Abstract
This article presents numerical investigations on accuracy and convergence properties of several numerical approaches for simulating steady state flows in heterogeneous aquifers.
Finite difference, finite element, discontinuous Galerkin, spectral, and random walk methods are tested on two-dimensional benchmark flow problems.
Realizations of log-normal hydraulic conductivity fields are generated by Kraichnan algorithms in closed form as finite sums of random periodic modes, which allow direct code verification by comparisons with manufactured reference solutions.
The quality of the methods is assessed for increasing number of random modes and for increasing variance of the log-hydraulic conductivity fields with Gaussian and exponential correlation.
Experimental orders of convergence are calculated from successive refinements of the grid.
The numerical methods are further validated by comparisons between statistical inferences obtained from Monte Carlo ensembles of numerical solutions and theoretical first-order perturbation results.
It is found that while for Gaussian correlation of the log-conductivity field all the methods perform well, in the exponential case their accuracy deteriorates and, for large variance and number of modes, the benchmark problems are practically not solvable with reasonably large computing resources, for all the methods considered in this study.
Authors
Cristian D. Alecsa
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Imre Boros
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Babes-Bolyai University, Romania
Florian Frank
Friedrich-Alexander University of Erlangen-Nuremberg, Germany
Peter Knabner,
Friedrich-Alexander University of Erlangen-Nuremberg, Germany
Mihai Nechita
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
University College London, United Kingdom
Alexander Prechtel
Friedrich-Alexander University of Erlangen-Nuremberg, Germany
Andreas Rupp
Friedrich-Alexander University of Erlangen-Nuremberg, Germany
Ruprecht-Karls-University, Germany
Nicolae Suciu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Keywords
Darcy flow; Accuracy; Convergence; Computational feasibility; Finite difference; Finite element; Discontinuous Galerkin; Spectral methods; Global random walk
Paper coordinates:
C.D. Alecsa, I. Boros, F. Frank, P. Knabner, M. Nechita, A. Prechtel, A. Rupp, N. Suciu, Numerical benchmark study for flow in highly heterogeneous aquifers, Adv. Water Res., 138 (2020), 103558
doi: 10.1016/j.advwatres.2020.103558
Please consult the extended version published on Arxiv.
Notes
- the paper appeared for some time on the journal website at the “Most popular” and “Most downloaded” categories (posted here);
- an extended version of this work is posted on Arxiv;
- the code is posted on GitHub.
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R. Ababou, D. McLaughlin, L.W. Gelhar, A.F. Tompson
Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media
Transp. Porous Media, 4 (6) (1989), pp. 549-565, 10.1007/BF00223627
Alecsa, C. D., Boros, I., Frank, F., Knabner, P., Nechita, M., Prechtel, A., Rupp, A., Suciu, N., 2020a. Benchmark for numerical solutions of fow in heterogeneous groundwater formations. arXiv:1911.10774.
Alecsa, C.D., Boros, I., Frank, F., Knabner, P., Nechita, M., Prechtel, A., Rupp, A., Suciu, N., 2020b. Git repository. https://github.com/PMFlow/FlowBenchmark. doi:10.5281/zenodo.3662401.
M.S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M.E. Rognes, G.N. Wells
The FEniCS project version 1.5
Arch. Numer. Softw., 3 (100) (2015), pp. 9-23, 10.11588/ans.2015.100.20553
J.L. Aurentz, L.N. Trefethen
Chopping a Chebyshev series
ACM Trans. Math. Softw. (TOMS), 43 (4) (2017), p. 33, 10.1145/2998442
A.A. Bakr, L.W. Gelhar, A.L. Gutjahr, J.R. MacMillan
Stochastic analysis of spatial variability in subsurface flows: 1. comparison of one-and three-dimensional flows
Water Resour. Res., 14 (2) (1978), pp. 263-271, 10.1029/WR014i002p00263
A. Bellin, Y. Rubin
HYDRO_GEN: a spatially distributed random field generator for correlated properties
Stoch. Hydrol. Hydraul., 10 (4) (1996), pp. 253-278, 10.1007/BF01581869
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
G.C. Bohling, G. Liu, P. Dietrich, J.J. Butler
Reassessing the MADE direct-push hydraulic conductivity data using a revised calibration procedure
Water Resour. Res., 52 (11) (2016), pp. 8970-8985, 10.1002/2016WR019008
M. Bollhöfer, J.I. Aliaga, A.F. Martín, E.S. Quintana-Ortí
ILUPACK. in padua, d.a. (ed)
Encyclopedia of Parallel Computing,, Springer, New York (2011), pp. 917-926, 10.1007/978-0-387-09766-4_513
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), pp. 163-171, 10.1016/j.advwatres.2011.10.001
O. Cainelli, A. Bellin, M. Putti
On the accuracy of classic numerical schemes for modeling flow in saturated heterogeneous formations
Adv. Water Resour., 47 (2012), pp. 43-55, 10.1016/j.advwatres.2012.06.016
V. Calo, Y. Efendiev, J. Galvis
A note on variational multiscale methods for high-contrast heterogeneous porous media flows with rough source terms
Adv. Water Resour., 34 (9) (2011), pp. 1177-1185, 10.1016/j.advwatres.2010.12.011
J. Carrayrou, J. Hoffmann, P. Knabner, S. Kräutle, C. De Dieuleveult, J. Erhel, J. Van Der Lee, V. Lagneau, K.U. Mayer, K.T. Macquarrie
Comparison of numerical methods for simulating strongly nonlinear and heterogeneous reactive transport problems–the momas benchmark case
Comput. Geosci., 14 (3) (2010), pp. 483-502, 10.1007/s10596-010-9178-2
J. Carrayrou, M. Kern, P. Knabner
Reactive transport benchmark of momas
Comput. Geosci., 14 (3) (2010), pp. 385-392, 10.1007/s10596-009-9157-7
C. Cordes, M. Putti
Accuracy of galerkin finite elements for groundwater flow simulations in two and three-dimensional triangulations
Int. J. Numer. Meth. Eng., 52 (4) (2001), pp. 371-387, 10.1002/nme.194
H. Cramér, M.R. Leadbetter
Stationary and Related Stochastic Processes
John Wiley & Sons, New York. (1967)
G. Dagan
Flow and Transport in Porous Formations
Springer, Berlin (1989)
T.A. Davis
Direct methods for sparse linear systems
Siam., 2 (2006), 10.1137/1.9780898718881
T.A. Davis
Algorithm 915, SuiteSparseQR: Multifrontal Multithreaded Rank-Revealing Sparse QR Factorization
ACM Trans. Math. Softw., 38 (1) (2011), 10.1145/2049662.2049670
F.W. Deng, J.H. Cushman
On higher-order corrections to the flow velocity covariance tensor
Water Resour. Res., 31 (7) (1995), pp. 1659-1672, 10.1029/94WR02974
V. Di Federico, S.P. Neuman
Scaling of random fields by means of truncated power variograms and associated spectra
Water Resour. Res., 33 (1997), pp. 1075-1085, 10.1029/97WR00299
D.A. Di Pietro, A. Ern
Mathematical Aspects of Discontinuous Galerkin Methods
Springer, Heidelberg (2012)
J.R. de Dreuzy, A. Beaudoin, J. Erhel
Asymptotic dispersion in 2d heterogeneous porous media determined by parallel numerical simulations
Water Resour. Res., 43 (2007), p. W10439, 10.1029/2006WR005394
J.R. de Dreuzy, G. Pichot, B. Poirriez, J. Erhel
Synthetic benchmark for modeling flow in 3d fractured media
Comput. Geosci., 50 (2013), pp. 59-71, 10.1016/j.cageo.2012.07.025
J.P. Eberhard, N. Suciu, C. Vamos
On the self-averaging of dispersion for transport in quasiperiodic random media
J. Phys. A, 40 (4) (2007), pp. 597-610, 10.1088/1751-8113/40/4/002
B. Flemisch, I. Berre, W. Boon, A. Fumagalli, N. Schwenck, A. Scotti, I. Stefansson, A. Tatomir
Benchmarks for single-phase flow in fractured porous media
Adv. Water Resour., 111 (2018), pp. 239-258, 10.1016/j.advwatres.2017.10.036
F. Frank, B. Reuter, V. Aizinger, P. Knabner
FESTUNG: A MATLAB / GNU Octave toolbox for the discontinuous Galerkin method, Part I: diffusion operator
Comput. Math. Appl., 70 (1) (2015), pp. 11-46, 10.1016/j.camwa.2015.04.013
L.W. Gelhar
Stochastic subsurface hydrology from theory to applications
Water Resour. Res., 22 (9S) (1986), pp. S135S-145S, 10.1029/WR022i09Sp0135S
L.W. Gelhar, C. Axness
Three-dimensional stochastic analysis of macrodispersion in aquifers
Water Resour. Res., 19 (1) (1983), pp. 161-180, 10.1029/WR019i001p00161
H. Gotovac, R. Andričevicć, B. Gotovac
Multi resolution adaptive modeling of groundwater flow and transport
Adv. Water Resour., 30 (5) (2007), pp. 1105-1126, 10.1016/j.advwatres.2006.10.007
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
C. Grenier, H. Anbergen, V. Bense, Q. Chanzy, E. Coon, N. Collier, F. Costard, M. Ferry, A. Frampton, J. Frederick, J. Gonçalvès, J. Holmén, A. Jost, S. Kokh, B. Kurylyk, J. McKenzie, J. Molson, E. Mouche, L. Orgogozo, R. Pannetier, A. Rivière, N. Roux, W. Rühaak, J. Scheidegger, J.-O. Selroos, R. Therrien, P. Vidstrand, C. Voss
Groundwater flow and heat transport for systems undergoing freeze-thaw: intercomparison of numerical simulators for 2d test cases
Adv. Water Resour., 114 (2018), pp. 196-218, 10.1016/j.advwatres.2018.02.001
Hoepffner, J., 2007. Implementation of boundary conditions. http://www.lmm.jussieu.fr/~hoepffner/boundarycondition.pdf.
T. Hou, X.H. Wu, Z. Cai
Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients
Math. Comput., 68 (227) (1999), pp. 913-943, 10.1090/S0025-5718-99-01077-7
C.T. Kelley
Iterative methods for linear and nonlinear equations
Siam, Philadelphia., 16 (1995)
P. Knabner, L. Angermann
Numerical Methods for Elliptic and Parabolic Partial Differential Equations
Springer, New York. (2003)
R. Kornhuber, D. Peterseim, H. Yserentant
An analysis of a class of variational multiscale methods based on subspace decomposition
Math. Comput., 87 (314) (2018), pp. 2765-2774, 10.1090/mcom/3302
R.H. Kraichnan
Diffusion by a random velocity field
Phys. Fluids, 13 (1) (1970), pp. 22-31, 10.1063/1.1692799
P.R. Kramer, O. Kurbanmuradov, K. Sabelfeld
Comparative analysis of multiscale gaussian random field simulation algorithms
J. Comp. Phys., 226 (2007), pp. 897-924, 10.1016/j.jcp.2007.05.002
O. Kurbanmuradov, K. Sabelfeld, P.R. Kramer
Randomized spectral and fourier-wavelet methods for multidimensional gaussian random vector fields
J. Comp. Phys., 245 (2013), pp. 218-234, 10.1016/j.jcp.2013.03.021
O.A. Kurbanmuradov, K.K. Sabelfeld
Stochastic flow simulation and particle transport in a 2d layer of random porous medium
Transp. Porous Media, 85 (2010), pp. 347-373, 10.1007/s11242-010-9567-y
S. Larsson, V. Thomée
Partial Differential Equations with Numerical Methods
Springer, Berlin Heidelberg (2009), 10.1007/978-3-540-88706-5
H. Li, D. Zhang
Probabilistic collocation method for flow in porous media: comparisons with other stochastic methods
Water Resour. Res., 43 (2007), p. W09409, 10.1029/2006WR005673
S. Li, D.B. McLaughlin
A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis
Water Resour. Res., 27 (7) (1991), pp. 1589-1605, 10.1029/91WR00881
W. Li, Z. Lu, D. Zhang
Stochastic analysis of unsaturated flow with probabilistic collocation method
Water Resour. Res., 45 (2009), p. W08425, 10.1029/2008WR007530
Z. Li, Z. Qiao, T. Tang
Numerical Solution of Differential Equations: Introduction to Finite Difference and Finite Element Methods
Cambridge University Press (2017), 10.1017/9781316678725
H.A. Liu, F.J. Molz
Multifractal analyses of hydraulic conductivity distributions
Water Resour. Res., 33 (11) (1997), pp. 2483-2488, 10.1029/97WR02188
G.J. Lord, C.E. Powell, T. Shardlow
An Introduction to Computational Stochastic PDEs
Cambridge University Press (2014)
A. Mantoglou, J.L. Wilson
The turning bands method for simulation of random fields using line generation by a spectral method
Water Resour. Res., 18 (5) (1982), pp. 1379-1394, 10.1029/WR018i005p01379
S.A. Mizell, A.L. Gutjahr, L.W. Gelhar
Stochastic analysis of spatial variability in two-dimensional steady groundwater flow assuming stationary and nonstationary heads
Water Resour. Res., 18 (4) (1982), pp. 1053-1067, 10.1029/WR018i004p01053
F.J. Molz, G.K. Boman
Further evidence of fractal structure in hydraulic conductivity distributions
Geophys. Res. Lett., 22 (18) (1995), pp. 2545-2548, 10.1029/95GL02548
F.J. Molz, H.H. Liu, J. Szulga
Fractional brownian motion and fractional gaussian noise in subsurface hydrology: A review, presentation of fundamental properties, and extensions
Water Resour. Res., 33 (10) (1997), pp. 2273-2286, 10.1029/97WR01982
W.L. Oberkampf, F.G. Blottner
Issues in computational fluid dynamics code verification and validation
AIAA J., 36 (5) (1998), pp. 687-695, 10.2514/2.456
M. Putti, C. Cordes
Finite element approximation of the diffusion operator on tetrahedra
SIAM J. Sci. Comput., 19 (4) (1998), pp. 1154-1168, 10.1137/S1064827595290711
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) (2017), pp. 884-920, 10.1093/imanum/drx032
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
H. Rajaram, L.W. Gelhar
Three-dimensional spatial moments analysis of the borden tracer test
Water Resour. Res., 27 (6) (1991), pp. 1239-1251, 10.1029/91WR00326
K.R. Rehfeldt, J.M. Boggs, L.W. Gelhar
Field study of dispersion in a heterogeneous aquifer: 3. geostatistical analysis of hydraulic conductivity
Water Resour. Res., 28 (12) (1992), pp. 3309-3324, 10.1029/92WR01758
R.W. RitziJr, M.R. Soltanian
What have we learned from deterministic geostatistics at highly resolved field sites, as relevant to mass transport processes in sedimentary aquifers?
J. Hydrol., 531 (2015), pp. 31-39, 10.1016/j.jhydrol.2015.07.049
B. Riviere
Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM, Philadelphia (2008), 10.1137/1.9780898717440.fm
P.J. Roache
Verification and Validation in Computational Science and Engineering
Hermosa Publishers, New Mexico (1998)
P.J. Roache
Code verification by the method of manufactured solutions
J. Fluids Eng., 124 (1) (2002), pp. 4-10, 10.1115/1.1436090
C.J. Roy
Grid convergence error analysis for mixed-order numerical schemes
AIAA J., 41 (4) (2003), pp. 595-604, 10.2514/2.2013
C.J. Roy
Review of code and solution verification procedures for computational simulation
J. Comput. Phys., 205 (2005), pp. 131-156, 10.1016/j.jcp.2004.10.036
A. Rupp
Simulating structure formation in soils across scales using discontinuous galerkin methods
Shaker, Düren (2019)
P. Salandin, V. Fiorotto
Solute transport in highly heterogeneous aquifers.
Water Resour. Res., 34 (5) (1998), pp. 949-961, 10.1029/98WR00219
L.J. Segerlind
Applied Finite Element Analysis
J. Wiley and Sons, New York (1987)
J. Shen, Y. Wang, J. Xia
Fast structured direct spectral methods for differential equations with variable coefficients, i. the one-dimensional case
SIAM J. Sci. Comput., 38 (1) (2016), pp. A28-A54, 10.1137/140986815
V. Srzic, V. Cvetkovic, R. Andricevic, H. Gotovac
Impact of aquifer heterogeneity structure and local-scale dispersion on solute concentration uncertainty
Water Resour. Res., 49 (6) (2013), pp. 3712-3728, 10.1002/wrcr.20314
N. Suciu
Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields
Phys. Rev. E, 81 (2010), p. 056301, 10.1103/PhysRevE.81.056301
N. Suciu
Diffusion in random velocity fields with applications to contaminant transport in groundwater
Adv. Water. Resour., 69 (2014), pp. 114-133, 10.1016/j.advwatres.2014.04.002
N. Suciu
Diffusion in random fields
Applications to Transport in Groundwater. Birkhäuser, Cham (2019), 10.1007/978-3-030-15081-5
N. Suciu, S. Attinger, F.A. Radu, C. Vamoş, J. Vanderborght, H. Vereecken, P. Knabner
Solute transport in aquifers with evolving scale heterogeneity
An. Sti. U. Ovid. Co-Mat., 23 (3) (2015), pp. 167-186, 10.1515/auom-2015-0054
N. Suciu, F.A. Radu, A. Prechtel, 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), pp. 27-37, 10.1016/j.cam.2012.06.027
N. Suciu, L. Schüler, S. Attinger, P. Knabner
Towards a filtered density function approach for reactive transport in groundwater
Adv. Water Resour., 90 (2016), pp. 83-98, 10.1016/j.advwatres.2016.02.016
N. Suciu, C. Vamoş, J. Vanderborght, H. Hardelauf, H. Vereecken
Numerical investigations on ergodicity of solute transport in heterogeneous aquifers
Water. Resour. Res., 42 (2006), p. W04409, 10.1029/2005WR004546
M.G. Trefry, F.P. Ruan, D. McLaughlin
Numerical simulations of preasimptotic transport in hetoregenous porous media: departures from the gaussian limit
Water Resour. Res., 39 (3) (2003), pp. 1063-1077, 10.1029/2001WR001101
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
J.A. Weideman, S.C. Reddy
A MATLAB differentiation matrix suite. ACM trans
Math. Software, 26 (4) (2000), pp. 465-519, 10.1145/365723.365727
A.M. Yaglom
Correlation theory of stationary and related random functions
Vol. 2: Supplementary Notes and References., Springer, New York (1987)
Y. Zhang, M. Person, C.W. Gable
Representative hydraulic conductivity of hydrogeologic units: insights from an experimental stratigraphy
J. Hydrol., 339 (2007), pp. 65-78, 10.1016/j.jhydrol.2007.03.007
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