Partial Learning Using Partially Explicit Discretization for Multicontinuum/Multiscale Problems with Limited Observation: Dual Continuum Heterogeneous Poroelastic Media Simulation

Abstract

In this paper, we consider the poroelasticity problem in heterogeneous media. The mathematical model is described by a coupled system of equations for displacement and pressure in the coupled dual continuum porous media. We propose a new method based on hybrid explicit–implicit (HEI) learning to solve the poroelasticity problem in dual continuum heterogeneous media. We use a finite element method with standard linear basis functions for spatial approximation. We apply the explicit–implicit time scheme, where the explicit scheme is used for the low-conductive continuum and the implicit scheme for the high-conductive. The fixed-strain splitting scheme is used to accelerate the computation and decouple the flow and mechanics problems. The main idea of the proposed method is partial learning of particular degrees of freedom of the high-conductive continuum’s pressure (implicit part of the flow). First, we train a deep neural network (DNN) to obtain values of the implicit part of the flow at some spatial points at some time moments. Then, we apply the Discrete Empirical Interpolation Method (DEIM) combined with Proper Orthogonal Decomposition (POD) to restore the complete implicit parts and perform linear interpolation over time. Consequently, we treat the high-conductive continuum’s pressure as a known function and use it to find the other continuum’s pressure and displacements. Numerical results for the two-dimensional model problem are presented. The results demonstrate that the proposed method provides fast and accurate predictions.