Embedded fracture model in numerical simulation of the fluid flow and geo-mechanics using Generalized Multiscale Finite Element Method

In this work, we consider a pororelasticity problem in fractured porous media. Mathematical model contains a coupled system of equations for pressure and displacements, for which we use an embedded fracture model. The fine grid approximation is constructed based on the finite volume approximation for the pressure in fractured media and finite element method for the displacements. Multiscale approximation is developed using a structured coarse grid and is based on the Generalized Multiscale Finite Element Method for pressures and displacements. The performance of the method is tested using a two-dimensional model problem with different number of the multiscale basis functions.


Introduction
Effective numerical simulations of the problems in fractured porous media is important for many real world applications, for example, in oil and gas reservoirs, geothermal fields and underground waste disposal [1,2]. Mathematical models of the flow problems in fractured porous media are constructed based on the mixed dimensional formulations [3,4]. Fine grid approximation depends on the mesh construction, where the discrete fracture model is used for conforming fracture and porous matrix grid [5,6,7], and in the case of separate independent construction of the fracture and porous matrix grids, an embedded fracture model is used [8,9,10]. Mixed dimensional formulation of the flow problem is similar to the dual continuum approaches [11].
For accurate numerical solution of the poroelasticity problem using embedded fracture model, we should use a fine grid which leads to the large discrete system of equations. To reduce the size of the discrete system, multiscale methods are used [12,13,14,15]. In this paper, we develop a Generalized Multiscale Finite Element method for solution of the poroelasticity problems in fractured media with embedded fracture model. We construct multiscale basis functions for pressures and displacement via solution of the local spectral problems. We numerically investigate our method using a two-dimensional model problems in fractured and heterogeneous poroelastic media.
The paper is organized as follows. In Section 2, we present the mathematical model and fine grid approximation of the poroelasticity problem with embedded fracture model. In Section 3, we construct a multiscale coarse grid solver using Generalized Multiscale Finite Element Method. Numerical results for two -dimensional model poroelasticity problems are presented in Section 4.

Mathematical model and fine grid approximation
Let Ω is computation domain for the porous matrix and γ is the lower dimensional fracture domain. We consider a mathematical model of flow and geo-mechanics in fractured poroelastic medium that described by a following system of equations for displacements, pressure in porous matrix and fractures with a linear relation between stress σ and strain ε tensors where λ, µ are the Lames parameters, u is the displacements, p is the pressure, α is the Biot coefficient, f is the source term, k α = κ α /ν, ν is the viscosity, κ α is the permeability for α = m, f , r αβ = η αβ r, r is the transfer coefficient, η αβ is the geometric factors, c α is the compressibility for α = m, f . Note that, we neglect the gravitational forces, suppose that effect of the mechanics to the flow is relatively small and consider effect of the porous matrix pressure to the displacements.
In this work, we use the two-dimensional problem for illustration of the presented method and consider an implicit scheme for approximation of time with given time step τ . Let T h = ∪ i ς i be a fine scale finite element partition of the domain Ω and E γ = ∪ l ι l is the fracture mesh. N m f is the number of cells in T h , N f f is the number of cell for fracture mesh E γ . For approximation of the flow problem, we use a finite volume approximation on the structured fine girds and obtain the following discrete system where T ij = k m |E ij |/∆ ij (|E ij | is the length of interface between cells ς i and ς j , ∆ ij is the distance between mid point of cells ς i and ς j ), W ln = k f /∆ ln (∆ ln is the distance between points l and n), |ς i | and |ι l | is the volume of the cells ς i and ι l , q il = r if ς i ∩ ι l = 0 and equals zero otherwise. Here (p m ,p f ) are solutions from the previous times step. For displacement, we use Galerkin method with linear basis functions. Therefore, we have a following computational algorithm for solution on the fine grid in the matrix form • Solve pressure system for p = (p m , p f ) T : where and A m , A f are the transmissibility matrices, Q is the tranfer term matrix between porous matric and fractures, where D is the elasticity stiffness matrix = Ω αp m · ε y (ψ j ) dx with linear basis functions ψ i .

Coarse grid approximation using GMsFEM
For coarse grid approximation of the poroelasticity problems in fractured porous media, we use the Generalized Multiscale Finite Element Method (GMsFEM). GMsFEM contains following steps: (1) construction of the coarse and fine meshes; (2) generation of the projection matrix using local multiscale basis functions; (3) construction of the coarse grid system using projection matrix; (4) solution of the coarse scale problem and reconstruction of the fine grid solution.
For construction of the multiscale basis functions for pressures and displacements, we solve a spectral problem in the local domain ω i where D ω i and A ω i are the restrictions of the global matrices D and A to the local domain ω i . Here We form the multiscale spaces for pressures and displacements using eigenvectors Lu } corresponding to the first smallest L p and L u eigenvalues, where λ p,1 ≤ λ p,2 ≤ ... ≤ λ p,Lp and λ u,1 ≤ λ u,2 ≤ ... ≤ λ u,Lu . Furthermore, for obtaining conforming basis functions we use linear partition of unity functions χ ω i . We construct transition matrices R u and R p from a fine grid to a coarse grid and use it for reducing the dimension of the problem Lp , }. where χ ω i is linear partition of unity functions, L p and L u are the number of basis functions for pressure and displacements, N c is the number of vertices of a coarse grid.
Then the system of equations can be translated into a coarse grid, and we have following computational algorithm in the matrix form • Solve pressure system for p c = (p c,m , p c,f ) T : where where D c = R u DR T u and B c = R u B. After obtaining of a coarse-scale solution, we can reconstruct fine-scale solution u ms = R T u u c and p ms = R T p p c . Next, we present numerical results for a test problems and compare multiscale solutions for pressures and displacements (p ms and u ms ) with the reference fine grid solutions (p and u).

Numerical results
In this section, we consider poroelasticity problem in fractured porous media and consider two test problems. We set parameters of model problem E = 10, ν = 0.3, k m = 10 −5 , k f = 1 for Test 1 . For Test 2, we use a heterogeneous coefficients, where elasticity modulus and heterogeneous permeability are presented in Figure 1. We set α = 1, σ = k m , c m = 0.1 and c f = 0.01. The calculation is performed by T max = 100 with time step τ = 10. Computational grids and fracture distribution are presented in Figure 1. Fine grid contains 40401 vertices and 40000 cells. For study of the presented multiscale method, we consider two coarse grids 10 × 10 and 20 × 20.
In Figures 2 and 3, distribution of pressure and displacement along X and Y directions at final time for 8 multiscale basis functions in homogeneous and heterogeneous media are presented. In  Table 1, we present the relative L 2 errors for different number of the multiscale basis functions on the coarse grid 10 × 10 and 20 × 20 between multiscale and fine grid solutions. Presented results of the poroelasticity problem show that proposed multiscale method can provide good accuracy for a small number of multiscale basis functions for embedded fracture model.