Peter Indelman

Nonlocal properties of nonuniform averaged flows in heterogeneous media

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 Darcys 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 Darcys law are developed to be applicable to large and small scales of heterogeneity. Several asymptotic expansions of tue two-dimensional and three-dimensional mean Green functions are derived for exponential and Gaussian correlations.

Steady Flow Toward Wells in Heterogeneous Formations: Mean Head and Equivalent Conductivity

We consider steady flow of water in a confined aquifer toward a fully penetrating well of radius rw (Figure 1). The hydraulic conductivity K is modeled as a three-dimensional stationary random space function. The two-point covariance of Y = In (K/KG) is of axisymmetric anisotropy, with I and Iυ, the horizontal and vertical integral scales, respectively, and KG, the geometric mean of K. Unlike previous studies which assumed constant flux, the well boundary condition is of given constant head (Figure 1). The aim of the study is to derive the mean head 〈H〉 and the mean specific discharge 〈q〉 as functions of the radial coordinate r and of the parameters σy2, e = I/Iυ and rw/I. An approximate solution is obtained at first-order in σy2, by replacing the well by a line source of strength proportional to K and by assuming ergodicity, i.e., equivalence between , , space averages over the vertical, and 〈H〉 〈q〉, ensemble means. An equivalent conductivity Keq is defined as the fictitious one of a homogeneous aquifer which conveys the same discharge Q as the actual one, for the given head Hw in the well and a given head in a piezometer at distance r from the well. This definition corresponds to the transmissivity determined in a pumping test by an observer that measures Hw, , andQ. The main result of the study is the relationship (19) Keq = KA(1 − λ) + Kefuλ, where KA is the conductivity arithmetic mean and Kefu is the effective conductivity for mean uniform flow in the horizontal direction in the same aquifer. The weight coefficient λ 10, λ has the simple approximate expression λ* = ln (r/I)/ In )r/rw). Near the well, λ ≅ 0 and Keq ≅ KA, which is easily understood, since for rw/I ≪ 1 the formation behaves locally like a stratified one. In contrast, far from the well λ ≅ 1 and Keq ≅ Kefu the flow being slowly varying there. Since KA > Kefu, our result indicates that the transmissivity is overestimated in a pumping test in a steady state and it decreases with the distance from the well. However, the difference between KA and Kefu is small for highly anisotropic formations for which e ≪ 1 . A nonlocal effective conductivity, which depends only on the heterogeneous structure, is derived in Appendix A along the lines of Indelman and Abramovich [1994].

Upscaling of permeability of anisotropic heterogeneous formations: 1. The general framework

In numerical simulations of flow through heterogeneous formations, the domain is partitioned into numerical elements. The solution requires assigning physical properties to each numerical block. The process of transferring information from the scale of actual heterogeneity to that of the numerical elements is known as upscaling. We consider steady, one-phase flow and the only property of interest is the permeability, which is regarded as a random space function, and the same is true for the upscaled conductivity. In this paper we establish the necessary and sufficient conditions to be satisfied by upscaling. The necessary conditions are expressed with the aid of the global response of the formation, e.g., the total flux caused by a constant pressure head drop applied on the boundaries. The requirement is that the expected value and the variance of the total flux are the same in the actual and upscaled domains. These are supplemented by local conditions which stipulate that space averages of flow variables over the numerical blocks in the upscaled domain tend to those in the actual formation when the elements size tends to 0. The result of this analysis is that the upscaled permeability must be such as to ensure the equality, in a statistical sense, between the space averaged dissipation in the two media. By assuming an unbounded domain, average uniform flow and stationarity, simplified relationships between the moments of the dissipation in the upscaled and actual media are derived. It is shown that the effective permeability of the two media is the same. The approach is illustrated for stratified formations for which exact composition rules are possible. Even in this simple case the upscaled permeability is a tensor, characterized at second order by two mean components and three covariance functions.

Solute transport in divergent radial flow through heterogeneous porous media

Radial flow takes place in a heterogeneous porous formation of random and stationary log-conductivity Y ( x ), characterized by the mean 〈 Y 〉, the variance σ 2 Y and the two- point autocorrelation ρ Y which in turn has finite and different horizontal and vertical integral scales, I and I v , respectively. The steady flow is driven by a head difference between a fully penetrating well and an outer boundary, the mean velocity U being radial. A tracer is injected for a short time through the well envelope and the thin plume spreads due to advection by the random velocity field and to pore-scale dispersion. Transport is characterized by the mean front r = R ( t ) and by the second spatial moment of the plume S rr . Under ergodic conditions, i.e. for a well length much larger than the vertical integral scale, S rr is equal to the radial fluid trajectory variance X rr . The aim of the study is to determine X rr ( t ) for a given heterogeneous structure and for given pore-scale dispersivities. The problem is more complex than the similar one for mean uniform flow. To simplify it, the well is replaced by a line source, the domain is assumed to be infinite and a first-order approximation in σ 2 Y is adopted. The solution is still difficult, being expressed with the aid of a few quadratures. It is found, however, that it can be derived quite accurately for a sufficiently small anisotropy ratio e = I v / I by retaining only one term of the velocity two-point covariance. This major simplification leads to simple calculations and even to analytical solutions in the absence of pore-scale dispersion. To compare the results with those prevailing in homogeneous media, apparent and equivalent macrodispersivities are defined for convenience. The major difference between transport in radial and uniform flow is that the asymptotic, large-time, apparent macrodispersivity in the former is smaller by a factor of 3 than in the latter. For a three-dimensional point source the reduction is by a factor of 5. This effect is explained by the rapid change of the mean velocity during the period in which the velocities of two particles injected at the source become uncorrelated. In contrast, the equivalent macrodispersivity tends to its value in uniform flow far from the well, where the flow is slowly varying in space.

Flow in Heterogeneous Media Displaying a Linear Trend in the Log Conductivity

The purpose of this study is to derive a general solution for the problem of flow in nonstationary geological formations where the nonstationary manifests itself in the form of a spatial trend in the mean log conductivity. A stochastic frame of reference is adopted to account for the spatial variability of the hydraulic conductivity. For a complete stochastic description we derive the expected values and spatial covariances of the hydraulic head and the fluid flux vector, as well as a relation between the expected values of the head and the fluxes. These expressions are obtained using a perturbation expansion of the log conductivity about its nonstationary mean, and they are correct to the first order in the variance of the log conductivity. The expressions we derive are applicable for any space dimensionality and for arbitrary orientation of the trend in space. A general methodology is outlined for derivation of these expressions for any type of spatial covariance; and for demonstration, explicit results are obtained for a Gaussian isotropic covariance. 16 refs., 6 figs.

A higher‐order approximation to effective conductivity in media of anisotropic random structure

Properties of the effective conductivity tensor Keff are studied by deriving the second-order terms in its expansion in the variance σ2 of normally distributed log conductivity. It is shown that for media of anisotropic structure, the components of the effective conductivity tensor are expressed by a functional of the log conductivity covariance; that is, it depends on the shape of the correlation function and not only on anisotropy ratios, variance σ2, and space dimensions. However, the trace of Keff is independent of the log conductivity autocovariance, and for a given mean conductivity depends only on σ2. The dependence of the effective conductivity on the correlation structure is illustrated for Gaussian and exponential autocovariances of log conductivity and for two- and three-dimensional flows.

Averaging of unsteady flows in heterogeneous media of stationary conductivity

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 Darcys Law relating the mean velocity to the mean head gradient. In the Fourier-Laplace domain the averaged Darcys 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 Darcys Law. It is shown that the transverse component of the effective conductivity tensor does not affect the mean flow characteristics. The effective Darcys 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 Darcys 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.

Correlation structure of flow variables for steady flow toward a well with application to highly anisotropic heterogeneous formations

The study, a continuation of that of Indelman et al. [1996], aims at deriving the second-order moments of flow variables such as hydraulic head, its gradient, and the specific discharge for steady flow toward a fully penetrating well in a confined heterogeneous aquifer. The log conductivity Y=ln K is modeled as a three-dimensional stationary function of Gaussian correlation of anisotropy ratio e. By using first-order approximations in σ2Y and e, we derive the variance and the vertical integral scale of the piezometric head H, of its radial gradient Er and of the radial component of the specific discharge qr. Owing to the nonuniformity of the average flow, these quantities are functions of the distance from the well. It is shown that the variances of the head σ2H and of its gradient σ2Er, as well as the crossvariance σE,Y between Er and Y vanish at the well, whereas the discharge variance σ2qr tends to the product between the log conductivity variance σ2Y and the squared mean discharge 〈qr〉2. This behavior pertains to a stratified formation surrounding the well. Far from the well (≈75 horizontal Y integral scales I) the head variance approaches a constant value. For r ≥ 10I the moments σ2Er, σ2qr and σErY tend to the corresponding values for uniform flow but with the local mean head gradient replacing the constant one. The head vertical integral scale grows indefinitely with r, whereas the vertical integral scale of the flux is larger by one log conductivity vertical scale than the one prevailing in uniform flow. This latter property is explained by the presence of the source line, which increases the correlations in the vertical direction. The present results may be used in identifying the log conductivity statistical parameters from flowmeter velocity measurements in piezometers surrounding pumping or injecting wells.

Stochastic analysis of unsaturated steady state flow through bounded heterogeneous formations

The effect of heterogeneity in saturated hydraulic conductivity (Ks) and in a pore-scale distribution parameter (α) on unsaturated steady state flow in bounded domains is studied. The properties are assumed to be random space functions having stationary means and covariances. An analytic model based on small-perturbation approximation in Ks and α for predicting the spatial moments of the pressure head from the moments of the input variables is developed. The solution applies to the entire unsaturated domain without invoking the unit mean hydraulic gradient assumption. Under conditions of small-scale variability the solution simplfies considerably resulting in closed-form expressions for the moments of the pressure head. These expressions are tested in synthetically generated random fields of α and Ks having prescribed statistical properties.

Upscaling of permeability of anisotropic heterogeneous formations: 2. General structure and small perturbation analysis

The general methodology developed in part 1 (Indelman and Dagan, this issue) of this study is applied to the detailed analysis of upscaled permeability. First, the structure of the dissipation function for the general conductivity, a tensor of stationary random components, is examined with the aid of dimensional analysis. It is shown that for arbitrary shapes of the numerical elements, the upscaled permeability has also this general structure, and the numbers of unknown parameters and equations match. This result suggests that the upscaling problem has an unique solution. In the particular case of scalar permeability of isotropic covariance of the actual formation, it is shown that a similar upscaled permeability is possible only for spherical (circular) numerical elements. Otherwise, the upscaled permeability has to be a tensor of anisotropic covariance. If the actual formation has a scalar permeability of axisymmetric covariance, upscaling preserves the last property only for axisymmetric partition elements, i.e., for a sphere, cylinder, and ellipsoid. Explicit expressions for the first moments of the upscaled permeability (mean, covariance) are derived at first order in the log permeability variance. The detailed computations for a scalar permeability of axisymmetric covariance and for axisymmetric numerical elements lead to simple results. The upscaled permeability expected values are components of an axisymmetric tensor, whereas the fluctuations are determined by a scalar random space function of anisotropic covariance. The general case of upscaling by a first-order approximation is examined in Appendix B. These results will be applied to a few particular cases in part 3 (Indelman, this issue).