General Reduction Technique for Numerical Simulation of Darcy Flow in Fractured Porous Media

Simulating fluid flow in fractured porous media has two main challenges: The simulta- neous of modeling the flow in multiple fractures and in the surrounding porous media; and resolving all scales to capture the effect of the fractures for more accurate approximation. The first challenge could be resolved by using the Discrete Fracture Model (DFM) since it couples Darcy flows in the fractures with the flow in the surrounding porous media (matrix domain) is suitable for this kind of simulation to overcome the first difficulty. The simplification of the DFM is based on modeling the fractures in one dimension while the matrix domain in two dimension. Various numerical methods such as Finite Element Method (FEM), Finite Difference Method (FDM), etc., have been utilized to simulate flow in fractured porous media. How- ever, using standard numerical methods is prohibitively expensive if the effect of the small scales is to be considered. Moreover, in some cases, multiple calculations have to be con- ducted using different parameters. In such cases, one needs to implement an appropriate reduction method that can inexpensively solve the resulting system, and capture the fine- scale effects using different parameters at the same time. One of the accurate and efficient reduction methods that has been widely used to identify small scale features is the Gener- alized Multiscale Finite Element Method (GMsFEM). In this proposed research, we combine GMsFEM with a reduced model based on DFM to resolve the aforementioned difficulties of simulating fluid flow in fractured porous media. The geometrical structure of the fractures is discretely resolved within the model using DFM. The advantage of using GMsFEM is to represent the fracture effects on a coarse grid via multiscale basis functions constructed using local spectral problem. Solving local problem leads to consider small scale information in each coarse grid. On another hand, the multiscale basis functions, generated following GMsFEM framework, are parameter independent and constructed once in what we call it offline stage. These basis functions can be re-used for solving the problem for any input parameter when it is needed. Combining GMsFEM and DFM has been introduced in other works assuming continuous pressure across the fractures interface. This continuity is obtained when the fractures are much more permeable than that in the matrix domain. In this work, we will consider the following cases for permeability; Case 1: the matrix is more permeable than the fractures. In this case we expect flow discontinuity along the fractures. Case 2: the matrix is less permeable than the fractures. In this case we expect flow continuity along the fractures. Moreover, we will apply this combination for mixed type problem where the conservation of mass is essential. In this case, Mixed-GMsFEM is used to approximate the coarse-scale solution. The proposed reduction technique has significant impact on enabling engineers and scien- tist to efficiently, accurately and inexpensively solve the large and complex system resulting from modeling flow in fractured porous media.