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 < 1 is derived explicitly in terms of two quadratures and is a function of e, rw/I and r/I . Hence Keq unlike Kefu, is not a property of the medium solely. For rw/I < 0.2 and forr/I > 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].

Radial flow takes place in a heterogeneous porous formation of random and stationary log-conductivity Y(x), characterized by the mean 〈Y〉, the variance σ2Y and the two- point autocorrelation ρY which in turn has finite and different horizontal and vertical integral scales, I and Iv, 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 Srr. Under ergodic conditions, i.e. for a well length much larger than the vertical integral scale, Srr is equal to the radial fluid trajectory variance Xrr.The aim of the study is to determine Xrr(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 σ2Y 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=Iv/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.

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.

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.

Abstract. Steady flow between a fully penetrating recharging and pumping well (doublet) takes place in a heterogeneous aquifer. The spatially variable hydraulic conductivity is modeled as a lognormal stationary random function, of anisotropic two-point covariance. The latter is characterized by the horizontal and vertical integral scales I and Iv respectively. A tracer, or a reactive solute obeying first-order kinetics, is injected as a pulse or continuously in the recharging well. Our aim is to determine the flux-averaged concentration (the breakthrough curve) in the pumping well as a function of tim, e and of the various parameters of the problem, i.e., o-• (the logconductivity variance), l = l/I (l is the distance between wells), and e = IvI (the anisotropy ratio). A simple solution of this difficult problem is achieved by adopting a few simplifying assumptions: (1) the wells are fully penetrating, of length much larger than I vand of radius r w much smaller than I, (2) a first-order solution in o-• of the flow and transport equations is sought, (3) the anisotropy ratio is small, say e < 0.2, and (4) neglect of the effect of pore-scale dispersion. After determining the travel time, mean and variance, the mean flux-averaged concentration is found by assuming that, is lognormal. In a homogeneous medium there is a large spreading of the solute signal in the pumping well owing to the variation of the travel time among the streamlines connecting the two wells. The effect of heterogeneity is similar to that of pore-scale dispersion; that is, it leads to enhanced spreading and in particular to an early breakthrough. The solution has potential applications to aquifer tests and to evaluation of efficiency of remediation schemes and may serve as a benchmark for numerical models. IntroductionWe consider steady aquifer flow between a recharging well and a pumping one (briefly a doublet, Figure 1), operating under a constant head difference. A solute is injected either for a short period (pulse) or continuously in the recharging well, and the concentration is monitored in the pumping one. Such a configuration is quite common in many applications, for example, as a testing method to determine aquifer properties or in remediation schemes. Therefore it is of interest to model the flow and transport for this type of application. The simplest case is that of a homogeneous formation, which was investigated in the past. Thus, in the recent work of Koplik et al. [1994], the problem is solved for transport of a tracer by advection and by pore-scale dispersion. For the high Peclet numbers characterizing natural formations, the spread of the pulse in the pumping well is dominated by the advective effect. This large effect is due to the nonuniformity of the flow, resulting in differences in the travel times along the streamlines connecting the two wells, the quickest and slowest paths being the ones in the wells plane [Kurowski et al., 1994]. With neglect of pore-scale dispersion, the transport problem can be solved In reality, natur...