An analysis of polynomial chaos approximations for modeling single-fluid-phase flow in porous medium systems

• Published in: Journal of computational physics Vol. 226; no. 2; pp. 2175 - 2205
• Main Authors: Rupert, C.P., Miller, C.T.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.10.2007; Elsevier
• Subjects: APPROXIMATIONS; CHAOS THEORY; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Computational techniques; Exact sciences and technology; FLOW MODELS; GROUND WATER; Groundwater flow; HYDRAULIC CONDUCTIVITY; Karhunen–Loève expansion; Mathematical methods in physics; Monte Carlo; MONTE CARLO METHOD; MULTIVARIATE ANALYSIS; Physics; POLYNOMIALS; POROUS MATERIALS; RANDOMNESS; SERIES EXPANSION; SIMULATION; STEADY-STATE CONDITIONS; Stochastic; STOCHASTIC PROCESSES; Stochastic; Monte Carlo; Groundwater flow; Karhunen–Loève expansion; Chaos; Karhunen-Loève expansion; Dimensionality; Polynomial approximation; Porous medium; Stochastic approximation; Calculation methods; Random field; Monte Carlo methods; Decoupling; Series expansion; Models; Modelling; Calculation; Infinite dimension; Single phase flow
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2007.07.001

We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method.