Stochastic Analysis of Transient Flow in Heterogeneous, Variably Saturated Porous Media: The van Genuchten–Mualem Constitutive Model

Early stochastic studies focused on steady-state, gravity-dominated unsaturated flow in unbounded domains In this study, on the basis of the van Genuchten–Mualem constitu(e.g., Yeh et al., 1985a,b; Russo, 1993, 1995a,b; Yang et tive relationship, we develop a general nonstationary stochastic model for transient, variably saturated flow in randomly heterogeneous meal., 1996; Zhang et al., 1998; Harter and Zhang, 1999). dia with the method of moment equations. We first derive partial Under these conditions, the unsaturated flow field is differential equations governing the statistical moments of the flow stationary, and hence analytical or semianalytical soluquantities by perturbation expansions and then implement these equations are possible. Recently, some researchers investitions under general conditions with the method of finite differences. gated the effects of boundary conditions on steady-state The nonstationary stochastic model developed is applicable to the flow and the consequent effects of flow nonstationarity entire domain of bounded, multidimensional vadose zones or intein one-dimensional semibounded domains (Andersson grated unsaturated–saturated systems in the presence of random or and Shapiro, 1983; Indelman et al., 1993) or two-dimendeterministic recharge and sink–source and in the presence of sional bounded domains (Zhang and Winter, 1998). It multiscale, nonstationary medium features. We demonstrate the has been found that the simpler, gravity-dominated flow model with some two-dimensional examples of unsaturated and integrated unsaturated–saturated flows. The validity of the developed models may provide good approximations for flow in stochastic model is confirmed through high-resolution Monte Carlo vadose zones of large thickness and/or coarse-textured simulations. We also investigate the relative contributions of the soil soils, although they may not be valid for vadose zones variabilities (KS, , and n ) as well as the variability in recharge Q to of fine-textured soils with a shallow water table. More the pressure head variance. It is found that the pressure head variance recently, a number of studies looked at transient unsatuis sensitive to these variabilities, in the order of n, , KS, and Q. rated flows (Protopapas and Bras, 1990; Unlu et al., Though the variability of and n is usually smaller than that of 1990; Mantoglou, 1992; Liedl, 1994; Zhang, 1999; FousKS and Q, their effect on the pressure head variance should not sereau et al., 2000) and transient unsaturated–saturated be ignored. flow (Li and Yeh, 1998; Ferrante and Yeh, 1999; Zhang and Lu, 2002). To describe unsaturated flow, the constitutive relaA geologic formations exhibit a high detionships of unsaturated hydraulic conductivity K vs. gree of spatial variability, medium properties, inpressure head and effective water content e vs. cluding fundamental parameters such as permeability must be specified. Three models are commonly used to and porosity, are usually observed only at a few locadescribe these functional relationships: the van Genuchtions because of the high cost associated with subsurface ten–Mualem model (van Genuchten, 1980), the Brooks– measurements. This combination of significant spatial Corey model (Brooks and Corey, 1964), and the Gardheterogeneity with a relatively small number of observaner–Russo model (Gardner, 1958; Russo, 1988). Most tions leads to uncertainty about the values of medium existing stochastic analyses utilize the Gardner–Russo properties and, thus, to uncertainty in predicting flow model because of its simplicity (e.g., Yeh et al., 1985a,b; and solute transport in such media. It has been recogYeh, 1989; Russo, 1993, 1995a,b; Yang et al., 1996; nized that the theory of stochastic processes provides a Harter and Yeh, 1996a,b; Zhang, 1999; Tartakovsky et natural method for evaluating flow and transport unceral., 1999; Lu et al., 2000; Zhang and Lu, 2002). On the tainties. Many stochastic theories have been developed other hand, the more complex van Genuchten–Mualem to study the effects of spatial variability on flow and and Brooks–Corey models usually fit measured K( ) transport in saturated zones (e.g., Dagan, 1989; Gelhar, and ( ) data better. Zhang et al. (1998) investigated 1993; Zhang, 2002) and in unsaturated zones (e.g., Dathe impact of different constitutive models on the results gan and Bresler, 1979; Bresler and Dagan, 1981; Andersof stochastic analyses of steady-state, gravity-dominated son and Shapiro, 1983; Yeh et al., 1985a,b; Hopmans et flow. It was found that the impacts of the constitutive al., 1988; Destouni and Cvetkovic, 1989; Polmann et al., models on the statistical moments of pressure head, 1991; Mantoglou, 1992; Indelman et al., 1993; Liedl, effective water content, unsaturated hydraulic conduc1994; Russo, 1993, 1995a,b; Harter and Yeh, 1996a,b; tivity, and velocity depend on saturation ranges. For Zhang and Winter, 1998; Zhang et al., 1998; Zhang, example, the mean head and the mean effective water 1999; Tartakovsky et al., 1999; Foussereau et al., 2000; content for the Brooks–Corey model differ greatly from Lu et al., 2000, 2002). In the unsaturated zones, the their counterparts for the Gardner–Russo model near nonlinearity of flow further complicates the problem. the dry and wet limits, while the differences are small at the intermediate range of saturation. Owing to its Zhiming Lu and Dongxiao Zhang, Hydrology, Geochemistry, and Geology Group (EES-6), MS T003, Los Alamos National Laboratory, mathematical complexity, the van Genuchten–Mualem Los Alamos, NM 87545. Received 31 Jan. 2002. *Corresponding author ([email protected]). Abbreviations: MC, Monte Carlo approach; ME, moment equation– based approach. Published in Vadose Zone Journal 1:137–149 (2002).