Abstract
Probability density function (PDF) methods are a promising alternative to predicting the transport of solutes in groundwater under uncertainty. They make it possible to derive the evolution equations of the mean concentration and the concentration variance, used in moment methods. The mixing model, describing the transport of the PDF in concentration space, is essential for both methods. Finding a satisfactory mixing model is still an open question and due to the rather elaborate PDF methods, a difficult undertaking. Both the PDF equation and the concentration variance equation depend on the same mixing model. This connection is used to find and test an improved mixing model for the much easier to handle concentration variance. Subsequently, this mixing model is transferred to the PDF equation and tested. The newly proposed mixing model yields significantly improved results for both variance modelling and PDF modelling.
Authors
L. Schüler
-Institute of Geosciences, Friedrich Schiller University Jena, Burgweg 11, 07749 Jena, Germany
-Department Computational Hydrosystems, Helmholtz Centre for Environmental Research – UFZ, Permoserstraße 15, 04318 Leipzig, Germany
N. Suciu
-Mathematics Department, Friedrich-Alexander University of Erlangen-Nuremberg, Cauerstraße 11, 91058 Erlangen, Germany
-Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
P. Knabner
-Mathematics Department, Friedrich-Alexander University of Erlangen-Nuremberg, Cauerstraße 11, 91058 Erlangen, Germany
S. Attinger
-Institute of Geosciences, Friedrich Schiller University Jena, Burgweg 11, 07749 Jena, Germany
-Department Computational Hydrosystems, Helmholtz Centre for Environmental Research – UFZ, Permoserstraße 15, 04318 Leipzig, Germany
Keywords
Cite this paper as:
L. Schüler, N. Suciu, P. Knabner and S. Attinger, A time dependent mixing model to close PDF equations for transport in heterogeneous aquifers, Adv. Water Resour., 96 (2016), 55-67
doi: 10.1016/j.advwatres.2016.06.012
References
Not available yet.
About this paper
Print ISSN
0169-3913
Online ISSN
1573-1634
[1] R. Andricevic, 1998. Effects of local dispersion and sampling volume on the evolution of concentration fluctuations in aquifers. Water Resour. Res. 34 (5), 1115–1129.
CrossRef (DOI)
[2] Andricevic, R., Cvetkovic, V., 1996. Evaluation of risk from contaminants migrating by groundwater. Water Resour. Res. 32 (3), 611–621.
CrossRef (DOI)
[3] de Barros, F.P.J., Fiori, A., 2014. First-order based cumulative distribution function for solute concentration in heterogeneous aquifers: theoretical analysis and implications for human health risk assessment. Water Resour. Res. 50 (5), 4018–4037.
CrossRef (DOI)
[4] de Barros, F.P.J., Fiori, A., Bellin, A., 2011. A simple closed-form solution for assessing concentration uncertainty. Water Resour. Res. 47 (12), 1–5.
CrossRef (DOI)
[5] Bellin, A., Tonina, D., 2007. Probability density function of non-reactive solute concentration in heterogeneous porous formations. J. Contam. Hydrol. 94 (1–2), 109–125.
CrossRef (DOI)
[6] Burr, D.T., Sudicky, E.A., Naff, R.L., 1994. Nonreactive and reactive solute transport in three-dimensional heterogeneous porous media: mean displacement, plume spreading, and uncertainty. Water Resour. Res. 30 (3), 791–815.
CrossRef (DOI)
[7] Celis, C., Figueira da Silva, L.F., 2015. Lagrangian mixing models for turbulent combustion: review and prospects. Flow, Turbul. Combust. 94 (3), 643–689.
CrossRef (DOI)
[8] Cirpka, O.A., de Barros, F.P.J., Chiogna, G., Nowak, W., 2011. Probability density function of steady state concentration in two-dimensional heterogeneous porous media. Water Resour. Res. 47 (11), 1–14.
CrossRef (DOI)
[9] Cirpka, O.A., Schwede, R.L., Luo, J., Dentz, M., 2008. Concentration statistics for mixing-controlled reactive transport in random heterogeneous media. J. Contam. Hydrol. 98 (1–2), 61–74.
CrossRef (DOI)
[10] Colucci, P.J., Jaberi, F.A., Givi, P., Pope, S.B., 1998. Filtered density function for large eddy simulation of turbulent reacting flows. Phys. Fluids 10 (2), 499–515.
CrossRef (DOI)
[11] Dagan, G., 1982. Stochastic modeling of groundwater flow by unconditional and conditional probabilities 1.conditional simulation and the direct problem. Water Resour. Res. 18 (4), 813–833.
CrossRef (DOI)
[12] Dentz, M., Kinzelbach, H., Attinger, S., Kinzelbach, W., 2000. Temporal behavior of a solute cloud in a heterogeneous porous medium 1 .Point-like injection. Water Resour. Res. 36 (12), 3591–3604.
CrossRef (DOI)
[13] Dentz, M., Kinzelbach, H., Attinger, S., Kinzelbach, W., 2002. Temporal behavior of a solute cloud in a heterogeneous porous medium 3. Numerical simulations. Water Resour. Res. 38 (7), 23–1–23–13.
CrossRef (DOI)
[14] Dentz, M., Tartakovsky, D.M., 2010. Probability density functions for passive scalars dispersed in random velocity fields. Geophys. Res. Lett. 37 (24), 1–4.
CrossRef (DOI)
[15] Dodoulas, I.A., Navarro-Martinez, S., 2013. Large Eddy simulation of premixed turbulent flames using the probability density function approach. Flow, Turbul. Combust. 90 (3), 645–678.
CrossRef (DOI)
[16] Dopazo, C., O’Brien, E.E., 1974. An approach to autoignition of a turbulent mixture. Acta Astronaut. 1, 1239–1266.
CrossRef (DOI
[17] Drummond, I.T., Duane, S., Horgan, R.R., 1984. Scalar diffusion in simulated helical turbulence with molecular diffusivity. J. Fluid Mech. 138, 75–91.
CrossRef (DOI)
[18] Eberhard, J.P., Suciu, N., Vamos¸ , C., 2007. On the self-averaging of dispersion for transport in quasi-periodic random media. J. Phys. A 40 (4), 597.
CrossRef (DOI)
[19] Fiori, A., 2001. The Lagrangian concentration approach for determining dilution in aquifer transport: theoretical analysis and comparison with field experiments. Water Resour. Res. 37 (12), 3105–3114.
CrossRef (DOI)
[20] Fiori, A., Bellin, A., Cvetkovic, V., de Barros, F.P.J., Dagan, G., 2015. Stochastic modeling of solute transport in aquifers: from heterogeneity characterization to risk analysis. Water Resour. Res. 51 (8), 6622–6648.
CrossRef (DOI)
[21] Fiorotto, V., Caroni, E., 2002. Solute concentration statistics in heterogeneous aquifers for finite Peclet values. Transp. Porous Media 48 (3), 331–351.
CrossRef (DOI)
[22] Fox, R.O., 2003. Computational Models for Turbulent Reacting Flows. Cambridge Series in Chemical Engineering. Cambridge University Press, New York.
[23] Gelhar, L.W., Axness, C.L., 1983. Three dimensional stochastic analysis of Macrodispersion in aquifers. Water Resour. Res. 19 (1), 161–180.
CrossRef (DOI)
[24] Heße, F., Prykhod’ko, V., Schlüter, S., Attinger, S., 2014. Generating random fields with a truncated power-law variogram. A comparison of several numerical methods with respect to accurary and reproduction of structural features. Environ. Model. Softw. 55, 32–48.
CrossRef (DOI)
[25] Im, H.G., Lund, T.S., Ferziger, J.H., 1997. Large eddy simulation of turbulent front propagation with dynamic subgrid models. Phys. Fluids 9 (12), 3826–3833.
CrossRef (DOI)
[26] Jones, W., Marquis, A., Prasad, V., 2012. LES of a turbulent premixed swirl burner using the Eulerian stochastic field method. Combust. Flame 159 (10), 3079–3095.
CrossRef (DOI)
[27] Kapoor, V., Gelhar, L.W., 1994a. Transport in three-dimensionally heterogeneous aquifers: 1. Dynamics of concentration fluctuations. Water Resour. Res. 30 (6), 1775–1788.
CrossRef (DOI)
[28] Kapoor, V., Gelhar, L.W., 1994b. Transport in three-dimensionally heterogeneous aquifers 2. Predictions and observations of concentration fluctuations. Water Resour. Res. 30 (6), 1789–1801.
CrossRef (DOI)
[29] Kapoor, V., Kitanidis, P.K., 1997. Advection-diffusion in spatially random flows: Formulation of concentration covariance. Stoch. Hydrol. Hydraul. 11 (5), 397–422.
CrossRef (DOI)
[30] Kraichnan, R.H., 1970. Diffusion by a Random Velocity Field. Phys. Fluids 13 (1), 22
CrossRef (DOI)
[31] Meyer, D.W., Jenny, P., Tchelepi, H.A., 2010. A joint velocity-concentration PDF method for tracer flow in heterogeneous porous media. Water Resour. Res. 46 (12), 1–17.
CrossRef (DOI)
[32] Pierce, C.D., Moin, P., 1998. A dynamic model for subgrid-scale variance and dissipation rate of a conserved scalar. Phys. Fluids 10 (12), 3041–3044.
CrossRef (DOI)
[33] Pope, S.B., 1985. PDF Methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11 (2), 119–192.
CrossRef (DOI)
[34] Popov, P.P., Pope, S.B., 2014. Implicit and explicit schemes for mass consistency preservation in hybrid particle/finite-volume algorithms for turbulent reactive flows. J. Comput. Phys. 257, 352–373.
CrossRef (DOI)
[35] Raman, V., Pitsch, H., 2007. A consistent LES/filtered-density function formulation for the simulation of turbulent flames with detailed chemistry. Proc. Combust. Inst. 31 (2), 1711–1719.
CrossRef (DOI)
[36] Sabel’nikov, V., Gorokhovski, M., Baricault, N., 2006. The extended IEM mixing model in the framework of the composition PDF approach: applications to diesel spray combustion. Combust. Theory Modell. 10 (1), 155–169.
CrossRef (DOI)
[37] Sanchez-Vila, X., Guadagnini, A., Fernàndez-Garcia, D., 2009. Conditional probability density functions of concentrations for mixing-controlled reactive transport in heterogeneous aquifers. Math. Geosci. 41 (3), 323–351.
CrossRef (DOI)
[38] Srzic, V., Andricevic, R., Gotovac, H., Cvetkovic, V., 2013a. Collapse of higher-order solute concentration moments in groundwater transport. Water Resour. Res. 49 (8), 4751–4764.
CrossRef (DOI)
[39] Srzic, V., Cvetkovic, V., Andricevic, R., Gotovac, H., 2013b. Impact of aquifer heterogeneity structure and local-scale dispersion on solute concentration uncertainty: impact of aquifer heterogeneity on concentration uncertainty. Water Resour. Res. 49 (6), 3712–3728.
CrossRef (DOI)
[40] Suciu, N., 2014. Diffusion in random velocity fields with applications to contaminant transport in groundwater. Adv. Water Resour. 69, 114–133.
CrossRef (DOI)
[41] Suciu, N., Radu, F.A., Attinger, S., Schüler, L., Knabner, P., 2015a. A Fokker-Planck approach for probability distributions of species concentrations transported in heterogeneous media. J. Comput. Appl. Math. 289, 241–252.
CrossRef (DOI)
[42] Suciu, N., Radu, F.A., Prechtel, A., Brunner, F., Knabner, P., 2013. A coupled finite element–global random walk approach to advection-dominated transport in porous media with random hydraulic conductivity. J. Comput. Appl. Math. 246, 27–37.
CrossRef (DOI)
[43] Suciu, N., Schüler, L., Attinger, S., Knabner, P., 2016. Towards a filtered density function approach for reactive transport in groundwater. Adv. Water Resour. 90, 83–98.
CrossRef (DOI)
[44] Suciu, N., Schüler, L., Attinger, S., Vamos, C., Knabner, P., 2015b. Consistency issues in PDF methods. An. St. Univ. Ovidius Constanta, Ser. Mat. 23 (3), 187–208.
CrossRef (DOI)
[45] Suciu, N., Vamos¸ , C., Vanderborght, J., Hardelauf, H., Vereecken, H., 2006. Numerical investigations on ergodicity of solute transport in heterogeneous aquifers. Water Resour. Res. 42 (4), 1–17.
CrossRef (DOI)
[46] Tartakovsky, D.M., Dentz, M., Lichtner, P.C., 2009. Probability density functions for advective-reactive transport with uncertain reaction rates. Water Resour. Res. 45 (7), 1–8.
CrossRef (DOI)
[47] Tennekes, H., Lumley, J.L., 1972. A First Course in Turbulence. MIT Press, Cambridge, Mass.
[48] Vamos¸ , C., Suciu, N., Vereecken, H., 2003. Generalized random walk algorithm for the numerical modeling of complex diffusion processes. J. Comput. Phys. 186 (2), 527–544.
CrossRef (DOI)
[49] Venturi, D., Tartakovsky, D., Tartakovsky, A., Karniadakis, G., 2013. Exact PDF equations and closure approximations for advective-reactive transport. J. Comput. Phys. 243, 323–343.
CrossRef (DOI)
[50] Villermaux, J., Devillon, J.C., 1972. Représentation de la coalescence et de la redispersion des domaines de ségrégation dans un fluide par un modèle d’interaction phénoménologique. In: Proceedings of the 2nd International Symposium on Chemical Reaction Engineering. Elsevier, New York, pp. 1–13. WWAP, 2012. The United Nations World Water Development Report 4: Managing Water under Uncertainty and Risk, no. Vol. 1 in World Water Assessment Programme. Unesco, Paris.
[51] Yee, E., Chan, R., 1997. Comments on ”Relationships between higher moments of concentration and of dose in turbulent dispersion”. Bound-Lay Meteorol 82 (2), 341–351.
CrossRef (DOI)