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Two-Dimensional Fluid Filtration Problems with Boundary Sources in an Anisotropic Inhomogeneous Layer

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Abstract

We study the first and second boundary value problems and the transmission problem for the complex potential of a two-dimensional filtration flow in an anisotropic and inhomogeneous (variable permeability and thickness) porous layer. The flow sources are arbitrary discrete and can generally be located both on the boundaries and outside the boundaries. The boundaries are modeled by arbitrary smooth (piecewise smooth) closed lines, and the flow sources are singularities (isolated singular points) of the complex potential. The presence of a system of sources on the boundaries leads to a fundamentally new generalization (complication) of the boundary conditions, which are characterized by singular functions with isolated singular points. In the case of an anisotropic homogeneous (constant permeability and thickness) layer and rectilinear boundaries, the solutions of the problems are presented in closed form. In the general case, when an arbitrary smooth closed curve models a boundary with sources located on it, a generalized Cauchy type integral for the complex flow potential is used. This permitted reducing the second boundary value problem and the transmission problem to boundary singular integral equations. The problems studied are mathematical models of two-dimensional filtration processes in layered porous media, which are of interest, for example, for the practice of extracting fluids (oil, water) from natural anisotropically heterogeneous soil layers.

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Notes

  1. The points \(\zeta _0=\{\zeta _{01},\zeta _{02},\ldots ,\zeta _{0n }\}\) are the images of the isolated singular points \(z_0=\{z_{01},z_{02},\ldots ,z_{0n}\} \) of location of sources given on the \(z \)-plane, which are related by the homeomorphic transformation \( \zeta _0=\zeta (z_0)\).

  2. The functions \(\varphi (\zeta ) \) and \(\psi (\zeta ) \) satisfy system (2.4) with given constant coefficients \(\sqrt {D(P_s)} \) and \(\sqrt {D(P_a)} \).

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Authors and Affiliations

  1. Turgenev Orel State University, Orel, 302026, Russia

    V. F. Piven’

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  1. V. F. Piven’
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Correspondence to V. F. Piven’.

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Translated by V. Potapchouck

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Piven’, V.F. Two-Dimensional Fluid Filtration Problems with Boundary Sources in an Anisotropic Inhomogeneous Layer. Diff Equat 59, 781–798 (2023). https://doi.org/10.1134/S0012266123060071

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