WATER RESOURCES RESEARCH, VOL. 17, NO. 2, PAGES 33?-350, APRIL 1981 Stochastic Models of Subsurface Flow: Infinite Versus Finite Domainsand Stationarity ALLAN L. GUTJAHR AND LYNN W. GELHAR New Mexico Institute of Mining and Technology, Socorro, New Mexico 87801 Stochastic solutions of the differential equation describing one-dimensional flow through a porous me- dium with spatially variablehydraulicconductivity which is represented by a stationary (statistically ho- mogeneous) randomprocess are developed usingseveral techniques. The analyticalapproximations us- ing first-order analysis (propagation of error), covariance differential equations, and variogram analysis all yield consistent results which demonstrate the important effects of boundaryconditions and condi- tioning.Using the theoryof intrinsicrandomfunctions, stochastic solutions are developed for the case whenthe logarithm of the hydraulicconductivity is a three-dimensional stationary randomfield. In con- trast tothe one-dimensional case, it isfound that the resulting thr•e-dimensional head perturbation will be locally stationary under very general conditions. Resultsfrom the one-dimensional analytical solu- tions are found to be in agreement with previous Monte Carlo simulations for a flow system of finite length.The solution based on linearization in the logarithmof hydraulicconductivity provedto be very robust, showing reasonable agreement with Monte Carlo results evenfor the largest input standard de- viation of ouar-- 4.6. INTRODUCTION Severalrecentpaperson stochastic analysis of subsurface flows have introduceda variety of stochastic models [Freeze, 1975;Smith and Freeze, 1979; Bakr et al., 1978;Gutjahr et al., 1978;Dagan, 1979].In this paper we concern ourselves with some of the relationships between the variousresults and the interpretation of the models. Specifically, the analyses of Bakr et al. [1978], Gutjahr et al. [1978], and Dagan [1979] treat infinitedomainmodels, while the Monte Carlo simulations of Freeze [1975] and Smith and Freeze [1979] consider finite domain models. Weinvestigat e the relationship between the infiniteand finite domainmodels in one dimension by deriving some explicitanalyticalapprox- imations for the finite case. Someapparentconflicts that arise are resolved by recognizing the important role of conditioning on fixed end point values. We also analyze nonstationary modelsusing variogram analysis and the theory of intrinsic random functions [Matheron, 1971, 1973; Delhomme, 1978, 1979], applying thesetechniques to one- and three-dimen- sional flow problems. Finally, a comparison of our one-di- mensional results and the simulations of Smith and Freeze is made. In making our comparisons, we concentrate on one-dimen- sional models. While thesecases may not be very practical, they do illustrate some of the problems encountered in the ap- plication and interpretation of random differential equations in hydrology and emphasize the importance of choosing a re- alisticphysicalmodel. ONE-DIMENSIONAL, FIRST-ORDER ANALYSIS In this section we assume K(x), the hydraulicconductivity, and W(x) -- 1/K(x), the hydraulic resistivity, are second-order stationary (alsocalledstatistically homogeneous [Lumley and Panofsky, 1964]) processes. We let !• -- E[W(x)] and R,,•(•) -- cov[W(x + •), W(x)]be th e mean andcovariance function, respectively, for the resistivity. Finally,we let q•(x) be the ran- dom process representing thehydraulic head andlet oh 2be the varianceof q•(x)in the stationary case. Model 1, q Fixed In Gutjahr et aL [1978]we considered stationary solutions to the flow equation •x'---qW(x) (1) whereq is the specific discharge, a constant. If Rha(•) is the co- variancefunction of q• -- H + h, whereH is nonrandomand h is presumed stationary, with E(h) -- O,then we showed and where Raa(O -- -q2 tR•(•- t) dt (2a) o• • -- Rha(0) -- -q2 •R•(0 d•-- q'ow'-/• (2b) is the square of the correlation length and • -- Rw•(0).[See Gutjahret al., 1978,equations (1), (3), (24), and (25).] For fur- ther reference we call this model 1. Note also that the final re- sult in (2b) assumes R•(•) -- a•(1 -[•l/l)e -I•l/1 (2c) Model 2, q Random A model similar to model 1 is the one whereboundary con- ditions are imposed thereby destroyingthe stationarity of q•(x). This model was investigated by Freeze [1975] and, in- cluding the importanteffects of correlations between values of W(x), by Smith andFreeze [1979]. We call the following situa- tion model 2: d IK(X) d•l -- 0 (3a) Copyright¸ 1981 by the AmericanGeophysical Union. Paper number 80W 1186. 0043-1397/81/080W- 1186501.00 337 q•(O) = 0 q•(L) -- JL (3b)