Generalized Multiscale Finite Element method for multicontinua unsaturated flow problems in fractured porous media

Abstract

In this paper, we present a multiscale method for simulations of the multicontinua unsaturated flow problems in heterogeneous fractured porous media. The mathematical model is described by the system of Richards equations for each continuum that coupled by the specific transfer term. To illustrate the idea of our approach, we consider a dual continua background model with discrete fractures networks that generalized as a multicontinua model for unsaturated fluid flow in the complex heterogeneous porous media. We present fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) approach. In this model, we construct an unstructured fine grid that take into account a complex fracture geometries for two and three dimensional formulations. Due to construction of the unstructured grid, the fine grid approximation leads to the very large system of equations. For reduction of the discrete system size, we develop a multiscale method for coarse grid approximation of the coupled problem using Generalized Multiscale Finite Element Method (GMsFEM). In this method, we construct a coupled multiscale basis functions that used to construct highly accurate coarse grid approximation. The multiscale method allowed us to capture detailed interactions between multiple continua. The adaptive approach is investigated, where we consider two approaches for multiscale basis functions construction:(1) based on the spectral characteristics of the local problems and (2) using simplified multiscale basis functions. We investigate accuracy of the proposed method for the several test problems in two and three dimensional formulations. We present a comparison of the relative error for different number of basis functions and for adaptive approach. Numerical results illustrate that the presented method provide accurate solution of the unsaturated multicontinua problem on the coarse grid with huge reduction of the discrete system size.

Constraint Energy Minimizing Generalized Multiscale Finite Element Method for multi-continuum Richards equations

“…At first glance, a coupled system of dual-continuum nonlinear Richards equations [71,79] has the form…”

“…In this paper, we consider a general multi-continuum model of such system as follows [79,85]: for each continuum i = 1, . .…”

“…Fine-scale discretization and Picard iteration for linearization. To tackle our problem's nonlinearity, we take advantage of an efficient Picard iterative scheme, as described in [61,79,42,62]. Over this section, such an iteration algorithm is presented for time-dependent multi-continuum systems.…”

“…To achieve the system (2.7)'s first goal of temporal discretization (see [61,79,12], for instance), we will use the following conventional backward Euler finite-difference algorithm: find…”

“…A variety of domain decomposition methods [43,56,55] have leveraged the idea of building local basis functions utilizing spectral problem. Recently, the GMsFEM has been successfully applied to a variety of problems [2,86,23,79,81,1,3,5,18,27,26,59].…”