We consider numerical solution of elliptic problems with heterogeneous diffusion coefficients containing thin highly conductive structures. Such problems arise e.g. in fractured porous media, reinforced materials, and electric circuits. The main computational challenge is the high resolution needed to resolve the data variation. We propose a multiscale method that models the thin structures as interfaces and incorporate heterogeneities in corrected shape functions. The construction results in an accurate upscaled representation of the system that can be used to solve for several forcing functions or to simulate evolution problems in an efficient way. By introducing a novel interpolation operator, defining the fine scale of the problem, we prove exponential decay of the shape functions which allows for a sparse approximation of the upscaled representation. An a priori error bound is also derived for the proposed method together with numerical examples that verify the theoretical findings. Finally we present a numerical example to show how the technique can be applied to evolution problems.
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）
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…
In this paper, we consider several possible ways to set up Heterogeneous Multiscale Methods for the Landau-Lifshitz equation with a highly oscillatory diffusion coefficient, which can be seen as a means to modeling rapidly varying ferromagnetic materials. We then prove estimates for the errors introduced when approximating the relevant quantity in each of the models given a periodic problem, using averaging in time and space of the solution to a corresponding micro problem. In our setup, the Landau-Lifshitz equation with highly oscillatory coefficient is chosen as the micro problem for all mod…
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 …
A Finite Volume Method for Elastic Waves in Heterogeneous, Anisotropic and Fractured Media
Center for Modeling of Coupled Subsurface, Dynamics, Department of Mathematics, University of Bergen（卑尔根大学耦合地下动力学中心数学系）；NORCE Norwegian Research Centre AS（挪威研究中心有限公司）
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 parabolic problem with time-dependent heterogeneous coefficients. Many applied problems have coupled space and time heterogeneities. Their homogenization or upscaling requires cell problems that are formulated in space-time representative volumes for problems with scale separation. In problems without scale separation, local problems include multiple macroscopic variables and oversampled local problems, where these macroscopic parameters are computed. These approaches, called Non-local multi-continua, are proposed for problems with complex spatial heterogeneities i…
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 Fr…
There is an intimate connection between numerical upscaling of multiscale PDEs and scattered data approximation of heterogeneous functions: the coarse variables selected for deriving an upscaled equation (in the former) correspond to the sampled information used for approximation (in the latter). As such, both problems can be thought of as recovering a target function based on some coarse data that are either artificially chosen by an upscaling algorithm, or determined by some physical measurement process. The purpose of this paper is then to study that, under such a setup and for a specific e…
We performed a set of numerical simulations to characterize the interplay of fracture network topology, upscaling, and mesh refinement on flow and transport properties in fractured porous media. We generated a set of generic three-dimensional discrete fracture networks at various densities, where the radii of the fractures were sampled from a truncated power-law distribution, and whose parameters were loosely based on field site characterizations. We also considered five network densities, which were defined using a dimensionless version of density based on percolation theory. Once the network…
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）
A priori error analysis for transient problems using Enhanced Velocity approach in the discrete-time setting
Center for Subsurface Modeling, Institute for Computational Engineering and Sciences, University of Texas at Austin（德克萨斯大学奥斯汀分校地下建模中心计算工程与科学研究所）；School of Science and Technology, Nazarbayev University（纳扎尔巴耶夫大学理学院）
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 …
