A stochastically and spatially adaptive parallel scheme for uncertain and nonlinear two-phase flow problems

Simulating in flow problems in porous media often requires techniques for uncertainty quantification in order to represent parameter values that are not given exactly. Straightforward Monte-Carlo (MC) methods have a limited efficiency due to slow convergence. Better convergence in low-parametric problems can be achieved with polynomial chaos expansion (PCE) techniques. The PCE approach yields a coupled deterministic system to be solved. The degree of coupling increases with the non-linearity of the considered equations and with the order of polynomial expansion. This fact increases the computational effort of PCE and significantly reduces the scalability in parallelization. We present an application of the hybrid stochastic Galerkin finite volume (HSG-FV) method to a two-phase flow problem in two space dimensions. The method extends the classical polynomial chaos expansion by a multi-element discretization in the probability space of the parameters. It leads to a deterministic system that is coupled to a lesser degree than in element-free PCE versions, respectively, fully decoupled in stochastic elements (SE). Therefore, the HSG-FV method allows for more efficient parallelization. For the further reduction of complexity, we present a new stochastic adaptivity method. We present numerical examples in two spatial dimensions with linear and nonlinear fractional flow functions in the two-phase flow problem. The flow functions might depend in a discontinuous manner on the unknown spatial position of porous-medium heterogeneities. Finally, we discuss the interplay of the new method with spatial adaptivity per SE for these problems.