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<p>In this paper, we study the following inverse PDE problem: to find a geometric parameter of the domain of the time-dependent problem that matches a numerical one. It is important that the discretization box under consideration contains source (fractures) generating transport in the porous media. From the industrial point of view, we construct a machinery for stitching the simulated pressure in the reservoir with analytical one. Our goal is to obtain the value of the pressure on the fracture (or near it) depending on the distance between multiple fractures (cf.&#xa0;[<CitationRef CitationID="CR13">13</CitationRef>]). To do this, we generalize the Einstein probabilistic method (see&#xa0;[<CitationRef CitationID="CR4">4</CitationRef>]) for the Brownian motion to study the fluid transport in a porous media. We generalize the Einstein paradigm to relate the average changes of the fluid density with the velocity of the fluid and derive an anisotropic diffusion equation in nondivergence form containing a convection term. Then we combine this with the Darcy and the constitutive laws for compressible fluid flows to derive nonlinear partial differential equations for the density function. We use the Bernstein transformation to reduce the original nonlinear problem to the linear one. This method allows us to use a steady state analytical solution to interpret the result of numerical time-dependent pressure function on the fracture that takes into account 1-D geometry of the flow towards a “long” fracture.</p>

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ABOUT STITCHING OF ANALYTICAL AND NUMERICAL SOLUTIONS OF THE PROBLEM ON A VIRTUAL BOUNDARY WITH DOMINANCE OF FLOW GEOMETRY IN A BOUNDED DOMAIN

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • A. I. Ibraguimov,
  • E. M. Varfolomeev,
  • E. S. Zakirov

摘要

In this paper, we study the following inverse PDE problem: to find a geometric parameter of the domain of the time-dependent problem that matches a numerical one. It is important that the discretization box under consideration contains source (fractures) generating transport in the porous media. From the industrial point of view, we construct a machinery for stitching the simulated pressure in the reservoir with analytical one. Our goal is to obtain the value of the pressure on the fracture (or near it) depending on the distance between multiple fractures (cf. [13]). To do this, we generalize the Einstein probabilistic method (see [4]) for the Brownian motion to study the fluid transport in a porous media. We generalize the Einstein paradigm to relate the average changes of the fluid density with the velocity of the fluid and derive an anisotropic diffusion equation in nondivergence form containing a convection term. Then we combine this with the Darcy and the constitutive laws for compressible fluid flows to derive nonlinear partial differential equations for the density function. We use the Bernstein transformation to reduce the original nonlinear problem to the linear one. This method allows us to use a steady state analytical solution to interpret the result of numerical time-dependent pressure function on the fracture that takes into account 1-D geometry of the flow towards a “long” fracture.