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Ground Water Models: Scientific and Regulatory Applications (1990)
Commission on Physical Sciences, Mathematics, and Applications (CPSMA)

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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

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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

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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

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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

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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

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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.

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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

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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

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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

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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

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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.

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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

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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.

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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

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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

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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

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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.

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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.

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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.

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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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158 GROUND WATER MODELS Commission, Washington, D.C., 148 pp. (available from National Technical Information Service, U.S. Department of Commerce, Springfield VA 22161~. Pinder, G. F., and L. M. Abriola. 1986. On the simulation of nonaqueous phase organic compounds in the subsurface. Water Resources Research 22~9), 109~-1 19~. Pinder, G. F., and A. Shapiro. 1979. A new collocation method for the solution of the convection-dominated transport equation. Water Resources Research 15~5), 1177-1182. Plummer, L. N., B. F. Jones, and A. H. Truesdell. 1976. WATEQF A FORTRAN IV version of WATEQ, a computer code for calculating chem- ical equilibria of natural waters. U.S. Geological Survey Water Resources Investigation 76-13, 61 pp. Rittmann, B. E., and P. L. McCarty. 1980. Model of steady-state-biofilm kinetics. Biotechnology and Bioengineering 22, 2343-2357. Rittmann, B. E., and P. L. McCarty. 1981. Substrate flux into biofilms of any thickness. Journal of Environmental Engineering 107, 831-849. Rubin, J. 1983. Transport of reacting solutes in porous media: Relation between mathematical nature of problem formulation and chemical nature of reactions. Water Resources Research 19~5), 1231-1252. Sa da Costa, A. A. G., and J. L. Wilson. 1979. A Numerical Model of Seawater Intrusion in Aquifers. Technical Report 247, Ralph M. Parsons Laboratory, Massachusetts Institute of Technology, Cambridge. Saez, P. B., and B. E. Rittmann. 1988. An improved pseudo-analytical solution for steady-state-biofilm kinetics. Biotechnology and Bioengineering 32, 379-385. Scheidegger, A. E. 1961. General theory of dispersion in porous media. Journal of Geophysical Research 66~10), 3273-3278. Schiegg, H. O. 1986. 1.5 Ausbreitung van Mineralol ale Fluessigkeit (Methode our Abschaetsung). In Berteilung und Behandlung van Mineralolschadens- fallen im Hinblick auf den Grundwasserschutz, Tell 1, Die wissenschaftlichen Grundlagen zum Verstandnis des Verhaltens van Mineralol im Untergrund. LTwS-Nr. 20. Umweltbundesamt, Berlin. [Spreading of Oil as a Liquid (Estimation Method). Section 1.5 in Evaluation and Treatment of Cases of Oil Damage with Regard to Groundwater Protection, Part 1, Scien- tific Fundamental Principles for Understanding the Behavior of Oil in the Ground. LTwS-Nr. 20. Federal Office of the Environment, Berlin.) Schwille, F. 1984. Migration of organic druids immiscible with water in the unsaturated zone. Pp. 27-48 in Pollutants in Porous Media, The Unsatu- rated Zone Between Soil Surface and Groundwater, B. Yaron, G. Dagan, and J. Goldshmid, eds. Ecological Studies Vol. 47, Springer-Verlag, Berlin. Segol, G., G. F. Pinder, and W. G. Gray. 1975. A Galerkin-finite element technique for calculating the transient position of the saltwater front. Water Resources Research 11~2), 343-347. Smith, L., and F. W. Schwartz. 1980. Mass transport, 1, A stochastic analysis of macroscopic dispersion. Water Resources Research 16~2), 303-313. Sposito, G., and S. V. Mattigod. 1980. GEOCHEM: A Computer Program for the Calculation of Chemical Equilibria in Soil Solutions and Other Natural Water Systems. Department of Soils and Environment Report, University of California, Riverside, 92 pp.

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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).

Representative terms from entire chapter:

solute transport