Abstract
Solute transport through heterogeneous porous media considered in environmental and industrial problems is often characterized by high Péclet numbers. In this paper we develop a new numerical approach to advection-dominated transport consisting of coupling an accurate mass-conservative mixed finite element method (MFEM), used to solve Darcy flows, with a particle method, stable and free of numerical diffusion, for non-reactive transport simulations. The latter is the efficient global random walk (GRW) algorithm, which performs the simultaneous tracking of arbitrarily large collections of particles on regular lattices at computational costs comparable to those of single-trajectory simulations using traditional particle tracking (PT). MFEM saturated flow solutions are computed for spatially heterogeneous hydraulic conductivities parameterized as samples of random fields. The coupling is achieved by projecting the velocity field from the MFEM basis onto the regular GRW lattice. Preliminary results show that MFEM–GRW is tens of times faster than the full MFEM flow and transport simulation.
Authors
Keywords
Random walk methods; Advection-dominated transport; Porous media
Cite this paper as:
N. Suciu, F.A. Radu, A. Prechtel, F. Brunner, P. Knabner, A coupled finite element-global random walk approach to advection-dominated transport in porous media with random hydraulic conductivity, J. Comput. Appl. Math., 246 (2013), 27-37, doi: 10.1016/j.cam.2012.06.027
References
About this paper
Journal
Journal of Computational and Applied Mathematics
Publisher Name
Elsevier
Online ISSN
0377-0427
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Not available yet.
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[1] U. Jaekel, H. Vereecken, Renormalization group analysis of macrodispersion in a directed random flow, Water Resour. Res. 33 (1997) 2287–2299.
CrossRef (DOI)
[2] H. Schwarze, U. Jaekel, H. Vereecken, Estimation of macrodispersivity by different approximation methods for flow and transport in randomly heterogeneous media, Transp. Porous Media 43 (2001) 265–287.
[3] N. Suciu, C. Vamos, J. Vanderborght, H. Hardelauf, H. Vereecken, Numerical investigations on ergodicity of solute transport in heterogeneous aquifers, Water Resour. Res. 42 (2006) W04409.
CrossRef (DOI)
[4] N. Suciu, C. Vamoş, J. Eberhard, Evaluation of the first-order approximations for transport in heterogeneous media, Water Resour. Res. 42 (2006) W11504.
CrossRef (DOI)
[5] N. Suciu, C. Vamos, F.A. Radu, H. Vereecken, P. Knabner, Persistent memory of diffusion particles, Phys. Rev. E 80 (2009) 061134.
CrossRef (DOI)
[6] N. Suciu, Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields, Phys. Rev. E 81 (2010) 056301.
CrossRef (DOI)
[7] F.A. Radu, N. Suciu, J. Hoffmann, A. Vogel, O. Kolditz, C.-H. Park, S. Attinger, Accuracy of numerical simulations of contaminant transport in heterogeneous aquifers: a comparative study, Adv. Water Resour. 34 (2011) 47–61.
CrossRef (DOI)
[8] P.E. Kloeden, E. Platen, Numerical Solutions of Stochastic Differential Equations, Springer, Berlin, 1999.
[9] C. Vamoş, N. Suciu, H. Vereecken, Generalized random walk algorithm for the numerical modeling of complex diffusion processes, J. Chem. Phys. 186 (2003) 527–544.
CrossRef (DOI)
[10] N. Suciu, C. Vamoş, Global random walk algorithm for diffusion processes, in: A. Skogseid, V. Fasano (Eds.), Random Walks: Principles, Processes and Applications, Nova Science Publishers, New York, 2012, pp. 341–388.
[11] A. Prechtel, P. Knabner, E. Schneid, K.U. Totsche, Simulation of carrier facilitated transport of phenanthrene in a layered soil profile, J. Contam. Hydrol. 56 (2002) 209–225.
CrossRef (DOI)
[12] F.A. Radu, I.S. Pop, P. Knabner, Order of convergence estimates for an Euler implicit, mixed finite element discretization of Richards’ equation, SIAM J. Numer. Anal. 42 (2004) 1452–1478.
CrossRef (DOI)
[13] F.A. Radu, M. Bause, A. Prechtel, S. Attinger, A mixed hybrid finite element discretization scheme for reactive transport in porous media, in: K. Kunisch, G. Of, O. Steinbach (Eds.), Numerical Mathematics and Advanced Applications, Springer-Verlag, 2008, pp. 513–520.
CrossRef (DOI)
[14] F.A. Radu, I.S. Pop, S. Attinger, Analysis of an Euler implicit-mixed finite element scheme for reactive solute transport in porous media, Numer. Methods Partial Differential Equations 26 (2009) 320–344.
CrossRef (DOI)
[15] F. Brunner, F.A. Radu, M. Bause, P. Knabner, Optimal order convergence of a modified BDM1 mixed finite element scheme for reactive transport in porous media, Adv. Water Resour. 35 (2012) 163–171.
CrossRef (DOI)
[16] S.E. Pashenko, K.K. Sabelfeld, D.A. Trubitsyn, Experimental and computer modeling study of stochastic porous filtration of polydispersed aerosols, in: EAC 2008: European Aerosol Conference, Thessaloniki, Greece, 24–29 August, 2008.
[17] F. Delay, P. Ackerer, C. Danquigny, Simulating solute transport in porous or fractured formations using random walk particle tracking: a review, Vadose Zone J. 4 (2006) 360–379.
CrossRef (DOI)
[18] F. Delay, H. Housset-Resche, G. Porei, G. de Marsily, Transport in a 2-D saturated porous medium: a new method for particle tracking, Math. Geol. 28 (1996) 45–71.
CrossRef (DOI)
[19] J. Eberhard, N. Suciu, C. Vamoş, On the self-averaging of dispersion for transport in quasi-periodic random media, J. Phys. A 40 (2007) 597–610.
CrossRef (DOI)
[20] N. Suciu, C. Vamoş, Evaluation of overshooting errors in particle methods for diffusion by biased global random walk, Rev. Anal. Numér. Théor. Approx. 35 (2006) 119–126.
paper on journal website
[21] S. Viswanathan, H. Wang, S.B. Pope, Numerical implementation of mixing and molecular transport in LES/PDF studies of turbulent reacting flows, J. Comput. Phys. 230 (2011) 6916–6957.
CrossRef (DOI)