Partial learning using partially explicit discretization for multicontinuum/multiscale problems. Fractured poroelastic media simulation

Abstract

The poroelasticity problem plays a vital role in various fields of science and technology. For example, it has applications in the petroleum industry, agricultural science, and biomedicine. The mathematical model consists of a coupled system of differential equations for mechanical displacements and pressure. In this paper, we propose a new approach for solving the poroelasticity problem in a fractured medium based on hybrid explicit–implicit learning (HEI). For spatial approximation, we use the finite element method with standard linear basis functions. We consider a poroelastic medium with large hydraulic fractures and use the Discrete Fracture Model. For the time approximation, we apply the explicit–implicit scheme. The explicit scheme is used for the porous matrix, which is a low-conductive continuum. While for fractures, we apply the implicit scheme due to its high conductivity. To facilitate the calculations, we use the fixed strain and fixed stress splitting schemes. The main idea of the proposed method lies in partial learning. Full training of the numerical solution is extremely difficult due to a large number of degrees of freedom, so we propose to train a part of the solution. Large hydraulic fractures have few degrees of freedom, but they are highly conductive and require a more costly implicit scheme. Therefore, we train the neural network to generate pressure in the fractures at several time points and then interpolate for other times. Thus, we treat the fracture pressure as a known function and use it to find the pressure in the porous matrix. Numerical results for a two-dimensional model problem are presented.