Efficient stochastic finite element methods for flow in heterogeneous porous media. Part 2: Random lognormal permeability

• Published in: International journal for numerical methods in fluids Vol. 92; no. 11; pp. 1626 - 1652
• Main Authors: Traverso, Luca, Phillips, Timothy Nigel
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 01.11.2020
• Subjects: AMG; Chaos; Computational fluid dynamics; conjugate gradient; Fields; Fields (mathematics); Finite element method; Fluid flow; groundwater flow; Iterative methods; Linear systems; MINRES; Permeability; Polynomials; Porous media; Porous media flow; preconditioners; Random variables; Robustness; stochastic finite element method; stochastic mixed finite element method; Stochastic processes
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.4842

Summary Efficient and robust iterative methods are developed for solving the linear systems of equations arising from stochastic finite element methods for single phase fluid flow in porous media. Permeability is assumed to vary randomly in space according to some given correlation function. In the companion paper, herein referred to as Part 1, permeability was approximated using a truncated Karhunen‐Loève expansion (KLE). The stochastic variability of permeability is modeled using lognormal random fields and the truncated KLE is projected onto a polynomial chaos basis. This results in a stochastic nonlinear problem since the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables. Symmetric block Gauss‐Seidel used as a preconditioner for CG is shown to be efficient and robust for stochastic finite element method. The uncertainty associated with permeability in porous media flow leads to a stochastic problem. This article explores iterative methods for solving the stochastic nonlinear problem arising from a representation of the permeability using a polynomial chaos basis. The use of a symmetric block Gauss‐Seidel is shown to be efficient and robust as a preconditioner for CG for the stochastic finite element method.