Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00003013/00001
## Material Information- Title:
- Adding microscale effects to a macroscale SDE model of NAPL flow in heterogeneous porous media
- Creator:
- Barnhart, Kevin S
- Place of Publication:
- Denver, Colo.
- Publisher:
- University of Colorado Denver
- Publication Date:
- 2005
- Language:
- English
- Physical Description:
- xii, 97 leaves : ; 28 cm
## Thesis/Dissertation Information- Degree:
- Master's ( Master of Science)
- Degree Grantor:
- University of Colorado Denver
- Degree Divisions:
- Department of Mathematical and Statistical Sciences, CU Denver
- Degree Disciplines:
- Applied Mathematics
- Committee Chair:
- Lodwick, Weldon
- Committee Members:
- Bennethum, Lynn
Dean, Dave
## Subjects- Subjects / Keywords:
- Nonaqueous phase liquids ( lcsh )
Porous materials -- Fluid dynamics ( lcsh ) Differential equations ( lcsh ) Granular materials ( lcsh ) Differential equations ( fast ) Granular materials ( fast ) Nonaqueous phase liquids ( fast ) Porous materials -- Fluid dynamics ( fast ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 93-97).
- General Note:
- Department of Mathematical and Statistical Sciences
- Statement of Responsibility:
- by Kevin S. Barnhart.
## Record Information- Source Institution:
- |University of Colorado Denver
- Holding Location:
- Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 66462116 ( OCLC )
ocm66462116 - Classification:
- LD1193.L622 2005m B37 ( lcc )
## Auraria Membership |

Full Text |

Wi ADDING MICROSCALE EFFECTS TO A MACROSCALE SDE MODEL OF NAPL FLOW IN HETEROGENEOUS POROUS MEDIA by Kevin S. Barnhart Bachelor of Science, Montana Tech, 2002 A thesis submitted to the University of Colorado at Denver and Health Sciences Center in partial fulfillment of the requirements for the degree of Master of Science Applied Mathematics 2005 This thesis for the Master of Science degree by Kevin S. Barnhart has been approved by / Dr. Weldon Lodwick fj/)V. 2$jMOS Date"7 Barnhart, Kevin S. (M.S., Applied Mathematics) Adding Microscale Effects to a Macroscale SDE Model of NAPL Flow in Het- erogeneous Porous Media Thesis directed by Professor Dr. Weldon Lodwick ABSTRACT A three-dimensional stochastic differential equation to model the flow of a non-aqueous phase liquid in a saturated heterogeneous porous media developed by D. Dean and T. Russell1 is explained. This model aims to closely describe the physics of the fluid flow at the macroscale while taking into account phenomena at the meso and micro scales. An additional microscale addition is presented which is intended to simulate irregularities in the pore structure. From empirical data, lognormal distributions of grain size are used to generate a probabilistic percolation at the macroscale. This methodology has a high degree of agree- ment with new experimental water retention curves suggesting a powerful link between the distribution of grain sizes and how the two-phases move through a porous media. A brief discussion is given on how the computational model is parallelizeable along with numerical results showing the efficacy of efforts in implementing the parallelized framework. 1D. Dean is a senior researcher at the UCDHSC Center for Computation Mathematics where T. Russell served as a tenured faculty (he is currently a director at the NSF). The model is still being refined by D. Dean in collaboration with T. Illangasekare, M. Mathews, T. Sakaki and others at the Colorado School of Mines Environmental Science and Engineering department. ill This abstract accurately represents the content of the candidates thesis. I recommend its publication. Signed Dr. Weldon Lodwick IV DEDICATION To my parents. This thesis is a production of their values and support. ACKNOWLEDGMENT I cannot give enough thanks to (Dr.) Dave Dean for his support. He has unselfishly given his time and expertise to help me to learn the construction of his model, overcome my mathematical deficiencies, guide my research, and explain many of the subtleties of doing applied mathematics. Working with him has been the most edifying experience that I have had as a graduate student at UCDHSC. I would like to thank Dr. Lodwick for his support and more than anything his extreme patience. It has been nice to work with someone who wishes me success in however I may define it-even if it doesnt coincide with his own desires. Many thanks to Dr. Bennethum whose enthusiasm and understanding of the science have been very helpful. I would like to acknowledge Dr. Cozzens who is at least somewhat respon- sible for my involvement in the Applied Mathematics graduate program. Dr. Illangasekare, Dr. Sakaki, and Dr. Mathews of the Colorado School of Mines have been helpful by providing feedback and supplying me with valuable data. Finally, my family and friends have made earning this degree a much easier (or even possible) process. I cannot express enough gratitude to them. CONTENTS Figures .............................................................. x Tables............................................................ xii Chapter 1. Introduction...................................................... 1 1.1 Overview ......................................................... 1 1.2 Summary of Results ............................................... 3 1.3 Nomenclature...................................................... 3 2. Preliminaries .................................................... 6 2.1 Fundamentals of Groundwater Fluid Mechanics.................... 6 2.1.1 Darcys Law.................................................... 6 2.1.2 Representative Elementary Volumes for Sands and Fluids .... 8 2.1.3 Saturation and Volumetric Fraction................ 11 2.1.4 Continuity Equation........................................... 13 2.1.5 Young-Laplace Equation........................................ 14 2.1.6 Water Retention Curves........................................ 17 2.1.6.1 Brooks-Corey Theory ....................................... 17 2.1.6.2 Other Water Retention Curve Theories........................ 18 2.2 Measurability................................................ 19 2.2.1 Measurable Sets .............................................. 19 2.2.2 Measurable Functions ......................................... 22 2.3 Probability Theory.................................................. 23 2.3.1 Random Variables and Distributions.................................. 27 2.3.2 Expectation and Conditional Expectation............................. 30 2.4 Stochastic Processes................................................ 33 2.4.1 Definitions ........................................................ 33 2.4.2 Martingales......................................................... 35 2.4.3 The Weiner Process ................................................. 36 2.5 Stochastic Differential Equations .................................. 40 2.5.1 The Ito Integral.................................................... 40 2.5.2 Ito Stochastic Differential Equations (SDEs)........................ 45 2.5.2.1 Itos Lemma....................................................... 46 2.5.3 Fokker-Planck Equation.............................................. 48 2.5.4 Fokker-Planck PDE to Ito SDE........................................ 50 2.6 Heaviside Step Functions and the Delta Function.................. 51 3. The SDE Model........................................................... 55 3.1 Motivation......................................................... 55 3.1.1 The Saturation-Probability Relationship............................. 58 3.2 Fokker-Planck Equation for Fluid Flow............................... 59 3.3 PDE to SDE ......................................................... 61 3.3.1 Specifics........................................................... 61 3.3.1.1 qt Velocity Field................................................. 63 3.3.1.2 Interface Control with the Jump Term......................... 64 3.3.2 Computational Model................................................ 69 4. Macro Percolation....................................................... 71 viii 4.1 Particle Size Distributions...................................... 71 4.2 Pore Size Distributions.......................................... 75 4.3 Lognormal Water Retention Curves................................. 76 4.4 Water-NAPL Water Retention Curves .............................. 78 4.4.1 Macro Percolation in the SDE Model............................. 79 4.5 Beowulf.......................................................... 85 4.6 Future Work ..................................................... 90 5. Conclusion......................................................... 92 References............................................................ 93 FIGURES Figure 1.1 An Application: Pollutant Spill................................. 2 2.1 Fluid Volume and Density Characteristics....................... 10 2.2 Contact Angle of Wetting and Non-Wetting Fluids................ 15 2.3 Young-Laplace Equation Example Setup........................... 16 2.4 Water Retention Curve Experimental Data and Brooks-Corey Model 18 2.5 Example Path of a Stochastic Process .......................... 33 3.1 Experimental Tank Setup........................................ 55 3.2 LNAPL Plume in Experimental Tank............................... 56 3.3 Premature Breakthrough of Plume from Numerical Diffusion .... 57 3.4 Microscale Effects that are Stochastic at the Macroscale....... 58 3.5 Dc Df versus Non-Wetting Phase Saturation ......... 66 3.6 Entry Pressure and Corresponding Saturation.................... 67 3.7 Dc Df with Brooks-Corey Theory .............................. 68 3.8 Example Run of Computational SDE Model......................... 70 4.1 Sands Used in Experiments...................................... 71 4.2 Lognormal CDF Curves Fit to Grain Size Data.................... 74 4.3 Lognormal PDF Curves Fit to Grain Size Data.................... 75 4.4 Lognormal WRC Compared with Experimental and Brooks-Corey WRCs........................................................... 79 4.5 Example Run of SDE Model with a Well-Defined Interfaces .... 80 4.6 Interfacial Layer Between Fine and Coarse Sands............... 81 4.7 Variation in Grain Size Causes Variation in Pore Size......... 81 4.8 Brooks-Corey and Lognormal Curves for #16 and #30 Sands ... 83 4.9 Dc Df with Brooks-Corey and Macro Percolation............... 84 4.10 Before and After Macro Percolation is Applied to the SDE Model . 85 4.11 UCDHSC CCM Beowulf Cluster ....................................... 86 4.12 Computation Time versus Number of Particles for Movement and Pressure Calculations........................................ 87 4.13 Grid for Finite Element Method............................... 88 4.14 Decrease in Computation Time from Parallelizing Pressure Field . 89 4.15 Fingering in LNAPL Plume..................................... 91 xi TABLES Table 4.1 Grain Size Data from Sand Suppliers................................ 72 4.2 Lognormal Distribution Parameters, SSE Fit, and Distribution Statis- tics ................................................................... 73 1. Introduction 1.1 Overview High standards of living in industrialized nations have a major negative impact on the environment. Consumerism is destroying habitat through mining, deforestation, and pollution at an alarming rate. The demand is unarguably unsustainable and as time goes on more resources will need to be spent on mitigating the most immediate of these issues. Beginning in the mid-to-late eighties there has been increased research in dense groundwater pollutants. This classification includes polychlorinated biphenyl oils, creosotes, and chlorinated solvents. These fluids have a wide va- riety of uses and are fairly inexpensive to produce, making them common in consumer products and in the production of consumer products. Because these fluids are so hazardous to biological life, expensive precautions are required for even controlled laboratory experiments. Field experiments are even more difficult to perform because of environmental concerns. Unfortunately such care is not always taken in the industry which sometimes leads to the contamination of groundwater systems (see Figure 1.1). 1 Figure 1.1: An Application: Pollutant Spill For the above reasons, there is an increasing need for hydrological researchers to produce models (both mathematical and computational) that describe how these contaminants will flow in (water) saturated porous soils. Despite twenty or more years of research on this particular problem and the similarity of this problem to the extraction of petroleum from depressurized oil reserves, there are very few sureties. Uncertainties in the geological structures from the size and shape of sand grains to the rock types in a large aquifer are certainly stumbling blocks, but, also, the very nature of the flow of two fluids is being re-thought. This may be somewhat surprising since hydrologists have been using some of these laws of fluid flow for over a century. 2 This thesis describes a three-dimensional probabilistic model by Dean and Russell [12] of two-phase flow in porous medium. While other models may be computationally faster, this one attempts to capture the physics of the situation as accurately as possible; resulting in a complex model because the scenario at hand is also quite complex. To this end, the author has made a contribution to the model motivated from microscale (i.e. pore scale) phenomena in order to better the multi-scale representation. 1.2 Summary of Results The proofs of Theorem 2.13, Lemmas 2.32 and 2.33, and Proposition 2.35 are created by the author, but are not novel constructions. Improvements to the model are given at the end of Section 3.3.1.2 and throughout Chapter 4. 1.3 Nomenclature The following is a list of commonly used symbols and what they represent. a.s. almost surely Bd the set of the Borel sets on Rd B(I) the Borel sets generated by the interval / Ac the complement of the set A 6 delta function 0 the empty set T event space Tt a filtration T* the natural filtration generated by the stochastic process X 3 9 gravitational constant [Length/Time2] 1 specific weight [Force/Length3] h hydraulic head [Length] hc capillary pressure head [Length] H Heaviside step function Vh hydraulic gradient i k (intrinsic) permeability [Length2] K hydraulic conductivity [Length/Time] A fitting parameter for Brooks-Corey model 9 viscosity [Force x Time/Length2] n porosity Q sample space 4>* piezometric head [Length] IX 3.14159... Pd entry pressure [Force/Length2] p pressure [Force/Length2] Pc capillary pressure [Force/Lengtli2] P probability measure/distribution function V(S) the power set (set of all subsets of S) Q specific discharge [Length3/Time] Q Darcy flux [Length/Time] Rd the d-dimensional space of real numbers P density [Mass/Volume] 4 r.v. random variable S saturation Se effective saturation Srw residual saturation Ac B A is a strict subset of B ACB A C B or A = B t time 9 volumetric fraction 0 contact angle * V velocity [Length/Time] V volume [Length3] w.r.t with respect to wt Weiner process at time t i 5 2. Preliminaries 2.1 Fundamentals of Groundwater Fluid Mechanics Within this section, many of the basics of fluid mechanics in porous media are reviewed. For a more comprehensive explanation, the reader is referred to the classic Dynamics of Fluid in Porous Media by Jacob Bear [3]. This study is only concerned with the flow of immiscible fluids or phases. These fluids are not soluble in each other and, consequently, a tension is created at their interface. This tension creates a pressure difference known as the cap- illary pressure between the two phases. Fundamental laws that govern the flow of these fluids in porous media are summarized below. 2.1.1 Darcys Law In 1855 and 1856, Henry Darcy formulated what is now called Darcys Law by experimenting with water flow in sand. The law is a constitutive equation.1 which describes the flow of fluid through a porous medium, and is still generally accepted when used to describe the viscous (slow) flow of fluids. Darcys Law is an expression of the conservation of momentum which makes it analogous to Ohms law in the theory of electrical networks. It is given by the equation: Q = ~KAha~kb (2.1) Â¥ where Q is known as the specific discharge (the volume of fluid flowing per unit time), K is the hydraulic conductivity (a value which describes the ease with 1A constitutive equation is specific to a material or substance. 6 which the fluid can move through the porous media), A is the cross-sectional area to the flow, and Vh = (ha hb)jL is the hydraulic head drop or hydraulic gradient where ha and h,b are the heights at points a and b, respectively, and L is the distance between the two points. See [3] for a derivation. Define h = ha hb as the hydraulic head which has the corresponding pressure P = hpg, (2.2) where p is the density of the fluid and g is the acceleration due to gravity. If (2.1) is divided by A then an equation for the Darcy flux is obtained: q = -KVh (2.3) which describes the flow per unit area. Since the fluid is only moving through the pores in the medium, the Darcy flux should be divided by the porosity of the medium in order to find a rule for the velocity that a tracer would actually experience. This relationship is known as the Dupuit-Forchheimer Equation, where n is the porosity: v = -. (2.4) n Another relationship is that between the hydraulic gradient and the piezometric head, 4>*\ S7h = -V0*. The piezometric head is defined as the elevation above some datum plus the pressure head, so it is expressed as: (j)* z-\-h = z + (2.5) 99 7 and its gradient as: Vfi* = (Vp pgz) where z = P9 v-v As mentioned above, the hydraulic conductivity measures the ease with which a fluid can move through a porous medium. To keep with the electrical circuit analogy, hydraulic conductivity is akin to 1/resistivity. It is related to the intrinsic permeability or just permeability, k, through the equation: K = (2.6) P where 7 = P9 (2.7) is the specific weight of the fluid and p is the viscosity. Thus, hydraulic con- ductivity depends on the properties of both the fluid and the porous medium, whereas permeability is a property of the porous media alone. Put simply, per- meability is just a measure of the ability of a material to transmit fluids through it. From the equivalences given above Darcys Law is restated as Q = (Vp 72). (2.8) ft 2.1.2 Representative Elementary Volumes for Sands and Fluids This section describes the concept of a Representative Elementary Volume (REV) used in Continuum Mechanics. The explanation follows that given by Bear in [3]. 8 t Fluids are composed of molecules which bounce off each other and the solids around them. While one may be able to deterministically solve the direction and momentum of a few molecules, we still lack the computing power to reliably predict the path of even a few hundred molecules. So, instead of trying to predict fluid flow using the properties at a molecular level, a statistical approach is adapted. Bear describes a fluid particle as ...an ensemble of many molecules contained in a small volume. Its size is much larger than the mean free path of a single molecule. It should, however, be sufficiently small as compared to the considered fluid domain that by averaging fluid and flow properties over the molecules included in it, meaningful values, i.e., values relevant to the description of bulk fluid properties, will be obtained. The fluid property most often used to determine the size of the fluid particle is density the ratio of the mass of fluid, m, and the volume, V, in which it is contained. Let x be a point in the fluid and consider a volume (e.g. a sphere), V, for which x is the centroid. The relationship between the size of the volume and the density can empirically be shown to be similar to Figure 2.1 9 Figure 2.1: Fluid Volume and Density Characteristics The point Vo in Figure 2.1 indicates a volume where the density is fairly stable. Let Vo to be the volume of the fluid particle. All points in the fluid are then associated with a fluid particle with this density, and, in this way, the fluid particle will have more definite dynamic and kinemetic properties which can greatly simplify fluid flow models. The same concept of the REV of a fluid (the fluid particle) can be applied to porous media. The analogous property to density in porous materials is the porosity the mass of solid per unit volume. The relationship between volume and porosity is quite similar to that of volume and density and so a REV is chosen in the same manner.2 2The notion of a REV is somewhat limited in application by the medium. For some materials it is very difficult to obtain a reliable REV for which the desired property is stable due to unreliable factors such as the packing and shape of the sand grains. 10 2.1.3 Saturation and Volumetric Fraction Consider a REV of a homogeneous porous material with total volume V. Let Vv represent the volume of the void in V, and Va represent the volume of the a-phase3 in V. Saturation is the percent of void space that is filled by the fluid. Accordingly, is the saturation of the ct-phase in V. When a 11011-wetting fluid displaces a wetting fluid (such as water) there will always exist some residual water content (or wetting fluid) in the soil because water becomes trapped in the crevices of the porous medium. The residual saturation, Srw, is related to the effective saturation, Se: S Sj' 1 Srw (2.9) where S is the measured saturation. The effective saturation is scaled to range from zero to one by treating the volume of the residual water as a in-displaceable substance (such as the porous media). The following equality is usually assumed to hold in a two-phase system: Smv + Sw 1- (2.10) The volumetric fraction is the fraction of total volume that is filled by a fluid: Va V ' 3The term a-phase is used in multi-phase discussions to indicate any one of the phases. 11 It is possible for the volume of the liquid to fill all the void space, so Sa is a value between 0 and 1. However, the fluid can never fill the entire volume since part of it is filled by the porous medium, and so 9a must vary between 0 and n (the porosity) v; = nV. Using this and the definitions given above, the saturation and volumetric fraction are related by Particular to this study, is how the concept of fluid particles move in porous media, so saturation and volumetric fraction is described with these in mind. Each REV of the porous media has a finite volume and can therefore only contain a finite number of fluid particles. Let VfP be the volume of a single fluid particle of the non-wetting phase. Given that p is constant, the number, Np, of a-phase particles needed to com- pletely saturate an REV is N P V So, if the REV contains x o-phase fluid particles its saturation is 9a = nSa. (2.11) and the corresponding volumetric fraction must be (2.12) 12 2.1.4 Continuity Equation The theory of continuum mechanics postulates the global balance law given by 4 f which states that the time rate of change of the property
of the fluxes, r, across the surface S(t) plus the sum of the forces, /, acting |