A multi-scale particle-tracking framework for dispersive solute transport modeling. (English) Zbl 1405.65137

Summary: Particle-tracking simulation offers a fast and robust alternative to conventional numerical discretization techniques for modeling solute transport in subsurface formations. A common challenge is that the modeling scale is typically much larger than the volume scale over which measurements of rock properties are made, and the scale-up of measurements have to be made accounting for the pattern of spatial heterogeneity exhibited at different scales. In this paper, a statistical scale-up procedure developed in our previous work is adopted to estimate coarse-scale (effective) transition time functions for transport modeling, while two significant improvements are proposed: considering the effects of non-stationarity (trend), as well as unresolved (residual) heterogeneity below the fine-scale model. Rock property is modeled as a multivariate random function, which is decomposed into the sum of a trend (which is defined at the same resolution of the transport modeling scale) and a residual ( representing all heterogeneities below the transport modeling scale). To construct realizations of a given rock property at the transport modeling scale, multiple realizations of the residual components are sampled. Next, a flow-based technique is adopted to compute the effective transport parameters: firstly, it is assumed that additional unresolved heterogeneities occurring below the fine scale can be described by a probabilistic transit time distribution; secondly, multiple realizations of the rock property, with the same physical size as the transport modeling scale, are generated; thirdly, each realization is subjected to particle-tracking simulation; finally, probability distributions of effective transition time function are estimated by matching the corresponding effluent history for each realization with an equivalent medium consisting of averaged homogeneous rock properties and aggregating results from all realizations. The proposed method is flexible that it does not invoke any explicit assumption regarding the multivariate distribution of the heterogeneity.


65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
76T20 Suspensions


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[1] Aronofsky, JS; Heller, JP, A diffusion model to explain mixing of flowing miscible fluids in porous media, Trans. AIME, 210, 345-349, (1957)
[2] Bear, J.: Hydraulics of Groundwater. McGraw-Hill, New York (1979)
[3] Becker, MW; Shapiro, AM, Interpreting tracer breakthrough tailing from different forced-gradient tracer experiment configurations in fractured bedrock, Water Resour. Res., 39, 1024, (2003)
[4] Benson, DA; Aquino, T; Bolster, D; Engdahl, N; Henri, CV; Fernàndez-Garcia, D, A comparison of eulerian and Lagrangian transport and non-linear reaction algorithms, Adv. Water Resour., 99, 15-37, (2017)
[5] Benson, DA; Wheatcraft, SW; Meerschaert, MM, Application of a fractional advection-dispersion equation, Water Resour. Res., 36, 1403-1412, (2000)
[6] Berentsen, CW; Kruijsdijk, CP; Verlaan, ML, Upscaling, relaxation and reversibility of dispersive flow in stratified porous media, Transport Porous Med., 68, 187-218, (2007)
[7] Berkowitz, B; Scher, H, Anomalous transport in correlated velocity fields, Phys. Rev., 81, 011128, (2010)
[8] Berkowitz, B; Cortis, A; Dentz, M; Scher, H, Modeling non-Fickian transport in geological formations as a continuous time random walk, Rev. Geophys., 44, rg2003, (2006)
[9] Berkowitz, B; Scher, H; Silliman, SE, Anomalous transport in laboratory-scale, heterogeneous porous media, Water Resour. Res., 36, 149-158, (2000)
[10] Bijeljic, B; Raeini, A; Mostaghimi, P; Blunt, MJ, Predictions of non-Fickian solute transport in different classes of porous media using direct simulation on pore-scale images, Phys. Rev. E, 87, 013011, (2013)
[11] Binning, P; Celia, MA, A forward particle tracking eulerian-Lagrangian localized adjoint method for solution of the contaminant transport equation in three dimensions, Adv. Water Resour., 25, 147-157, (2002)
[12] Boso, F; Bellin, A; Dumbser, M, Numerical simulations of solute transport in highly heterogeneous formations: a comparison of alternative numerical schemes, Adv. Water Resour., 52, 178-189, (2013)
[13] Carrera, J; Sánchez-Vila, X; Benet, I; Medina, A; Galarza, G; Guimerà, J, On matrix diffusion: formulations, solution methods and qualitative effects, Hydrobiol. J., 6, 178-190, (1998)
[14] Chiogna, G; Eberhardt, C; Grathwohl, P; Cirpka, OA; Rolle, M, Evidence of compound-dependent hydrodynamic and mechanical transverse dispersion by multitracer laboratory experiments, Environ. Sci. Technol., 44, 688-693, (2009)
[15] Cortis, A., Emmanuel, S., Rubin, S., Willbrand, K., Berkowitz, B.: The CTRW Matlab toolbox v3.1: a practical user’s guide. Retrieved from http://www.weizmann.ac.il/ESER/People/Brian/CTRW (2010)
[16] Cortis, A; Gallo, C; Scher, H; Berkowitz, B, Numerical simulation of non-Fickian transport in geological formations with multiple-scale heterogeneities, Water Resour. Res., 40, w04209, (2004)
[17] Dagan, G, Stochastic modeling of groundwater flow by unconditional and conditional probabilities: 1. conditional simulation and the direct problem, Water Resour. Res., 18, 813-833, (1982)
[18] Dagan, G, Solute transport in heterogeneous porous formations, J. Fluid Mech., 145, 151-177, (1984) · Zbl 0596.76097
[19] Dagan, G, Theory of solute transport by groundwater, Annu. Rev. Fluid Mech., 19, 183-213, (1987) · Zbl 0687.76091
[20] Dagan, G.: Flow and Transport in Porous Formations. Springer, New York (1989)
[21] Dentz, M; Cortis, A; Scher, H; Berkowitz, B, Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport, Adv. Water Resour., 27, 155-173, (2004)
[22] Deutsch, C.V., Journel, A.G.: Geostatistical Software Library and Users Guide. Oxford University Press, New York (1998)
[23] Durlofsky, LJ, Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water Resour. Res., 27, 699-708, (1991)
[24] Efendiev, Y; Durlofsky, LJ; Lee, SH, Modeling of subgrid effects in coarse-scale simulations of transport in heterogeneous porous media, Water Resour. Res., 36, 2031-2041, (2000)
[25] Fanchi, JR, Multidimensional numerical dispersion, SPE J., 23, 143-151, (1983)
[26] Fernàndez-Garcia, D; Sanchez-Vila, X, Optimal reconstruction of concentrations, gradients and reaction rates from particle distributions, J. Contam. Hydrol., 120-121, 99-114, (2011)
[27] Fernàndez-Garcia, D; Illangasekare, TH; Rajaram, H, Differences in the scale-dependence of dispersivity estimated from temporal and spatial moments in chemically and physically heterogeneous porous media, Adv. Water Resour., 28, 745-759, (2005)
[28] Fernàndez-Garcia, D; Llerar-Meza, G; Gómez-Hernández, JJ, Upscaling transport with mass transfer models: mean behavior and propagation of uncertainty, Water Resour. Res., 45, w10411, (2009)
[29] Fleurant, C; Der Lee, J, A stochastic model of transport in three-dimensional porous media, Math. Geol., 33, 449-474, (2001) · Zbl 1011.86001
[30] Fogedby, HC, Langevin equations for continuous time Lévy flights, Phys. Rev. E, 50, 1657, (1994)
[31] Gelhar, LW, Stochastic subsurface hydrology from theory to applications, Water Resour. Res., 22, 135s-145s, (1986)
[32] Gelhar, LW; Axness, CL, Three-dimensional stochastic analysis of macrodispersion in aquifers, Water Resour. Res., 19, 161-180, (1983)
[33] Gelhar, LW; Welty, C; Rehfeldt, KR, A critical review of data on field-scale dispersion in aquifers, Water Resour. Res., 28, 1955-1974, (1992)
[34] Gomez-Hernandez, JJ.: A stochastic approach to the simulation of block conductivity fields conditioned upon data measured at a smaller scale. Ph.D. thesis, Stanford University (1991)
[35] Gylling, B; Moreno, L; Neretnieks, I, The channel network model—a tool for transport simulations in fractured media, Ground Water, 37, 367-375, (1999)
[36] Haajizadeh, M., Fayers, F.J., Cockin, A.P., Roffey, M., Bond, D.J.: On the importance of dispersion and heterogeneity in the compositional simulation of miscible gas processes. In; SPE Asia Pacific Improved Oil Recovery Conference. Society of Petroleum Engineers, Kuala Lumpur (1999)
[37] Haggerty, R; Gorelick, SM, Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity, Water Resour. Res., 31, 2383-2400, (1995)
[38] Hassan, AE; Mohamed, MM, On using particle tracking methods to simulate transport in single-continuum and dual continua porous media, J. Hydrol., 275, 242-260, (2003)
[39] Holden, L; Lia, O, A tensor estimator for the homogenization of absolute permeability, Transp. Porous Media, 8, 37-46, (1992)
[40] Jha, RK; Bryant, S; Lake, LW, Effect of diffusion on dispersion, SPE J., 16, 65-77, (2011)
[41] John, A.K.: Dispersion in large scale permeable media. Dissertation, University of Texas at Austin (2008)
[42] Journel, A.G., Huijbregts, C.J.: Mining Geostatistics. Academic Press, London (1978)
[43] Kinzelbach, W., Uffink, G.: The random walk method and extensions in groundwater modelling. In: Bear, J., Corapcioglu, M. Y. (eds.) Transport Processes in Porous Media, pp 761-787. Kluwer Academic Publishers, The Netherlands (1991)
[44] Kitanidis, PK, Prediction by the method of moments of transport in a heterogeneous formation, J. Hydrol., 102, 453-473, (1988)
[45] Kitanidis, PK, Analysis of macrodispersion through volume-averaging: moment equations, Stoch. Hydrol. Hydraul., 6, 5-25, (1992) · Zbl 0743.76083
[46] Kleinhans, D; Friedrich, R, Continuous-time random walks: simulation of continuous trajectories, Phys. Rev. E, 76, 061102/1-6, (2007)
[47] LaBolle, EM; Fogg, GE; Tompson, AF, Random-walk simulation of transport in heterogeneous porous media: local mass-conservation problem and implementation methods, Water. Resour. Res., 32, 583-593, (1996)
[48] Lake, LW; Srinivasan, S, Statistical scale-up of reservoir properties: concepts and applications, J. Pet. Sci. Eng., 44, 27-39, (2004)
[49] Lantz, RB, Quantitative evaluation of numerical diffusion (truncation error), Soc. Pet. Eng. J., 11, 315-320, (1971)
[50] Le Borgne, T; Gouze, P, Non-Fickian dispersion in porous media: 2. model validation from measurements at different scales, Water Resour. Res., 44, w06427, (2008)
[51] Le Borgne, T; Bolster, D; Dentz, M; Anna, P; Tartakovsky, A, Effective pore-scale dispersion upscaling with a correlated continuous time random walk approach, Water Resour. Res., 47, w12538, (2011)
[52] Le Borgne, T; Dentz, M; Carrera, J, Lagrangian statistical model for transport in highly heterogeneous velocity fields, Phys. Rev. Lett., 101, 090601, (2008)
[53] Leung, JY; Srinivasan, S, Analysis of uncertainty introduced by scaleup of reservoir attributes and flow response in heterogeneous reservoirs., SPE J., 16, 713-724, (2011)
[54] Leung, JY; Srinivasan, S, Scale-up of mass transfer and recovery performance in heterogeneous reservoirs, J. Pet. Sci. Eng., 86-87, 71-86, (2012)
[55] Leung, JY; Srinivasan, S, Effects of reservoir heterogeneity on scaling of effective mass transfer coefficient for solute transport, J. Contam. Hydrol., 192, 181-193, (2016)
[56] Levy, M; Berkowitz, B, Measurement and analysis of non-Fickian dispersion in heterogeneous porous media, J. Contam. Hydrol., 64, 203-226, (2003)
[57] Li, X; Koike, T; Pathmathevan, M, A very fast simulated re-annealing (VFSA) approach for land data assimilation, Comput. Geosci., 30, 239-248, (2004)
[58] Lichtner, P.C., Kelkar, S., Robinson, B.: New form of dispersion tensor for axisymmetric porous media with implementation in particle tracking. Water Resour. Res. 38(8) (2002). https://doi.org/10.1029/2000WR000100
[59] Lu, Z; Stauffer, PH, On estimating functional average breakthrough curve using time-warping technique and perturbation approach, Water Resour. Res., 48, w05541, (2012)
[60] Metzler, R; Klafter, J, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77, (2000) · Zbl 0984.82032
[61] Nash, J; Sutcliffe, JV, River flow forecasting through conceptual models part I—a discussion of principles, J. Hydrol., 10, 282-290, (1970)
[62] Neuman, SP, Generalized scaling of permeabilities: validation and effect of support scale, Geophys. Res. Lett., 21, 349-352, (1994)
[63] Neuman, SP; Tartakovsky, DM, Perspective on theories of non-Fickian transport in heterogeneous media, Adv. Water Resour., 32, 670-680, (2009)
[64] Neuman, SP; Zhang, YK, A quasi-linear theory of non-fickian and Fickian subsurface dispersion: 1. theoretical analysis with application to isotropic media, Water Resour. Res., 26, 887-902, (1990)
[65] Neuman, SP; Winter, CL; Newman, CM, Stochastic theory of field-scale Fickian dispersion in anisotropic porous media, Water Resour. Res., 23, 453-466, (1987)
[66] Pedretti, D; Fernàndez-Garcia, D, An automatic locally-adaptive method to estimate heavily-tailed breakthrough curves from particle distributions, Adv. Water Resour., 59, 52-65, (2013)
[67] Pedretti, D; Fernàndez-Garcia, D; Sanchez-Vila, X; Bolster, D; Benson, DA, Apparent directional mass-transfer capacity coefficients in three-dimensional anisotropic heterogeneous aquifers under radial convergent transport, Water Resourv Res., 50, 1205-1224, (2014)
[68] Perkins, TK; Johnston, OC, A review of diffusion and dispersion in porous media, SPE J., 3, 70-84, (1963)
[69] Pickup, GE; Ringrose, PS; Jensen, JL; Sorbie, KS, Permeability tensors for sedimentary structures, Math. Geol., 26, 227-250, (1994)
[70] Pyrcz, M.J., Deutsch, C.V.: Geostatistical Reservoir Modeling. Oxford University Press, Oxford (2014)
[71] Riva, M; Guadagnini, A; Fernandez-Garcia, D; Sanchez-Vila, X; Ptak, T, Relative importance of geostatistical and transport models in describing heavily tailed breakthrough curves at the lauswiesen site, J. Contam. Hydrol., 101, 1-13, (2008)
[72] Rolle, M; Hochstetler, D; Chiogna, G; Kitanidis, PK; Grathwohl, P, Experimental investigation and pore-scale modeling interpretation of compound-specific transverse dispersion in porous media, Transp. Porous Media, 93, 347-362, (2012)
[73] Rubin, Y.: Applied Stochastic Hydrogeology. Oxford University Press, Oxford (2003)
[74] Salamon, P; Fernàndez-Garcia, D; Gómez-Hernández, JJ, A review and numerical assessment of the random walk particle tracking method, J. Contam. Hydrol., 87, 277-305, (2006)
[75] Salamon, P; Fernàndez-Garcia, D; Gómez-Hernández, JJ, Modeling mass transfer processes using random walk particle tracking, Water Resour. Res., 42, w11417, (2006)
[76] Salamon, P; Fernandez-Garcia, D; Gómez-Hernández, JJ, Modeling tracer transport at the MADE site: the importance of heterogeneity, Water Resour. Res., 43, w08404, (2007)
[77] Scheidegger, A.E.: An evaluation of the accuracy of the diffusivity equation for describing miscible displacement in porous media. In: Proceedings of the Theory of Fluid Flow in Porous Media Conference, pp. 101-116 (1959)
[78] Scheidegger, AE, General theory of dispersion in porous media, J. Geophys. Res., 66, 3273-3278, (1961)
[79] Schulze-Makuch, D; Cherkauer, DS, Variations in hydraulic conductivity with scale of measurement during aquifer tests in heterogeneous, porous carbonate rocks, Hydrobiol. J., 6, 204-215, (1998)
[80] Schulze-Makuch, D; Carlson, DA; Cherkauer, DS; Malik, P, Scale dependency of hydraulic conductivity in heterogeneous media, Ground Water, 37, 904-919, (1999)
[81] Srinivasan, G; Tartakovsky, DM; Dentz, M; Viswanathan, H; Berkowitz, B; Robinson, BA, Random walk particle tracking simulations of non-Fickian transport in heterogeneous media, J. Comput. Phys., 229, 4304-4314, (2010) · Zbl 1334.76113
[82] Sund, NL; Porta, GM; Bolster, D, Upscaling of dilution and mixing using a trajectory based spatial Markov random walk model in a periodic flow domain, Adv. Water Resour., 103, 76-85, (2017)
[83] Tompson, AF; Gelhar, LW, Numerical simulation of solute transport in three-dimensional, randomly heterogeneous porous media, Water Resour. Res., 26, 2541-2562, (1990)
[84] Vishal, V., Leung, J.Y.: Statistical scale-up of dispersive transport in heterogeneous reservoir. In: Geostatistics Valencia 2016 (pp. 733-743). Springer International Publishing (2017)
[85] Vishal, V; Leung, JY, Modeling impacts of subscale heterogeneities on dispersive solute transport in subsurface systems, J. Contam. Hydrol., 182, 63-77, (2015)
[86] Wang, J; Kitanidis, PK, Analysis of macrodispersion through volume averaging: comparison with stochastic theory, Environ. Res. Risk Assess., 13, 66-84, (1999) · Zbl 0973.76085
[87] White, C.D., Horne, R.N.: Computing absolute transmissibility in the presence of fine-scale heterogeneity. In: SPE Symposium on Reservoir Simulation. Society of Petroleum Engineers, San Antonio (1987)
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