Averaging of unsteady flows in heterogeneous media of stationary conductivity

Abstract

A procedure for deriving equations of average unsteady flows in random media of stationary conductivity is developed. The approach is based on applying perturbation methods in the Fourier-Laplace domain. The main result of the paper is the formulation of an effective Darcy's Law relating the mean velocity to the mean head gradient. In the Fourier-Laplace domain the averaged Darcy's Law is given by a linear local relation. The coefficient of proportionality depends only on the heterogeneity structure and is called the effective conductivity tensor. In the physical domain this relation has a non-local structure and it defines the effective conductivity as an integral operator of convolution type in time and space. The mean head satisfies an unsteady integral-differential equation. The kernel of the integral operator is the inverse Fourier-Laplace transform (FLT) of the effective conductivity tensor. The FLT of the mean head is obtained as a product of two functions: the first describes the FLT of the mean head distribution in a homogeneous medium; the second corrects the solution in a homogeneous medium for the given spatial distribution of heterogeneities. This function is simply related to the effective conductivity tensor and determines the fundamental solution of the governing equation for the mean head. These general results are applied to derive the effective conductivity tensor for small variances of the conductivity. The properties of unsteady average flows in isotropic media are studied by analysing a general structure of the effective Darcy's Law. It is shown that the transverse component of the effective conductivity tensor does not affect the mean flow characteristics. The effective Darcy's Law is obtained as a convolution integral operator whose kernel is the inverse FLT of the effective conductivity longitudinal component. The results of the analysis are illustrated by calculating the effective conductivity for one-, two- and three-dimensional flows. An asymptotic model of the effective Darcy's Law, applicable for distances from the sources of mean flow non-uniformity much larger than the characteristic scale of heterogeneity, is developed. New bounds for the effective conductivity tensor, namely the effective conductivity tensor for steady non-uniform average flow and the arithmetic mean, are proved for weakly heterogeneous media.

Quantifying flow and remediation zone uncertainties for partially opened wells in heterogeneous aquifers

“…Similar conditions were considered by previous investigations (e.g. Dagan, 1989;Indelman, 1996Indelman, , 2002Severino et al, 2008 ). The comparison of different boundary types for well locations can be found in the study of Indelman and Dagan (2004).…”

Effective Conductivity of an Isotropic Heterogeneous Medium of Lognormal Conductivity Distribution

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php the stationary K(x) has a finite integral scale, (ii) the ratio between the latter and the domain size tends to zero (unbounded domain), and (iii) the flow is uniform in the mean. The effect of domain finiteness was discussed recently by Dagan (2001), whereas the impact of mean flow nonuniformity was analyzed by Indelman (1996). Only under the above conditions does the common relationship V = −K ef ∇ H hold.…”

Multiscale Description and Upscaling of Fluid Flow in Subsurface Reservoirs

“…This linear equation has the structure of an integro differential equation, and was derived independently by Indelman (1996), and Noetinger and Gautier (1998) who used a Feynman graph approach. The memory kernel Σ(r, t) (a second order symmetric tensor) depends explicitly only on the correlation functions of the permeability of arbitrary order.…”

“…It shows that this kernel has a typical spatial range equal to l c , and a time range of l c 2 /D which is the typical diffusion time over l c (D = < k > /φµc t ). Equation (19) can be written under a conservation equation form when introducing a local velocity V(r, t) as: (21) (22) Indelman (1996) showed that this velocity V (r, t) is exactly the average of the local Darcy flux defined by V(r, t) = < -k/µ ∇p(r, t) >. This means that the average local rate at point r is a weighted average of the pressure gradients into the "correlation" zone of r. In the Fourier-Laplace domain, defined by it is possible to generalise the concept of effective permeability by introducing a wave vector dependant effective permeability tensor K eff, (q, s).…”