On the stitching of analytical and numerical solutions of the problem on a virtual boundary with the dominance of flow geometry in a bounded domain

We are studying the following inverse PDE problem: to find geometric parameter of the domain of the time-dependent problem that match numerical one. Important feature is that discretization box of the interest contains source (fractures) generating transport in the porous media. From industrial point of view, we are building a machinery of the sewing the simulated pressure in the reservoir with analytical one. The goal is to obtain the value of the pressure function on the fracture (or near fracture) depending on the distance between multiple fractures (cf. [14]). For that, we generalize Einstein's probabilistic method (see [5]) for the Brownian motion to study the fluids transport in porous media. We generalize Einstein's paradigm to relate the average changes in the fluid density with the velocity of fluid and derive an anisotropic diffusion equation in nondivergence form that contains a convection term. This is then combined with the Darcy and the constitutive laws for compressible fluid flows to yield a nonlinear partial differential equations for the density function. Bernstein's transformation is used to reduce the original nonlinear problem to the linear one. The method which we employ allow us to use a steady state analytical solution to interpret the result of numerical time-dependent pressure function on the fracture which takes into account 1-D geometry of the flow towards “long” fracture.