Nonlinear nonlocal multicontinua upscaling framework and its applications

In this paper, we discuss multiscale methods for nonlinear problems. The main idea of these approaches is to use local constraints and solve problems in oversampled regions for constructing macroscopic equations. These techniques are intended for problems without scale separation and high contrast, which often occur in applications. For linear problems, the local solutions with constraints are used as basis functions. This technique is called Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM). GMsFEM identifies macroscopic quantities based on rigorous analysis. In corresponding upscaling methods, the multiscale basis functions are selected such that the degrees of freedom have physical meanings, such as averages of the solution on each continuum. This paper extends the linear concepts to nonlinear problems, where the local problems are nonlinear. The main concept consists of: (1) identifying macroscopic quantities; (2) constructing appropriate oversampled local problems with coarse-grid constraints; (3) formulating macroscopic equations. We consider two types of approaches. In the first approach, the solutions of local problems are used as basis functions (in a linear fashion) to solve nonlinear problems. This approach is simple to implement; however, it lacks the nonlinear interpolation, which we present in our second approach. In this approach, the local solutions are used as a nonlinear forward map from local averages (constraints) of the solution in oversampling region. This local fine-grid solution is further used to formulate the coarse-grid problem. Both approaches are discussed on several examples and applied to single-phase and two-phase flow problems, which are challenging because of convection-dominated nature of the concentration equation

Multiscale simulations for upscaled multi-continuum flows

We consider in this paper a challenging problem of simulating fluid flows, in complex multiscale media possessing multi-continuum background. As an effort to handle this obstacle, model reduction is employed. In \cite{rh2}, homogenization was nicely applied, to find effective coefficients and homogenized equations (for fluid flow pressures) of a dual-continuum system, with new convection terms and negative interaction coefficients. However, some degree of multiscale still remains. This motivates us to propose the generalized multiscale finite element method (GMsFEM), which is coupled with the dual-continuum homogenized equations, toward speeding up the simulation, improving the accuracy as well as clearly representing the interactions between the dual continua. In our paper, globally, each continuum is viewed as a system and connected to the other throughout the domain. We take into consideration the flow transfers between the dual continua and within each continuum itself. Such multiscale flow dynamics are modeled by the GMsFEM, which systematically generates either uncoupled or coupled multiscale basis (to carry the local characteristics to the global ones), via establishing local snapshots and spectral decomposition in the snapshot space. As a result, we will work with a system of two equations coupled with some interaction terms, and each equation describes one of the dual continua on the fine grid. Convergence analysis of the proposed GMsFEM is accompanied with the numerical results, which support the favorable outcomes.Comment: 35 pages, 6 figures, 4 tables, submitted to Journal of Computational and Applied Mathematic

Non-Local Multi-Continuum method (NLMC) for Darcy-Forchheimer flow in fractured media

This work presents the application of the non-local multicontinuum method (NLMC) for the Darcy-Forchheimer model in fractured media. The mathematical model describes a nonlinear flow in fractured porous media with a high inertial effect and flow speed. The space approximation is constructed on the sufficiently fine grid using a finite volume method (FVM) with an embedded fracture model (EFM) to approximate lower dimensional fractures. A non-local model reduction approach is presented based on localization and constraint energy minimization. The multiscale basis functions are constructed in oversampled local domains to consider the flow effects from neighboring local domains. Numerical results are presented for a two-dimensional formulation with two test cases of heterogeneity. The influence of model nonlinearity on the multiscale method accuracy is investigated. The numerical results show that the non-local multicontinuum method provides highly accurate results for Darcy-Forchheimer flow in fractured media