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3 Flow Processes INTRODUCTION As suggested by the discussion in the previous chapter, most ap- plications of ground water models to aid decisionmaking begin with the potential energy and flow of water alone. First of all, ground water flow models, with their focus on the prediction of head, vol- umes, and velocity of flow, can be important tools in the assessment and development of water resources. For example, predictions of the economic yield of an aquifer, or of the impacts of new or increased pumping on existing wells, or of ground water recharge below ir- rigated agriculture, all require an understanding and prediction of ground water head and flow. Second, ground water flow models are a crucial component of all analyses of contaminant transport because of the need to define the ground water velocity field. As noted in a number of sections of this report, advection with the flow field is often the dominant process controlling the direction, if not the rate, of transport. In the absence of significant density differences caused by contaminant concentration differences, the velocity field is independent of chemical and biological transport processes. Thus transport modeling studies usually begin with a prediction of the velocity field based on a ground water flow model. Historically, the earliest ground water models were developed to 79
80 GROUND WATER MODELS predict head and volumetric flow in fully saturated, porous (nonfrac- tured) geologic environments. Because of this relatively long history, saturated continuum flow models have been investigated extensively and are quite well understood in the context of a wide variety of prob- lems. The modeling of flow in unsaturated, nonfractured settings has a shorter history and is largely dominated by problems of understand- ing and predicting infiltration from rainfall, irrigation, rivers, canals, and ponds. Our understanding of such models is less sophisticated than our understanding of saturated flow models. Least well devel- oped, and indeed only in its infancy, is our understanding of recent attempts to mode! head and flow in both saturated and unsaturated fractured environments. For all three cases saturated continuum flow, unsaturated continuum flow, and fracture flow most of our understanding of flow modeling has been gained for problems requir- ing predictions of head and volumetric flow rates. The demands of contaminant transport prediction, in which the critical flow variable is velocity, are more challenging and have only recently become the focus of attention in ground water flow modeling. Although ground water flow modeling ~ older and more ad- vanced than ground water transport modeling, many issues and un- certainties remain in the application of flow modem to decisionmaking problems, particularly those involving transport. This chapter sum- marizes the committee's sense of the state of the art of ground water flow modeling and of the issues related to the current and future use of flow models in decisionmaking. SATURATED CONTINUUM FLOW As mentioned in the introduction to this chapter, the earli- est models of ground water were saturated continuum flow models. There are several reasons for this early interest in fully saturated flow First, the dominant problems 30 years ago were problems of water resources development. Attention was focused on questions of avail- able ground} water resources ant! on the impacts of the installation of wells on these resources (and surface water resources). Variables of interest were head and volumetric flow rate. Relevant spatial scales were large aquifers and aquifer systems. Ground water quality was assumed to be high, except in areas of saltwater intrusion. A second impetus for the early focus on fully saturated flow was the relative ease with which the physical processes of greatest
FLOW PROCESSES 81 importance to water resources questions can be represented. Gov- erning equations are linear or nearly so; assumptions of spatial ho- mogeneity and temporal steady state are often justified; one- and two-dimensional formulations yield relevant, useful information; and relatively unsophisticated numerical approximation techniques are adequate. Put simply, the easiest types of ground water problems to mode! are fully saturated flow problems. This ease is only relative, however. fully saturated ground water flow modeling remains quite challenging. This is especially true for problems of contaminant transport below the water table. For such problems, the role of ground water flow modeling is to provide an estimate of the flow velocities. Head predictions are of little direct interest. Velocity estimates, however, are usually based on hydraulic head differences and therefore are much more sensitive to modeling errors than are estimates of hydraulic head alone. In addition, satisfactory predictions of transport often require that the velocity field be well predicted on fine spatial grids. The use of large-scale average velocities, which are usually very adequate for water supply problems, can place high demands on the dispersive component of a transport model, demands that we are only just beginning to understand (see Chapters 2 and 4~. This need for high spatial resolution presents formidable challenges for data collection, parameter estimation, mode} formulation, numerical methods, and computational power and speed. State of the Art The physical processes controlling the flow of water through fully saturated porous rock or soil are well understood, both theoretically and experimentally. The mathematical statements of the funda- mental physical laws governing general fluid motion-conservation of mass, momentum, and energy which are collectively known as the Navier-Stokes equations, are universally accepted (White, 1974~. More important, the simplifications of these equations, which lead to Darcy's law for fully saturated flow through porous media (equation t2.43), have been investigated both in the laboratory and in theory. The Darcy equation is known to yield good predictions of head and flow under a wide range of conditions encountered in the subsurface (cf. Freeze and Cherry, 1979~. The conditions under which the Darcy equation is not adequate for prediction have been reasonably well delineated. Darcy's law
82 GROUND WATER MODELS is known to fad! for high-velocity flows, which might occur in very porous grave! or boulder deposits, karat terrain, or the immediate vicinity of pumping wells (Bear, 1972, pp. 125-127~. Darcy's law is suspected to fail for flow through extremely small pores under low- pressure head gradients (Freeze and Cherry, 1979, p. 72), conditions that might occur at great depth, for example, in the vicinity of potential radioactive waste repositories. Considerable controversy about this behavior exits. In neither case of failure is there adequate support for universally accepted alternative formulations (short of the Navier-Stokes equations), although a number of models have been proposed (Bear, 1972,pp. 170184~. Prediction uncertainty for these flows must be considered larger than for the vast majority of flows for which Darcy's law is a valid approximation. The mathematical properties of the governing partial differential equation for fully saturated continuum ground water flow (Equation t2.5~) are well understood. The form of the equation is typical of a wide variety of physical problems and so has been studied extensively in many contexts. Because the equation is linear, many powerful tools of mathematical analysis are applicable. Exact, analytical so- {utions are available for a wide variety of problems characterized by very simple geometries, boundary conditions, initial conditions, and parameter fields (usually homogeneous). These analytical solutions are essential in testing and verifying approximate numerical solution techniques, and they often provide considerable insight into more complex problems. In cases where prediction of detailed velocity or concentration fields is unnecessary, analytical solutions often provide adequate precision for certain problems and goals. For those problems in which the simplifications necessary for at- taining analytical solutions are inappropriate (there are many), nu- merical approximation techniques are highly developed and widely available. A sophisticated literature exits, including several texts devoted exclusively to ground water flow problems (Huyakorn and Pinder, 1983; Remson et al., 1971; Wang and Anderson, 1982~. Nu- merical accuracy and its control are well understood. A number of well-documented, robust, and flexible computer codes are readily available (Bachmat et al., 1980; see also information from the Inter- national Ground Water Modeling Center, Indianapolis, Indiana). In addition, in the last few years a variety of computational and graph- ical tools have been introduced, such as pre- and post-processors and expert systems, designed to aid in the application of such codes. Proper use of numerical codes, however, still requires considerable
FLOW PROCESSES 83 training and experience, and it is unlikely that solution procedures will ever be fully automated. Until recently, most numerical methods have focused on efficient and accurate computation of ground water heads for one- and two- dimensional problems. However, in response to the increased avail- ability of affordable computational power, the last few years have seen significant progress in three-dimensional solution techniques. Such techniques are no longer experimental and are beginning to be used in practice (e.g., Ward et al., 1987~. In addition, researchers are now focusing attention on the accurate computation of head gradients (velocities), a task much more challenging than accurate computation of heads (Bear and Verrujt, 1987~. The nature of the parameters appearing in the various forms of the fully saturated ground water flow equation ~ reasonably well understood. Both the hydraulic conductivity and the specific storage are empirical parameters that arise from the simplifications leading to Darcy's law and a workable statement of continuity (see discussion on ground water flow in Chapter 2~. While they are not directly measurable, theoretical and experimental studies have clarified how these parameters depend on the properties of the rock and of the fluid when used to predict flow in laboratory columns and boxes (Bear, 1972, pp. 132-136~. Less well understood are the natures of these parameters when used to predict average flows over large distances through heterogeneous geologic deposits. Theoretical studies have explored the relationship between large-scale conductivity (and/or transm~ssivity) and the variability of local conductivity (cf. Dagan, 1986; Gelhar, 1986), but our understanding remains limited. The state of the art of fully saturated continuum flow modeling is least well developed in the area of hydrologic characterization. The magnitude of flow parameters and their spatial variability currently remain unpredictable a priori. As discussed in detail in Chapter 6, this unpredictability is a major source of uncertainty in ground water flow modeling today. It is of course impractical to fully characterize an aquifer's perme- ability distribution via small-scale permeameter testing, since such tests are conducted on disturbed samples of material and are ex- pensive. In addition, simple correlations between more readily mea- surable geophysical and soil physical parameters have proven elusive (e.g., Lake and Carroll, 1986, pp. 181-221~. Several investigators have carried out detailed studies of the spatial structure of per- meability and porosity in ground water environments (e.g., Byers
84 GROUND WATER MODELS and Stephens, 1983; Hoeksema and Kitanidis, 1985; Smith, 1981; Sudicky, 1986~. Petroleum engineers have devoted considerable at- tention to of! and gas reservoir characterization, developing both techniques and insight that are useful to hydrogeologists. These studies have shown that ground water geologic environments are highly variable, but in general, the quantitative knowledge remains very limited. Parameter values must in general be inferred from field obser- vations of head response to stress. Well tests are the most obvious example. In most applications of ground water flow models, param- eter values are obtained via calibration using some type of "inverse technique," leavened by well test estimates and geologic knowledge. Parameter values are chosen that yield satisfactory predictions of observed head at selected observation points (usually few in number) under known conditions. A large body of theoretical literature has grown up around the ground water "inverse problem" (Yeh, 1986~. It is known from these studies that parameter values estimated in this way are nonunique and are very sensitive to errors in measured head data. A number of automated techniques have been suggested for dealing with these problems and for quantifying the resulting uncertainties. There is, however, no agreement on the best approach to this problem, nor is there reason to expect such agreement. Automated techniques remain experimental in practice, and most calibrations proceed by trial-ancI-error fitting procedures with no quantification of uncer- tainty. In the hancis of an experienced, knowledgeable hydrogeologist, trial-and-error techniques can yield satisfactory results for problems requiring modest spatial resolution. Parameter estimation remains, however, one of the crucial challenges in successful saturated contin- uum flow modeling, especially for problems of contaminant transport. Because of the inherent uncertainty in defining the parameter fields for ground water models, a number of investigators have begun exploring the uncertainty in head and velocity predictions that results from parameter uncertainty (e.g., Dagan, 1982; Smith and Freeze, 1979a,b). While limited by a number of restrictive assumptions, these studies are showing how predictive uncertainties may be related to uncertainties in parameters and boundary conditions. They also suggest that prediction uncertainties can be large, especially for velocities. While not yet common in practice, techniques for the quantification of uncertainty are gradually becoming more accepted and accessible.
FLOW PROCESSES 85 Implications for the Use of Saturated Continuum Flow Models in Decisionmak~ng Because of the relatively long history of development and use of saturated continuum flow models, the issues surrounding their application to modern decisionmaking problems are generally not conceptual or theoretical, but are practical. As noted in the previ- ous section, the physical processes controlling saturated flow are well understood, and the mathematical models describing these processes have been studied extensively. The challenges posed by practical application arise in situations where it is not feasible to mode} a flow system at the spatial and/or temporal scale appropriate to con- ceptual and mathematical understanding. Saturated continuum flow models rest on fluid mechanical principles and laboratory column val- idation of Darcy's law. Field application at this scale is not possible; we lack complete data sets, and if we had such data sets, modeling costs in time and computer resources would be extraordinarily high. Therefore, successful application of ground water flow models rests on the skill and art of the hydrogeologist in understanding when, where, and how to simplify and respond to a lack of information. The next few paragraphs summarize the most important issues that must be addressed in applying saturated continuum flow models to practical ground water problems, given the current state of the art. Spatial Dimensionality Many ground water flow problems may be successfully addressed by assuming that flow occurs in only one or two dimensions, i.e., in a single direction or in a plane. Several texts provide careful discussions of the implications of such an assumption (e.g., Bear, 1972; Freeze ant] Cherry, 1979~. The computational savings are obvious. The cost of such a simplification is that mode} parameters are defined as spatial averages of the "fundamental" parameters (e.g., transmissivity is a depth-average of hydraulic conductivity) and that the predicted responses (head and/or velocity) are similarly averaged. The utility of a reduced dimension mode! (from a three-dimensional reality) is generally greatest for problems focusing on spatially averaged predictions (volumetric flow rates and/or heads in wells with long screens) and away from boundaries and stresses (welIs, for example). E`ully three-dimensional flow models typically justify their expense only for problems requiring significant resolution in the vicinity of
86 GROUND WATER MODELS complex geometries (of boundaries or heterogeneities) or physically small sources or sinks. Boundary Conditions The uniqueness of any particular ground water flow problem is expressed in a mode} in part by locating mode! boundaries and defining conditions along those boundaries (see Chapter 2~. Often the boundaries correspond to physical boundaries in the environment along which conditions are known or can be estimated by the use of data. In many other situations, mode} boundaries must be defined on the basis of practicality-physical boundaries are unknown or are at great distance from the region of interest. In either case, boundary condition specification ~ extremely important in many problems and requires a thorough understanding of the mathematical role of boundary conditions as well as the hydrogeolog~c environment. Boundary condition Specification is an often overlooked source of significant error (Franke et al., 1987~. No matter how complex the model, proper application will always depend on a knowledgeable, trained user working with data that have been collected in such a way as to shed light on boundary conditions. Transient Versus Steady State Another valuable assumption in the application of saturated continuum flow models ~ that of steady state, i.e., that conditions remain constant over time. Because the stresses that drive ground water flow often vary only slowly in time much more slowly than the system requires to respond steady state assumptions are often justified. However, there are situations where transients must not be ignored. For example, Sykes et al. (1982) have suggested that ignoring seasonal periodicities in ground water flow direction can lead to otherwise unexpected dispersion during transport. Pulsed- pumping remedial schemes, which are becoming more common, also may demand explicit consideration of transient effects if accurate prediction of contaminant breakthrough is required. Discretization The accuracy of numerical approximations to the mathematical equations used to represent ground water flow depends on the size
FLOW PROCESSES 87 of the discretization used relative to the rate at which the gradi- ent of hydraulic head changes. In other words, if the gradient of head varies rapidly because of hydraulic property heterogeneity or boundary conditions, then discretization must be fine to achieve com- parable accuracy to coarse discretization in regions of slowly varying gradient. The choice of grid discretization (in space and/or in time) is further complicated by the averaging incorporated into numerical models. Most models make some very simple assumption about how parameter values vary between computational nodes. For example, many models assume parameters to be constant over a grid block. This averaging, in the face of geologic heterogeneity, requires careful consideration of the resolution required to answer a particular ques- tion. Details of flow behavior below the scale of the discretization are usually lost. A general rule of thumb is that the discretization must be at least as dense as the data available for defining param- eter heterogeneity. Because data are usually scarce, considerations of numerical accuracy will often determine the grid discretization necessary in many practical problems. Velocity Computation More and more problems of saturated continuum flow focus on the prediction of ground water flow velocities. As noted earlier, the accurate prediction of head gradients, on which velocities directly depend, is much more difficult than the accurate prediction of head alone. Computed head gradients are more sensitive to numerical errors of approximation. In addition, head gradients are more sensi- tive to parameter values. Thus problems requiring accurate velocity prediction, e.g., transport problems, may require more sophisticated numerical methods and may require more careful specification of parameter values for sufficient accuracy. Parameter Values The dominant problem in the application of saturated continuum flow models, given today's state of the art, is the specification of pa- rameter values, i.e., the characterization of the geologic environment. Direct measurement is at best costly. Complete characterization is in any case impossible because nondestructive measurement methods are not available. Parameter identification via inversing techniques is computationally difficult. However, when appropriate data are avail- able, the approach provides a practical way to characterize large-scale
88 GROUND WATER MODELS distributions, for example, hydraulic conductivity distributions from hydraulic head data. While a tremendous amount of current research is focused on these problems, simple panaceas do not seem to be on the horizon. Parameter evaluation will remain a challenging task demanding education, experience, skill, and wisdom. FLOW IN THE: UNSATURATED ZONE This report is primarily devoted to a discussion of various is- sues related to modeling water flow and contaminant transport in the saturated zone. A detailed discussion of flow and transport in the unsaturated zone would seem out of place and not necessary. However, the unsaturated zone is the region through which contam- inants must pass to reach the saturated zone. The various processes occurring within this region, therefore, play a major role in deter- mining both the quality and the quantity of water recharging into the saturated zone. It is necessary to understand the role played by the unsaturated zone in ground water contamination and how the processes in this zone are either similar to or different from those in deeper flow systems. Characterization of the Unsaturated Zone Of course, the major feature of the unsaturated zone that dis- tinguishes it from the saturated zone, as the terms clearly indicate, is the degree of saturation of the pore spaces, in this case by wa- ter. In the saturated zone, all of the pores are filled with water (or other water-miscible or immiscible liquids) and the volumetric water content (~) is equal to porosity Ail. In contrast, the fluid phase occupying the pore spaces in the unsaturated zone may be liquids (mostly water and sometimes nonaqueous-phase liquids iNAP Es]) and gases. The degree of liquid saturation at a given time varies considerably depending on the soil's physical properties (primarily pore-size distribution, which is related to soil texture and structure), and the pattern of inputs and losses of water at the soil surface. A brief examination of what happens as a completely saturated soil gradually becomes unsaturated is necessary. It is the largest pores that become air-fi~led first, and removal of water from the smaller pores becomes increasingly more difficult (i.e., requires more work or energy). This phenomenon may be explained by considering the capillary forces that are responsible for water retention in pores.
FLOW PROCESSES r2r3 .__ a J 1 l -120 h1 ~ ~ ~ -100 h ~ ~- ~-80 ~-~: ~I ::::::::::::::: ::::::::::::::::::::~::: ::::::::::::::: :::::::::::::::::::::::: -60 .~-~-~- ~ ~- ~- ~In -:- h Oh - 3 LL -40 -20 r1 < r2 ' r3 h1 > h2 ~ h3 89 Capillary Rise Equation \ 0 50 100 150 200 DIAMETER OF TUBULAR PORE, r (,um) FIGURE 3.1 Relationship between pore size (r) on capillary rise and pressure head (h). When capillaries of different sizes are placed in water in a beaker, water will rise to different levels above the free water surface. If these capillary tubes are then lifted out of the water, they will not drain unless external pressure is applied. The capillary (or suction) forces that hold water inside the capillary against the gravitational forces arise from the attraction of water molecules for each other (cohesion) and the attraction of water molecules to the walls of the capillaries (adhesion). The height to which water rises in a capillary is indeed a measure of the capillary forces. These forces are stronger in the smaller capil- laries, as reflected by the higher rise of water there than in the larger capillaries. In fact, the capillary forces are inversely proportional to the radius (r) of the capillary (see Figure 3.1~. Water inside a capillary tube (and, by analogy, in soil pores) is under suction and is further illustrated by the concave curvature of the water-air interface at both ends of the capillary tube when it has been taken out of the water. By convention, free water is taken as the reference and is assigned a value of zero for the capillary forces; thus capillary forces are assigned a value less than zero, which is why they are referred to as "suction" forces. If a porous medium can be thought of as a random network of capillary tubes of varying sizes, it can be seen that the suction force with which water is held in different pore sequences varies inversely
go GROUND WAlrER MODELS 50 ~ 40 a) Cal a, Q - - ~ 30 o > m z ,_ 20 o CC llJ 3: 10 o \ \ Fine Sandy Loam Fine Sand , _ 0 -50 -100 -150 -200 PRESSURE HEAD, h (cm) FIGURE 3.2 Examples of soil-water characteristic curares (3(h), for several soils. The vertical arrow at h = -100 cm indicates soil water content at field capacity (0fc). with pore radius (r). Thus sandy soils with larger pores retain less water at a given suction than do clayey soib that have smaller pores. This relationship between soil water content (~) and negative capillary pressure or suction (h) is an important physical property of soils and is commonly referred to as the soil-water characteristic curve (~(h)~; typical curves for several soils are shown in Figure 3.2. Another important physical property of unsaturated soils is their ability to transmit fluids, in particular, water. This property, the hy- draulic conductivity (K), has a maximum value, called the saturated hydraulic conductivity (Ke), in a completely saturated soil and de- creases dramatically with decreasing soil water content. Again, by using the analogy of capillaries, it is known that for a given hydraulic
FLOW PROCESSES 91 (or potential) gradient, the flux of water (q) through a capillary is di- rectly proportional to the radius squared (~2); this principle is known as Poiseuille's law. Thus larger capillaries conduct water much faster than do smaller ones. At saturation, sandy soils will have larger values of Ke than do clayey soils. As a soil becomes unsaturated, the larger pores drain and water flow is restricted to increasingly smaller pore sequences, which conduct water at much lower rates. Based on Poiseuille's law, soil hydraulic conductivity is expected to decrease. This is indeed the case, as shown in Figure 3.3 for the relationship between K and h for several soils. It should be noted that even though sandy soils are much more permeable than clayey soils at saturation, the reverse may be true when these soils are unsaturated. The knowledge of soil-water characteristic curves (~thi), and soil hydraulic conductivity curves (Kthi), is essential for describing water flow and contaminant transport in the unsaturated zone. Because K and ~ are both functions of h, K can be stated as a function of ~ or h. These relationships have been experimentally measured for a large number of soils, and empirical or theoretically based equations for O(h), K(h), and K(~) have been derived to predict them. Some of these are listed in Tables 3.1 and 3.2. CONCEPTS OF WATER FLOW IN THE UNSATURATED ZONE As in a saturated soil, the rate and the direction of water flow in an unsaturated soil also depend upon the magnitude of the soil hydraulic conductivity (Kth]) and the magnitude and direction of the hydraulic potential gradient (AH). Darcy's law which states that soil water flux (q) is directly proportional to potential gradient is also applicable to unsaturated soils. It should be recognized, however, that the soil hydraulic conductivity now is a strong nonlinear function of h (see Figure 3.3 and Table 3.2) and not a constant value as is the case for saturated flow. In unsaturated soils, the rate of soil water movement may be small even though there may be a large potential gradient, because the hydraulic conductivity (Kthi) is small. Another distinction in water flow between saturated and unsat- urated zones must be appreciated. While water flow in the saturated zone usually occurs at a steady rate (except for short periods of time when pumping is initiated), water flow in the unsaturated zone is unsteady (or transient). That is, soil water flux (q) is constant in space (x,y,z) and tune (t) for saturated water flow, but may vary
92 1 000 100 >` - 10 it 8 1 tar C) I 0.1 o id GROUND WATER MODELS \ Sandy loam \ - - \\ Clay .. 1 1 0 -60 -120 PRESSURE HEAD, h (cm) FIGURE 3.3 Examples of K(h) relationships for several soils. dramatically with both space and time for unsaturated, transient water flow. A detailed treatment of unsaturated water flow is beyond the scope of the present discussion, the reader is referred to several text- books on the topic (e.g., Campbell, 1985; Hanks and Ashcroft, 1980; Hillel, 1980a,b; Koorevaar et al., 1983) for a thorough analysis of is
FLOW PROCESSES TABLE 3.1 Some (3(h) Relationships Reported in the Literature 93 Function Source A(().- E)r)l;h<~ (I) (h) = (I) (h) = Or + [a + His; h ' O = (or+ [a + (l |h~)b]; h < - 1 his; h- - 1 ()(h) = (9r+ L1 + a(|h~)h]m; h ~ O (I) (h) = (his; h-O m = 1 - Brutsaert (1966) Haverkamp et al. (1977) van Genuchten (1980) Ash) = (9r+ [8s- Or] [all]; h ~ a C)(h) = (his; h 2 0 NOTE: Us is saturated water content, equal to porosity; Or iS "residual" water content; h is soil- water metric potential; and a, b, and m are constants. the conceptual basis and theoretical approaches. Here the focus is on a qualitative, phenomenological description of unsaturated water flow. It may be convenient to think of the following three sequential phases in transient water flow: (1) infiltration, (2) redistribution, and (3) static. Phase 1 begins with the input of water at the ground surface (as a result of pending water, for example), and water begins to infiltrate the soil. The rate of water intake at the ground surface, the infiltration rate (i), is controlled by the method and the rate of water input and the antecedent soil-water conditions. In early stages of phase 1, the infiltration rate is controlled primarily by the water input (application) rate, but in later stages the soil hydraulic properties control the infiltration rate. For infiltration into a "dry" soil, the infiltration rate is initially large because the large hydraulic potential gradients can sustain a large soil water flux. However, as the soil becomes saturated and the wetting front penetrates deeper, the potential gradient driving water flow decreases, and the infiItra- tion rate asymptotically approaches the saturated conductivity (K,) value. Several equations derived to describe the time dependence of infiltration are listed in Table 3.3. The above scenario and the equations listed in Table 3.3 are applicable only when water is ponded at the surface. The changes in infiltration rate with time during a water application at a steady rate (as in sprinkler irrigation or during a steady rain) are slightly
94 GROUND WAltER MODELS TABLE 3.2 Some K((~) or K(h) Relationships Reported in the Literature Source Function K(h) K(h) K(h) K(h) K(h) K((~) = K(~) = K(h) = Ks[exp(bh)]; h < 0 K(h)(Ks); h 2 0 K(h) = a[(~(h))b] NOTE: K(h) is hydraulic conductivity at h; K((~) is hydraulic conductivity at (a; E) is volumetric water content; h is hydraulic potential (and has negative values); Ks is saturated hydraulic conductivity; Os is saturated water content; (fir is "residual" water content; and a, b, her, and hi are empirical constants. alhl -b Ks[exp(-a|hl)] [~|h|/a)b + 1]-1 [|h|/|hCrl] b; |h| C |hCr Ks; |h| ~ Ihcrl [K (Gil - (err) ] b 3 5 K(h) = exp{a[|h| - |hCrl]}; thy| C |h| is |hCr K(h) = Ks; |hl 2 Ihcr K(h) = Kl[lhl/lhll] b; |hl ~ Ih ( ' sea + his] |h| ~ O a[exp~b)~BJ(h)~]; ~ ~ C's Ks~ = es Wind (1955) Gardner (1958) Gardner (1958) Brooks and Corey (1966) Averjanov (1950) Rijtema (1965) Haverkamp et al. (1977) van Genuchten (1980) TABLE 3.3 Some Expressions Developed to Describe the Time Dependence of Infiltration Rate (i) Following Ponding of Water Equation i = ic + (blI) i = Bt-n i = is + (io- ie)e k i = ie + (312)t l/2 i = ic + a(M-I)n; I ~ M i = ic;I >M NOTE: i is the infiltration rate (cm3 of water per cm2 area per hour); 1 is the cumulative volume (cm3) of water infiltrated in time t; a, B. M S. n and k are constants; ic is the steady-state infiltration rate; ie is the initial infiltration rate; and lo is the final infiltration rate. Source Green and Ampt (191 1) Kostiakov (1932) Horton (1940) Philip (1957) Holton (1961)
FLOW PROCESSES a _~ \~ Ponding (0 Steady Application \ \ 95 -it - LL o A b ra= i l 1 tp TIME (fur) ic= Ks FIGURE 3.4 Examples of the changes in infiltration rate (i) with time for different water application scenarios. Vertical arrows indicate initiation of pending and surface runoff. different, although the physical principles that govern flow are the same. The capacity of the soil profile to take in water is initially large enough such that the infiltration rate is equal to the application rate Steak. If tea is larger than K,,, soil near the ground surface becomes saturated after some time [p and water begins to pond at the surface. The infiltration rate begins to drop off and finally reaches
96 GROUND WATER MODELS the asymptotic value of K`,, as is the case for pending. In Figure 3.4, several examples illustrating this sequence of events are presented. The physical significance of tp should be recognized, because it is the time after which the infiltration rate is smaller than the application rate, and the excess water that is ponded on the surface is lost as surface runoff. Note that (1) surface runoff is initiated only when the application rate exceeds the infiltration rate; (2) [p is small when Fa ~ Ks and gets larger as the value of Fa approaches Ks; and (3) the soil will remain unsaturated (43 < e) if ra < Ks, and there will be no surface runoff. Phase 2 starts when water application at the surface has ceased, and soil water content decreases as water begins to redistribute deeper within the soil profile. During this phase, soil water flux decreases with time to zero in an exponential manner until a unit hydraulic gradient (i.e., only gravitational forces operative) is es- sentially achieved. This may occur within a few hours in sandy soil but may take several days or even weeks in a slowly permeable clay soil. The soil water content at this time is referred to as the "field-capacity" value in the soil science literature. The presence of a shallow water table, or impermeable soil layers, has a strong impact on the duration of phase 2. During phase 3, the soil water flux is practically zero (i.e., not measurable) and any decreases in soil water content below the "field- capacity" value are primarily due to losses of water by evaporation and plant uptake (transpiration). Phase 3 continues until the next event of water input (e.g., rainfall, irrigation). In the presence of shallow water table and/or as evapotranspiration proceeds, there may actually be upward water flow, rather than downward flow as assumed for phase 2. The foregoing scenario, simplified for the present discussion, should illustrate that transient water flow in the unsaturated zone, particularly in the top several meters, is episodic as determined by water input events at the ground surface (see Figure 3.5~. Each of these episodes, in turn, has at least three extinct phases where the soil water content and soil water flux vary considerably. Each episode does not have to go through all three successive stages described above; another water input event (e.g., rain, irrigation) may interrupt a given stage. This episodic nature of water flow contrasts with the steady water flow in the saturated zone. At lower depths within the unsaturated zone, beyond the influence of transient conditions at the ground surface, soil water flow is likely to occur under unsaturated
FLOW PROCESSES es go A o ~ e fc o co Opw 97 1 \ it Saturation Field Capacity -- -Permanent Wilting TIME (days) FIGURE 3.5 Variations in soil water content as a result of several episodes of water inputs at the surface. Water inputs are indicated by vertical arrows. Water contents at saturation, field capacity, and permanent wilting are indicated by horizontal dashed lines. Water depletion below Ofc is the result of losses via evapotranspiration. conditions, but the soil water flux may be steady and the soil water content constant. The steady flux is equivalent to the annual rate of ground water recharge, which is dependent on site conditions (rainfall patterns, soil physical properties, land use, and so on). The major differences between saturated and unsaturated flow are summarized In Table 3.4. As noted in Chapter 2, by coupling Darcy's law with the conservation of mass principles, the governing differential equation for transient water flow can be derived; this equation is known as the Richards equation. FRACTURE: FLOW The term fracture is a general one referring to the various types of discontinuities that can break a medium into blocks (Torsaeter et al., 1987~. In the most general case, fracturing adds secondary porosity to some original porosity. The rock blocks contain pores having lengths and widths of about the same dimension and a highly tortuous pattern of interconnection (Shapiro, 1987~. The fractures
98 GROUND WATER MODELS provide more continuous openings with lengths far in excess of their widths. The most widely used conceptual mode} for fractured media (Figure 3.6) usually considers that discontinuities are represented by joints, fracture zones, and shear zones. As the figure shows, joints are usually discontinuous in their own plane or, in other words, have a definite length and width. When several closely spaced families (sets) of joints are present, they can form a highly interconnected, three-dimensional network for flow (Gale, 1982~. Within a given rock or sediment a relatively large number (e.g., five or six) of joint sets can be present, each with its own unique orientation in space. Gale (1982) defines fracture zones as zones of closely spaced and highly interconnected discrete fractures that are generally not filled with clay or other material. The typical width of a fracture zone is from 1 to 10 m. Shear zones provide another example of a large-scale discontinuity with a permeability that can be either higher or lower than the rock mass depending upon filling materials, age, and stress (Gale, 19823. The picture of fracturing presented so far is one end member in a more complicated hierarchy of multiple-porosity systems. In the case of soluble bedrock like limestone, dolostone, or evaporites, conduit flow can develop as original fracture systems are enlarged by solution. The important feature of conduit flow, when it is able to develop, is the integration of the drainage network (QuinIan and Ewers, 1985~. In many ways, the network is analogous to a river system with smaller tributaries supplying water to a succession of TABLE 3.4 Major Differences Between Saturated arid Transient Unsaturated Water Flow in Porous Media Parameter Saturated ~ = Os H = h + z (h 2 0) Gravity (z) dHI fix Y order ( 1 ) K = Ks Unsaturated Water content (I) Total potential (H) Dominant force Driving force Hydraulic conductivity (K) Flux (q) Governing flow equation q = -Ks6Hl~x Laplace equation V2h = 0 O ~ ~ < Os H = h + z ( - °° ~ h ~ 0) Matric potential (h) dHIdx Y order (1-104) K = K(h) or K(~) Range 3 to 8 orders of magnitude q = -K(h) dHI3x Richards equation d = V[K(h) VH] NOTE: Appreciation is extended to D. Nofziger, Oklahoma State University, Stillwater, for assistance in preparing this table.
FLOW PROCESSES SOLID ROCK 99 , SHEAR ZONE JOINTS ~-~-FRACTURE ZONE FIGURE 3.6 Conceptualization of discontinuities in a fractured medium. SOURCE: Gale, 1982. larger and larger conduits. As a result of the integration, both the conduit system and the individual conduits can become large. The famous karst system at Mammoth Cave, Kentucky, is a good example. The relatively large number of geologic processes that can give rise to fracturing (e.g., tectonism, weathering, glacial stresses, or thermal stresses), coupled with the tendency for older fractures to be propagated into unfractured units, means that fractures are a dominant element controlling fluid migration in all kinds of geologic settings. Similarly, the abundance of carbonate bedrock in North America implies that cases of conduit flow are also common. Theory of Flow in Fractures The presence of fractures or conduits in geologic units adds a further complexity to understanding fluid flow. Many of the funda- mental principles developed in the previous sections of this chapter need to be extended to deal with fractured media. The first major question of how one conceptualizes flow in a single fracture can be addressed with the help of the so-called parallel plate mode! (Figure 3.7~. A fracture is idealized as a planar opening having a constant thickness or aperture. Flow in the fracture is assumed to obey Darcy's law, or q = Kf tH/§I, (3.1), where q is the fracture flux (volume flow per unit time per unit length of cross-sectional fracture area t2b x I]), Kf is the hydraulic
100 GROUND WATER MODELS x3 Lit 2b roll FIGURE 3.7 Idealization of a natural fracture as parallel plates with an aperture of fib. SOURCE: Gale, 1982. conductivity of the fracture, and [H/§I is the hydraulic gradient along the fracture (Gale, 1982~. From fundamental principles of fluid dynamics, Kf can be expressed in terms of fluid and fracture properties as Kf = P9 (2b)2 (3.2) where p is the fluid density, 9 is gravity, ,u is dynamic viscosity, and 2b is the fracture aperture. On the basis of (3.2), by keeping in mind that flow occurs through an area (2b x D, it can be shown that the quantity of flow passing through a fracture is a function of the aperture cubed (Gale, 1982~. This relationship, referred to as the cubic law, is generally valid in describing flow through fractures (Gale et al., 1985~. However, indications are that deviations from this general behavior will occur when the fracture surfaces are rough or the fracture surfaces are in contact with one another. Another feature of fractures that makes them difficult to deal with is the fact that the apertures, and hence the hydraulic conduc- tivity, depend on the stress within the medium. In other words, a fracture can be opened or closed simply by reducing or increasing the forces applied to it. For example, pumping a well in a fractured medium reduces the pore pressure, effectively causing the fracture aperture to decrease. Figure 3.S, taken from Gale (1982), presents experimental data illustrating how fluxes of water through a frac- ture change with changing stress conditions. Gale (1982) describes a number of empirical-theoretical approaches designed to mode! the stress coupling to hydraulic conductivity. In representing the hydraulic conductivity of a fractured medium, it must be considered that two systems of porosity are present. The hydraulic conductivity of the blocks is a straightforward} porous
FLOW PROCESSES 101 medium type-similar to that discussed in Chapter 2. Conceptual- izing the hydraulic conductivity of the fractures is more complicated because of the stress dependence. Recently, there has been interest in formulating and modeling multiphase flow in fractured media. A significant motivation is the assessment of the fractured and unsaturated Yucca Mountain turf in Nevada as a potential host rock for nuclear waste (Evans and Nicholson, 1987~. Further, several of the most serious hazardous waste sites in the United States (e.g., Love Canal and Hyde Park Landfill) involve the migration of NAPEs in fractured media. As before, the theory for multiphase flow is a generalization of single-phase theory to account for the presence of more than one fluid in the pores and fracture networks. A complication with fractured systems is that relative permeability relationships need to be developed for both the blocks and the fractures. Providing these data is sufficiently difficult that in most cases an attempt is made to approximate these dual-porosity systems as an equivalent single-porosity system. 0.075 0.050 X - LL 0.025 o AP = + 0.28 MPa 1 ,m' O Injection flow rate · VVithdrawal flow rate · A2b, nonuniform fracture-center borehole a ~ WP1 Fracture ~ :2 1.83 m 1 0.97 m| In_ 0 1.38 2.76 EFFECTIVE STRESS, a0 = a' - P2 (MPa) 3.0 2.4 -8 0 - z _ 1.8 o LU Cat 1.2 Lo 0.6 o FIGURE 3.8 Experimental data that illustrate how flux in the fracture and aperture change as a function of effective stress for a sandblasted, sawcut fracture surface. SOURCE: Gale, 1982.
102 GROUND WATER MODELS So far nothing has been said about the flow of ground water in conduit systems. In general, the character of flow is so very different often turbulent and partially saturated that most texts do not treat the basic theory. A few books and papers (LeGrand and Stringfield, 1973; Milanovic, 1981; White, 1969) treat ground water flow in karst and provide a starting point for readers interested in this fascinating topic. Strategies for Modelmg To date, single-phase or multiphase flow in fractured media has been modeled using one of three possible conceptualizations: (1) an equivalent porous continuum, (2) a discrete fracture network, and (3) a dual-porosity medium. With the first of these approaches, it is assumed that the medium is fractured to the extent that it behaves hydraulically as a porous medium. Under this condition, the con- tinuum equations for porous medium flow developed in Chapter 2 describe the problem mathematically. The actual existence of frac- tures is reflected in the choice of values for the material coefficients (e.g., hydraulic conductivity, storativity, or relative permeability). Often these parameters take on values significantly different from those used for modeling a porous medium (Shapiro, 1987~. Exam- ples of this approach an cited by Shapiro (1987) include Elkins (1953), Elkins and Skov (1960), and Grisak and Cherry (1975~. With the discrete fracture approach, most or all of the ground water moves through a network of fractures. This approach assumes that the geometric character of each fracture (e.g., position in space, length, width, and aperture) is known exactly as well as the pattern of connection among fractures. In the simplest theoretical treatment, the blocks are considered to be impermeable. Figure 3.9a is an ide- alization of a two-dimensional network of fractures consisting of two different sets. Note how each fracture, represented on the figure by a line segment, has a definite position in space, length, and aperture. The hydraulic characteristics of the fracture system develop as a consequence of the intersection of the individual fractures. In three dimensions, the network can be described in terms of intersecting planes that could be rectangular (Figure 3.9b) or circular in shape (Figure 3.9c). Examples of the discrete fracture treatment of flow in networks include Long et al. (1982, 1985), Robinson (1984), Schwartz et al. (1983), and Smith and Schwartz (1984~.
FLOW PROCESSES a ~ ~~3; 4;Z~ ~1 ~. . FIGURE 3.9 Three different conceptualizations of fracture networks: (a) a two-dimensional system of line segments (from Shimo and Long, 1987~; (b) a three-dimensional system of rectangular fractures (from Smith et al., 1985~; and (c) a three- dimensional system of `'penny- shaped~ cracks (from Long, 1985~. 103 b c '··· ., A:::::::::::: ;;; ;' :::::::::: ..:: · ~ :~:~:~:~:~:~:~ ·...~..... a::::.:.:.:: :; __ ............... - :::: A:: :::::: .._ .....~..... . ~ ........... .. an.... V :::::::::' · · · · ~ ·e ~ ~e ~ :~:~:~:~:~:~:~: · ~ ~ .... · ~ ::::::> . 4: r l ~ . ~. The dual-porosity conceptualization of a fractured medium con- siders the fluid in the fractures and the fluid in the blocks as separate continua. Unlike the discrete approaches, no account is taken of the specific arrangement of fractures with respect to each other there is simply a mixing of fluids in interacting continua (Shapiro, 1987~. In the most general formulation of the dual-porosity model, the possi- bility exists for flow through both the blocks and the fractures with a transfer function describing the exchange between the two continua. Mathematically then, one flow equation is written for the fractures and one for the blocks, with the equations coupled by the source-sink terms. Thus a loss in fluid from the fracture represents a gain in fluids in the blocks (Shapiro, 1987~. In most applications involving multiphase flow, a more restrictive approach is followed. Fluids are assumed to flow in the fracture network, with the blocks acting as sources or sinks to the fractures (Torsaeter et al., 1987~. Examining the mathematical details of the dual-porosity for- mulation is beyond the scope of this overview. To understand the
104 GROUND WATER MODELS equation development or to locate the available references, readers can refer to Huyakorn and Pinder (1983), Shapiro (1987), and Tor- saeter et al. (1987~. ISSUES IN MODELING The modeling approaches just described are subject to significant complexities of both a theoretical and a practical nature that affect the modeling process. Three main issues are discussed here: (1) whether fractured media can even be approximated as continua, (2) computational constraints on discrete network models, and (3) the uncertainty in establishing the network geometry. A Fractured Medium as a Continuum For a porous medium, it is not difficult to believe in the existence of what is termed a representative elemental volume for various con- trolling parameters. Consider a parameter like hydraulic conductiv- ity as an example. The representative elemental volume is a sample volume for which the hydraulic conductivity is independent of sample volume or averaging volume. In other words, the representative ele- mental volume exists when a small change in the sample volume does not result in a change in hydraulic conductivity. This concept can be demonstrated using Figure 3.10. When the volume of a porous medium is small (e.g., a few pores), even a slight change in the sam- ple or averaging volume can cause appreciable changes in hydraulic conductivity. As the sample size increases, there comes a point when the hydraulic conductivity is not sensitive to the averaging volume. In modeling a porous medium as a continuum, it is assumed implicitly that the domain or individual cells within the domain for which the flow equation is written satisfy a representative elemental volume condition. For most cases, this assumption is reasonable. In modeling a fractured medium using continuum approaches' the sample assumption is required. However, in the case of a fractured medium there is much less certainty in the assumption of a represen- tative elemental volume being valid (Schwartz and Smith, 1987~. The first major problem with fractured media is that a represen- tative elemental volume can only be defined when fracture densities are above some critical density. The critical density is defined as that density of fractures that provides connectivity of the network (Fig- ure 3.11~. Below the critical density, the network is not connected
FLOW PROCESSES o C) At, I o 105 air REV I I ~ AVERAGING VOLUME FIGURE 3.10 Variation in hydraulic conductivity as a function of the av- eraging volume. The dashed lines point to volume where the assumption of a representative elemental volume (REV) is valid. SOURCE: Modified from Shapiro, 1987. (nonpercolating) and the mean hydraulic conductivity will be zero no matter how large the averaging volume (Schwartz and Smith, 1987~. Thus in modeling a fractured system, simply choosing a large volume of rock for a cell will not necessarily guarantee that the assumption of a representative elemental volume is met. Situations also exist in which the concept of a representative ele- mental volume is either impractical or invalid. Consider the following two examples. By assuming a network to be connected but sparsely fractured, the averaging volume necessary to obtain a representative elemental volume of the medium could be much larger than the scale of interest. For example, the minimum averaging volume might be a block of rock 200 m on a side, while the scale of interest is 100 m, which makes the concept impractical. Further, a network could be connected, but with a hierarchy of fracture types. In this case, as the sampling volume expands, hydraulic conductivity might keep increasing without necessarily becoming constant. The concept of a representative elemental volume probably does not hold for some fractured media. Thus there are going to be systems that cannot be modeled by using continuum approaches. Without relatively detailed analyses, it will probably be difficult to identify these systems in advance. When the continuum approaches to modeling are not appropriate, one must turn to discrete modeling approaches, which are clearly not without their own problems, as the next two sections show.
106 GROUND WATER MODELS i// / _ Nonpercolating Critical Density Percolating FIGURE 3.11 Examples of percolating and nonpercolating networks in two dimensions. The critical density is the point where infinite clusters of fractures appear and connectivity is achieved. SOURCE: Schwartz and Smith, 1987. Computational Constraints on Discrete Network Models Modeling fluid flow in a network of discrete fractures does not require that the fractures behave as a continuum. All of the problem cases discussed in the previous section can be modeled without the- oretical constraints. Unfortunately, there are practical constraints that severely limit the capability of modeling discrete fracture net- works. The most serious is the number of fracture intersections, because describing flow in the network requires that hydraulic head be calculated at each intersection. In two dimensions, a network with 50,000 intersections will require a major computational effort and will be expensive. Yet the size of a network would in many cases be much smaller than the size of the region of interest. One probably cannot create a discrete fracture system in two dimensions that is large enough to solve intermediate and regional problems. The situation is even more pessimistic when one tries to account for the presence of the rock blocks, or the three-dimensionality of the fracture network. With these more complex flow conditions, one
FLOW PROCESSES 107 has to further reduce the number of discrete fractures that can be incorporated in the model. Discrete models are of limited practical value, although they are potentially a theoretically more powerful approach to modeling fractured systems. Their main use to date has been to explore the fundamentals of flow and mass transport in fractured media. Uncertainty In Establishing the Network Geometry One further limitation in the use of discrete fracture models is the requirement to specify the exact geometry of the network. There will never be a situation where the geometry of a natural fracture network is exactly known. At best, hydraulic testing can provide estimates of apertures, and fracture mapping in tunnels or on the surface may provide indications of fracture orientation, fracture lengths, and the pattern of connection. However, no tests can provide a definitive description of the network within a rock or sediment mass. Uncertainty in describing a network ultimately translates into uncertainty in mode] predictions made for flow in the system. Es- sentially, the less one knows about a system, the less confident one can be in predicting system response. Stochastic modeling methods (e.g., Smith and Freeze, 1979a) offer a possible approach to making predictions and establishing the potential range of uncertainty in the face of uncertain data. The complexity of natural fracture networks and the difficulty in making measurements practically guarantee that predictions made by using discrete fracture models will be relatively uncertain. The same situation will probably hold for continuum models as well. However, so little work has been conducted on natural systems that it will require years to fully assess how uncertain predictions in fractured rock systems might be. Adequacy of Modeling Technology In most practical problems involving saturated flow in fractured media, there has never been much hesitation in applying continuum- type models. For example, many would argue that the classical methods of well hydraulics appear to mode! the response of fractured systems to the extent necessary for design. The question remains, however, as to how realistically such models account for the fractured flow processes. Experience from the petroleum industry does suggest
108 GROUND WATER MODELS that in some cases more sophisticated flow formulations (e.g., dual- porosity models) will be required. Further, issues of stress coupling and the validity of the cubic law will require further study. In addition to the theoretical questions that remain to be resolved, there is a significant gap in practical knowledge about flow in fractured media. For example, only very limited testing has been carried out with well-characterized media, except on the laboratory scale. The modeling tools exist to deal with fractured media, but at present, results should be interpreted with caution. Systems are often complex and extraordinarily difficult to characterize, especially with the level of effort considered normal for most site investigations. The state of the art in field testing provides a relatively rudimentary esti- mate of values for some parameters like hydraulic conductivity, while other parameters, like storativity, must be established through fitting simple theoretical models (usually of the porous medium type). Unsaturated flow modeling of fractured systems is a subject of increasing interest, particularly in light of work at Yucca Mountain in Nevada to assess the feasibility of disposing of high-level nuclear waste in an engineered repository. Most of the same theoretical and practical concerns that were discussed for flow in saturated and fractured media hold for unsaturated media as well. The greatest ad- ditional problem lies with the increased difficulty in measuring perti- nent parameters down boreholes. According to Evans and Nicholson (1987), a lack of data has restricted the validation of models to a few very simple systems. This problem of the unavailability of data or a large range in variability of existing data was also identified by Pruess and Wang (1987) as an impediment to progress in modeling. Thus the capability in modeling again exceeds the ability to fully establish the validity of the model. Not much has been written here concerning the multiphase transport of fluids in ground water systems. Notwithstanding the significant capability of solving these problems in petroleum-related applications, there is much less research and overall experience with contaminant-related, multiphase modeling. In the case of dense non- aqueous-phase liquids (DNAPEs), especially the contaminant and petroleum types, problems are sufficiently different that not all of the of! field capabilities are directly transferable. Again, the commit- tee would consider the capability of modeling to exist but without the theoretical and practical experience with the models to consider these applications in any sense routine. As was the case with unsaturated flow modeling in fractured media, limitations in the data provide
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