as before would be established and the resulting velocity measured. By maintaining a constant hydraulic gradient in both measurements, the effect of permeability on velocity can be determined. Because we want to know only the effect of permeability on velocity, the hydraulic gradient and conductivity covariance will be assumed to be constant in the following calculations. 2.4.1 Local Grid Block Dispersivity Coefficients The seepage velocity is given by ___, K V= --\7 n And, the mean seepage velocity is given by E[V] =E[K]V n Let r be the position vector of a fluid particle at time t. This vector is given by the kinematic relationship lot ___, 1 1 lot K 1 r= V(t )dt =-Vdt o o n The deviation of the position vector r from its expected value is given by r-E[f1, and the matrix of covariances of these deviations is given by 75

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X E [(f-E[rj)(f-E[rj)t] E [ (fo' {K-E[K]) : dt") (!,' {K-E[K]) : dt') tl E [l /,' ( K' :) ( K' : r dt" dt'] But, since ( E[V'(x)V'\i7)l) ij = Pij(x, y), this is basically the same form as Equation[ 2.14], page 59. In order to generalize the one dimensional result, we can argue as follows: E [ (!,' >,Pdt'E [fa' (fa' >,Pdt'E [fa' >dt']) tl E [ { {K-E[K]) t:} { {K-E[K]) t: rJ Letting K = K-E[K], and noting that ___, t E[rj = tE[V] = E[K]\7 n which implies that So Let A be an r x s matrix and B be a t x u matrix, then the Kronecker product is the rt x su matrix 76

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A0B= In particular, if if E z E Rs, then if is r x 1 and z is s x 1 so that if0 Is is rs x s and given by Ylls if0 Is= Y2ls Yrls and Ir 0 z is rs x s and given by z 0 0 0 ..... z ..... ..... 0 0 0 Ir 0z= ..... ..... ..... z 0 0 0 Given that X= E [(f-E[rj)(f-E[rj)t] then using the differentiation formula from Marlow[66] that if if E Rr, z E Rs, if ERn, then where and d ( ..... ;yf) (..... I ) dz (I ;:;'\ dif dx y z -y 0 s dx + r 0 z} dx dif dx 81Y1 82Y1 81Y2 82Y2 77

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In our case, we let r = s = n and iJ = z = KE[Kt1 E[rj = KE[f1 = L-j= 1 K1jE[rj] L-j=1K2jEhl L-j=lKnjEh] which is ann x 1 vector. It has been shown that the dispersivity tensor contains the term According to Marlow[66], the matrix can be written as dXn djt12 dt dX1n dX ------a:t dt dXnl dt dXn2 dt dXnn dt which is n 2 x 1. Also, is an n 2 x n matrix and E[V] is ann x 1 vector so dX __, 2 that dE[TJ E[V]1s also n x 1. Now, dX dE[f1 Since iJ In has dimension n 2 x n and In z has dimension n 2 x n and and have dimension n x n, the matrix has dimension n 2 x n. The components of this equation are evaluated as follows: 78

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And, ..... 0 0 z 0 z 0 Inz= ..... ..... z 0 0 where ..... r L-j=l 1 Z-: L-j=l KnjE[rj] and Furthermore, Kn K12 K13 dz dif K21 K22 K23 dE[rj dE[rj Knl Kn2 Kn3 So, we can write dZ r 1 Kn K21 (y In) dE[rj A LJ=l KnjEh]In Knl which is an n2 x n matrix. And, 79 K12 K22 Kn2 0 0 0 Kln K2n Knn K13 K23 Kn3

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0 L,7=l KnjElrJ] 0 0 0 [ K12 K13 K21 K22 K23 0 Knl Kn2 Kn3 0 L,7=1 KljElrJ] 0 0 0 0 L,7=l KnjE[rj] which is also an n2 x n matrix. The result will be demonstrated using a two dimensional example: Local Grid Block Example: n = 2 In this example we can write 1 dXn 2 ----a:t .!_dX12 2 dt Expanding this yields 80 l K2n Knn

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2 2 A A l::j=l KnKljE[rj] 2 A A l::j=l K21K1jE[rj] 2 A A l::j=l KnK2jE[rj] 2 A A l::j=l K21K2jE[rj] 2 dt 2 A A l::j=l K12K1jE[rj] 2 A A l::j=l K22K1jElrJ] 2 A A l::j=l K12K2jE[rj] 2 A A l::j=l K22K2jElrJ] 2 A A l::j=l KnKljE[rj] 2 A A l::j=l K12K1jE[rj] 2 A A l::j=l KnK2jElrJ] 2 A A l::j=l K12K2jE[rj] + 2 A A l::j=l K21K1jE[rj] 2 A A l::j=l K22K1jE[rj] 2 A A l::j=l K21K2jElrJ] 2 A A l::j=l K22K2jE[rj] Using the first order approximation of the expected value of the fluid particle's position, viz., E[f1 = E[V]t and assuming that Kij = 0 for i =I j, it follows that 1 d --X= 2 dt E[Ki 1]E[Vlj2t E[K11K22]E[V1]E[V2]t E[K11K22]E[V1]E[V2]t The presence of the expected velocity vector in this expression is key since it can be changing from numerical grid block to numerical grid block. Figure 3 shows the effect of a non-constant expected velocity on the path of a fluid particle. This figure shows the result of solving Equation[ 2.17] first with E[V] = (0, -0.1), the dotted line, and then allowing E[V] = (0, -0.1) for the first 25 time steps, E[V] = (0, 0.0) for the next 25 time steps, E[V] = (0, 0.1) for the next 25 time steps, and finally, E[V] = (0, -0.1) for the last 25 time steps, the solid line in Figure 3. 81

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y 0 .02 0 -0.02 -0.04 -0.06 -0.08 -0.1 0 0.02 Figure 3 -Effect Of Velocity Changes 0.04 82 0.06 ..... 0.08 0.1 X

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2.4.2 Global Dispersivity Coefficients The presence of heterogeneities in the porous medium will cause the velocity field to be non-uniform. To maintain dispersive symmetry, the disper sion tensor should be recalculated on a grid block by grid block basis, taking into consideration the expected velocity on the grid block. The preceding for mulation can be extended to this case in the following manner: Let Xr be the trajectory of a fluid particle. Then, the position vector of the particle is given by lntK r= V(Xr(t'))dt' =-\lcpdt' o o n It is assumed that \1 is steady state so that it does not depend on t'. Fur thermore, it is assumed that the porous medium is locally homogeneous, i.e., in the sense of Section 1.1 and Section 1.5, and that this local homogeniety applies to numerical grid blocks. Suppose that Xr spends from t0 to t1 on grid block 1, from t1 to t2 on grid block 2, and from tn-l to tn on grid block n. Then since \1 does not depend on time and the statistics of K do not depend on time on individual grid blocks, we can write n-llt+l \1 ,.i,(i+l) E[rj = t;' E[K(i+l)]dt' '+'n then, Using the formulas it follows that 83

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X E[(f-E[f1)(r-E[f1)t] E [{fti+l (K(i+l)-E[K(i+llJ) dt' \7/i+l)} z=O lt, n { lti+l (K(J+l)-E[K(J+llJ) dt" \7U+I) }t] J=O n E [{1ti+l I((i+Ildt' \7(i+l)} {fti+l I(U+Ildt" \7U+l) }t] z=O t, n j=O ltj n E [{1ti+l I((i+Ildt' \7(i+l)} 0 { ftHl I(U+Ildt" \7U+l) }t] z=O J=O t, n ltj n E [{1ti+l I((i+Ildt' \7(i+l)} { ftHl I(U+Ildt" \7U+1) }t] z=O J=O t, n ltj n E [ {t, {<+' Vr1 }{t; t' V;+1l rJ Define the vectors a and b such that for u v = 1 2 and then n-ln-1 x = L L E[a(i+ll("bU+ll)tl i=O j=O where 84

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If it is assumed that the components are constant over blocks, then ( X) = '"' '"'E '"' '"' Llt 'I-'m Llt n-l n-l [ 2 2 V' V' l uv ul vm J+l i=O j=O l=l m=l n n And, using the relationship that The displacement covariance matrix becomes or E [ (:K(i+l)E[K(i+l)tlE[r\i+l)J) (:K(j+l)E[K(i+l)tlE[r\i+l)J) t] i=O j=O To get an adequate representation of exactly where in the domain different magnitudes of dispersion are to be expected, ensembles of particles must be tracked. Such a representation is given in Figure 4. In this example, 5 particles are tracked from two adjacent numerical grid blocks located in the center of the domain. These particle paths are used to identify the numerical grid blocks that are most likely to be reached by a tracer plume that emanates from the grid blocks containing the origin of the plume. Once the grid blocks most likely to be reached have been identified, the previous formulation can be applied on a grid block by grid block basis to estimate how dispersion will develop over time. From Equation[ 2.24], page 71, Dij =lot Pij (E[V]t',E[VJt) dt' = latE [v\ (E[V]t') i\ (E[VJt)J dt' hr Pijds 85

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As time t' increases from 0 to t, the particle moves from its original position at time t = 0 to its position at time t, in block ( m). As it moves, it traverses different grid blocks on its way to block (m). Using the arguments given previously, Duu E [!,."+> { (K(Hl)E [Kl'+'i]) vq,:+l)} u x { (KCmlE [KCml]) dt'l m-1 [ ( \7 A.(i+l) ) ( \7 A.(m)) l E i((i+l) 'f' n u i((m) : v 1:1 E [(:K(i+1)E [K(i+1)r1 E [r(i+l)J) (:K(m)E [K(m)r1 E [v(m)J)] z=O u v Or, in terms of the full tensor D = Y:1 E [ (:K(i+1)E [f(i+1)]) (:K(m)E [v(m)]) t] z=O Since it may be difficult to identify exactly the numerical grid blocks where a local source is originating, or if more than one point source is involved, an entire column of grid blocks can be used to determine the particle paths. Figure 5 shows the result of tracking a particle from each of the grid blocks in a column in the center of the domain. Of course, in a simulation, more than one particle from each grid block would be tracked. 2.4.3 Molecular Diffusion It is reasonable to have some level below which these dispersivity coef ficients cannot fall. This level would, in effect, represent the level of molecular diffusion. In order to specify such a level, an example from Batchelor[9] is used. For a solute of NaCl in water, the coefficient of diffusion is found to be 1.1 x 10-5 cm2 /sec at 15 C and for any concentration. For molecules such as potassium permanganate, KMn04 which are much larger than water molecules, the coefficient of diffusion is found to vary with the level of concentration. Since this is not a problem with sodium chloride, the level of 86

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Figure 4 -Ensemble Of Particle Paths 1 0 8 6 -..... .---: 4 F 2 12 14 16 18 20 Figure 5 -Tracks From A Column Of Grid Blocks 1 0 --..... 8 -r-.... 6 / ....... .,.,. ............._ -..::::: .......... 4 ............... ............... I'--' --2 --.............. ___. 12 14 16 18 20

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10-5 cm 2 /sec, or the equivalent level of 6 x 10-4 cm 2 /min, will be used in the examples. 2.5 Random Variable Generation 2.5.1 Independent Random Variables In order to generate the sample path of the stochastic process, it is as sumed that the stochastic process is Gaussian and stationary, i.e., at each time t, the random variable X ( t, w) has the same mean and variance and that these two moments are sufficient to describe the random variables's distribution. A sample path can then be generated by sampling from a Gaussian distribution with the specified mean and variance. The following analysis shows that by generating two independent uniform random variables on the interval [0, 1] and using the Box-Muller transformation, Equation[ 2.25], two independent Gaussian random variables can be produced. Let U 1 and U 2 be two independent uniform random variables with the same density function on the interval [0, 1]. Define the random variables The inverse relations are given by exp 1 2 [-(X2 + X2)] 2 1 (x2) U2 = 27r arctan X 1 Then, by taking derivatives, it follows that cos(21rU2) 88 (2.25) (2.26)

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And, that sin(27rU 2 ) (-2ln(UI))tU1 The Jacobian of the transformation is then J -27r cos 2(27rU2 ) 27r sin 2(27rU2 ) -27r u1 u1 u1 From Equation[ 2.26], Hence, If g(UI, U2) is the joint density function of ul and u2, then the joint density function of xl and x2 is given by From, Equation[ 2.26], it follows that g (exp [_l_ arctan (x2)) f (X X ) = 2 27!" X 1 1' 2 27r [ (x2+x2)] exp-12 2 89

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And, we have that and, 1 (x2) 27r arctan xl E (0, 1) Since g is the joint density of two independent uniform random variables on (0, 1), it follows that Hence, exp [-(XI+ Xi)] 27r 2 1 [ x 2 ] 1 [ x2] J21r exp ----;fJ21r exp 22 which shows that X1 and X2 are independent and each has the standard normal distribution. Hence, the two U(O, 1) independent random variables are used to produce two independent N(O, 1) random variables. Once an N(O, 1) distribution has been produced, an N(f.t, (}2 ) distri bution can be produced by the transformation Y=f.t+(}X X""' N(O, 1) Figure 6 illustrates this method by showing the results of generating 20000 samples from a Gaussian distribution with mean 3.0 and standard deviation 0.3. Plots of the two normal samples are shown along with their associated lognormal plots. The sample lognormal points are found from the formula lgn[i] = exp(n[i]) where n[i], i = 1, ,mare the sample points from the normal distribution. 90

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700 600 500 400 300 200 100 0 Figure 6 -Sample Distributions Normal Normal 20 40 60 80 100 20 40 60 LogNormal LogNormal .A '\ f \ I \ I \ I \ j \._ ) '-v i'---800 I \ \ I \ r\. ) 600 400 200 0 0 20 40 60 80 100 0 20 40 60 91 80 100 80 100

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2.5.2 Correlated Random Variables Once two independent normal random variables have been produced, it is possible via a linear transformation to produce two correlated random variables. Correlated random variables will be required in order to model an anisotropic porous medium. Given two independent standard normal random variables, X1 rv N(O, 1) and X2 rv N(O, 1), two new random variables can be defined by letting a, b, c E lR1 d, e, f E lR1 ad+ be+ cf Example: Suppose we want to define two random variables X3 and X4 such that then a = d = 0 and b2 + c2 1 e2 + !2 1 -3 be+ cf 1 -2 92

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Letting c = 1 ::::} b = 0 ::::} f = ::::} e = so that 1 (d+e)2+f2=-3 1 ad+ be+ cf = 2 as required. Chapter 7 of Law and Kelton[61] gives many approaches to gen erating random variables, both correlated and uncorrelated. If a multivariate normal distribution is to be used, then a particularly simple algorithm exists for generating a multivariate normal vector The reason for this is that in the joint density function, Equation[ 2.19], page 64, the covariance matrix, V, is symmetric and positive definite. Hence, it can be factored as V=cct where C is lower triangular. The algorithm consists of the following two steps: (1) Generate Z1 Z 2 Zn as independent identically distributed N(O, 1) random variables following the procedure in Section 2.5.1. (2) Fori= 1, 2, n, set Xi = Mi + Cijzj. It then follows that In the study of porous media systems, the measurement of the physical properties of the system at each point of the domain is a practical impossibil ity. In the case of these systems, it is common to assume that the physical properties of the system, i.e., hydraulic conductivity, etc., are realizations of a underlying random field. Since the development of our dispersion estimates is based on assumptions about the covariance function describing the hydraulic conductivities in the spatial domain, a method of generating random fields that takes into consideration the degree of variance, correlation lengths, cross correlations, anisotropies, etc. of the hydraulic conductivity is necessary. Such 93

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a method is provided by the Spectral Turning Bands method. In this method, simulations are performed along several lines using a unidimensional covariance function, CI ( ), that corresponds to the 2 or 3-dimensional covariance function given for the spatial domain. Given two spatial points XI, x2 in the domain, this correspondence is given by c(xi, x2) = C(h) = h ci (h. u) f(u) du where represents the unit circle or unit sphere, f ( u) is the probability density function of u, and h = XI x2 The value assigned to a point in the domain is given by an average of the values generated for the projection of the point onto the various lines used in the simulation. Finally, the algorithm described in Robin, et al[80] is capable of co generating pairs of three dimensional cross correlated random variables. 2.6 Summary In this section, the all important dispersion tensor component of the transport equation was investigated under some simplifying assumptions that allow a better understanding of the concept. Section 2.1 looked at the disper sion concept from an Eulerian point of view which lead to its characterization as half the time rate of change of the second spatial moment around the centroid. Section 2.3 introduced the Lagrangian approach. Here the motion of a fluid particle was described by a stochastic differential equation, Equation[ 2.17], page 60, and dispersion was characterized as describing the uncertainty sur rounding the path of a fluid particle as it proceeds through the porous medium. Under the assumption that the trajectory of the fluid particle has a Gaussian probability density function, it was shown that the dispersion tensor is equal to half the time rate of change of the covariances of the displacements in the fluid particle's position, which was then related to the velocity covariances of the particle. Special consideration was given to a method due to Dagan that allows the specification of the dispersion on a grid block by grid block basis. In Section 2.4, a one dimensional result that describes the dispersiv ity as a distance dependent entity was extended to higher dimensions. This method shows that the symmetry of the dispersion tensor is with respect to the average velocity vector. If the dispersion tensor is changed on a grid block by grid block basis, then the dispersion tensor must be recomputed to take into consideration changes in the expected velocity on the block. Section 2.4.2 ap plies the results of Section 2.4.1, which are applicable to locally homogeneous 94

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grid blocks, to the calculation of global dispersivity coefficients. This is done by using a particle tracking algorithm to identify the grid blocks that will be visited by fluid particles along their paths and applying the results of Section 2.4.1 on a grid block by grid block basis. Finally, Section 2.5 reviewed some methods being used to generate random variables. These methods will be used extensively in our numerical calculations. 95

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3. Methods Of Solution 3.1 General In this section, methods of solving the groundwater flow /pollution problem are discussed. Whenever two miscible fluids come together to form one phase, there is the potential for either the density of the phase or the vis cosity of the phase to change. This change can be brought about by changes in the concentration and/ or changes in the pressure. In the pollution problem, a change in the density can be caused by the pollutant mixing with the water in an aquifer. In the Enhanced Oil Recovery (EOR) problem, a change in the viscosity can be brought on by a surfactant mixing with the oil in the reser voir. The solution of an EOR problem may assume a viscosity/concentration relationship such as the quarter power rule, Russell and Wheeler[84] where cis the concentration of the surfactant and M is the ratio of the viscosity of oil to the viscosity of surfactant, ie, the mobility ratio. In the case that the density is dependent on the concentration of the pollutant, a coupled system of PDE's is obtained that must be solved simul taneously. However, there are probably many cases where the density is not dependent on the concentration or that this dependence is weak and it can be assumed that there is no dependence. In this case, the flow and concentration equations may be solved separately. The groundwater flow equation yields esti mates of the piezometric head and velocity estimates are derived from Darcy's law. This is the case in tracer experiments. Then, by using either relation ships between the velocity covariances and dispersion or between permeability covariances and dispersion, dispersivities can be developed for use in the con centration equation. By starting with the simpler uncoupled case, the total system can be tested for reasonableness without having to deal immediately with the more difficult coupled case. But, since the ultimate goal is to even tually handle the coupled case, the next section investigates some methods of handling the simultaneous solution. 3.1.1 Simultaneous Solution The groundwater flow equation used in tracer experiments has the form 96

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s8* = v. TV* at where S is the storativity of the aquifer, T is the transmissivity and is the piezometric head. If this equation is not used to determine velocities, then an alternative expression must be used to obtain the velocities. These velocities can be obtained from Darcy's law in the following way: In general, Darcy's law can be written as if=Kf where if is the specific discharge, the volume of fluid flowing per unit time through a unit cross-sectional area normal to the flow, K is the hydraulic conductivity tensor and J = -V* is the hydraulic gradient. Because the cross-sectional area used in the definition is a unit, the vector if has the dimension ( and so is considered a velocity. The Dupuit-Forchheimer equation accounts for the fact that the flow is only through the void part of the solid matrix by dividing the specific discharge by the porosity, n. so that _, if V= n Hubbert defined the potential of a compressible fluid as 1p dp = Z + Po gp(p) where z is the elevation, p is the pressure and g is the gravitational constant. In general, the density p depends on temperature, concentration of dissolved matter and pressure. So, the potential is for a compressible fluid under isothermal conditions. The coefficient of compressibility of a fluid is expressed as, Bear[12], f3 =! op pop 97

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and its solution is given by p =Po exp[fJ(pPo)] so that from the Hubbert potential z+ 1p dp Po 9Po exp[jJ(pPo)] 1 1 z + -----e-f3(p-po) 9Pof3 9Pof3 1 1 z+---9Pof3 gj3p Using this relationship and differentiating, which leads to so that V*=Vz+ ; 2Vp and Vp=j3pVp 9tJP 1 Vcp*=Vz+-Vp gp ___, -1 J = -V* =(Vp + pgVz) gp Taking Z to be depth, Z = -z, it follows from Darcy's law that, ___, -K if= KJ = -KV* =(Vp-gpVZ) gp The tensor K can be expressed as which means that V = !!_ = -K (Vp-gpVZ) = -k (Vp-gpVZ) n ngp nM where k is the permeability tensor and M is the viscosity. 98 (3.1)

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In Chapter 2, the basic form of the concentration equation was illustrated using assumptions about the stochastic nature of the variables. Here the concentration equation is developed from a more traditional point of view using the conservation of mass law while at the same time retaining a stochastic flavor. Let c(x, t) represent the concentration of the pollutant, ie, c(x, t) is the mass of the pollutant per unit volume of the single phase. The conservation of mass law then states that a a r nc(x, t)dx = -r c(x, t)u. iJdx + r qdx t ln lan ln where n =Porosity, The Fraction Of The VolumeD Available For Flow u = Darcy Velocity or Volumetric Flow Rate Across A q = Mass Flow Rate Per Unit Volume Injected Into D Using the Divergence Theorem, :t k nc(x, t)dx = k \7. c(x, t)ildx + k qdx The total flux of the pollutant is given by the product c(x, t)V(x, t) (3.2) On a representative elementary volume, REV, the concentration and velocity can be represented as c(x, t) E[c(x, t)] + c' (x, t) E[c' (x, t)] = o V(x, t) = E[V(x, t)] + v' (x, t) E[v' (x, t)J = o then by multiplying and taking expected values, E[cV] = E[c]E[V] + --------convective flux E[c'v'] .........__.... dispersive flux Assuming, at least locally, e.g., on a grid block, that the dispersive flux is Fickian, then 99

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E[c'v'] = -DY'E[c] The velocity V represents the velocity of the fluid in the void part of the matrix, and so we can write, on average, that E[cil] = nE[cV] = n (E[c]E[V]-DY'E[cJ) Hence, by taking expected values in Equation[ 3.2], it follows that :t k nE[c(x, t)]dx =k V' n (E[c]E[V]-DV'E[cJ) dx + k E[q]dx Since this equation is in terms of volume averaged variables, the integrals can be removed, see Gray[49], so that :t nE[c(x, t)] = V' n (E[c]E[V] DV'E[cJ) + E[q] So, for the case of the density being dependent on the concentration, p = p(c) the system of equations can be written as (dropping expected value symbols) o(np) at -V' n (cV-DY'c) + q -k -(V'p-gpV' Z) nj.J, -V' npV (3.3) Clearly, this system assumes that the porosity is given by n(x, t) and that the four unknowns are c(x, t), V(x, t), p(x, t) and p(x, t). 3.1.2 The Mixed Model Using the mixed method, the velocity and pressure can be solved for at the same time. The Mixed Model introduces the space with norm 100

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lltJ11H(div;!1) = (lltJ116,n + lldiv(q') 116,n)112 The Mixed Variational Formulation is then defined as: Find a pair of functions (u,p) E H(div; 0) x L2(0) such that 't:!ij E H(div; 0), k u ijdx + kP div(q')dx = 0 and 't:/v E L2(0), k v(div(u) + f)dx = 0, f E L2(0) The Discrete Mixed Formulation requires that there exist two finite dimensional spaces Qh and vh such that Qh C H(div; 0) and a pair ( uh, Ph) E Qh X Vh is sought such that and A key element in the existence theorem of the mixed finite element analysis is the Babuska-Brezzi Condition. Theorem (Existence): Assume that iJh E Qh and that 't:/vh E Vh, k Vh div(ifh)dx = 0 =* div(ifh) = 0 and that :3 a constant a > 0 such that 101

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Then the problem (Ph) has a unique solution (uh,Ph) E Qh x Vh and ::J a constant T > 0, depending on a, such that Starting from the second line in Equation[ 303], page 99, ___, -k V =(\lp-gp'VZ) nj.J, and taking the divergence, which is written as Letting ..... (-k ) V 0 V = V 0 nj.J, \lp + V 0 (gp\l Z) -k ah = -Vph nj.J, and multiplying this equation by any ifh E Qh and integrating over 0, k nj.J,k-1uh 0 ifhdx + k \lph 0 ifhdx = 0 Integrating the second integral by parts using the Green's formula f pV 0 ifdi! + f Vp 0 ifdi! = 1 pifo iJd1 ln ln Ian (3.4) and assuming no flow boundary conditions, ifh 0 iJ = 0 on an, it follows that 102

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In nJ-Lk-1uh ifhdx-In Ph V ifhdx = 0 And, from Equation[ 3.4] with vh E Vh (3.5) (3.6) Hence, Equations[ 3.5] and [ 3.6] represent the Discrete Mixed Formulation for the pressure equation. 3.1.3 The Spaces Qh And Vh In order to define bases for the RaviartThomas finite element spaces Qh and vh for the 3 dimensional case, first let n be the 3-simplex [0, Ll] X [0, L2 ] x [0, L3 ] and define the meshes: : 0 = Xo < x1 < < xk = L1 : 0 = Yo < Y1 < < Yj = L2 : o = zo < z1 < < zj = L3 And, define the piecewise polynomial space = { v E Cq[O, L] : v is a polynomial of degree :=::; ron each subinterval where q = -1 refers to discontinuous functions. Since in the error analysis of RaviartThomas[77] there is no require ment that the elements of the subspace Vh be continuous across inter-element boundaries, the finite dimensional subspace for pressure used in Russell and Wheeler[84] for the 2-dimensional case is extended to the 3-dimensional case It is further assumed in RaviartThomas[77] that given any vh E Vh there exists a ifh E Qh such that 103

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div iJh = vh which leads to the definition of the subspace Qh as So that, Theorem 5 of RaviartThomas[77] then gives the error estimate where K is a constant independent of h, and from which it follows that for r=O and liP-Phllo,n O(h) so that the error estimates for the pressure and velocity are of the same order. For the general case where density and viscosity are dependent on the concentration, the following set of coupled equations holds: 8(np) at -Vn(cV-DVc)+q k -(Vp-gp\7 Z) nJ-L -V (npV) 104

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In the tracer case, where the density is constant, the equations can be written in the following uncoupled form: sa* = v. T\7 at :t ( nc) = -\7 n ( c V -D \7 c) + q where S is the storativity of the aquifer, T is the transmissivity, and is the piezometric head. 3.2 2D Finite Element Solution 3.2.1 General The goal of this current investigation is to study the system of equa tions specified by Equation[ 3.3], page 99, with the coefficients of the disper sivity tensor calculated in terms of the velocity covariances or permeability covariances. The dispersivity tensor would then be a piecewise constant tensor specified on a grid block basis. Clearly, the system specified by Equation[ 3.3] is a coupled system and must be solved simultaneously. This is because the density of the single phase is considered to be a function of the concentration of the pollutant. It is conceivable that in many cases the density may be con sidered to be constant, in which case the flow equation and the concentration equation may be solved separately. It is this case that will be studied first. 3.2.2 Finite Elements A finite element in Rn is defined, Ciarlet[24], as a triple (T, '11, such that (1) Tis a closed subset of Rn with nonempty interior and Lipshitz-continuous boundary. (2) w is a finite dimensional space ofreal valued functions defined over the set T with dim W = N. (3) is a set of linear functionals, (}i, 1 :::; i :::; N defined on the space w such that :3 '1/Jj E w, 1 :::; j:::; N, with the property that Also, the following holds 105

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N = i=l The sets { and { are dual bases. Suppose that there are two finite elements (D, and (De, 'll, that are related by the invertible affine mapping F : D ----+ De 3 F(x) Ax+b then the finite elements are affinely equivalent if (1) F(D) =De (2) = 0 p-l (3) iii = F(ai) where the iii are the element nodes. The following relationship between mappings is defined G D )Rl Figure 7A The mapping G* in Figure 7 A is called a pull-back since it pulls a function defined on D back to a function defined on De. It follows from (2) that if G = p-l, then = o G = where G* is a pull-back. Similarly, the mapping G* : 'l1 -----+ 'l1 allows the definition of a pushforward mapping G* : -----+ according to the following diagram: 106

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G* Figure 7B The function G pushes a functional defined on W forward to a functional defined on 3.2.3 Rectangular Elements In implementing the uncoupled version of the 2D finite element model, the elements are assumed to be rectangular with their edges aligned with the x and y axes as shown in Figure 8: i1=(0,1) iJ= (-1,0) iJ= (1,0) iJ=(0,-1) Figure 8 Rectangular Element 107

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Here iJ = vxi + vy) = cos(a)i + sin(o:)J where a is the angle between the outward normal to the boundary and the x-axis. For integration purposes, using Gaussian formulas, it is convenient to use as a reference element the following rectangle: (-1,1) (1, 1) 4 3 D 1 2 (-1,-1) (1, -1) Figure 9 Reference Element The numbers in the corners of the element in Figure 9 denote the numbering of the local nodes. On the element D, the first order (linear) Lagrange shape functions are given by 1 1/Jl -(177) 4 1/J2 1 4(1 + 77) 1/J3 1 4(1 + + 77) 1/J4 1 4(1+ 77) These reference functions can be transformed to any rectangular element whose side is of length a in the x-direction and of length b in the y-direction. Letting (x1 y1 ) represent the global coordinate of node 1, the transformation is given by 108

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[ l = F ( [ l) = [ l [ l + [ l For example, the 4 corners of the reference rectangle map in the following way: Node 1: (-1,-1) ---+ (x1, Yl) Node 2: (1, -1) ---+ (a+xl,Yl) Node 3: (1, 1) ---+ (x1 +a, Y1 +b) Node 4: (-1,1) ---+ (x1, Y1 +b) The inverse mapping then becomes [ l = p-l ( [ l ) = [ 8 l [ l + [ l Using this to transform the linear Lagrangian shape functions to the trans formed element De, results in '1/Jl ---+ ( 1 ( 1 = '1/Jl 'l/J2 ---+ ( ( 1 = 'l/J2 'l/J3 ---+ ( ( y-t) = 'l/J3 'l/J4 ---+ ( 1 ( = 'l/J4 3.2.4 Numerical Integration Let D be the reference element whose corners are given by {( -1, -1), (1, -1), ( -1, 1), (1, 1)} and let such that [ l = F ([ ]) = [ l [ l + [ l then 109

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1 r l8(x,y) I !1e f(x, y)drle = }D TJ), TJ)) TJ) The quadrature formulas for evaluating integrals over the reference rectangle, D, are derived from the quadrature formulas for the 1D case. Letting then 18(x,y)l TJ) = TJ), TJ)) TJ) ill (ill TJ)dTJ) "' L (t, ry;) W;) m n 2:2: TJj)wiwj i=lj=l where m and n are the number of quadrature points in the and TJ directions, TJj) are the Gauss points and the Wi and Wj are the Gauss weights. The number of Gauss points to use is based on the result that if there are n + 1 Gauss points, then the formula n 2:ad(xi) i=l is exact for a polynomial of degree 2n + 1. If the polynomial has degree p, then p+1 n+1=--2 2n + 1 = p ::::} so that the number of Gauss points (n9p) is given by if p + 1 is even if p + 1 is odd For the linear Lagrange functions, the highest degree in both the and TJ directions is 2. So, in the numerical integrations m = n = 2. 110

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3.2.5 Groundwater Flow Equation The 2D groundwater flow equation is given by sa* = v. T\7 at where S is the storativity of the aquifer, T is the transmissivity, and is the piezometric head. The semi-discrete variational formulation takes the form with respect to a four-node rectangular element ne The second integral on the right side can be integrated via Green's formula r if. \7pdn + r p\7. ifdn = 1 pif. iJdry ln ln !an to give Assuming a uniform time step, llt, and a backward Euler estimate of the time derivative, acp* c/J*n c/J*n-1 at llt the fully discrete variational formulation is 0 foe vS ( c/J*n dOe+ foe Tn\7 cjJ*n \7vd0e 1 vTn\7 c/J*n iJdry lane where the superscript n represents the nth timestep. Then making the substi tutions r z= ; ( tn)?/Jj ( x) x= (x,y) j=1 111

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and letting The system of equations for the eth element becomes i = 1,2,,r The surface integral in this equation takes the form (3.7) On an!, iJ = (v1 v2 ) = (0, -1) this becomes And, with the no flow boundary condition on an!, this becomes 112

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On an;, iJ = (v1 v2 ) = (1, 0) Equation[ 3.7] becomes And, since is constant along this boundary, this becomes On iJ = (v1 v2 ) = (0, 1) Equation[ 3.7] becomes And, with the no flow boundary condition, this becomes On an!, iJ = (v1 v2 ) = ( -1, 0) Equation[ 3.7] becomes And, since is constant along this boundary, this is where iJ = vxi+vy] = cos(o:)i+sin(o:)J and o; is the angle between the outward normal to the boundary and the x-axis. Furthermore, it is not necessary to compute the boundary integrals on a boundary between two elements in the interior of the global domain. Consider the following two elements that have a common boundary that lies in the interior of the global domain. 113

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4 5 6 4 3 4 3 1 2 1 2 1 2 3 Figure 10Boundary Between Elements Along this common boundary in element D1 we have the integral And, along the common boundary in element D2 we have These integrals are to be interpreted as unoriented flux integrals, so that 8Di = and the same tensor Tn is used in both. This means that in the case that boundary coincides with the interface of two physical blocks of differing permeabilities, sufficient mixing has occurred along the interface to insure that there is not a discontinuity of permeabilities along the interface. In the element D1 of Figure 10, the shape functions have the following form on the common boundary: 1/Jl 0 1/J2 1-(y-YI) b 'l/J3 Y -Y1 b 1/J4 0 114

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And, in the element D2 of Figure 10, the shape functions have the following form on the common boundary: '1/JI 1-(y-YI) b 'I/J2 0 'l/J3 0 'I/J4 Y -Y1 b So that, and, Making the substitutions r L j(tn)'l/Ji(x) x= (x,y) j=l and letting it follows that (3.8) 115

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where ;(tn) represents global point 2, ;(tn) represents global point 5 and for non-zero results, 1/Ji can be either 'lj;2 or 'lj;3 And, (3.9) where i(tn) represents global point 2, (tn) represents global point 5 and for non-zero results, 1/Ji can be either 'lj;2 or 'lj;3 It is clear that these fluxes are equal but of opposite sign. Consequently, the contributions to either global point 2 or global point 5 cancel each other. Hence, it is only necessary to compute the boundary integrals when the local boundary coincides with the global boundary. Letting G21 i 0/, O'lj;j d ij = "( 80e UX then letting 116

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and then in matrix form it follows that (3.10) 3.2.6 Matrix Assembly This section describes the assembly of the system of equations on the local rectangular elements into a global system for the entire domain. On a rectangular element De, the local nodes will be designated Uj, j = 1, 2, 3, 4. On the global domain, the mesh nodes will be designated Uj, j = 1, nnp, where nnp is the number of global mesh nodes. The element nodal data is stored in the element nodes array, nod, which relates local node numbers to global node numbers by nod[e][a] =A where e is the element number, a is the local node number and A is the global node number. This array is set up from the input data. Figure 11 illustrates the subdivision of a rectangular domain into 4 rectangular sub-domains, D1 D2 D3 and D4 In each sub-domain the corners of the rectangle are labeled 1, 2, 3, 4 in a counter-clockwise way. These numbers represent the local node numbers. And, the mesh nodes of the global domain are labeled 1 -9 in a left to right and bottom to top way. 117

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7 8 9 4 3 4 3 n3 n4 1 2 1 2 4 6 4 3 4 3 n1 n2 1 2 1 2 1 2 3 Figure 11 Subdivision Of A Rectangular Domain The element nodes array is then given by nod[1 ][1] = 1 nod[1] [2] = 2 nod[1 ][3] = 5 nod[1][4] = 4 nod[2] [1] = 2 nod[2] [2] = 3 nod[2] [3] = 6 nod[2][4] = 5 (3.11) nod[3] [1] = 4 nod[3] [2] = 5 nod[3] [3] = 8 nod[3][4] = 7 nod[4][1] = 5 nod[4] [2] = 6 nod[4][3] = 9 nod[4][4] = 8 In this way each local node of each local rectangular element is associated with a global node in the mesh for the global domain. Equation[ 3.10], page 116, describes the system of equations on a rectangular element De. If we let 118

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s t1t [Mij] + [Kij]-[Gij] [Mij] {
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etc. By either reading the associations off the previous diagram or by using the element nodes array, Equation[ 3.11], page 117, along with the associations just described, the coefficient matrix of the global system has the form: Al1 Ab 0 Al4 Al3 0 0 0 Al A + Ai1 Ai2 A A + Ai4 Ai3 0 0 0 Al A 0 A A 0 0 Ai1 Ai2 0 Ai4 + Arl Ai3 + Ar2 0 Ai4 Ar3 Al A + A + A3 + + + Ai 1 + Ai2 + Ai4 0 0 + + 0 0 0 0 0 0 0 0 + A11 A12 + A14 0 0 0 0 Ajl Aj2 0 Aj4 3.2.7 Velocity Field Once the piezometric head has been computed using the flow equation, the associated velocity field can be computed from the equation V =KV* n If K is the hydraulic conductivity and B is the average depth of the tank, then we can write K1 = !K = -1-T n nB Hence, V_, K11 K12 a u ax 12 !!v. [ 1 1 l [ a l [ K1 !lit_ K1 !lit_ l = I Kl ffl_ = -Kl !!__ -Kl !!.P':_ K21 22 ay 21 ax 22 ay Here, on a given element, De is given by the equation r = L U(lPi i=1 where 120 0 0 0 0 Ai3 0 A13 Aj3

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( 1 (x x1)) ( 1 (y Yl)) (X ( 1 (y YI)) XI) (y YI) = ( 1 (x (y YI) In these equations, a is the horizontal length of the element, b is the vertical height of the element and the point (xi, YI) is the coordinate of the lower left-hand corner of the element. 3.2.8 The Transport Equation The 2D transport equation for an incompressible porous medium is given by au at \7 (D\7u) + V \7u = f (x, t) E 0 x T where Dis the dispersion tensor which may depend on time or distance from the source and may have a component that is a stochastic process. The velocity vector, V, is assumed to be in (L00(0 x T))2, but may also have a random component. The semi-discrete variational formulation takes the form with respect to a four-node rectangular element ne The second integral on the right side can be integrated via Green's formula r if. \7pdn + r p\7. ifdn = J pif. iJdry ln ln !an to give 121

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0 f v aau dfle + f DV'u V'vdfle Jne t Jne 1 vDV'u iJd[ lane Assuming a uniform time step, tlt, and a backward Euler estimate of the time derivative, the fully discrete variational formulation is o Lev ( un drle + Le nnvun. vvdfle + f vv. vundnef vfndne Jne Jne where the superscript n represents the nth timestep. Then making the substi tutions r L ui(tn)'l/Ji(x) x= (x,y) j=l and letting The system of equations for the eth element becomes 122

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J vDn\7un. iJ&y fane t Uj(tn-l) { '1/Ji'l/JjdO.e j=l f1t Jne + { '1/Jdn dOe i = 1, 2, r Jne 3.2.9 Imposition Of The Boundary Condition In our problem, no flow boundary conditions are assumed to exist along the sides of the tank and zero concentrations are assumed to exist at the ends of the tank. In addition, a pulsed-input is allowed to take place at one of the injection ports in the tank for a specified period of time. The following example illustrates how the system of equations is modified to handle constant concentrations at grid points. Suppose that the concentration is to remain constant at a grid point on the left hand boundary, i.e., Suppose that the system to be solved is 123

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Then by letting A11 = 1, A1i = 0, i = 2, n and F1 =a, u1 is forced to be equal to a and the following system emerges But, since u1 = a, the known terms can be moved to the right hand side to yield Direction Of Flow Injection Sampling Figure 12-Boulder's Experimental Tank 124

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3.2.10 The Boulder Experiments Next, consider the test environment used in the Boulder experiments. It is similar to a confined aquifer into which a tracer (pollutant) is injected. It is assumed that the tracer immediately mixes locally with the water in the aquifer to form one phase. The Boulder experiment's horizontal tank has 45 port locations where tracer injections can be made or samples can be taken. Constant head conditions are assumed at the ends of the tank and no flow conditions are assumed on the sides of the tank. Figure 12 illustrates the layout of the tank. The tank can be packed with various types of sand. In the homogeneous experiments, the tank is packed with a single type of sand. In the heterogeneous experiments, the tank is packed in a block arrangement with each block having the dimensions of 12.2 em x 12.2 em, and there are 200 blocks in a 20 x 10 rectangular array. Five types of sand are used with the following hydraulic conductivities: Sand # HydraulicConductivity (m/day) 1 6.05 2 20.74 3 125.28 4 371.52 5 1036.5 Figure 13 shows conceptually the locations of the five different types of sands by relative level of hydraulic conductivity. The actual hydraulic con ductivity levels were not used to produce this plot because if the actual levels were used, the resulting plot would not capture all of the features clearly be cause of the wide spread between the lowest and highest values. Figure 14 is a contour plot of Figure 13, and probably shows in better detail the arrangement of the 5 sand types. In this figure, the white areas represent the sands with the highest hydraulic conductivity and the darker shades of gray representing progressively lower hydraulic conductivities with the black areas the lowest hydraulic conductivity. With this arrangement of the sand blocks there is a significant amount of heterogeneity in the tank. Using the finite element model explained in Chapter 3, a simulation of the Boulder horizontal tank is conducted that is essentially described by Figure 15. This methodology is a Monte Carlo method. The first thing that is done is 125

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40 30 20 10 0 Figure 13 -Hydraulic Conductivity Levels 80 Figure 14 -Hydraulic Conductivity Contours ----------20 1 1-40 60 126 80

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for each of the 5 sands a sample hydraulic conductivity distribution is calculated according to the method explained in Section 2.5.1. The samples generated are then distributed over the domain of the tank by sand type. Following this, the flow equation is solved for the steady state piezometric heads according to the finite element scheme described in Section 3.2.5. 127

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Flow /Concentration Solution Method Create Permeabilities (5 Sands) N Distribute Over Domain Solve Flow Equation For Piezometric Head Compute Element Velocities Sample Size OK ? y Compute Element Dispersivities Solve For Concentration Figure 15 Solution Methodology 128

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Figure 16 shows the computed pressure contours for one of these runs, which agrees with the pressure contours as measured in one of the tank exper iments. The contours show very similar patterns. Figure 17 overlays the computed velocity field, Section 3.2. 7, on the computed pressure contours showing that the velocity vectors are orthogonal to the pressure contours. Flow lines from the tank experiments demonstrate velocity patterns similar to the computed velocities shown in Figure 17. Following the calculation of the velocities, the program cycles back to generate another sample distribution of hydraulic conductivities. This process is continued until a predetermined number of sample runs have been made. The program then computes the dispersion coefficients for the numerical grid blocks according to the methodology specified in Section 2.4. These dispersion coefficients are used along with the average velocities to compute tracer plumes created by injecting a tracer into the tank at a selected point. The finite element scheme for this is outlined in Section 3.2.8. Figure 18 shows the development of a computed plume. The dog-leg appearance is to be expected as can be inferred from looking at the flow lines of the velocity field. 3.3 Summary Chapter 3 begins by defining the two basic forms of the flow /transport problem. In the general case, either the density or the viscosity of the sin gle phase created by two mixing fluids is dependent on the concentration and/ or the pressure. In this case, the flow /transport problem takes the form of a coupled system of PDE's that must be solved simultaneously. The pres sure/velocity pair can be solved for together using mixed finite elements, followed by a solution of the transport equation by some method. The code that is currently available to solve the coupled system in this fashion is the SEGMIX code. It uses the mixed method to solve for the pressure/velocity pair and the modified method of characteristics (MMOC) to solve the transport equation. However, in order to use this code to study the horizontal tank experiments, the treatment of the boundary conditions would have to be modified. In the tracer case, the problem is less complicated, and the uncoupled system can be solved using the standard Galerkin finite elements. Section 3.2 contains descriptions of the implementation of the various components of the finite element method. Section 3.2.2 to 3.2.4 explain the numerical grid and the numerical integration scheme used to calculate the integrals arising 129

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Figure 16 -Computed Pressure Contours 20 15 10 5 Figure 17 -Computed Velocity Field On Pressure Contours .L IJ..L/ I i L I I --15 \'"I 10 .. ... ... -.: 5 .... it.. 0 10 20 30 40 Figure 18 -Computed Tracer Plume 20 15 10 5 10 30 40

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from the variational formulation. Section 3.2.5 contains a derivation of the local system of equations for the flow equation, and Section 3.2.6 describes the assembly of the local systems into a global system. The derivation of the velocity field from the computed piezometric heads is contained in Section 3.2. 7. Section 3.2.8 shows the derivation of the system of equations that follow from the variational formulation of the transport equation. Since there are constant boundary conditions at the ends of the tank, and a pulsed-input is allowed to take place at an injection point, it is necessary to allow constant concentration conditions to exist at some grid points. The modification of the global system of equations to allow certain grid points to maintain a constant level of concentration is explained in Section 3.2.9. Because of the pulsed nature of the injection process, additional discritization is required around the injection point in order to obtain acceptable Peclet numbers. Section 3.2.10 describes in more detail the horizontal test tank used in the Boulder experiments. Two types of experiments are conducted in the tank. The homogeneous experiments are those in which the tank is packed with a single type of sand as rated by its hydraulic conductivity. In the heterogeneous experiments, the tank is packed in a block arrangement with 5 different types of sand. The hydraulic conductivities of the sands range from 6.05 m/day for Sand #1 to 1036.5 m/day for Sand #5. With this wide span of hydraulic conductivities, a significant amount of heterogeneity is represented in the tank. The block arrangement of the sands in the tank is represented graphically in Figures 13 and 14 in Section 3.2.10. Figure 15 provides a flowchart of the basic program components used and how they interact. Comparisons of computer simulation results shown in Figures 16 and 17 to actual tank measurements show very good agreement. Figure 18 illustrates a computed tracer plume. 131

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4. Moment Equations Permeability, density and viscosity are related through hydraulic con ductivity. So, both soil properties and fluid properties are represented in hy draulic conductivity. Furthermore, hydraulic conductivity determines the ve locity field of the water in the aquifer, and if a solute is introduced into the aquifer, the path a solute particle takes through the aquifer is determined by two components. First, the path has a component that is due to molecular dif fusion and, secondly, a component that is due to the mechanical mixing that results from the convective transport. This means that the developing plume is dispersing about a path that is changing due to the influence of the con vective transport determined by the large-scale heterogeneities of the aquifer's domain. In Chapter 2, one method of determining the dispersion used in the transport equation was discussed, and in Chapter 3 that method was imple mented to estimate the expected value of the concentration of a tracer injected into the tank. The entire approach was based on stochastic descriptions of the hydraulic conductivities of the 5 different types of sands with which the tank is packed. As explained in Chapter 3, the domain is subdivided into rectangular grid blocks, each of which is assigned its own average velocity and dispersion coefficient computed according to the methodology explained in Chapter 2. This approach provides the domain of the aquifer with a velocity field that mimicks large-scale changes in the permeabilities of the different sections. However, only the expected value of the concentration is predicted. What is needed in addition to this is a system that will provide information on higher order moments. In general, the more moments that are known, the better the probability density can be described. It would be desirable to at least know something about the second moments. The purpose of this section is to analyze three methods of providing information on higher moments. In the first method, the second moments are derived from the transport equation by a method of distributed parameters. The second and third methods both involve the theory of stochastic differential equations. For the second method, an approach using the Ito calculus, specifi cally Ito's lemma, is used. Finally, the third method seeks to find the solution in terms of the evolution operator. In certain circumstances, this method is a semigroup method. 132

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4.1 Moments Derived From Distributed Parameters The first method is based on the work of Graham And McLaughlin[48] who derive unconditional ensemble moments, ie, moments that do not depend on concentration observations. Starting from the transport equation, and letting oc --+ -+ \7 (cV)\7 [D\7c] = 0 at (4.1) c = E[c] + c' V = E[V] + V' E[c'] = 0 E[V'] = 0 (4.2) we get by substituting Equation[ 4.2] into Equation[ 4.1] + + \7. [E[c]E[V] + E[c]V' + E[V]c' + c'V'] \7 [D\7E[c] + D\7c'] = 0 (4.3) and by taking expectations and using E[c'] = E[V'] = 0, we get the mean concentration equation + \7. (E[c]E[VJ) \7. (D\7E[c]) + \7 E[c'V'] = 0 (4.4) Subtracting Equation[ 4.4] from Equation[ 4.3] we get an equation that involves the perturbations of the concentration and the velocity field oc' --+ at+ \7 E[c]V' + \7 E[V]c' + \7. (c'V') \7 D\7c'-\7 E[c'V'] = o (4.5) Multiply Equation[ 4.5] by the perturbed velocity vector at a point :2 different from x. Since the velocity perturbation depends only on the spatial variable and :2 is different from x and the derivatives are taken at x, it follows that o(c'V'(:d)) ot + (v X. E[c]V'(x)) vt2) + \7 X. E[V]c'V'(:d) + \7 x ( c'V'(x)) V'(:d) \7 x D\7 xc'V'(:d) \7 x E[c'V'(x)]V'(:d) = o 133

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The ith component of this equation is written ( a ) at + ax/E[c]Vj(x) + (E[V]c') V/(Xi) + (c'V'(x)) V'(Xi) ax. J z ax. J z J J ( Djk (c')) (E[c'Vj(x)J) o Since the components are evaluated at Xi, they are constants with respect to the operator, and so we can write J Define J + (E[V]c'V'(Xi)) + (c'V!(x)V'(Xi)) ax. J z ax. J z J J (4.6) = o CV;c(Xi, X, t) cV;VJ (Xi, x) CcV;Vj (Xi, X, t) Ccc(Xi, X, t) t)] E[c'(x, E[c'(Xi, t)c'(x, t)] Taking expectations of Equation[ 4.6] yields the velocity-concentration covari ance equation. :tCcV;(Xi,x,t) + J a ___, + axE[Vj]CcV;(x',x,t) (4.7) J + E[c'(x, J ( Djk CcV; (Xi, x, t)) = 0 134

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Note: The last term in Equation[ 4.6] vanishes because = 0. Taking Equation[ 4.5] and multiplying by c'(Xi, t) where Xi-# x and using the assumption that E[c'(Xi, t)] = 0 and the covariance definitions we get E [:t(c'(x,t))c'(Xi,t)] + E[c(x, t)]Cv-c(Xi, x, t) UXj J + E[Vj(x)]Ccc(Xi, x, t) J + E[c'(x, t)c'(Xi, t)Vj(x)] J =0 Interchanging the roles of Xi and x we get [ 0 1-+ 1 l E ot(c(x',t))c(x,t) a .... .... + ox' E[c(x', t)]CVjc(x, x', t) J a .... .... + ox' E[Vj(x')]Ccc(x, x', t) J a .... .... + ox' E[c'(x', t)c'(x, t)Vj(x')] J =0 (4.8) (4.9) Adding Equations [ 4.8] and [ 4.9] and using the product rule of differentiation :tE[c'(Xi, t)c'(x, t)] [ 0 1-+ 1 l [0 1 1-+ l E ot (c (x', t))c (x, t) + E ot (c (x, t))c (x', t) the following equation for the concentration covariance is obtained :tCcc(Xi,x,t) + J 135

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+ E[Vj(Xi)]Ccc(x, Xi, t)[njk Gee( Xi, x, t)] uXj UXj UXk [njk Gee( X', Xi, t)] + E[c(x, t)]CVjc(Xi, x, t) uxj uxk uxJ + E[c(Xi, t)]CVjc(X', Xi, t) + E[c'(x, t)c'(Xi, t)Vj(x)] uxj uxJ + E[c'(Xi, t)c'(x, t)Vj(Xi)] = 0 (4.10) J The mean concentration equation can be written using the covariance notation as fJE[c] + t at i=l axi 2:-LDij-E[c] n a [n a l i=l OXi j=l OXj n a + = 0 i=l UXi This equation has the form of a transport equation with a forcing term that involves the concentration-velocity covariance. The velocity-concentration equation[ 4. 7] has the form of a transport equation that involves a forcing term that consists of one term that contains the product of the mean concentration and the velocity covariances and one term that involves the expected value of the product of the perturbation of the concentration and velocities. The mean concentration equation and the velocity-concentration covariance are coupled through the E[c] variable and the Ccv; variable. The concentration-covariance equation[ 4.10] also has the form of a transport equation with a forcing term consisting of the last four terms in Equation[ 4.10]. The coupling to the other two equations is through the terms E[c] and Ccv; In order to solve this system, the mean velocities and the velocity covariances are required as inputs. The terms, then, that have to be dealt with to form a closed system are 136

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E[c'(x, t)Vj(x)Vi'(Xi)] J t)c'(Xi, t)Vj(x)] (4.11) E[c'(Xi, t)c'(x, t)Vj(Xi)] These terms are considered to be small, and therefore neglected. To say that these terms can be considered to be small and therefore can be neglected does not seem convincing. By saying that the perturbations are small would cer tainly imply that the expectations of the products of the perturbations are small, but these terms involve the spatial derivatives of the perturbations and there is no reason to believe that they are small. These terms can be eliminated if the assumption is made that they come from a multivariate Gaussian distribution. The multivariate Gaussian probability density function for n dependent random variables is given by f( ) { 1(_, ..... )t( -1)( .......... )} Z1, Z2, Zn = ( 27r) exp 2 z jj V z jj where V= ( ai a 1:; .. 0"21 (}"2 O"nl O"n2 So that V is the variance-covariance matrix. The multivariate characteristic function is given by, Springer[93], page 75, 137

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Applying this theory to our problem, consider the trivariate case where z = (c'(x, t), c'(:?, t), Vj(x))t Since the variations are assumed to have zero means, it follows that jl=O And, the trivariate characteristic function has the form The expression (tv (is a quadratic form, and when expanded is equal to 3 3 = L:L:aij(i(j i=l j=l So, the trivariate characteristic function is given by The reason for introducing the multivariate characteristic function is that mo ments can be generated from it by taking derivatives. In particular, On taking the partial derivatives and using the condition that (1 = (2 = (3 = 0, it follows that So, with the assumption of a joint Gaussian distribution the terms in Equa tion[ 4.11] can be removed. 138

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4.2 An Ito Calculus Approach 4.2.1 System Definition A solution is sought for the following system: au ot + A(x, t, w)u g(x,t,w) (x,t,w) E G X [O,t] X n Q(x, t, w) J(w) (x,t,w) E oG x [O,T] x D (4.12) u(x, 0, w) = u0(x, w) (x,w) E G x D where g E L2(D, B, P) the space of second order random functions. G c lRn is an open domain with a Lipshitz continuous boundary, oG, and t E (0, oo). The operator A is defined as Au= L ( -1)1k1Dk(Pkz(x, t, w)D1u) lkl,lli:S:m The operator D represents weak differentiation and the solution u E L2(0, T; V), where L2(0, T; V) = {t: [0, T]-+ v : lot < 00} The Hilbert space V represents an mth order Sobolev space of L2(D)-valued random functions on the set G. The space V will be more completely specified in the sub-section entitled Existence Theory. 4.2.2 Types Of Problems The following is a list of the different types of problems that can potentially be handled using the stochastic evolution equation formulation: The random initial value problem; u0 is random 139

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The random boundary value problem; J is random The random forcing problem; g is random The random operator problem; A or Q is random The random geometry problem; G is random Combinations of the above In this report, the groundwater flow problem will be treated as a random forcing term problem, ie, g is allowed to be random. And, the groundwater transport problem will be treated as a random operator problem, ie, the operator A will be allowed to have a random component. The interest here is in techniques for solving the stochastic evolution equations and in determining their first and second moment equations. First, the problem of existence of solutions has to be addressed. It is necessary to be able to state conditions under which solutions will exist, and be able to specify the spaces that will contain the solutions. 4.2.3 Existence Theory The existence theory in this section is compiled from Becus[13], Sawaragi, Soeda, Omatu[85], Serrano, Unny, Lennox[90] and Oden and Reddy[72]. Let (0, B, P) be a complete probability space and define to be the space of second order random functions on 0. A probability space is complete if the measure Pis complete, i.e., if any subset of a set, BE B, with P(B) = 0, also belongs to B. The space L2(0) is a Hilbert space with inner product (!, g)n = k fgdP = E[fg] Next the following set M is defined 140

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to be the set of second order random functions on G c lRn. Using the set M, the following spaces are defined: where L2(G) are the square-integrable functions on G. H is a Hilbert space with inner product (j, g)H And, for m 0, fc(f,g)ndG = fcE[fg]dG fcfnJgdPdG Hm is a Hilbert space with inner product (!, g)Hm = L (Da j, Dag)H Hence, Hm is the mth order Sobolev space of L2(D)-valued functions on G. Let V be a real separable Hilbert space such that and the injection i:V----+H is continuous. It then follows that the following diagram can be established ., V' ?-H' 141

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where the mappings Zv and Z H are the Riesz maps between the Hilbert spaces V and its dual V' and between H and its dual H', respectively. And, by identifying H with its dual, H', it can be shown that Vc H=H' c V' and that H' is densely embedded in V'. Using the Hahn-Banach theorem, the duality pairing on V' x V can be identified with the unique extension of the duality pairing on H' x H, < q, u >H And, by the Riesz Representation theorem, \:fq E H', :J Vq E H such that VuE H where (, )H is the inner product on H. So, the duality pairing on V' x V can be identified with the unique extension of the inner product on H. Given this, the norm on V' can be represented as llv' I< ,u >vI sup -'------:-:----:-:-----'uEV llullv uo;to For 0 < T < oo, define L2(0, T; V) = { f: [0, T]---+ V : for < oo} And, iff E L2(0, T; V), then Dtf is the derivative off in the sense of V-valued distributions, ie, DtfEV' Define W(O,T) = {f E L2(0,T;V) W(O, T) is a Hilbert space with norm 142

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=loT + dt Becus[13] recasts the stochastic evolution equation in its variational form, and letting (A(x,t,w)u,v)H = a(u,v) satisfy la(u, v)l:::; Mllullvllvllv ::l M > 0 and the ellipticity condition that ::J). such that Vv E V and for some a > 0 a(v, v) + for almost all t E [0, T], proves the following existence theorem. Theorem: There exists a unique stochastic process u E W(O, T) as a solution of the system [ 4.12]. Also, this solution is continuously dependent on the data, ie, the mapping {g, u0}---+ u is continuous from 2(0, T; V') x H to 2(0, T; V). D 4.2.4 Stochastic Integration At this point I will return to the concept of stochastic integration that was discussed briefly in Chapter 2. That discussion characterized the stochastic integral in terms of a Wiener or Brownian motion process. Doob[36] general izes this somewhat to define the stochastic integral in terms of a martingale. This term is not very descriptive. In fact, the primary definition in Webster's dictionary is that of a part of a harness for a horse. However, it is also used to describe a system of betting strategies. Of course, probability theory makes this form of the definition more precise. Following Doob[36], Burrill[17] and Jazwinski[54], the major ideas are outlined below. As a matter of convenience the Radon-Nikodym Theorem is stated as found in Burrill[17]. Radon-Nikodym Theorem Let the measure f.t and the absolutely continu ous additive function be O"-finite. Then there is a finite valued measurable function g such that 143

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for each measurable set E. D Given a probability space (D, P) and an integrable random variable X on D, define the function which is an additive function and absolutely continuous relative toP, (cjJ(E) = 0 if P(E) = 0). The set E belongs to a O"-algebra F contained in So, by the Radon-Nikodym Theorem there is an F-measurable function denoted by E[XIF] such that for each E E F and called the conditional expectation of X given F. In fact, E[XIF] represents an equivalence class of integrable random variables such that any member of the equivalence class is measurable with respect to F and has the same integral as X over any E E F. Next, let T C I U { -oo, +oo} where I is the set of integers, and let { Ft : t E T} be a collection of O"-algebras such that Fs C Ft C for s < t and, finally, let {X(t) : t E T} be a collection on integrable random variables such that X(t) is measurable relative to Ft for each t. In probability theory, the sets in Ft are called events, and the measurability of X(t) with respect to the O"-algebra Ft can be interpreted to mean that the values of X(t) are detectable by the events in Ft. Definition: The collection {X ( t) : t E T} is a martingale relative to { Ft : t E T} if 144

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X(s) = E[X(t)IFs] s < t and a semi-martingale relative to { Ft : t E T} if X(s):::; E[X(t)IFs] s < t Furthermore, the following theorem holds Theorem: The collection {X(t) : t E T} of integrable random variables is a martingale iff for all s, t E T with s < t and all E E Fs k X(s)dP = k E[X(t)IFs]dP = k X(t)dP and a semi-martingale iff for all s, t E T with s < t and all E E Fs k X(s)dP:::; k E[X(t)IFs]dP:::; k X(t)dP o The relationship between martingales and the Wiener process is given by the following theorem from Doob[36] Theorem: Let {X ( t), Ft, a :::; t :::; b} be a real martingale, and suppose that almost all sample paths of the process are continuous. Suppose that and that for each pair s, t with s < t E[(X(t)X(s))2]1Fs] = t-s with probability 1. Then it follows that the X(t) process has independent increments and is a Wiener process. D Doob[36] defines the stochastic integral k CI>( t, w )df3( t) by assuming that the process f3(t) is a martingale (evidently it can be extended to include f3(t) as a semi-martingale). The Ito integral discussed in Chapter 2 follows as a special case from the preceding theorem. 145

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Jazwinski[54] defines the Ito integral as the mean square limit of step function processes in the following manner: Definition A step function, g(t, w), is defined as where a1 < < an, and gj(w) is measurable with respect to Fai and E [lgj(w)l2 ] < oo and gj(w) is independent of This is a condition of nonanticipativeness. One way of interpreting this is that the function gj is independent of the Wiener process in future time t. In other words, the values of gj are observable only by events prior to aj. Let {gn ( t, w)} be a sequence of step function processes converging to the process g(t, w) in the sense that then the Ito integral of the process g(t, w) with respect to the Wiener process {3(t,w) is defined to be r g(t,w)d{3(t) = (m2 ) lim r gn(t,w)d{3(t) Jr Stochastic integrals are defined in the sense of mean squared convergence which implies convergence in measure P, because if E > 0 and then Hence, from the mean convergence it follows that 146

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P(On) < El2 kn lfrgn(t,w)d(3(t)hg(t,w)d(3(t)l2 dP < E12 k lfrgn(t,w)d(3(t)hg(t,w)d(3(t)l2 dP --+ 0 as n --+ oo In order to extend these results to a Hilbert space, H, it is necessary to define a Wiener process in a Hilbert space, Falb[37], Curtain and Falb[26], [27], and Sawaragi, Soeda, Omatu[85]. If W(t) is an H-valued Wiener process, then there are complex random processes such that 00 W(t) = L f3i(t)ei i=O almost everywhere in (t,w). Here, is an orthonormal basis of H. And, R(f3i ( t)) and ( t)) are real Wiener processes. Ito stochastic integration is extended to the Hilbert space setting as follows: First, in Section 1.3, a complex-valued second order random variable was defined in terms of the modulus function, I I In the case of a Hilbert space valued random variable, the H-valued random variable, X(w), is second order if E < oo where the modulus function is now replaced by the H-norm, II IIH Secondly, the mean squared convergence is done in terms of the II II H norm instead of the I I function. Let H be a Hilbert space and W(t) an H-valued Wiener process. Also, let g(t, w) be a step function from T into (H, H) where a1 < < an, and if gn ( t, w) is a sequence of step functions converging to g(t,w), then 147

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h g(t, w)dW(t) Or, Because the Ito integral is applicable to a wider class of functions, it is used in this analysis even though new rules of Ito calculus must be devised. 4.2.5 Ito's Lemma In Hilbert Space The most important new rule required for the solution of the stochastic evolution equations is Ito's lemma in Hilbert space. It is a kind of change of variable formula. The one-dimensional version of the Ito formula was described in Section 1.1, where the relationship dW(t)2 = dt was used to develop it. Using the relationship where is an m-dimensional Wiener process, and the mappings the d-dimensional vector stochastic differential equation is dX(t) = a(t, X(t)) + b(t, X(t)) dW(t) Then, Jazwinski[54], Kloeden, et al[59], for a sufficiently smooth function 148

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g : [to, T] x lRd --+ lRk of the solution X(t), t0 t T, there is a k-dimensional process Y(t) = g(t, X(t)) such that for the pth component of the vector process Y(t), p = 1, ,k where all terms are evaluated at the points (t, X(t)). This is the finite dimen sional vector version of Ito's formula. As for the infinite dimensional version, let JL1 (H, K) {S(t,w): S(t,w) E C(H,K) and for < oo} Ito's Lemma Let H, K, and G be Hilbert spaces and let W(t) be an H-valued Wiener process. Suppose that g(t, c) is a continuous map of [0, T] x K into G and that u(t) is a K-valued stochastic process with stochastic differential du(t) = A(t, w)dt + C(t, w)dW(t) such that gt(t, c) is continuous on [0, T] x K g(t, ) is twice continuously differentiable on K for each fixed t E [0, T]. gc(t, c) and gcc(t, c) are continuous in (t, c) on [0, T] x K. A(t,w) is a K-valued stochastic process which is measurable relative to :Ft, t E [0, T], and integrable on [0, T], with probability 1. C(t,w) E P,1(H,K) and J;[ E[IIC(t,w)ll4dt < oo W(t) is real, and gt and gc denote the partial and Frechet derivatives. 149

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Then z(t) = g(t, u(t)) has the G-valued stochastic differential dz(t) { 9t(t, u(t)) + gc(t, u(t))[A(t, w)] + dt + 9c(t, u(t))[C(t)]dW(t) Here fr represents a trace operator which is defined as 00 fr(gcc(t, L 9cc(t, (4.13) i=l where = y');;ei and the { ei} is an orthonormal basis of H of eigen vectors of Q, the covariance operator associated with the Wiener process, W(t), with corresponding eigenvalues {.\i} The existence of these eigenvalues and eigenvectors follows from the definition of an H-valued Wiener process, Falb[37], Curtain and Falb[26, 27] and Sawaragi, Soeda and Omatu[85], where the covariance operator, Q, is assumed to be compact. A Corollary that will be more useful is: Corollary: Suppose that in addition to the hypothesis of the theorem that we let G = Then dz(t) can be written as dz(t) {gt(t,u(t)) + (A(t,w), Vcg(t,u(t))) 1 + 2tr(C(t, w)Q(t)C*(t, w)8ccg(t, u(t)))}dt + ( C*(t, w )V cg(t, u(t) ), dW(t)) where V c9 and 8cc9 are the gradient and Hessian of g with respect to c. Versions of these results on Ito's lemma are found in Curtain and Falb[26], Bensoussan[14] and Sawaragi, Soeda, Omatu[85]. 150

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4.2.6 Small o Notation Let X be a B-space with dual X', then < y', x > is the duality pairing of y' and x. Let x1 EX, EX' and define the mapping such that (x1 o = x1 < > Vx EX Theorem: Let X be a B-space and let 'ljJ be the mapping of X E9 X' into .C(X, X) defined by Then 'ljJ has the following properties: 'ljJ is continuous 'ljJ is linear in both x1 and (xi o = o XI if X is reflexive Note: If X= Rn, then XI 0 Yl can be identified with the matrix XIYt. The small o notation can be used to define the concept of covari ance on a Hilbert space H. The inner product (h, X(w))H is a linear random functional on H'. So, if X ( w) E H, then and (h, X(w)H is a real random variable. Hence, E[(hi, XI)H(h2 X2)H] repre sents the covariance of XI and X2. Let hi, h2 E H' and let XI, X2 E H, then by identifying H = H' it follows that Since (XI o E H, by taking expectations it follows that 151

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Assuming for the moment that E[X1 ] = E[X2 ] = 0 and that the mapping 3 : H' X H' -----+ R1 Then, by the Riesz Representation theory, there is a unique Riesz map A E (H', H) such that where A is called the covariance operator, and since A is unique, A= E[X1 oX;]. In the case that the expected values of X1 and X2 are not zero, the covariance operator of X1 and x; is defined as, Falb[37], 4.2. 7 Hilbert Space Structures Kadison[55] gives the following definitions for a Direct Sum Of Hilbert Spaces and for a Direct Sum Of Operators: Let 1-l1 1-l2 1-ln be Hilbert spaces and }( be the set of all n-tuples { x1 x2 Xn} with Xi E 1-li. Then there is a Hilbert space structure on }( with the following definitions: Algebraic Operations: Inner Product: Norm: The resulting Hilbert space }( is called a Hilbert direct sum of 1-l1 1-ln and is denoted 152

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n 1-l1 EB EB 1-ln L EB1-li i=l Let 1-li and /Ci be Hilbert spaces and Ti E B(1-li, JCi), i equation defines a linear operator T such that n n T : L EB1-li ---+ L EB/Ci i=l i=l where The following notation will be used: n {xl,,xn} LEBXi i=l The direct sum of operators has the following properties: i=l i=l Frechet Derivatives In order to derive the first moment equation, let g(t, v) = (h, v)H hEH' So that 153 1, n, then the

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g : [0, T] x H ---+ lR And, for a fixed t, g:H-+lR If the Frechet derivative exists, then the Gateaux derivative exists and the two are equal. Denote the Frechet derivative by the symbol og ov E (H, lR) By the Riesz Representation theorem, the Frechet derivative can be represented by the inner product on H. So, for a fixed h, the Frechet derivative is defined as fJg TJ = (f)g' TJ) =lim (h, V + G:TJ)H(h, v)H ov ov H E--+0 a From this definition it follows that og ov =hE (H, lR) Since hand TJ are fixed, (h, TJ)H is a constant. So, the second Frechet derivative of g is zero. To derive the second moment equation, let Then, l (hi, v + ETJ)(h2, v + ETJ)-(hi, v)(h2, v) E--+0 E Or, in operator notation if we let 154

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and Then, T = T1 EB T2 : H EB H --+ H EB H and, using the definitions for the inner products we have, So, we can make the identification For the second derivative, we can write a2g rv av2 E .C(H, .C(H, R)) = .C(H EB H, R) where rv represents an isometry. Again, by the Riesz Representation theorem, the second derivative is given by a2g (a2g ) a 2((,77)= a 2(EB(,77EB77 V V HtBH =lim [(h1, v + E()(h2, 77) + (h2, v + E()(hl, 77)][(h1, v)(h2, 77) + (h2, v)(h1 77)] E =lim [(h1, v + E()(h2, 77)(h1, v)(h2, 77)] + [(h2, v + E()(hl, 77)(h2, v)(h1 77)] E And, if we identify H with its dual H' we can write this operator in terms of the small o notation, so that With this notation we can write 155

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Hence, (hi, ()(h2, TJ) + (h2, ()(hi, TJ) < (h2 0 hi) E9 (hi 0 h2)( E9 (, TJ E9 TJ > [J2g -= hl 0 h2 E9 h2 0 hi 8v2 4.2.8 Moment Equation Derivation In this section, equations for which the forcing term has a Gaussian white noise component are discussed, References for this section are Astrom[7], Bensoussan[14], Chow[22] and Serrano, Unny, Lennox[90]. In the finite dimensional case, Astrom[7] shows that the linear stochas tic differential equation dx = -Axdt + f + dv where dv is a white noise process, has moment equations and dMI -=-AM1+f dt dM2 t t dt = -(AM2 + M2A ) + f MI + Md + RI (4.14) where E[v(t)v(t)] = R1t. A result similar to this will be derived for the infinite dimensional case. Consider the equation du -=-Au+f+( dt where u is a function of ( t, x, w) and belongs to a Hilbert space H; A is a spatially elliptic operator; f is deterministic; and ( is a Gaussian white noise process. In integral form, this equation is 156

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u(t) = u(O) +lot (-Au+ f)ds +lot ((s)ds Then, W(t) = JJ ((s)ds is an H-valued Wiener process and the original equa tion can be written as du(t) = (-Au+ f)dt + dW(t) For more generality, introduce the stochastic operator ? (t) such that ? (t) E (H,H) and So that we have du(t) = (-Au+ f)dt +? (t)dW(t) Using Ito's lemma, dg(t, u(t)) {ag ( ag ) at (t, u(t)) + -Au+ j, au (t, u(t)) (4.15) 1 ?Q( )?*a2g} d (ag ?dw) + 2tr. t au2 t + au'. And, Q(t) E 00(0, T; (H, H)) and is called the covariance operator. Allowing g(O, u(O)) = 0, Equation[ 4.15] can be interpreted in the stochastic differential equation sense as g(t,u(t)) lot { (s, u(s)) + (-Au+ j, (s, u(s))) + 1tr? Q(s)? ds +lot ( (s, u(s)),? dW(s)) where dW(s) is to be interpreted as a Gaussian white noise. Now, if g = (h, u(t))H, hE H', we have from the Frechet derivative 157

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8g = h au Furthermore, since g depends on t only through u, Taking expectations and using the result that E[(h, u)H] = (h, E[u])H, it follows that (h, E[u])H =lot E [(-Au+ j, h)H] ds +lot E [(h,? dW(s))H] To evaluate the last integral, lot E[(h,? dW(s))H] = (m2 ) t E [ ( h, gj(w)[W(tj+l)-W(tj)) H] j=l 0 since gj(w) and W(tj+1)-W(tj) are independent which follows from the nonan ticipativeness of the operator gj(w) with respect to W(tj+1 ) W(tj) So, if M1 = E[u], then assuming that A is deterministic ( dM1) h, dt = -(h, AMI)+ (h, f) Or, in this weak sense, the first moment equation is dM1 -=-AM1+f dt To obtain the moment equation for the second moment, let From the Frechet derivatives we have And, 158 (4.16)

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[J2g -= h2 o h1 EB h1 o h2 8u2 Hence, it follows from Ito's lemma d dt ( hl' u) ( h2' u) -< (h1, u)h2 EB (h 2 u)h1 Au EB Au> + < (h1, u)h2 EB (h2, u)h1, fEB f > 1 + 2tr[? *(h2 o h 1 ) EB (h 1 o h 2)? Q(t)] + < (h1, u)h2 EB (h2, u)h1,? dW EB? dW > Expanding this equation we get d dt ( hl' u) ( h2' u) -[(h1 u)(h2 Au)+ (h2 u)(h1 Au)] + (h1, u)(h2, f)+ (h2, u)(h1, f) + *(h 2 o h 1 ) EB (h 1 o h 2)? Q(t)] + (h1, u)(h2,? dW) + (h2, u)(h1,? dW) ( 4.17) Taking expectations and using the following definition of the correlation oper ator, where Rxy = M2 if X= Y, it follows that for M2 = M; and A deterministic, E[(h1 u)(h2 Au)] E[(h2 u)(h1 Au)] E[(h1, u)(h2, f)] And, E[(h1, u)(A*h2, u)] = (h1, M2A*h2) E[(h2, u)(A*h1, u)] = (h2, M2A*h1) = (h1, AM2h2) (h1, Rufh2) = (h1, Mdh2) -[(h1, M2A*h2) + (h1, AM2h2)] + (h1, Mdh2) + (h1, Mdh2) (4.18) 1 + "2 E [tr? *(h2 o h1) EB (h 1 o h 2)? Q(t)] 159

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Interchanging the roles of hi and h2 we get -[(h2, M2A*hi) + (h2, AM2hi)] + (h2, Mdhi) + (h2, Mdhi) (4.19) 1 + 2 E [tr? *(hi o h 2 ) E9 (h 2 o hi)? Q(t)] Adding these Equations [ 4.18] and [ 4.19] together and using the definition of the inner product on a direct sum of Hilbert spaces, (hi E9 h2, -(AM2 E9 M2A*)h2 E9 hi) + (hi E9 h2, (Md E9 (Md)*)h2 E9 hi) 1 + 2 E [tr? *(h2 o hi) E9 (hi o h 2 )? Q(t)] 1 + 2 E [tr? *(hi o h 2 ) E9 (h 2 o hi)? Q(t)] where, as before, the last term of Equation[ 4.17] vanishes on taking expecta tions. Even though this equation has a weak sense formulation, it has a form similar to the simpler case Equation[ 4.14], page 155, above. Using Equation[ 4.13], page 149, and the fact that Qei = Aiei, the trace term can be put into a more usable form by expanding and using Parseval's relation and the definition of the small o notation 1 -([J2g) 1 00 [J2g [ h h ] 2tr ou2 [? = 2 ou2 ? y Aiei,? y Aiei = 1 o h2) E9 (h2 o hi) [? = 1 + (h2,? 160

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(4.20) This theory will be demonstrated in the next two examples. The first example will use the theory to derive the mean concentration equation, Equation[ 4.4], page 132, the velocity-concentration equation, Equation[ 4. 7], page 133, and the concentration-covariance equation, Equation[ 4.10], page 135. Consider the transport equation oc _, ot + \7 (cV)\7 (D\7c) = 0 Suppose that the tensor D has been specified in a deterministic manner such as specified in Chapter 2, and the velocity and concentration are expressed as V(x,w) = E[V(x)] + V'(x) c(x,t) = E[c(x,t)] +c'(x,t) E[c'(x, t)] = o In terms of Equation[ 4.12], page 138, A(x, t, w) \7 (()(E[V(x,w)] + V'(x,w)))-\7 (D\7()) g(x,t,w) = o From now on, w will not be specifically stated. Then, from Equation[ 4.16] page 157, (h, = -(h,Ac) -(h, \7 [(E[c] + c')(E[V] + V')]-\7 (D\7(E[c] + c'))) -( h, \7 [ (E[c]E[V] + E[c]V' + c'E[V] + c'V'] \7 (D\7(E[c] + c'))) Taking expectations, and using E[c'] = E[V'] = 0, ( h, =-(h, \7. [(E[c]E[V] + E[c'V'J]-\7 (D\7(E[c]))) 161

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So that in the weak sense, + \7. (E[c]E[V]) \7 (D\7E[c]) + \7 E[c'V'] = 0 which agrees with Equation[ 4.4], page 132. For the second moment equations, let x and :2 be two different coordinate systems, then from Equation[ 4.17], page 158, with hi, h2 E H', :t(hi,c(x,t))(h2,c(:d,t)) =[(hi,c(x,t))(h2,Ax-;c(:d,t)) +(h2, c(:d, t))(hi, Ax-c(x, t) J ..... ( ac(:d, t)) ..... ( ac(x, t)) =(hi, c(x, t)) h2, at + (h2, c(x', t)) hi, at (4.21) as would be expected since the Wiener process is excluded from playing a role in this example. Expanding the left hand side of Equation[ 4.21], tak ing expectations, using E[c'(x, t)] = E[c'(:d, t)] = 0 and letting Ccc(x, :2, t) = E[c'(x, t)c'(:d, t)] E [ :t (hi, c(x, t)) (h2, c(:d, t))] = E [ :t (hi, E[c(x, t)] + c' (x, t)]) x (h2 E[c(:d, t)] + c' (:2, t)) J = :t [E(hi,E[c(x,t)])(h2,E[c(:d,t)]) + (hi,Ccc(x,:d,t)h2)] = :t [(hi, E[c(x, t)]E[c(:d, t)]h2) +(hi, Ccc(x, :2, t)h2) J = (h1 iJE[:, t)]E[c(J?, t)]h2 ) + ( h1 iJE[ct, t)] E[c(X, t)]h2 ) + :t (hi, Ccc(x, :2, t)h2) Setting this equal to the expected value of the right hand side of Equation[ 4.21] a ( ..... 1 ) ( [aE[c(x, t)] --; aE[c(:d, t)] ..... ] ) at hi, Ccc(x, X' t)h2 + hi, E at c(x 't) + at c(x, t) h2 162

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( [ oc(:?,t) _, ] ) ( [oc(x,t) _, ] ) = hi, E ot c(x, t) h2 + hi, E at c(x', t) h 2 so that on rearranging terms, From the first moment equation, t)] = -\7 x(E[c(x, t)]E[V(x)])+\7 x(D\7 xE[c(x, t)])\7 xE[c'(x, t)V'(x)] and, t)] = -\7x-; (E[c(:?, t)]E[V(:?)]) + \7x-; (D\7x-;E[c(:?, t)]) \7x-; E[c'(:?, t)V'(:?)] Since Ax()= \7x (()(E[V(x)] + V'(x))-\7x (D\7x()) it follows from the first term on the right hand side of Equation[ 4.22] and by letting c(x, t) = E[c(x, t)] + c'(x, t) that ( [ _, _, oE[c(x, t)] _, ] ) hi, E c(x', t)Axc(x, t) + ot c(x', t) h2 -(hi, E [c(:?, t) [v X. (E[c(x, t)]V'(x) + c'(x, t)E[V(x)] + c'(x, t)V'(x))-\7x (D\7xc'(x, t))\7x E[c'(x, t)V'(x)J]] h2) letting c(:?, t) = E[c(:?, t)] + c'(:?, t) and expanding the previous result, 163

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-( hl, E [v X. (E[c(Xi, t)]E[c(x, t)]V'(x) + c'(?, t)E[c(x, t)]V'(x) + E[c(Xi, t)]c'(x, t)E[V(x)] + c'(Xi, t)c'(x, t)E[V(x)] + E[c(Xi, t)]c'(x, t)V'(x) + c'(Xi, t)c'(x, t)V'(x)) V x (DV xc' (x, t)E[c(Xi, t)]) V x (DV xc' (x, t)c' (Xi, t))) Vx E[c'(x, t)V'(x)]E[c(Xi, t)]Vx E[c'(x, t)V'(x)]c'(Xi, t)] h2) taking expectations, cancelling terms and using E[c'] = E[V'] = 0 -( h1, [v x E[c(x, t)]Ccv(x, Xi, t) + V x E[V(x)]Ccc(x, Xi, t) ( 4.23) + V x E[c'(Xi, t)c'(x, t)V'(x)] V x (DV xCcc(x, Xi, t)) J h2) Also, from the second term on the right hand side of Equation[ 4.22] an equation exactly like this one can be derived in the same way only with the vectors x and Xi interchanged. (4.24) Equations[ 4.22], [ 4.23] and [ 4.24] give the result for the concentration covari ance equation as :t (h1, Ccc(x, Xi, t)h2) = -( h1, [v x E[c(x, t)]Ccv(x, Xi, t) + V x E[V(x)]Ccc(X', Xi, t) + V x E[c'(Xi, t)c'(x, t)V'(x)] V x (DV xCcc(x, Xi, t)) J h2) Comparing this with Equation[ 4.10], page 135, it is seen that this equation is the vector form of Equation[ 4.10]. 164

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Finally, returning to Equation[ 4016], page 157, and multiplying both sides by (h, where is the ith compoment of the vector V(Xi), we have Next, write gt (hl, c'(x, t))(h2, V'i(Xi) = gt (hl, c(x, t)E[c(x, t)])(h2, = gt [(hl, c(x, t))(h2, (h1, c(x, t))(h 2 -(h1, E[c(x, t)])(h2, (h1, E[c(x, t)])(h2, Taking expectations and differentiating, a ( --+ ) a --+ --+ at h1, CcV;(x, x', t)h2 = atE[(h1, c(x, t))(h 2 Vi(x')] -E [(h1, gtc(x,t)) = ( h1, [ E [ gt c(x, ]gt E[c(x, h2) Substituting expressions for gtc(x, t) and gtE[c(x, t)] yields gt ( hl, ccV; (x, Xi, t)h2) = ( hl, [E [-{ \7 X 0 c(x, t)(E[V(x)] + V'(x)) \7x 0 (D\7xc(x, t))} + { \7 x 0 (E[c(x, t)]E[V(x)]) \7 x 0 (D\7 xE[c(x, t)]) + \7x 0 E[c'(x, t)V'(Xi)J} h 2 ) Let c(x, t) = E[c(x, t)] + c'(x, t) and expand 165

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+V X. E[c(x, t)]V'(x) + v X. c'(x, t)V'(x) Vx (DVxE[c(x, t)])Vx (DVxc'(x, t))} + {vx. (E[c(x, t)]E[V(x)])-Vx. (DVxE[c(x, t)]) + Vx E[c'(x, t)V'(Xi)J} h2) And, finally, by letting = + V'i(Xi) and expanding again :t ( hl, ceil; (x, Xi, t)h2) = ( hl, [E [-{ v X. E[c(x, +V x E[c(x, t)]E[V(x)]V'(Xi) +V X. c'(x, + v X. c'(x, t)E[V(x)]V'i(Xi)] -Vx (DVxE[c(x, Vx (DVxE[c(x, t)]V'i(Xi)) v x (DV xc'(x, v x (DV xc'(x, t)V'i(Xi)}] + { v x (E[c(x, t)]E[V(x)]) v x (DV xE[c(x, t)]) + Vx E[c'(x, t)V'(Xi)J} h2) So, by taking expectations, cancelling terms and using the conditions E[V'] = E[c'] = 0, the final equation for the velocity-concentration equation is 166

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+ \7 x E[c(x, t)]CV;v(x, :?, t) \7 x (D\7 xCcV; (x,:?, t) )h2) which agrees with the previously obtained velocity-concentration Equation[ 4. 7], page 133. The second example involves measurement uncertainty. This example will shows how the trace term in Equation[ 4.17] can be used. Consider ac ..... -+ \7 (cV)-\7 (D\7c) = 0 at ( 4.25) Suppose that there is measurement uncertainty in the laboratory experiment, and that this uncertainty is random. Then, in the laboratory, the experimenters will record the results c, and the variables u and c will be related by c(x,t,w) = c(x,t) +E(t,w) Using Equation[ 4.26] in Equation[ 4.25], a(c(x, t,w)-E(t,w)) r7. (-(-+ )V-+)-r7. (Dr7_(_, ) = 0 at + v c x, t, w v v c x, t, w ( 4.26) (4.27) The derivatives in Equation[ 4.27] have to be interpreted in the mean-square sense, ie, if x(t, w) is a random function, then ( ) ( 2 )1 x(t+h,w)-x(t,w) x t,w = m 1m h h--+0 '* E [lx(t + h, wlx(t, w) X(t,w)l'] /, lx(t + h, wlx(t, w) (t,w)l' dP 167

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And, from the inequality it follows that Using this, Equation[ 4.27] can be written as oc(x,t,w) n.(-( ..... )V ..... )-n(Dn-( ..... )=de(t,w) at + v c x, t, w v v c x, t, w dt where is a stochastic process. Hence, the introduction of measurement uncertainty is equivalent to applying a stochastic forcing term to the equation. The situation can be characterized by the following diagram similar to one found in Gelb[43]: System Measurement Uncertainty Uncertainty A Priori Sources Sources Information System Observation System State State u(t) u(t) Estimate System Measurement Model u(t) In this diagram, the System Uncertainty Sources are represented by any un certainty that may exist in the specification of D and V, the Measurement Uncertainty Sources are represented by E. The random forcing term is assumed to be a Gaussian white noise process. This is equivalent to assuming that the process, e( t, w), is a Wiener process which can be defined as the limit of a random walk, or as the integral of a Gaussian white noise process with zero mean. W(t) =lot ((s)ds 168

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The following is a block diagram representation of this equation: (( t) w (t) E[((t)((T)] = q(t)6(tT) A second feature of the Boulder experiments that must be modeled is the pulsed input feature of the experiment. This means that in the tracer experiment, the tracer, benzene, is injected at a rate of 5 ml/min for a period of 4 hours and then the injection pump is turned off. However, samples are taken for a period of 8 hours. This means that at a specified measuring point, the sampling device will see the concentrations of benzene first increase, then level off, and finally decrease to zero. In the finite element model, this allowed for by imposing a non-zero boundary condition at the origin for a specified number of time steps, and then imposing a zero boundary condition at the origin for the remainder of the time steps of the simulation. The following is a segment of code that performs this task: 1*************************************************************1 I* Impose The Left Hand Boundary Condition *I 1*************************************************************1 if (nt <= bctimesteps) impose_bndy_cond(); else { } lbdy = 0.0; impose_bndy_cond(); Here, the variable lbdy is originally input to the program with a non-zero value. Once the specified number of timesteps for injection of the tracer, bctimesteps, has passed, lbdy is set to zero and the boundary condition function imposes a zero boundary condition on each succeeding time step. The two graphs on the next page illustrate the output from the finite 169

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element program with a pulsed input. The surface shown in Figure 19 is a space-time representation of the concentration. Figure 20 shows a time-slice of this surface. This curve has the same shape as the actual measurements when they are plotted. Figure 21 entitled Comparison Time Profile shows the Pulsed Input Time Profile with and without the effects of a random forcing term (mea surement uncertainty). The dotted line represents the time profile without measurement uncertainty, and the solid line shows the time profile with mea surement uncertainty taken into consideration. Returning to the equation ac(x, t,w) r7. (-(--+ )V--+) r7. (Dr7-(--+ ) = dE(t,w) at + v c x, t, w v v c x, t, w dt For the sake of simplicity, assume that the parameters D and V are deter ministic. This means that there will be no need of a velocity-concentration covariance equation as in the previous example. The equation for the expected value of the concentration takes the form t)] + v. (E[c(x, t)]V(x))-\7. (D\7E[c(x, t)]) = o For the equation of the concentration covariance, let x and :? be two different coordinate systems, then from Equation[ 4.17], page 158, + (hi, c(:?, t)(h2, Ax-c(x, t))] 1 + 2tr [(h2 o hi) EB (hi o h 2 )Q(t)] where ? =I. And, by proceeding as in the previous example, 170

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1 0.75 1 0.8 0 6 0.4 0.2 0 Figure 19 -1D Pulsed Input O ver Time Figure 20 -Pulsed Input Time Slice / \ I I \ I \ I \ I \ '-........ 0 20 40 6 0 80 100 120 140 171

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Figure 21 -Comparison Time Profile 1 ..... .. _.,... ... v-............ '\ .... ; I ( I \. I \. I \ 0.8 0.6 0.4 0 2 0 ....... 0 20 40 60 80 100 120 140 172

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\7 xl (D\7 x-;Ccc(Xi, X, t)) J h2) 1 + 2tr [(h2 o h1) EB (h 1 o h 2)Q(t)] Finally, substituting for the tr term from Equation[ 4.13], page 149, it follows that 4.3 Summary Chapter 4 actually starts the second part of the thesis. The previous sections have investigated the components of the equations and the forms of the equations. However, only the expected or mean value of the concentration is predicted. Because of the uncertainties involved in specifying the physical characteristics of the porous medium, the concentration of a solute at a given point in time is a random variable, and over a period of time it is a stochastic process. Consequently, in order to more accurately characterize the distribu tion of the solute concentration, higher order statistical moments such as the variance need to be estimated also. In theory, the more moments that can be predicted, the better this characterization will be. But, in practice, it is usually a difficult problem just to obtain information on the variance or covariance of variables in the system. A much referenced paper in this area is the Graham and McGlaughlin[48] paper which specifies a set of three equations that are to be solved for the mean concentration, the velocity-concentration covariance and the concentration covariance. These equations were presented in Section 4.1 for the purpose of comparison with mean and covariance equations derived from other methods. Randomness can enter the boundary value problem in many different ways. Equation[ 4.12], page 138, is a statement of the stochastic boundary value problem, and the discussion following that equation specifies the various ways in which randomness can enter the picture. Existence theory for the 173

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stochastic boundary value problem was covered in Section 4.2.3 and found to be not unlike the nonstochastic case. Stochastic integration is again addressed in Section 4.2.4, this time from the more general perspective of a martingale. The Ito integral then follows from this more general definition as a special case. The use of the Ito integral requires that the rules of calculus have to be modified. The most important new rule is that of Ito's lemma. It is a change of variable formula. The reason the change of variable formula has to be modified is due to changes in differential relationships that were covered in Section 1.1. The Ito formula is a stochastic calculus chain-rule. It can also be extended to martingale type processes, Karatzas[57]. Curtain and Falb[26] have extended Ito's lemma to infinite dimensional Hilbert spaces. It is this form that is used to derive weak forms of the moment equations in Section 4.2.8. For the purpose of illustrating this theory, the key equation is Equation[ 4.17], page 158, which is applied to two examples. The first example uses this theory to derive mean and covariance equations that in the weak form are identical to those used by Graham and McLaughlin[48]. The second example is cast in terms of accounting for the effects of measurement error that is assumed to enter the experiment as a random perturbation that takes the form of a Wiener process. 174

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5. Stochastic Evolution Equations 5.1 General Theoretical Foundations As was observed in Chapter 2, the dispersion tensor can be considered a time stochastic process. That result along with the stochastic nature of the velocity field allows the transport equation to be written in a form that allows the separation of the deterministic components from the stochastic compo nents. That is, the stochastic PDE that represents the transport equation can be separated into a sum of a deterministic operator and a stochastic operator as the following 1-D example shows: ou -E[D(t)] fJ2u + E[V] ou m ox 2 ox u(x, 0) = u 0 '( )82u '( )au D t, w ox 2 V w ox u(oo, t) = 0 The left hand side of this equation is the standard form of the transport equa tion, while the random components associated with dispersion and velocity have been moved to the right hand side in the form of a random operator. It is well known that there is a correspondence between the Cauchy problem and the abstract boundary value problem similar to the left hand side of the equation stated above. The Cauchy problem can be stated as: Let H be a Hilbert space, D(A) a subspace of Hand let the operator A an unbounded, linear operator from D(A) to H. Then let u' ( t) + Au ( t) = f ( t) (5.1) u(O) = uo and the Cauchy problem is to find a function u(t) such that for t > 0, u(t) E D(A) and satisfies Equation[ 5.1], so that u(t) is an H-valued function. Hence, there must exist a correspondence between the H-valued func tion u(t) and the solution, u(x, t) to the boundary value problem. And, a cor responding relationship between the derivative u'(t) and the partial derivative 175

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gtu(x, t). Also, the operator A must be related to the boundary value problem. Following Showalter[91], let I = [a, b] be a closed, bounded interval in lR and let 0 be a bounded, measurable subset of lRn. Let u(t) E C[I, 2(0)]. Let a = t0 < t1 < < tn = b be a uniform partition of I such that ( ..... t) { uo(i!, tk) Un X, -( ..... b) u0 x, tk ::; t < tk + 1, k = 0, 1, ... n-1 t = tn Here u(ti) = u0(x, ti) E D(A) is a representation of u(t). Then Un(x, t) : 0 X I-+ lR Since each u0(x, tk) E 2(0), fort E I, the t-section of the function un(x, t) is given by = un(i!, t) And, since the t-section E 2(0), it is measurable. And if x E 0 is fixed we get a step function. So, the t-section defines the following mapping: If a E lR, then the set p = {(x, t): >a} is either empty or of the form where E is a measurable subset of 0. In either case, P is a measurable subset of 0 xI. And, since un(x, t) is a finite sum of these functions, it is a measurable function on the product measure space 0 x I. Since the partition of I is uniform and since u(t) is uniformly contin uous on I, then forE > 0 :J 6E > 0 such that if the partition n is large enough, then k = 0,1,,n1 Hence, if t E I, then :J k such that t E [tk, tk+1 ) and so 176

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::::?-llun(x, t) u(t) llo,n = lluo(X', tk) u(t) llo,n < E This means that llun(x, t)u(t)llo,n = 0 uniformly on I. Hence, using Fubini's theorem, lim llun(X', t) u(t) lli2(flxi) = lim r llun(x, t) u(t) 116 ndt = 0 n--+oo n--+oo } I Clarkson's inequality states that if j, g E 2(0), then Using this, it follows that llum(X', t)un(x, t)116,n 2 (llum(X', t)u(t)116,n + llun(x, t)u(t)116,n) And, since this limit is uniform on I, it follows that lim { llum(X, t) Un(X, t) 116 ndt = lim llum(X, t) Un(X, t) lli2(!1xi) = 0 n,m--+oo } I n,m--+oo So, the sequence { un(x, t)} is a Cauchy sequence in 2(0 xI), and since 2(0 x I) is complete, :J u(x, t) E 2(0 x I) such that lim llun(X', t) u(x, t) lli2(0xi) = 0 n--+oo Hence, for E > 0, llu(x, t) u(t) 11(nxi) llu(x, t) un(x, t) 11(nxi) + llun(x, t) u(t) 11(nxi) for n large enough. This implies then that llu(x, t) u(t) ll(nxi) = 0 Hence, u(x, t) = u(t) a.e. on I. And by changing u(t) to u0(x, t) on a set of measure zero, the correspondence between u(t) and u(x, t) is established. 177

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To show the relationship between u'(t) and gtu(x, t), let cjJ E C0[J, L2(D)] and let (x, t) E C0[D x I] be a representation of cjJ(t). Also. let u E C1[J, L2(D)] and let v(x, t) be a representation ofu'(t) for almost all t E J. Then, integrating by parts, h k u(t)cjJ'(t)dDdt h [fan u(t)cjJ(t)dD-k u'(t)cjJ(t)dn] dt h k u'(t)cjJ(t)dDdt Then, using the representations, it follows that h k u(x, t) :t (x, t)dDdt = h k v(x, t)(x, t)dDdt And, this means that in the weak or distributional sense that u'(t) = v(x, t) = :t u(x, t) In order to show the relationship of the operator A to the boundary value problem, suppose that V, H, and B are Hilbert spaces, that "( is a linear surjection of V onto B with kernel Vo, that Vo = H, that i is a continuous injection of Vo into H, that H is a pivot space, and that H is identified with its dual, H = H', then the following diagram can be established Vo H Zvo t t ZH ., ?-H' This means that the following embeddings exist: I I Voc......tH=H c......tV0 Similarly, suppose that V = H, that i is a continuous injection of V into H, that H is a pivot space, and that H is identified with its dual, H = H', then the following diagram can be established 178

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V H Zv t t ZH ., V' ?-H' This means that the following embeddings exist: VYH=H'YV' Hahn-Banach Theorem: Let X be a normed linear spece, M a linear sub space of X and h a continuous linear functional on M. Then there exists a continuous linear functional f defined on X such that h(u) = f(u) VuE M ll llhll = 11!11 If v E V and v' E V', then v'(v) =< v', v >v D and since V is a linear subspace of H, there is an h' E H' such that h' is an extension of v'. Hence, the duality pairing on V' x V can be identified with the duality pairing on H. By the Riesz Representation Theorem, 3 vh' E H such that Hence, the duality pairing on V' x V can be identified with the inner product on H. The trace operator 1 maps the space V onto the Hilbert space B. For example, we might have Vo = HJ(D); 1 B = H2(8D) 179

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Suppose there is a continuous bilinear form ai : V x V --+ If (VI, v2 ) E V x V and VI is considered fixed and v2 E Vo, then a I (VI, v2 ) is a continuous linear operator on Vo. So, for each vi E V we can write The linear functional Av1 depends linearly and continuously on VI. This de pendence is given formally by Hence, And, the operator A is a continuous linear operator from V to ie, A E [V, Let Note: The reason that VA is required can be seen by the following example: Let V = HI(D), then by the trace theorem the operator 'Yo can be extended by continuity to a mapping of HI(D) onto but it says nothing about the mapping 'YI In fact, 'YI cannot be extended to all of HI(D). A smaller space is required, and VA is that space. Since we can write A I I VIE H = H c......t Yo Now, let v2 E V and define the operator If v2 E Vo, it follows that 180

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and this means that The trace operator 1 is a continuous linear operator that maps the space V onto the boundary value space B. At this point the following theorem can be applied: Theorem: Let X, Y be B-spaces, and letT E .C(X, Y), and T' E .C(Y', X') its transpose, then the following hold: N(T') = R(T)11 R(T) is dense in Y iff T' is 1 -1 lll R(T) is closed in Y iff R(T') is norm-closed m X' IV If R(T) is closed, then R(T') = N(T)-D By letting X= V, Y = B, T = 1, T' = 1', and 1: V onto Bit follows from part (iii) of the Theorem that since B is closed the transpose 1' maps B' onto R(1'). Furthermore, by part (ii) of the Theorem, since R(1) = B, 1' is 1-1. Hence, 1' is an isomorphism of E' onto R( 1'). Since the N ( 1) = Vo and since R( 1) is closed, part (iv) of the theorem gives This means that Define the function Then, 181

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Hence, if GAvl E H y then since Jv1 E B' and 1v2 E B. So, for v1 E VA it follows that from the definition of ( G AVI, v2 ) H that (5.2) This equation has the general form of a Green's formula. Also, by specifying the bilinear form a I (VI, v2 ) the operator A can be derived. For example, given the equation + A(x)u = o where ( ) [ au (x) ai2(X') ] Then ai VI, v2 can be wntten as, w1th A = ( --+) ( --+) a 2I x a22 x ai(vi, v2) = k A(x)V'vi Y'v2dD + k v2b(x)Y'v1dD + k c(x)vi v2dD And, if the first term on the right-hand side is integrated by parts it follows that 182

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By comparing this equation to Equation[ 5.2], and by making the following associations, So, that A=A It is known that if the operator A generates a strongly continuous semigroup, 7t, then the mild solution of the Cauchy problem is given by u(t) = Ttuo +lot 1t-sf(s)ds (5.3) Pazy[74] defines a strong solution as a function u which is differentiable a.e. on [0, T] such that u' E L1(0, T: H), u(O) = u0 and u'(t) = Au(t) + f(t) a.e. on [0, T]. Furthermore, Pazy[74] Corollary 2.10 and Corollary 2.11, since H is reflexive iff is Lipshitz continuous on [0, T], then the Cauchy problem has a unique strong solution given by Equation[ 5.3]. In the above discussion, the operator A does not depend on the time variable t. In our case, the situation is more complicated because the operator A not only depends on t, e.g. is temporally inhomogeneous, but also has a random component w. As far as the temporally inhomogeneous case is concerned, the funda mental solution, as characterized in Tanabe[94], is an evolution operator U(t, s) which has the following properties: (1) U(t, s) is a strongly continuous function, defined on 0 :::; s :::; t :::; T, and is bounded (2) U(t, r)U(r, s) = U(t, s) for 0:::; s:::; r:::; t:::; T 183

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(3) U(s, s) =I for each s E [0, T] (4) s) = A(t)U(t, s) (5) 88U(t, s) = -U(t, s)A(t) And, the solution to the initial value problem can be written as du(t) dt A(t) u(t) + f(t) u(O) uo u(t) = U(t, O)u0 +lot U(t, s)f(s)ds In Curtain and Falb[27], the authors extend this result to evolution equations of the form du + A(t)u(t)dt = dW(t) u(O) = uo where for a separable Hilbert space Hand a Hilbert space K, A(t) is a closed, possibly unbounded linear operator on K, (,) E M 2 (H, K) where M2(H, K) = { S(t, w) : S(t, w) E .C(H, K) and loT IIS(t, < oo wp 1} W(t) is an H-valued Wiener process and u0 is a K-valued random variable. Theorem 3.6 of Curtain and Falb[27] gives the solution as u(t) = U(t, O)u0 +lot U(t, s)(s)dW(s) where U(t, s) is an evolution operator generated by -A(t). Letting (s) =I, the identity operator, u(t) = U(t, O)u0 +lot U(t, s)dW(s) From the definition of the stochastic integral, 184

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Consequently, E[u(t)] = U(t, O)E[u0], because U(t, 0) is deterministic and u0 is a random variable. From Section 4.2.6, Cov[u(t), u(t)] = E[u(t) o u'(t)]-E[u(t)] o E[u'(t)] and, in the weak sense, (5.4) -(h1 E[u(t)] o E[u(t)]h2 ) Expanding the first term in the right hand side of Equation[ 5.4], (h1, U(t, O)E[u0]) (h2, U(t, O)E[u0]) E [(h 1 U(t, O)E[u0]) (U*(t, O)h 2 u0)] -(h1 U(t, O)E[u0]E[u0]u*(t, O)h 2 ) Assuming nonanticipativeness of u0 with W ( s) for s > 0, the second term on the right hand side of Equation[ 5.4] is (h1 E[u(t) o u(t)]h2 ) = E [ ( h 1 U(t, O)u0 +lot U(t, s)dW(s)) x (h2,U(t,O)u0+ lotU(t,s)dW(s))] E [(h1 U(t, O)u0)(h2 U(t, O)u0 ) + ( h1 1t U(t, s)dW(s)) ( h2 lot U(t, s)dW(s)) J -(h1 U(t, O)E[u0u0]U*(t, O)h 2 ) + ( h1 E [lot U(t, s)dW(s) o lot U(t, s)dW(s)] h2 ) But, from Sawaragi[85], Lemma 2.3, 185

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E [lot U(t, s)dW(s) o lot U(t, s)dW(s) J =lot U(t, s)Q(t)U*(t, s)ds where Q(t) is the covariance operator associated with the Wiener process W(t). Hence, in the weak sense of Equation[ 5.4] (h1 Cov[u(t), u(t)]h2 ) = ( h1 [ U(t, O)Cov[u0 u0]U* (t, 0) +lot U(t, s )Q(t)U*(t, s )ds J h2 ) which is in agreement with results from the finite dimensional case, Astrom[8]. It should be noted that in these cases, the operator A(t), although allowed to depend on t, is not allowed to have random components. Hence, this theory would apply to the situation where the boundary value problem is allowed to have a random forcing term. 5.2 Application To Transport And Scale-Up As presented in Section 1.7, the spectral methods used to treat scale up resulted in analytical expressions of a scaled-up dispersivity tensor. And, as commented on in Section 1.8, these scaled-up dispersivity tensors were all subject to the constraint that the medium involved was assumed to be only mildly heterogeneous. Another way of saying this is that the variance of the log-hydraulic conductivities, Y, is subject to the condition < 1. This assumption allowed linearization techniques to be used in the development of the scaled-up dispersivity tensors. The approach that makes the most sense in describing dispersion is the Lagrangian framework used by Dagan[29, 30, 31, 32, 33, 33]. In this approach, transport is developed in terms of indivisible solute particles which are convected by the fluid. The second spatial moment about the centroid of the plume, sij, characterizes how the solute plume is dispersed about the centroid. The actual dispersivity coefficients are defined as half the time rate of change of the second spatial moment about the centroid, D (t) = dSij(t) ZJ 2 dt 186

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The validity of this definition was verified for a special case in Chapter 2. There are two sources of uncertainty in the expression for Sij, the exact position of the centroid of the plume, R(t), and the exact level of solute concentration, C(x, t). Since the concentration has a random component to it, the second spatial moment about the centroid is a stochastic process in time. Because of this, Dagan, Section 2.3.5, defines effective dispersivity coefficients as the expected values of the Dij 's. D ( ) dE[Sij(t)] ZJ t -2 dt Figure 22 conceptualizes this idea of dispersivity. As the plume spreads out, a new average dispersivity tensor applies at each time step. But in the types of models that are considered in this section, Monte Carlo meth ods will be used to include both the dispersivity and the velocity as random components of a random differential operator. .--------t---------.Dt4 Figure 22 Conceptual Dispersivity 187

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D. dE[Sij] ZJ-2 dt In the following sections, the time dependent dispersivity coefficient will be incorporated into the stochastic partial differential equation model. In keeping with the basic notion of dispersion being a stochastic quantity, both the dispersion coefficients and the velocity will be allowed to have random components. The approach used is to treat the stochastic partial differential equation as a stochastic evolution equation. The solution of which requires an iterative process. References for this section are Adams[1], Adomian[3], Butzer and Berens[18], Serrano[88, 89], Tanabe[94], Tang and Pinder[95] and Yosida[99]. 5.3 Stochastic Parameters In Section 4.2.1, the stochastic partial differential equation was in troduced, Equation[ 4.12], page 138. And, in Section 4.2.2, one of the many problems that can be treated using the stochastic partial differential equation was identified as the stochastic operator equation. The operator A is stochastic if one or more of its components is a stochastic process. Consider the equation au at (x, t, w) + A(x, t, w)u = g(x, t, w) u lac= 0 U (X, 0) = Uo (X) Let A(x, t,w)u (E[V] + V'(t, w))\7u-\7 ((E[D(t)] + D'(t, w))\7u) = E[V]\7u-\7 (E[D(t)]\7u) + V'(t,w)\7u-\7 (D'(t,w)\7u) So, we can write au --+ at + E[V]\7u-\7 (E[D(t)]\7u) = g-Ru u lac= 0 u(x, 0) = uo 188

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where R V'(t, w)\7()-\7 (D'(t, w)\7()) Notice that in order to simplify matters, the velocity is assumed to be strictly homogeneous (strictly stationary). Since the solution u appears on both sides of this equation, it cannot be represented explicitly. The solution can, however, be formally represented using an iterative process. The next section applies this series approach to represent the solution. 5.4 Formal Solution In this section we formally develop a series representation for the inverse of the partial differential operator. Consider the equation where Lt X = Lt X + Rt X and Lt,x is a deterministic partial differential operator and Rt,x is a zero mean stochastic partial differential operator. If and exist, then Lt,xU g-Rt,xU u L-l L-lR t,x9t,x t,xU Hence, r-l L-l L-lR r-l U = J..-t X g = t X g -t X t X J..-t X g ' So, the operator equation is r-l L-l L-lR r-l J..-t X = t X -t X t xJ..-t X ' Parametrizing with A we get Substituting r-l L-1 'L-lR r-l J..-t X = t X /\ t X t xJ..-t X ' 189 (5.5)

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00 = L AiHi i=O we have that 00 00 LAiHi = LAiHi i=O i=O Equating powers of A, it follows that Ho So, Hi can be expressed in terms of Hi_1 and since 00 = L AiHi i=O it follows with A = 1 that 00 .c-1 = "(-1 )i(L -1 R )i L -1 t,x L.....J t,x t,x t,x i=O Hence, 00 u = "'( i=O Since Lt,x is the deterministic part of the equation, we can write its inverse in terms of the evolution operator as Lt]; = U(t, O)u0 + {t U(t, s)()ds lo then from Equation[ 5.5], the solution can be written as 190

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u = U(t, O)u0 + {t U(t, s)g(s)ds-{t U(t, s)Rs xu(s)ds lo lo and parametrizing with A as u = U(t, O)uo +lot U(t, s)g(s)dsA lot U(t, s)Rs,xu(s)ds The two integrals in this expression are stochastic since the integrands are stochastic processes. By selecting sample paths of the random components, the integrals become ordinary integrals. Letting 00 u = L AiHig = i=O then, in operator notation it follows that 00 L AiHi() i=O U(t, O)uo +lot U(t, s)()ds-A lot U(t, s)Rs,x AiHi()ds = U(t, O)uo +lot U(t, s)()ds-lot U(t, s)Rs,x Ai+1 Hi()ds Equating powers of A, i=O ::::} Ho() = U(t, O)u0 +lot U(t, s)()ds i=1 ::::} H1(-) =-lot U(t, s)Rs xHo()ds 0 i=2 ::::} H2(-) =-lot U(t, s)Rs xHl()ds 0 = n ::::} Hn() =-lot U(t, s)Rs xHn-l()ds 0 Expanding H 1 ( ), 191

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H1() -lot U(t, s)Rs,x [u(s, O)uo +los U(s, ds -{t U(t, s)Rs xU(s, O)u0ds{t t U(t, s)Rs xU(s, lo lo lo Expanding H2(), -lot U(t, s)Rs,x [-los U(s, los ds ft t U(t, s)Rs xU(s, lo lo Hence, in this way all of the terms of the series can be expanded. The procedure is demonstrated in the following example: Example Transport Equation Consider the 1-D problem in which the parameters E[D] and E[V] are con stants. OU o2u OU ot -(E[D] + D'(t, w)) ox2 + (E[V] + V'(t, w)) ox = 0 -oo :::; x :::; oo t > 0 u(x, 0) = uo, u( -oo, t) = 0, u(oo, t) = 0 This equation can be rewritten as 192

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Before proceeding, the next two definitions are required. Definition: Let X be a B-space. If 7t is an operator such that 7t : R+ ---+ B(X) and satisfies (1) 7i+s =TiTs t 0, S 0 (2) To= I (3) limt--to 117tx-xllx = 0 Vx EX then 7t is called a strongly continuous semigroup. Definition: The infinitesimal generator of a semigroup 7t is defined by Ax= lim !(Ttxx) t--+0+ t 'D(A) is the set x E X for which the limit exists. In order to illustrate the meaning of this definition, the following Theorem and Proposition from Rudin[83] regarding bounded or unbounded self-adjoint operators are helpful: Theorem: To every self-adjoint operator A in H there corresponds a unique resolution E of the identity, on the Borel sets of the real line, such that (Ax, y) = /_: >.dEx,y(>.) = /_: >.d(E;..x, y) (x E 'D(A), y E H) Also, E is concentrated on a(A) C ( -oo, oo) in the sense that E(a(A)) =I D Proposition: Let A be self-adjoint. (Ax, x) :::; 0 if and only if a(A) C ( -oo, 0]. From the Theorem, it is clear that A = >.dE;.., and from the symbolic calculus for operators, Rudin[83], Friedman[39], if E is a spectral decomposition of the operator A and the spectrum of A is such that 193

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O"(A) c ( -oo, 0] then etA can be represented as 7t =etA= 1 et>.dE;.. u(A) Then, from the definition of the integral it follows that if t = 0, then e0 = I and fu(A) et>.dE;..x -X hm------'-----'---------t--+0+ t 1 (et>.1) lim dE;..x t--+0+ u(A) t -et>. dE;..x 1 d I u(A) dt t=O 1 )..dE;..x u(A) = Ax Therefore, (7t)'lt=O = A, and in this sense, the operator A is the infinitesimal generator of the semigroup Tt. Since E[D] is a constant, the solution can be expressed in terms of a semigroup as So, if we let '( ) fP '( ) a Rs,x = D s,w OX2 -V t,w OX the solution can be written as t ( a) u(x, t) = Ttu0 + fo Tt-s D'(s, w) ox2 V'(t, w) ox u(x, s)ds 194

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The semigroup 7t is the one generated by the operator a a2 A= -E[V]+ E[D]-ax ax2 To find the semigroup, start from the problem au+ E[V]auE[D]a2u = 0 at ax ax2 (5.6) And, u( -oo, t) = 0 u(oo, t) = 0 u(x, 0) = u0 In Section 2.3, it was shown that the solution to this problem was given, with the change of variables X= xE[V]t and T = t, by v(X, T) = j_: K(Xwhere 1 K(XE[D]T) = e-4E[DJT Substituting for X and T yields 100 e-4E[D]t v(X, T) = V -oo 2 1rE[D]t And, the candidate for the semigroup operator becomes (x-E[V]t-02 100 e 4E[D]t Tt() -oo 2J1rE[D]t (5.7) The following verify the semigroup properties: Let u0(x) E LP( -oo, oo) Property 1: To = I 195

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(x-E[V]t-02 Joo e-4E[D]t Ttuo(x) = V -oo 2 1rE[D]t Let E[V]t + 2JE[D]t s Then, 1 !00 2 Ttuo(x) = y7F -oo e-s u0(xE[V]t + 2y E[D]t s)ds And, it follows that as t ----+ 0, 1 !00 2 Ttuo(x) ----+ y7F -oo e-s uo(x)ds = u0(x) Hence, To= I Property 2: TiTs = h+s t 0, s 0 Tt'Tsuo(x) !00 e-(x-E[V]t-p)2 /4E[D]t [!00 /4E[D]s l ---------r==--Uo ( dp -oo 2y7FjE[D]t -oo 2y7FjE[D]s Joo 1 Joo [-(x-E[V]t-p)2 ] -oo 47rE[D]y'tS -oo exp 4E[D]t [-(p-E[V]sx exp 4 E[D]s dp Let g = p-E[V]sthen the inner integral becomes 1 joo [-(x-E[V](t + s)g)2 ] [ -g2 l exp exp dp 47rE[D]y'tS -oo 4E[D]t 4E[D]s Then using the following property of the Gaussian distribution 196

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it follows that 1 Joo [-(xE[V](t + s)g)2 ] [ -g2 l -oo exp 4E[D]t exp 4E[D]s dp 47rE[D]vtS 1 (-(xE[V](t + s)-r=====exp 2yf1rE[D](t + s) 4E[D](t + s) Hence, -(x-E[V](t+s)-02 Joo e 4E[D](t+s) TtTsuo(x) = J = Ti+suo(x) -oo 2 1rE[D](t + s) Property 3: limt--to 117tx-xllx = 0 Vx EX Starting from the definition (x-E[V]t-02 !00 e-4E[D]t Ttuo(x) = J -oo 2 1rE[D]t If we let s = x E[V]t and Then it follows that Ttuo(x) = j_: G(t, sFrom Adams[1] we have the following Theorem 4.30 due to Young: Theorem: Let 1 ::; p < oo and let u E L1(Rn) and v E LP(Rn). Then the convolution products u v(x) = r u(x-y)v(y)dy, }Rn v u(x) = r v(x-y)u(y)dy }Rn are well defined and equal for almost all x E Rn. Moreover, u v E V (Rn) and 197

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Applying this to our case, it follows that II'Jtuo(x) IILP(-oo,oo) ::S: IIG(t, S-11(-oo,oo) IILP(-oo,oo) But, (x-E[V]t-02 Joo e-4E[D]t IIG(t, S-11(-oo,oo) = J = 1 -oo 2 1rE[D]t Hence, II'Jtuo(x) IILP(-oo,oo) ::S: lluoiiLP(-oo,oo) ::::} II Ttl I ::S: 1, so that each 7t is continuous. (x-E[V]t-02 ( ) oo -4E[D]t ( ) Clearly, Uo X = Loo e Uo X so that 2 1rE[D]t Letting y = = then 2JE[Dlt 2JE[Dlt E[V]t2JE[D]ty And, l1tuo(x)uo(x)l = Jn li: e-Y2(uo(x-E[V]t-2JE[D]ty)uo(x))dyl Letting p' be the conjugate exponent of p, ie, l + -lr = 1, then p p l1tuo(x)uo(x)l ::S: Jn j_: e=fe -lluo(xE[V]t-2JE[D]ty)uo(x)ldy Applying Holder's inequality to the right hand side integral 1 ITtuo(x)uo(x)l < Jn (/_: e -;2luo(xE[V]t-2JE[D]ty)uo(x)IPdy) 1> (!00 2 ) -1,x -oo e-Y dy p 1 K (/_: e=flu0(xE[V]t-2JE[D]ty)u0(x)IPdy) 1> 198

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So that, !()() =.r_ l1tuo(x)-uo(x)IP:::; KP -oo e p luo(xE[V]t-2yE[D]ty)uo(x)IPdy Integrating with respect to x and using Fubini's theorem, 117tuo(x) -uo(x) lliP(-oo,oo) :S !()() !()() =.r_ KP -oo -oo e P luo(xE[V]t-2yE[D]ty)uo(x)IPdydx !()() =.r_ !00 KP -oo e P -oo lu0(xE[V]t-2yE[D]ty)u0(x)IPdxdy Then, using Fatou's lemma, lim sup 117tuo(x) -uo(x) lliP(-oo oo) :S t.j,.O j_: j_: lu0(xE[V]t2VE[D]ty)-u0(x)IPdx) dy But, limsupj00 lu0(xE[V]t2VE[D]ty)-uo(x)IPdx = 0 t.j,.O -oo since this is true for continuous functions with compact support and the inte grand can be arbitrarily closely approximated by such functions. So, lim 117tuo(x)-uo(x)lliP(-oo oo) = 0 t.j,.O Hence, 7t is a strongly continuous semigroup. The above argument is based on similar arguments given in Tanabe[94] and Yosida[99]. This argument can be extended to lRn. Next, a uniqueness argument can be used to show that 7t is a semigroup generated by the operator A. To do this we need the following Theorem: Let 7t be a strongly continuous semigroup on a B-space X with infinitesimal generator A. If x0 E 'D(A), then (1) Ttxo E 'D(A) Vt 0 (2) -it(Ttxo) = A1txo = TtAxo t > 0 D 199

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If Yt is a strongly continuous semigroup generated by A, then by the theorem d dt YtXo = A9txo then by forming the product Ys-t 1tx0 and differentiating Since Yt is the semigroup generated by A, the theorem gives that AQ s-t Ttxo = Q s-tA Ttxo so that Hence, Ys-t 1tx0 is constant with respect tot. Letting t = 0 and s = tit follows that Therefore, for all s > 0 and x0 E 'D(A) the following holds: Since 'D(A) = X, if x E X and { C 'D(A) such that Xn ----+ x, then ----+ 0 as Xn ----+ x since Ys, Ts E B(X). This means that YsX = Tsx, Vx E X so that the semigroups are equal. Hence A is the infinitesimal generator of 7t. 200

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Using the expression for the solution where for g = 0, llo(O) 7tuo ll1 (0) ht ?t-sRTsuods ll2(0) = ht hs ?t-sRTs-rRT,.uodrds And, the solution is given by t ( a2 a) oo u(x, t) = 7tuo + { ?t-s D'(s, w) a 2 -V'(t, w)-a L lli(O)ds Jo X X i=O with -(x-E[V]t-02 Joo e 4E[D]t 7t() -oo 2yi1rE[D]t And, for the infinite interval, the solution to the problem [ 5.6] page 194, is given by Joo e 4E[D]t 7tuo(x) = V -oo 2 1rE[D]t Also, if this operator is restricted to the functions u(x) for which u( -x) = -u(x) for x > 0 then by a change of variables and rearranging terms, it follows that (x-E[V](t-s)-02 Joo _e_-r==4=E=[D=](t=-=s) =U ( -oo 2yi1rE[D](ts) [ (x-E[V](t-s)-02 (x-E[V](t-s)+02 ] 100 e 4E[D](t-s) -e 4E[D](t-s) o 2yi1rE[D](ts) 201

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So, on this restricted domain, the operator [ (x-E[V](t-s)-02 (x-E[V](t-s)+e)2 ] r::o e 4E[D](t-s) e 4E[D](t-s) U(t, s) Tt-s() = Jo J o 2 1rE[D](ts) is our original semigroup. Restricting the domain should not effect us, since we are only interested in the non-negative x axis. The solution for a semi-infinite interval is then given by Guenther and Lee[51] as 100 e 4E[D]t e 4E[D]t [ (x-E[V]t+02 ] YtUo = J o 2 1rE[D]t which is the solution for the problem v(x, 0) v(O, t) uo 0 where x = x = E[V]t, i = t and v(x, t) = u(x, t). Li[62] uses a Laplace transform to show that the solution is also given by a(x, t) { erfc ( + exp erfc ( {5.8) One numerical routine that makes this formulation of the solution easy to work with is a routine for calculating the error function erf(x), which is computed by an algorithm due to Hastings[52] which is given by where 202

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1 t = 1 + 0.3275911x, a1 = 0.254829592, a 2 = -0.284496736 a3 = 1.421413741, a4 = -1.453152027, a5 = 1.061405429 According to Greenberg[50] it is supposed to be uniformly accurate over 0 :::; x < oo to within .5 x 10-7 The function u(x, t), which is given by rt ( rP a) u(x, t) = YtUo + lo Yt-s D'(s, w) ox2 -V'(t, w) ox u(x, s)ds is then the solution of the problem of a long uniform channel which is initially uncontaminated, and at t = 0, a contaminant is introduced whose concentra tion at x = 0 is maintained at u0 Taking this equation and parameterizing it with A, u(x, t) = YtUo +A fat Yt-s ( D'(s, w) ::2 -V'(t, w) :x) u(x, s)ds (5.9) Clearly, as long as YtUo #-0, the operator t ( 82 a) YtUo +A Ia Yt-s D'(s,w) ox2V'(t,w) ox ()ds is not linear. So. by writing 00 u (X' t) = L A i Hi (g (X' t)) i=O where g(x, t) is the forcing term, on substituting, it follows that Equating powers of A, 203

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i=O ::::} Ho(g) i=1 ::::} lot ( a2 a) HI(g) = 0 Yt-s D'(s,w) ox2V'(t,w) ox Ho(g)ds i=2 ::::} t ( a2 a) H2(g) =fa Yt-s D'(s,w) ox2V'(t,w) ox HI(g)ds (5.10) = n ::::} lot ( a2 a) Hn(g) = 0 Yt-s D'(s,w) ox2 V'(t,w) ox Hn-I(g)ds H0(g) has already been determined. Since it is assumed that the solute injec tion at the left hand boundary is held constant at u0 the formula Equation[ 5.8] will be used as a representation of H0(g). In order to have a specific realization for Hi(g), it is necessary to fix w. This will make V'(t, w) and D'(s, w) sample paths. Additional questions that have to be answered before the feasibility of this type of approach can be addressed clearly involve the convergence of the series. Some type of convergence analysis has to be made to determine the number of terms that are needed to be retained for a desired accuracy. Also, the additional complexity of extending this approach to 2 and 3 dimensions has to be considered. And, numerical implementations need to be worked out. 5.5 Convergence This section will consider convergence questions surrounding the in finite series solutions developed above. As discussed earlier, recall that the solution to the problem is given as an infinite series in the form 00 u(x, t) = L Hi(g) i=O where the Hi(g) terms are given by Equations[ 5.10]. In particular, H0(g) is g1ven as H0(g) = a(x, t) First off, define the operators 204

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lot looo e 4E[D](t-s) e 4E[D](t-s) [ (x-E[V](t-s)H)2 ] o o 2yl1rE[D](ts) And, '( ) a2 '( ) a = D s, w ae V t, w So that rt r::o e 4E[D](t-s) e 4E[D](t-s) [ (x-E[V](t-s)-02 (x-E[V](t-s)+e)2 ] lo lo 2yl1rE[D](ts) x (D'(s,w) :;2 V'(t,w) Fix X and T so that (x, t) E [0, X] x [0, T] Then, using Equations[ 5.10] as a reference, the Hi(g)'s can be written in the following way: H0(g(x, t)) a(x, t) H1 (g(x, t)) Hence, u1(x,t) H0(g(x, t)) + H1(g(x, t)) 205

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And, continuing in this way, we can write So that, H0(g(x, t)) + H1 (g(x, t)) + H2(g(x, t)) In general, the nth approximation can be written as n n Un = L Hi(g(x, t)) = a(x, t) + s) i=O i=l The solution is given by the limit n u(x, t) = lim un(x, t) = a(x, t) + lim "[ci;Rs s) (5.11) n-+oo n-+oo L.....J i=l Consider the difference for n > m ( n+l 1 n+l ) II a(x, t) + s) -( a(x, t) + s)) II [ 1 J n+l II s)ll < s)ll 206

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So, if we let a= then More generally, But, So, it follows that ll(unUn-1) + (un-1-Un-2) + + (um+lUm)ll < llun-Un-111 + llun-1-Un-211 + + llum+l-Umll oo 1 1 o;m o;m =-. 1-o; 1-o; 1-o; Now, if a< 1, ie is a contraction, then given E > 0, 3 NE suxh that if n > m > NE, then and, 207

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The sequence { un} is a Cauchy sequence and by the completeness of the Hilbert space converges to a unique limit, namely, u(x, t). Referring to Equation[ 5.9] on page 202, and as shown on that page, Equa tion[ 5.9] is parametrized with A to obtain t ( a2 a) u(x, t) = (;ltuo +A h Yt-s D'(s, w) ax2 V'(t, w) ax u(x, s)ds(5.12) This equation has the form of a Volterra integral equation of the second kind. Recalling that rt r::o e 4E[D](t-s) e 4E[D](t-s) [ (x-E[V](t-s)-02 (x-E[V](t-s)+e)2 ] lo lo 2yf1rE[D](ts) X (D'(s,w) :;2 -V'(t,w) Equation[ 5.12] can be put in the form Therefore, the equation to solve is given by Tu=u with If fo is some initial estimate of the solution, then 208

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In general, + ... + An[_c-1 R c]n t,x s,., J I and if is a contraction then the solution is given by u lim rn fo n--too 00 a+ L i=l with A= 1 then u becomes m u(x, t) = a(x, t) + lim "'[.Ct;Rs s) i=l which is the same as Equation[ 5.11], page 205. So, the convergence theory is the same as that for the Volterra integral equation of the second kind. Following methods similar to the one illustrated in Figure 12, Section 3.2.10, Monte Carlo methods can be used to construct concentration profiles from which concentration means and variances can be constructed. As a test problem, consider the 1-D example of a long channel of uniform sand which has an established flow through it. The channel is initially uncontaminated and at t = 0 a contaminant or tracer is introduced whose concentration at x = 0 is maintained at u0 The physical parameters used in the model are as follows: 209

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Distance From Origin 1, 2, 5, 10 Meters Time 10 Days Velocity 0.3 m/day Dispersion 0.1 m2/Day Concentration At x = 0 1.0 mgrm/liter 11t 0.05 !1x 0.05 The following tables give some results of applying this iterative method to this problem. The numbers in the columns labeled Concentration and In crement are interpreted as follows: In row i, add the number in the Increment column to the number in the Concentration column. This will produce the number in the Concentration column in row i + 1. 210

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Example 1: Time = 10.0 Days Distance From Origin Meters Iteration Concentration Increment 1.0 1 0.968878174061 0.000018964773 2 0.968897138835 0.00000084 7 497 3 0 0 968897986332 0 0 000000338 709 4 0 0 968898325040 0 0 000000002449 5 0.968898327489 -0 0 000000004363 6 0.968898323127 2.0 1 0.843295693603 -0.000014472127 2 0.843281221477 -0 0 000002613105 3 0.843278608371 0 0 000000035 7 4 7 4 0.843278644118 0 0 000000044050 5 0.843278688168 0 0 000000002064 6 0.843278690232 5.0 1 0.101972946853 -0.000002073268 2 0.101970873586 0.000000148862 3 0.101971022447 -0.000000034350 4 0.101970988097 0 0 000000005056 5 0.101970993154 0.000000000575 6 0.101970993728 10.0 1 0.000000211885 -0.000000141955 2 0.000000197690 0 0 000000000034 3 0.000000197724 0 0 000000000000 4 0.000000197724 0 0 000000000000 5 0.000000197724 0 0 000000000000 6 0.000000197724 In this example, the dispersiOn IS allowed to have a random component and the velocity is not. The sample path of the stochastic process D' ( t, w) was generated from a Gaussian distribution with a mean of zero and a standard deviation of 0.03. 211

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Example 2: Time = 10.0 Days Distance From Origin Meters Iteration Concentration Increment 1.0 1 0.971451709277 0.000027765168 2 0.971479474446 -0 0 000000288321 3 0.971479186125 0 0 000000008 773 4 0.971479194898 -0 0 000000000252 5 0.971479194646 0.000000000007 6 0.971479194653 2.0 1 0.835760450146 -0.000031500611 2 0.835728949535 0.000001117351 3 0.835730066886 -0.000000023517 4 0.835730043368 0.000000000398 5 0.835730043767 -0 0 000000000004 6 0.835730043762 5.0 1 0.101035253556 0.000023792755 2 0.101059046311 0.000000049911 3 0.101059096222 -0.000000007834 4 0.101059088388 0 0 000000000358 5 0.101059088746 -0 0 000000000013 6 0.101059088733 10.0 1 0.000000249522 -0.000000007750 2 0.000000241772 -0 0 000000003842 3 0.000000237930 -0 0 000000000018 4 0.000000237912 0 0 000000000000 5 0.000000237912 0 0 000000000000 6 0.000000237912 0 0 000000000000 In this example, the velocity IS allowed to have a random component and the dispersion is not. The sample path of the stochastic process V' (t, w) was generated from a Gaussian distribution with a mean of zero and a standard deviation of 0.03. 212

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Example 3: Time = 10.0 Days Distance From Origin Meters Iteration Concentration Increment 1.0 1 0.972000869383 0.000001256451 2 0.972002125834 -0 0 000002943310 3 0.971999182524 -0 0 000000029912 4 0.971999152612 0 0 000000008036 5 0 0 971999160648 0 0 000000000603 6 0.971999161252 2.0 1 0.836701222644 -0 0 000060724077 2 0.836640498567 0.000001026103 3 0.836641524670 -0.000000493583 4 0 0 836641031086 -0.000000062218 5 0 0 836640968868 0 0 000000005893 6 0.836640974761 5.0 1 0.099088557 4 72 0.00002544 7894 2 0.099114005366 0.000001381694 3 0 0 09911538 7060 -0.000000291936 4 0.099115095124 0.000000027379 5 0.099115122503 0 0 000000005205 6 0.099115127708 10.0 1 0.000000089299 -0.000000236624 2 -0.000000147325 0 0 000000001784 3 -0.000000145541 -0 0 000000000044 4 -0.000000145585 -0 0 000000000001 5 -0.000000145585 0 0 000000000000 6 -0.000000145585 In th1s example, both the veloCity and the d1spers10n are allowed to have ran dom components. The sample paths of the stochastic processes V' ( t, w) and D' ( t, w) were generated from a Gaussian distribution with a mean of zero and a standard deviation of 0.03. At the 10.0 meter mark, the process has the pathological behavior of converging to a negative concentration. 213

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In Section 2.3.3, the transport equation was solved using the Fourier transform. As part of that process, a first order equation, Equation[ 2.21], page 66, was obtained. Since the dispersion tensor contained constant coefficients, the autonomous first order equation had an easy solution in terms of the Fourier transform, However, the situation changes considerably if the dispersion tensor is allowed to depend on time. The first order equation, 2.21] is now a nonau tonomous equation because the dispersion term D depends on ti:ne. In order to solve this equation, we need to know the analytical form of D. Referring to Equation[ 1.13], page 37, it can be seen that in the case of steady state or ergodic flow, the dispersion is esentially constant and so can be handled by the semigroup approach presented earlier. In the case of non-ergodic flow, the problem becomes much more difficult since the dispersion tensor is now time dependent. The equation that needs to be solved for the fundamental solution is now a nonautonomous equation and has the form du(t) = A(t) u(t) dt where A(t) -E[V] gt + E[D(t)] ::2. This case can be reduced to the autonomous case by creating the system { ::\') l A(T) u(t) dt u(O) = uo T(O) = 0 Solving this system yields the solution !00 e 4E[D(t)]t 'Ttu0(x) = V -oo 2 1rE[D(t)]t If the operator U(t, 0) is defined as -(x-E[V]t-02 !00 e 4E[D(t)]t U(t, 0)() V -oo 2 1rE[D(t)]t Then this integral operator can be used to study the case of the expected dispersion being time dependent. 214

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5.6 Summary Chapter 5 investigated the application of the theory of stochastic evo lution equations to the problems of flow and transport. Section 5.1 reviewed the connections between the boundary value problem and the abstract evo lution equation for both the autonomous and nonautonomous cases. Curtain and Falb[27] extend these results to the case where the forcing term of the abstract evolution equation contains an H-valued Wiener process. From the Curtain and Falb result, weak forms of the mean and covariance equations are found. Section 5.2 considered the time dependent forms of the dispersion tensor and Section 5.3 gave the form of the stochastic PDE in terms of a sum of deterministic and stochastic operators. Section 5.4 represented the solution of the stochastic PDE as an in tegral equation involving an evolution operator that derived from treating the deterministic part of the stochastic PDE as an abstract differential equation. An iterative approach to solving this integral equation was presented and the convergence property of the series solution was given in Section 5.5. A test problem was also solved in order to investigate the speed of convergence of the series. 215

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6. Future Research 6.1 General The Neumann expansion procedure outlined in Sections 5.4 and 5.5 can be extended using the Karhunen-Loeve expansion for a stochastic pro cess and the Galerkin method, thus turning it into a stochastic finite element method, Ghanem and Spanos[46]. However, the method continues to be subject to the convergence criterion In order to avoid this restriction, a method using the Homogeneous Chaoses of Wiener is outlined next. The Karhunen-Loeve expansion is a Fourier-type expansion of the form 00 w) = k=l The { ,\k} is a sequence of constants, the { k ( x)} is a sequence of orthonormal deterministic functions and the { ( w)} is a sequence of random variables given by The details of the existence of the constants { Ak} and the orthonormal se quence { k(x)} is covered in Loeve[63]. Basically, the covariance function of the process w) can be written as 00 c(x1, x2) = L -\kk(xl)k(x2) k=O And, since the k's form an orthonormal sequence, it follows that so that the constants { -\k} and the functions { k ( x)} are solutions of this Fredholm type two integral equation. And, the random variables are determined by these solutions. 216

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In Section 5.4, the formal solution of the equation was developed by first splitting the operator .Ct,x into a deterministic part and a zero mean stochastic part, i.e., For example, the stochastic transport equation can be written as + :x [E[V(x)]u]-D(t) + w)) = 0 or, + E[V(x)] -D(t) + w)) = E[V(x)]u Since txE[V(x)] can be computed and u(t, x, w) can be estimated from the previous time step, t flt, by letting a f(t, x, w) = ox E[V(x)]u(t, x, w) the following holds approximately, ou + E[V(x)] ou D(t) o2u + o(uV' (x, w)) f ot ox ox2 ox where f = txE[V(x)]u(t-flt, x, w). In this example, it is assumed that the dispersion tensor, D, has been dealt with as explained in Section 2.4. This leaves as the only random coefficient the random velocity term, V' ( x, w) which does not depend on t. By the KarhunenLoeve expansion, v' (X' w) can be expanded as m v' (x, w) k=l where the sum has been truncated after m terms, the are random variables and { k(x)} are deterministic functions of x. Then, it follows that m Lt,xU + f k=l 217

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or, m m Lt,xU + J= J k=I k=I where a a a2 -() + E[V(x)]-()-D(t)-() at ax ax2 R() () 6.2 Homogeneous Chaos Clearly, the Karhunen-Loeve expansion requires a knowledge of the covariance function. This is can be done for the random coefficients in the operator equation, but not for the solution process since its covariance function is not known. What is needed is a way of representing the solution process that does not require knowledge of its covariance function. In order to circumvent this problem, the Homogeneous Chaoses first introduced by Wiener in 1938 can be used. First, the following results are provided for convenience. Definition: Let D be a subset of lRn. Let X denote the complete inner product space of functions defined on D. A function of two variables XI and x2 in d, K(xi, x2 ) is called a Reproducing Kernel Function for the space X if for each fixed x2 ED, K(xi, x2 ) considered as a function of xi is in X. for each function f(xi) E X and every point x2 E D, the reproducing property where (, )x is the inner product on X and x2 is held constant. The following results due to Aronszajn provide existence and uniqueness: Theorem: A necessary and sufficient condition that X have a reproducing kernel function is that for each fixed x2 E D, the linear functional is bounded IL(f) I ::; cllfll V f EX 218

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Theorem: If X possesses a reproducing kernel, it is unique. Let w), x E D be a real Gaussian process defined on a probability space (D, B, P). Define the O"-algebra Bf. to be the completion with respect to the measure P of O" { ( x) : x E D} where O"{ : x E D} denotes the smallest O"-algebra with respect to which the are measurable. Assume that = o and Then, H(C) is the Hilbert space determined by the kernel C(x1 x2). H(C) is called the Reproducing Kernel Hilbert Space (RKHS) of C(x1 x2 ) or of the process For each xED, C(, x2 ) E H(C) and Vf E H(C) The following notation for tensor products is required for what follows. Let Hi, i = 1, p be Hilbert spaces, then the tensor product Hilbert space is given by H1 Hp or the shorthand Let be a complete orthonormal sequence in H(C). Also, let be the closed linear subspace of L2(D, Bf., P) spanned by all finite, real linear combinations n i=l then it can be shown, Kallianpur[56], that there is an isometric isomorphism between L1 ( and H (C). Clearly, ei1 eip as i1 ip range independently from 1 to oo form a complete orthonormal system for Q9P H (C). And, if h1 hp E Q9P H, then the tensor 219

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where 1r = (1r1 1rp) is a permutation of the integers (1, p), is a com pletely symmetric tensor. The Hilbert space a (&;;PH) is then taken to be the closed linear sub space of &;;PH generated by elements of the form n L cka ( h;) k=l Let H = H (C) be the RKHS of the Gaussian process ( x, w), x E D. Further more, let { be a sequence of random variables over (0, P) such that is the element in 1 ( which corresponds to ei in H (C) by the isometry above. Then, from Kallianpur[56, Lemma 6.5.1], a complete orthonormal system de noted { exists for the space a (&;;PH (C)) where AI, Ar are distinct integers in the sequence ii, ip with Ai occurring ni times and I ni = p. Next, define the following linear subspaces of 2(0, P). First, define 1 P to be the space of all polynomials in { ( w)} of degree not exceeding p. Then let ? P to be the set of all polynomials in 1 P orthogonal to 1 p-l, sometimes written? P = 1 P 81 p-I Finally, let ?P be the space spanned by? p The subspace ?P of 2(0, P) is called the pth Homogeneous Chaos. The set ? P is called the Polynomial Chaos of Order p. It can then be shown, Kallianpur[56, Lemma 6.6.1], that the following representation of elements of ? P holds: Lemma: A random variable 1 belongs to ? P (p > 1) iff it is of the form for some choice of distinct integers AI,, Ar The summation is over mi 0 and AI,, Ar are fixed, hn(x) is the nth normalized Hermite polynomial and the coefficients satisfy r am1,,mr = o if L mi =1 p i=I 220

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The desired homogeneous chaos decomposition of the L2-space of a Gaussian process is given in the following theorem from Kallianpur[56, Theorem 6.6.1]: Theorem: L2(0, P) = L EB?p For any u E L2(0, P), the expansion U=L L L (6.1) n1 ++np=P .>.1 <<>-r holds where Fp E a(PH(C)). Polynomial chaoses of any order p consist of all orthogonal polyno mials of order p involving any combination of the random variables { ( w)} Ghanem and Spanos[47, 46] rewrite the previous expression for u(w) E L2(0, P) as 00 u(w) aio?o + L il=l 00 il + L L 00 il i2 + L L L ai1i2i3? (w), + il=l i2=1 i3=1 Each polynomial chaos is a function of the countably infinite set { ( w)} and is therefore an infinite dimensional polynomial. For practical purposes, this must be reduced to a finite dimensional subspace. The n-dimensional polynomial chaos of order p is the subset of the polynomial chaos of order p which involves only n of the uncorrelated random variables ( w). The convergence properties will then depend on n and the choice of the subset { 6.; }i=I In Ghanem and Spanos[47, 46] the polynomial chaoses for orders 0 to 3 are found to be 221

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?a 1 where 6ij is the Kronecker delta. 6.3 Stochastic Finite Elements Starting from the equation m Lt,xu + j k=l with boundary conditions of the form x, w) = 0 x E an (6.2) assume that for a fixed t, u(t, x, w) is a second order random variable, hence its Karhunen-Loeve expansion can be expressed as where l u(t, x, w) = L ej(t)xj(t, w)bj(t, x) j=l Xj(t,w) = /, u(t,x,w)bj(t,x)dx ej t D (6.3) Since at time t the covariance function is not known, Equation[ 6.3] is of little use in this form. However, using the polynomial chaoses, the random variable x( t, w) can be expressed as 00 o + L il=l 222

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where .. ip (t) are deterministic constants and ? is the pth order polynomial chaos. Truncating after the pth polynomial gives p Xj(t,w) = (6.4) i=O where and Substituting Equation[ 6.4] into Equation[ 6.3] and letting Cj ( t, x) = ej ( t)bj ( t, x) yields where u(t,x,w) p L di(t, }] i=O l di(t,x) = 2:xP)(t)cj(t,x) j=l Then, substituting Equation[ 6.5] into Equation[ 6.2] gives Next, expanding di(t, x) in the space C2 as n di(t, x) L dji(t)gj(x) j=l and substituting it into Equation[ 6.6] results in 223 (6.5) (6.7)

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P n P m n L L }]Lt,xdji(t)gj(x)+ L }] L dji(t)Rgj(X)(fJk(x) j i=O j=l i=O k=l j=l The terms can be rearranged to form the following Multiplying both sides by g1(x) and integrating over D gives the system of equations where l = 1, ,nand Jn gj(x)gz(x)dx !, [E[V(x)] ogj(x) D(t) 02gj(x)l gz(x)dx D OX OX2 Jn R(gi(x))gz(x)(!Jk(x)dx fz = Jn Jgz(x)dx In this system of equations, i spans the number of polynomial chaoses and j spans the number of basis functions from C2 In order to find the vector of entries {dji(t)} at timet, the polynomial chaoses have to be replaced with numbers. This is done by multiplying by \]! m [ { taking expected values and using the orthonormal relationships and the computable quantities This yields an nP x nP system. Once this system is solved for the vector {dji(t)}, the di(t,x) coefficients can be calculated from Equation[ 6.7], and u(t, x, w) can then be subsequently represented by Equation[ 6.5]. 224

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6.4 The Covariance Function The success of the procedure outlined above depends in large part in describing the covariance function and obtaining its eigenvalues and eigenfunc tions. The problems of flow and transport as presented in this thesis assume a knowledge of the porous medium that is very difficult to obtain in actual prac tice. For that reason, special assumptions regarding the stochastic properties of the hydraulic conductivities are often made. In particular, Equation[ 1.21], page 43, relates the spectrum of the velocity to the spectrum of the log hydraulic conductivity. By taking Fourier transforms, a relationship between the covariance of the velocity and the covariance of the log-hydraulic conduc tivity can then be established. The problem is then in choosing a spectrum that can be representative of the log-hydraulic conductivity. In order to illustrate the procedure, one choice that has been used is the Whittle spectrum for two-dimensional spatial processes. This spectrum has the form, Mizell[67], 7r o:=-2A where A is the integral scale and a2 the variance of the process. Graham and McLaughlin[48] use a slightly modified form of this spectrum called Spectrum A by Mizell[67] which has the form S(k) = 2a2o:2 (ki + 1r (ki + + o:2 ) 3 Substituting this for SY'Y' (k) in Equation[ 1.21], page 43, gives an expression for the velocity spectrum of In order to obtain the velocity covariance matrix, the Fourier transform of this expression must be taken, which yields for this two dimensional example c2 (Illkll-2fft) E[t}1E[qjt (Illkll-2kft) 2 a 2o:2 (ki + ei[kdk Jilt 7r (kr + + a2)3 where as described in Section 1.6, the vector (is the separation vector extending from xl to x2. This particular 2 X 2 symmetric velocity covariance matrix 225

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is evaluated in Graham and McLaughlin[48]. As noted in Section 1.6, this relationship is the result of the first order approximation Equation[ 1.19], page 42. However, higher order estimates have been calculated, one of the more recent studies in this area is given in Deng and Cushman[35]. Also, instead of analytical representations of the velocity covariance, numerical estimates can be established by Monte Carlo methods. In order to obtain estimates of the eigenvalues and eigenfunctions for the problem >.(i!t) = fn C(x\, if2)(if2)dif2 a Galer kin approach is used. Let {hi (if)} be a complete set of functions in the space of square integrable functions on D. Each eigenfunction of the kernel C(x\, i!2 ) can be approximated by the finite sum N k(x) L hi(x) i=l On substituting this finite sum into the preceding equation produces an error. The error, EN, is then assumed to be orthogonal to the approximating subspace, or equivalently, (EN, hj(x)) = fn ENhj(x)dx = o which yields the following system of equations: j = l,,N Letting BZJ AZJ j = 1,2,,N fn fn C(x1, x2)hi(x2)hj(xl)dx1dx2 fn hi(xi)hj(xi)dxl d(j) z 226

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yields the system (CAB)D = 0 which is a generalized eigenvalue-eigenfunction problem. If the { hi(x)} is an orthonormal sequence, then an ordinary eigenvalue-eigenfunction problem is the result. 227

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