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2 Modeling of Processes INTRODUCTION This chapter describes what models are and how they work. It begins by explaining the processes that control ground water flow and contaminant transport. To understand models, it is neces- sary to describe these processes by using certain mathematical equa- tions that quantitatively describe flow and transport. The mathe- matical aspects of modeling are critical. The precise language of mathematics provides one of the best ways to integrate and express knowledge about natural processes. By developing an awareness of the natural processes, the mathematics should be understandable. Also, where the process is not well understood, this awareness pro- vides an appreciation of the limits of the mathematics. Methods of solving the mathematical expressions are presented at the end of the chapter. Subsurface movement- whether of water, contaminants, or heat is affected by various processes. These processes can be related to three different modeling problems: ground water flow, multiphase flow (e.g., soil, water, and air; water and gasoline; or water and a dense nonaqueous-phase liquid (MAPLE, and the flow of contami- nants dissolved in ground water. 28

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MODELING OF PROCESSES 29 Ground Water Flow Of these three problems, ground water flow is the simplest to characterize and understand In most cases, models need to consider only two ground water flow processes: flow in response to hydraulic potential gradients, and the loss or gain of water from sinks or sources, recharge, or pumping from wells. Hydraulic potential gradi- ents simply represent the difference in energy levels of water and are generated because precipitation that is added to a ground water sys- tem at high elevations has more potential energy or hydraulic head than water added at a lower elevation (Figure 2.~. The result of these potential differences is that water moves from areas of high po- tential to areas of lower potential. As rainfall or other recharge keeps supplying water to the flow system, ground water continues to flow. On a cross section, it is possible to represent the spatial variability in hydraulic potential existing along a flow system by using what are called equipotential lines (see Figure 2.1~. The equipotential lines are contours of hydraulic potential within some area of interest. In some simple situations, the direction of ground water flow is perpendicular to these equipotential lines, as shown in Figure 2.~. The actual distribution of hydraulic head observed for an area depends mainly on two factors, how much and where water is added and removed, and the hydraulic conductivity distribution that exists in the subsurface. Consider a few examples. Figure 2.2 illustrates the hydraulic head distribution for two different water table config- urations. The water table effectively represents the top boundary of the saturated ground water system, and its configuration reflects different recharge conditions. In both cases, the bottom and sides of the section are considered to be impermeable (no flow). With a linear water table and recharge mainly at the right end of the sys- tem, a relatively smooth regional flow system develops (see Figure 2.2a). The second water table, representing significant local areas of recharge and discharge at three locations, shows a much different flow pattern (see Figure 2.2b). Instead of a broad regional trend, several small, local flow systems have developed. Ground water flow patterns also depend on the hydraulic con- ductivity distribution. Figure 2.3 compares the pattern of ground water flow along a cross section where all properties except the hy- draulic conductivity for each layers are kept constant. Each of the two layers shown is defined in terms of a hydraulic conductivity in the horizontal direction (Kh) and in the vertical direction (Kit), with the ratio Kh/KV describing the degree of directional dependence in

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MODELING OF PROCESSES 0.2S co m > 0.2S up a: 31 i ' ' 1 ---Equipotential Lines a ~ Ground Surtace , Water Table O _ OK = 1i ~ 1 ~'\ \~ W~/ _ - ~I _ _ 1 ~1, I I K = 10 1 1 ~ , , 1, , 1, , 1, , 1, , 1, , 1, , 1, , 1, , 1, , 1, , 1, . . \. S ~I ~ l ~I l I I I I I I I I ~i ~I b o O S RELATIVE BASIN LENGTH FIGURE 2.2 Dependence of the pattern of ground water flow on the recharge rate, as reflected by the configuration of the water table. All other parameters are the same in the two sections (from Freeze, 1969b). 0.2S ~ O id 0.2S UJ o a o ~I ~ ~ ~ ---Equipotential Lines C~rolint] Pilirf~r!- WAter Ts~hl- I ~11 | I K = lo ~I j 11 ~71 S 1 ~: o RELATIVE BASIN LENGTH S FIGURE 2.3 Dependence of the pattern of ground water flow on the hydraulic conductivity distribution. The only difference in the two diagrams is the pattern of geologic layering defined in terms of the relative hydraulic conductivities shown (from Freeze, 1969b).

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32 GROUND WATER MODELS hydraulic conductivity. Examination of the two flow patterns shows how changes in the hydraulic conductivity distribution can change the character of ground water flow. Adding or removing water also can have a significant impact on the pattern of flow. The most important sources and sinks in a ground water flow system are pumping or injection wells (i.e., point sources/~nks). These are considered internal flows of water (fluxes). Other possibilities such as recharge or evaporation are most often considered as boundary fluxes. Pumping lowers the hydraulic potential at the well and in its immediate vicinity, creating what is known as a cone of depression. The result of decreasing hydraulic potential toward the well is the flow of water to the well. Injection does the opposite and results in flow away from a well. So far, only steady-state flow, or flow that does not change as a function of tune, has been discussed. Often, however, flow systems are transient, which means that hydraulic heads change with time, leading to variations in flow rates. For example, water leveh decline when a pumping well is first turned on, providing an early transient response. In many instances when sources of recharge are available, water levels will eventually stabilize, providing a new equilibrium or steady-state flow system. The most important feature of a transient flow system is the ability of water to be removed from or added to storage in individual layers. The parameter describing the water storage capabilities of a geologic unit is called the Unspecific storage." For transient flow problems, its value contributes to determining the distribution of hydraulic head at a given tune. Note that the smaller the specific storage, the faster the ground water system will seek a new equilibrium. Readers wishing a more detailed explanation of this parameter and aspects of ground water flow should see Freeze and Cherry (1979~. M~tiphase Flow Multiphase flow occurs when fluids other than water are mov- ing ~ the subsurface. These other fluids can include gases found in the soil zone or certain organic solvents that do not appreciably dissolve In water (i.e., immiscible liquids). Examples of fluids that are immiscible with water include many different manufactured or- ganic chemical such as the cleaning solvent trichioroethylene and preservatives such as creosote. Petroleum products such as crude oil, heating oil, gasoline, or jet fuel are also examples.

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MODELING OF PROCESSES 33 The process causing all of these phases to flow ~ again movement in response to a potential gradient. Now, however, the situation is more complicated because the potential causing each fluid to move is not necessarily the same as that for water. Thus each fluid can be moving in a different direction and at a different rate. Another complexity Is that many characteristic parameters are no longer constant when several fluids are present together and competing for the same pore space. For example, the relative permeability of a geologic unit to a particular fluid like water will be small if the proportion of water present in a given volume of porous medium is small and will tend to increase as the amount of water increases. As discussed previously for water, a fluid's potential also depends on any sources or sinks that add or remove fluid. The same idea ap- plies to multifluid systems, except that now the number of processes increases because the effects of pumping/injection and evaporation (volatilization) affect each of the fluids present and, in addition, there can be transfers of mass between fluids. An example of this latter mechanism is that some portion of a gasoline spin might dissolve in water. To illustrate these concepts about the theory of multiphase sys- tems, consider two problems of particular interest to this report-the flow of water in the unsaturated zone and the migration of organic contaminants that are either more or less dense than water. When studying the problem of water movement in the presence of soil gas in the unsaturated zone, it is sometimes assumed that only the water moves. The only effect of the gas on water movement is the variabil- ity in the parameters caused by the presence of several fluids. For example, hydraulic conductivity varies as a function of the quantity of water in the pores. Figure 2.4 shows a relationship between hydraulic conductivity (K) and pressure head (fib). According to Freeze and Cherry (1979), pressure head is one component of the total energy water possesses at a point. Several features should be noted. As the pressure head becomes smaller (more negative), the soil becomes drier and the hy- draulic conductivity decreases. Much less water wall move through a dry soil than through a wet soil. Another feature ~ that if the soil is drying out there is one -K relationship and if it is wet- ting there is another. Further, repeated wetting and drying cause the relationship to be defined by the scanning curves that join the wetting and drying curves at intermediate points (Figure 2.43. In

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34 GROUND WATER MODELS 0.03 c) I > 0.02 it 8 0.01 Cot o I -300 -200 -100 0 100 200 300 Unsaturated Saturated Drying ~;- . I I ~ K o -Scanning Curves Wetting PRESSURE HEAD, ~ (cm H2O) FIGURE 2.4 Example of the relationships between pressure head and hydraulic conductivity for an unsaturated soil (modified from Freeze, 1971a). most multifluid systems, hydraulic conductivity and other parame- ters commonly exhibit this kind of "hysteretic" behavior, and yet for many applications, these types of site-specific data are not available. The progress of a wetting front moving into a dry soil can be described in terms of either potentials or volumetric water content, 0.14 - t _ ~ MOISTURE CONTENT FIGURE 2.5 The distribution of water in the unsaturated zone can be de- scribed in terms of pressure head and moisture content. Results presented for a combined saturated and unsaturated flow system illustrate how pressure head in particular is continuous across the water table (from Freeze, 1971b).

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MODELING OF PROCESSES 35 defined as the ratio of the volume of water in the voids to the total volume of voids. Figure 2.5 illustrates how both are used to define a wetting zone near the top of the ground. The water table is clearly illustrated by the zero pressure contour and the total porosity contour (complete saturation). Given that moisture contents are easier to measure than potentials, the former are used more frequently to describe real systems. A more complicated case to consider is a flow involving an im- miscible fluid and water in the subsurface. Eventually, a distinction has to be made between a fluid that is less dense than water and one that is more dense. However, where an organic liquid is spilled on the ground surface, both fluids will move much the same way through the unsaturated zone (Figure 2.6a and b). The free organic liquid in a homogeneous medium moves vertically downward, leaving a residual trail of organic contaminants. Each pore through which the free organic liquid moves retains some of the contaminant (resicl- ual saturation) in a relatively immobile state. Thus, if the volume of spilled liquid ~ small and the unsaturated zone is relatively thick, no free liquid may reach the water table. Of course, free liquid may reach the water table over extended periods of time, and dissolved organic liquid may be conducted by water flow. It is when the free liquids begin to approach the top of the capillary fringe above the water table that the differences in density begin to affect transport. The capillary fringe is a zone above the water table where the pores are completely saturated with water but the pressure heads are less than atmospheric (Freeze and Cherry, 1979~. A contaminant that is lighter than water will mound and spread, following the dip of the water table (Figure 2.6a). A fluid that is heavier than water will spread slightly and keep moving downward. This fluid will ultimately mound on the bottom of the aquifer or on a low-permeability bed within the aquifer and move in whatever direction the unit is dipping (Figure 2.6b). Thus water and the organic liquid need not move in the same direction. To understand the details of multicomponent flow, it is essential to study the concepts of wettability, imbibition and drainage, and relative permeability. A discussion of these topics is, however, beyond the scope of this report. Readers can refer to Bear (1972) and Greenkorn (1983) for en overview of the basic theory. The key point to remember is that, as in the case of water, the permeability of the material through which these fluids are moving plays a major role in controlling the direction and rate of flow. In the case of multiphase

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36 GROUND WATER MODELS a Ground Surface . _ _ Oil Phase -Oil Components Unsaturated Zone Capillary Fringe Dissolved in Water~ --__ Saturated Zone ~/////'~0~: b Ground Surface _v _ ~- ~ , ~ ~ ~ 1: .~#~. 1 t..~ :! ~ CHC Phase Unsaturated Zone Capillary Fringe K2 K1 CHC Dissolved in Water Saturated _ _ _ _ _ _ _ _ ' Zone l _ I_ ~////////////////////////////////////////i/////~, FIGURE 2.6 The flow of a nonaqueous-phase liquid that is (a) less dense than water (oil) and (b) more dense than water (chlorohydrocarbon, CHC) in the unsaturated and saturated zones. In both cases the contaminants are also transported as dissolved compounds in the ground water (from Schwille, 1984~. systems, a relative permeability is defined for each fluid with values ranging between zero and one as the relative abundance of each fluid (i.e., saturation) changes. The key point here is that as more fluids are introduced to the pore space, more of the pore space is devoted to the relatively immobile state of each fluid and therefore less pore space is devoted to liquid flow.

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MODELING OF PROCESSES 37 The distribution of an NAP L in the subsurface is described quan- titatively in terms of the relative saturation of the NAPL, which is given by the ratio of the volume of the NAPL to the total pore vol- ume. In other words, it describes what proportion of the pore volume is filled with the NAPL. A relative saturation can be defined for each one of the organic liquids and water. This kind of description is generally not used in field settings because of the detailed study that is necessary. Instead, presence/absence indications are used, as illus- trated in Figure 2.6. The results of computer simulations normally characterize relative fluid saturations. Expressing the distribution of fluids in terms of relative saturation is analogous to expressing the moisture content in terms of unsaturated flow. Dissolved Contanunant Passport One of the reasons why problems involving dissolved contam- inants are so difficult to model is the number and complexity of controlling processes. The processes can be divided into two groups: (1) those responsible for material fluxes and (2) sources or sinks for the material. For the problem of contaminant migration these are the mass transport and mass transfer processes, respectively (Table 2.1~. A brief discussion of each of the processes listed in Table 2.1 follows, with a general assessment of its impact on contaminant transport. Advection Advection is the primary process responsible for contaminant migration in the subsurface. Mass is transported simply because the ground water in which it is dissolved is moving in a flow system. In most cases, it can be assumed that dissolved mass is transported in the same direction and with the same velocity as the ground water itself. For example, given the conditions of flow described by the equipotential lines and flowlines of Figure 2.7a, it is a simple matter to define the plume of dissolved contaminants in terms of the streamtubes that pass through the source. A streamtube is defined an the area between two adjacent Bowlines. When Bowlines are equally spaced, the discharge of water through each is the same (Freeze and Cherry, 1979~. This simple approach assumes that the density of the contaminated fluid is about the same as that of the ground water. The mean velocity of contaminant migration can also be assumed to be the same as the mean ground water velocity (or seepage velocity).

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38 t-TROUND WATER MODELS TABLE 2.1 A Summary of the Processes Important in Dissolved Contaminant Transport and Their Impact on Contaminant Spreading Process Definition Impact on Transport Mass transport 1. Advection Movement of mass as a Most important way of consequence of ground transporting mass away water flow. from source. 2. Diffusion Mass spreading due to An attenuation mechanism molecular diffusion in of second order in most response to concentration flow systems where gradients. advection and dispersion dominate. 3. Dispersion Fluid mixing due to effects An attenuation mechanism of unresolved hetero- that reduces contaminant geneities in the per- concentration in the meability distribution. plume. However, it spreads to a greater extent than predicted by advection alone. Chemical mass transfer 4. Radioactive decay 5. Sorption 6. Dissolution/ . . . precipitation Irreversible decline in the activity of a radionuclide through a nuclear reaction. Partitioning of a contaminant between the ground water and mineral or organic solids in the aquifer. The process of adding contaminants to, or removing them from, solution by reactions dissolving or creating various solids. 7. Acid/base Reactions involving a reactions transfer of protons (H+). An important mechanism for contaminant attenuation when the half-life for decay is comparable to or less than the residence time of the flow system. Also adds complexity in production of daughter products. An important mechanism that reduces the rate at which the contaminants are apparently moving. Makes it more difficult to remove contamination at a site. Contaminant precipitation is an important attenuation mechanism that can control the concentration of contaminant in solution. Solution concentration is mainly controlled either at the source or at a reaction front. Mainly an indirect control on contaminant transport by controlling the pH of ground water.

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68 G IIJ m z ~ 10 o a: 1d GROUND WATER MODELS , Constant Head Nodes it. . . ~Kh= ~=00 Kh = v = 10-5 cm/s Kh = 10-7 cm/s; Kv = 10~ cm/s Kh = It = 10 3 cm/s COLUMN NUMBERS FIGURE 2.13 Model system from the previous figure subdivided by a rect- angular grid system. Nodes are defined in the center of each grid cell. By assigning Kh = Kv = 0.0 in the area at the top left, it is effectively excluded from the calculation. 2.7. Included are a brief description of the methods and a few key references that can be used to obtain more detailed information. The last topic that needs to be addressed in this section involves the mathematical techniques for solving matrix equations. In most models, the major computational effort comes in solving the system of mode! equations. In general, there are two basic methods. In one approach the entire system of equations is solved simultaneously with direct methods, providing a solution that is exact, except for machine round-off error. In the second approach, iterative methods obtain a solution by a process of successive approximation, which involves making an initial guess at the matrix solution and then improving this guess by some iterative process until an error criterion is satisfied. Direct methods have two main disadvantages. The first is that a computer may not be able to store the large matrices or solve the system in a reasonable time when the number of nodes is large. Sometimes this problem can be dealt with to some extent by us- ing sparse matrix solvers and various node-numbering schemes. The second problem with direct methods is the round-off errors. Be- cause many arithmetic operations are performed, round-off errors can accumulate and significantly influence results for certain types of matrices. Iterative schemes avoid the need for storing large matrices. This

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69 o EM cat cat cd a' c~ cat ._ cd 3 o .= o cat - o o cat . _ au o o v, o o an o m 2 co Do car ~I Go ~^ ~_ '_ ~ C)d _ ~ c ~ ~ ~x 2 ~ ~ = ~ ~ ~_ ~ jet ~ _ o |, ~ e ~ 3 ~ 3 ~ 3- ~ v: cat c or .o ~,3 ~ s ~ c u: Em 3 o sit a' cd 3 0

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70 s is in He o au o 4- o . . ~ a) ~ .= 4- ~ o=o - o ~ - ~ ^ ^ - ~ oo E Go, ~ ~ ox X on X w ~ =,^ X ~ ~ . id ~ 0N 0) 0 ~ ~ ~ ~ ~ O ~ ~ X ~0 E o ~ E ~ ~ E E ~ ~ ~ ~ ~ it_ ~ ~ ~ it. ~ ~ 3 o ~ v o ~ ~ ILL C:: (O ~ ~ E~ ~ C ~, E ~ ~ ~ " 3 ~ o con ~ ~ ~ OCR for page 28
71 Go - ~ oo To 'x ~ - ~ ~ -= .= - ~ ~ c == ^ 3 E hi, ~ ~ 0 ~ ~ O ~ ~ ~ ~ m ~ ~ of; ._ ._ ct v, so . ~ a' > Q Ct Go Do cry - . ^ ~ 00 3 car $, _ ~ Is: 0 4- ~ so ~ ct i, cat c~ it,) ~0 In ~ ~ ~ ~ , c, ~ O &.5 9 Y . ~ ~ y _ m ~0 ~ se c,0 ~ s,.( cat ~0 ~ 3 ,~ if: (=,, ~ c ~ .= ~ ~ cat ~ = S ~ S O O - C) ~ O > O Cal ~ _

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72 GROUND WATER MODELS feature makes them attractive for solving problems with many un- knowns. Numerous schemes have been developed; a few of the more commonly used ones include successive overrelaxation methods (Varga, 1962), the alternating-direction implicit procedure (Douglas and Rachford, 1956), the iterative alternating-direction implicit pro- cedure (Wachpress and Habetler, 1960), and the strongly implicit procedure (Stone, 1968~. Because operations are performed many times, iterative methods also suffer from potential round-off errors. The efficiency of iterative methods depends on an initial estimate of the solution. This makes the iterative approach less desirable for solving steady-state problems (Narasimhan et al., 1978~. To speed up the iterative process, relaxation and acceleration factors are used. Unfortunately, the definition of best values for these factors commonly is problem dependent. In addition, iterative approaches require that an error tolerance or convergence criterion be specified to stop the iterative process. This, too, may be problem dependent. All of these parameters must be specified by the mode} user. According to Narasimhan et al. (1977) and Neuman and Nara- simhan (1977), perhaps the greatest limitation of the iterative schemes is the requirement that the matrix be well conditioned. An ill-conditioned matrix can drastically affect the rate of conver- gence or even prevent convergence. An example of an ill-conditioned matrix is one in which the main diagonal terms are much smaller than other terms in the matrix. More recently, a semi-iterative method has gained popularity (Gresho, 1986~. This method, or class of methods known as conju- gate gradient methods, was first described by Hestenes and Stiefel (1952~. It is widely used to solve linear algebraic equations where the coefficient matrix is sparse and square (Concus et al., 1976~. One advantage of the conjugate gradient method is that it does not require the use or specification of iteration parameters, thereby elim- inating this partly subjective procedure (Manteuffe} et al., 1983~. Kuiper (1987) compared the efficiency of 17 different iterative meth- ods for the solution of the nonlinear three-dimensional ground water flow equation. He concluded that, in general, the conjugate gradient methods did the best. Numerical methods by their very nature, yield approximate so- lutions to the governing partial differential equations. The accuracy of the solution can be significantly affected by the choice of numerical parameters, such as the size of the spatial discretization grid and the length of time steps. Those using ground water models and those

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MODELING OF PROCESSES 73 making management decisions based on mode! results should always be aware that trade-offs between accuracy and cost will always have to be made. If the grid size or time steps are too coarse for a given problem, it is possible to generate a numerical solution that converges on an answer that has an excellent mass balance but is still inaccu- rate. furthermore, if iteration parameters are not properly specified, the solution may not converge. It is hoped that this will show up as a mass balance error, which will be noted by the user. This indicates, however, the importance of a mass balance in numerical models. REFERENCES 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. Ahlstrom, S. W., H. P. Foote, R. C. Arnett, C. R. Cole, and R. J. Serne. 1977. Multicomponent Mass Transport Model: Theory and Numerical Im- plementat ion (discrete-parcel-random-walk version). BNWL-2 127. Battelle Northwest Laboratories, Richland, Wash. Alexander, M. 1985. Biodegradation of organic chemicals. Environmental Science and Technology 19, 106. Anderson, M. P. 1979. Using models to simulate the movement of contaminants through ground water flow systems. Critical Reviews in Environmental Control 9~2), 97-156. Anderson, M. P. 1984. Movement of contaminants in groundwater: Groundwa- ter transport-advection and dispersion. Pp. 37-45 in Groundwater Con- tamination. Studies in Geophysics. National Academy Press, Washington, D.C. Atlas, R. M. 1981. Microbial degradation of petroleum hydrocarbons: An environmental perspective. Microbiological Reviews 45, 180. Baehr, A. L., and M. Y. Corapcioglu. 1987. A compositional multiphase model for groundwater contamination by petroleum products, 2. Numerical solution. Water Resources Research 23~1), 201-214. Bates, J. K., and W. B. Seefeldt. 1987. Scientific Basis for Nuclear Waste Management X. Materials Research Society, Symposium Proceedings, Vol. 84, 829 pp. Bear, J. 1972. Dynamics of Fluids in Porous Media. Elsevier, New York, 764 PPe Bear, J. 1979. Hydraulics of Groundwater. McGraw-Hill, New York, 569 pp. Bredehoeft, J. D., and G. F. Pinder. 1973. Mass transport in Bowing ground- water. Water Resources Research 9, 194-210. Cherry, J. A., R. W. Gillham, and J. F. Pickens. 1975. Contaminant hydroge- ology: Part 1, Physical processes. Geoscience Canada 2~2), 76-84. Cleary, R. W., and M. J. Ungs. 1978. Groundwater Pollution and Hydrology, Mathematical Models and Computer Programs. Rep. 78-WR-15, Water Resources Program, Princeton University, Princeton, N.J.

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74 GROUND WATER MODELS Concus, P., G. Golub, and D. O'Leary. 1976. Sparse Matrix Computations. Academic Press, New York, pp. 300332. 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. DeMarsily, G. 1986. Quantitative Hydrogeology. Academic Press, Orlando, Fla., 440 pp. Domenico, P. A., and G. A. Robbins. 1985. A new method of contaminant plume analysis. Ground Water 23~4), 476-485. Douglas, J., Jr., and H. H. Rachford, Jr. 1956. On the numerical solution of heat conduction problems in two and three space variables. Transactions of the American Mathematics Society 82, 421-439. Faust, C. R. 1985. Transport of immiscible druids within and below the un- saturated zone: A numerical model. Water Resources Research 21~4), 587-596. Freeze, R. A. 1969a. Regional ground water Bow Old Wives Lake drainage basin, Saskatchewan. Inland Waters Branch, Department of Energy, Mines, and Resources, Canada, Scientific Series, No. 5, 245 pp. Freeze, R. A. 1969b. Theoretical analysis of regional ground water Bow. Inland Waters Branch, Department of Energy, Mines, and Resources, Canada, Scientific Series, No. 3. Freeze, R. A. 1971a. Influence of the unsaturated flow domain on seepage through earth dams. Water Resources Research 7~4), 929-941. Freeze, R. A. 1971b. Three-dimensional, transient, saturated-unsaturated Bow in a groundwater basin. Water Resources~Research 7~2), 347-366. Freeze, R. A., and J. A. Cherry. 1979. Ground Water. Prentice-Hall, Englewood Cliffs, N.J., 604 pp. E`reyberg, D. L. 1986. A natural gradient experiment on solute transport in a sand aquifer: II. Spatial moments and the advection and dispersion of non-reactive tracers. Water Resources Research 22~13), 2031-2046. Fried, J. J. 1975. Ground Water Pollution: Theory, Methodology, Modeling, and Practical Rules. Elsevier, Amsterdam, 330 pp. Frind, E. O. 1987. Simulation of ground water contamination in three dimen- sions. Pp. 749-763 in Proceedings of Solving Ground Water Problems with Models. National Water Well Association, Denver, Colo. Galloway, W. E., and D. K. Hobday. 1983. Terrigenous Clastic Depositional Systems. Springer-Verlag, New York, 423 pp. Greenkorn, R. A. 1983. Flow Phenomena in Porous Media: Fundamentals and Applications in Petroleum, Water and Food Production. Marcel Dekker, New York, 550 pp. Gresho, P. M. 1986. Time integration and conjugate gradient methodsfor the incompressible Navier-Stokes equations. Pp. 3-27 in Finite Elements in Water Resources, Proceedings of the 6th International Conference, Lisbon, Portugal. Springer, Berlin. Hanks, R. J., A. Klute, and E. Bresler. 1969. A numeric method for estimating infiltration, redistribution, drainage, and evaporation of water from soil. Water Resources Research 5, 1064-1069. Hantush, M. S. 1964. Hydraulics of wells. Advances in Hydroscience 1, 281-432. Hestenes, M., and E. Stiefel. 1952. Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards 49~6), 409-436.

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