Prediction of transient flow in random porous media by conditional moments

Transcription

1 Prediction of transient flow in random porous media by conditional moments Item Type Dissertation-Reproduction (electronic); text Authors Tartakovsky, Daniel. Publisher The University of Arizona. Rights Copyright is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 04/09/ :51:59 Link to Item

2 PREDICTION OF TRANSIENT FLOW IN RANDOM POROUS MEDIA BY CONDITIONAL MOMENTS by Daniel Tartakoysky A Dissertation Submitted to the Faculty of the DEPARTMENT OF HYDROLOGY AND WATER RESOURCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY WITH A MAJOR IN HYDROLOGY In the Graduate College THE UNIVERSITY OF ARIZONA 1996

3 2 THE UNIVERSITY OF ARIZONA & GRADUATE COLLEGE As members of the Final Examination Committee, we certify that we have read the dissertation prepared by Daniel M.Tartakovsky entitled Prediction Of Transient Flow In Random Porous Media By Conditional Moments and recommend that it be accepted as fulfilling the dissertation requireme or the Degre of Doctor of Philosophy 42, et-el Date 6f Date Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requireme

4 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgement the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED:

5 4 ACKNOWLEDGMENTS This dissertation was supported by the U.S. Nuclear Regulatory Commission under contract NRC I am greatly indebted to Dr. Shlomo P. Neuman, my advisor and the principal investigator of this project, for his generous financial support. I am grateful to my mentor Dr. Shlomo Neuman, from whose ingenious ideas I profited so much; who was always able to convey to me sometimes incomprehensible concepts of stochastic hydrology; whose thorough reading of this manuscript has enriched and unrecognizably changed (for the better) the original draft. I am thankful to Dr. Shlomo Neuman for his almost parental moral support he has provided me in the past three years. I would like to express my gratitude to my committee members, Drs. Tom Maddock III, Arthur W. Warrick, T.-C. and Jim Yeh for their editing of an original draft of this dissertation and their stoic struggle with sleep during my defense. I thank my friends in our office who endured my noisy being every day: Colleen Filippone, Kuo-Chin Hsu, and Velirnir Vessilinov. Their assistance helped me at different stages of my research. I am also thankful to all my friends in Tucson and in Russia who immeasurably enriched my life.

6 5 DEDICATION To my parents

7 6 TABLE OF CONTENTS LIST OF FIGURES 10 ABSTRACT 13 1 INTRODUCTION Why Stochastic Classification of stochastic methods Transient flow in heterogeneous formations Steady-state flow in bounded domains On differentiability of random functions Stochastic partial differential equations Green's function as an operator's kernel Decomposition of linear stochastic partial differential equations Nonhomogeneous random initial and boundary conditions 42 2 CONDITIONAL NON-LOCAL FORMALISM FOR TRANSIENT FLOW General formulation Residual flux Properties of the kernels Second conditional moments of the prediction errors Perturbation analysis 58

8 7 TABLE OF CONTENTS - Continued 3 EFFECTIVE HYDRAULIC CONDUCTIVITY Effective hydraulic conductivity Uniform flow in infinite domain Uniform flow under the presence of Dirichlet boundaries Asymptotic analysis Numerical evaluation Uniform flow under the presence of Dirichlet and Neumann boundaries NONLOCAL FORMALISM IN LAPLACE SPACE The flow image in Fourier-Laplace space (I ndelman, [1995]) Non-local formalism in Laplace space Effective hydraulic conductivity Analysis of the "slow time-variation" assumption GENERALIZATION TO STRONGLY HETEROGENEOUS MEDIA Effective hydraulic conductivity Residual flux CONCLUSIONS AFTERWORDS 146 APPENDICES: A GREEN'S FUNCTION 148

9 8 TABLE OF CONTENTS - Continued B OPERATIONAL ANALYSIS OF STOCHASTIC FLOW EQUATIONS 153 C IMPLICIT RESIDUAL FLUX, r AND SECOND CONDITIONAL MOMENTS OF THE PREDICTION ERRORS 158 Cl Residual flux 158 C2 Second conditional moments 161 D PERTURBATION ANALYSIS 164 D1 General relationships 164 D2 Kernels (2.26) 165 D3 Deterministic hydraulic head hc(x, t) 169 D4 Second conditional moments 170 D5 Mean flow equation (2.13) 178 E TWO EXPANSIONS OF A GREEN'S FUNCTION 180 F EVALUATION OF DOMAIN INTEGRALS D(t) 183 Fl Parseval's identity 183 F2 Integral over domain bounded by Direchlet boundaries 183 F3 An alternative representation of (F28) 191 F4 An alternative representation of (F37) 194 G ASYMPTOTIC ANALYSIS 209 H FLOW DOMAINS IN THE PRESENCE OF NEUMANN BOUNDARIES (L I ALTERNATIVE REPRESENTATION OF A TRIPLE SUMMATION IN (3.59) 230 J DERIVATION OF NONLOCAL KERNELS 234

10 9 TABLE OF CONTENTS - Continued J1 One-dimensional flow 234 J2 Three-dimensional flow 237 K FINITE INTEGRAL TRANSFORMATIONS 241 K1 General theory 241 K2 Examples of finite integral transformations 247 LIST OF REFERENCES 250

11 10 LIST OF FIGURES Figure 1.1, Permeability and porosity of cores collected at 1-ft intervals from borehole (IL056) in the Mt. Simon aquifer in Illinois 16 Figure 3.1, Dependence of the effective hydraulic conductivity on time for isotropic 1-D, 2-D, and 3-D heterogeneous media (after Dagan [1982]) 75 Figure 3.2, The flow domain, Q, under presence of Dirichlet boundaries 77 Figure 3.3, Influence of the separation distance, 2p, between two Dirichlet boundaries on the steady-state component of the domain integral, D, 89 Figure 3.4, Influence of the separation distance, 2p, on a transient behavior of the effective hydraulic conductivity, K, 90 Figure 3.5, Dependence of a steady-state part of the domain integral, D,, on an anisotropie ratio, c 91 Figure 3.6, Influence of a degree of anisotropy on a transient behavior of Ke 92 Figure 3.7, The flow domain, Q, under presence of Dirichlet and Neumann boundaries 94 Figure 3.8, Box-shaped domain. Domain integral, D, as a function of dimensionless time, tp, and dimensionless domain's size, pe p, = P2 = p3 101 Figure 3.9, Box-shaped domain. Steady-state part of the domain integral, 13, 1, vs. dimensionless domain's size, pe pi = p2 = P3 102 Figure 4.1, 1-D flow. Nonlocal vs local models../(td) = 1 + a td, o

12 11 LIST OF FIGURES - Continued Figure 4.2, 1-D flow. Nonlocal vs local models. J(tD) = 1 + a te, 02y = Figure 4.3, 1-D flow. Nonlocal vs local models. J(tD) = 1 + sin (ate), o, = Figure 4.4, 1-D flow. Nonlocal vs local models. J(tD) = 1 + sin (ate), cr2y = Figure 4.5, 1-D flow. Nonlocal vs local models. J(tD) = exp (- a td), cfj", = Figure 4.6, 1-D flow. Nonlocal vs local models. J(tD) = exp (- a td), cry = Figure 4.7, 1-D flow. Nonlocal vs local models. J(tD) = 1 + a te, oy = Figure 4.8, 1-D flow. Nonlocal vs local models. J(tD) = 1 + sin (a td), c = Figure 4.9, 1-D flow. Nonlocal vs local models. J(tD) = exp (- a td), cry = Figure 4.10, 3-D flow. Nonlocal vs local models. J(tD) = 1 + a td 126 Figure 4.11, 3-D flow. Nonlocal vs local models. J(tD) = 1 + sin (a td) 127 Figure 4.12, 3-D flow. Nonlocal vs local models. J(tD) = exp (- a td) 128 Figure 4.13, Three-dimensional flow in highly random porous media. Nonlocal vs local models. J(tD) = 1 + sin (atd) 129 Figure 5.1, Transient part of the effective hydraulic conductivity, Ke(tD): linearized expression vs. generalized one (infinite domain) 134 Figure 5.2, Transient part of the effective hydraulic conductivity, Ke(tD): linearized expression vs. generalized one (domain with Dirichlet boundaries) 135

13 12 Ladies and gentlemen, attention, please! Come in close where evelyone can see! I got a tale to tell, it isn't gonna cost a dime! (And if you believe that, we're gonna get along just fine.) - Steve Earle "Snake Oil"

14 13 ABSTRACT This dissertation considers the effect of measuring randomly varying local hydraulic conductivity K(x) on one's ability to predict transient flow within bounded domains, driven by random sources, initial head distribution, and boundary functions. The first part of this work extends the steady state nonlocal formalism by Neuman and Orr [1992] in order to obtain the prediction of local hydraulic head h(x, t) and Darcy flux q(x, t) by means of their ensemble moments <h(x, t)> and <q(x, t)>e conditioned on measurements of K(x). These predictors satisfy a deterministic flow equation which contains a nonlocal in space and time term called a "residual flux". As a result, <q(x, t)> is nonlocal and non-darcian so that an effective hydraulic conductivity K. does not generally exist. It is shown analytically that, with the exception of several specific cases, the well known requirement of "slow time-space variation" in uniform mean hydraulic gradient is essential for the existence of Ke. In a subsequent chapter, under this assumption, we develop analytical expressions for the effective hydraulic conductivity for flow in a three dimensional, mildly heterogeneous, statistically anisotropic porous medium of both infinite extent and in the presence of randomly prescribed Dirichlet and Neumann boundaries. Of a particular interest is the transient behavior of K, and its sensitivity to degree of statistical anisotropy and domain size. In a bounded domain, Ke(t) decreases rapidly from the arithmetic mean KA at t = 0 toward the effective hydraulic conductivity corresponding to steady state flow, Ks, IC, exhibits similar behavior as a function of the dimensionless separation distance p between boundaries. At p = 0, K, = KA and rapidly

15 14 decreases towards an asymptotic value obtained earlier for an infinite domain by G. Dagan. Our transient nonlocal formalism in the Laplace space allows us to analyze the impact of other than slow time-variations on the prediction of <q(x, t)>,. Analyzing several functional dependencies of mean hydraulic gradient, we find that this assumption is heavily dependent on the (relaxation) time-scale of the particular problem. Finally, we formally extend our results to strongly heterogeneous porous media by invoking the Landau-Lifshitz conjecture.

16 15 CHAPTER 1 INTRODUCTION "Models are to be used, but not to be believed." H. Theil 1.1 Why Stochastics? Any model of a real system is an approximation; often, its inputs and outputs are measurable variables and its structure is a set of differential equations linking the inputs and outputs. Consequently, models may be classified by the ways in which this approximation is accomplished and by the consistency of its output with actual system's behavior. Such consistency is a prerequisite for the acceptance of a model. A deterministic model does not consider randomness; a given input always produces the same output. A stochastic model has outputs that are at least partially random. Although all ground water phenomena involve some randomness, the resulting variability in the output may be quite small when compared to the variability resulting from known factors. In such cases, a deterministic model is appropriate. If the random variation is large, a stochastic model is more suitable, because the actual output could be quite different from the single outcome that a deterministic model would produce [Chow et al., 1988]. The recognition of the ubiquitous presence of heterogeneities in natural formations (see Figure 1.1) and of their effect upon flow and transport has prompted the development

17 16 porosity (percent) permeabddy (rrid) Figure Li: Permeability and porosity of cores collected at 1-ft intervals from borehole (IL056) in the Mt. Simon aquifer in Illinois (reproduced from Gelhar [1986], after Bakr [1976]).

18 17 of new approaches in subsurface hydrology. The most straightforward response to the necessity of dealing with spatial variability of hydrogeologic parameters is a desire to collect more experimental data and to construct a finer net for numerical simulations of the deterministic boundary-value problem. Although having raison d' etre, this approach has its distinct drawbacks. It may require prohibitively large computer time and, more importantly, it does not eliminate uncertainty! First, one cannot cover a field with too dense a network of wells and, second, experimental data obtained from wells are random due to measurement errors. Another problem stems from a recognition of the fact that in porous media flow takes place through a complex network of interconnected pores. Although the fluid itself may be thought of as a continuum, its flow paths are limited to the pore space and do not extend over the entire volume of the porous medium. To treat flow as if it occurred over a continuum, it must be spatially averaged over some representative elementary volume (REV). Unfortunately, the latter is rarely defined in a rigorous manner. Consider the definition of de Marsily [de Marsily, p. 15, 1986]: "The size of the REV is defined by saying that it is (1) sufficiently large to contain a great number of pores so as to allow us to define a mean global property, while ensuring that the effect of the fluctuations from one pore to another are negligible. (2) sufficiently small so that the parameter variations from one domain to the next may be approximated by continuous functions, in order that we may use the infinitesimal calculus". It remains unclear just how small/ large is "sufficiently small/ large". In some media, such as

19 18 fractured rocks, an REV may be very large or not exist at all [Neuman, 1987, 1990]. An approach that was fashionable in mathematical physics during the 1980s is that based on "homogenization theory" (see, for example, Bensoussatz et al. [1978], Sanchez- Palencia [1980], Berryman [1986], Levy [1987], etc.). Given a medium with a microstructure on a scale much smaller than the macroscopic scale of interest, homogenization is the process of passing from a microscopic to a macroscopic description of the problem of interest through a process akin to space- averaging. The basic idea according to Levy [p.64, 1987] is that if "the pore configuration has a scale length I which is small compared to a typical macroscopic scale length L which may be the dirrension of a specimen of the porous solid", the small ratio c 11L is introduced to account for this disparity in scales. "In the study of physical or mechanical processes in media with microstructure, known and unknown quantities are depending on e and an asymptotic analysis is used to determine the unknown field quantities. Furthermore, to make precise the fact that the medium varies rapidly on the small scale land may also vary slowly on the large scale L, we assume that every property of the medium is of the form f(x, y), where y = x/e....we shall look for each unknown field quantity uc(x) in the form of a double-scale asymptotic expansion: te(x) = u'(x, y) + c u l (x, y) + e 2 u 2(x, y) +. The twoscale process introduced in the partial differential equations of the problem produces equations in x and y variables....this leads to a rigorous deductive procedure for obtaining the macroscopic equations (in x) for the global behavior of the medium". Although based on elegant mathematics enriched by rigorous theorems of convergence, stability and precision, homogenization theory fails to address some of the practical problems relating to

20 19 groundwater hydrology. It is rigorously applicable only to periodic microstructures; mathematical operations (such as differentiation and integration) with space-averaged functions are complicated; it fails to deal with heterogeneities in situations where there is no clear separation of scales; and, last but not least, homogenization does not leave a place for the conditioning of the results on available data. Stochastic modeling seems to be free of these disadvantages but, as for anything in this world, there is a price to be paid: one must hypothesize ergodicity and stationarity. The mean ergodic hypothesis states that [Gelhar, p.36, 1993] "expected values or ensemble averages can be replaced by appropriate time or space averages". Stationarity in the wide sense is described by Yaglom [p.56, 1987] as follows: "a random function X(t) is called stationary provided that its mean value <X(t)> is a constant and its correlation function <X(t)X(s)> depends only on t - s". The requirement of stationarity can sometimes be relaxed through the introduction of deterministic trends or generalind covariance functions; a recent example of the former is the work of hide/man and Rubin [1995] who consider flow in a log hydraulic conductivity field with a given spatial trend. Li and McLaughlin [1991] describe a nonstationary spectral technique which is based on an extension of the Fourier representation commonly used to describe stationary random functions. Conditioning of random variables of interest (such as hydraulic conductivity, for example) on available data is an additional source of nonstationarity. In this case nonstationarity is a desirable goal, since conditional statistics generally provide a better description of a field. There are several ways to analyze this kind

21 20 of nonstationarity in flow and transport problems. These include Monte Carlo methods [Smith and Schwartz, 1980, 1981], moment equation methods based on small perturbation assumptions [Hoeksema and Kitanidis, 1985; Graham and McLaughlin, 1989], and, possibly, a nonstationary spectral method [Li, personal communications, 1995]. Ergodicity is always required to infer field statistics from spatial measurements; it cannot be proven, only hypothesized, unless falling under the purview of well established ergodic theorems such as the law of large numbers, which states that the time [space] averages over an infinite time interval [space domain] coincide with probability 1 (almost surely), for some classes of random functions, with the probabilistic mean values. It can be shown that [Yaglom, p.216, 1987] "in any application, non-ergodicity usually just means that the random function concerned is, in fact, an artificial union of a number of distinct ergodic stationary functions". Above, we defined ergodicity in its weakest form. Sometimes one needs a stronger statement. It is possible for a process to be ergodic for some statistical measures and not for others. Specifically, for purposes of our research it is enough to assume ergodicity of the mean and second statistical moments of the random function involved. Ultimately, predictions rendered by stochastic theories must be judged on the extent to which they correspond to independently established results. For example, the stochastic prediction that, in a layered medium, flow normal to the layers is controlled by a harmonic mean hydraulic conductivity is known to be "true" outside the scope of stochastic theory. Another approach based on a stochastic representation of hydrological properties is commonly referred to as geostatistics. This is the methodology of inferring the spatial and

22 21 ensemble statistics of presumed random fields from samples representing a single realization. The results of geostatistical inference form the input into stochastic flow and transport analyses. Two important aspects of geostatistics are (1) identification of the spatial structure of the variable (variogram estimation, trend estimation, etc.) and (2) interpolation or estimating the value of a spatially distributed variable from neighboring values taking into account spatial structure of the variable I Yeh, p.102, 1989]. Other aspects of interest are (3) averaging of available data, and (4) conditional simulation of equally likely spatial field realization. In the past, due to well-known reasons, science in general and stochastic theory in particular have been developing along sometimes parallel, sometimes intersected paths in Russia and in the rest of the World. Avoiding fruitless discussion on the theme "Who's been first", let us mention that the first analysis of seepage processes carried out by statistical methods we have been able to find is given in B.B. Devisons seminal paper published in Russian in In this paper, the concepts (introduced considerably earlier) of seepage velocity, porosity, and hydraulic conductivity are interpreted as mathematical expectations of some statistical ensembles, defined within a sufficiently small volume of the porous medium containing sufficiently many irregular interstitial channels. Such a homogenization of the flow and of the medium in the "small" has allowed him to consider the averaged parameters of the seepage process to be continuous, sufficiently smooth functions of the space coordinates and of time. A description of Devison's work together with a brief overview of Russian achievements in stochastic hydrology can be found in the book by M.1. Shvidler [1964].

23 22 R.A. Freeze is considered by many to be a "Western pioneer" in applying stochastic methods to subsurface hydrology. In his classical paper [Freeze, 1975], he has identified the need for inclusion of uncertainty analysis into groundwater modeling in the following way: "at the very least, we must recognize the uncertainties associated with our deterministic predictions due to the inherent nonuniformity of the porous media and to our uncertainty as to the exact nature of these nonuniformities. We also ought to investigate the further uncertainties introduced into the solutions by the uncertainties in initial and boundary conditions. And perhaps most importantly, we ought to examine the underlying assumption of the deterministic approach, one that is scarcely thought about as such by most groundwater hydrologists; namely, that it is possible to define an equivalent uniform medium that will act in every sense like the actual nonuniform one". These results, the developments of geostatistics (especially, spatial correlation) by Matheron [1967] and its subsequent adoption to hydrological problems by Delhomme [1979], Clifton and Neuman [1982], de Marsily [1986] and others have brought about a rapid growth of the literature on stochastic subsurface hydrology in recent years ( a review of the development of stochastic subsurface hydrology is given by Neuman [1982], Dagan [1989], Gelhar [1993], and Dagan and Neuman [19961). 1.2 Classification of stochastic methods Once one has decided to devote his/her scientific zeal to stochastic hydrology, there is a great variety of stochastic approaches to chose from. In this section, we shall classify

24 23 some of the existing methods. To solve any boundary-value problem (including those for a stochastic differential equation subject to random boundary and initial conditions), both numerical and analytical approaches are widely used. We shall briefly discuss Monte Carlo simulation as an example of a numerical method dealing with stochastic problems and such analytical and semianalytical methods as lumped-parameter models, perturbation analysis (as applied to stochastic problems), the self consistent technique, and the conditional non-local formalism. (a) Monte Carlo Simulation (MCS) The method of Monte Carlo Simulation (MCS) is a powerful and versatile mathematical tool capable of handling situations where other methods fail. This name was applied to a class of mathematical methods first by scientists working on the development of nuclear weapons in Los Alamos in the 1940s. Warren and Price [1961] were among the first to apply MCS to problems of fluid flow in heterogeneous oil reservoirs. In 1975, Freeze introduced the concept into subsurface hydrology. The implementation of the method consists of numerically simulating a population corresponding to random quantities in the physical problem (hydraulic conductivity distribution, in our case), solving the deterministic problem associated with each member of that population, and obtaining a sample of the random response quantities. This sample is used to estimate statistics of the response variables. Despite the fact that MCS is a very powerful method, it has drawbacks peculiar to all

25 24 numerical schemes: it provides limited insight into the overall behavior of the system modeled, and makes it difficult to draw general properties and conclusions from the solution obtained. Moreover, MCS has limitations due to the very random nature of the problem it is called upon to solve: there are no theoretical and practical criteria to control and measure convergence. A priori it is virtually impossible to guarantee convergence and to tell whether or not it has been achieved. A good introductory reference book on MCS is that by Kalos and Whitlock [1986]. (b) Lumped-Parameter Models Application of this method to a linear reservoir model of a stream-connected phreatic aquifer with a natural recharge modeled as a stationary random process is discussed by Gelhar [p.64, 1993]. The main idea of the approach is to write an overall water balance in terms of an ordinary differential equation (with time as the only independent variable). "Models of this type are appropriate to treat situations where the time variation of aquifer conditions is the primary concern. Such an approach often will be appropriate when addressing the problems of overall policy and management decisions relating to the behavior of the aquifer over extended periods of time. This kind of model, obviously, cannot be used to address questions of the spatial distribution. Also, lumped-parameter models are often consistent with the kind of limi ted data that is available for analyzing such problems".

26 25 (c) Perturbation Method The perturbation approach as applied to problems of random media is an extension of the method used in nonlinear analysis. Given certain smoothness conditions, the functions and operators involved are expanded in a Taylor series about their respective mean values. In doing so, hydraulic conductivity function, hydraulic head function, their covariances and cross-covariances are expanded (under assumption that these expansions are permissible!) in an asymptotic sequence in a small parameter o 2y (to be defined below). Then, partial differential equations satisfied by the various statistical moments of head are derived at each order. Unfortunately, the presence of higher moments precludes solving these equations without a closure approximation [Dugan, p.183, 1989]. The small perturbation analysis has been applied extensively to the study of flow through heterogeneous formation. Shvidler [1962] and Matheron [1967] are among the first who applied it to petroleum and groundwater problems, respectively. Many more references for, and thorough description of, the perturbation method can be found in the book of Dagan [1989]. The main disadvantages of the approach are (i) its limitation to a special and narrow class of "mildly heterogeneous" media so that the variance, G, of natural log hydraulic conductivity, Y = in K, is much smaller than unity ( o 2y << 1); and (ii) the question of validity of the perturbation approximation, i.e. of the convergence of expansion series and of the limiting value of cf y which ensures the accuracy of low-order approximations, which is hard to answer.

27 Nevertheless, the perturbation analysis is a universal and practically-oriented technique (when it is applicable!) that can be used in conjunction with other methods described below. 26 (d) Self-Consistent or Renormalization Technique (after Dagan [1989]) This method has been applied to determining the effective properties of multi-phase, heterogeneous, media (for review, see Beran, 1968, and Landauer, 1978). Let us consider, following G. Dagan, a formation made up from a collection of blocks of different conductivities K "In the first-order perturbation approximation, the head field can be determined as the sum of the head residuals caused by flow in the formation of uniform conductivity equal to [geometrical mean of hydraulic conductivity] K a in which each time a block of logconductivity Y inserted. At next order, two blocks have to be considered simultaneously and the procedure can be extended in principle to any order" [Dagan, p.186, 1989]. According to this approach, the discrete probability distribution function (p.d.f.) [Dagan, p.164, 1989] f(k; x) = n i (x) 8(K -Ki), (1.1) is used. In (1.1), M is the number of phases present, and nj is the volume fraction of blocks Ki such that for M n / f(k1)dk. and f(k) becomes the continuous p.d.f.. In the self-.) consistent approximation, the head field is that caused by an isolated block in a formation of constant but unknown, conductivity Kef under otherwise uniform mean flow. This constant is determined by the requirement that Kef is defined as the effective conductivity of the

28 27 composite medium. The advantage of the self-consistent approximation (SCA) over the first-order perturbation approximation is said to be its applicability to large variances o 2y. It has been shown by Dagan [1981] that the effective hydraulic conductivity thus derived is [Dagan, p.186, accurate for a "completely random" medium, i.e. one in which there is no correlation between the conductivities of neighboring blocks, and for a large number of phases, for which this is a plausible assumption". However, later studies (see, for example, Kitanidis [19901) have shown that SCA performs poorly for correlated fields at large o 2y. The disadvantage of the SCA comes from the need to introduce a p.d.f. as above, and from the assumption that blocks with constant conductivities K1 exist. Even assuming that they do, the following questions arise: What scale should one assign to a block? How does a choice of scale influence the value of Kej? Moreover, the assumption that fields created by individual block do not interfere seems to be artificial. (e) Conditional Non-Local Formulism This is a new approach proposed by Neuman and Orr [1993]. They considered steady state flow within a bounded domain, driven by random source and boundary conditions. Starting from the premise that Darcy's law applies locally within a random hydraulic conductivity field K(x), on the scale of a bulk volume (support) V centered about the point x, they have rigorously derived equations for unbiased conditional ensemble moments (i.e. ensemble mean and second moments) of hydraulic head and water flux. The ensemble mean

29 28 equation has the form of a standard deterministic flow equation subject to ensemble mean boundary conditions with an additional term called "residual flux" by the authors. The residual flux is an unknown mixed moment of hydraulic conductivity and gradient fluctuations about their respective conditional means. In some previous investigations (Bakr et al., [1978]; Mizell et al., [1982], Sun and Yeh, [1992], Gracharn and Tankersley, [1994], etc.), the products of fluctuations (residual flux) have been routinely omitted from the governing equation on the shaky ground that this term "is hard to evaluate and, therefore, is impossible to deal with". Loaicigia and Marino [1990] analyzed the error in neglecting some of the usually dropped perturbation terms in the case of steady flow. They presented a necessary and sufficient condition for the smallness of these terms rigorously applicable only to steady state flow. Residual flux has been shown to form a compact integral expression which is rigorously valid for a broad class of K(x) fields, including fractals above some cutoff, under arbitrary steady state flow regimes in either bounded or unbounded domains. The expression for the residual flux consists of spatial convolution (for nonconditional cases) integrals with kernels that cannot be evaluated quantitatively (they include Green's function for random partial differential equation) without either high-resolution Monte Carlo simulation or additional approximations (such as low order perturbation analysis for mildly heterogeneous porous media or "weak approximation of the residual flux"). Nevertheless, these kernels are sufficiently well defined to reveal some of the most interesting fundamental properties of a medium such as nonlocal nature of flow in random media, conditions required for the

30 29 existence of the effective hydraulic conductivity, influence of boundaries, etc. Unlike previous results (Dagan [1989], Gelhar [1993], etc.), this approach allows one to deal with boundary effects and formal conditioning of the random functions (hydraulic conductivity and hydraulic head) on local measurements. We shall extend this approach to transient flow in the present study. 1.3 Transient flow in heterogeneous formations While the direct problem of steady state flow, i.e., determining the head field for a given random structure, has been covered quite thoroughly by the aforementioned approaches, less attention has been paid to the more difficult direct problem of transient flow in heterogeneous formations. Applying Monte Carlo simulations to the conceptually simple problem of 1-D flow without spatial correlation for the case of constant initial head and a sudden drop to a new uniform value, Freeze [1975] has questioned the very possibility of determining effective parameters for unsteady flow: "The results... throw into question the validity of the hidden assumption that underlies all deterministic groundwater modeling; namely, that it is possible to select a single value for each flow parameter in a homogeneous but nonuniform medium that is somehow representative and hence defines an "equivalent" uniform porous medium. For transient flow there may be no way to define an equivalent medium". Since then, few numerical Monte Carlo simulations of transient flow have been published due, in part, to the large amount of computer time and storage required to run such simulations in two and three dimensions. Available published results [Sagar, 1978; Dettinger

31 30 and Wilson, 1981] have so far not dispelled the doubts raised by Freeze. Clearly, there is a need to address the issue analytically. The first attempt to address transient aspects of groundwater flow through heterogeneous porous media analytically has been undertaken by Mizell et al. [1980]. In their analysis, temporal variations of hydraulic head are restricted to its mean, and the storage coefficient is taken to be deterministic. Using a first-order perturbation analysis in which second-order perturbation products are neglected, and working with spectral representations of random functions (log-hydraulic conductivity and hydraulic head), the authors have obtained analytical expressions for the transient head variance, the transient head covariance function, and the transient head correlation function. Questions about the existence of effective hydraulic conductivity, or its form, have not been addressed. Although of great importance as a first step toward a better understanding of transient flow in random media, this analysis is not entirely rigorous as some terms in the equation are dropped that may have the same order as terms retained [Loaic.igia and Marino, 1990]. Application of Green's functions has enabled Dagan [1982] to obtain a simple, closedform solution for transient flow by means of a first-order perturbation approximation. He has considered two problems: (i) the general transient case, in which the head is initially constant and subsequently starts to change, and (ii) the periodic transient case, in which the average head gradient varies harmonically with time. In doing so, the following restrictions have been adopted: (1) the variance of the log hydraulic conductivity is smaller than one, (2) the average head gradient varies slowly in space and time. Dagan has found that effective parameters for

32 31 the medium, such as hydraulic conductivity and storativity, can be defined in a way which depends on the random structure of the medium and on time. The main result, from the author's viewpoint, is the derivation of criteria to decide under what transient regimes may the simpler and better-developed steady state analytical results be applied. Sun and Yen [1992] developed a stochastic approach to the inverse problem of transient groundwater flow. Using perturbation analysis and neglecting perturbation products, they derived a stochastic partial differential equation relating transient head and log hydraulic conductivity perturbations to each other. Based on the first-order approximation, the authors developed an algorithm to compute the variances of prediction errors. The algorithm involves calculating sensitivity coefficients with respect to a distributed In K field and cokriging these coefficients on a grid. Using variance estimates of prediction errors, the authors estimate the confidence region of their model predictions. hidelawn [1995] used perturbation analysis in the Fourier-Laplace (FL) domain to derive equations of average transient flow. The advantage of this approach is that in the FL domain dependence of the mean hydraulic gradient and mean velocity on the heterogeneity structure on one hand, and on the initial head and the distribution of sources on the other hand, are decoupled and have a local structure. This enables one to introduce an effective conductivity tensor in the FL domain which depends only on the statistical properties of the random K(x) field and acts as the proportionality coefficient in an average transformed version of Darcy's law. The inverse FL transform yields an average flux term that is nonlocal, forming a convolution integral in space- time of a kernel with the mean head gradient. This

33 kernel is the inverse FL transform of the effective conductivity tensor as defined in the FL domain. In addition to being valid only for a mildly heterogeneous porous medium with small Y, Indelman's results do not account for domain boundaries and are not applicable under conditioning. The present investigation is intended to extend the more general conditional nonlocal formalism of Neuman and OIT, discussed earlier, to transient flow in a bounded domain under the action of random source, boundary, and initial functions. As all results and approaches discussed above deal with derivatives of random functions and with stochastic differential equations, we shall start by discussing briefly the issue of differentiability of random functions Steady-state now in bounded domains In all investigations mentioned so far, flow in unbounded domains has been analyzed. Such an analysis simplifies considerably the computations, for instance, allowing the Fourier transform methodology. It is clear, however, that natural formations are of bounded extent, and the question is whether or not boundary effects influence flow analysis, in general, and effective parameters such as effective hydraulic conductivity, in particular. Analytical studies of bounded domains have primarily focused on the effects that boundaries may have on second moments involving head [Nail and Vecchia, 1986; Rubin and Dagan, 1988, 1989]. To estimate boundary effects on the effective hydraulic conductivity, Dagan [p.204, 1989] modified his results, obtained for an infinite domain, to account for flow in a half-space

34 33 bounded by an impervious lower boundary. He concluded that [Dagan, p.206, 1989] "for a uniform flow in a formation of stationary random structure, the presence of an impervious boundary has a small effect upon the effective conductivity, which is slightly different from its bulk value in a thin layer near the boundary". Nevertheless, the following questions remain: Is Dagan's conclusion valid for the effective hydraulic conductivity of the geological formations with more complicated geometry? What are the effects of boundaries and different boundary conditions? Evaluation of effective parameters for bounded domains is closely related to the issue of upscaling (a process of assigning equivalent properties to the grid blocks of a numerical model within which these properties are known to vary). Apart from numerous numerical investigations related to upscaling [e.g., Journel et al., 1986; Deutsch, 1989; Desbarcus and Dimitrakopoulos, 1990; Dur/ofsky, 1991; Fenton and Griffiths, 1993], the only rigorous theoretical approach, we could find, was that due to Indelman and Dagan [1993a, b]. However, their upscaling is rigorously valid only in a region sufficiently far from the boundary. Using the Neuman and OIT theory, Paleologos et al., [1996] developed a firstorder approximation for the effective conductivity of a domain bounded by two parallel boundaries on which head is constant in the mean. Their results compare favorably with those obtained numerically by A. Desbarats [Desbarats, 1992], and are rigorously valid in every point of the domain of interest. In the present investigation we shall extend their results to address a problem of evaluating the effective conductivity for a box-shaped domain, i.e. assigning an upscaled parameter for a rectangular grid.

35 On differentiability of random functions Darcy's law and the mass conservation equation describing groundwater flow in a porous medium are partial differential equations. When hydrologic variables and parameters are random functions, the former become stochastic differential equations. It is therefore important to consider the question of the differentiability of random functions. The question is of particular importance in the choice of an appropriate covariance model for a random field. The formal deterministic limit for partial differentiation, ii rn F(x /- F(x l,...,x (1.2) is not defined for a random function Rx i, xd. An alternative is to talk about mean square differentiation. A random function F is said to be differentiable in mean square, and has a partial derivative, F i -OF I a xi, with respect to x, if RX I +8,,Xn).....x ) af, lim ax, 2 (1.3) where < > represents ensemble mean. It can be shown that [Yaglom, 1962; Priestley, p.154, F'exists if and only if the second derivative of the correlation function, p"(x), exists. It is clear that "mean square convergence involves only quadratic functions of F(x), and thus depends only on the second order properties of the process [field]. Consequently, conditions for, e.g., stochastic continuity or stochastic differentiability are immediately translatable into corresponding conditions on the autocorrelation function, p" [ Priestley, p.154, 1981]. For

36 statistically homogeneous (stationary) field F(x) with <F(x)> = 0, the existence of F' implies [Loaiciga and Marino,1990; Priestley, p.156, 1981] 35 acf(x=o) _ 0 x, for all j, (1.4) where x is the separation vector between any two locations x 1 and x, defined as =-- X i - X2, X = lx1 = (1.5) As has been pointed out, for example, by Loaiciga and Marino [1990], the widely used exponential, isotropic covariance model for log hydraulic conductivity, Y = In K, C y(x) = cr 2y e x ix (1.6) where o the variance of log hydraulic conductivity and A. is its correlation scale, satisfies condition (1.4) in three-dimensions (3-D) but not in two (2-D) and one (1-D) dimensions (n = 1 and 2, respectively). Although of theoretical importance, condition (1.4) does not provide a practical tool for solving the problem of differentiability. To deal with this issue, Gelhar [p.53, proposed weaker conditions to be satisfied by the spectrum, S yy, and the covariance function, C. If the partial derivative, Z = Y is a stationary field with finite variance, oz2, these conditions have the following form

37 36 a2 cyoc = 0 2z = f(02syy(w)do) - < a xj2 (1.7) where co is an n-dimensional wave number vector. The spectrum of Y must go to zero more rapidly than 16.)-3 1 for this to be satisfied. In 2-D and 1-D, one achieves this by truncating the spectrum beyond a certain maximum wave number, co according to 0 2 x Syy(6)) = { TE (1 + X 2 W2) ' o, 1601 ^ lw 1 J nzi 1 6) 1 > 1 6) 1 '.1 In..1 for all j, (1.8) where a' is an approximate variance of Y resulting from truncation (1.8). Gelhar further concludes that the random field Y is differentiable and "the variance of this process is found by integrating (1.8), which for the case of a large cutoff frequency is _ TC CO X CO >> 1. (1.9) The variance of Y is practically unchanged by the truncation, but the microscale, which is found by using (1.8) in (1.7), is now finite and is given by = TEX I (il (A)» 1 nz' (1.10) We see that the differentiability of a process is sensitive to the high-frequency behavior of the spectrum, or to the behavior of the covariance function at small separation. The kind of truncation that is presented by (1.8) is reasonable from a physical point of view, especially for

38 37 the case of a spatially variable process. In that situation, it is clear, that the differential equations describing flow, and the parameters such as hydraulic conductivity that occur in the differential equations, are applicable only down to some minimum spatial scale". Throughout this investigation, we define such a scale as a volume, V, of medium samples necessary for measuring hydrologic parameters. It is important to note that V need not constitute a representative elementary volume (REV); in fact, an REV need not even exist. The only requirement is that all the quantities which enter the flow equations be defined and/ or measured on the same support, V [Neuman, It is clear that satisfying condition (1.7) does not guarantee that condition (1.4) holds true and, therefore, that the random function of interest is really differentiable. The following argument by Dagan may be helpful [Dagan, p.165, 1989]; "We shall often adopt a function Cy of discontinuous slope at the origin, even for continuous Y, for the sake of convenience. This is permissible if Y serves as an input for integration, the detailed behavior of Cy near the origin being then of little consequence upon results. Furthermore, in practice Cy is inferred from measurements and the points are not that close to warrant an accurate description of Cy near the origin, and the shape there is generally obtained by extrapolating from finite [x] toward Ix = 01". 1.6 Stochastic partial differential equations Once the question of stochastic differentiability has been explored, one can consider stochastic differential equations. Stochastic modeling of groundwater flow uses the same

39 38 governing differential equations as deterministic models. However, the former differs from the latter in that its input functions are uncertain and, therefore, the model output must be described in probabilistic rather than deterministic terms. The following is a brief review of a theory of stochastic differential operators which is used in Chapter 2 of this investigation Green's function as an operator's kernel Let us consider a deterministic differential equation L 1x h(x, t) = fix, t), subject to homogeneous initial and boundary conditions. Here L is a linear differential operator acting on a space of functions h(x, t). Suppose that there exists an inverse operator such that LL' = L'L = I, where / is the identity operator. For general differential operators L, is an integral operator with a deterministic kernel G o(y, x, t,t), so that h(x, t) = L i, xl f(x, t) f f Go(y, t, -c) fly, dy elt (1.12) 0 SI Operating with L gives Li,xh(x, t) = Li, x f f G0(y, t, t)f(y, dy (1.13) 0 S2 Using the theory of distributions it can be shown that the operation of differentiation, L, and

40 the operation of integration can be legitimately interchanged [Adomian, p.4,1983]. Then, considering (1.11), 39 fix, t) = f fl1. xgo(y, x, t,t)f(y, dy (1.14) 0 0 Hence xgo(y,, x, t, = 8(y - x) 8(t -, (1.15) where 8 is the Dirac delta function and the kernel of the inverse operator L1 coincides with the corresponding deterministic Green's function G o Decomposition of linear stochastic partial differential equations In the following two sections of this chapter, we present a brief overview of a decomposition method due to Adomian [1983]. Generalizing equation (1.11), we consider a stochastic differential equation h = f,(1.16) where g r. is a stochastic linear partial differential operator acting on a space of functions h which satisfy homogeneous initial and boundary stochastic conditions. Decomposing it into a deterministic partial differential operator L and a zero-mean random partial differential operator ;sr leads to

41 40 L1,x tt,x (1.17) Here and in the sequel, script letters denote stochastic operators and block letters - deterministic ones. If g,, x1 and L,,x' exist, then L,, x h = f - at,. x h or h = - L R1h. (1.18) Recalling (1.16) yields the following operator identity co - - _ L -1 - L ,x 1,x 1,x 1,x (1.19) where, in analogy to (1.12), the stochastic inverse operator g-i is given by g i,"xf(x, t) -= f f G(y, x, t, t)f(y, t) dy dt, 0 S2 (1.20) and G(y, x, t - 7) is a stochastic Green's function. The identity (1.19) arises quite naturally from decomposition (1.17) of the stochastic operator Y. Let us parameterize this operator identity with the parameter X, by writing [Adomian, p.135, 1983] "- I _ -' (1.21) 1,X We shall later set X = 1; it is not a small parameter. This is identical to parameterizing the decomposition h = h + h l + h +... as h = h o + + X Analogously, parameterizing and decomposing into a series of partial differential operators 41 as