Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media


In this article, we present new random walk methods to solve flow and transport problems in saturated/unsaturated porous media, including coupled flow and transport processes in soils, heterogeneous systems modeled through random hydraulic conductivity and recharge fields, processes at the field and regional scales. The numerical schemes are based on global random walk algorithms (GRW) which approximate the solution by moving large numbers of computational particles on regular lattices according to specific random walk rules. To cope with the nonlinearity and the degeneracy of the Richards equation and of the coupled system, we implemented the GRW algorithms by employing linearization techniques similar to the L-scheme developed in finite element/volume approaches. The resulting GRW  L-schemes converge with the number of iterations and provide numerical solutions that are first-order accurate in time and second-order in space. A remarkable property of the flow and transport GRW solutions is that they are practically free of numerical diffusion. The GRW solvers are validated by comparisons with mixed finite element and finite volume solvers in one- and two-dimensional benchmark problems. They include Richards’ equation fully coupled with the advection-diffusion-reaction equation and capture the transition from unsaturated to saturated flow regimes.


Nicolae Suciu
Mathematics Department, Friedrich-Alexander University of Erlangen-Nurnberg, Erlangen, Germany
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania

Davide Illiano
Department of Mathematics, University of Bergen, Norway

Alexander Prechtel
Department of Mathematics, University of Bergen, Norway

Florin A. Radu
Department of Mathematics, University of Bergen, Norway


Richards equation; Coupled flow and transport; Linearization; Iterative schemes; Global random walk


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Extended version published in Arxiv:2011.12889v3

Cite this paper as:

N. Suciu, D. Illiano, A. Prechtel, F.A.Radu, Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media, Advances in Water Resources, 152 (2021), 103935,

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[1] W. Abeele, Hydraulic Testing of Crushed Bandelier Tuff, Technical Report LA 10037-MS, Los Alamos National Laboratory, Los Alamos, New Mexico (1984) Google Scholar

[2] C.D. Alecsa, I. Boros, P. Knabner Frank, M. Nechita, A. Prechtel, A. Rupp, N. Suciu, Numerical benchmark study for fow in heterogeneous aquifers, Adv. Water Resour., 138 (2019), p. 103558, 10.1016/j.advwatres.2020.103558

[3] W. Alt, H. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (3) (1983), pp. 311-341, 10.1007/BF01176474

[4] D.G. Anderson, Iterative procedures for nonlinear integral equations, J. ACM, 12 (4) (1965), pp. 547-560, 10.1145/321296.321305

[5] M. Bause, P. Knabner, Numerical simulation of contaminant biodegradation by higher order methods and adaptive time stepping, Comput. Visual. Sci, 7 (2) (2004), pp. 61-78, 10.1007/s00791-004-0139-y

[6] A. Bellin, A. Fiori, G. Dagan, Equivalent and effective conductivities of heterogeneous aquifers for steady source flow, with illustration for hydraulic tomography, Adv. Water Resour., 142 (2020), p. 103632, 10.1016/j.advwatres.2020.103632

[7] A. Bellin, P. Salandin, A. Rinaldo, Simulation of dispersion in heterogeneous porous formations: statistics, first-order theories, convergence of computations, Water Resour. Res., 28 (9) (1992), pp. 2211-2227, 10.1029/92WR00578

[8] J.W. Both, K. Kumar, J.M. Nordbotten, F.A. Radu, Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media, Comput. Math. Appl., 77 (6) (2019), pp. 1479-1502, 10.1016/j.camwa.2018.07.033

[9] D. Caviedes-Voullième, P. Garci, J. Murillo, Verification, conservation, stability and efficiency of a finite volume method for the 1D Richards equation, J. Hydrol., 480 (2013), pp. 69-84, 10.1016/j.jhydrol.2012.12.008

[10] E. Cătinaş, A survey on the high convergence orders and computational convergence orders of sequences, Appl. Math. Comput., 343 (2019), pp. 1-20, 10.1016/j.amc.2018.08.006

[11] E. Cătinaş, How many steps still left to x*? SIAM Rev (2020)

[12] H.J. Hendricks Franssen, A. Alcolea, M. Riva, M. Bakr, N. Van der.Wiel, F. Stauffer, A. Guadagnini, A comparison of seven methods for the inverse modelling of groundwater flow. Application to the characterisation of well catchments, Adv. Water Resour., 32 (6) (2009), pp. 851-872, 10.1016/j.advwatres.2009.02.011

[13] H. Fujita, The exact pattern of a concentration-dependent diffusion in a semi-infinite medium, Part II, Textil Res. J., 22 (12) (1952), pp. 823-827, 10.1177/004051755202201209

[14] W.R. Gardner, Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table, Soil Sci., 85 (4) (1958), pp. 228-232, CrossRefView Record in ScopusGoogle Scholar,

[15] H. Gotovac, V. Cvetković, R. Andričevicć, Adaptive Fup multi-resolution approach to flow and advective transport in highly heterogeneous porous media: methodology, accuracy and convergence, Adv. Water Resour., 32 (6) (2009), pp. 885-905, 10.1016/j.advwatres.2009.02.013, ArticleDownload PDFView Record in ScopusGoogle Scholar

[16] H. Hajibeygi, M.B. Olivares, M. Hosseini Mehr, S. Pop, M. Wheeler, A benchmark study of the multiscale and homogenization methods for fully implicit multiphase flow simulations, Adv. Water Resour., 143 (2020), p. 103674, 10.1016/j.advwatres.2020.103674, ArticleDownload PDFView Record in ScopusGoogle Scholar

[17] Haverkamp, Vauclin, Touma, Wierenga, Vachaud, 1977, R. Haverkamp, M. Vauclin, J. Touma, P.J. Wierenga, G. Vachaud, A comparison of numerical simulation models for one-dimensional infiltration 1, Soil. Sci. Soc. Am. J., 41 (2) (1977), pp. 285-294, 10.2136/sssaj1977.03615995004100020024x, CrossRefView Record in ScopusGoogle Scholar

[18] D. Illiano, I.S. Pop, F.A. Radu, Iterative schemes for surfactant transport in porous media, Comput. Geosci. (2020), 10.1007/s10596-020-09949-2, Google Scholar

[18] P. Knabner, L. Angermann, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer, New York (2003), Google Scholar

[19] P. Knabner, S. Bitterlich, R. Iza Teran, A. Prechtel, E. Schneid, Influence of surfactants on spreading of contaminants and soil remediation Jäger W., Krebs H.J. (Eds.), Mathematics – Key Technology for the Future, Springer, Berlin, Heidelberg (2003), 10.1007/978-3-642-55753-8_12, Google Scholar

[20] D. Kuzmin, Explicit and implicit FEM-FCT algorithms with flux linearization, J. Comput. Phys., 228 (7) (2009), pp. 2517-2534, 10.1016/, ArticleDownload PDFView Record in ScopusGoogle Scholar

[21] K.A. Lie, An Introduction to Reservoir Simulation Using MATLAB/GNU Octave: User Guide for the MATLAB Reservoir Simulation Toolbox (MRST), Cambridge University Press (2019), 10.1017/9781108591416, Google Scholar

[21] F. List, F.A. Radu, A study on iterative methods for solving Richards’ equation, Comput. Geosci., 20 (2) (2016), pp. 341-353, 10.1007/s10596-016-9566-3, CrossRefView Record in ScopusGoogle Scholar

[22]  F. Liu, Y. Fukumoto, X. Zhao, Stability analysis of the explicit difference scheme for Richards equation, Entropy, 22 (3) (2020), p. 352, 10.3390/e22030352, CrossRefView Record in ScopusGoogle Scholar

[23] D. Pasetto, A. Guadagnini, M. Putti, POD-based monte carlo approach for the solution of regional scale groundwater flow driven by randomly distributed recharge, Adv. Water Resour., 34 (11) (2011), pp. 1450-1463, 10.1016/j.advwatres.2011.07.003, ArticleDownload PDFView Record in ScopusGoogle Scholar

[24] J.R. Philip, Theory of infiltration, Adv. Hydrosci., 5 (1969), pp. 215-296, 10.1016/B978-1-4831-9936-8.50010-6, ArticleDownload PDFView Record in ScopusGoogle Scholar

[25] K.K. Phoon, T.S. Tan, P.C. Chong, Numerical simulation of richards equation in partially saturated porous media: under-relaxation and mass balance, Geotech. Geol. Eng., 25 (5) (2007), pp. 525-541, 10.1007/s10706-007-9126-7, CrossRefView Record in ScopusGoogle Scholar

[26] I.S. Pop, F.A. Radu, P. Knabner, Mixed finite elements for the Richards’ equation: linearization procedure, J. Comput. Appl. Math., 168 (1) (2004), pp. 365-373, 10.1016/, ArticleDownload PDFView Record in ScopusGoogle Scholar

[27] F.A. Radu, K. Kumar, J.M. Nordbotten, I.S. Pop, A robust, mass conservative scheme for two-phase flow in porous media including hölder continuous nonlinearities, IMA J. Numer. Anal., 38 (2) (2018), pp. 884-920, 10.1093/imanum/drx032, CrossRefView Record in ScopusGoogle Scholar

[28] 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), pp. 47-61, 10.1016/j.advwatres.2010.09.012, ArticleDownload PDFView Record in ScopusGoogle Scholar

[29] F.A. Radu, W. Wang, Convergence analysis for a mixed finite element scheme for flow in strictly unsaturated porous media, Nonlinear Anal. R. World Appl., 15 (2014), pp. 266-275, 10.1016/j.nonrwa.2011.05.003, ArticleDownload PDFView Record in ScopusGoogle Scholar

[30] G.C. Sander, J.Y. Parlange, V. Kühnel, W.L. Hogarth, D. Lockington, J.P.J. O’kane, Exact nonlinear solution for constant flux infiltration,J. Hydrol., 97 (3–4) (1988), pp. 341-346, 10.1016/0022-1694(88)90123-0, ArticleDownload PDFView Record in ScopusGoogle Scholar

[31] E. Schneid, Hybrid-gemischte finite-elemente-diskretisierung der richards-gleichung, Naturwissenschaftliche Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg (2000), Doctoral dissertation, Google Scholar, Schneid

[32] E. Schneid, A. Prechtel, P. Knabner, A comprehensive tool for the simulation of complex reactive transport and flow in soils, Land Contam. Reclamat., 8 (2000), pp. 357-365, 10.2462/09670513.570, View Record in ScopusGoogle Scholar

[33] H. Schwarze, U. Jaekel, H. Vereecken, Estimation of macrodispersion by different approximation methods for flow and transport in randomly heterogeneous media,Transp. Porous Media, 43 (2) (2001), pp. 265-287, 10.1023/A:1010771123844, View Record in ScopusGoogle Scholar

[34] J. Simunek, M. Sejna, H. Saito, M. Sakai, M. van Genuchten, The HYDRUS-1D Software Package for Simulating the Movement of Water, Heat, and Multiple Solutes in Variably Saturated Media, Version 4.0, (2008), Google Scholar

[35] M. Slodicka, A robust and efficient linearization scheme for doubly non-linear and degenerate parabolic problems arising in flow in porous media, SIAM J. Numer. Anal., 23 (5) (2002), pp. 1593-1614, 10.1137/S1064827500381860, View Record in ScopusGoogle Scholar

[36] R. Srivastava, T.C.J. Yeh, Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous and layered soils, Water Resour. Res., 27 (5) (1991), pp. 753-762, 10.1029/90WR02772, View Record in ScopusGoogle Scholar

[37] J.C. Strikwerda, Finite difference schemes and partial differential equations, SIAM (2004), 10.1137/1.9780898717938, Google Scholar

[38] N. Suciu, Diffusion in Random Fields. Applications to Transport in Groundwater, Birkhäuser, Cham (2019), 10.1007/978-3-030-15081-5, Google Scholar

[39] N. Suciu, Global Random Walk Solutions for Flow and Transport in Porous Media, Springer Nature, Switzerland (2020), 10.1007/978-3-030-55874-1_93, Google Scholar, Numerical Mathematics and Advanced Applications ENUMATH 2019, Lecture Notes in Computational Science and Engineering 139

[40] Suciu, N., Illiano, D., Prechtel, A., Radu, F. A., 2020. Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media (extended version). arXivpreprint: 2011.12889., Google Scholar

[41] Suciu, N., Illiano, D., Prechtel, A., Radu, F. A.2021. Git repository. doi:10.5281/zenodo.4709693., Google Scholar

[42] C. Vamoş, N. Suciu, H. Vereecken, Generalized random walk algorithm for the numerical modeling of complex diffusion processes, J. Comput. Phys., 186 (2003), pp. 527-544, 10.1016/S0021-9991(03)00073-1, ArticleDownload PDFView Record in ScopusGoogle Scholar

[43 H.F. Walker, P. Ni, Anderson acceleration for fixed-point iterations, SIAM J. Numer. Anal., 49 (4) (2011), pp. 1715-1735, 10.1137/10078356X, CrossRefView Record in ScopusGoogle Scholar

[44] A.W. Warrick, D.O. Lomen, S.R. Yates, A generalized solution to infiltration, Soil. Sci. Soc. Am. J., 49 (1) (1985), pp. 34-38, 10.2136/sssaj1985.03615995004900010006x, CrossRefView Record in ScopusGoogle Scholar

[45] K.K. Watson, V.A. Sardana, G.C. Sander, Comparison of analytical and numerical results for constant flux infiltration, J. Hydrol., 165 (1–4) (1995), pp. 101-112, 10.1016/0022-1694(94)02580-5, Google Scholar

[46]  K.S. Zadeh, A mass-conservative switching algorithm for modeling fluid flow in variably saturated porous media, J. Comput. Phys., 230 (3) (2011), pp. 664-679, 10.1016/, Google Scholar

[47] C.E. Zambra, M. Dumbser, E.F. Toro, N.O. Moraga, A novel numerical method of high-order accuracy for flow in unsaturated porous media, Int. J. Numer. Meth. Eng., 89 (2) (2012), pp. 227-240, Google Scholar, 10.1002/nme.3241

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