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Sparse Grids and Applications - Munich 2018 pp 101–126Cite as

Propagation of Uncertainties in Density-Driven Flow

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 144)

Abstract

Accurate modeling of contamination in subsurface flow and water aquifers is crucial for agriculture and environmental protection. Here, we demonstrate a parallel algorithm to quantify the propagation of uncertainty in the dispersal of pollution in subsurface flow. Specifically, we consider the density-driven flow and estimate how uncertainty from permeability and porosity propagates to the solution. We take a two-dimensional Elder-like problem as a numerical benchmark, and we use random fields to model our limited knowledge on the porosity and permeability. We use the well-known low-cost generalized polynomial chaos (gPC) expansion surrogate model, where the gPC coefficients are computed by projection on sparse tensor grids. The numerical solver for the deterministic problem is based on the multigrid method and is run in parallel. Computation of high-dimensional integrals over the parametric space is done in parallel too.

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Notes

  1. 1.

    https://www.hpc.kaust.edu.sa/content/shaheen-ii.

  2. 2.

    https://github.com/UG4/ughub.

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Acknowledgements

This work was supported by the King Abdullah University of Science and Technology (KAUST) and by the Alexander von Humboldt Foundation. We used the resources of the Supercomputing Laboratory at KAUST, under the development project k1051. We would like to thank the KAUST core lab for the assistance with Shaheen II parallel supercomputer, developers of the ug4 simulation framework from Frankfurt University, two anonymous reviewers, and the associate editor for their careful reading and suggestions.

Author information

Authors and Affiliations

  1. RWTH Aachen, Aachen, Germany

    Alexander Litvinenko

  2. KAUST, Thuwal/Jeddah, Saudi Arabia

    Dmitry Logashenko & David Keyes

  3. RWTH Aachen, Aachen, Germany

    Raul Tempone

  4. KAUST, Thuwal/Jeddah, Saudi Arabia

    Raul Tempone

  5. KAUST, Thuwal/Jeddah, Saudi Arabia

    Gabriel Wittum

  6. G-CSC, Frankfurt University, Frankfurt, Germany

    Gabriel Wittum

Authors
  1. Alexander Litvinenko
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  2. Dmitry Logashenko
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  3. Raul Tempone
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  4. Gabriel Wittum
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  5. David Keyes
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Corresponding author

Correspondence to Alexander Litvinenko .

Editor information

Editors and Affiliations

  1. Institut für Informatik, TU München, Garching, Germany

    Prof. Dr. Hans-Joachim Bungartz

  2. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    Prof. Dr. Jochen Garcke

  3. Institut für Parallele und Verteilte Systeme (IPVS), Universität Stuttgart, Stuttgart, Germany

    Prof. Dr. Dirk Pflüger

Appendix A: Difficulties in Computing Statistics

Appendix A: Difficulties in Computing Statistics

We consider the following uncertain porosity

(25)

with 3 stochastic parameters θ 1, θ 2, θ 3 ∼ U[−1, 1], x ∈ [0, 600], y ∈ [0, 150]. Figure 8 shows four different realizations of the solution (the mass fraction). These realisations correspond to four different Clenshaw-Curtis quadrature points. The number of fingers, their shapes, location and propagation are different. This fact makes it difficult to compute and to interpret the mean value, the variance and other statistics.

Fig. 8
figure 8

Four different realizations of the mass fraction: with (a) 5, (b) 4.5, (c) 4, (d) 4 fingers. For all realisations c ∈ [0, 1]

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The porosity fields, corresponding to Fig. 8a,b are shown in Fig. 9a,b.

Fig. 9
figure 9

Four realisations of the porosity field in (25)

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Litvinenko, A., Logashenko, D., Tempone, R., Wittum, G., Keyes, D. (2021). Propagation of Uncertainties in Density-Driven Flow. In: Bungartz, HJ., Garcke, J., Pflüger, D. (eds) Sparse Grids and Applications - Munich 2018. Lecture Notes in Computational Science and Engineering, vol 144. Springer, Cham. https://doi.org/10.1007/978-3-030-81362-8_5

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