Benchmark for numerical solutions of flow in heterogeneous groundwater formations

TL;DR

This paper evaluates the accuracy and convergence of various numerical methods for simulating steady flow in heterogeneous aquifers, highlighting their performance limitations with increasing heterogeneity and complexity.

Contribution

It introduces a benchmark using Kraichnan algorithms for generating log-normal conductivity fields, enabling direct verification of numerical methods in groundwater flow simulations.

Findings

Gaussian correlation yields good performance across methods

Exponential correlation reduces accuracy, especially with high variance

High heterogeneity challenges computational tractability for all methods

Abstract

This article presents numerical investigations on accuracy and convergence properties of several numerical approaches for simulating steady state flows in heterogeneous aquifers. Finite difference, finite element, discontinuous Galerkin, spectral, and random walk methods are tested on one- and two-dimensional benchmark flow problems. Realizations of log-normal hydraulic conductivity fields are generated by Kraichnan algorithms in closed form as finite sums of random periodic modes, which allow direct code verification by comparisons with manufactured reference solutions. The quality of the methods is assessed for increasing number of random modes and for increasing variance of the log-hydraulic conductivity fields with Gaussian and exponential correlation. Experimental orders of convergence are calculated from successive refinements of the grid. The numerical methods are further validated by comparisons between statistical inferences obtained from Monte Carlo ensembles of numerical solutions and theoretical first-order perturbation results. It is found that while for Gaussian correlation of the log-conductivity field all the methods perform well, in the exponential case their accuracy deteriorates and, for large variance and number of modes, the benchmark problems are practically not tractable with reasonably large computing resources, for all the methods considered in this study.