In this paper, we develop and analyze a rigorous multiscale upscaling method for dual continuum model, which serves as a powerful tool in subsurface formation applications. Our proposed method is capable of identifying different continua and capturing non-local transfer and effective properties in the computational domain via constructing localized multiscale basis functions. The construction of the basis functions consists of solving local problems defined on oversampling computational region, subject to the energy minimizing constraints that the mean values of the local solution are zero in all continua except for the one targeted. The basis functions constructed are shown to have good approximation properties. It is shown that the method has a coarse mesh dependent convergence. We present some numerical examples to illustrate the performance of the proposed method.
Nonlocal multicontinuum (NLMC) upscaling of mixed dimensional coupled flow problem for embedded and discrete fracture models
Texas A&M University（德克萨斯农工大学）；North-Eastern Federal University（东北联邦大学）；The Chinese University of Hong Kong (CUHK)（香港中文大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
In this paper, we consider a poroelasticity problem in heterogeneous multicontinuum media that is widely used in simulations of the unconventional hydrocarbon reservoirs and geothermal fields. Mathematical model contains a coupled system of equations for pressures in each continuum and effective equation for displacement with volume force sources that are proportional to the sum of the pressure gradients for each continuum. To illustrate the idea of our approach, we consider a dual continuum background model with discrete fracture networks that can be generalized to a multicontinuum model for …
Upscaling method for problems in perforated domains with non-homogeneous boundary conditions on perforations using Non-Local Multi-Continuum method (NLMC)
Texas A&M University（德克萨斯农工大学）；North-Eastern Federal University（东北联邦大学）；The Chinese University of Hong Kong（香港中文大学）
In this work, we present a novel nonlocal nonlinear coarse grid approximation using a machine learning algorithm. We consider unsaturated and two-phase flow problems in heterogeneous and fractured porous media, where mathematical models are formulated as general multicontinuum models. We construct a fine grid approximation using the finite volume method and embedded discrete fracture model. Macroscopic models for these complex nonlinear systems require nonlocal multicontinua approaches, which are developed in earlier works. These rigorous techniques require complex local computations, which in…
Skin contraction is an important biophysical process that takes place during and after the recovery of deep tissue injury. This process is mainly caused by fibroblasts (skin cells) and myofibroblasts (differentiated fibroblasts) that exert pulling forces on the surrounding extracellular matrix (ECM). Modelling is done in multiple scales: agent-based modelling on the microscale and continuum-based modelling on the macroscale. In this manuscript, we present some results from our study of the connection between these scales. For the one-dimensional case, we managed to rigorously establish the lin…
A hierarchical approach for multicontinuum homogenization in high contrast media
School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）；National Center for Applied Mathematics in Hunan（湖南应用数学中心）；Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学物理与数学学院数学科学系）
In this paper, we develop a space-time upscaling framework that can be used for many challenging porous media applications without scale separation and high contrast. Our main focus is on nonlinear differential equations with multiscale coefficients. The framework is built on nonlinear nonlocal multi-continuum upscaling concept and significantly extends the results in the proceeding paper. Our approach starts with a coarse space-time partition and identifies test functions for each partition, which plays a role of multi-continua. The test functions are defined via optimization and play a cruci…
Generalized Multiscale Multicontinuum Model for Fractured Vuggy Carbonate Reservoirs
Texas A&M University（德州农工大学）；The Chinese University of Hong Kong（香港中文大学）；North-Eastern Federal University（东北联邦大学）；Texas A&M University at Qatar（卡塔尔德州农工大学）
Nonlocal multicontinua upscaling for multicontinua flow problems in fractured porous media
Institute for Scientific Computation, Texas A&M University（德州农工大学科学计算研究所）；Department of Computational Technologies, North-Eastern Federal University（东北联邦大学计算技术系）；Department of Mathematics, The Chinese University of Hong Kong (CUHK)（香港中文大学数学系）；Department of Mathematic, Texas A&M University（德州农工大学数学系）；Multiscale model reduction laboratory, North-Eastern Federal University（东北联邦大学多尺度模型降维实验室）
We study homogenization of a locally periodic two-scale dual-continuum system where each continuum interacts with the other. Equations for each continuum are written separately with interaction terms (exchange terms) added. The homogenization limit depends strongly on the scale of this continuum interaction term with respect to the microscopic scale. In J. S. R. Park and V. H. Hoang, {\it Hierarchical multiscale finite element method for multicontinuum media}, arXiv:1906.04635, we study in details the case where the interaction terms are scaled as $O(1/\ep^2)$ where $\ep$ is the microscale of …
Simulation in media with multiple continua where each continuum interacts with every other is often challenging due to multiple scales and high contrast. One needs some types of model reduction. One of the approaches is multi-continuum technique, where every process in each continuum is modeled separately and an interaction term is added. Direct numerical simulation in multi scale media is usually not practicable. For this reason, one constructs the corresponding homogenized equations. The paper develops a hierarchical approach for solving cell problems at a dense network of macroscopic points…
In this work, we introduce a time memory formalism in poroelasticity model that couples the pressure and displacement. We assume this multiphysics process occurs in multicontinuum media. The mathematical model contains a coupled system of equations for pressures in each continuum and elasticity equations for displacements of the medium. We assume that the temporal dynamics is governed by fractional derivatives following some works in the literature. We derive an implicit finite difference approximation for time discretization based on the Caputo time fractional derivative. A Discrete Fracture …
Stochastic agent-based models can account for millions of cells with spatiotemporal movement that can be a function of different factors. However, these simulations can be computationally expensive. In this work, we develop a novel computational framework to describe and simulate stochastic cellular processes that are coupled to the environment. Specifically, through upscaling, we derive a continuum governing equation that considers the cell density as a function of time, space, and a cumulative variable that is coupled to the environmental conditions. For this new governing equation, we consi…
