×

zbMATH — the first resource for mathematics

Convergence analysis of macro spreading in 3D heterogeneous porous media. (English) Zbl 1329.76325
Summary: Models of hydrogeology must deal with both heterogeneity and lack of data. We consider in this paper a flow and transport model for an inert solute. The conductivity is a random field following a stationary log normal distribution with an exponential or Gaussian covariance function, with a very small correlation length. The quantities of interest studied here are the expectation of the spatial mean velocity, the equivalent permeability and the macro spreading. In particular, the asymptotic behavior of the plume is characterized, leading to large simulation times, consequently to large physical domains. Uncertainty is dealt with a classical Monte Carlo method, which turns out to be very efficient, thanks to the ergodicity of the conductivity field and to the very large domain. These large scale simulations are achieved by means of high performance computing algorithms and tools.

MSC:
76S05 Flows in porous media; filtration; seepage
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65C05 Monte Carlo methods
Software:
hypre; Wesseling
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] D. Allard. Statistiques spatiales : introduction ‘a la g\'eostatistique. Lecture notes, University of Montpellier, 2012.
[2] I. Babuska, F. Nobile, and R. Tempone. A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Review, 52(2):317–355, 2010. · Zbl 1226.65004
[3] I. Babuska, R. Tempone, and G. Zouraris. Solving elliptic boundary value problems with uncertain coeffcients by the finite element method: the stochastic formulation. Computer methods in applied mechanics and engineering, 194:1251–1294, 2005. · Zbl 1087.65004
[4] A. Barth, C. Schwab, and N. Zollinger. Multi-level monte carlo finite element method for elliptic pdes with stochastic coefficients. Numer. Math., 119(1):123–161, 2011. · Zbl 1230.65006
[5] A. Beaudoin and J.-R. de Dreuzy. Numerical assessment of 3D macrodispersion in heterogeneous porous media. WRR, 49:2489– 2496, 2013.
[6] A. Beaudoin, J-R. de Dreuzy, and J. Erhel. An efficient parallel particle tracker for advection-diffusion simulations in heterogeneous porous media. In A.-M. Kermarrec, L. Boug\'e, and T. Priol, editors, Euro-Par 2007, LNCS 4641, pages 717–726. Springer-Verlag, Berlin, Heidelberg, 2007.
[7] A. Beaudoin, J. Erhel, and J.-R. de Dreuzy. A comparison between a direct and a multigrid sparse linear solvers for highly heterogeneous flux computations. In Eccomas CFD 2006, volume CD, 2006.
[8] J. Charrier. Numerical analysis of the advection-diffusion of a solute in random media. Research Report RR-7585, INRIA, March 2011.
[9] J. Charrier. Strong and weak error estimates for elliptic partial diffrential equations with random coefficients. SIAM Journal on numerical analysis, 50(1):216–246, 2012. · Zbl 1241.65011
[10] J. Charrier. Numerical analysis of the advection-diffusion of a solute in porous media with uncertainty. Technical report, University of Aix-Marseille, 2013. · Zbl 1329.76326
[11] J. Charrier and A. Debussche. Weak truncation error estimates for elliptic pdes with lognormal coefficients. Stochastic Partial Differential Equations : Analysis and Computations, 1:63–93, 2013. · Zbl 1273.65166
[12] J. Charrier, R. Scheichl, and A. Teckentrup. Finite element error analysis for elliptic pdes with random coefficients and applications. SIAM Journal on Numerical Analysis, 51:322–352, 2013. · Zbl 1273.65008
[13] G. Dagan, A. Fiori, and I. Jankovic. Flow and transport in highly heterogeneous formations: 1. conceptual framework and validity of first-order approximations. Water Resources Research, 9, 2003.
[14] J.-R. de Dreuzy, A. Beaudoin, and J. Erhel. Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations. Water Resource Research, 43(W10439, doi:10.1029/2006WR005394), 2007.
[15] J.-R. de Dreuzy, A. Beaudoin, and J. Erhel. Reply to comment by a. fiori et al. on ”asymptotic dispersion in 2d heterogeneous porous media determined by parallel numerical simulations”. Water Resources Research, 44(W06604, doi:10.1029/2008WR007010), 2008.
[16] Ghislain de Marsily, Fr\'ed\'eric Delay, J. Gon\ccalvez, Philippe Renard, Vanessa Teles, and S. Violette. Dealing with spatial heterogeneity. Hydrogeology Journal, 13:161–183, 2005.
[17] C.R. Dietrich and G.N. Newsam. Fast and exact simulation of stationary gaussian processes through circulant embedding of the covariance matrix. SIAM J. Sci. Comput., 18:1088–1107, 1997. · Zbl 0890.65149
[18] J. Erhel. Stochastic groundwater simulations for highly heterogeneous porous media. In B. Amaziane, D. Barrera, M. Fortes, M. Ibanez, M. Odunlami, A. Palomares, M. Pasadas, M. Rodriguez, and D. Sbibih, editors, Proceedings of the third international conference on approximation methods and numerical modelling in environment and natural ressources, MAMERN’09, volume 1, pages 419–422. EUG, 2009. invited plenary talk.
[19] J. Erhel. Computational Technology Reviews, volume 3, chapter Some Properties of Krylov Projection Methods for Large Linear Systems, pages 41–70. Saxe-Coburg Publications, 2011.
[20] J. Erhel, J.-R. de Dreuzy, A. Beaudoin, E. Bresciani, and D. Tromeur-Dervout. A parallel scientific software for heterogeneous hydrogeology. In Ismail H. Tuncer, Ulgen Gulcat, David R. Emerson, and Kenichi Matsuno, editors, Parallel Computational Fluid Dynamics 2007, volume 67 of Lecture Notes in Computational Science and Engineering, pages 39–48. Springer, 2009. invited plenary talk.
[21] J. Erhel, J.-R. de Dreuzy, and E. Bresciani. Multi-parametric intensive stochastic simulations for hydrogeology on a computational grid. In D. Tromeur-Dervout, G. Brenner, D. Emerson, and J. Erhel, editors, Parallel Computational Fluid Dynamics 2008, Lecture Notes in Computational Science and Engineering (LNCSE), Lyon, May 2010. accepted contribution. · Zbl 1398.76187
[22] J. Erhel, Z. Mghazli, and M. Oumouni. Calcul de l’esp\'erance de la solution d’une edp stochastique unidimensionnelle ‘a l’aide d’une base r\'eduite. Comptes Rendus de l’Acad\'emie des Sciences de Paris (CRAS), s\'erie I, 349:861–865, 2011. · Zbl 1225.60102
[23] R.D. Falgout, J.E. Jones, and U.M. Yang. Numerical Solution of Partial Differential Equations on Parallel Computers, chapter The Design and Implementation of Hypre, a Library of Parallel High Performance Preconditioners, pages 267–294. SpringerVerlag, 2006. · Zbl 1097.65059
[24] P. Frauenfelder, C. Schwab, and R.A. Todor. Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Engrg., 194:205–228, 2005. · Zbl 1143.65392
[25] L. Gelhar. Stochastic Subsurface Hydrology. Engelwood Cliffs, New Jersey, 1993.
[26] A. George. Nested dissection of a regular finite element mesh. SIAM Journal on Numerical Analysis, 10:345–363, 1973. · Zbl 0259.65087
[27] R. G. Ghanem and P. D. Spanos. Stochastic finite elements : a spectral approach. Springer, 1991. · Zbl 0722.73080
[28] I. Graham, F. Kuo, D. Nuyens, R. Scheichl, and I. Sloan. Quasi-monte carlo methods for elliptic pdes with random coefficients and applications. Journal of Computational Physics, 230:3668–3694, 2011. · Zbl 1218.65009
[29] H. Hoteit, P. Ackerer, R. Mos\'e, J. Erhel, and B. Philippe. New two-dimensional slope limiters for discontinuous galerkin methods on arbitrary meshes. International Journal of Numerical Methods in Engineering, 61:2566–2593, 2004. · Zbl 1075.76575
[30] H. Hoteit, J. Erhel, R. Mos\'e, B. Philippe, and P. Ackerer. Numerical reliability for mixed methods applied to flow problems in porous media. Computational Geosciences, 6:161–194, 2002. · Zbl 1079.76581
[31] O.P. Le Ma\hatıtre and O.M. Knio. Spectral Methods for Uncertainty Quantification With Applications to Computational Fluid Dynamics. Scientific Computation. Springer, 2010. · Zbl 1193.76003
[32] G. Meurant. Computer solution of large linear systems. North Holland, Amsterdam, 1999. · Zbl 0934.65032
[33] M. Oumouni. Analyse num\'erique de methodes performantes pour les EDP stochastiques mod\'elisant l’\'ecoulement et le transport en milieux poreux. PhD thesis, University of Rennes 1 and University of Kenitra, 2013.
[34] E. Pardo-Iguzquiza and M. Chica-Olmo. The Fourier integral method: an efficient spectral method for simulation of random fields. Mathematical Geology, 25(2):177–217, 1993.
[35] G. Pichot. About the generation of a log-normal correlated field using spectral simulation. Technical report, Inria, 2012.
[36] F. Ramasomanana and A. Younes. Efficiency of the eulerian lagrangian localized adjoint method for solving advection-dispersion equation on highly heterogeneous media. International Journal for Numerical Methods in Fluids., 69:639–652, 2012.
[37] Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, 2003. · Zbl 1031.65046
[38] P. Salandin and V. Fiorotto. Solute transport in highly heterogeneous aquifers. Water Resources Research, 34:949–961, 1998.
[39] P. Wesseling. An Introduction to Multigrid Methods. Edwards, 2004. · Zbl 0760.65092
[40] D. Xiu. Fast numerical methods for stochastic computations:a review. Communications in Computational Physics, 5:242–272, 2009. · Zbl 1364.65019
[41] T. Yao. Reproduction of the mean, variance, and variogram model in spectral simulation. Mathematical Geology, 36:487–506, 2004. · Zbl 1081.86008
[42] D. Zhang. Stochastic Methods for Flow in Porous Media: Coping with Uncertainties. Academic Press, San Diego, 2002.
[43] C. Zheng and G. D. Bennett. Applied Contaminant Transport Modeling; second edition. John Wiley & Sons, New-York, 2002.
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.