Stochastic Galerkin Framework with Locally Reduced Bases for Nonlinear Two-Phase Transport in Heterogeneous Formations

TL;DR

This paper introduces a local basis reduction approach within the stochastic Galerkin framework to efficiently solve nonlinear two-phase transport problems in heterogeneous formations, significantly reducing computational costs while maintaining accuracy.

Contribution

It develops a basis reduction method that adaptively identifies significant modes locally without rewriting the entire system, enabling efficient stochastic Galerkin simulations.

Findings

Effective local basis reduction reduces computational cost.

Numerical convergence demonstrated with Monte Carlo reference solutions.

Method applied successfully to 1D and 2D problems with realistic permeability fields.

Abstract

The generalized polynomial chaos method is applied to the Buckley-Leverett equation. We consider a spatially homogeneous domain modeled as a random field. The problem is projected onto stochastic basis functions which yields an extended system of partial differential equations. Analysis and numerical methods leading to reduced computational cost are presented for the extended system of equations. The accurate representation of the evolution of a discontinuous stochastic solution over time requires a large number of stochastic basis functions. Adaptivity of the stochastic basis to reduce computational cost is challenging in the stochastic Galerkin setting since the change of basis affects the system matrix itself. To achieve adaptivity without adding overhead by rewriting the entire system of equations for every grid cell, we devise a basis reduction method that distinguishes between locally significant and insignificant modes without changing the actual system matrices. Results are presented for problems in one and two spatial dimensions, with varying number of stochastic dimensions. We show how to obtain stochastic velocity fields from realistic permeability fields and demonstrate the performance of the stochastic Galerkin method with local basis reduction. The system of conservation laws is discretized with a finite volume method and we demonstrate numerical convergence to the reference solution obtained through Monte Carlo sampling.