## Abstract

Evolution equations for probability density functions (PDFs) and filtered density functions (FDFs) of random species concentrations weighted by conserved scalars are formulated as Fokker–Planck equations describing stochastically equivalent processes in concentration-position spaces. This approach provides consistent numerical PDF/FDF solutions, given by the density in the concentration-position space of an ensemble of computational particles governed by the associated Itô equations. The solutions are obtained by a global random walk (GRW) algorithm, which is stable, free of numerical diffusion, and practically insensitive to the increase of the number of particles. The general FDF approach and the GRW numerical solution are illustrated for a reduced complexity problem consisting of the transport of a single scalar in groundwater. Randomness is induced by the stochastic parameterization of the hydraulic conductivity, characterized by short range correlations and small variance. The objective is to infer the statistics of the random concentration sampled at the plume center of mass, integrated over the transverse dimension of a two-dimensional spatial domain. The PDF/FDF problem can therefore be formulated in a two-dimensional domain as well, a spatial dimension and one in the concentration space. The upscaled drift and diffusion coefficients describing the PDF transport in the physical space are estimated on single-trajectories of diffusion in velocity fields with short-range correlations, owing to their self-averaging property. The mixing coefficients describing the PDF transport in concentration spaces are parameterized by the trend and the noise inferred from the statistical analysis of an ensemble of simulated concentration time series, as well as by classical mixing models. A Gaussian spatial filter applied to a Kraichnan velocity field generator is used to construct coarse-grained simulations (CGS) for FDF problems. The purposes of the CGS simulations are two-fold: first to understand the significance of the FDF approach from a practical point of view and its relation to the PDF approach; second to investigate the limits of the mixing models considered here and the desirable features of the mixing models for groundwater systems.

## Authors

## Keywords

PDF/FDF methods; Mixing; Global random walk; Groundwater

## Cite this paper as:

N. Suciu, L. Schüler, S. Attinger, P. Knabner (2016), *Towards a filtered density function approach for reactive transport in groundwater*, Adv. Water Resour., 90, 83–98,

doi: 10.1016/j.advwatres.2016.02.016

## About this paper

##### Print ISSN

0309-1708 |

##### Online ISSN

1573-1634

##### MR

?

##### ZBL

?

##### Google Scholar Profile

?

## References

## Paper in html format

## References

[1] Pope SB., *The probability approach to the modelling of turbulent reacting ﬂows*. Combust Flame 1976; 27:299–312.

CrossRef (DOI)

[2] Pope SB., *PDF methods for turbulent reactive ﬂows.* Prog Energy Combust Sci 1985;11(2):119–192.

CrossRef (DOI)

[3] Pope SB., *Turbulent Flows*. Cambridge: Cambridge University Press: 2000.

[4] Fox RO., *Computational Models for Turbulent Reacting Flows*. New York: Cambridge University Press; 2003.

[5] Haworth DC., *Progress in probability density function methods for turbulent reacting ﬂows*. Prog Energy Combust Sci 2010; 36:168-259.

CrossRef (DOI)

[6] Haworth DC, Pope SB., *Transported probability density function methods for Reynolds-averaged and large-eddy simulations. *In: EchekkiT, Mastorakos E, editors. Turbulent combustion modeling. Fluid mechanics and its applications, vol. 95. Dordrecht: Springer; 2011. p.119–42.

CrossRef (DOI)

[7] Colucci PJ, Jaberi FA, Givi P., *Filtered density function for large eddy simulation of turbulent reacting ﬂows*. PhysFluids 1998; 10(2):499–515.

CrossRef (DOI)

[8] Jaberi FA, Colucci PJ, James S, Givi P, Pope SB., *Filtered mass density function for large-eddy simulation of turbulent reacting ﬂows*. J. Fluid Mech. 1999; 401:85–121.

CrossRef (DOI)

[9] McDermott R, Pope SB., *A particle formulation for treating differential diffusion in filtered density function methods*. J Comput Phys 2007; 226:947–993.

CrossRef (DOI)

[10] Heinz S., *Uniﬁed turbulence models for LES and RANS, FDF and PDF simulations*. Theor Comput Fluid Dyn 2007;21:99118.

CrossRef (DOI)

[11] Dodoulas IA, Navarro-Martinez S., *Large eddy simulation of premixed turbulent ﬂames unsing probability density approach*. Flow Tur-bulence Combust 2013;90:645–678.

CrossRef (DOI)

[12] Schwede RL, Cirpka OA, Nowak W, Neuweiler I., *Impact of sampling volume on the probability density function of steady state concentration*. Water Resour Res 2008;44(12):W12433.

CrossRef (DOI)

[13] Sanchez-Vila X, Guadagnini A, Fernandez-Garcia D., *Conditional probability density functions of concentrations for mixing-controlled reactive transport in heterogeneous aquifers*. Math Geosci 2009;41:32351.

CrossRef (DOI)

[14] Dentz M, Tartakovsky DM., *Probability density functions for passive scalars dispersed in random velocity ﬁelds*. Geophys Res Lett 2010;37:L24406.

CrossRef (DOI)

[15] Meyer DW, Jenny P, Tchelepi HA., *A joint velocity-concentration PDF method for tracer ﬂow in heterogeneous porous media*. Water Resour Res 2010;46:W12522.

CrossRef (DOI)

[16] Cirpka OA, de Barros FPJ, Chiogna G, Nowak W., *Probability density function of steady state concentration in two-dimensional heterogeneous porous media*. Water Resour Res 2011;47:W11523.

CrossRef (DOI)

[17] Venturi D, Tartakovsky DM, Tartakovsky AM, Karniadakis GE., *Exact PDF equations and closure approximations for advective reactive transport.* J Comput Phys 2013;243:32343.

CrossRef (DOI)

[18] Suciu N., *Di**ﬀusion in random velocity ﬁelds with applications to contaminant transport in groundwater.* Adv Water Resour 2014;69:114–133.

CrossRef (DOI)

[19] Suciu N, Radu FA, Attinger S, Schuler L, Knabner P., *A Fokker-Planck approach for probability distributions of species concentrations transported in heterogeneous media*. J Comput Appl Math 2015;289:241–252.

CrossRef (DOI)

[20] Suciu N, Schuler L, Attinger S, Vamos¸ C, Knabner P., *Consistency issues in PDF methods*. An Sti U Ovid Co-Mat 2015;23(3):187–208.

CrossRef (DOI)

[21] Beckie R, Aldama AA, Wood EF., *Modeling the large-scale dynamics of saturated groundwater ﬂow using spatial ﬁltering theory: 1.Theoretical development*. Water Resour Res 1996;32(5):1269–1280.

CrossRef (DOI)

[22] Beckie, R, Aldama AA, Wood EF., *Modeling the large-scale dynamics of saturated groundwater ﬂow using spatial ﬁltering theory: 2.Numerical Evaluation*. Water Resour Res 1996;32(5):1281–1288.

CrossRef (DOI)

[23] Efendiev YR, Durlofsky LJ, Lee SH., *Modeling of subgrid e**ﬀects in coarse scale simulations of transport in heterogeneous porous media*.Water Resour Res 2000;36:2031–2041.

CrossRef (DOI)

[24] Efendiev Y, Durlofsky LJ., *A Generalized convection-di**ﬀusion model for subgrid transport in porous media*. Multiscale Model Simul 2003;1(3):504–526.

CrossRef (DOI)

[25] Attinger S., *Generalized Coarse Graining Procedures for Flow in Porous Media*. Computational Geosciences 2003;7(4):253–273.

CrossRef (DOI)

[26] Heße F, Radu FA, Thullner M, Attinger S., *Upscaling of the advection di**ﬀusion reaction equation with Monod reaction*. Adv Water Resour2009;32:1336–1351.

CrossRef (DOI)

[27] Minier JP, Peirano E., *The PDF approach to turbulent and polydispersed two-phase ﬂows*. Phys Rep 2001;352:1–214.

[28] Klimenko AY, Bilger RW., *Conditional moment closure for turbulent combustion*. Progr Energ Combust Sci 1999; 25:595–687.

CrossRef (DOI)

[29] Morales-Casique E, Neuman SP, Gaudagnini A., *Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: theoretical framework.* Adv Water Resour 2006;29:1238-1255.

CrossRef (DOI)

[30] Morales-Casique E, Neuman SP, Gaudagnini A*., Nonlocal and localized analyses of nonreactive solute transport inbounded randomly heterogeneous porous media: computational analysis*. Adv Water Resour 2006;29:1399–1418.

CrossRef (DOI)

[31] Pope SB., *A Monte Carlo method for the PDF equations of turbulent reactive ﬂow*. Combustion Science and Technology 1981;25:159–174.

CrossRef (DOI)

[32] Mobus H, Gerlinger P, Brggemann D., *Comparison of Eulerian and Lagrangian Monte Carlo PDF methods for turbulent di**ﬀusion ﬂames*. Combust Flame 2001;124:519–534.

CrossRef (DOI)

[33] Jones WP, Marquis AJ, Prasad VN., *LES of a turbulent premixed swirl burner using the Eulerian stochastic ﬁeld method*. Combust Flame 2012;159:3079–3095.

CrossRef (DOI)

[34] Valino L., *A Field Monte Carlo Formulation for Calculating the Probability Density Function of a Single Scalar in a Turbulent Flow*. Flow Turb Combust 1998;60(2):157–172.

CrossRef (DOI)

[35] Sabelnikov V, Soulard O., *Rapidly decorrelating velocity-ﬁeld model as a tool for solving one-point Fokker-Planck equations for probability density functions of turbulent reactive scalars*. Phys Rev E 2005;72(1):016301.

CrossRef (DOI)

[36] Waclawczyk M, Pozorski J, Minier JP., *New Molecular Transport Model for FDF/LES of Turbulence with Passive Scalar*. Flow Turbulence Combust 2008;81:235–260.

CrossRef (DOI)

[37] Wang H, Popov PP, Pope SB., *Weak second-order splitting schemes for Lagrangian Monte Carlo particle methods for the composition PDF/FDF transport equations*. J Comput Phys 2010;229:1852–1878.

CrossRef (DOI)

[38] Kloeden PE, Platen E., *Numerical solutions of stochastic di**ﬀerential equations*. Berlin: Springer; 1999.

[39] Herz M., *Mathematical modeling and analysis of electrolyte solutions*. 2014; PhD thesis. http://www.mso.math.fau.de/ﬁleadmin/am1/projects/PhD Herz.pdf.

[40] Rubin Y, Sun A, Maxwell R, Bellin A., *The concept of block e**ﬀective macrodispersivity and a uniﬁed approach for grid-scale and plume-scale-dependent transport*. J Fluid Mech 1999;395:161–180.

CrossRef (DOI)

[41] de Barros FPJ, Rubin Y., *Modelling of block-scale macrodispersion as a random function*. J Fluid Mech 2011;676:514–545.

CrossRef (DOI)

[42] Bellin A, Tonina D., *Probability density function of non-reactive solute concentration in heterogeneous porous formations*, J. Contam. Hydrol. 2007;94:109–125.

CrossRef (DOI)

[43] Minier J-P, Chibbaro S, Pope SB., *Guidelines for the formulation of Lagrangian stochastic models for particle simulations of single-phase and dispersed two-phase turbulent ﬂows*. Phys Fluids 2014;26:113303.

CrossRef (DOI)

[44] Klimenko AY., *On simulating scalar transport by mixing between Lagrangian particles*. Phys Fluids 2007;19:031702.

CrossRef (DOI)

[45] Vamos¸ C, Suciu N, Vereecken H., *Generalized random walk algorithm for the numerical modeling of complex di**ﬀusion processes*. J. Comput Phys 2003;186(2):52744.

CrossRef (DOI)

[46] Suciu N, Radu FA, Prechtel A, Brunner F, Knabner P., *A coupled ﬁnite element-global random walk approach toadvection-dominated transport in porous media with random hydraulic conductivity.* J Comput Appl Math 2013;24627–37.

CrossRef (DOI)

[47] Suciu N, Vamos¸ C, Vanderborght J, Hardelauf H, Vereecken H., *Numerical investigations on ergodicity of solute transport in heterogeneous aquifers*. Water Resour Res 2006;42:W04409.

CrossRef (DOI).

[48] Kraichnan RH., *Di**ﬀusion by a random velocity ﬁeld*. Phys Fluids 1970;13(1):2231.

CrossRef (DOI)

[49] Vamos¸ C, Craciun M., *Separation of components from a scale mixture of Gaussian white noises*. Phys Rev E 2010;81:051125.

CrossRef (DOI)

[50] Suciu N, Vamos C., *Ergodic estimations of upscaled coe**ﬃcients for di**ﬀusion in random velocity ﬁelds.* In: L’Ecuyer Pierre, Owen Art20 B, editors. Monte Carlo and quasi-Monte Carlo methods 2008. Berlin: Springer; 2009. p. 617-626.

CrossRef (DOI)

[51] Dagan G., *Upscaling of dispersion coe**ﬃcients in transport through heterogeneous porous formations*. In: Peters A et al., editors. Computational Methods in Water Resources X. Norwell, Mass: Kluwer Acad; 1994. p. 431–439.

[52] Schwarze H, Jaekel U, Vereecken H., *Estimation of Macrodispersion by Di**ﬀerent Approximation Methods for Flow and Transport in Randomly Heterogeneous Media*. Transport Porous Med 2001;43:265–287.

CrossRef (DOI)

[53] Dentz M, Kinzelbach H,Attinger S,Kinzelbach W., *Temporal behavior of a solute cloud in a heterogeneous porous medium*. 3. Numerical simulations. Water Resour Res 2002;38:1118.

CrossRef (DOI)

[54] Kurbanmuradov OA, Sabelfeld KK., *Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium *Transp Porous Med 2010;85:347–373.

CrossRef (DOI)

[55] Heße F, Prykhodko V, Schlter S, Attinger S., *Generating random ﬁelds with a truncated power-law variogram: A comparison of several numerical methods*. Environ Model Software 2014;55:32–48.

CrossRef (DOI)

[56] Bronstein I.N, Semendjajew KA, Musiol G, Muhlig H., *Taschenbuch der Mathematik*. Frankfurt am Main: Verlag Harri Deutsch; 2006.

## Paper in html format

soon