Intrusive uncertainty quantification for hyperbolic-elliptic systems governing two-phase flow in heterogeneous porous media

Abstract

We model random locations of spatial interfaces of heterogeneities in porous media by means of the hybrid stochastic Galerkin (HSG) approach. This approach extends the concept of the generalized polynomial chaos (PC) expansion for a multi-element decomposition of the multidimensional stochastic space. In this way, the physically two-dimensional hyperbolic-elliptic fractional flow formulation for two-phase flow in heterogeneous porous media is transformed from a random partial differential equation into a deterministic system for the coefficients of the PC expansion of the primary unknown saturation, total velocity, and global pressure. The hyperbolic part is discretized based on a central-upwind finite volume scheme along with a mixed finite element method for the elliptic part. The latter partly uses the tensor product structure of the saddle point system together with appropriate preconditioning to speed up the computations. Since we use the sequential implicit pressure explicit saturation (IMPES) approach, the elliptic part is solved in every time step. The proposed method is particularly well-suited for parallel computations and allows for the consideration of a huge variety of complex flow problems. We illustrate the power of the method by means of several striking numerical examples of different complexities including their numerical convergence analysis.