Random Porous Media Flow on Large 3-D Grids: Numerics, Performance, and Application to Homogenization

Abstract

Subsurface flow processes are inherently three-dimensional and heterogeneous over many scales. Taking this into account, for instance assuming random heterogeneity in 3-D space, puts heavy constraints on numerical models. An efficient numerical code has been developed for solving the porous media flow equations, appropriately generalized to account for 3-D, random-like heterogeneity. The code is based on implicit finite differences (or finite volumes), and uses specialized versions of preconditioned iterative solvers that take advantage of sparseness. With Diagonally Scaled Conjugate Gradients, in particular, large systems on the order of several million equations, with randomly variable coefficients, have been solved efficiently on Cray-2 and Cray-Y/MP8 machines, in serial mode as well as parallel mode (autotasking). The present work addresses, first, the numerical aspects and computational issues associated with detailed 3-D flow simulations, and secondly, presents a specific application related to the conductivity homogenization problem (identifying a macroscale conduction law, and an equivalent or effective conductivity). Analytical expressions of effective conductivities are compared with empirical values obtained from several large scale simulations conducted for single realizations of random porous media.

This paper was written while its author was at the Commissariat a 1’Energie Atomique, CEN-Saclay, France. However, parts of this work are based on previous research conducted while the author was at the Center for Nuclear Waste Regulatory Analyses, San Antonio, Texas. Logistic help for some of the computations was provided by the NASA Ames Research Center, and Cray Research Inc. The 3-D graphics views of potential surfaces were produced with Dynamic Graphics IVM package. The author is indebted to C. Hempel for assistance with Cray timing analyses, to L.W. Gelhar for advice on several aspects of stochastic modeling, and to A.C. Bagtzoglou, G.W. Wittmeyer, T.J. Nicholson, B. Sagar, and M. Durin for help or encouragements on various aspects of this work. The views expressed here are solely those of the author.