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4 Transport INTRODUCTION Ground water contamination occurs when chemicals are detected where they are not expected and not desired. This situation is a result of movement of chemicals in the subsurface from some source (per- haps unknown) that may be located some distance away. Ground water contamination problems are typically advection dominated (see "Dissolved Contaminant Transport" in Chapter 2), and the pri- mary concerns in defining and treating ground water contamination problems must initially focus on physical transport processes. If a contaminant is chemically or biologically reactive, then its migration tends to be attenuated in relation to the movement of a nonreactive chemical. The considerations of reaction add another order of mag- nitude of complexity to the analysis of a contamination problem, in terms of both understanding and modeling. Regardless of the reac- tivity of a chemical, a basic key to understanding and predicting its movement lies in an accurate definition of the rates and direction of ground water flow. The purpose of a mode} that simulates solute transport in ground water is to compute the concentration of a dissolved chemical species in an aquifer at any specified place and time. Numerical solute trans- port models were first developed about 20 years ago. However, the modeling technology did not have a long time to evolve before a 113
114 GROUND WATER MODELS great demand arose for its application to practical and complex field problems. Therefore the state of the science has advanced from the- ory to practice in such a short time (considering the relatively small number of scientists working on this problem at that time) that a large base of experience and hypothesis testing has not accumulated. It appears that some practitioners have assumed that the underly- ing theory and numerical methods are further beyond the research, development, and testing stage than they actually are. Most transport models include reaction terms that are math- ematically simple, such as decay or retardation factors. However, these do not necessarily represent the true complexities of many reactions. In reality, reaction processes may be neither linear nor equilibrium controlled. Rubin (1983) discusses and classifies the chemical nature of reactions and their relation to the mathematical problem formulation. Difficult numerical problems arise when reaction processes are highly nonlinear, or when the concentration of the solute of interest is strongly dependent on the concentration of numerous other chemical constituents. However, for field problems in which reactions sig- nificantly affect solute concentrations, simulation accuracy may be limited less by mathematical constraints than by data constraints. That is, the types and rates of reactions for the specific solutes and minerals in the particular ground water system of interest are rarely known and require an extensive amount of data to assess accurately. Mineralogic variability may be very significant and may affect the rate of reactions, and yet be essentially unknown. There are very few documented cases for which deterministic solute transport mod- els have been applied successfully to ground water contamination problems involving complex chemical reactions. Many contaminants of concern, particularly organic chemicals, are either immiscible or partly miscible with water. In such cases, processes in addition to those affecting a dissolved chemical may significantly affect the fate and movement of the contaminant, and the conventional solute transport equation may not be applicable. Rather, a multiphase modeling approach may be required to rep- resent phase composition, interphase mass transfer, and capillary forces, among other factors (see Pinder and Abriola, 1986~. This would concurrently impose more severe data requirements to de- scribe additional parameters, nonlinear processes, and more complex geochemical and biological reactions. Faust (1985) states, "Unfortu- nately, data such as relative permeabilities and capillary pressures
TRANSPORT 115 for the types of fluids and porous materials present in hazardous waste sites are not readily available." Well-documented and efficient multiphase models applicable to contamination of ground water by immiscible and partly miscible organic chemicals are not yet gener- ally available. TRANSPORT OF CONSERVATIVE SOLUTES Much of the recently published research literature on solute transport has focused on the nature of dispersion phenomena in ground water systems and whether the conventional solute transport equation accurately and adequately represents the process causing changes in concentration in an aquifer. In discussing the development and derivation of the solute transport equation, Bear (1979, p. 232) states, "As a working hypothesis, we shall assume that the dispersive flux can be expressed as a Fickian type law." The dispersion process is thereby represented as one in which the concentration gradient is the driving force for the dispersive flux. This is a practical engineer- ing approximation for the dispersion process that proves adequate for some field problems. But, because it incorrectly represents the actual physical processes causing observed dispersion at the scale of many field problems, which is commonly called macrodispersion, it is inadequate for many other situations. The dispersion coefficient is considered to be a function both of the intrinsic properties of the aquifer (such as heterogeneity in hydraulic conductivity and porosity) and of the fluid flow (as rep- resented by the velocity). Scheidegger (1961) showed that the dis- persivity of a homogeneous, isotropic porous medium can be defined by two constants. These are the longitudinal dispersivity and the transverse dispersivity of the medium. Most applications of trans- port models to ground water contamination problems documented to date have been based on this conventional formulation, even when the porous medium is considered to be anisotropic with respect to flow. The consideration of solute transport in a porous medium that is anisotropic would require the estimation of more than two disper- sivity parameters. For example, in a transversely isotropic medium, as might occur in a horizontally layered sedimentary sequence, the dispersion coefficient would have to be characterized on the basis of six constants. In practice, it is rare that field values for even the two constants longitudinal and transverse dispersivity can be
116 GROUND WATER MODELS defined uniquely. It appears to be impractical to measure as many as six constants in the field. If just single values of longitudinal and transverse dispersivity are used in predicting solute transport in an anisotropic medium when the flow direction ~ not always parallel to the principal directions of anmotropy, then dispersive fluxes will be either overestimated or underestimated for various parts of the flow system. This can sometimes lead to significant errors in predicted concentrations. Dispersion and advection are actually interrelated and are de- pendent on the scale of measurement and observation and on the scale of the model. Because dispersion is related to the variance of velocity, neglecting or ignoring the true velocity distribution must be compensated for in a mode! by a correspondingly higher value of dispersivity. Domenico and Robbins (1984) demonstrate that a scaling up of dispersivity will occur whenever an (n-l) dimensional mode} is calibrated or used to describe an e-dimensional system. Davis (1986) used numerical experiments to show that variations in hydraulic conductivity can cause an apparently large dispersion to occur even when relatively small values of dispersivity are assumed. Similarly, Goode and Konikow (1988) show that representing a tran- sient flow field by a mean steacly-state flow field, as is commonly done, inherently ignores some of the variability in velocity ancI must also be compensated for by increased values of dispersivity. The scale dependence of dispersivity coefficients (macrodisper- sion) is recognized as a limitation in the application of conventional solute transport models to field problems. Anderson (1984) and Gelhar (1986) show that most reported values of longitudinal dis- persivity fall in a range between 0.01 and 1.0 on the scale of the measurement (see Figure 4.~. Smith and Schwartz (1980) conclude that macrodispersion results from large-scale spatial variations in hydraulic conductivity and that the use of relatively large values of dispersivity with uniform hydraulic conductivity fields is an inapt propriate basis for describing transport in geologic systems. It must be recognized that geologic systems, by their very nature, are com- plex, three-dimensional, heterogeneous, and often anisotropic. The greater the degree to which a mode! approximates the true hetero- geneity as being uniform or homogeneous, the more must the true variability in velocity be incorporated into larger dispersion coeffi- cients. We will never have so much hydrogeologic data available that we can uniquely define all the variability in the hydraulic properties of a geologic system; therefore, assumptions and approximations are
TRANSPORT 117 Sand, Gravel, Sandstone Limestone, Basalt · Granite, & Schist 100 Oh llJ In 10 z At o 1 . I` · / ~ / / / · ·~. By a/ · ~/ ~- . A_ ·, / in)/ · / · / . / - . 1 0 1 00 1 ,000 DISTANCE (m) FIGURE 4.1 Variation of dispersivity with distance (or scale of measurement). SOURCE: Modified from Anderson, 1984. always necessary. Clearly, the more accurately and precisely we can define spatial and temporal variations in velocity, the lower will be the apparent magnitude of dispersivity. The role of heterogeneities is not easy to quantify, and much research is in progress on this problem. An extreme but common example of heterogeneity is rocks that
118 GROUND WATER MODELS exhibit a dominant secondary permeability, such as fractures or so- lution openings. In these types of materials, the secondary perme- ability channels may be orders of magnitude more transmissive than the porous matrix of the bulk of the rock unit. In these settings, the most difficult problems are identifying where the fractures or so- lution openings are located, how they are interconnected, and what their hydraulic properties are. These factors must be known in or- der to predict flow, and the flow must be calculated or identified in order to predict transport. Anderson (1984) indicates that where transport occurs through fractured rocks, diffusion of contaminants from fractures to the porous rock matrix can serve as a significant retardation mechanism, as illustrated in Figure 4.2. Modeling of flow and transport through fractured rocks is an area of active research, but not an area where practical and reliable approaches are readily available. Modeling the transport of contaminants in a secondary permeability terrain is like predicting the path of a hurricane with- out any knowledge of where land masses and oceans are located or which way the earth is rotating. Because there is not yet a consensus on how to describe, account for, or predict scale-dependent dispersion, it is important that any conventional solute transport mode} be applied to only one scale of a problem. That is, a single model, based on a single value of dispersivity, should not be used to predict both near-field (near the solute source) and far-field responses. For example, if the clispersivity value that is used in the mode} is representative of transport over distances on the order of hundreds of feet, it likely will not accurately predict dispersive transport on smaller scales of tens of feet or over D I f FU POROU S ROCK MATRIX FRACTURE F LOW FIGURE 4.2 Flow through fractures and diffusion of contaminants from frac- tures into the rock matrix of a dual-porosity medium. SOURCE: Anderson, 1984.
TRANSPORT C nJection well Land surface A C {B+D 1- Flow ~ B 1 o 119 Fully penetrating observation well /A _ ~ - `. Point', samples C ~ C: D:Br FIGURE 4.3 Effect of sampling scale on estimation of dispersivity. SOURCE: L. F. Konikow, U.S. Geological Survey, Ralston, Va., written communication, 1989. larger scales of miles. Warning flags must be raised if measurements of parameters such as dispersivity are made or are representative of some scale that is different from that required by the mode! or by the solution to the problem of interest. Similarly, the sampling scale and manner of sampling or measur- ing dependent variables, such as solute concentration, may affect the interpretation of the data and the estimated values of physical pa- rameters. For example, Figure 4.3 illustrates a case in which a tracer or contaminant is injected into a confined and stratified aquifer sys- tem. It is assumed that the properties are uniform within each layer but that the properties of each layer differ significantly. Hence, for injection into a fully penetrating injection well, as shown at the left of Figure 4.3, the velocity will differ between the different layers. Arrival times will then vary at the sampling location. Samples col- lected from a fully penetrating observation well will yield a gentle breakthrough curve indicating a relatively high dispersivity. How- ever, breakthrough curves from point samples will be relatively steep, indicating low dispersivity in each layer. The finer scale of sampling
120 GROUND WATER MODELS yields a more accurate conceptual mode} of what is really happening, and an analogous mode! should yield more reliable predictions. Because advective transport and hydrodynamic dispersion de- pend on the velocity of ground water flow, the mathematical simu- lation mode! must solve at least two simultaneous partial differential equations. One is the flow equation, from which velocities are calcu- lated, and the other ~ the solute transport equation, which describes the chemical concentration in ground water. If the range in concen- tration throughout the system is small enough that the density and viscosity of the water do not change significantly, then the two equa- tions can be decoupled (or solved separately). Otherwise, the flow equation must be formulated and solved in terms of intrinsic per- meability and fluid pressure rather than hydraulic conductivity and head, and iteration between the solutions to the flow and transport equations may be needed. Ground water transport equations, in general, are more diffi- cult to solve numerically than are the ground water flow equations, largely because the mathematical properties of the transport equa- tion vary depending upon which terms in the equations are dominant in a particular situation (Konikow and Mercer, 1988~. The transport equation has been characterized as "schizophrenic" in nature (Pin- der and Shapiro, 1979~. If the problem is advection dominated, as it is in most cases of ground water contamination, then the govern- ing partial differential equation becomes more hyperbolic in nature (similar to equations describing the propagation of a shock front or wave propagation). If ground] water velocities are relatively low, then changes in concentration for that particular problem may re- sult primarily from diffusion and dispersion processes. In such a case, the governing partial differential equation is more parabolic in nature. Standard finite-difference and finite-element methods work best with parabolic and elliptic partial differential equations (such as the transient and steady-state ground water flow equations). Other approaches (including method of characteristics, random walk, and related particle-tracking methods) are best for solving hyperbolic equations. Therefore no one numerical method or simulation mode! will be ideal for the entire spectrum of ground water contamination problems encountered in the field. Mode! users must take care to use the mode} most appropriate to their problem. Further compounding this clifficulty is the fact that the ground water flow velocity within a given multidimensional flow field will normally vary greatly, from near zero in low-permeability zones or
TRANSPORT 121 near stagnation points, to several feet per day in high-permeability areas or near recharge or discharge points. Therefore, even for a single ground water system, the mathematical characteristics of the transport process may vary between hyperbolic and parabolic, so that no one numerical method may be optimal over the entire domain of a single problem. A comprehensive review of solute transport modeling is pre- sented by Naymik (1987~. The mode! survey of van der Heij~e et al. (1985) reviews a total of 84 numerical mass transport models. Currently, there is much research on mixed or adaptive methods that aim to minimize numerical errors and combine the best features of alternative standard numerical approaches because none of the standard numerical methods is ideal over a wide range of transport problems. In the development of a deterministic ground water transport mode! for a specific area and purpose, an appropriate level of com- plexity (or, rather, simplicity) must be selected (Konikow, 1988~. Finer resolution in a mode} should yield greater accuracy. However, there also exists the practical constraint that even when appropriate data are available, a finely subdivided three-dimensional numerical transport mode] may be too large or too expensive to run on available computers. This may also be true if the mode! incorporates nonlinear processes related to reactions or multiphase transport. The selection of the appropriate mode} and the appropriate level of complexity will remain subjective and dependent on the judgment and experience of the analysts, the objectives of the study, and the level of prior information on the system of interest. In general, it is more difficult to calibrate a solute transport mode! of an aquifer than it is to calibrate a ground water flow model. Fewer parameters need to be defined to compute the head distribu- tion with a flow mode} than are required to compute concentration changes using a solute transport model. A mode} of ground water flow is often calibrated before a solute transport mode} is developed because the ground water seepage velocity is determined by the head distribution and because advective transport ~ a function of the seepage velocity. In fact, in a field environment, perhaps the single most important key to understanding a solute transport problem is the development of an accurate definition (or model) of the flow system. This is particularly relevant to transport in fractured rocks where simulation is based on porous-media concepts. Although the
122 GROUND WATER MODELS head distribution can often be reproduced satisfactorily, the required velocity field may be greatly in error. It is often feasible to use a ground water flow mode} alone to analyze directions of flow and transport, as well as travel times, because contaminant transport in ground water ~ so strongly (if not predominantly) dependent on ground water flow. An illustrative example is the analysis of the Love Canal area, Niagara Fails, New York, described by Mercer et al. (1983~. Faced with inadequate and uncertain data to describe the system, Monte CarIo simulation and uncertainty analysis were used to estimate a range of travel times (and the associated probabilities) from the contaminant source area to the Niagara River. Similarly, it is possible and often useful to couple a particle-tracking routine to a flow mode! to represent advective forces in an aquifer and to demonstrate explicitly the travel paths and travel times of representative parcels of ground water. This ignores the effects of dispersion and reactions but may nevertheless lead to an improved understanding of the spreading of contaminants. Figure 4.4 illustrates in a general manner the role of models in providing input to the analysis of ground water contamination problems. The value of the modeling approach lies in its capability to integrate site-specific data with equations describing the relevant processes as a basis for predicting changes or responses in ground water quality. There is a major difference between evaluating existing contaminate sites and evaluating new or planned sites. For the former, if the contaminant source can be reasonably well defined, the history of contamination itself can, in effect, serve as a surrogate long-term tracer test that provides critical information on velocity and Aspersion at a regional scale. However, it is more common that when a contamination problem is recognized and investigated, the locations, tinning, and strengths of the contaminant sources are for the most part unknown, because the release to the ground water system occurred in the past when there may have been no monitoring. In such cases it is often desirable to use a mode} to determine the characteristics of the source on the basis of the present distribution of contaminants. That is, the requirement is to run the mode! backward in time to assess where the contaminants came from. Although this is theoretically possible, in practice there is usually so much uncertainty in the definition of the properties and boundaries of the ground water system that an unknown source cannot be uniquely identified. At new or planned sites, historical data are commonly not available to provide a basis for mode! calibration and to serve as a control
TRANSPORT P NN NO FOR FUTURE ASSESSMENTS OF EXISTING WASTE DISPOSAL 1 | CONTAM NATED S TES. · SITE SELECTION · SOURCES OF CONTAMINATION · OPERATIONAL DESIGN · FUTURE CONTAMINATION · MONITORING NETWORK · MANAGEMENT OPTIONS ~1- ~ 1 1 L 123 | MODEL PREDICTIONS| rid - | MODEL APPLICATION AND CALIBRATION ~ 1 NUMERICAL MODEL' I OF GROUND WATER 1 FLOW AND CONTAMINANT TRANSPORT I I COLLECTION AND INTERPRETATION OF SITE-SPECIFIC DATA CONCEPTUAL MODELS OF GOVERNING PHYSICAL, CHEMICAL, AND BIOLOGICAL PROCESSES ~1 FIGURE 4.4 Overview of the role of simulation models in evaluating ground water contamination problems. SOURCE: Konikow, 1981. On the accuracy of predictions. As indicated in Figure 4.4, there should be allowances for feedback from the stage of interpreting mode! output both to the data collection and analysis phase and to the conceptualization and mathematical definition of the relevant governing processes. NONCONSERVATIVE SOLUTES The following sections assess the state of the art for modeling abiotic transformations, transfers between phases, and biological pro- cesses in the subsurface. Descriptions of all of these processes are provided in Chapter 2. The focus for this assessment is an examina- tion of what reactions are important and to what extent they can be described by equilibrium and kinetic models.
124 GRO AND WATER MODELS Equilibrium and Kinetic Models of Reactions Much of the discussion in this section refers to inorganic species. Generally speaking, reactions can be described from an equilibrium and/or kinetic viewpoint. As an example of an equilibrium descrip- tion, consider the following reversible reaction: A + B = 2C. (4.1) At equilibrium, the reaction is described by the following mass law: K = ~ ~ ~tB' At equilibrium, (4.2) where K = the equilibrium constant, which is temperature depen- dent A,B = reactant species C = product species -activity, a thermodynamic property that is proportional to the aqueous-phase concentration for dissolved species and to the partial pressure for a gas This equation implies that at equilibrium the activities of the reactants and products should be related in the relative proportions indicated by (4.2~. When this relationship is not achieved, mass is transferred forward in the reverse reaction. Another way of looking at a reaction involves using a kinetic approach. Unlike the equilibrium description, the kinetic approach describes how the concentration of a constituent changes with time. Kinetics usually are expressed by a rate law of the form rA =-kV [Ai~ tB]Y, where (4.3) rA = rate of mass accumulation of species A (MAT- ~ ~ Y = volume of water in the system being modeled (~3) k = a rate coefficient that depends on the mechanisms of the reaction, the temperature, and other environmental con- ditions and has units giving MAT-i for FA x, y = reaction-order exponents
TRANSPORT 125 Expressions like ?.A can be employed directly as source-sink terms in the mass balance equations used in a solute transport model. Whether one adopts an equilibrium or a kinetic mode] clearly depends on the character of the reaction. For example, irreversible reactions cannot be described using the equilibrium concept because they continue until all of the reactions are used up. Another factor determining how a reaction can be discussed is the rate of the reaction relative to the physical transport process. For example, when a reversible reaction is fast in relation to advection and dispersion, an equilibrium description is appropriate. When the reaction is slower, a kinetic viewpoint is more appropriate. Thus the same reaction can be described in different ways depending upon the conditions of transport. Abiotic Reactions Table 4.1 summarizes the equilibrium relationships for each of the abiotic transformations and the status of the thermodynamic databases describing these reactions. Except for radioactive decay, which is an irreversible reaction and is not describable using equi- librium concepts, a simple mass law expression describes the reac- tion. further, the databases of thermodynamic parameters (i.e., En, Ka, Kgo, K'~b' and Kaa`) are relatively complete and accessible only for oxidation/recluction and acid/base processes. Therefore funda- mental knowledge must be generated to extend the thermodynamic databases. Table 4.2 provides an assessment of the kinetic relationships for the abiotic transformations. Two mechanisms, radioactive decay and acid/base processes, are well understood and have well-defined databases. The decay coefficients for all the major radionuclides have been known for some time. For acid/base processes, the reactions normally are so fast that instantaneous equilibrium can be assumed. The term for Fab in Table 4.2 reflects that the formation of A- ion is proportional to the rate of change in concentration of the sum of acidic (HA) and basic (A-) species multiplied by the fraction of the total composed by A-. For three mechanisms-dissolution, complexation, and substitu- tion/hydrolysis the means to describe the rate of reaction have been identified, but the kinetic parameters needed to quantify the rates (i.e., A, k``, kcom, kH, KOH, and KN ) are poorly defined for many of the relevant species and coalitions. Fundamental information
126 GROUND WATER MODELS TABLE 4.1 Summary and Evaluation of Thermodynamics of Abiotic Transformation Mechanisms for Aqueous Species Mechanisms/Reaction Form Status Thermodynamic Relation Codea Not applicable Radioactive decay, P ~ D + nuclear particle Oxidation/reduction. N+ + R > N + R+ E = E°+ RT in Acid/base processes, {H }{A } = K {H4} Precipitation/dissolution, C+ + A- = CA(solid) {C+}{A } = Kso Complexation, C+ + L- = CL Substitution/hydrolysis, RX + N = RN + X {R+}{N} | {R}{N } I {CL} -K {C}{L- } st {RN}{X} _ K {RX}{N} su 2 2 a``l" indicates that the thermodynamic database is well established, and "2" that the database is incomplete. NOTE: Definition of parameters: P = parent radionuclide D = daughter product of decay, R = reductant or electrophile, N+ = oxidant or nucleophile, R+ = oxidized reductant, N = reduced oxidant, HA = acid, H + = hydrogen ion, A = conjugant base of HA or anion, C + = cation, CA(solid) = precipitate, L = ligand7 CL = complex, X = leaving group, E = potential (volts), E° = standard potential (volts), Ka = acid/base dissociation constant, KSo = solubility product, Ks~ = stability constant, KSU = substitution constant, R = universal gas constant = 1.99 x 10 kcal/mole K, T = temperature (in kelvin units). needed to implement the kinetic relationships in models awaits future research. The kinetic formulations for two mechanisms, oxidation/reduc- tion and precipitation, are not firmly established. The equations pre- sented in Table 4.2 are reasonable first approaches, but considerably more research is needed before the correct forms are known for these two reactions. It is likely that the correct forms will not be univer- sally generalizable and that the correct form will vary depending on the species involved and on environmental conditions. It also is clear that many oxidation/reduction reactions are irreversible and are at
TRANSPORT TABLE 4.2 Summary and Evaluation of Kinetics of Abiotic Transformation Mechanisms 127 Status Mechanism Kinetic Expression Codea _ Radioactive decay rid = -A[P]V 1 Oxidation/reduction Fred = -Fred [N+ ] [R] V 3 Acid/base processes Instantaneous equilibrium 1 ~ Ka \\ {d([HA] + [A ])\ V rate- \[~] + [A-]J ~dt J Precipitation rp = -kpA(1 - Q/Kso)[c+]n 3 Dissolution rat= k4A(1 - Q/Kso)n 2 Complexation room = kCom[c+][L ]V 2 Substitution/hydrolysis rSub = -kT[RX] V, 2 where kT= kH[H+] + koH[OH ] + kN - a" 1 " indicates that the kinetic expression is well understood and the database of kinetic parameters is well established; "2" indicates that the kinetic expression is well understood and the database of kinetic parameters is incomplete; "3" indicates that the kinetic-expression is poorly understood and the kinetic parameters are incomplete. NOTE: Definition of parameters: rrd = rate of loss of parent isotope by radioactive decay (MT ~ ), A = decay constant (T 1), rred = rate of loss of reactants (N+ and R) due to oxidation/reduction (MT- '), heed = oxidation/reduction rate coefficient (L3M ~ T ~ ), rate = rate of formation of A ion due to acid/base reaction (MT i), rp = rate of loss of C+ due to precipitation (MT i), kp = rate coefficient for precipitation (units depend on n), n = exponent-1, A = surface area onto which solid forms (L2), A, = rate of formation of C + due to dissolution (MT ~ ), k,l = dissolution rate coefficient (ML IT I), rCOm = rate of formation of complex, Mom = complexation rate coefficient (L3M IT I), rSub = rate of loss of original electrophile (MT I), kT = total sub- stitution rate coefficient (J i), kid = acid-catalyzed rate coefficient (L3M IT I), koH = base- catalyzed rate coefficient (L3M T I), and kN = neutral rate coefficient (T I). disequilibrium in ground waters at low temperatures (Lindberg and RunnelIs, 1984~. Geochemical Models In reality, many different abiotic transformations occur simulta- neously. Equations describing all the different reactions and all the participating chemical species must be solved together because one chemical species can participate in several different reactions of the same type or of different types. Several geochemical models, such as MINTEQ (Felmy et al., 1984), PHREEQE (Parkhurst et al., 1980), GEOCHEM (Sposito and Mattigod, 1980), and WATEQF (Plummer et al., 1976), are designed to set up and solve simultaneous thermo- dynamic equations for many different reactions and species. These
128 GROUND WATER MODELS models were first used with purely inorganic chemical systems, but they also are being applied to systems with organic chemicals. The geochem~cal models begin with a thermodynamic database for the normally dominant aqueous species present at the normal pH range of waters. A computer code then poses and solves a mass balance problem, subject to the thermodynamic constraint that an equilibrium be reached for all reactions. The equations describing the thermodynamic system are posed in a matrix format in which the stoichiometric coefficients of the chemical reactions form the elements of the matrix. The use of activity coefficients, computed by the model, allows the thermodynamic and mam balance equations to be solved together. The solution proceeds by successive approximations in a stepwise manner from the measured concentrations. Successive iterations continue until ad equilibrium expressions are true (e.g., until Q = K for all reactions) and all elemental mass balances sum to the original concentrations within acceptable tolerances. The solution describes the equilibrium makeup of the water if all known reactions proceed to equilibrium. Most geochemistry models do not mode} reactions kinetically. They assume that equilibrium models apply, and therefore none of the information contained in Table 4.2 is contained in most geochem- istry modem. A notable exception is the mode} Code EQ6, which in- corporates some kinetic expressions for dissolution and precipitation of minerals (Delany et al., 1986; Wolery et al., 1988~. Because many of the reactions (particularly oxidation/reduction, precipitation, dis- solution, substitution/hydrolysis, and some complexation reactions) are slow and often cannot be modeled with instantaneous equilib- rium, the output of a geochemistry mode} provides information only on possible trends. The actual transformations that occur and the times and distances over which they occur are not specified by geo- chemical models; research in this area is badly needed for predictive modeling. Incorporation of Abiotic Transformations into Solute Transport Modeb Geochem~cal codes can be used independently of transport codes to provide estimates of species mobility. The Incorporation of only one abiotic reaction into a solute transport mode} normally does not pose extraordinary difficulties, as long as the kinetic expression and its parameters can be specified. Incorporation requires that
TRANSPORT 129 mass balances be written for all species of interest; in most cases, the species of interest include all reactants and any products of special interest. The mass balance must then be solved, subject to the flow conditions provided externally or from a coupled flow model. Except for very simple cases, such as radioactive decay in a homogeneous aquifer, solution involves a numerical technique. The most difficult aspect usually is keeping mass balances on all reacting species and maintaining electrical charge balance in the solution, while concentrations change over space and time. For the more complicated and often more realistic situation in which many transformation reactions are possible, the logical step is to link a geochemical mode! with a mass balance model. The models CHEMTRN (Miller and Benson, 1983) and TRANQL (Cederberg et al., 1985) have achieved operational linkage between chemical equilibrium calculations and modeling of transport through porous media. However, and this is very important, the geochemical models are very complex and computationally demanding to solve for only the equilibrium relationships and relatively simple kinetic expres- sions. The linking of geochemical models, in their present forms, into solute transport models is difficult because of the computing demand. Therefore most of the currently available geochemistry models may be inappropriate for solute transport modeling, even if kinetics can be included. Instead, simpler versions perhaps involv- ing only the species and reactions of known importance to the site or to specific problems being studied need to be developed. This is the approach that has been taken in the new FASTCHEM_ mode! (HostetIer et al., 1988), in which a minimal chemical database is used in the MINTED portion (Felmy et al., 1984; Peterson et al., 1987) of a package of computer codes that include linkage between equip rium geochemical modeling and hydrologic flow and transport. An- other possible approach would be to provide a general framework for equilibrium computations (e.g., some form of simultaneous-equation solver) that could be coupled to appropriate kinetic expressions to provide a flexible source-sink subroutine for chemical reactions. Before geochem~cal models can be routinely incorporated into solute transport modem, the models must be tested In controlled field studies. So far, researchers have validated only sections or portions of the geochemical codes against field or laboratory data, such as the state of saturation of the water with respect to calcium carbonate. The lack of extensive validation is not the fault of the codes; instead, it points out the surprising lack of laboratory and
130 GROUND WATER MODELS field studies that are designed for, or are suitable for, testing of theoretical models. Modelers tend to go their own way, building impressive computer codes, while experimentalmts tend to gather data for purposes other than evaluating models. The resolution of this problem must eventually come from a close interaction among modelers, experunentalists, and field scient~te. We have probably reached the point at which it ~ now imperative to gather laboratory and field data to evaluate the validity and utility of the geochemical codes. One of the unique aspects of solute transport modeling for sub- surface waters is the very large amount of solid surface area to which the water is exposed. Surfaces often behave as reactants in the types of reactions described above. In particular, functional groups on solids can act as oxidants or reductants, acids or bases, complexing ligands, and dissolution or precipitation sites. In general, the ther- modynamics and kinetics for surface reactions are similar to those for reactions in solution. However, transport of dissolved species to and from the surface often needs to be taken into account; thus the kinetics often are controlled by diffusion processes. In addition, sur- face reactions are unique because the surface reactants and products need not move with the water phase, but often remain fixed on the solid phase. Phase ~ansfere The transfer of chemical species between two different phases can be a major source-sink term. The major transfers are between the following pairs of phases: solid/liquid, liquid/liquid, liquid/gas, and solid/gas. The simplest way of ~nodeling phase transfers is as equilibrium processes (Table 4.3), in which the concentration or density in one phase is proportional to the concentration or density in the other phase. The exception to this general rule is the transfer of colloids, which normally Is not described in terms of equilibrium. While the forms for the partitioning expressions are well established, only for gas/liquid partitioning is there a relatively complete set of partition coefficients. For the other transfers, the partition parameters are not complete, because they depend on site-specific characteristics (e.g., Ksorp' Rex, Qm, and b) or have not been systematically studied yet. Therefore database expansion and the means to characterize local
TRANSPORT TABLE 4.3 Summary and Evaluation of the Thermodynamics of Phase Transfers 131 Status Phases Involved Partitioning Expressions Codea ~ . . . Solid/liquid Organic solutes Q = K~[C]N 2 · {C+ ~(C2 - X) Inorganic torts KeX = ICE )(C -X) 2 Colloids Not applicable - Liquid/liquid [C]O= K*[C] 2 Liquid/gas [C]g-H[C] 1 Sold/gas Q m g = 2Q b[C] 1 + b[C]g a`` 1 " indicates that partitioning parameters are well established and "2" that partitioning parameters are incomplete. NOTE: Definitions of parameters: Q - sorption density of the solute on and/or in the solid phase (MMSo~ i), Kit- sorption constant (units depend on N), N = sorption exponent for Preundlich isotherms, [C]-concentration of solute in liquid phase (usually water) (ML 3), Ken = ion-exchange coefficient, {C+ ~) and {C + 2) = solution activities of two exchanging cations, (Cat - X) and (C2 - X) = surface densities of two exchanging cations (My 2 or MM,r I), [C]O = concentration of species in second (usually organic) liquid phase (MLo~3), K* - phase distribution coefficient (L3Lo 3), [C]g = concentration of volatile species in gas phase (MLg3), H = Henry's constant (L3Lg 3), Em-monolayer sorption density to a solid (MLSo1 2 or MMSol i), b = Langmuir energy constant (Lg3M ~ ). sites are key needs for the successful modeling of subsurface phase transfers. Assessment of Kinetics As noted previously, phase transfers are usually modeled using equilibrium reaction models. Kinetic expressions containing a km parameter multiplied by some sort of concentration difference nor- mady are used to describe filtration of colloids but, when necessary, can be used to describe most of the other transfers as well. Ki- netic models do require additional information about the system of interest, namely, estimates or measurements of km, the interfacial surface area, and concentrations or densities In both phases. These parameters often are difficult to estimate. It is for this reason that the equilibrium models are used to mode} phase transfers, but even so the parameters needed to quantify them K&' Kerr, K*, Qm, and are difficult to estimate for field conditions. Most mass transport expressions, not including colloid filtration
132 GROUND WATER MODELS and gas/liquid transfer, are only first approximations. Considerable research on transfer mechanisms will be necessary before reliable expressions are available. Such research is necessary because the in- stantaneous equilibrium approaches are not appropriate when solute advection is significant, which occurs in highly porous media and near wells and trenches used for remediation. Incorporation of Phase Transfers into Solute Transport Models The incorporation of phase transfers into solute transport models is relatively straightforward, as long as the rate term is available. Modeling with instantaneous equilibria is especially easy, because the movement of the solute can be modeled as simple advection at a velocity that is a fraction of the liquid flow velocity. For example, a sorbing organic solute moves at velocity v': v' = v/~1 + paqK&/~, where (4.4) v = water flow velocity (LT-~) v, = velocity of movement of the center of mass of the solute (L.T-~) Paq = mass of aquifer solids per unit volume of aquifer (M~o~~3) = porosity, or volume of liquid per unit volume of aquifer t~3~-3) K,` = linear partition coefficient (~3M~o~~~) (1 + pucks/= retardation factor Similar relationships can be derived for the other transfers. Ton exchange is a special case of instantaneous equilibrium, be- cause two competing ions must be modeled in the liquid and solid phases. The task of following two aqueous species and two solid phase species increases the computational burden but has been achieved successfully. When a mass transport approach is necessary, the computations become more cumbersome because concentrations or densities in both phases must be modeled. However, as long as mass balances on species in both phases are set up and linked via the transport rates, the mass transport approach creates no special modeling difficulties. The main difficulty with implementing any of the approaches to solute transport with phase transfers is characterization of the subsurface medium in terms of partition coefficients, gas flows, and
TRANSPORT 133 nonaqueous liquid contents. Because the solid, gas, and second liquid phases do not move with the water, a solute being transported in the water can encounter many different environments for phase transfers. This spatial heterogeneity of nonaqueous phases in subsurface porous media can play an important role in determining the fate of chemical species that transfer across phase boundaries. Gathering the data to characterize the heterogeneous subsurface in terms of its nonaqueous phases is expensive and difficult. The difficulty is compounded when the nonaqueous phase is changing or moving over time, as could be the case when a gas is generated or is in multiphase liquid flow. Biological Reactions On the one hand, modeling biological reactions involves all the same considerations and approaches described for abiotic re- actions. This similarity occurs because microorganisms are cata- lysts for the transformations described under abiotic transforma- tions. Microorganisms are especially associated with the oxida- tion/reduction and substitution/hydrolysis reactions, but they also can catalyze acid/base, precipitation/dissolution, and complexation reactions. Microorganisms can catalyze chemical reactions, but they cannot cause reactions that are not thermodynamically possible; thus microorganisms affect only the rate of reactions. On the other hand, modeling biological reactions involves fea- tures that are not part of abiotic transformations. The two most crit- ical features are (1) that the microorganisms have to grow through the utilization of requ*ed substrates and (2) that most of the mi- croorganisms are attached to the solid-medium particles. The first feature means that, in modeling of the biological processes, mass balances often are needed for required substrates, even when they are not the chemical species of interest. The second feature means that the reactions by the microorganisms and the mass balance for the microorganisms must be posed in terms of a solid phase that does not move with the water. Microbiological Kinetics For any biodegradable compound in the water, its removal from the water is described by a flux from the liquid and to the microor- ganisms attached to the solids: rbio =-JA, (4.5)
134 where GROUND WATER MODELS Fbio= rate of substrate loss from the pore liquid by biological transformation (MT-~) ~ = substrate flux to the attached microorganisms (ML-2T-~) A = surface area of nucrobial biofiIm or m~crocolonies (~2) The flux (~) can be computed by simultaneous solution of two equations: one for mass transport of substrate to the surface of the biofiIm or microcolony and the other for simultaneous diffusion and utilization of the substrate within the film or colony. Relatively sim- ple techniques are available for obtaining ~7 (Rittmann and McCarty, 1981), as long as the accumulation of attached biomass is known and the transformation kinetics can be characterized, both of which are difficult to determine in a field situation. The most common kinetic expression for microbial utilization by individual cells is the Monod relation: kXaSf V Ut Ks + Sf ' where (4.6) rut = rate of substrate utilization by an individual cell (MT-~) Xa = concentration of active cells (MxL-3) Sf = concentration of rate-limiting substrate (ML-3) in contact with the cell k = maximum specific rate of substrate utilization (MM -1T-~) Ke = concentration at which the specific rate is one-half of k (MI,-3) V = volume containing cells (~3) The key parameters characterizing the kinetics of utilization of a substrate are k and Ks. A large value of k and a small value of Ks are associated with a rapid rate of biodegradation. The kinetic parameters are not well known for many of the organic chemicals that commonly pollute ground water. The Ks parameter seems to vary widely (e.g., from as low as about 1 ,ug/} to hundreds of milligrams per liter). Within the biofiIm, substrates must be transported by molecular diffusion if they are to penetrate beyond the outer surface of the biofihn or microcolony. Diffusion is described by Fick's second law,
TRANSPORT 135 Fdiff =-Df d 2fV, (4.7) where rain = rate of substrate accumulation due to diffusion (MT-~) Df = molecular diffusion coefficient of the substrate in the film or colony (LIT- ~ ~ z = distance dimension normal to the surface of the film or colony (~) The simultaneous utilization and diffusion of substrate within the film or colony are usually represented as a steady-state mass balance on Sf: d2sf dz2 Sf i+ Sf. (4.8) Because the microorganisms are attached to a surface, substrates must be transported to the surface. This external mass transport is represented in the conventional manner by J-Km(S-Se), (4.9) where S = substrate concentration in the pore liquid (ML-3) S., = substrate concentration at the interface between the liquid and the biofiIm or colony surface (ML-3) Equations (4.8) and (4.9) are the ones that must be solved si- multaneously to give ~1 (see t4.5] and Rittmann and McCarty, 1981~. Provided the amount of attached biomass is known, equation (4.5) and a solution for ~ can be employed for any type of rate-limiting substrate. The microorganisms must be grown and sustained. At a mini- mum, they must consume an electron donor and an electron acceptor; nutrients (e.g., nitrogen and phosphorus) are also needed if cells are accumulating. One of these materials is "growth rate limiting" and must be modeled if the amount of active biomass is to be described. For attached biomass, the linkage of limiting-substrate utilization can be made by solving a mass balance equation on cell mass for a limiting electron donor, often called the primary substrate (Rittmann and McCarty, 1980; Saez and Rittmann, 1988~. The mass balance (O) is given by
136 GROUND WATER MODELS 0=JY-b'XfLf, (4.10) where Y = true yield of cell mass per unit of primary substrate con- sumed (MEMO) [' = biofiIm or microcolony thickness (~) Xf = density of active cells in the biofiIm or microcolony (M2I.-3) b' = overall biomass loss rate (TV) In summary, the rate term for biodegradation is given by (4.9~. However, predicting .7 for a given compound requires knowledge of the amount of active biomass (X, If ~ and the rate parameters (k, Kit, km, Df) for that compound. Because the amount of active biomass depends on the utilization of the growth-rate-limiting substrate, that compound must be modeled. Often that limiting material is the electron donor or acceptor, but it need not be the compound of primary interest. If the contaminant of interest is not the growth-rate-limiting substrate, its utilization does not affect the cell accumulation. The kinetics for non-growth-rate-limiting substrates fall into one of two classes, a nonlimiting necessary substrate and a secondary substrate. A nonlimiting necessary substrate is an electron donor, electron acceptor, or nutrient that is required for cell growth or maintenance but is present at a concentration sufficiently high that it does not limit the overall rate of cell metabolism. The flux for such a material is proportional to '7 for the rate-limiting material times a stoichiometric ratio. A secondary substrate is an organic compound whose utilization contributes negligible energy, electrons, or carbon for cell growth or maintenance. A secondary substrate contributes negligibly toward the accumulation of cells because of its low concentration, transient presence, or inability to support any growth. The last situation is known as co-metabolism. The rate of secondary-substrate utilization is determined by its intrinsic kinetic parameters, its concentration, and the amount of active biomass, which is controlled by the long- term availability of primary substrate.
TRANSPORT 137 Incorporation of Biological Processes into Solute Transport Modem Adding a biological reaction term, in the form of equation (4.9), to the solute mass balance presents no conceptual challenge to mod- eling subsurface transport. Because algorithms are available for com- puting ~7 as a function of 5 and kinetic parameters, the biological reaction term can be treated as a pseudoconstant that is computed as needed by pseudoanalytical solutions (e.g., Rittmann and McCarty, 1981; Saez and Rittmann, 1988~. This approach, using a pseudo- constant .7, has been applied successfully many times in a research setting but is not yet common in field practice. The application of the pseudoanalytical solutions can encounter four complications and practical problems. The first occurs when the system being modeled becomes large or spatially complicated. Then, the nonlinearity of the biological reaction terms (e.g., ?.bio iS not a first-order function of 5) makes the computations very expensive if an accurate solution is to be attained. New techniques are needed for making tractable the solution of mass balance equations containing highly nonlinear reaction terms. A second practical difficulty with modeling biological systems is that several components should be modeled. At a minimum, the active biomass needs to be estimated, but that task may require modeling the fate of one or more necessary substrates. If the com- pound of interest is not one of the necessary substrates, it needs to be modeled separately. The tools to mode} all of the components are available, but they must be combined properly. Clear distinctions must be made among primary substrates, nonlimiting necessary sub- strates, and secondary substrates. Again, all the tools have been properly combined for research investigations but are not being used routinely in practice. A third complication Is that increased accumulations of biomass in an aquifer can lead to loss of permeability, or clogging. The most obvious mechanism is growth of bacteria into the pores, thereby reducing the pore area available for flow. In addition, microbial action can reduce permeability through formation of gas pockets, precipitation of solids, or increase in the viscosity of the liquids from excretion of polymers. Incorporation of clogging into solute transport modeling ~ difficult for two reasons. First, the mechanisms of clogging are not yet well enough understood to allow formulation of quantitative expressions. Second, clogging of the pores alters the flow paths; thus water flow and solute transport models must become
138 GROUND WATER MODELS interactively coupled. Much research must still be done on all aspects of the clogging phenomenon. A fourth complication involves substrates that are poorly solu- ble. Examples are organic solvents that form separate liquid phases, sorb strongly to solids, or volatilize to a gas phase. Incorporation of biodegradation into a mode} that already contains one or more transfers between phases is a very challenging problem. The main difficulties are two: (~) substrate mass balances are required in two or more phases, which intensifies computational demands, and (2) concentration gradients probably occur on a scale (e.g., micrometers to centimeters) much smaller than the mode} grid. The eject is to require a microscale in the direction normal to the phase interface. This microscale may force addition of another space dimension to the model, greatly increasing computational demands. The four complications and practical problems can be accentu- ated when in situ bioremediation strategies are to be modeled. The addition and extraction of water through wells add to the local non- homogeneities of flow velocity and solute concentration. The input of oxygen and nutrients is a hydraulics (delivery) problem that is often the limiting factor in the bioremediation strategy. The input of stimulating substrates or nutrients also induces significant and localized microbial growth, which can eject clogging. Hence, the modeling difficulties are made more intense by the localized and non- homogeneous microbial activity created by bioreciamation practices. This adds to the problems already associated with heterogeneity in the permeability distribution. TRANSPORT IN THE UNSATURATED ZONE Reactive mass transport ~ of particular interest within the un- saturated zone. In cases where we are dealing with contaminants that are released at the soil surface (e.g., application of fertilizers and pesticides, land treatment of hazardous wastes, accidental spills of wastes, leaky storage tanks, lagoons or ponds used for storing waste liquids), the unsaturated zone may be thought of as a buffer zone that offers protection to the underlying aquifer. The unsatu- rated zone thickness may vary from a few to several hundred meters; water and vapors (and the contaminants dissolved in these fluids) must travel through the unsaturated zone and arrive in sufficient quantities at the water table to be an environmental or a health
TRANSPORT 139 concern. These issues focus attention on the ability to quantify and mode} these processes in the unsaturated zone. A large number of the processes, discussed earlier, will alter the nature and quantities of the contaminants arriving at the water table as a function of the travel time within the unsaturated zone. The unsaturated zone, particularly the top ~ or 2 m, is characterized by high microbial activity, which promotes biodegradation. This zone is also high in organic matter and clay content, which promotes sorption, biological degradation, and transformation. Of particular significance are the differences in the rates and magnitudes of these processes in the unsaturated zone as compared to the saturated zone. The conventional view of mass transport in the unsaturated zone is simply that of advection moderated by both retardation and at- tenuation. Overall then, the presence of the unsaturated zone should in theory generally lead to decreased loadings of contaminants to the ground water. Vapor-phase transport, which occurs only in the unsaturated zone, can contribute to gaseous losses of the volatile contaminants. The unsaturated zone provides a buffer, either pre- venting or minimizing ground water contamination from chemicals applied at the ground surface. However, detection of a large num- ber of volatile and nonvolatile contaminants (e.g., pesticides used widely in agriculture and organic contaminants derived from spills or leaks of gasoline and industrial solvents) in both shallow and deep ground water has raised questions as to the validity of what has been called the "filter fantasy," i.e., that the unsaturated zone acts as a protective buffer . An alternative scenario is equally probable. The unsaturated zone might actually serve as a "source zone" for contamination of ground water. Pesticides and fertilizers sorbed on mineral and or- ganic constituents of the solid matrix, as well as the residual amounts of gasoline or other organic solvents entrapped within the soil pores, may in fact be released slowly over a long time period, leading to long-term loadings of contaminants into the saturated zone. Thus short-term measures to remediate ground water, for example by pump-and-treat methods, may fad! because of the long-term "bleed- ing~ of contaminants from the unsaturated zone overlying the water table. It is evident that the role of the unsaturated zone, either as a buffer or as a source, must be carefully evaluated in assessing ground water contamination and in selecting remedial measures. Coupling of simulation models developed for the unsaturated zone to those for
140 GROUND WATER MODELS the saturated zone ~ an essential element of ground water modeling and is necessary for devising appropriate management/regulatory · ~ pot .lcles. The modeling of contaminant behavior in the unsaturated zone is designed generally to answer the following three questions, listed in order of priority and increasing complexity. First, when might a contaminant arrive at a specified depth? This requires a prediction of the travel lime (fir) for the contaminant to arrive at the specified depth (Zc) of interest: for example, the bottom of the root zone, the bottom of the treatment zone at a hazardous waste land treatment (HWLT) facility, or the water table. Second, how much of the surface- applied (or spilled) contaminant might arrive at Zc? This requires an estimation of the mass loadings (M') of the contaminant beyond the depth Zc as influenced by retardation resulting from sorption and attenuation as a consequence of various biotic/abiotic trans- formations during contaminant transport through the unsaturated zone. Finally, it might also be necessary to predict the concentration distribution (Ciz,ti) of the contaminant within the unsaturated zone such that the time changes in contaminant concentrations as well as fluxes (~7c~ti) at Zc may be evaluated in addition to M'. The spatial and temporal scales at which these questions need to be addressed and the ability to provide the necessary data characterizing the un- saturated zone and the contaminant determine the complexity of the mode} used and the reliability of the predictions provided by the models. In answering the questions posed above, it is important to un- derstand the coupling between the physical processes of pow and storage, the chemical processes of retention and reaction, en c} the biological processes of degradation (complete breakdown to nontoxic products) and transformation (partial decomposition that may or may not lead to the production of toxic by-products). It is also necessary to examine the differences in the rates and magnitudes of these processes as they occur in the unsaturated zone in contrast to what happens in the saturated zone. As water infiltrates and redistributes within the unsaturated soil, various solutes dissolved in it are carried along. The advective and hydrodyna~nic transport of solutes is discussed elsewhere. Here it is sufficient to recognize that when the soil water flow is transient, solute transport is also transient. The advective velocity (v) at which a nonadsorbed solute is transported in an unsaturated soil is given by q/e, (recall that both q and ~ vary with space and time
TRANSPORT 141 during transient water flow). Thus the transport of a nonadsorbed, conservative solute (e.g., chIoride) can be described, given only the knowledge of how water flows in a soil profile. For an adsorbed or nonconservative solute, however, retardation of transport, because of sorption and attenuation owing to transformations and reactions, must also be taken into account. Because water flow in the unsaturated zone is episodic, so is contaminant transport. This feature is illustrated schematically in Figure 4.5 for vertical, downward leaching of nonsorbed and sorbed contaminants in a sandy soil as a result of rainfall over a 1-yr period. The progressive downward leaching, in a stepwise manner, of the contaminant pulse is clearly evident. Note that periods during which there is no contaminant transport (indicated by horizontal lines in Figure 4.5), even though rainfall occurs at the ground surface, are the periods when the soil-water depletion above the contaminant pulse, because of evapotranspiration losses, was not overcome by a given sequence of water input events. Further downward leaching of the contaminant (indicated by short vertical lines in Figure 4.5) can occur only when this soil-water deficit is overcome. Attenuation is defined for the present discussion as the decrease in the total amount of the contaminant present within the unsat- urated zone. Attenuation therefore includes all losses via various transformations but does not include the decrease in the contami- nant concentration (i.e., Dilutions resulting from hydrodynamic dis- persion. Near the ground surface, where microbial activity is likely to be highest, losses due to microbial degradation will be the largest. Microbial activity declines rapidly with increasing soil depth, and losses are primarily due to chemical transformations (e.g., hydrol- ysis). The residence time within this biologically active zone is of paramount importance in determining the extent of attenuation and, hence, mass loadings to ground water. For the hypothetical case pre- sensed in Figure 4.5, the attenuation of three contaminants with time as they travel through the unsaturated zone Is depicted in Figure 4.6. Note the changes in the slope of the decay curve, each change be- ing coincidental with the movement of the contaminant pulse from a soil horizon of high microbial activity (faster rate of attenuation) to a deeper horizon with lower microbial activity and, hence, slower attenuation. With this consideration, the significance of the episodic nature of contaminant transport within the unsaturated zone now becomes more apparent. The pattern of water input at the ground surface
142 4.0 0.0 o 1 50 300 o 30 - I 60 cam 90 120 150 GROUND WATER MODELS 8.0 ,1 1 1~ a L b Rooting Depth | Contaminant 1 ~ (sorbed) Contaminant 3 (nonsorbed) 1 \ ~ 1 1 1 1; 1 ~ I ~ - I Contaminant 3 I (nonsorbed) C Contaminant \ (sorbed) I Rooting Depth 0 20 40 60 80 100 EL APSED TIME (days) FIGURE 4.5 Sequential leaching of two sorbed contaminants (1,2) and one nonsorbed contaminant (3) through the root zone as a result of rainfall shown in (a). Note that contaminant 2 is sorbed to a greater extent than contaminant 1 and, as a consequence, is leached to a lesser extent.
TRANSPORT IL J CL TIC 0.1 _ o 143 Contaminant 3 ~ (nondegrading) 1 7\ , ~Contaminant 1 i\ l (slowly degrading) o z O 0.01 .001 \ Contaminant 2 (rapidly degrading) TIME (days) FIGURE 4.6 Attenuation of three contaminants during their transit through the unsaturated zone. The shifts in attenuation rates, indicated by arrows, coincide with leaching from one soil horizon to the next. Note the logarithmic scale used to show the amount of contaminant remaining in the unsaturated zone. and the soil's physical properties control the temporal and depth variations in soil water flux which, in turn, dictates the residence time of the contaminant in each depth increment. Both retardation and the rate of attenuation vary with soil depth; therefore the extent of attenuation occurring within a given zone is dependent on the residence time in that zone. The actual mass loadings of a contami- nant beyond the root zone (or to the water table) are also episodic. An example, based on simulations using an unsaturated zone mode} called PRIM, is shown in Figure 4.7 for the loadings of the nema- tocide aldicarb to shallow ground water beneath citrus groves. Note the variations in tinning and amount of daily pesticide inputs into the shallow water table. These episodes (or loading events) are con- trolled by the dynamics of unsaturated water flow in the citrus root zone. Such mode! outputs are then used as inputs to a mode} that simulates pesticide behavior in the saturated zone (see Figure 4.8~. Unsaturated Flow and Transport in Structured Soile The classical Richards equation for transient water flow and the advective-dispersive solute transport mode! may be adequate in homogeneous soils, but may not be appropriate for describing flow and transport in structured soils (van Genuchten, 1987~. Structured
144 GROUND WATER MODELS 10 1 0.1 z 0.01 LIJ C:) C) In ~L' 1 0 HI: C) 1 0.1 0.01 - a Water Table Depth - 3.0 m - b Water Table Depth = 7.3 m 50 100 150 200 250 300 350 400 DAYS AFTER APPLICATION FIGURE 4.7 (a) Episodes of pesticide loadings to shallow water table located beneath a citrus grove. (b) Daily pesticide loadings predicted by PROM and used in saturated zones simulations. SOURCE: Jones et al., 1987. soils are characterizes! by large, more or less continuous voids often referred to as macropores (Luxmoore, 1981~. Voids in porous media in which water is essentially not subjected to capilIarity (capillary potential greater than -0.! kPa) and that therefore may be wider than about 3 mm have been defined as macropores (Germane and Beven, 19813. A few examples of such macrop ores are interaggregate pores; interpedal voids; earthworm or gopher holes; decayed-root
TRANSPORT 145 channels; and drying cracks and fissures in clay soils. It must be noted that designation of macropores based on size only is still disputed (see Germann, 1989~. During an infiltration event, water and solutes can preferentially flow into and through these macropores and bypass a major portion of the soil matrix. In this regard, the conceptual problem of predict- ing macropore flow is somewhat similar to that of describing solute transport in saturated, fissured and fractured media. In both cases, rapid flow in the macropores (or fissures) is accompanied by much slower infiltration or diffusion-controlled mass transfer into the soil matrix. The major distinction is, of course, that unlike the case for fissures and fractures in aquifers, flow and transport in macropores occur only when specific conditions are satisfied. Identification of these conditions and modeling of macropore flow and transport have been the main focus of recent investigations by soil scientists and subsurface hydrologists. The occurrence of macropore flow is deter- mined by, among other factors, the antecedent soil-water conditions, hydraulic properties of the soil matrix, the rate of water input at the soil surface, and the spatial distribution (i.e., density) and in- terconnectedness of the macropore sequences. The impact of such preferential flow on solute transport Is further detertn~ned by the rate of diffusive mass transfer into the soil matrix and the sorptive properties of the macropore and matrix regions. While the impacts of preferential water flow on subsurface hy- drology have been more thoroughly investigated, only recently have efforts been initiated to investigate the influence of macropore flow on solute transport in structured soils. Beven and Germann (1982) and White (1985) have reviewed the available experimental evidence for preferential flow and bypassing. One major impact of macropore flow is that of accelerated movement of surface-applied solutes (e.g., fertil- izers, pesticides, and salts) through the vadose zone. Macropore flow is probably responsible for the frequent reports that field-measured dispersion coefficients are much larger (by an order of magnitude or more) than those measured in packed laboratory columns. Germann (1989) summarized the attempts to mode! transient flow and transport in structured soils and grouped them into three basic approaches: (1) macroscopic averaging of flow and transport based on the "mobile-immobile" zone concept; (2) flow and transport based on various routing procedures along presumed stream lines; and (3) transfer function models based on a continuous velocity dis- tribution. One of the major limitations in predicting macropore flow
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148 GROUND WATER MODELS and bypassing is perhaps not our inability to develop comprehensive mathematical models, but a limitation in characterizing the geomet- ric and hydraulic features (e.g., size, length, spatial distributions, permeability, and interconnectedness) of the macrop ores and in pro- viding the values for the required mode} parameters. In this sense, the problems of a ground water hydrologist dealing with flow and transport in fractured media are like those of a soil scientist. The added complexities faced by soil scientists are those of transient flow domain and uncertainty as to when macrop ore flow is dominant and how it should be modeled. MULTIPHASE TRANSPORT Two classes of multiphase flow and contaminant transport arise most often in ground water studies: seawater intrusion and organic fluid migration. The standard conceptual mode! of the vadose zone also includes two fluid phases air and water but this case has been treated previously. Codes that simulate the two classes of problems exist but have not been so heavily involved in water quality regula- tion or litigation as standard ground water flow and miscible solute transport codes. This is for three reasons: (~) the applications are primarily concerned with the resource (e.g., water or oil) quantity, (2) the technology is new and relatively untested, and (3) insufficient data exist to employ multiphase principles. In the following para- graphs, an outline of typical problems governing equations, necessary data, and typical codes is presented. Seawater Intrusion Freshwater supplies located adjacent to bodies of saltwater can be affected by water use strategies adopted by the public and indus- try. A typical setting for seawater intrusion is shown in Figure 4.9. Other settings can be envisioned, e.g., a confined aquifer instead of an unconfined aquifer, and saline contamination that completely un- derlies an overlying freshwater body. Clearly, the degree to which the freshwater and saltwater interface is dispersed could be important to a study. The extent to which the interface is smeared longitudinally may also be important in deciding upon the conceptual model. The proper choice of a conceptual mode! and method of analysis will be determined by the operative physics of the system and the behavior of interest in the study (doss, 1984~. If the area considered is large,
TRANSPORT V SEA 149 Accretion Fresh water ~ Ground Surface Water Table /~e ~Impervious Seawater =~ c ~Boundary "'a_ Toe FIGURE 4.9 Seawater intrusion in an unconfined aquifer. SOURCE: After Sa da Costa and Wilson, 1979. then the problem scale is such that an interface approximation is valid and saltwater and fresh water may be treated as immiscible. The analysis of aquifers containing both fresh water and salt- water may be based on a variety of conceptual models (doss, 1984~. The range of numerical models includes dispersed interface mod- els of either cross-sectional or fully three-dimensional fluid-density- dependent flow and solute transport simulation. Sharp interface models are also available for cross-sectional or areal applications. Of the sharp interface models, some account for the movement of both fresh water and saltwater, while others account only for fresh- water movement. The latter models are based on an assumption of instantaneous hydrostatic equilibrium in the saltwater environment. In the majority of cases involving seawater intrusion, water qual- ity is viewed as good or bad; either it is fresh water or it is not a resource. Therefore many studies seek to determine acceptable lev- els of pumping or appropriate remediation or protection strategies. These resource management questions are resolved through fluid flow simulation and do not require solute transport simulations. Organic FIllid Contamination The migration and fate of organic compounds in the subsurface are of significant interest because of the potential health effects of these compounds at relatively low concentrations. A significant body of work exists within the petroleum industry regarding the move- ment of organic compounds, e.g., of} and gas resources. However, this capability has been developed for estimating resource recovery
150 GROUND WATER MODELS or production and not contaminant migration. To compound prob- lems, the petroleum industry's computational capability is largely proprietary and is oriented toward deep geologic systems, which typ- ically have higher temperatures and higher pressure environments than those encountered in shallow contamination problems. Within the past decade, a considerable effort has been made to establish a capability to simulate immiscible and miscible or- ganic compound contamination of ground water resources. Migration patterns associated with immiscible and miscible organic fluids are schematically described by Schwille (1984) and Abriola (1984~. Fig- ure 4.10 depicts one possible organic liquid contamination event. If not remediated, the migration of an immiscible organic liquid phase is of interest because it could represent an acute or chronic source of pollution. Movement of the organic liquid through the vadose zone is governed by the potential of the organic liquid, which in turn depends upon the fluid retention and relative permeability properties of the air/organic/water/solid system. As an organic liquid flows through a porous medium, some Is adsorbed to the medium or trapped within the pore space. Specific retention defines that fraction of the pore space that will be occupied by organic liquid after drainage of the bulk organic liquid from the soil column. This organic contamina- tion held within the soil column by capillary forces (at its residual saturation) represents a chronic source of pollution because it can be leached by percolating soil moisture and carried to the water table. If the organic liquid is lighter than water, it may migrate as a distinct immiscible contaminant (the acute source) within the cap- ilIary fringe overlying the water table. The soluble fraction of the organic liquid will also contaminate the water table aquifer and mi- grate as a miscible phase within ground water. This is the situation shown in Figure 4.10. If the organic liquid is heavier than water, it will migrate vertically through the vadose zone and water table to directly contaminate the ground water aquifer. It may also pene- trate water-confining strata that are permeable to the organic liquid and, consequently, contaminate underlying confined aquifers. The organic contaminant may form a pool on the bedrock of the aquifer and move in a direction defined by the bedrock relief rather than by the hydraulic gradient. Contamination of ground water occurs by dissolution of the soluble fraction into ground water contacting either the main body of the contaminant or the organic liquid held by specific retention within the porous medium.
TRANSPORT Ground Surfacer Capillary Fringe Water Table 151 'my ,.::::.. i. :$...////// Oil Zone ~ l - - .' lo,!. . of ~ CUnsaturated Zone Gas Zone (evaporation envelope) ~ ", ~.~;-, Oil Core \ Diffusion Zone (soluble components) Figure 4.10 Organic liquid contamination of unsaturated and saturated porous media. SOURCE: After Abriola, 1984. Governing Equations for M~tiphase Flow The region of greatest interest in seawater intrusion problems is the front between fresh water and seawater. The problem of salinity as a miscible contaminant in ground water is addressed with standard solute transport models. In reality, seawater is miscible with fresh water, and the front between the two bodies of water Is really a transition zone. The density and salinity of water across the zone gradually vary from those of fresh water to those of seawater (Bear, 1979~. A sharp or abrupt interface Is assumed if the width of the transition zone is relatively small. Fresh water is buoyant and will float above seawater. The balance struck among fresh water (i.e., ground water) moving toward the sea, seawater contaminating the approaching fresh water by miscible displacement, and fresh water overlying seawater results in a nearly stationary saline wedge. Figure 4.9 illustrates the stationary saline wedge conceptual model. This wedge will change if influenced by pumping or changes in recharge. While one can pose and solve the seawater intrusion problem as a single fluid having variable density (e.g., Begot et al., 1975; Voss, 1984), the most common approach has been to simulate fresh water
152 GROUND WATER MODELS and seawater as distinct liquids separated by an abrupt interface. Along the interface, the pressures of both liquids must be identical. Sharp interface methods are applied to both vertical cross sections (e.g., Volker and Rushton, 1982) and areal models (e.g., Sa da Costa and Wilson, 1979~. The equations used to formulate the problem are the same as those used for the standard ground water flow problem. The only differences are that two equations are used (i.e., freshwater and seawater versions) and that their joint solution is conditioned to the pressure along the interface. Assuming that the response of the seawater domain is instantaneous and hence that hydrostatic equi- librium exists in the seawater domain, one can mode} the intrusion problem with a stanciard transient ground water flow mode} (doss, 1984~. Pinder and Abriola (1986) provide a broad overview of the prob- lem of modeling multiphase organic compounds in the subsurface. Abriola (1984) grouped models of multiphase flow and transport into two categories, those that address the migration of a miscible contaminant in ground water and those that address two or more distinct liquid phases. The former category of models addresses the far-field problem of chronic miscible contamination. Standard ground water flow and solute transport codes can be applied to these organic compound contamination problems. However, standard codes may require modifications to address biodegradation or sorption charac- teristics of a specific organic compound. As in the case of seawater intrusion, the region of greatest interest is the region exhibiting multiphase behavior. The problem of organic contamination is more complex for two reasons: (~) in general, a stationary interface will not exist, and (2) one is often interested in contamination of unsaturated soil deposits as a precursor to contam- ination of a ground water aquifer. Interest in the migration and fate of organic compounds has required that transient analysis methods be developed. Such methods enable one to simulate the movement of bulk contamination through the vadose zone and into a ground water aquifer. One is also able to estimate the mass of contamina- tion held in the media by specific retention. Because the front is not stationary, one must mode} liquid/solid interactions that govern the movement of each fluid in the presence of others. The equations describing multiphase flow and transport are similar to those previ- ously described for simulating water movement and solute transport in unsaturated soils. One fluid flow (e.g., fluid mass conservation) equation is required for each fluid phase simulated (e.g., gas, organic
TRANSPORT 153 liquid, water). Rather than simulate distinct fluid regions separated by abrupt interfaces, one simulates a continuum shared by each of the fluids of interest. The equation set is coupled by the fluid retention and relative permeability relationships of the multiphase system. Miscible displacement of trace quantities of an organic fluid can occur within the water and gas phases. This is a common occurrence; however, it greatly complicates the ma" balance equation for the organic fluid. The statement of mass conservation must now account for organic mass entrained in the water and gas phases as well as the organic mass held in the immiscible fluid phase. Transport processes are introduced into the conservation equations, and the exchange of organic mass between fluid phases must be accounted for through partition coefficients. Abriola (1988) and Allen (1985) review models available for the simulation of multiphase problems. A variety of solutions have been published for multiphase contamination problems. This is due to the complexity of the overall problem and the variety of approaches that can be taken to provide an approximate solution. A useful hierarchy of modeling approaches is as follows: sharp interface approxima- tions, immiscible phase flow modem incorporating capiliarity, and compositional models incorporating interphase transfer. Examples of models based on sharp interface approximations are those of Hochmuth and Sunada (1985), Schiegg (1986), and van Dam (1967~. Immiscible phase models incorporating capilIarity al- low a more realistic sunulation of the specific retention phenomena but do not address hysteresis in the fluid-soi! interaction. Examples of these models are presented by Faust (1985), Kuppusamy et al. (1987), and Osborne and Sykes (1986)0 Compositional models in- corporating interphase transfer are extremely complex and require the most data, many of which are not routinely available for con- taminants of interest. Examples of these models are presented by Abriola and Pinder (1985a,b), Baehr and Corapcioglu (1987), and Corapcioglu and Baehr (1987~. Parameters and Titian and Boundary Conditions for Multiphase Flow The physical complexity exhibited by multiphase flow models consumes all available computer resources. This strain on computer resources has precluded acknowledgment, in models, of the com- plexities of heterogeneous media that are spatially distributed in
154 GROUND WA173R MODELS the real environment. At the present time, computational resources restrict fully three-dimensional problems to homogeneous, porous media. Realistically, currently available computational resources are best suited to addrem conceptual modem Mode} parameters necessary for the simulation of seawater in- trusion are basically identical to those required for the simulation of ground water flow; however, two-fluid models require duplicate pa- ran~eters for fresh water and saltwater. A great many more mode} pa- rameters are necessary for a complete analysis of immiscible organic contaminant migration in the subsurface. While the seawater intru- sion problem is restricted to saturated porous media, organic fluid migration often occurs in the unsaturated zone. Consequently, fluid retention and relative permeability properties are required for the air/organic/water/solid system. Other standard data requirements for multiphase fluid flow simulation include porosity, compressibility of liquids and porous media (or storage coefficient), fluid densities and viscosities, and the intrinsic permeability tensor. As in the case of the fluid flow simulations, mode} parameters for transport simulations are more detailed for the organic fluid mi- gration problem than for the seawater intrusion problem. Mode} parameter requirements for solute migration within variable-density seawater intrusion are very similar to the requirements of any single- phase saturated zone model; however, duplicate data sets are re- quired for freshwater and seawater domains. Parameters necessary for detailed analysis of organic liquid transport phenomena include macroscopic diffusion and dispersion coefficients for each fluid phase (e.g., gas, water, or organic liquid), partition coefficients for water-gas and water-organic phases, sorption mode} parameters for alternative sorption models, and degradation mode! parameters for the organic fluid. Certainly, the more complex and complete models of multiphase contaminant problems require more data. If one considers only the immiscible flow problem in an attempt to estimate the migration of the bulk organic plume, then one will not require any of the miscible displacement (transport) parameters. If one assumes that the gas phase is static, one greatly reduces the data requirement in terms of both flow and transport phenomena. Key data for any analysis of multiphase migration are the fluid retention and relative permeability characteristics for the fluids and media of interest. The media porosity and intrinsic permeability, as well as fluid densities and viscosities, are also essential.
TRANSPORT 155 All comments made regarding boundary and initial conditions for flow and transport of a single-phase contamination analysis also apply to a multiphase analysis. Aspects of transient analysis can be important in seawater intrusion problems because of seasonal pumping stresses. Transient analyses are also essential for organic fluid migration simulation because of interest in the migration and fate of these potentially harmful substances. Spatial dunensionality of a multiphase analysis can influence re- sults. For example, in the real, fully three-dimensional environment, a heavier-than-water organic fluid can move vertically through the soil profile and form a continuous distinct fluid phase from the wa- ter table to an underlying impermeable medium. Ground water will simply move around the immiscible organic fluid as though it were an impermeable object. Attempts to analyze such a situation in a ver- tical cross section with a two-dimensional multiphase model will fad! because the organic fluid will act as a dam to laterally moving ground water. Thus only an intermittent source of immiscible organic fluid can be analyzed. Note that such an analysis will be flawed for most real-worId applications because it will represent a laterally infinite intermittent source rather than a point source of pollution. Problems Associated with Mult~phase Flow The problems associated with modeling multiphase flow include the following: . complexities; magnitude of computational resources required to address all data requirements of the multiphase problem that are inde- pendent of consideration of spatial variability, paucity of data specific to soils and organic contaminants of interest, and no way to address the problem of mixtures of organics; . absence of hysteresis submodels needed to address retention capacity of porous media and to enable one to simulate purging of the environment; . virtual orn~ssion of any realistic surface geochemistry or mi- crobiology submodels necessary to more completely describe the as- sim~lative or attenuative capacity of the subsurface environment; and . viscous fingering and its relationship to spatial variability occurring in the natural environment.
156 GROUND WATER MODELS REFERENCES Abriola, L. M. 1984. Multiphase Migration of Organic Compounds in a Porous Medium: A Mathematical Model, Lecture Notes in Engineering, Vol. 8. Springer-Verlag, Berlin. Abriola, L. M. 1988. Multiphase Flow and Transport Models for Organic Chemicals: A Review and Assessment. EA-5976, Electric Power Research Institute, Palo Alto, Calif. Abriola, L. M., and G. F. Pinder. 1985a. A multiphase approach to the mod- eling of porous media contamination by organic compounds, 1. Equation development. Water Resources Research 21~1), 11-18. Abriola, L. M., and G. F. Pinder. 1985b. A multiphase approach to the modeling of porous media contamination by organic compounds, 2. Numerical simulation, Water Resources Research 21~1), 19-26. Allen, III, M. B. 1985. Numerical modeling of multiphase flow in porous media. In Proceedings, NATO Advanced Study Institute on Fundamentals of Transport Phenomena in Porous Media, July 14-23, J. Bear and M. Y. Corapcioglu, eds. Martinus Nijhoff, Newark, Del. Anderson, M. P. 1984. Movement of contaminants in groundwater: Ground- water transport Advection and dispersion. Pp. 37-45 in Groundwater Contamination. National Academy Press, Washington, D.C. Baehr, A. L., and M. Y. Corapcioglu. 1987. A compositional multiphase model for ground water contamination by petroleum products, 2. Numerical solution. Water Resources Research 23~1), 201-213. Bear, J. 1979. Hydraulics of Groundwater. McGraw-Hill, New York, 567 pp. Beven, K., and P. F. Germann. 1982. Macropores and water flows in soils. Water Resources Research 18, 1311-1325. Cederberg, G. A., R. L. Street, and J. O. Leckie. 1985. A Groundwater mass- transport and equilibrium chemistry model for multicomponent systems. Water Resources Research 21~8), 1095-1104. Corapcioglu, M. Y., and A. L. Baehr. 1987. A compositional multiphase model for Groundwater contamination by petroleum products, 1. Theoretical considerations. Water Resources Research 23~1), 191-200. Davis, A. D. 1986. Deterministic modeling of a dispersion in heterogeneous permeable media. Ground Water 24~5), 609-615. Delany, J. M., I. Puigdomenech, and T. J. Wolery. 1986. Precipitation kinetics option of the EQ6 Geochemical Reaction Path Code. Lawrence Livermore National Laboratory Report UCR~53642, Livermore, Calif. 44 pp. Domenico, P. A., and G. A. Robbins. 1984. A dispersion scale effect in model calibrations and field tracer experiments. Journal of Hydrology 70, 123-132. Faust, C. R. 1985. Transport of immiscible Guide within and below the un- saturated zone: A numerical model. Water Resources Research 21~4), 587-596. Felmy, A. R., S. M. Brown, Y. Onishi, S. B. Yabusaki, R. S. Argo, D. C. Girvin, and E. A. Jenne. 1984. Modeling the transport, speciation, and fate of heavy metals in aquatic systems. EPA Project Summary. EPA- 600/53-84-033, U.S. Environmental Protection Agency, Athens, Gal, 4 PP Gelhar, L. W. 1986. Stochastic subsurface hydrology from theory to applica- tions. Water Resources Research 22~9), 135s-145s.
TRANSPORT 157 Germann, P. F. 1989. Approaches to rapid and far-reaching hydrologic processes in the vadose zone. Journal of Contamination Hydrology 3, 115-127. Germann, P. F., and K. Beven. 1981. Water flow in soil macropores, 1, An experimental approach. Journal of Soil Science 32, 1-13. Goode, D. J., and L. F. Konikow. 1988. Can transient Bow cause appar- ent transverse dispersion? (abst.~. Eos, Transactions of the American Geophysical Union 69 (44), 1 184-1 185. Hochmuth, D. P., and D. K. Sunada. 1985. Ground-water model of two-phase immiscible flow in coarse material. Ground Water 23~5), 617-626. Hostetler, C. J., R. L. Erikson, J. S. Fruchter, and C. T. Kincaid. 1988. Overview of FASTCHEMTM Code Package: Application to Chemical Transport Problems, Report EQ-5870-CCM, Vol. 1. Electric Power Re- search Institute, Palo Alto, Calif. Jones, R. L., A. G. Hornsby, P. S. C. Rao, and M. P. Anderson. 1987. Movement and degradation of aldicarb residues in the saturated zone under citrus groves on the Florida ridge. Journal of Contaminant Hydrology 1, 265-285. Konikow, L. F. 1981. Role of numerical simulation in analysis of groundwater quality problems. Pp. 299-312 in The Science of the Total Environment, Vol. 21. Elsevier Science Publishers, Amsterdam. Konikow, L. F. 1988. Present limitations and perspectives on modeling pollution problems in aquifers. Pp. 643-664 in Groundwater Flow and Quality Modelling, E. Custudio, A. Gurgui, and J. P. Lobo Ferreira, eds. D. Reidel, Dordrecht, The Netherlands. Konikow, L. F., and J. M. Mercer. 1988. Groundwater flow and transport modeling. Journal of Hydrology 100~2), 379-409. Kuppusamy, T., J. Sheng, J. C. Parker, and R. J. Lenhard. 1987. Finite-element analysis of multiphase immiscible flow through soils. Water Resources Research 23~4), 625-631. Lindberg, R. D., and D. D. Runnells. 1984. Groundwater redox reactions: An analysis of equilibrium state applied to Eh measurements and geochemical modeling. Science 225, 925-927. Luxmoore, R. J. 1981. Micro-, meso- and macro-porosity of soil. Soil Science Society of America Journal 45, 671. Mercer, J. M., L. R. Silka, and C. R. Faust. 1983. Modeling ground-water flow at Love Canal, New York. Journal of Environmental Engineering ASCE 109~4), 924-942. Miller, D., and L. Benson. 1983. Simulation of solute transport in a chemically react~ve heterogeneous system: Model development and application. Water Resources Research 19, 381-391. Naymik, T. G. 1987. Mathematical modeling of solute transport in the subsur- face. Critical Reviews in Environmental Control 17~3), 229-251. Osborne, M., and J. Sykes. 1986. Numerical modeling of immiscible organic transport at the Hyde Park landfill. Water Resources Research 22~1), 25-33. Parkhurst, D. L., D. C. Thorstenson, and L. N. Plummer. 1980. PHREEQE-A computer program for geochemical calculations. U.S. Geological Survey Water Resources Investigation 80-96, 210 pp. Peterson, S. R., C. J. Hostetler, W. J. Deutsch, and C. E. Cowan. 1987. MINTEQ User's Manual. Report NUREG/CR-4808, PN~6106, Prepared by Battelle Pacific Northwest Laboratory for U.S. Nuclear Regulatory
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TRANSPORT 159 van Dam, J. 1967. The migration of hydrocarbons in a water-bearing stratum. Pp. 55-96 in The Joint Problems of the Oil and Water Industries, P. Hepple, ed. The Institute of Petroleum, 61 New Cavendish Street, London. van der Heijde, P. K. M., Y. Bachmat, J. D. Bredehoeft, B. Andrews, D. Holtz, and S. Sebastian. 1985. Groundwater management: The use of numerical models. Water Resources Monograph 5, 2nd ed. American Geophysical Union, Washington, D.C., 180 pp. van Genuchten, M. Th. 1987. Progress in unsaturated flow and transport modeling. U.S. National Report, International Union of Geodesy and Geophysics, Reviews of Geophysics 25~2), 135-140. Volker, R. E., and K. R. Rushton. 1982. An assessment of the importance of some parameters for sea-water intrusion in aquifers and a comparison of dispersive and sharp-interface modeling approaches. Journal of Hydrology 56~3/4), 239-250. Voss, C. I. 1984. AQUIFEM-SALT: A Finite-Element Model for Aquifers Containing a Seawater Interface. Water-Resources Investigations Report 84-4263, U.S. Geological Survey, Reston, Va. White, R. E. 1985. The influence of macropores on the transport of dissolved suspended matter through soil. Advances in Soil Science 3, 95-120. Wolery, T. J., K. J. Jackson, W. L. Bourcier, C. J. Bruton, B. E. Viani, and J. M. Delany. 1988. The EQ3/6 software package for geochemical modeling: Current status. American Chemical Society, Division of Geochemistry, 196th ACS National Meeting, Los Angeles, Calif., Sept. 25-30 (abstract).
