LA4111/ch17 Page 357 Wednesday, December 27, 2000 2:54 PM CHAPTER 17 Groundwater Modeling in Health Risk Assessment Jeanette H Leete CONTENTS I II III IV V Introduction .357 Groundwater Modeling Reports 358 Technical Aspects of Groundwater Modeling 358 A Definition of “Model” 358 Technical Aspects of Contaminant Transport 365 A Physical and Chemical Forces Influencing Movement 365 B Model Misuse, Limitations, and Sources of Error 366 C Groundwater Quality Monitoring .367 Conclusion 367 References 367 I INTRODUCTION Groundwater modeling refers to the construction and operation of a model that can mimic the actual behavior of groundwater in an aquifer system There are several kinds of groundwater models: electrical analog, physical (most physical models look like ant farms packed with layers of sand and clay), and mathematical For this primer, we use “groundwater model” to mean a mathematical model A mathematical model is a set of equations and assumptions chosen to represent a groundwater system Computer programs then solve these sets of equations Groundwater modeling is extremely useful for developing credible risk assessments where groundwater is a potential exposure pathway Groundwater modeling is employed during the risk assessment process in the hazard evaluation and exposure assessment steps Modeling is used to evaluate the possible contaminant transport 357 © 2001 by CRC Press LLC LA4111/ch17 Page 358 Wednesday, December 27, 2000 2:54 PM 358 A PRACTICAL GUIDE TO ENVIRONMENTAL RISK ASSESSMENT REPORTS pathways (so that exposure potential can be evaluated) Groundwater modeling can provide information about changes in concentration from source to discharge or withdrawal point Assessing the risk to humans or to the environment of a constituent of groundwater requires the ability to predict exposure to this groundwater This requires knowledge of the direction and amount of groundwater movement; the chemical nature of the water; concentration of the undesirable constituent at the point or area of entry into groundwater; possible interactions with the aquifer material and natural groundwater; interconnections to other water sources (discharge to springs, pumping from wells, hydraulic connections between aquifers); and potential for transformations during transport, such as adsorption, dilution, and dispersion Direct measurement of this information is generally impossible because of limited access to subsurface information Movement of water or contaminated water in the subsurface cannot be directly observed, nor can continuous measurements over an area be taken Available information is always limited to point information at a limited number of locations If done efficiently and well, groundwater modeling can combine sparse data into a coherent representation of the workings of a hydrogeologic system That information can then be used to predict the current and future extent of contamination and pathways of exposure II GROUNDWATER MODELING REPORTS Groundwater modeling reports are typically produced as one large deliverable This format is acceptable for simple physical and geochemical settings, and for situations where previous work has created a credible understanding of the geology of the area, and has defined the existing hydrogeochemistry and extent of contamination A groundwater modeling report should be broken into several interim deliverables for complex or poorly understood settings Examples of complex settings include multiaquifer problems, flow in fractured formations, and situations where groundwater withdrawals are variable in amount, timing, and location Possible logical subreports include Site Geology and Conceptual Hydrogeologic Setting; Ground Water Flow Model Calibration and Verification; and Ground Water Transport Modeling A series of smaller reports allows the project manager to review intermediate results and ensure that the project is on “solid ground” before authorizing subsequent work If necessary, the project manager may arrange for peer review by a second consultant, selected to review the specific report segment III TECHNICAL ASPECTS OF GROUNDWATER MODELING A Definition of “Model” A model is a characterization of a real system In hydrogeology, as mentioned in the introduction, there are several classes of models These classes are discussed below © 2001 by CRC Press LLC LA4111/ch17 Page 359 Wednesday, December 27, 2000 2:54 PM GROUNDWATER MODELING IN HEALTH RISK ASSESSMENT 359 Conceptual Models Conceptual models describe and offer an explanation of “how groundwater works” in a given system Conceptual models should always precede data collection For example, the regional geology would be described in a conceptual model along with the locations and nature of the bounding conditions on the aquifer (which might be rivers, discharge areas, recharge areas, faults, and areas where the aquifer is not present) An example of a conceptual model could read: The groundwater system in the study area consists of a stack of three regional aquifers, within a vaguely bowl-shaped basin of horizontally layered Paleozoic sedimentary rock, over a crystalline bedrock surface The uppermost unit consists of varying thicknesses of glacial materials Where these materials are sandy and of sufficient thickness, they too can serve as local aquifers Preglacial drainage systems have cut through all but the deepest of the aquifers Recharge to the system is focused where aquifers subcrop beneath sandy glacial deposits, and where aquifers appear at the surface A major river system bisects the study area The valley is incised from 100 to 300 feet below the general surface elevations, and forms the major discharge zone for the regional aquifer system, and thus a major boundary to the system From such a model (i.e., the description and accompanying geologic cross-sections and maps), the risk assessment professional can form a mental picture of regional groundwater flow directions and groundwater/surface water interactions General opinions of cause and effect are given in a conceptual model, but for predictions and analysis of local conditions, dynamic models are necessary Dynamic Models A dynamic model can be changed to reflect changing conditions, that is, it can be manipulated Physical models, scale models of the groundwater system, can be built in aquariums or narrow plexiglass “ant farms” of sand, gravel, and clay or other porous materials The surface topography, complete with lakes and/or streams, can be represented — wells can be built of acrylic or other clear tubing (so that water levels can be observed); leaky underground storage tanks can be made from empty film canisters with pinholes and an access pipe made from a straw With some imagination, patience, and visits to the hardware store, most types of groundwater problems can be built into a physical model A physical model can show groundwater movement in response to regional flow, and can show response to pumping of the model’s wells Food coloring can be added to the recharge water or to water at a contaminant source in order to reveal the flow paths of the water Because of the difficulty in deriving quantifiable results from such models, and the amount of time needed to rebuild it every time a change is needed, these models are rarely used today to solve groundwater problems They have, however, proven to be very useful in the public meeting forum where they can be used to demystify the concepts of groundwater flow and contaminant transport The flow of electricity through a conductor is analogous to the flow of groundwater through an aquifer, the realization of which was the breakthrough which © 2001 by CRC Press LLC LA4111/ch17 Page 360 Wednesday, December 27, 2000 2:54 PM 360 A PRACTICAL GUIDE TO ENVIRONMENTAL RISK ASSESSMENT REPORTS allowed the development of mathematical solutions to groundwater flow problems Accordingly, some of the first groundwater models were built as networks of resistors and capacitors The aquifer characteristics were scaled into the model by using different resistors to represent the transmission of water and different capacitors to represent the storage of water When such a model was finished, current represented the flow of water and voltage represented the hydraulic head (which can be understood as the water level in wells which penetrate the aquifer) These models are called electrical analog models Electrical analog models can take months to build and adjust so that they represent the aquifer system under study, and they tend to take up quite a bit of lab space Three-dimensional flow can be modeled by connecting two or more horizontal models to each other with the appropriate electronics to represent leakage between the layers Electrical analog models are rarely built today because other models are easier to work with Today’s uses still include permanent museum and public education displays Stochastic models are statistical models Much recent research, and possibly hundreds of recent papers, have explored the use of stochastics in the modeling of groundwater flow and contaminant transport, but the method has not gained wide acceptance among practitioners This is almost certainly due to the complexity of the concepts employed and the fact that none of the many modeling approaches presented has become a standard It is possible that rapid progress toward an accepted standard could be made in the next several years Mathematical models are derived from the physical laws that govern the situation (e.g., conservation of mass, conservation of momentum, and Darcy’s equation) with simplifying assumptions about the aquifer and about the edges of the modeled area Analytical models can be used to solve very simple problems (e.g., the aquifer can be assumed to be the same in every direction and only one value for each parameter is needed) Equations are set up which represent the system variables (e.g., hydraulic head) over the domain of the model The resulting analytical model of groundwater flow will be a set of partial differential equations that can be solved directly using calculus Analytical models for solute transport can be created in a similar fashion The results of more than one analytical model run can be combined to produce a solution to a more complicated situation One could, for example, set up an analytical model which produced a solution for the hydraulic gradient over the area of concern, then use a different analytical model to predict the movement of contaminants in response to those gradients The data requirements for an analytical model are not extensive, because after all, only one number can be used for each system parameter Analytical models can be solved quickly with an inexpensive programmable calculator or personal computer Graphical solution of some of the less complicated flow equations is possible For example, flow nets combine lines which describe flow paths and lines which represent equal hydraulic head to provide a visualization of the groundwater flow field Once constructed, a flow net can be used for prediction of flow directions and amounts It is clear that many real world problems are not simple enough to be accurately assessed with simple analytical or graphical models Where enough is known about © 2001 by CRC Press LLC LA4111/ch17 Page 361 Wednesday, December 27, 2000 2:54 PM GROUNDWATER MODELING IN HEALTH RISK ASSESSMENT 361 a hydrogeologic or contaminant transport problem to be able to characterize the system with variable aquifer parameters and detailed boundary conditions, the groundwater flow equations cannot be directly solved with calculus; rather, they must be approximated by systems of algebraic equations Groundwater models using this technique are termed numerical models Calculations must be carried out repeatedly over the entire system of equations until a solution is reached The process is repeated every time a change in any of the model data is made Mathematical models have replaced other types of models as the speed of computers has increased and the cost of computers has decreased As few as 15 years ago, the best high speed computers the major universities had to offer often took hours to complete one run of a numerical model Because of computing costs, these model runs were often done overnight at lower rates The results were picked up in the morning (if indeed the program had run without fatal errors), and during the day necessary changes were made in model input for the next night’s run Today’s personal computers provide the speed and flexibility needed to handle many model runs, of even very complex models, in one day, and advanced workstations allow the calculation of detailed three-dimensional models, and provide graphical color output of the results in seconds Model Selection The particular problem at hand will determine which of the methods is appropriate Each of the modeling approaches discussed above has its limitations, advantages, and disadvantages The essential question is: Can this method answer my question most efficiently? There is a tendency in the groundwater profession to turn to the numerical models without consideration of the less elegant methods To counteract this bias, the following questions should be posed as part of the model selection process: • What is the model’s purpose? The scope of the study may be such that answers can be obtained from analytical models or from graphical solutions • What data are available to characterize the aquifer system? If the aquifer system can only be described in general terms, what is the justification for the use of a complex model? • Is the collection of additional data to be part of the study? If so, then a preliminary model can be constructed to guide data acquisition and eventual construction of a full model If the decision is made that a numerical model is indeed necessary, an appropriate numerical method should be selected There are three basic approaches to numerical modeling: finite difference, finite element, and analytic element As mentioned above, the continuous equations that describe conditions in the aquifer at every point can only be directly solved for very simple situations To accommodate more complex aquifer characteristics, the study area is divided up into smaller pieces In both the finite difference and finite element approaches, each aquifer segment is described by an algebraic equation or set of equations, all of © 2001 by CRC Press LLC LA4111/ch17 Page 362 Wednesday, December 27, 2000 2:54 PM 362 A PRACTICAL GUIDE TO ENVIRONMENTAL RISK ASSESSMENT REPORTS which must be solved for each time step In the analytic element method, the aquifer is divided into segments (elements), each of which can be simplified so that analytic solution techniques can be used The finite difference method superimposes a grid system over the study area The method has developed to the point where the grid need not be regular, nor even rectangular A finer mesh can be located over the area of greatest concern so that more detail can be obtained Within each cell or aquifer segment there is a node point at which the equations are solved The nodes can be in the centers of the cells or at the intersections of the grid lines Any pumping occurring within the cell, or any water added to the cell, will be treated as if it were added at the node All water levels are calculated at the node and applied over the whole cell To avoid a “stairstep” effect in predicted water levels, areas of concern should have finer meshes A first approximation of the head at each of the nodes is the starting point The computer recalculates heads at each node (some nodes may have fixed heads as part of the boundary conditions), based on the heads of adjacent nodes, until the changes between successive recalculations is less than the predetermined error limit The solution of the finite difference model is an iterative process Sometimes during iterations, errors can start to build on each other and the resulting solution may be nonsense Only by comparing the computer’s answer to reality can you know if a real solution has been reached This problem is called numerical dispersion The finite element method divides the aquifer into polygonal elements (often triangular) by connecting irregularly placed nodes into a mesh where each element has multiple nodes, termed “discretization,” this allows more accurate representation of irregular areas than does the finite difference model, even though the finite element model will usually have fewer nodes Values of system variables are interpolated over the element by basis functions The basis functions are specified in terms of the node coordinates and the results are combined into an integral system which is then approximated using finite difference techniques Be aware that these methods are also subject to numerical dispersion If you divide any groundwater system into small enough pieces, you reach a point where the pieces are internally simple enough that analytical solutions can be used Superposition (the adding together of analytical solutions) is used to deal with complexity Relatively uniform portions of the aquifer can be turned into model elements, and because there are no restrictions on element size or shape, the model can be built with exactly the level of detail needed to meet the modeling requirements with no excess elements As this solution is continuous over the model domain, it is independent of scale (this means that detailed local formation and comprehensive regional information can easily be obtained from the same solution) Use of the model has been limited in the past, because it required more computing power than did equivalent finite difference or finite element models, and because some types of situations could not be simulated as analytical solutions had not yet been developed The analytic element technique shows great promise and is quickly gaining in popularity as more analytical solutions are added to the code to handle more types of situations, and the typical computer available to the groundwater modeler gains speed and memory © 2001 by CRC Press LLC LA4111/ch17 Page 363 Wednesday, December 27, 2000 2:54 PM GROUNDWATER MODELING IN HEALTH RISK ASSESSMENT 363 Most groundwater problems can be addressed using models that have been created and tested by others The advantage to this, aside from not having to devise a solution method and write the computer program, is that the model will have undergone peer review This could be very important if the issue might go to court By the time a model is published, whether in the public domain (e.g, a model produced by the U S Geological Survey [U.S.G.S.]) or a commercial code, it should run without internal errors and should produce accurate results A recent survey of groundwater modelers revealed that widely used models are those developed by the U.S.G.S (Geraghty and Miller, 1992) The International Ground Water Modeling Center at the Colorado School of Mines in Golden, Colorado, a clearinghouse for information on models, offers information on types of groundwater models and computer codes Modeling Process The conceptual model is the first description of the aquifers’ nature, relationships with surrounding water resources, and boundaries of the system It forms the framework for the mathematical model The ideas in the conceptual model are then quantified by data collection and located in space through mapping The first step is to decide where the logical boundaries of the study area should be The boundaries referred to, in this sense, are typically hydrologic boundaries, and will not necessarily coincide with boundaries used in any other context Faulty assumptions about the conditions at boundaries, and oversimplified boundary conditions, are among the most common problems in setting up a valid model Once the boundaries are set, and the regional extent of the model is determined, the grid or mesh network can be designed, or the domain of the model can be divided into elements It makes sense that the finer the divisions, the more accurate the solution; it is also logical that finer divisions mean more work for the modeler and for the computer A typical approach is to use finer spacing or smaller elements near areas of concern to provide both accuracy and efficiency Finite difference grids should be arranged so that boundaries are represented as accurately as possible Changes in grid spacing, or finite element sizes, should be gradual When a well is part of the model, a node should be located close to the actual location of the well(s) When the model structure is complete, the process of data collection and preparation begins A listing of data requirements for predictive models is given in Table Data must be formatted and structured for computer input Much of the information starts out as maps where different parameter values are portrayed as areas of different colors or as areas between contour lines Such data must be discretized so that parameter values are known for each of the grid cells, or elements, of the models The discretization process could entail overlaying the scaled grid or mesh network on the map and interpolating values from map data, or it could be done by computer The above tasks are part of an iterative process Information gained during data collection may lead to changes in the conceptual model, and it may be necessary to change how the domain of the model is divided, or to revise the boundaries as the system is better understood © 2001 by CRC Press LLC LA4111/ch17 Page 364 Wednesday, December 27, 2000 2:54 PM 364 Table A PRACTICAL GUIDE TO ENVIRONMENTAL RISK ASSESSMENT REPORTS Data Requirements for Predictive Models Groundwater flow Topographic base map of the study area (adequate cultural features to identify and understand project location; streams, rivers, wetlands) Cross sections showing three-dimensional relationships between aquifers Maps of surficial aquifer: water table contours; saturated thickness and transmissivity distribution; boundaries and boundary conditions Hydrogeologic maps of all other aquifers: area underlain by aquifer; Hydraulic head distribution; boundaries and boundary conditions; Transmissivity and storage coefficient distribution Hydrogeologic maps of confining beds: areal extent; transmissivity and specific storage map Hydraulic connections between surface water and aquifers and between aquifers Solute transport Background information on natural water quality in the study area Effective porosity distribution Estimates of the hydrodynamic dispersivity Estimates of the variation in and distribution of fluid density Boundary conditions for the concentrations of any groundwater quality constituents of concern Constituent dispersion, adsorption, desorption, ion exchange, biological or chemical degradation, oxidation, reduction, complexation, dissolution, and precipitation Groundwater Precipitation and evapotranspiration Natural and cultural recharge areas Timing and volumes of stream discharge Timing and volumes of stream water withdrawals Timing and volumes of groundwater withdrawals Solute transport Contaminant sources and concentrations Ambient water quality distribution (areal and temporal) in the aquifer Quality of streamflow and of any imported recharge water As soon as an initial set of input data is ready, trial runs of the model can be carried out to see if the model can match the observed conditions for one of the data sets Comparison of observed and modeled heads gives the modeler an idea about the accuracy of the data that was entered into the model Adjustments to the model data are made until the comparison is satisfactory The model is then run with a different set of initial conditions or stressors and adjustments are made to the input data These model adjustments are called calibration or history matching Sensitivity analysis can be carried out to guide the collection of new aquifer data This analysis involves changing model parameters in a systematic fashion to learn which parameters the calculated heads are most sensitive to For example, if order-of-magnitude changes in transmissivity in one part of the model have very little effect on changes in head at your area of concern, then that area is not where you want to spend money on an aquifer test © 2001 by CRC Press LLC LA4111/ch17 Page 365 Wednesday, December 27, 2000 2:54 PM GROUNDWATER MODELING IN HEALTH RISK ASSESSMENT 365 The ultimate test of the quality of the model is whether it can match real conditions in a data set that the model was not calibrated with If the model passes this test, it is said to have been verified and can be used with greater confidence for predicting similar conditions The prediction phase of groundwater modeling is a process of assessing planned changes in water and land use for their possible effects on groundwater Sensitivity analysis can also be performed in this stage and can quantify the range of possible outcomes IV TECHNICAL ASPECTS OF CONTAMINANT TRANSPORT A Physical and Chemical Forces Influencing Movement Contaminant transport models calculate the movement of constituents of concern in groundwater as a function of their movement with (or in proportion to) the movement of groundwater (advection), and spreading or mixing of the contaminant with the natural groundwater Movement of contaminants in the subsurface is either under saturated or unsaturated conditions (above the water table) Unsaturated conditions are difficult to model because of the effects of periods of wetting and drying, and because of the addition of a gas phase into the problem This discussion will cover contaminant transport in saturated media Dissolved substances in water will move from areas of high concentration to areas of low concentration by diffusion The water does not have to be moving for diffusion to occur because the driving force is the concentration difference The flux of dissolved material is proportional to the concentration gradient When the water is in pore spaces of an aquifer, the diffusion process cannot work as fast because the dissolved material has to move around the matrix of solid matter Advection carries dissolved substances along with flowing water It relates to the average linear velocity and the effective porosity of the matrix If the water is not moving, no advection occurs When a contaminant is moved through an aquifer by advection, some of the water will travel faster than the rest This could be due to different flow path lengths, to water movement through pores of different sizes, and to friction which slows water movement adjacent to the wall of the pore These differences will ultimately bring water with the contaminant into contact with other water and create a diluted mixture of the two This mixing is mechanical dispersion At the same time, diffusion will be occurring between the contaminated water and the other water due to the concentration gradient The combined effect is called hydrodynamic dispersion In addition to the above, the concentrations of solutes in groundwater may also change due to transformation by biological or chemical processes, and due to adsorption of the contaminant to the matrix Appropriate parameters for many common contaminants have been derived from laboratory and limited field tests All of these processes are relatively well understood Where the hydraulic characteristics of the aquifer are well-known, a model based on the advection-dispersion equations will be useful Analytical solutions of the advection-dispersion contaminant transport equations are possible for less complex problems, and numerical © 2001 by CRC Press LLC LA4111/ch17 Page 366 Wednesday, December 27, 2000 2:54 PM 366 A PRACTICAL GUIDE TO ENVIRONMENTAL RISK ASSESSMENT REPORTS solutions are available for more complex problems Solution methods are analogous to those discussed for groundwater flow, with the exception that grid sizes must be kept small to avoid numerical problems This requirement for small grids or meshes may make these solution methods too inefficient for practical use Models for contaminant transport incorporate all of the uncertainties of the groundwater flow models plus all of the uncertainties involved in the movement of contaminants For this reason, several contaminant transport models rely on statistical approaches to describe both the aquifer materials and movement of solutes This approach acknowledges the reality that there is, in fact, a range of possible starting conditions and a range of possible outcomes Many popular contaminant transport models follow theoretical particles of contaminant along flow paths These methods are called particle tracking methods Each particle tracked represents a certain amount of contaminant, and by figuring out how many particles are in a given volume of the aquifer, the concentrations of contaminants are calculated One simple method, which accounts for both advection and dispersion, is called the Random Walk Model (Prickett et al., 1981) In this model, particles are moved along their flow paths by advection, and a statistical function is used to add an additional movement to each step, which represents dispersion B Model Misuse, Limitations, and Sources of Error Modeling in general is subject to several types of errors First, one might have started the process with an erroneous conceptual model, or have used an underlying groundwater flow equation that cannot handle the specific site conditions Second, roundoff error can accumulate during internal calculations, and truncation errors may have happened during the translation of the flow equations into algebraic computer code Third, your input data may be wrong Mathematical modeling of the fate and transport of contaminants in the subsurface is used to simulate the transport and behavior of substances when monitoring data are inadequate Obviously, this means that many times model results cannot be physically verified Trust in the results of these models can only come from an understanding of the underlying assumptions and confidence that the best possible data underlie the computer’s calculations Direct sampling of groundwater at exposure points (ambient monitoring) can and should be used in conjunction with groundwater modeling to determine concentrations of a substance of concern at a particular location Data from ambient monitoring is then used to refine the predictive value of the groundwater model Limitations to any model include uncertainties in input data, uncertainties in the simplifying assumptions, validity of the computer code, ability of the model to handle complexity, and the adequacy of model calibration, sensitivity analysis, and verification In some cases, the amount of underlying real data does not justify the use of a complex computer model (because most of the input data would have been created or extrapolated from the few real data points) It is thus possible that a relatively simple analytical model may incorporate less uncertainty than a complex numerical model in certain situations © 2001 by CRC Press LLC LA4111/ch17 Page 367 Wednesday, December 27, 2000 2:54 PM GROUNDWATER MODELING IN HEALTH RISK ASSESSMENT 367 C Groundwater Quality Monitoring Groundwater monitoring results, both for assessment of natural water quality and water quality impacted by the constituents of concern, must be collected and analyzed under very specific, predefined conditions in order for the data to be representative of exposure concentrations In any sampling study there is variation in concentration over time, there are errors introduced by sampling methodology and analysis, and there is the potential for study design errors Groundwater quality monitoring study plans must carefully consider the placement, depth, construction method, construction materials, and design of monitoring wells In addition, samples must be collected and handled following a specified field and laboratory QA/QC protocol V CONCLUSION When logical and scientifically supportable assumptions, mathematical methods, and professional judgments are used in a groundwater model, the model will add to the strength of the whole report Also, a model that correctly characterizes the physical setting, and that can be manipulated to assess the effect on groundwater flow and transport, due to a given set of environmental conditions, will facilitate efficient monitoring of contaminated sites and ensure that monitoring is done in the proper locations, at the proper depths, and at reasonable intervals in time and space If the hydrogeologic setting of a contamination problem is misunderstood through faulty or hasty analysis, money and time may be wasted and opportunities lost, in an effort to protect people and the environment from nonexistent hazards In the worst case, real pathways of exposure will remain unrecognized, putting people and the environment at risk Obviously, a weak groundwater flow and transport model will weaken the entire risk assessment report, and it may be judged unacceptable Unfortunately, weaknesses in groundwater modeling reports are not always evident A polished report, with impressive graphics derived from powerful computer programs, may mask an inadequate assessment of the geology and contaminant hydrogeology of the site Weaknesses may be discovered through peer review, however, or through detailed and diligent tracking by the project manager REFERENCES Anderson, M.P and Woessner, W.W., Applied Ground Water Modeling Simulation of Flow and Advective Transport, Academic Press, Inc., San Diego, 1992 Bear, J and Verruijt, A., Modeling Groundwater Flow and Pollution, D Reidel Publishing, Dordrecht, Holland, 1987 Bredehoeft, J.D and Pinder, G.F., Mass transport and flowing groundwater, Water Resources Research, 9, 194–210, 1973 Driscoll, F., Groundwater and Wells, Johnson Division UOP, St Paul, Minnesota, 1986 Fetter, C.W., Contaminant Hydrogeology, Macmillan, New York, 1993 © 2001 by CRC Press LLC LA4111/ch17 Page 368 Wednesday, December 27, 2000 2:54 PM 368 A PRACTICAL GUIDE TO ENVIRONMENTAL RISK ASSESSMENT REPORTS Fetter, C.W., Applied Hydrogeology, Macmillan, New York, 1994 Freeze, R.A and Cherry, J.A., Groundwater, Prentice-Hall, Engelwood Cliffs, NJ, 1979 Geraghty and Miller, Inc., Survey of groundwater model use, Geraghty and Miller Software Newsletter, Summer 1992 Hem, J.D., Study and Interpretation of the Chemical Characteristics of Natural Water, U.S Geological Survey Water-Supply Paper 2254, 1985 Konikow, L.F and Bredehoeft, J.D., Computer model of two-dimensional solute transport and dispersion in groundwater, Chapter 2, in U.S Geological Survey, Techniques of Water Resources Investigations, Book 7, 1978 McDonald, H.R.M.G and Harbaugh, A.W., A Modular Three-dimensional Finite Difference Groundwater Flow Model, U.S Geological Survey, 1984 Moore, J.E., Contribution of groundwater modeling to planning, J Hydrology, 43, 121–128, 1979 Pinder, G.F and Gray, W.G., Finite Element Simulation in Surface and Subsurface Hydrology, Academic Press, Inc., New York, 1977 Prickett, T.A and Lonnquist, C.G., Selected Digital Computer Techniques for Groundwater Resources Evaluation, Illinois State Water Survey Bulletin 55, 1971 Prickett, T.A., Naymik, T.G., and Lonnquist, C.G., A Random-Walk Solute Transport Model for Selected Ground Water Quality Evaluations, Illinois State Geological Survey Bulletin 65, 1981 Strack, O.D.L., Groundwater Mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1989 Trescott, P.C., Pinder, G.F., and Larson, S.P., Finite differences model for aquifer simulation in two dimensions with results of numerical experiments, Chapter 1, in U.S Geological Survey Techniques of Water Resources Investigations, Book 7, 1976 van der Heide, P and Beljin, M., Model Assessment for Delineating Wellhead Protection Areas, U.S Environmental Protection Agency, Washington, 1988 © 2001 by CRC Press LLC ...LA4111/ch17 Page 358 Wednesday, December 27, 2000 2:54 PM 358 A PRACTICAL GUIDE TO ENVIRONMENTAL RISK ASSESSMENT REPORTS pathways (so that exposure potential... breakthrough which © 2001 by CRC Press LLC LA4111/ch17 Page 360 Wednesday, December 27, 2000 2:54 PM 360 A PRACTICAL GUIDE TO ENVIRONMENTAL RISK ASSESSMENT REPORTS allowed the development of mathematical... equations, all of © 2001 by CRC Press LLC LA4111/ch17 Page 362 Wednesday, December 27, 2000 2:54 PM 362 A PRACTICAL GUIDE TO ENVIRONMENTAL RISK ASSESSMENT REPORTS which must be solved for each time step