Solution of the 3D density-driven groundwater flow problem with uncertain porosity and permeability. (English) Zbl 1434.76016

Summary: The pollution of groundwater, essential for supporting populations and agriculture, can have catastrophic consequences. Thus, accurate modeling of water pollution at the surface and in groundwater aquifers is vital. Here, we consider a density-driven groundwater flow problem with uncertain porosity and permeability. Addressing this problem is relevant for geothermal reservoir simulations, natural saline-disposal basins, modeling of contaminant plumes and subsurface flow predictions. This strongly nonlinear time-dependent problem describes the convection of a two-phase flow, whereby a liquid flows and propagates into groundwater reservoirs under the force of gravity to form so-called “fingers”. To achieve an accurate numerical solution, fine spatial resolution with an unstructured mesh and, therefore, high computational resources are required. Here we run a parallelized simulation toolbox ug4 with a geometric multigrid solver on a parallel cluster, and the parallelization is carried out in physical and stochastic spaces. Additionally, we demonstrate how the ug4 toolbox can be run in a black-box fashion for testing different scenarios in the density-driven flow. As a benchmark, we solve the Elder-like problem in a 3D domain. For approximations in the stochastic space, we use the generalized polynomial chaos expansion. We compute the mean, variance, and exceedance probabilities for the mass fraction. We use the solution obtained from the quasi-Monte Carlo method as a reference solution.


76-10 Mathematical modeling or simulation for problems pertaining to fluid mechanics
76S05 Flows in porous media; filtration; seepage
35Q35 PDEs in connection with fluid mechanics
65M22 Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs
65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
Full Text: DOI arXiv


[1] Askey, R.; Wilson, Ja, Some Basic Hypergeometric Orthogonal Polynomials that Generalize Jacobi Polynomials (1985), Providence: American Mathematical Society, Providence · Zbl 0572.33012
[2] Babuška, I.; Nobile, F.; Tempone, R., A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45, 1005-1034 (2007) · Zbl 1151.65008
[3] Babuška, I.; Tempone, R.; Zouraris, Ge, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Numer. Anal., 42, 800-825 (2004) · Zbl 1080.65003
[4] Barthelmann, V.; Novak, E.; Ritter, K., High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math., 12, 273-288 (2000) · Zbl 0944.41001
[5] Bear, J., Dynamics of Fluid in Porous Media (1972), New York: Dover Publications, INC, New York
[6] Bear, J., Hydraulics of Groundwater (1979), Mineola: Dover Publications. Inc., Mineola
[7] Bear, J.; Bachmat, Y., Introduction to Modeling of Transport Phenomena in Porous Media (1990), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 0743.76003
[8] Blatman, G.; Sudret, B., An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis, Probab. Eng. Mech., 25, 183-197 (2010)
[9] Bode, F.; Ferré, T.; Zigelli, N.; Emmert, M.; Nowak, W., Reconnecting stochastic methods with hydrogeological applications: a utilitarian uncertainty analysis and risk assessment approach for the design of optimal monitoring networks, Water Resour. Res., 54, 2270-2287 (2018)
[10] Bompard, M., Peter, J., Désidéri, J.-A.: Surrogate models based on function and derivative values for aerodynamic global optimization. In: Fifth European Conference on Computational Fluid Dynamics, ECCOMAS CFD 2010, Lisbon, Portugal, (2010)
[11] Bungartz, H-J; Griebel, M., Sparse grids, Acta Numer., 13, 147-269 (2004) · Zbl 1118.65388
[12] Caflisch, Re, Monte Carlo and quasi-Monte Carlo methods, Acta Numer., 7, 1-49 (1998) · Zbl 0949.65003
[13] Cameron, Rh; Martin, Wt, The orthogonal development of non-linear functionals in series of fourier-hermite functionals, Ann. Math., 48, 385 (1947) · Zbl 0029.14302
[14] Carrera, J.: An overview of uncertainties in modelling groundwater solute transport. J. Contam. Hydrol., 13, 23 - 48 (1993). Chemistry and Migration of Actinides and Fission Products
[15] Chkifa, A.; Cohen, A.; Schwab, Ch, Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, Journal de Mathematiques Pures et Appliques, 103, 400-428 (2015) · Zbl 1327.65251
[16] Clenshaw, Cw; Curtis, Ar, A method for numerical integration on an automatic computer, Numer. Math., 2, 197-205 (1960) · Zbl 0093.14006
[17] Conrad, P.; Marzouk, Y., Adaptive smolyak pseudospectral approximations, SIAM J. Sci. Comput., 35, A2643-A2670 (2013) · Zbl 1294.41004
[18] Constantine, Pg; Eldred, Ms; Phipps, Et, Sparse pseudospectral approximation method, Comput. Methods Appl. Mech. Eng., 229-232, 1-12 (2012) · Zbl 1253.65117
[19] Costa, A., Permeability-porosity relationship: A reexamination of the kozeny-carman equation based on a fractal pore-space geometry assumption, Geophys. Res. Lett., 33, 1-5 (2006)
[20] Cremer, Cjm; Graf, T., Generation of dense plume fingers in saturated-unsaturated homogeneous porous media, J. Contam. Hydrol., 173, 69-82 (2015)
[21] Crestaux, T.; Le Maıtre, O.; Martinez, Jean-Marc, Polynomial chaos expansion for sensitivity analysis, Reliab. Eng. Syst. Saf., 94, 1161-1172 (2009)
[22] Diersch, H-Jg; Kolditz, O., Variable-density flow and transport in porous media: approaches and challenges, Adv. Water Resour., 25, 899-944 (2002)
[23] Dolgov, S., Khoromskij, B.N., Litvinenko, A., Matthies, H.G.: Computation of the response surface in the tensor train data format. arXiv preprint arXiv:1406.2816 (2014) · Zbl 1329.65271
[24] Dolgov, S.; Khoromskij, Bn; Litvinenko, A.; Matthies, Hg, Polynomial chaos expansion of random coefficients and the solution of stochastic partial differential equations in the tensor train format, SIAM/ASA J. Uncertain. Quantif., 3, 1109-1135 (2015) · Zbl 1329.65271
[25] Elder, J.; Simmons, C.; Diersch, H-J; Frolkovič, P.; Holzbecher, E.; Johannsen, K., The Elder problem, Fluids, 2, 1, 11 (2017)
[26] Elder, Jw, Steady free convection in a porous medium heated from below, J. Fluid Mech., 27, 29-48 (1967)
[27] Elder, Jw, Transient convection in a porous medium, J. Fluid Mech., 27, 609-623 (1967)
[28] Ernst, Og; Mugler, A.; Starkloff, H-J; Ullmann, E., On the convergence of generalized polynomial chaos expansions, ESAIM Math. Modell. Numer. Anal., 46, 317-339 (2012) · Zbl 1273.65012
[29] Espig, M.; Hackbusch, W.; Litvinenko, A.; Matthies, Hg; Waehnert, Ph, Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats. High-order finite element approximation for partial differential equations, Comput. Math. Appl., 67, 818-829 (2014) · Zbl 1350.65005
[30] Fan, Y.; Duffy, Cj; Oliver, Ds, Density-driven groundwater flow in closed desert basins: field investigations and numerical experiments, J. Hydrol., 196, 139-184 (1997)
[31] Frolkovič, P.: Consistent velocity approximation for density driven flow and transport. In: R. Van Keer, et al. (eds.) Advanced Computational Methods in Engineering, Part 2: Contributed papers. Shaker Publishing, Maastricht, pp. 603-611
[32] Frolkovič, P., Maximum principle and local mass balance for numerical solutions of transport equation coupled with variable density flow, Acta Mathematica Universitatis Comenianae, 1, 137-157 (1998) · Zbl 0940.76039
[33] Frolkovič, P., Knabner, P.: Consistent velocity approximations in finite element or volume discretizations of density driven flow. In: Aldama,A.A. et al. (eds.) Computational Methods in Water Resources XI, pp. 93-100. Computational Mechanics Publication, Southhampten
[34] Gerstner, Th; Griebel, M., Numerical integration using sparse grids, Numer. Algorithms, 18, 209-232 (1998) · Zbl 0921.65022
[35] Ghanem, R.; Owhadi, H.; Higdon, D., Handbook of Uncertainty Quantification (2017), Berlin: Springer, Berlin · Zbl 1372.60001
[36] Ghanem, R.; Spanos, P., Stochastic Finite Elements: A Spectral Approach (1991), New York: Springer, New York · Zbl 0722.73080
[37] Ghili, S.; Iaccarino, G., Least squares approximation of polynomial chaos expansions with optimized grid points, SIAM J. Sci. Comput., 39, A1991-A2019 (2017) · Zbl 1371.41005
[38] Giraldi, L.; Litvinenko, A.; Liu, D.; Matthies, Hg; Nouy, A., To be or not to be intrusive? the solution of parametric and stochastic equations-the “plain vanilla” galerkin case, SIAM J. Sci. Comput., 36, A2720-A2744 (2014) · Zbl 1310.65132
[39] Giunta, A.A., Eldred, M.S., Castro, J.P.: Uncertainty quantification using response surface approximation. In: 9th ASCE Specialty Conference on Probabolistic Mechanics and Structural Reliability. Albuquerque, New Mexico, USA (2004)
[40] Griebel, M.: Sparse grids and related approximation schemes for higher dimensional problems. In: Foundations of Computational Mathematics, Santander 2005, vol. 331 of London Mathematical Society Lecture Note Series, pp. 106-161. Cambridge University Press, Cambridge (2006) · Zbl 1106.65332
[41] Hackbusch, W., Multi-Grid Methods and Applications (1985), Berlin: Springer, Berlin · Zbl 0585.65030
[42] Hackbusch, W., Iterative Solution of Large Sparse Systems of Equations (1994), New York: Springer, New York
[43] Heppner, I.; Lampe, M.; Nägel, A.; Reiter, S.; Rupp, M.; Vogel, A.; Wittum, G.; Nagel, We; Kröner, Dh; Resch, Mm, Software framework ug4: parallel multigrid on the hermit supercomputer, High Performance Computing in Science and Engineering 2012, 435-449 (2013), Berlin: Springer, Berlin
[44] Joe, S.; Kuo, F., Constructing sobol sequences with better two-dimensional projections, SIAM J. Sci. Comput., 30, 2635-2654 (2008) · Zbl 1171.65364
[45] Johannsen, K., On the validity of the boussinesq approximation for the Elder problem, Comput. Geosci., 7, 169-182 (2003) · Zbl 1134.76471
[46] Karhunen, K.: Zur Spektraltheorie stochastischer Prozesse, Ser. A : 1, Math. Phys., 34, Annales Academiae Scientiarum Fennicae (1946) · Zbl 0063.03144
[47] Keese, A.; Mattthies, Hg, Hierarchical parallelisation for the solution of stochastic finite element equations, Comput. Struct., 83, 1033-1047 (2005)
[48] Khoromskij, Bn; Litvinenko, A.; Matthies, Hg, Application of hierarchical matrices for computing the Karhunen-Loève expansion, Computing, 84, 49-67 (2009) · Zbl 1162.65306
[49] Kim, Nh; Wang, H.; Queipo, Nv, Efficient shape optimization under uncertainty using polynomial chaos expansions and local sensitivities, AIAA J., 44, 1112-1116 (2006)
[50] Klimke, A.: Sparse grid interpolation toolbox. https://sparsegrids.org (2008)
[51] Kobus, Helmut, Soil and Groundwater Contamination and Remediation Technology in Europe, Groundwater Updates, 3-8 (2000), Tokyo: Springer Japan, Tokyo
[52] Le Maître, O.P., Knio, O.M.: Introduction: Uncertainty Quantification and Propagation. In: Spectral Methods for Uncertainty Quantification, pp. 1-13. Springer, Berlin (2010)
[53] Litvinenko, A., Logashenko, D., Tempone, R., Wittum, G., Keyes, D.: Propagation of uncertainties in density-driven flow. arXiv preprint arXiv:1905.01770 (2019)
[54] Litvinenko, A.; Matthies, Hg, Uncertainties quantification and data compression in numerical aerodynamics, PAMM, 11, 877-878 (2011)
[55] Litvinenko, A., Matthies, H.G., El-Moselhy, T. A.: Sampling and low-rank tensor approximation of the response surface. In: Dick, J., Kuo, F. Y., Peters, G. W., Sloan, I. H. (eds.) Monte Carlo and Quasi Monte Carlo Methods 2012, vol. 65 of Springer Proceedings in Mathematics & Statistics, pp. 535-551. Springer, Berlin (2013) · Zbl 1302.65030
[56] Liu, Dishi; Görtz, Stefan, Efficient Quantification of Aerodynamic Uncertainty due to Random Geometry Perturbations, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 65-73 (2014), Cham: Springer International Publishing, Cham
[57] Liu, D.; Litvinenko, A.; Schillings, C.; Schulz, V., Quantification of airfoil geometry-induced aerodynamic uncertainties-comparison of approaches, SIAM/ASA J. Uncertai. Quantif., 5, 334-352 (2017) · Zbl 06736506
[58] Loeven, G. J. A., Witteveen, J. A. S., Bijl, H.: A probabilistic radial basis function approach for uncertainty quantification. In: Proceedings of the NATO RTO-MP-AVT-147 Computational Uncertainty in Military Vehicle Design Symposium (2007)
[59] Matthies, Hg; Stein, E.; De Borst, R.; Hughes, Trj, Uncertainty quantification with stochastic finite elements, Encyclopedia of Computational Mechanics (2007), Chichester: Wiley, Chichester
[60] Matthies, Hg; Keese, A., Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Comput. Methods Appl. Mech. Eng., 194, 1295-1331 (2005) · Zbl 1088.65002
[61] Matthies, Hg; Zander, E.; Rosić, Bv; Litvinenko, Alexander, Parameter estimation via conditional expectation: a Bayesian inversion, Adv. Model. Simul. Eng. Sci., 3, 24 (2016)
[62] Matthies, Hermann G.; Zander, Elmar; Rosić, Bojana V.; Litvinenko, Alexander; Pajonk, Oliver, Inverse Problems in a Bayesian Setting, Computational Methods in Applied Sciences, 245-286 (2016), Cham: Springer International Publishing, Cham
[63] Najm, Hn, Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics, Ann. Rev. Fluid Mech., 41, 35-52 (2009) · Zbl 1168.76041
[64] Nobile, F., Tamellini, L., Tesei, F., Tempone, R.: An adaptive sparse grid algorithm for elliptic PDEs with log-normal diffusion coefficient. MATHICSE Technical Report 04, (2015) · Zbl 1339.65016
[65] Nobile, F.; Tempone, R.; Webster, Clayton G., A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46, 2309-2345 (2008) · Zbl 1176.65137
[66] Novak, E.; Ritter, K., High dimensional integration of smooth functions over cubes, Numerische Mathematik, 75, 79-97 (1996) · Zbl 0883.65016
[67] Novak, Erich; Ritter, Klaus, The Curse of Dimension and a Universal Method For Numerical Integration, Multivariate Approximation and Splines, 177-187 (1997), Basel: Birkhäuser Basel, Basel · Zbl 0889.65016
[68] Novak, E.; Ritter, K., Simple cubature formulas with high polynomial exactness, Constr. Approx., 15, 499-522 (1999) · Zbl 0942.41018
[69] Nowak, W.; Litvinenko, A., Kriging and spatial design accelerated by orders of magnitude: combining low-rank covariance approximations with fft-techniques, Math. Geosci., 45, 411-435 (2013) · Zbl 1321.86027
[70] Panda, M.; Lake, W., Estimation of single-phase permeability from parameters of particle-size distribution, AAPG Bull., 78, 1028-1039 (1994)
[71] Pape, H.; Clauser, C.; Iffland, J., Permeability prediction based on fractal pore-space geometry, Geophysics, 64, 1447-1460 (1999)
[72] Post, Vea; Houben, Gj, Density-driven vertical transport of saltwater through the freshwater lens on the island of baltrum (germany) following the 1962 storm flood. Investigation of coastal aquifers, J. Hydrol., 551, 689-702 (2017)
[73] Radović, I.; Sobol, Im; Tichy, Rf, Quasi-monte carlo methods for numerical integration: comparison of different low discrepancy sequences, Monte Carlo Methods Appl., 2, 1-14 (1996) · Zbl 0851.65015
[74] Reiter, S.; Logashenko, D.; Vogel, A.; Wittum, G., Mesh generation for thin layered domains and its application to parallel multigrid simulation of groundwater flow, Comput. Vis. Sci., 16, 151-164 (2017) · Zbl 1380.65463
[75] Reiter, S.; Vogel, A.; Heppner, I.; Rupp, M.; Wittum, G., A massively parallel geometric multigrid solver on hierarchically distributed grids, Comput. Vis. Sci., 16, 151-164 (2013) · Zbl 1380.65463
[76] Rosić, Bv; Kučerová, A.; Sýkora, J.; Pajonk, O.; Litvinenko, A.; Matthies, Hg, Parameter identification in a probabilistic setting, Eng. Struct., 50, 179-196 (2013)
[77] Rubin, Yoram, Applied Stochastic Hydrogeology (2003), Oxford: Oxford University Press, Oxford
[78] Schneider, A.; Kröhn, K-P; Püschel, A., Developing a modelling tool for density-driven flow in complex hydrogeological structures, Comput. Vis. Sci., 15, 163-168 (2012) · Zbl 1388.76006
[79] Schwab, Ch; Todor, Ra, Sparse finite elements for stochastic elliptic problems-higher order moments, Computing, 71, 43-63 (2003) · Zbl 1044.65006
[80] Schwab, Ch; Todor, Ra, Karhunen-Loève approximation of random fields by generalized fast multipole methods, J. Comput. Phys., 217, 100-122 (2006) · Zbl 1104.65008
[81] Sinsbeck, M.; Nowak, W., An optimal sampling rule for nonintrusive polynomial chaos expansions of expensive models, Int. J. Uncertai. Quantif., 5, 275-295 (2015)
[82] Smolyak, Sa, Quadrature and interpolation formulas for tensor products of certain classes of functions, Sov. Math. Dokl., 4, 240-243 (1963)
[83] Sudret, B., Global sensitivity analysis using polynomial chaos expansions, Reliab. Eng. Syst. Saf., 93, 964-979 (2008)
[84] Tartakovsky, Dm, Assessment and management of risk in subsurface hydrology: a review and perspective, Adv. Water Resour., 51, 247-260 (2013)
[85] Thimmisetty, Ch; Tsilifis, P.; Ghanem, R., Homogeneous chaos basis adaptation for design optimization under uncertainty: application to the oil well placement problem, Artif. Intell. Eng. Des. Anal. Manuf., 31, 265-276 (2017)
[86] Tipireddy, R.; Ghanem, R., Basis adaptation in homogeneous chaos spaces, J. Comput. Phys., 259, 304-317 (2014) · Zbl 1349.60058
[87] Todor, Ra; Schwab, Ch, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMA J. Numer. Anal., 27, 232-261 (2007) · Zbl 1120.65004
[88] Tsilifis, P.; Ghanem, Rg, Bayesian adaptation of chaos representations using variational inference and sampling on geodesics, P. R. Soc. Math. Phys. Eng. Sci., 474, 20180285 (2018) · Zbl 1407.62087
[89] Tsilifis, P.; Huan, X.; Safta, C.; Sargsyan, Kh; Lacaze, G.; Oefelein, Jc; Najm, Hn; Ghanem, Rg, Compressive sensing adaptation for polynomial chaos expansions, J. Comput. Phys., 380, 29-47 (2019)
[90] Vereecken, H., Schnepf, A., Hopmans, J.W., Javaux, M., Or, D., Roose, T., Vanderborght, J., Young, M.H., Amelung, W., Aitkenhead, M., Allison, S.D., Assouline, S., Baveye, P., Berli, M., Brüggemann, N., Finke, P., Flury, M., Gaiser, T., Govers, G., Ghezzehei, T., Hallett, P., Hendricks Franssen, H.J., Heppell, J., Horn, R., Huisman, J.A., Jacques, D., Jonard, F., Kollet, S., Lafolie, F., Lamorski, K., Leitner, D., McBratney, A., Minasny, B., Montzka, C., Nowak, W., Pachepsky, Y., Padarian, J., Romano, N., Roth, K., Rothfuss, Y., Rowe, E.C., Schwen, A., Šim \({\mathring{u}}\) nek, J., Tiktak, A., Van Dam, J., van der Zee, S.E.A.T.M., Vogel, H.J., Vrugt, J.A., Wöhling, T., Young, I.M.: Modeling soil processes: Review, key challenges, and new perspectives. Vadose Zone J., 15, p. vzj2015.09.0131 (2016)
[91] Vogel, A.: Flexible und kombinierbare Implementierung von Finite-Volumen-Verfahren höherer Ordnung. Dissertation, Universität Frankfurt, (2014)
[92] Vogel, A.; Reiter, S.; Rupp, M.; Nägel, A.; Wittum, G., Ug 4: a novel flexible software system for simulating pde based models on high performance computers, Comput. Vis. Sci., 16, 165-179 (2013) · Zbl 1375.35003
[93] Voss, Ci; Souza, Wr, Variable density flow and solute transport simulation of regional aquifers containing a narrow freshwater-saltwater transition zone, Water Resour. Res., 23, 1851-1866 (1987)
[94] Wiener, N., The homogeneous chaos, Am. J. Math., 60, 897-936 (1938) · JFM 64.0887.02
[95] Xie, Y.; Simmons, Ct; Werner, Ad; Diersch, H-Jg, Prediction and uncertainty of free convection phenomena in porous media, Water Resour. Res., 48, 2, 2535 (2012)
[96] Xiu, D., Fast numerical methods for stochastic computations: a review, Commun. Comput. Phys., 5, 2-4, 242-272 (2009) · Zbl 1364.65019
[97] Xiu, D., Numerical Methods for Stochastic Computations: A Spectral Method Approach (2010), Princeton: Princeton University Press, Princeton · Zbl 1210.65002
[98] Xiu, D.; Karniadakis, Ge, Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Physics, 187, 137-167 (2003) · Zbl 1047.76111
[99] Xiu, D.; Karniadakis, Ge, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, Comput. Meth. Appl. Mech. Eng., 191, 4927-4948 (2002) · Zbl 1016.65001
[100] Xiu, D.; Karniadakis, Ge, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24, 619-644 (2002) · Zbl 1014.65004
[101] Zhaojun, B.; Demmel, J.; Dongarra, J.; Ruhe, A.; Van Der Vorst, H., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide (2000), Philadelphia: SIAM, Philadelphia · Zbl 0965.65058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.