# Steady-State Source Flow in Heterogeneous Porous Media

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## Abstract

The flow of fluids in heterogeneous porous media is modelled by regarding the hydraulic conductivity as a stationary random space function. The flow variables, the pressure head and velocity field are random functions as well and we are interested primarily in calculating their mean values. The latter had been intensively studied in the past for flows uniform in the average. It has been shown that the average Darcy's law, which relates the mean pressure head gradient to the mean velocity, is given by a local linear relationship. As a result, the mean head and velocity satisfy the local flow equations in a fictitious homogeneous medium of effective conductivity. However, recent analysis has shown that for nonuniform flows the effective Darcy's law is determined by a nonlocal relationship of a convolution type. Hence, the average flow equations for the mean head are expressed as a linear integro-differential operator. Due to the linearity of the problem, it is useful to derive the mean head distribution for a flow by a source of unit discharge. This distribution represents a fundamental solution of the average flow equations and is called the mean Green function *G*_{d} (**x**). The mean head *G*_{d}(**x**) is derived here at first order in the logconductivity variance for an arbitrary correlation function ρ(**x**) and for any dimensionality *d* of the flow. It is obtained as a product of the solution *G*_{d}^{(0)}(*x*) for source flow in unbounded domain of the mean conductivity *K*_{A} and the correction Ψ_{d} (**x**) which depends on the medium heterogeneous structure. The correction Ψ_{d} is evaluated for a few cases of interest.

Simple one-quadrature expressions of Ψ_{d} are derived for isotropic two- and three-dimensional media. The quadratures can be calculated analytically after specifying ρ (*x*) and closed form expressions are derived for exponential and Gaussian correlations. The flow toward a source in a three-dimensional heterogeneous medium of axisymmetric anisotropy is studied in detail by deriving Ψ_{3} as function of the distance from the source *x* and of the azimuthal angle θ. Its dependence on *x*, on the particular ρ(**x**) and on the anisotropy ratio is illustrated in the plane of isotropy (θ=0) and along the anisotropy axis (θ = π/2).

The head factor *k*^{*} is defined as a ratio of the head in the homogeneous medium to the mean head, *k*^{*}=*G*_{d}^{(0)}/*G*_{d}=Ψ_{d}^{−1}. It is shown that for isotropic conductivity and for any dimensionality of the flow the medium behaves as a one-dimensional and as an effective one close and far from the source, respectively, that is, lim_{x→0}*k*^{*}(*x*) = *K*_{H}/*K*_{A} and lim_{x→∞}*k*^{*}(*x*) = *K*^{efu}/*K*_{A}, where *K*_{A} and *K*_{H} are the arithmetic and harmonic conductivity means and *K*^{efu} is the effective conductivity for uniform flow. For axisymmetric heterogeneity the far-distance limit depends on the direction. Thus, in the coordinate system of ρ(**x**) principal directions the limit values of *k*^{*} are obtained as \(\begin{gathered} \tilde H_{{\text{cont}}}^ * (\mathcal{A},\mathcal{A}) = H(K_{\text{h}}^{{\text{efu}}} ) \subset _{{\text{diff}}}^ * (\mathcal{A},\mathcal{A}) = \mathcal{V}^ * ,{\text{ where }}\mathcal{A}\mathcal{U}\mathcal{R}q(\mathfrak{g})\mathfrak{k}\mathcal{U}_q (gl(2N + 1))\mathcal{R}\mathcal{B}(N) \subset \mathcal{U}_q \mathcal{F} \hfill \\ (K_{\text{h}}^{{\text{efu}}} K_{\text{v}}^{{\text{efu}}} )^{1/2} /K_{\text{A}} {\text{ for }}\theta = 0{\text{ and }}K_{\text{h}}^{{\text{efu}}} /K_{\text{A}} {\text{for }}\theta = \pi /2 \hfill \\ \end{gathered} \). These values differ from the corresponding components \(K_{\text{h}}^{{\text{efu}}} /K_{\text{A}} {\text{ and }}K_{\text{v}}^{{\text{efu}}} /K_{\text{A}} \) of the effective conductivities tensor for uniform flow for θ = 0 and π/2, respectively. The results of the study are applied to solving the problem of the dipole well flow. The dependence of the mean head drop between the injection and production chambers on the anisotropy of the conductivity and the distance between the chambers is analyzed.

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