Retrospective Theses and Dissertations Iowa State University Capstones, Theses andDissertations 2002 Investigation of NAPL dissolution and instability using a pore-scale network model: sensitivity analysis of porous medium characteristics Yanhui Hu Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Chemical Engineering Commons This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact digirep@iastate.edu. Recommended Citation Hu, Yanhui, "Investigation of NAPL dissolution and instability using a pore-scale network model: sensitivity analysis of porous medium characteristics " (2002). Retrospective Theses and Dissertations. 379. https://lib.dr.iastate.edu/rtd/379 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bteedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Photographs included in the original manuscript have been reproduced xerographicaUy in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 Investigation of NAPL dissolution and instability using a pore-scale network model: Sensitivity analysis of porous medium characteristics by Yanhui Hu A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major. Chemical Engineering Program of Study Committee: Rodney O. Fox, Major Professor James C Hill Robert Horton W. Robert Stephenson R Dennis Vigil Robert P. Ewing Iowa State University Ames, Iowa 2002 UMI Number: 3051470 UMT UMI Microform 3051470 Copyright 2002 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 ii Graduate College Iowa State University This is to certify that the doctoral dissertation of Yanhui Hu has met the dissertation requirements of Iowa State University Ma r Professor For the Major Pro; Signature was redacted for privacy. Signature was redacted for privacy. iii Table of Contents CHAPTER 1. INTRODUCTION I 1.1 NAPL movement and dissolution 1 1.2 Dissertation organization 4 1.3 References 4 CHAPTER 2. LITERATURE REVIEW 7 2.1 Macroscopic scale studies 9 2.2 Pore scale studies 20 2.3 References 29 CHAPTER 3. INVESTIGATION OF NONAQUEOUS PHASE UQUID DISSOLUTON AND INSTABILITY USING A NETWORK MODEL 40 Abstract 40 Nomenclature 41 1. Introduction 42 2. Model development 44 3. Simulation results 51 4. Conclusions 58 References 60 Captions for figures 66 CHAPTER 4. INVESTIGATION OF POROUS MEDIUM CHARACTERISTICS EFFECTS ON NONAQUEOUS PHASE UQUID DISSOLUTION AND INSTABILITY 79 Abstract 79 1. Introduction 80 2. Methodology 83 3. Simulation results and discussion 87 4. Conclusions 97 References 99 Captions for figures 105 Appendix 114 CHAPTER 5. CONCLUSIONS 116 1. Summary 116 2. Recommendations 119 ACKNOWLEDGMENTS 122 I CHAPTER 1. INTRODUCTION 1.1 NAPL movement and dissolution Multiphase flow in porous media is a complicated yet common phenomenon in many fields of practical interest. An example in chemical reaction engineering is the trickled bed reactor, in which solid catalyst pellets are packed to form a porous medium. Two phases, usually one gas and one liquid, are introduced into the reactor, where they mix in the void space and react on the surface of the catalyst. Another example lies in the area of petroleum engineering, in which residual oil is recovered from reservoir rock via water flooding. Agricultural drainage and irrigation, in which water and air compete to occupy pores via capillary and buoyancy forces, is also a multiphase flow process. Multiphase flow processes can also be encountered in the environment, for example when organic solvents enter an aquifer due to leakage from underground tanks or accidental surface spills. Organic chemicals can cause extensive water contamination. Many organic pollutants are hydrophobic, forming a separate nonaqueous phase rather than immediately dissolving into the groundwater, so they are referred to as NAPLs (nonaqueous phase liquids). The solubilities of many NAPLs are low, but still high enough to severely degrade water quality, causing long-term contamination. For example, the solubility for trichloroethylene (TCE, a commonly used degreaser and solvent) in water is around 103 mg/L, while its maximum concentration limit (MCL) in drinking water is around 5*103 mg/L, about 2*10* times less (Pankow and Cherry, 1996). The low solubility means that a spill takes a long time to dissolve; the high toxicity means that even extensive dispersion does not dilute it to safe levels. A list of the common organic groundwater pollutants and their MCLs can be found in Fetter (1994). 2 Following surface spills or underground tank leakage, NAPLs migrate into the underground porous media. NAPLs first infiltrate through the vadose zone (unsaturated zone), leaving disconnected residual blobs or ganglia behind their flow path when interfacial forces overcome viscous forces and gravity forces. Upon reaching the groundwater table, NAPLs lighter than water, known as LNAPLs, spread horizontally to form a floating pool. On the other hand, NAPLs denser than water, known as DNAPLs, migrate below the water table and into the saturated zone, leaving residual ganglia behind and forming a pool above an impermeable layer. As groundwater flows by these trapped ganglia and pools, NAPLs dissolve slowly into water, creating dilute contaminant plumes (Fig. I). In the study of residual ganglia behavior, buoyancy forces resulting from density differences are generally insignificant at the scale of a few pores (Ioannidis et al., 1996), therefore they are ignored in this study. When groundwater is contaminated, a decision must be made to either actively remediate the site, or to monitor it while natural processes accomplish the cleaning (PNNL, 2001). Both procedures are expensive and require a solid understanding of how contaminants move in the subsurface. Remediation used to be built around the processes of contaminant dissolution and recovery: the so-called pump-and-treat method, in which contaminated water is pumped out of the subsurface, and undergoes aboveground treatment, after which the clean water is released at the surface. However, pump-and-treat has been found to be ineffective in many practices (Pankow and Cherry, 1996), both because dissolution is a slow process when NAPL solubility is low, and because diffusion of dissolved contaminant from a water-NAPL interface to a region of higher-velocity water flow field is seriously rate limiting. Recent remediation methods also attempt to remobilize 3 the trapped NAPL ganglia or pools, with efforts concentrating on reducing capillary forces through the use of surfactants, or increasing the water velocity so the viscous forces will override the capillary forces that hold ganglia in place. Examples of these remediation techniques include water flooding and chemical-enhanced recovery using cosol vents and surfactants to reduce interfacial tension and/or increase NAPL solubility. The NAPL entrapment process and the fate of the ganglia are affected by many factors, including porous medium characteristics, fluid properties, boundary conditions, and the NAPL spill (or leakage) and migration history. Due to the complexity and uncertainty in the nature of the NAPL dissolution and migration processes, investigations have been conducted on several scales. The largest scale is the field scale, at which a wide range of remediation techniques has been tested. However, in order to have a better understanding of the underlying processes, experimental investigations and numerical simulations typically are conducted at much smaller scales. Until recently, most experimental studies have focused on the measurement of continuous fluid properties on a macroscopic or laboratory scale (-10 cm). Along with these experiments, mathematical models have also been developed and relationships between continuous properties and transport characteristics have been proposed to describe multiphase flow and interfacial mass transfer in porous media. Since the mass transfer process occurs at the NAPL-aqueous phase interfaces, investigation on the pore scale is useful for obtaining an accurate description of the transport processes. When the rocks are water-wet, capillary forces tend to restrict the residual NAPL blobs to the larger pores. Because these larger pores are often the locus of higher-velocity groundwater flow, the interplay between capillary and viscous forces acting on a ganglion is locally conditioned by the exact configuration of the ganglion, as determined by the porous 4 medium: size, structure and heterogeneity of the pores, aspect ratio, wetting angle, and so on. Among all the pore scale studies, pore-scale modeling is one of the most used investigation tools, since these models can explicitly take into account the effect of the porous medium on NAPL movement and dissolution, as well as provide an opportunity to test hypotheses that are difficult or expensive to test experimentally. This research investigates, via a pore-scale model, various aspects of NAPL dissolution and movement that have been overlooked in the past. 1.2 Dissertation Organization The goals of this research are to better understand the mechanisms of stranded NAPL dissolution and remobilization using a pore scale model, and to characterize the effects of the porous medium, flow rates and fluid properties on those processes. Chapter 2 contains a thorough review of previous work. Chapter 3 describes the development and testing of a network model for investigating NAPL dissolution and instability. Chapter 4 presents a sensitivity analysis of various factors influencing NAPL dissolution and instability. Chapter 5 gives the conclusions from this research and recommendations for future work. 1J References Fetter, C. W., Applied hydrogeology, 3rd ed.. New York, Prentice-Hall, Inc., 438-440,1994. Ioannidis, M. A., I. Chatzis, and F. A. L. Dullien, Macroscopic percolation model of immiscible displacement: Effects of buoyancy and spatial structure, Water Resour. Res. 32, 3297-3310,1996. 5 Pankow, J. F., and J. A. Cherry, Dense chlorinated solvents and other DNAPLs in groundwater, Portland, Oregon, Waterloo Press, 1996. PNNL (Pacific Northwest National Laboratory), Subsurface Contaminant Focus Area: Monitored Natural Attenuation (MNA) - Programmatic, Technical, and Regulatory Issues, Paper PNNL-13569, Richland, Washington, 2001. 6 Field Scale Spill Water Flow Water Table Macro/Lab Scale 'm Vadoze Zone -Rock Ganglion water Micro/Pore Scale DNAPL Pool Saturated Zone Impermeable Layer Figure 1. Schematic diagram of NAPL migration and research scales 7 Chapter 2. LITERATURE REVIEW Nomenclature: Symbol Description Dimension a Specific interfacial area [L L ] ki Mass transfer coefficient of NAPL in aqueous phase [LT1] kf Effective mass transfer coefficient [L Tl] km Relative permeability of fluid phase a [-] lc Characteristic length [L] qa Darcy or superficial velocity of fluid phase a [LT1] C NAPL concentration [M L"3] C„w NAPL concentration at the interfaces [M L"3] C, NAPL solubility in aqueous phase [M L"3] D Dispersion tensor [L2 Tl] Dm Molecular diffusion coefficient [L2T'] Fa Sources or sinks of fluid phase a [M L 3 T1] G Gravitational acceleration vector [L T2] J Mass flux of NAPL [M L"2T'] K Permeability tensor [L2] Ki Mass transfer rate coefficient [T1] Pa Pressure of fluid phase a [M L"1 T2] Sa Saturation of fluid phase a, fluid volume/void volume [-] Or Transverse dispersivity [L] Ol Longitudinal dispersivity [L] S Diffusion length [L] Sjt Kronecker delta [-] 0 Porosity of the porous medium, void volume/total volume [-] p Density of fluid phase [M L 3] lAa Viscosity of fluid phase a [M L"1 T1] at Mass fraction of NAPL in water [-] Subscript a= n, w n: nonaqueous phase w: water phase As nonaqueous phase liquids (NAPLs) migrate through a porous medium, a portion of the NAPL is left behind as residual ganglia. The complicated NAPL migration and dissolution process, as well as the fate of the ganglia, are frequently characterized using several dimensionless groups. One of the most important dimensionless numbers to describe ganglion mobilization is the capillary number Ca, as the ratio of viscous force to capillary 8 force. In general, for a specific system of porous medium and multiphase liquids, at some Ca, the viscous force will overcome the resistance posed by capillary force, and ganglia will move downstream. Another commonly used dimensionless group is the Bond number Bo (Bo- (pw - pa )glc2/^ 5*10 \ nearly all ganglia were displaced. Ng and Payatakes (1980) and Payatakes (1980) proposed a stochastic model to predict the fate of the residual ganglia in porous media during an immiscible displacement. Hundreds of one-step ganglion motions (rheons) were simulated using a two-dimensional network model with random sized unit cells, and the probabilities of mobilization, breakup, and stranding per rheon as functions of capillary number and ganglion sizes were obtained. It was found that larger ganglia can be mobilized 9 or broken up at lower Ca. For example, ganglia with a size of 15 pore units or larger are likely to be mobilized at Ca > 10"3 while singlets cannot be mobilized until Ca > 10"2. Later, experiments were performed on a two-dimensional chamber-and-throat etched glass apparatus by Avraam and Payatakes (1995a) and the effects of capillary number, flow-rate ratio, viscosity ratio and non-aqueous phase saturation on different flow regimes were investigated. Four main flow regimes were observed and mapped: large ganglion dynamics (LGD), small-ganglion dynamics (SOD), drop-traffic flow (DTF) and connected pathway flow (CPF). LGD (when the mean size of the moving ganglion larger than 10 chambers) corresponds to the small Ca and moderate-to-high water saturation. On the other hand, SGD corresponds to the larger value of Ca and/or smaller water saturation. Besides the effect of the capillary number on the ganglion mobilization, porous medium characteristics such as pore sizes, shape, neck/body correlations and aspect ratios also have a significant impact on the NAPL movement mechanics. This is especially evident in the constitutive relationships, such as capillary pressure-saturation-relative permeability relationship (Reeves, 1997). These constitutive relationships play an important role on determining the final distribution and configuration of the NAPL ganglia (Mayer and Miller, 1992; Lowry and Miller, 1995). 2.1. Macroscopic scale studies Various mathematical models have been developed and used to quantify multi-phase flow and NAPL dissolution into the aqueous phase (Kueper and Frind, 1991a, b; Powers et al., 1991; VanderKwaak and Sudicky, 1996). Many of the existing models are limited to simplified situations, such as only one NAPL species is considered, the porous medium is 10 homogeneous with only or two one spatial dimensions under consideration, local equilibrium of the NAPL aqueous phase concentration is assumed, or NAPL mass loss due to dissolution is neglected. The exceptions to these assumptions include a multi-region mass transfer model to describe more than one NAPL species (Garg and Rixey, 1999), a two-film model to incorporate surfactant interactions (Reitsma and Kueper, 1998), a rate-limited mass transfer model (Brusseau, 1992; Zhu and Sykes, 2000), a two-dimensional model (Mayer and Miller, 1996), and a NAPL pool dissolution and shrinkage model (Lee and Chrysikopoulos, 1998). A comprehensive review on multiphase flow and mass transport is given by Miller et al. (1998). In general, it can be assumed that the microscopic balance law of continuum mechanics holds for each phase. Thus, a macroscopic mass balance law for isothermal multiphase flow in porous media can be formed by applying volume averaging techniques over a microscopic mass balance equation (Bear, 1972). This gives a mass balance equation for each fluid phase: + ? (1) where the subscript or denotes the fluid phase, $)is the porosity, S is the saturation, q is the Darcy velocity, and F is the source term. Equation 1 can be used for an arbitrary number of fluid phases, with changing porosity or fluid densities. However, it is widely used for the common case of two-phase flow (aqueous and non-aqueous phases) with the solid phase being inert and immobile. Special cases like solid phase sorption and three-phases flow were studied by Abriola and Pinder (1985a,b) and Brusseau (1992). 11 For immiscible two-phase flow in a homogeneous and isotropic porous media, an extended version of Daisy's law can be used: kk -f~ (vr„-p.g) (2) r*a where k is the intrinsic permeability tensor, describing the absolute permeability of the porous media, and kra\s the relative permeability of phase a. A single governing equation for each fluid phase in multiphase flow can be attained by combining Eq. I and 2: -V kk Pa = Fa (3) dt For two-phase flow systems, Eq. 3 is needed for each fluid phase. In order to close the system, the following constraints must be satisfied: Sw+Sn = 1 (4a) P r = p . ( s J ( 4 b ) *„=*„($«) (4c) Eq. 4b gives the definition of capillary pressure. Constitutive relationships like Eq. 4b and 4c should account for the influences of all subscale phenomena and processes on immiscible fluid flow in porous media, such as porous medium characteristics and heterogeneity, wettability, and fluid properties. Both Eq. 4b and 4c are complex and hysteretic (Jerauld and Salter, 1990; Avraam et al., 1994; Avraam and Payatakes, 1995a), and the resultant flow equations are highly nonlinear and difficult to solve. Due to the complex nature of the constitutive equations, many researchers try to find a more complete relationship for relative permeability. The relative permeability coefficient 12 not only depends on fluid saturation but also on other system parameters, such as the capillary number, flow rate ratio, viscosity ratio and contact angle. A generalized two-phase relative permeability coefficient form proposed by Avraam and Payatakes (1995b) explicitly takes into account the viscous coupling effects: kk kk 1. ™ (W. - P.g) — ( V P , -p,g) (5a) Mw /'n kk kk (VP.-p.*)--f=-(V/>„-Ag) (5b) Mw /*« where Anm and £rww are the generalized relative permeabilities for water and nonaqueous phases, and krm and £mw are the viscous coupling effects. Note that original Darcy's law was originally developed to describe steady, saturated single-phase flow through a homogenous, isotropic porous medium, and the extension of Darcy's law to multiphase systems is for convenience rather than based on basic principles. Under certain conditions, the use of the extended Darcy's law leads to contradictions (Hassanizadeh and Gray, 1993). Therefore, caution is required when applying Darcy's law to build mathematical models. Dissolution of NAPL in a porous medium can be characterized using the following mass balance equation: j -( fr?wSwûj)+V-(p„qwaj)-V (0pwSwD V w ) = F (6) where m is the mass fraction of NAPL in the aqueous phase and D is the dispersion tensor. The first term represents the accumulation of NAPL in water, and the second term represents the advective transport of NAPL due to bulk flow. The third term describes the transport due to dispersion. The right-hand-side term accounts for all the sources, including interfacial mass transfer. 13 The dispersion tensor accounts for both mechanical dispersion and molecular diffusion (Bear, 1972). For an isotropic medium, j,k-component of the dispersion tensor can be expressed as: Dj* +(at~ar>ë^+D"8* <7) where o^is the transverse dispersivity, % is the longitudinal dispersivity, and Dm is the molecular diffusion coefficient of NAPL in the aqueous phase. Since most NAPLs of interest are only slightly soluble in the aqueous phase, the spatial variation of aqueous phase density can be neglected, and can be combined with the mass fraction to yield solute mass concentration. This simplifies the mass transport equation (Eq. 6) to: ^{*SwC)+V(qwC)~V(W>VC)= F (8) where C is the NAPL concentration in the aqueous phase. The source/sink term F has to contain all the generation and consumption terms, which may include chemical transformation due to reaction and the interface exchange of chemical species. The mass flux of a species across an interface results from the different chemical potentials of that species in the respective phases. A general assumption for mass transfer of hydrophobic chemicals from the organic to the aqueous phase is that diffusion through a boundary layer from the interface to the bulk aqueous phase is the rate-limiting step. Based on this assumption, a single resistance, linear driving force model is widely used to describe the interfacial mass transfer process (Miller et al., 1990; Powers et al., 1991). 14 V=^=-(C„-C) = t,(C„-C) (9) where J is the mass flux of NAPL into the aqueous phase, Cniv is the NAPL concentration at the NAPL/water interface, Ô is the diffusion length, and ki is the mass transfer coefficient. Under most circumstances, concentrations at interfaces are not practical to measure. Therefore, it is convenient to define the mass transfer coefficient based on the solubility of NAPL, C„ in the aqueous phase: J = k f ( C s - C ) (10) where kf is the effective mass transfer coefficient. In the situations where the only source or sink is interface mass transport, we have: F = Ja (11) where a is the specific interfacial area defined as the interfacial area per unit volume of porous media. Combining Eq. 8, 10, and 11 yields: l(*S„C)+V (».C)-V (i|iS„D VC)=t/a(C, -C) (12) The governing equations for flow (Eq. 3) and mass transport (Eq.12) are coupled through the volumetric flux term. Due to the strong influence of flow behavior on the interfacial mass transfer, it is difficult to solve Eq.12 and obtain an accurate prediction. Besides this linear mass transfer model, there are some other mass transfer models developed to describe the effect of a moving boundary on the mass transport, such as the pore-diffusion model (Baldwin and Gladden, 1996) and the shrinking-core model (Geller and Hunt, 1993). Baldwin and Gladden (1996) compared their experimental results with all three mass-transfer models, and found that the pore-diffusion model fit their data the best. 15 However, it remains an open question which model best describes any given physical situation. In order to solve Eq.12, the value of kf must be determined. Many investigators have proposed quantifying kf through the use of correlation models, which describe the dependence of mass transfer on flow and diffusion. Those models generally involve dimensionless groups, such as Sherwood number (Sh), Reynolds number (Re), Schmidt number (Sc), and Peclet number (Pe). Sh = k f l c / Dm ratio of interface mass transfer to molecular diffusion. Re = l cqwpw / ratio of inertial force to viscous force. Sc = nw /{Dmpw ) ratio of viscous force to molecular diffusional force. Pe = qw l c /Dn ratio of mass transfer by convection to diffusion. Table 1.1 provides a list of mass transfer coefficient correlations. Friedlander (1957) studied the mass transfer around a single solid sphere and a cylinder at low Reynolds numbers using boundary layer theory, and derived an expression for the Sherwood number in term of the Peclet number. His prediction is about 10 to 40% lower than the experimental data. Bowman et al. (1961) examined the mass transfer around a solid or fluid single sphere in Stokes flow using a theoretical approach similar to Friedlander's. Using a fluid sphere gave the opportunity to study the effect of internal circulation on the rate of mass transfer. He found that internal circulation could accelerate mass transfer at high Peclet number but not at low Peclet number. Both Pfefferet al. (1964) and Nelson et al. (1975) predicted the mass transfer rate in a packed bed at low Reynolds number analytically, and found good agreement with published experimental results. They 16 also showed that the Sherwood number in a packed bed is a function of both porosity and the Peclet number, and the effect of Peclet number decreases with decreasing porosity in the low range of Peclet number. Wakao and Funazkri (1978) collected data of gas-liquid mass transfer in packed beds from literature. They corrected those data for axial fluid dispersion and then proposed a correlation for Sherwood number in the range 3 < Re < 10000. Other investigators have studied this issue from an experimental standpoint, including Wilson and Geankoplis (1966) and Dwievedi and Upadhyay (1977). Their correlation relationships were developed from experimental results obtained from packed beds. They found that mass transfer coefficients for multi-sphere systems were typically lower than those from the extrapolation of correlations developed from single-sphere systems. Researchers in the field of environmental engineering have also done experimental work on different NAPL-water systems and derived similar correlation relationships. Due to the complex nature of the NAPL-water porous media system, such as the changing of the entrapped NAPL geometry and aqueous phase saturation due to NAPL dissolution, some dimensionless numbers need to be modified or added into the correlations. Two widely used dimensionless groups are the modified Sherwood number and modified Reynolds number Sh=K,l/ /Dm Re 'Ï.p.â/M'-S.)] where Ki = k fa is the mass transfer rate coefficient, and Sn is the nonaqueous phase saturation. The reason for using Ki instead of Ay in the modified Sherwood number is that unlike idealized chemical engineering systems in which interfacial area is well defined, NAPL/water interfacial area in a geological setting is typically very difficult to measure 17 accurately. It also should be noted that the mass transfer rate coefficient is not suitable for prediction of long-term dissolution in which the interfacial area changes with time. Miller et al. (1990) investigated the mass transfer rate coefficient as a function of aqueous phase velocity, NAPL saturation, and porous media characteristics, and found that local equilibrium can be achieved rapidly over a wide range of NAPL saturation and aqueous phase velocities in a homogeneous system. Later Mayer and Miller (1996) developed a two- dimensional multiphase transport model to investigate the NAPL movement and dissolution in heterogeneous porous media, and found that changes in the variances of the porous media properties lead to significant variations of the mass transfer rate, which can be attributed to the existence of immobile regions of NAPL pools in the heterogeneous porous media. Geller and Hunt (1993) proposed a two-dimensional NAPL dissolution model using mass transfer coefficient correlation reported in the literature. Their model treats NAPL as discrete mono- sized spheres and accounts for the water flowing by the residual NAPL. They were able to reproduce the effluent NAPL concentration using mass transfer zone propagation. Powers et al. (1991) investigated five different idealized blob shapes (singlets, doublets, fingers, large sphere, and cylinder), and assumed the effective surface area to be 1- 50% of the blob surfaces (ignoring the diffusive mass transfer through immobile water). They developed a one-dimensional convection-dispersion mass transfer model to examine the steady state and transient behavior of the system. Sensitivity analysis of the model shows that the nonequilibrium effects are significant for situations with large blobs, relative high velocities and low NAPL saturation. Powers et al. (1992) studied initial stage ("steady state") residual NAPL dissolution for 7.3* 10"* < Ca < 1.5* 10'7 and proposed a correlation model between Sh\ Re and grain size distribution measure t/,-, consistent with the 18 experimental results of Geller and Hunt (1993). The effects of porous medium characteristics, NAPL type and aqueous phase velocity on dissolution rates were investigated. The results showed that the effluent NAPL concentration was below solubility in most cases, indicating nonequilibrium conditions. The results also revealed that the dissolution rate depends on the residual NAPL distribution, which is strongly affected by different porous medium types. Similar conclusions about the influences of blob size and aqueous phase velocities on mass transfer rate coefficient were drawn by Imhoff et al. (1994), who included an extra term in their correlation model to describe the location of NAPL blobs within the column. Later Powers et al. (1994a) extended their work to transient NAPL dissolution, and proposed a correlation model using the volume fraction of NAPL to indicate the change in specific surface area of the shrinking NAPL blobs. The correlation between the volume fraction and modified Sherwood number is a function of median grain size and grain size distribution (Table 1). Powers et al. (1994b) proposed a mathematical residual NAPL transient dissolution model, which treats mass transfer coefficient and specific surface area independently. The specific surface area is obtained by assuming spherical NAPL blobs with a distribution of diameters. This model relies on calibration of the porous medium, which may not be obtained easily in practice. A recent review (Khachikian and Harmon, 2000) discussed correlation relationships from both chemical engineering and environmental engineering literatures and concluded that the two disciplines arrived at similar descriptions of mass transfer behavior. 19 Table 1. Mass transfer correlations from the literature. System Correlation Valid for Reference Theoretical work: Single sphere Sh =0.89Pel/3 Pe>1000 Friedlander [1957] Single sphere Sh=0.978Pel/3 Pe>10 Bowman et al. [1961] Sh=2+9/16Pe+— Pe<10 Packed bed Sh=2.4Re*033Sc042 0.08)Pel/3 [1963] Packed bed l)"2/3-( l-0 Nelson and Galloway Sh=2+1.1 Re°'6Sc1/3 [1975] Packed bed 3 water NAPL Ganglia Toluene-water Sh*= 12Re,0759„06Sc05 TCE-water Sh*=3.18e„054Re098Sc°J TCE-water Sh*=57.7Re06,d5o064Ui041 ' TCE-water Sh*=340e„087Re07,(dpZx)0Jl * TCE-water Sh*=4.13Re*0-598S0673Ui0369(6V6n Styiene-water * P4=0.518+0.1148+0. lOUj Naphthalene Sh=77.6Re0-658 Solid sphere 0.0016l, the NAPL concentration in crevice waters reaches solubility after passing through a ganglion. Pore throats in the current network model have a circular cross-section, with channel flow superimposed onto it. The ratio of channel volume to ganglion volume is 0, and channel aqueous phase conductance of a throat is f? times the original conductance. Thus, from Zhou's definition (2000), the equivalent dimensionless time rcan be expressed as: -3h When t < 1, the solute concentration increase after passing a ganglion is approximately Vr times the maximum possible concentration increase had equilibrium been attained. When t =1, the NAPL aqueous phase concentration increase reaches 90% of maximum possible concentration increase. Further increase in t brings the NAPL aqueous phase concentration up to the solubility limit after passing a ganglion (Fig. 4, Dillard et al., 2000). It is expected that at the same Capillary number, water flow through a larger ganglion will approach NAPL solubility at the exit; while at a lower Capillary number water flow through 51 any size ganglion will reach NAPL solubility limit. T is calculated for each case, and for those cases with high flow rate and large P, therefore small t, the aqueous phase solute concentration is adjusted. This approach is an approximation of the channel dissolution process, with the purpose of accounting for the effect of water channel flow rate on the NAPL dissolution. 3 Simulation results 3.1 Single ganglion dissolution Table 1 lists the parameters used in the single ganglion numerical experiments. The porous medium parameters represent a coarse-grained sand sample, and the NAPL was chosen to be styrene. Both the porous medium characteristics and the NAPL properties match the study settings of Powers et al., (1992) and Dillard et al. (2000). Therefore the results of this study can be compared to their published results. Because the model introduces ganglia of pre-determined size into the porous medium starting from a randomly chosen node, the configuration of a ganglion results from the local porous medium configuration and the local hydraulic gradient. A stable ganglion is obtained when NAPL/water pressure drops across all interfaces are balanced by capillary pressures. Plotting the entrapped ganglion interfacial area as a function of the pre-determined sizes reveals a linear relationship on a log-log scale (Fig. 3). The regression slope is 0.7, slightly greater than 0.667, the slope for a perfect sphere. It indicates that the ganglion in the porous medium is slightly ramified, and thus, in column studies the common assumption of spherical ganglia may underestimate the area/volume ratio, particularly for large ganglia. Further investigation of the stranded ganglia configuration shows that they are slightly elongated in 52 the direction of the pressure gradient (Fig. 4). The differential elongation is not statistically significant, which indicates that the effect of hydraulic pressure drop across the entire ganglion on the configuration of the entrapped ganglion is less than the local effect of the porous medium. In order to compare these two effects, the ratio of the two pressures is considered. At Ca = 10"4, the highest Capillary number tested in this study, the hydraulic pressure drop across the entire domain (about 2 cm) is 180 kg m'1 s"2. Therefore, for a ganglion of size 40, spanning about six pore units in the direction of pressure gradient, the hydraulic pressure drop is around 27 kg m"1 s"2. On the other hand, the change of capillary pressure in two throats, assuming one radius is one standard deviation above mean and the other one is one standard deviation below the mean, is about 400 kg m'1 s"2. This change in capillary pressure between two throats is 15 times larger than the possible hydraulic pressure drop across a ganglion of size 40. At lower Capillary number, or for a smaller ganglion, this ratio would be even larger. Therefore, at the range of Capillary numbers in this study, the configuration of stable ganglion is mainly controlled by porous medium characteristics rather than by aqueous flow rate. Numerous studies have shown that high aqueous flow can induce ganglion movement. For example, Ng and Payatakes's two-dimensional simulations (1980) found that ganglia were not stable when Ca > 10'3, and may occasionally move even when Ca = 10'5. Since an objective of this study is to find the dissolution effect on stable ganglia, a range of Ca from 10"7 to 10"4 was chosen to ensure stable ganglia. In order to study the effect of flow and ganglion size on the stability of ganglia, the proportion of different instability types were recorded (Fig. 5). In the range of Capillary numbers studied, the aqueous flow effect on the 53 ganglion instability type is not significant. This seemingly surprising result can be explained again by the relatively small effects of aqueous flow compared to the effect of porous medium on ganglion stability. Fig. 5 also reveals that ganglion size plays a significant role on the initial ganglion stability. It is found that both initial tail-retreat and snap-off occurrences decrease with increasing ganglion sizes, and small ganglia are more likely to have initial tail-retreat than initial snap-off. Since the relative effect of hydraulic pressure drop is insignificant compared to the effect of capillary force, a growing ganglion tends to expand into a region with bigger throat sizes which pose less capillary pressure resistance, and the resultant average pore body radius of a larger ganglion is larger. Tail-retreat (water imbibition) is controlled by the size of the pore bodies (Eq. 3) and snap-off is controlled by the size of throats (Eq. 4). The smaller the pore body, the more likely it will be the node of retreat, and the smaller a ganglion-occupied-throat is, the more likely it will be the throat to snap off. Therefore, a larger ganglion is less likely to have tail-retreat or snap-off. Although both probabilities of tail-retreat and snap-off decrease with increasing ganglion sizes, the rate of decrease for tail-retreat occurrences is higher, therefore a larger ganglion is more likely to have snap-off. A stranded ganglion undergoes dissolution due to diffusion at the water/NAPL interfaces and convection in the water channels passing the ganglion. Dissolution rates for the stagnant diffusion and channel convection mechanisms are dissimilarly affected by aqueous flow rate. Fig. 6 shows the effect of water channel on ganglion dissolution rate, given ganglion size 10. Three water channel volume fractions were investigated, {5=0.05,0.1 and 0.25. Neither the volume of the water channel nor the Capillary number strongly affects the stagnant layer diffusion rate. This result is reasonable since the stagnant layer diffusion is 54 a function of concentration gradient, which is not strongly affected by the flow in the water channel. However, when the Capillary number decreases to 10'7, the stagnant layer diffusion rates for all cases drops slightly, due to the resultant smaller concentration gradient. On the other hand, both the water channel volume and water flow rate affect the water channel dissolution rate greatly. It is shown that the channel dissolution rates increase about 10 times for every factor of 10 increase in the Capillary number except at high Ca when x « 1.0. Fig. 6 also suggested a transition Capillary number at which the dissolution process switches from diffusion limited to channel dissolution limited. In order to determine whether this transition between dominant mechanisms is a function of ganglion size, we examined the initial dissolution rate for both stagnant layer diffusion and channel flow dissolution as a function of ganglion size (Fig. 7). Only the case of P = 0.1 was examined here. The initial dissolution rates for both diffusion and channel dissolution are comparable at Ca - 10 s, with channel dissolution dominating at larger Ca and diffusion dominating at lower Ca. Therefore, in column ganglion dissolution studies, both diffusion through stagnant layer of water and convection in mobile water channel need to be taken into account when Capillary number is about 10 s. Only when aqueous flow rate is very high or very low can the dissolution process be reasonably approximated by a single mechanism. Figure 7 shows not only the switch of control from diffusion to convection with increasing Capillary number, but also the interaction effect between aqueous flow rate and ganglion size on initial dissolution rate. The regression slope of initial dissolution rate on ganglion size on the log-log scale was obtained and plotted (Fig. 8). It is evident that the effect of ganglion size on diffusion is more pronounced at high than at low Capillary number. 55 Diffusion rate is defined as the product of diffusion flux and interfacial area, and interfacial area per unit volume is essentially the same when Capillary number increases from 10"7 to 10"4 (Fig. 4). Therefore, the regression slope would be fixed at 0.7, the slope of the interfacial area versus volume relationship, if the diffusion flux effect was constant across ganglion sizes and flow rates. However, flow rate affects the NAPL concentration in the aqueous phase, i.e., low aqueous flow rate causes higher NAPL concentration in the nodes around ganglia, which in turn decreases the diffusion flux. The larger the ganglion, the stronger is this effect. Thus, aqueous flow rate changes the effect of ganglion size on dissolution rate, and the curve in Fig. 8 is observed. 3.2 Dissolution induced instability As dissolution occurs ganglion volume is reduced and NAPL/water interfaces retreat. Stability considerations indicate that ganglion pressure will change to accommodate changes in meniscus position. When a ganglion's internal pressure decreases to the critical pressure at which an interface will undergo an irreversible relocation, the ganglion becomes unstable. Depending on the type of the unstable interface, the ganglion may retreat, be invaded or have snap-off in a throat. In the range of Capillary number 10"7 < Ca < 10"4, dissolution-induced instabilities are mainly controlled by porous medium characteristics. It is found that the majority of the instability occurrences after dissolution are snap-off, with only a few occurrences of tail-retreat (Fig. 5). The volume loss and dissolution time prior to instability are recorded in Fig. 9 and 10, respectively. The fraction of volume loss vs. ganglion size shows a descending trend, with no effect of flow rate within the range 10 7 (S, )*/($,) (8) (=1 In Eq. 8, the dissolution rate for regression is based on pure water inflow, which does not hold for an arbitrary ganglion in the middle of the column. Thus, the dissolution rate needs to be adjusted to account for the inflow NAPL concentration. The column is discretized into m small sections. For each section, the outlet concentration is c„ = c„+-£-( l-£=-) (9) mQ C, The number of sections is increased until the effluent NAPL concentration at the end of the column approaches a constant, which is the "true" NAPL effluent concentration C(L). The steady state dissolution process in a column is described by q,^-=Ka{C, -C) (10) ax With the boundary condition C(0)=0, effluent concentration is C(L) = C, (1 - exp(- KaL/qw )) (11) Solving for Ka yields Ka=4^ ||nfl-£kl I f J , C , , (12) Thus, the effluent concentration C(L) from discretization can be used in Eq. 12 to obtain Ka, and the modified Sherwood number Sh ' is also acquired assuming that the characteristic length, /c, is equal to one unit length minus the mean diameter of the pore. The relationship between Sh ' and Pe can be derived using the above population analysis (Fig. 11). Three curves from the simulation are shown: stagnant layer diffusion only, channel flow dissolution only, and the two mechanisms combined. The combined 58 curve is very close to the results of Dillard (2000) and Powers et al. (1992). Since Pe is the ratio of convection to diffusion, when Pe» I, convection is dominant, and when Pe « 1, diffusion is dominant. The curve for Pe » 1 appears to be log-linear with a similar slope as the ones from Dillard's and Power's; but the curve transitions to a larger slope at 1 < Pe< 10. Figure 12 shows the relationship between Sh' and Re. Again, the simulation result from this study is within the range given by previous work (Miller et al., 1990; Parker et al., 1990; Powers et al., 1992; Guamaccia et al., 1992). The curve also shows a smaller slope for high Re (Re » 0.01) and a larger slope for low Re (Re « 0.01). Therefore, while the dissolution process in this study is based on individual ganglion, the population analysis still leads to results similar to those from various column experiments. 4.Conclusions A pore-scale network model was developed to study the dissolution of residual NAPL blobs and the instabilities induced by dissolution. Residual ganglia with pre-determined sizes were introduced into the flow field and the instantaneous instability occurrences were recorded. Over the range of Capillary numbers investigated in this study (107 2 R-b\ + &bl ( I ) ,V2'VÏ' 2a r where Rbi and Rbi are the radii of the two adjacent pore bodies. Under laminar flow conditions, the conductance of a biconical throat can be theoretically defined (Reeves and Celia, 1996), and pressures at each node (pore body) can then be found based on mass balance equations using an iterative successive over-relaxation algorithm. A stable interface at equilibrium satisfies the Young-Laplace equation (2) Kl where Pc is the stable capillary pressure, P„ and Pw are the pressures of nonaqueous and aqueous phases, 99%). It is also found that high aspect ratio increases initial instability by approximately 1.6 times, that is, ganglia are more likely to be unstable right from the outset. The initial instabilities in this case are mostly retreat or snap-off, with initial advance being very rare. The final regression model (Table 4) indicates that the frequency of initial tail retreat at high aspect ratio is affected by the main effects of ap, ganglion size, and correlation length as well as two-way interactions between these factors. The model of initial retreat at high aspect ratio gives similar trends of effects as those given by the model at low aspect ratio. With other factors being held at their low levels, the odds of having initial retreat increase about 6.7 (« e1'9) times with increasing ap, and this effect is more pronounced for a large 94 ganglion in a porous medium with a large correlation length. The effect of correlation is negative (odds ratio = 0.34) for a small ganglion in a porous medium with low crp; however, this negative effect is reduced with increasing ganglion size or r>r>]. where y, is the mean of the 300 observations under treatment i, and the coding of treatment i denotes the levels of each factor treatment i has (1 at low level and 2 at high level). The effects are defined as follows. The intercept is /, the main effect of factor A is a, the two-way interaction of factor and B is ab, the three-way interaction of A, B, and C is abc, and the six-way interaction is abcdef. The 1x64 coefficient vector 9 is then defined as &'=[I a .../ ab ...ef ...abcdef\. For a treatment i, the coefficient of a factor at its low level is I, else -I. and the coefficient for an effect is the product of those individual coefficients, with intercept being 1 always. For example, for treatment 112122, the 64 coefficients are as follows. The coefficient of intercept is I,a— l,c 1, ab 1(= -1*-1), ac -1 (=-1*1), and abcdef-\. Then, the 64x64 model matrix M is defined; each row of which corresponds to a treatment, and contains the 64 coefficients defined as before. The model Y = M0+ & where fis the independent normal distributed error vector, can be solved as: Ô = 2~6 M Y. The variance of any individual term of 0 is â2/N, where a2 = s2p is the pooled variance and JV(= 26*300) is the total number of observation. The t- Q statistic for an individual term, 0;, is t = . ' . This f-statistic can be compared to /(0.025, V47ïv 2**299) to test the significance of the effect. 115 2. Logistic regression model Logistic regression model is a type of generalized linear models, used to find the association between the response probability and a set of explanatory variables. The logistic model used in this research is: logit(^) = logf = XI, A*, P ) where x, is the i-th term of the explanatory variables (with value 0 or 1), and /?, is the regression coefficient. The logit transformation maps the probability interval (0,1) onto the real value interval (-<», co). The coefficient of a main effect x, can be interpreted as the log of the odds ratio when x, changes from 0 to I and other explanatory variables are held at 0: A = l°g| - log z*0 c f \ p \ u - P ) .r-l z X P V l l - P ) .t*0 J = log {odds ratio) 116 Chapter 5. CONCLUSIONS 5.1 Summary Nonaqueous phase liquids (NAPLs) in underground aquifer pose a long lasting contamination problem. Many factors affect the transport and fate of NAPLs in water- saturated porous media. Understanding NAPL behavior requires knowledge of the physical and chemical processes taking place at the NAPL/water interfaces, at the pore scale. A three- dimensional pore-scale network model was therefore developed to investigate ganglion dissolution and instability. The proposed network model uses biconical shaped pore throats to accommodate stable interfaces everywhere within a throat. After the porous medium is decorated and the water flow field established, residual ganglia with predetermined sizes are introduced into the medium and tested for stability. Once a stable ganglion is found, dissolution takes place until dissolution-induced instability occurs. Monte Carlo simulations are conducted to obtain data with statistical meaning. Two dissolution mechanisms were investigated: stagnant layer diffusion and channel flow dissolution. The former mechanism describes the slow diffusive transfer of NAPL from interfaces to the flowing aqueous phase due to a concentration gradient across an immobile water region, while the latter describes the faster convective transfer of NAPL to surrounding mobile water. Channels of mobile water penetrating the ganglion area were introduced into the model in order to capture the second mechanism. Several variables describing the residual ganglion configuration, dissolution and instability are generated from the simulation. It is found that the stable ganglion is slightly ramified, with higher specific interfacial area than a sphere. Furthermore, within the range of Capillary numbers being investigated 117 (10"7 ~ 10"4), stable ganglia are slightly elongated in the direction of the hydraulic pressure gradient. Analysis of dissolution rates reveals that both Capillary number and channel volume fraction have strong effects on the channel flow dissolution rates: larger channel volume fraction and higher Capillary number lead to higher channel flow dissolution rates. However, this effect is not as strong when Ca > 10"5 and channel volume fraction >0.1, because at this point the effluent concentration of NAPL after passing through a ganglion is no longer at the solubility limit due to the high flow rate. The effect of Capillary number on the diffusion dissolution rate is not noticeable until Ca < 10 *. This is attributed to dissolution being diffusion limited when aqueous flow rates are not sufficient to maintain low NAPL concentrations in the water surrounding the ganglion. The big picture is that, with increasing Capillary number, dissolution switches from stagnant layer diffusion dominant to channel flow dissolution dominant. Simulation results also show that normalized dissolution volume decreases with increasing ganglion size. Analysis of ganglion instabilities reveals that the Capillary number does not affect the type of instability, but ganglion size does. Large ganglia initially more stable: both initial tail-retreat and snap-off occurrences decrease with increasing ganglion sizes. This ganglion size effect is due to how small ganglia sample the medium in an unbiased manner, but large ganglia are biased as to which pores they grow into. This indicates the importance of porous medium characteristics on ganglion instability. A population analysis conducted on results from the single ganglion numerical experiments compares simulation data with the results from column studies in the literature. Using percolation theory, the ganglion size distribution was obtained, and relationships between the commonly used dimensionless groups (modified Sherwood number, Peclet 118 number, and Reynolds number) were generated. Relationships from the population analysis were in good agreement with relationships from various published experiments. The importance of porous medium characteristics on ganglion dissolution and instability suggested further investigation. A six-factor two-level full factorial design was used to conduct a simulation experiment to examine the importance of pore body mean, body size standard deviation, aspect ratio, correlation length, Capillary number and ganglion size. Statistical analysis enabled us to limit the significant effects to main effects and several two- way interactions only. It is found that correlation length did not affect stable ganglion configuration or dissolution, and that Capillary number also did not affect ganglion configuration, but had a strong positive effect on the dissolution flux. Aspect ratio is the most important parameter on normalized dissolution volume due to its destabilizing effect. Pore size and ganglion size all pose negative effect on the specific interfacial area. The most important factor on instability is the aspect ratio: snap-off is suppressed when aspect ratio is 1.7, and snap-off dominates when aspect ratio is 3. Initial retreat is favored in a more heterogeneous porous medium (high pore size standard deviation and no correlation). It is also found that the likelihood of initial retreat and initial snap-off decreases with increasing ganglion size, but the likelihood of initial advance increases with increasing ganglion size. A further set of simulations focused on the influence of aspect ratio was conducted. With increasing aspect ratio, initial advance decreases but both initial snap-off and initial retreat increases. It is found that a ganglion has the minimum initial instability occurrences when aspect ratio is around 2.2. At low aspect ratio, the dissolution-induced instability is mainly retreat than breakup. The effect of aspect ratio on instability is subjected on the 119 levels of body size standard deviation and correlation length. The most heterogeneous porous medium leads to the most initial instability. The work completed thus far provides a good simulator of ganglion dissolution and instability. This work also demonstrated that sensitivity analysis could be performed on this simulator to acquire valuable information and draw reasonable conclusions concerning the interactions between multiple variables, such as the porous medium characteristics, fluid and flow properties and the fate of the ganglia. The work clearly demonstrates that dissolution does induce instabilities, and this process is complex and affected by many factors. It also shows that detailed knowledge of the porous medium characteristics is needed to properly understand the NAPL dissolution and movement processes and so to improve remediation techniques. 5.2 Recommendations Recommendations resulting from this work address several areas including the model itself, model applications, and model validation. The objective of this work was to investigate the possibility of dissolution-induced instability of residual NAPL ganglia. Pursuant to this objective, the three-dimensional pore scale model used in this research attempted to capture those features of a real porous medium, which are related to the configuration of NAPL/water interfaces. However, the model we initially developed did not take water channels and channel flow dissolution into account. Although conceptual water channels were later introduced into the model, the model would be more robust if the water channels were a built-in feature of the model's geometry, as for example, using pore throats with a square or triangular cross-section. 120 The work has shown that porous medium properties play a significant role in NAPL/water transport processes; however, the parameters used in this model are typical of a consolidated sandstone, a very specific porous medium. Given detailed knowledge of a particular porous medium, this model can be tuned to simulate a real system and analysis can be more porous medium-specific. The model output can then be used to parameterize a larger-scale continuum model. The model developed in this work lays a foundation for future work. Chapter 4 discussed the effects of some specific porous medium characteristics on ganglion dissolution and instabilities. Other medium characteristics may also impact the dissolution rate and instability types. For example, contact between NAPL and the rock surface over time can change the solid surface wettability. In fact, the rocks in an oil reservoir oftentimes change from water-wet to oil-wet. This change in wettability can significantly alter the behaviors of the two interacting liquid phases in the porous medium. Since the network model keeps track of local contact angle, with slight modification this model could be used to investigate NAPL dissolution and instability in mixed-wettability porous media. A practical application related to wettability is surfactant-based remediation. By changing the NAPL solubility and solid contact angle over time, this model could be used to mimic the NAPL drainage-water flushing-surfactant remediation process. Up to this point, the model has been used to investigate stranded ganglia behavior only. Stranded ganglia are only a small part of the whole NAPL migration and dissolution processes. Ganglia are stranded as a result of NAPL migration through the porous medium, and the dissolution of stranded ganglia can cause a string of events, including instability (breakup and migration downstream), small ganglia coalescence, small NAPL blobs flushing 121 out the system, and NAPL cleanup through dissolution (pump-and-treat). Since the NAPL transport history affects the formation of stranded ganglia and thus the subsequent dissolution, and since dissolution can affect the stability and fate of the ganglia, it is recommended that both the upstream and downstream processes be incorporated into this model in order to present a whole picture of the fate of NAPL after entering the underground aquifer. The author is greatly concerned about model validation. This model emphasizes residual ganglia dissolution and instabilities, which have not been systematically investigated either experimentally or numerically. Validation of this model through comparison with results from the literature is therefore both indirect and incomplete. Due to this limitation, the analysis in this work emphasizes sensitivity for a general case, rather than the simulation of a real system. In order to have a good validation of the model, single ganglion experiments in a porous medium with known characteristics would be very helpful. Any further investigation using this model should follow an accompanying experimental setup, and use experimental data to calibrate and validate the model. 122 ACKNOWLEDGMENTS First and foremost, I thank Dr. Robert Ewing for being my mentor. Without his advice in both my academic and personal life, I would not have been able to conquer the difficulties I faced in the last five years or to finish this program. I would also like to thank my major professor Dr. Rodney Fox for taking on this project and for helping me meet the requirements for this project. To all committee members, I thank you for your services and support through this project. I acknowledge Mirrya Fontenot, my officemate and former research groupmate, for her encouragement and many insightful discussions. I would also like to acknowledge my friends and family for their support and happy times, which I will cherish. Finally, I am deeply grateful to my fiancé, Brian, for his love, patience, encouragement and support. You make me realize that sharing the joy and difficulties with you in life is the best gift I have ever received.

Simulation results for NAPL dissolution

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Simulation results for NAPL dissolution

Abstract

Extracted data

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