# Benchmark for numerical solutions of flow in heterogeneous groundwater formations

## 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.

## Authors

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

Imre Boros
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Department of Mathematics, Babes-Bolyai University, Mihail Kogalniceanu, 1, 400084 Cluj-Napoca, Romania

Florian Frank
Mathematics Department, Friedrich-Alexander University of Erlangen-Nuremberg, Cauerstraße. 11, 91058 Erlangen, Germany

Peter Knabner,
Mathematics Department, Friedrich-Alexander University of Erlangen-Nuremberg, Cauerstraße. 11, 91058 Erlangen, Germany

Mihai Nechita
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom

Alexander Prechtel
Mathematics Department, Friedrich-Alexander University of Erlangen-Nuremberg, Cauerstraße. 11, 91058 Erlangen, Germany

Andreas Rupp
Mathematics Department, Friedrich-Alexander University of Erlangen-Nuremberg, Cauerstraße. 11, 91058 Erlangen, Germany
Interdisciplinary Center for Scientific Computing, Ruprecht-Karls-University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany

Nicolae Suciu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

## Keywords

flow; accuracy; convergence; computational tractability; finite difference; finite elements; discontinuous Galerkin; spectral methods; global random walk

### Paper coordinates:

This is the extended, complete version of the following paper published in Adv. Water Res.

The coordinates of the extended preprint are:

C. D. Alecsa, I. Boros, F. Frank, P. Knabner, M. Nechita, A. Prechtel, A. Rupp, N. Suciu, Benchmark for numerical solutions of ﬂow in heterogeneous groundwater formations, arxiv 1911.10774.

## References

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