# Numerical benchmark study for flow in highly heterogeneous aquifers

## 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 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 solvable with reasonably large computing resources, for all the methods considered in this study.

## Authors

Cristian D. Alecsa
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Imre Boros
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Babes-Bolyai University, Romania

Florian Frank
Friedrich-Alexander University of Erlangen-Nuremberg, Germany

Peter Knabner,
Friedrich-Alexander University of Erlangen-Nuremberg, Germany

Mihai Nechita
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
University College London, United Kingdom

Alexander Prechtel
Friedrich-Alexander University of Erlangen-Nuremberg, Germany

Andreas Rupp
Friedrich-Alexander University of Erlangen-Nuremberg, Germany
Ruprecht-Karls-University, Germany

Nicolae Suciu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

## Keywords

Darcy flow; Accuracy; Convergence; Computational feasibility; Finite difference; Finite element; Discontinuous Galerkin; Spectral methods; Global random walk

### Paper coordinates:

C.D. Alecsa, I. Boros, F. Frank, P. Knabner, M. Nechita, A. Prechtel, A. Rupp, N. Suciu, Numerical benchmark study for flow in highly heterogeneous aquifers, Adv. Water Res., 138 (2020), 103558

## Notes

• the paper appeared for some time on the journal website at the “Most popular” and “Most downloaded” categories (posted here);
• an extended version of this work is posted on Arxiv;
• the code is posted on GitHub.

Elsevier

0309-1708

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