Parallel finite element algorithm for transient flow in bounded randomly heterogeneous domains

Abstract

We consider the effect of randomness of hydraulic conductivities K(x) on numerical predictions, without resorting to either Monte Carlo simulation, of transient flow in bounded domains driven by random source, initial and boundary terms. Our aim is to allow optimum unbiased prediction of hydraulic heads h(x, t) and fluxes q(x,t) by means of their respective ensemble moments, c and < q(x,t)>c, conditioned on measurements of K(x). These predictors have been shown by Tartakovsky and Neuman (1998) to satisfy exactly a space-time nonlocal (integro-differential) conditional mean flow equation in which < q(x,t)>c is generally non-Darcian. Exact nonlocal equations have been obtained for second conditional moments of head and flux that serve as measures of predictive uncertainty. The authors developed recursive closure approximations for the first and second conditional moment equations through expansion in powers of a small parameter σᵧ , which represents the standard estimation error of ln K(x). The authors explored the possibility of localizing the exact moment equations in real, Laplace- and/or infinite Fourier-transformed domains. In this paper we show how to solve recursive closure approximations of nonlocal first and second conditional moment equations numerically, to first order in σ²ᵧ, in a bounded two-dimensional domain. Our solution is based on Laplace transformation of the moment equations, parallel finite element solution in the complex Laplace domain, and numerical inversion of the solution from the Laplace to the real time domain. We present a detailed comparison between numerical solutions of nonlocal and localized moment equations, and Monte Carlo simulations, under superimposed mean-uniform and convergent flow regimes in two dimensions. The results are shown to compare very well for variances σ²ᵧ as large as 4. The degree to which parallelization enhances computational efficiency is explored.