Efficient stochastic FEM for flow in heterogeneous porous media. Part 1: random Gaussian conductivity coefficients

Abstract

SUMMARYThis paper is concerned with the development of efficient iterative methods for solving the linear system of equations arising from stochastic FEMs for single-phase fluid flow in porous media. It is assumed that the conductivity coefficient varies randomly in space according to some given correlation function and is approximated using a truncated Karhunen-Loève expansion. Distinct discretizations of the deterministic and stochastic spaces are required for implementations of the stochastic FEM. In this paper, the deterministic space is discretized using classical finite elements and the stochastic space using a polynomial chaos expansion. The highly structured linear systems which result from this discretization mean that Krylov subspace iterative solvers are extremely effective. The performance of a range of preconditioned iterative methods is investigated and evaluated in terms of robustness with respect to mesh size and variability of the conductivity coefficient. An efficient symmetric block Gauss-Seidel preconditioner is proposed for problems in which the conductivity coefficient has a large standard deviation.The companion paper, herein, referred to as Part 2, considers the situation in which Gaussian random fields are transformed into lognormal ones by projecting the truncated Karhunen-Loève expansion onto a polynomial chaos basis. This results in a stochastic nonlinear problem because the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables.

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“…Our research group has carried out extensive laboratory and field-based investigations on the degradation of various organic compounds in aqueous environment using dominantly fingerprinting techniques of analytical chemistry (Ahmad et al 2020, Song et al 2018, Wen et al 2015, Traverso et al 2014). These provided a sound foundation for further DFT modelling practice.…”

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

“…where L k are deterministic functions derived from (14) and for which closed forms can be obtained algebraically (see [24,25,26,6,4]) and χ k are multi-dimensional chaos polynomials in d random variables ξ 1 , . .…”

“…To convert the stochastic primal and mixed formulations into deterministic problems, we need to represent the stochastic variability of the permeability tensor by an appropriate set of independent random variables {ξ 1 (ω),…,ξ d (ω)}. In Part 1, 14 two approaches were described to represent the stochastic variability of . The first approach, herein referred to as coloured noise , assumes that the permeability varies randomly throughout D according to a given correlation function.…”