Nonlocal multicontinua with Representative Volume Elements. Bridging separable and non-separable scales

Recently, several approaches for multiscale simulations for problems with high contrast and no scale separation are introduced. Among them is the nonlocal multicontinua (NLMC) method, which introduces multiple macroscopic variables in each computational grid. These approaches explore the entire coarse block resolution and one can obtain optimal convergence results independent of contrast and scales. However, these approaches are not amenable to many multiscale simulations, where the subgrid effects are much smaller than the coarse-mesh resolution. For example, the molecular dynamics of shale gas occurs in much smaller length scales compared to the coarse-mesh size, which is of orders of meters. In this case, one can not explore the entire coarse-grid resolution in evaluating effective properties. In this paper, we merge the concepts of nonlocal multicontinua methods and Representative Volume Element (RVE) concepts to explore problems with extreme scale separation. The first step of this approach is to use sub-grid scale (sub to RVE) to write a large-scale macroscopic system. We call it intermediate scale macroscale system. In the next step, we couple this intermediate macroscale system to the simulation grid model, which are used in simulations. This is done using RVE concepts, where we relate intermediate macroscale variables to the macroscale variables defined on our simulation coarse grid. Our intermediate coarse model allows formulating macroscale variables correctly and coupling them to the simulation grid. We present the general concept of our approach and present details of single-phase flow. Some numerical results are presented. For nonlinear examples, we use machine learning techniques to compute macroscale parameters.