Multiscale Methods for Stochastic Collocation of Mixed Finite Elements for Flow in Porous Media

摘要

This thesis contains methods for uncertainty quantification of flow in porous media through stochastic modeling. New parallel algorithms are described for both deterministic and stochastic model problems, and are shown to be computationally more efficient than existing approaches in many cases.First, we present a method that combines a mixed finite element spatial discretization with collocation in stochastic dimensions on a tensor product grid. The governing equations are based on Darcy's Law with stochastic permeability. A known covariance function is used to approximate the log permeability as a truncated Karhunen-Loeve expansion. A priori error analysis is performed and numerically verified.Second, we present a new implementation of a multiscale mortar mixed finite element method. The original algorithm uses non-overlapping domain decomposition to reformulate a fine scale problem as a coarse scale mortar interface problem. This system is then solved in parallel with an iterative method, requiring the solution to local subdomain problems on every interface iteration. Our modified implementation instead forms a Multiscale Flux Basis consisting of mortar functions that represent individual flux responses for each mortar degree of freedom, on each subdomain independently. We show this approach yields the same solution as the original method, and compare the computational workload with a balancing preconditioner.Third, we extend and combine the previous works as follows. Multiple rock types are modeled as nonstationary media with a sum of Karhunen-Loeve expansions. Very heterogeneous noise is handled via collocation on a sparse grid in high dimensions. Uncertainty quantification is parallelized by coupling a multiscale mortar mixed finite element discretization with stochastic collocation. We give three new algorithms to solve the resulting system. They use the original implementation, a deterministic Multiscale Flux Basis, and a stochastic Multiscale Flux Basis. Multiscale a priori error analysis is performed and numerically verified for single-phase flow. Fourth, we present a concurrent approach that uses the Multiscale Flux Basis as an interface preconditioner. We show the preconditioner significantly reduces the number of interface iterations, and describe how it can be used for stochastic collocation as well as two-phase flow simulations in both fully-implicit and IMPES models.