Nonlocal properties of nonuniform averaged flows in heterogeneous media

Abstract

The properties of nonuniform average potential flows in media of stationary random conductivity are studied. The mathematical model of average flow is derived as a system of governing equations to be satisfied by mean velocity and mean head. The averaged Darcy's law determines the effective conductivity as an integral operator of the convolution type, relating the mean velocity to the mean head gradient in a nonlocal way. In Fourier domain the mean velocity is proportional to the mean head gradient. The coefficient of proportionality is referred to as the effective conductivity tensor and is derived by perturbation methods in terms of functionals of the conductivity spatial moments. It is shown that the effective conductivity cannot be defined uniquely for potential flows. However, this nonuniqueness does not affect the spatial distribution of the mean head and velocity. Analytical expressions of the effective conductivity tensor are derived for two-and three-dimensional flows and for exponential and Gaussian correlations of isotropic conductivity. The fundamental solution of the governing equations in effective media (mean Green function) is calculated for the same cases. Two new asymptotic models of the averaged Darcy's law are developed to be applicable to large and small scales of heterogeneity. Several asymptotic expansions of the two-dimensional and three-dimensional mean Green functions are derived for exponential and Gaussian correlations.

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“…Where simplifjring assumptions concerning the spatial structure of the porous medium can be made and flows are uniform, upscaled permeability estimates are derived directly through analytical solutions (e.g., Gutjahr et al, 1978, Dagan, 1981Gelhar and Axness, 1983;Deutsch, 1989;Rubin and Gomez-Hernandez, 1990;Desbarats, 1992a). Where non-uniform flows are encountered a single effective permeability value can no longer be defined that is dependent soIely on the statistical properties of the permeability; rather, upscaling must explicitly account for the flow conditions unique to the problem (e.g., Ababou and Wood, 1990;Desbarats, 1992b;Indelman and Abramovich, 1994;Sanchez-Vila, 1997). For cases involving complicated heterogeneities andor flows numerical solutions to the upscaling problem are necessitated (e.g., White and Home, 1987;I Kitanidis, 1990;Durlofsky, 1991).…”

Experimental and theoretical memory diffusion of water in sand

“…Darcy's law, which states that the flux is proportional to the pressure gradient, has led many authors to set rigorous equations representing the interaction between the porous media and the fluid flow (Bear, 1972;Sposito, 1980;Steefel and Lasaga, 1994;Dewers and Ortoleva, 1994;Indelman and Abramovici, 1994;Cushman and Moroni, 2001). In spite of this, some data exhibit properties which may not be inCorrespondence to: G. Iaffaldano (giampiero@geophysik.uni-muenchen.de) terpreted neither with the classical theory of propagation of pressure and fluids in porous media (Bell and Nur, 1978;Roeloffs, 1988) nor adequately with many of the new theories.…”

Simple closed form formulas for predicting groundwater flow model uncertainty in complex, heterogeneous trending media

“…[4] Stochastic approaches for analyzing groundwater flow in heterogeneous trending media generally divide into (1) analytical techniques [e.g., Loaiciga et al, 1993;Rubin and Seong, 1994;Li and McLaughlin, 1995;Neuman and Orr, 1993;Indelman and Abramovich, 1994], which provide convenient closed form expressions for quantities such as the velocity variances and effective hydraulic conductivities but depend on restrictive assumptions (e.g., linear trends) and (2) numerical techniques [e.g., Smith and Freeze, 1979;Mclaughlin and Wood, 1988;Li et al, 2003Li et al, , 2004a, which make fewer assumptions but are more difficult to apply in practice. Here we present simple, closed form analytical solutions that relax the assumptions required in existing analytical theories of stochastic groundwater flow.…”