Multiscale modelling of heat and mass transfer in fractured media for enhanced geothermal systems applications. Applied Mathematical Modelling.

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.

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In this paper, we present an upscaling method for problems in perforated domains with non-homogeneous boundary conditions on perforations. Our methodology is based on the recently developed Non-local multicontinuum method (NLMC). The main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non-local effects. We will construct multiscale basis functions for the coarse regions and additional multiscale basis functions for perforations, with the aim of handling non-homogeneous boundary conditions on perforations. We start with describing our method for the Laplace equation, and then extending the framework for the elasticity problem and parabolic equations. The resulting upscaled model has minimal size and the solution has physical meaning on the coarse grid. We will present numerical results (1) for steady and unsteady problems, (2) for Laplace and Elastic operators, and (3) for Neumann and Robin non-homogeneous boundary conditions on perforations. Numerical results show that the proposed method can provide good accuracy and provide significant reduction on the degrees of freedoms.