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Numerical simulation of fractured-vuggy porous media based on gamblets

Numerical simulation of a fractured-vuggy porous medium is a challenging problem. One reason is the coexistence of matrix, fractures and vugs on multiple scales that need to be coupled, and the other reason is that the high-resolution fractured-vuggy model may contain up to several millions of gridcells in applications, which brings severe computational challenges into the numerical methods. Therefore, the requirement for accurate and efficient technique is widely increasing. Fractured-vuggy porous medium is generally represented by triple-continuum model in which the matrix system, fracture system and vug system each are treated as a parallel continuous system. Although triple-continuum model is widely used because of its easy-implementation and high efficiency, it fails to capture the detailed flow patterns of reservoir with disconnected long fractures. Discrete fracture-vug network (DFVN) model can precisely model the fluid flow in fractures and vugs. However, the simulation of this model is deemed intractable even with the advent of supercomputers because of the large amount of calculation. In view of the fact that the multigrid method is now well known as one of the fastest method of solving elliptic problems, in this paper we introduce a nearly linear complexity multiresolution decomposition method for fluid flow in a fractured-vuggy reservoir. The detailed flow patterns are described by combing the advantages of continuum model and discrete model. That is, the homogenization theory is used to construct an equivalent permeability in each coarse grid block in which the vugs and small-scale fractures are represented by discrete fracture-vug network model. We decompose the solution space into several subspaces and then we compute the corresponding solutions of heterogeneous discrete fracture network model in each subspace. Gamblets are constructed and they are elementary solutions of hierarchical information games associated with the process of computing with partial information and limited resources. These gamblets have a natural Bayesian interpretation under the mixed strategy emerging from the game theoretic formulation. This method could realize its fast simulation by decomposing the solution space into a direct sum of linear subspaces that are orthogonal to each other. Finally, the pressure difference distribution of fractured-vuggy porous medium is obtained by combing the DFVN solutions of all subspaces. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed multiresolution decomposition method. The results show that this method is a promising method of numerically simulating the fractured-vuggy porous medium.