Two-phase numerical simulation of discrete fracture model based on multiscale mixed finite element method

Abstract

Fractured reservoirs are characterized by complex fractures on multiple scales and are quite difficult to model. Numerical simulation of fractured media is usually done based on dual-porosity model and equivalent continuum model. Dual-porosity model treats matrix system and fracture system as two parallel continuous systems coupled by crossflow function. This model is only valid for reservoirs with highly developed fractures. Equivalent continuum model treats fractured media as a continuum media and equivalent absolute permeability tensors for each grid block are calculated to describe the heterogeneity of the reservoir. This model is efficient only when there exist representative element volume and the equivalent permeability are difficult to decide. Both models treat the fractured media as a simplified model and cannot describe the multiscale flows exactly, because they cannot precisely consider the diversion effect of the fractures. Although the discrete fracture network (DFN) model can provide a detailed representation of flow characteristic, traditional numerical method does not suitable for DFN. The major difficulty is the size of the computation. A tremendous amount of computer memory and CPU time are required, and this can exceed limit of today’s computer resources. Upscaling methods are generally used to reduce the computational cost. However, it is not possible to have a priori estimates of the errors that are present when complex flow processes are investigated using coarse models constructed via simplified settings. In this paper, multiscale mixed finite element method (MsMFEM) is proposed to simulate water/oil two phase flow in discrete fracture media. By combining MsMFEM with the discrete fracture model, we aim towards a numerical scheme that facilitates fractured reservoir simulation without upscaling. The MsMFEM uses a standard Darcy model to approximate pressure and fluxes on a coarse grid. The multiscale basis functions are constructed numerically by solving local differential equations on the fine-scale grid. The advantage of MsMFEM is that the basis functions capable of reflecting information about fractures within elements. Therefore, this method can capture the fine-scale effects on the coarse grid, that is, multiscale method can reduce the computational cost and keep high calculation accuracy at the same time. Traditional numerical methods generally difficult to deal with complex grid element, in this paper, mimetic finite difference (MFD) method is used to construct the multiscale basis functions due to its local conservativeness and applicability of complex grids. Compared with traditional multiscale mixed finite element methods, this method is suitable for arbitrary complex grid system. This paper introduced fundamental principles of the multiscale mixed finite element method and described the numerical scheme of discrete fracture model based on MsMFEM in detail. Then we deduced discrete fracture model computing formulate for the multiscale basis function by using mimetic finite difference method. Oversampling technique is applied to get more accurate small-scale details. IMPES scheme is used in the two-phase flow simulation. Physical experiment is used to prove the validity the multiscale method. The numerical results show that compared with traditional numerical method, the MsMFEM can represent the fine-scale flow in fracture networks exactly and meanwhile has a higher computational efficiency.