Abstract
We consider a nonlinear mixed-dimensional model for simulating gas transport in shale formation. The mathematical model consists of a coupled system of nonlinear equations, where flow within fractures is represented using a lower-dimensional representation. For the numerical solution of the coupled transport problem, we construct an unstructured mesh that resolves lower dimensional fractures on the grid level and use the finite element approximation to build a discrete system. To construct an efficient scheme for the resulting nonlinear problem, we use an explicit-implicit method for time integration, where we carefully choose an additive partition of the nonlinear operators to separate the stiff linear component and integrate it implicitly to ensure the stability of the time integration. Next, we invert the linear partition of the operator by constructing an efficient two-grid preconditioner for shale gas transport in fractured porous media. We use a local pointwise smoother on the fine grid and carefully design an adaptive multiscale space for coarse grid approximation based on local generalized eigenvalue problems. We utilize an adaptive thresholding to automatically identify local dominant modes which correspond to the very small eigenvalues in local domains. We remark that such spatial features are automatically captured through our local spectral problems, and connect these to fracture information in the global formulation of the problem. Approximation properties of the local spectral space with convergence of the proposed two-grid algorithm are given. Numerical results are presented for two-dimensional fracture distributions with 30 and 160 fractures and a three-dimensional test case with 30 fractures, demonstrating iterative convergence independently of the contrast of fracture and porous matrix permeability.