Models and Modeling of Hydrogeologic Processes

Models and Modeling of Hydrogeologic ProcessesT. N. Narasimhan*

There is no doubt that the Western world no longer holds inthe highest esteem the one-time self-evident virtues of the lifeof the mind, the pursuit of understanding and the love of ideas.

J.R. Philip, 1991

ABSTRACTJ.R. Philip recently articulated a concern of many earth scientists

that computer-based mathematical models are impacting soil sciencepractice and soil science education in an undesirable way. Unrealisticfaith in the ability of these models to predict the future has encouragedoverzealous use of models at the expense of the observational enterprise.These real concerns draw attention to the fact that much needs to belearned about the proper use of models in general and computer-basedmodels in particular in the earth sciences. I was impressed by Philip'sthoughts, and here reflect on the current status and the role of modelsof hydrogeologic processes. While agreeing with Philip's concernsabout the improper use of models, I advance a perspective that models(analytical or numerical) are tools with inherent limitations. Despitetheir overenthusiastic use, computer-based models are potentially ca-pable of helping us advance our knowledge of earth processes inunprecedented ways. As we seek to exploit this tool to its full potential,we may be challenged to reexamine and refine our conceptual founda-tions so that hydrologic processes are described more precisely thanhas hitherto been possible.

Two BROAD APPROACHES are now available for solvingproblems involving hydrogeologic processes (de-

fined here to include processes of soil physics and soilhydrology). The first is the classical one of obtainingalgebraic, closed-form solutions to the governing differ-ential equations. The second is that of computationallysolving specific problems described in detail with numeri-cal values. Each of these methods merely constitutes ameans of evaluating the consequences of the underlyingphysical statement embodied in the governing equation.Thus, the governing equation, incorporating the laws ofNewtonian mechanics, constitutes the model and consti-tutes an abstract representation of the real system forpurposes of rational analysis. Yet, the word model isnow loosely taken by many to denote computationalalgorithms designed to obtain solutions by the numericalmethod.

Modeling, in this sense of computer-based numerics,is being enthusiastically pursued by many students andresearchers by way of model development and modelapplication. This enthusiasm is partly due to the fact thatsome agencies that provide research support perceiveresults generated by computer-based models to be reliablepredictions of the behavior of a complex world thatotherwise defies our capabilities of comprehension. Thecomputer is considered to be a gift to management from

Department of Environmental Science, Policy, and Management and Dep.of Materials Science and Mineral Engineering, 467 Evans Hall, Univ. ofCalifornia, Berkeley, CA 94720-1760. Received 8 Mar. 1993. *Corre-sponding author ([email protected]).

Published in Soil Sci. Soc. Am. J. 59:300-306 (1995).

a world of high technology. Moreover, in a generalatmoshpere of shrinking research budgets, it is perceivedby some that understanding complex system behavior canbe achieved less expensively through modeling exercisesthan through field work and experimentation, which tendto be more expensive.

Even as one part of the scientific community is enthusi-astically pursuing the modeling venture, taking advantageof the growing advances in computing technology, an-other part of the same community is concerned with theadverse impact of the same venture on the way wedo science. The intrinsic worth of the computationalapproach to the overall scientific method is being seri-ously contemplated. Concerns exist that the computa-tional approach is being given more importance thanwhat it merits and that it has an undesirable impact onthe way we do science and on the nature of scienceeducation.

Among those who have articulated concerns about theimpact of the new wave of modeling, Philip (1991)has been particularly eloquent. Commencing with thecommonly understood ethos of modern science as thebasis, he argued that the modeling venture is beingpracticed without necessary attention to the limitationsof natural science and that modeling is taken to be aninexpensive substitute for the observational venture. Anupshot is that students of natural sciences are movingaway from observation, experimentation, and data gath-ering, which are fundamental components of the scientificenterprise.

Philip has enriched the soil science literature throughsolving an impressive array of interesting problems per-taining to the soil, as well as through reflections onthe methodology of science in general. Following thethoughts exressed by Philip (1991), my purpose is toprovide some perspectives on modeling of hydrogeologicprocesses, their value, and their limitations. While recog-nizing the concerns of Philip on the unreasonable useof computer-based models, a perspective is advancedthat the computer constitutes a remarkably powerful toolfor us to use toward an improved understanding of hydro-geologic processes in unprecedented ways. Hydrogeol-ogy, as used here, refers generally to processes of theearth influenced by the flow of water. During the past twodecades, our insights into the workings of hydrogeologicsystems have been enhanced through the application ofa variety of mathematical models. As a background, Ibriefly review the status of mathematical modeling inhydrogeology, focusing attention on subsurface flow andtransport processes. This is followed by some thoughtson how these models may best be used to serve theultimate cause of understanding hydrogeologic pro-cesses. I end with a discussion of some of the challengesthat confront our ability to mimic the behavior of hydro-geologic systems through models.

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NARASIMHAN: MODELING HYDROGEOLOGIC PROCESSES 301

MODELS FOR HYDROGEOLOGIC PROCESSESSoil scientists in particular and earth scientists in general

are concerned with understanding the dynamic interactionsbetween the flow of fluids through earth materials, the deforma-tions induced by flowing fluids, the transport of energy, thechemical interactions between the fluids and solid phases, andthe transport of chemical species by flowing fluids. Tradition-ally, each of these processes is expressed in the form of apartial differential equation. These equations, coupled witheach other to account for the interactions between individualprocesses, constitute the overall basis for quantifying the evolu-tion of hydrogeologic systems.

The task of modeling, as is commonly understood, is totranslate the individual equations as well as their coupling intoa set of sequential instructions that can be implemented on adigital computer. Such an instruction set is frequently referredto as a program or a code. The earliest codes, which weredeveloped some 25 yr ago, addressed each process separately,in keeping with the limited speed and storage capacity of theearly computers. With the increasing speed of computation andvirtually unlimited computer storage, codes have progressivelybeen extended to handle the coupling between interactive pro-cesses. It is instructive, therefore, to begin by taking a brieflook at the nature of the governing equations and the tasksinvolved in translating them into computational codes. In whatfollows, attention is restricted to hydrologic processes of theearth's subsurface. Surface processes relating to stream flow,sediment transport, and the like are not included, although Irecognize that these processes are part of hydrogeology in thebroad sense.

Governing Equations and their Physical ContentThe governing equations of fluid flow, heat conduction,

and chemical diffusion are all mathematically described by adiffusion-type equation involving second-order derivatives inspace and a first-order derivative in time. This equation isfamiliarly known as a parabolic equation. The equation describ-ing stress-strain relations in a deforming porous medium isalso described by a diffusion-type equation, except that it doesnot involve the time component, being restricted to instanta-neous equilibrium between stress and strain. The parabolicequation has the general form

[1]V • AH V B ± S = C—at

where V is the gradient operator; Ay stands for the tensorialform of the conductivity parameter representing hydraulic con-ductivity (Kij, flow of water), thermal conductivity (KT, flowof heat), molecular diffussion coefficient (Dv, chemical trans-port), or the stress tensor (o</, deformation); B is the appropriatepotential, representing hydraulic head (cp, isothermal fluidflow), temperature (T, heat transport), strain (e, deformation),or concentration (c, chemical transport); S stands for appro-priate sources or sinks; C is the capacitance term representingspecific storage (Ss, flow of water), volumetric specific heat(cv, heat flow), or the retardation factor (R, chemical transport);and t is time.

In addition to diffusive transport, to handle the advectivetransport of heat or dissolved constituents, the parabolic equa-tion has to be augmented by the addition of a hyperboliccomponent to give rise to the advective dispersive equation:

11 dtwhere v denotes the pore velocity of the fluid and Ay combinescoefficients of molecular diffusion and hydrodynamic disper-sion. The left-ha,nd sides of Eq. [1] and [2] represent theaccumulation of appropriate mass or energy per unit bulkvolume within a vanishingly small volume of the geologicmaterial. On the right-hand side, the accumulating mass orenergy is converted into a change in appropriate potentialthrough the capacitance parameter. It is appropriate here tobriefly describe each of the processes of interest.

Isothermal Flow of WaterRichards (1931) formulated the general nonlinear parabolic

equation to represent the movement of water in an unsaturatedsoil. By extending Richards' equation to include the deformationof a porous medium in terms of Terzaghi's (1925) one-dimensional consolidation theory, one may obtain a governingequation that unifies the disciplines of soil physics, groundwaterhydrology, and geotechnical engineering (Narasimhan andWitherspoon, 1977). This equation, in addition to providingfor the dependence of hydraulic conductivity on soil waterretention, also provides for its dependence on porosity, andfor three distinct mechanisms for the release of water fromstorage, namely, desaturation of pores, expansion of water,and change in pore volume due to deformation. The work ofBishop (1955) helps account for the weak coupling betweenstress and water retention in the unsaturated state. This general-ized equation can also account for the changes in pore pressuregenerated by external stress changes (e.g., earth tides or baro-metric tides) with the help of Skempton's parameter B (Skemp-ton, 1954) implemented through appropriate source-sinkterms.

Isothermal Flow and Multidimensional DeformationTerzaghi's (1925) one-dimensional deformation assumption

is strictly applicable to laterally infinite systems. To handleporous medium deformation in response to fluid pressurechanges in laterally finite systems, geotechnical engineers pion-eered, during the late 1960s (e.g., Sandhu and Wilson, 1969),the coupling of the fluid flow equation with the three-dimensional consolidation equations of Biot (1941). Later ex-tensions by other researchers provided a basis for transientfluid flow and deformation in general three dimensions in thepresence of unsaturated-saturated flow and for failure inducedby fluid flow. The coupling between transient flow of waterand equilibrium deformation has successfully accounted fortime-dependent deformation such as land subsidence observedin the field. Explanations of time-dependent features have alsobeen attempted with the help of viscoelastic idealizations.

Flow of Multiple Fluid PhasesEnvironmental problems arising from spills of organic chem-

icals have motivated active research in recent years on thesimultaneous flow of two or more immiscible fluids, the flowof each being governed by Darcy's law. Governing equationshave been formulated by many researchers (e.g., Abriola andFinder, 1985; Baehr and Corapcioglu, 1987; Faust et al.,1989). The governing equations consist of a set of conservationequations for each of the fluid phases and account for capillarypressure effects at fluid-fluid interfaces, mass transfer dueto volatilization-condensation, and density-induced buoyancy

302 SOIL SCI. SOC. AM. J., VOL. 59, MARCH-APRIL 1995

effects. A key feature of these models is the dependence ofphase permeabilities on phase saturation (relative perme-ability).

Fluid Flow and Heat TransportThe coupled transport of water and heat is of fundamental

importance in understanding evapotranspiration processes inthe vadose zone. Philip and de Vries (1957) were among theearliest to systematize the analysis of water and heat transportin the vadose zone. Using a single governing equation in onedimension, they accounted for the transfer of heat by flowingwater as well as by water vapor and for the movement ofwater driven by temperature gradients. Following this concep-tual lead, more elaborate models have since been assembled(e.g., Pruess, 1988) that consist of a separate conservationequation for the liquid phase, the vapor phase, an air phasethat includes all the gas phases, and energy transport. Inaddition to accounting for latent heat effects involved in evapo-ration and condensation, the equations also account for capillarypressure vs. saturation relations, relative permeability effects,density effects, and segregation of wetting and nonwettingphases in pores of different sizes (heat-pipe effect).

Chemical TransportSome of the most exciting developments in mathematical

modeling are currently underway in the area of reactive chemi-cal transport. Modeling of chemical transport has attracted theattention of many researchers because of the impetus providedby concerns about groundwater pollution. Early transport mod-els developed during the late 1960s focused attention on themechanics of transport, notably hydrodynamic dispersion.However, the early 1980s witnessed the recognition that fluid-solid interactions are extremely important in evaluating andremediating contaminated groundwaters. An impressive litera-ture now exists on processes involving adsorption, ion ex-change, redox-controlled precipitation and dissolution, andbiologically mediated transformations. Literature is also grow-ing on the behavior of dissolved organic compounds, whichmay undergo degradation along a variety of pathways, con-trolled by biotic or abiotic factors.

Developments in reactive chemical transport modeling hasdone much to bring disciplines together. In the field of geo-chemistry, Garrels and Christ (1965) paved the way for quanti-fied geochemical calculations by systematizing the analysis ofmineral equilibria. Following this, Helgeson et al. (1970)developed the early algorithms for calculating mass transferin aqueous solutions. With the rapidly increasing power ofcomputational devices, the late 1970s saw the appearance inthe literature of a number of equilibrium speciation models(Wolery, 1979; Parkhurst et al., 1980; Felmy et al., 1984; Balland Nordstrom, 1991) and have enabled the use of geochemicalmodels for assessing the equilibrium chemistry of multicompo-nent systems. Similarly, models for sorption, ion exchange,and other surface processes have also been developed (e.g.,Sposito and Mattigod, 1980).

Even as the geochemists were developing models just toevaluate the geochemical equilibria, the 1970s also saw theinitiation of attempts by some earth scientists to develop modelsfor advective-dispersive transport of more than one chemicalspecies, giving consideration to chemical reactions. This ap-proach to chemical transport modeling continues. Notableamong these are the contributions of Dutt and Tanji (1972),Grove and Wood (1979), Walsh et al. (1982), Rubin (1983),Miller and Benson (1983), Cederberg et al. (1985), Lichtner(1985), and Carnahan (1987). The early 1980s saw an alternateapproach being taken by some researchers to reactive transport

modeling (e.g., Narasimhan et al., 1986; Liu and Narasimhan,1989). In this approach, an existing comprehensive geochemi-cal model (e.g., PHREEQE by Parkhurst et al., 1980) is usedas a module and coupled with a transport computation modulefor advective-dispersive transport of several components.

These models demonstrate the feasibility of coupling thegeochemical models with chemical transport models so as tounderstand a variety of earth processes including groundwatercontamination, evolution of soil profiles, dynamics of weather-ing processes, formation of mineral deposits, and the evolutionof petroleum reservoirs. Currently attention is being directedtoward extending the equilibrium models to incorporate thekinetics of chemical reactions and the transient changes in thegroundwater flow field influenced by chemical changes in thesystem.

More General Coupling of ProcessesEven with the currently available impressive speed of com-

putations and large computer storage, there are limits to thenumber of processes that can be coupled dynamically to obtainquantitative solutions. It is commonplace to find models thatcombine two processes (e.g., flow and deformation, flow andheat transport, flow and chemical transport). Algorithms tohandle three interacting processes have begun appearing in theliterature only within the past few years (e.g., fluid flow, heattransport, and deformation; fluid flow, heat transport, andchemical reactions; fluid flow, chemical transport, and compre-hensive geochemical modeling). The limitations of computa-tional time restrict the scope of many of these models tosteady-state fluid flow or a single space dimension. The cou-pling of all the four processes (i.e., fluid flow, deformation,heat transport, and chemical transport) is a task for the future.

Numerical ImplementationMany researchers have developed computer programs to

simulate the various types of earth systems described above.Well-documented codes for use by the technical communityare available through clearinghouses such as the InternationalGround Water Modeling Center at the Colorado School ofMines or the Energy Science and Technology Software Centerat the Oakridge National Laboratory; In addition, useful re-views of available models have been compiled, especially inconnection with the radioactive waste disposal efforts of theU.S. Department of Energy (Mangold and Tsang, 1991).

Broadly, the solution task consists of two parts. The firstconsists of subdividing the flow domain into discrete subdo-mains and associating the mean intensive properties (potentialssuch as hydraulic head, temperature, concentration, and stress)of each subdomain with a representative nodal point. Thesecond task is to implement the governing equation of eachof the processes and compute the change in the potential ateach nodal point. To this end, the accumulation of mass orenergy is first evaluated using the appropriate equation ofmotion and then the net accumulation of mass or energy isconverted, with the help of an appropriate capacitance term,into a mean change in potential at the nodal point.

Clearly, the computational effort involved will increase withthe number of subdomains involved. The equation of motion(Darcy's law, Pick's law, the Fourier law) involves the evalua-tion of spatial derivatives of potential. Therefore, improvedaccuracy in the evaluation of gradients will need to be achievedthrough an increase in the number of subdomains (equivalently,closer spacing between spatial points). Superimposed on this,the computational effort also increases with the number ofdependent variables (also referred to as degrees of freedom)


to be solved for at each spatial point. Thus, when severalprocesses are coupled, the requirements of optimal spatialdiscretization and the number of dependent variables lead toexceptionally large matrices, which are very cumbersome tohandle even with very powerful computing devices.

Another aspect of coupled systems, stemming from theirintrinsic physical nature, relates to the wide variations in thereaction times of individual processes. In complex systems,the transient migration fluid pressure, thermal fronts, andchemical interactions may proceed at very different time scales.Simultaneous handling of these processes with diverse timeconstants leads to the mathematically difficult task of solvingstiff matrices.

These logistic difficulties of solving problems related tocoupled earth systems are being overcome by researchersthrough various innovations such as moving grids (for improv-ing accuracy of evaluating steep local gradients), quasi-linearization (decoupling processes within small time steps),and improved iterative techniques such as conjugate gradientmethods to speed up solution of stiff matrices.

During the past decade, these emerging computational toolshave been applied to test hypotheses and gain insights intoa variety of hydrogeological problems including infiltration,evaporation, and transpiration; rainfall-runoff relationships inwatersheds; land subsidence, slope failure, and triggered earth-quakes; barometric pumping of geogases; topographically con-trolled airconvection and moisture circulaton in hills; weather-ing of rocks and secondary enrichment of ore minerals;evolution of ground water contaminant plumes; regionalgroundwater motion; hydrothermal convection systems; andso on. The successes so gained have encouraged regulatorsand managers of water and natural resources to see in thesemodels a means by which answers can be generated to helpin the most difficult environmental management decisions.Consequently, mathematical models are becoming the pre-ferred tool of analytical support for most resource managementprojects, large and small. As a result, substantial financialresources are being diverted to support mathematical modelingventures.

With the increased use of models, some researchers arebeginning to recognize that despite their success in generatinginsights, models are fallible and their role in modern scienceand engineering is subject to limitations. Indeed, some, suchas Philip (1991), argue that too much is being expected of thecomputer-based models to the detriment of science and scienceeducation.

ROLE OF MODELS IN HYDROGEOLOGYModels are intrinsic to the scientific method. They

help recognize patterns in the structure of nature andthe patterns so recognized are used to extrapolate andto predict. This very basis of modern science has aparadoxical constraint: a given pattern cannot be ex-tended uniquely. It is well known in mathematics thatgiven n members of a sequence, one cannot uniquelypredict the n + 1 member. This constraint, nevertheless,does not negate our ability to use models. We are simplyrequired to be modest in their usage. Models in thissense include the classical analytical solutions to partialdifferential equations (used so well by Philip throughouthis career) as well as those that have arisen from thescience-computer symbiosis. As Philip (1991) put it,"Perhaps, after all, Newton and Einstein were simplymodelers, and it may be that what sets them apart is

that they were especially wise and especially humble."In a way, the situation is similar to the fact that Godel'stheorems have not stopped the enterprise of mathematicsbut merely moderated it. These theorems assert thattheorems exist in axiomatic set theory, which can neitherbe proved nor disproved, and that no constructive proce-dure can prove that axiomatic set theory is consistent(Ballentyne and Lovett, 1980).

Against this backdrop, it is worth reflecting the concernamong some earth scientists that, in the wake of thecomputer-science symbiosis, modeling is being misinter-preted and models are being misused. Philip (1991)believes that this is partly due to differences in the natureof the ventures of science and professional practice.

Despite the spectacular achievements of classical andmodern physics in predicting the behavior of certainphysical systems, the applicability of the principles ofphysics to the earth is limited. The thoughts of Weinberg(1972) (as cited by Philip, 1991) are especially appro-priate. He has discussed a category of questions that"can be stated in the language of science [but which]are unanswerable by science."

Hydrogeologic models, as we know them, are dataintensive. Even if we assume that the conceptual basisand the computer algorithm are entirely correct, theparameters and the geometry of the system cannot becompletely characterized to uniquely apply physical"laws." This inability to completely characterize the sys-tem is not due to resource limitations but, rather, it isphilosophical; even with unlimited resources we cannotcompletely characterize the subsurface system.

Testability of models is fundamental to the scientificmethod. In the groundwater literature, the venture oftesting model outputs against observations of naturalsystems is referred to as validation. With much experi-ence in using mathematical models to predict the behaviorof groundwater systems, Konikow and Bredehoeft (1992)arrived at the conclusion that groundwater models cannotbe validated but only invalidated. That is, these modelsare not testable against real systems. They can be usedto test hypotheses and develop insights.

Despite these intrinsic limitations, hydrogeologic mod-els are perceived by some research sponsors to be capableof providing precise answers to aid decision making.With this expectation, research funds are made availablefor modeling exercises. Zeal for using models has led toexercising models even when data are extremely sparse.Indeed, model developers often find themselves dismayedby the misuse or abuse of models by end users withsketchy knowledge of the processes involved, compatibil-ity between available data and model sophistication, andthe intrinsic worth of the models.

The past decade has seen significant reductions inresearch budgets for science, severely impacting theavailability of funds for research. The shrinking re-sources have caused a reduced availability of resourcesfor field work and experimental observations. In thisatmosphere, computer-based modeling exercises are per-ceived to be attractive substitutes for field work andexperiments. In this context, Philip's (1991) comment isrelevant, "A disturbing aspect is that computer modeling


has largely supplanted laboratory experimentation andfield observation as the research activity of students."

In view of the noteworthy limitations detailed above,how useful are computer-implemented models in hydro-geology? It appears that the model-computer symbiosishas to be accompanied by a concurrent educational pro-cess of properly using its outputs. The thoughtful con-cerns of Philip, frequently echoed by other earth scien-tists, warn us that we are lagging behind in an importanteducational need. Nevertheless, the computer is not avillain.

As a tool available to science, the computer is unlikeanything humanity has seen before. It is hard to speculatewhat course mathematical physics might have taken hadNewton and Gauss had access to the computer; verydifferent mathematical techniques might have emerged.The concern about the computer-model symbiosis is notto be construed as a negation of the computer as a tool.Within the philosophical bounds of its role in the scientificframework, the computer can help evaluate complexinteractions and visualize complex objects in ways thathave never been possible with the classical analyticalmethods of solution. The availability of computers pro-vides exciting opportunities to rephrase the governingequations in a manner that is more general than whatwe have now.

Observational lore, innovative experimentation, andinterpretation are fundamental components of modernscience. They are orthogonal and independent. Meredata gathering will be philately without interpretationand pattern recognition. Without an abstract model inthe human mind, neither pattern recognition nor experi-mentation will be possible. Therefore, the three venturesof field observation, experimentation, and modelingshould proceed in parallel, complementing each other.Here, modeling includes the use of the computer as wellas the solution of idealized partial differential equationsby classical methods.

Perhaps the woes of the computer-model symbiosisis an inevitable outcome of how classical mathematicalphysics has been dominated by the language of the differ-ential equation. The language of the differential equationis abstract. To be constantly aware of the links betweenthe partial differential equation and its physical basisrequires a high degree of talent and training. Moreover,obtaining solutions to the partial differential equationsrequires considerable skills in mathematical manipula-tions, not necessarily based on physical intuition. Withthe advent of the computer, we continue to carry theclassical legacy. The paradigm of numerical modelingis that numerical models are merely approximate solversof the partial differential equation. The prevailing percep-tion is that once the governing equations (i.e., the models)are set up, obtaining the solution is purely a mathematicaltask, whether classical methods or computer methodsare used. As a result, we have a new breed of technicianscalled modelers. A discernible separation exists betweenresearchers of the "real world" and the modelers. Expen-sive projects are supported to provide data "for the mod-elers."

On reflection, it appears that a well-rounded science-

engineering education must have a balance between ob-servation and interpretation. Although at any given timeone may be devoting primary attention to any one ofthese ventures, one must be cognizant of the worth of theparticular venture in the larger context. Observationalistsmust have a sense of the framework of interpretation,and computationalists must have a sense of the physicalsystems they are trying to understand.

Even as the computer has burst on the scene and hasshaken up the classical ways of problem solving, oneimportant thought is probably being overlooked. Thecomputer may be providing us with the most powerfulmeans yet of simplifying the governing equations andreducing the gap between the model and the real world.Perhaps the computer may provide a way to reduce thedistinction between Observationalists and modelers.

SOME CHALLENGES TO MODELINGRecognizing that the modeling venture, especially the

computer-based kind, is a valuable component of modernscience, it is worthwhile to look at some of its challenges.The gamut of modeling extends from its own worth asa tool to how it is practically applied to solve problems.Models can be used to great advantage for developinginsights and testing alternate hypotheses. They may alsobe applied to data from specific sites to calibrate and toextrapolate. The discussion below is restricted to theproblem of assuring that the tool is internally consistentso that it can be applied with confidence to problems ofrelevance. Internal consistency is used to denote not onlysomething as obvious as mass conservation but also lessobvious aspects such as the nature of flow geometry(disposition of flow paths and their relation to surfacesof potential) and how it optimizes itself to the functionsthat force it.

A fundamental issue is that the model, before itsapplication, must be ascertained to be internally consis-tent. In the case of a numerical model in hydrogeology,this means that the solution generated by the model toany problem must be internally consistent. This assuranceof the veracity of the solution is often referred to asmodel verification.

When one obtains a closed-form solution to the para-bolic equation, one can readily verify the logical consis-tency of the solution by evaluating the second derivativein space and the first derivative in time and establishingthat they are equal. However, no such logic is availablefor verifying a solution generated by a numerical model.As a result, solutions generated by numerical modelsare seldom verified. They are treated as approximatesolutions. What is done instead, in practice, is merelyto show that the algorithm is potentially capable of solvinga category of problems. Accordingly, verification exer-cises are carried out in which test problems for whichanalytical solutions are already available are solved bythe numerical model to demonstrate that the numericalresults agree closely with the analytical solution. Thisdemonstration, nevertheless, does not guarantee that thesolution generated by the algorithm for any arbitrary


problem (for which no analytical solution is known toexist) is internally consistent.

Testability of solutions is a fundamental scientific re-quirement. Therefore, it is necessary to develop methodsby which any numerical solution from a hydrogeologicmodel can be verified for internal consistency. One wayto meet this challenge would be to reexamine the basisand the format of the governing equations. Some thoughtsalong this line are presented below.

One of the basic challenges in numerically solvingthe parabolic equation is the task of evaluating spatialgradients of potential. Because gradient is defined as theslope over an infinitesimally small distance, the onlyway to assure the accuracy of an estimated gradient isto have spatial points located infinitesimally close to-gether. It is for this reason that numerical solutions ofhydrogeologic systems are qualified by the conventionalwisdom, "the finer the discretization, the better the solu-tion."

On reflection, one could question this wisdom. Onecould argue that this situation primarily exists becauseDarcy's experimental observation is cast in the formatof a gradient to enable representation with a differentialequation. Suppose one chooses to write Darcy's observa-tion in the form of Ohm's Law:

— A(pR [3]

where R is the flow resistance through a flow tube offinite length. We know that R is a function of the propertyof the material in the tube as well as the geometricalattributes of the flow tube. Equation [3] expresses fluxcorrectly regardless of whether the gradient is constantor variable within the flow tube. Conversely, one couldreason out that if the geometry of the flow tube (that is,its converging or diverging shape) is known then onecould calculate gradients exactly. The implication is thatinability to numerically evaluate gradients exactly reflectsa lack of knowledge of flow geometry; if flow geometry isknown, infinitesimal discretization is unnecessary. Here,flow geometry denotes the individual and collective attri-butes of diverging and converging flow paths. Indeed,in a heterogeneous medium subject to nontrivial boundaryconditions, the flow problem is described by two indepen-dent variables, potential and flow geometry. If the com-puter can be used to facilitate the simultaneous handlingof potential and flow geometry in innovative ways, wemay be in a position to supersede the paradigm thatnumerical models approximately solve differential equa-tions. In fact, we should be able to generate fully self-consistent, accurate solutions with numerical models. Anumerical model will then exist in its own right, insteadof being a surrogate to the differential equation.

The transient aspect of the parabolic equation arisesbecause of the capacitance parameter. If we consider afinite mass of material (say, water in a calorimeter), itscapacitance (for example, its heat capacity) is uniquelydefined if and only if it is well mixed and it changesfrom one uniform temperature state to another. Supposewe have an ill-mixed calorimeter; then a true capacitance

is not definable because the change in temperature mayvary from location to location within the calorimeter.In a sense, an elemental volume in a nonsteady flowsystem is like a poorly mixed calorimeter. Within it, themagnitude of change in temperature during an intervalof time will be different at different locations. A capaci-tance in the classical sense is not definable for this volumeunless perhaps the volume is infinitesimally small. Evenso, if one wishes to carry out an integration of capacitancefor a finite elemental volume, one still has to contendwith the fact that capacitance is not definable. This is aconceptual issue of basic significance when dealing withdynamic systems and needs to be resolved.

There is yet another philosophical issue concerningthe physical content of the parabolic equation. Note thatthe equation of motion (Darcy's law, Pick's law, theFourier law) is a statement of an empirical observationof a steady-state system. The statement is valid for aflow tube within which flux is constant in time andno accumulation of mass or energy (as appropriate) isoccurring. So also, the capacitance term, stemming fromthe equation of state, pertains to changes that occur whenthe system changes from one state of equilibrium betweenthe state variables to another. If we take a segment ofa flow tube through which flow is transient, flux is notconstant through the tube because accumulation occurswithin the tube. If so, how may we justify the applicationof the equation of motion, which requires that inflow beequal to outflow? Presumably, implicit in the parabolicequation is the assumption that, in a vanishingly thinelement of the flow tube, one can simultaneously justifythe assumption of steady flow as well as accumulation.As long as one is preoccupied with obtaining closed-formsolutions, this latent paradox never surfaces. However,if one wishes to carry out numerical integration of theflow process through the flow tube, this paradox cannotbe ignored. This perhaps is an expression of the well-known fact in mathematical physics that a true variationalprinciple does not exist for a nonsteady system. In otherwords, we do not know if a unique criterion exists bywhich the transient system optimizes itself as it evolvessuccessively in time.

The issues raised above show that our conceptualfoundations are not as complete as one might assume,despite their long usage. In fact, deeper questions willarise when we extend these arguments to multiphasesystems involving complex interactions. The availabilityof the computer provides us with the motivation to re-formulate our conceptual foundations so that any solutionthe computer generates, however complex the problem,can be independently verified for its consistency.

CONCLUSIONSThe digital computer is a remarkably powerful tool

available to modern science and technology. Throughcomputer-based models, we can now gain insights intothe behavior of complex hydrogeologic systems that de-fied our analytical abilities only a decade ago. Yet,as a tool of the scientific method, the computer-basedmathematical model has inherent limitations. The model,


it appears, will best serve its purpose as a complementto the observational venture and not as a substitute forit.

ACKNOWLEDGMENTSThis work was ppartly supported by the Regional Research

Funds of the U.S. Department of Agriculture through theCooperative State Research Service.

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Models and Modeling of Hydrogeologic Processes