Abstract
A two-dimensional numerical model is used to study the nonlinear behavior of density gradients on transverse dispersion. Numerical simulations are conducted using d 3 f, a computer code for simulation of density-dependent flow in porous media. Considering a density-stratified horizontal flow in a heterogeneous porous media, a series of simulations is carried out to examine the effect of the density gradient on macro-scale transverse dispersivity. Changing salt concentration significantly affects fluid properties. This physical behavior of the fluid involves a non-linearity in modeling the interaction between salt and fresh water. It is concluded that the large-scale transport properties for high density flow deviate significantly from the tracer case due to the spatial variation of permeability, described by statistical parameters, at the local-scale. Indeed, the presence of vertical flow velocities induced by permeability variations is responsible for the reduction of the mixing zone width in the steady state in the case of a high density gradient. Uncertainties in the model simulations are studied in terms of discretization errors, boundary conditions, and convergence of ensemble averaging. With respect to the results, the gravity number appears to be the controlling parameter for dispersive flux. In addition, the applicability and limitations of the nonlinear model of Hassanizadeh (1990) and Hassanizadeh and Leijnse (1995) (Adv Water Resour 18(4):203–215, 1995) in heterogeneous porous media are investigated. We found that the main cause of the nonlinear behavior of dispersion, which is the interaction between density contrast and vertical velocity, needs to be explicitly accounted for in a macro-scale model.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bastian P. et al.: UG—a flexible software toolbox for solving partial differential equations. Comput. Visual. Sci. 1, 27–40 (1997)
Bellin A., Salandin P., Rinaldo A.: Simulation of dispersion in heterogeneous porous formations: statistics, first-order theories, convergence of computations. Water Resour. Res. 28(9), 2211–2227 (1992)
Burr D.T., Sudicky E.A., Na R.L.: Nonreactive and reactive solute transport in three-dimensional heterogeneous porous media: mean displacement, plume spreading, and uncertainty. Water Resour. Res. 30(3), 791–815 (1994)
Chaudhuri A., Sekhar M.: Stochastic modeling of solute transport in 3-D heterogeneous porous media with random source condition. Stoch. Environ. Res. Risk Assess. 21, 159–173 (2006)
Cirpka O.A., Attinger S.: Effective dispersion in heterogeneous media under random transient flow conditions. Water Resour. Res. 39(9), 1257 (2003)
Dagan G.: Time-dependent macrodispersion for solute transport in anisotropic heterogeneous aquifers. Water Resour. Res. 24(9), 1491–1500 (1988)
Dagan G.: Flow and Transport in Porous Formations. Springer, Berlin (1989)
Diersch H.-J., Kolditz O.: variable-density flow and transport in porous media: approaches and challenges. Adv. Water Resour. 25(8–12), 899–944 (2002)
Egorov A.G., Demidov D.E., Schotting R.J.: On the interaction between gravity forces and dispersive brine fronts in micro-heterogeneous porous media. Adv. Water Resour. 28(1), 55–68 (2005)
Fein E.: d 3 f—ein Programmpaket zur Modellierung von Dichtestromungen. Tech. Rep. GRS-139, GRS, Braunschweig, Germany (1998)
Frolkovic, P.: Consistent velocity approximation for density driven flow and transport. In: Advanced Computational Methods in Engineering, pp. 603–611. Shakeer Publishing, Maastrict (1998)
Gelhar L.W., Axness C.L.: Three dimensional stochastic analysis of macrodispersion in aquifers. Water Resour. Res. 19(1), 161–180 (1983)
Gelhar L.W.: Stochastic subsurface hydrology from theory to applications. Water Resour. Res. 22, 1355–1453 (1986)
Grane F.E., Gardner G.H.F.: Measurements of transverse dispersion in granular media. J. Chem. Eng. Data 6(2), 283–287 (1961)
Ham P.A.S., Schotting R.J., Prommer H., Davis G.B.: Effects of hydrodynamic dispersion on plume lengths for instantaneous biomolecular reactions. Adv. Water Resour. Res. 27, 803–813 (2004)
Hassanizadeh S.M.: Derivation of basic equations of mass transport in porous media. Part 2. Generalized Darcy’s and Fick’s laws. Adv. Water Resour. 9, 207–222 (1986)
Hassanizadeh, S.M.: Experimental study of coupled flow and mass transport: a model validation exercise. In: Calibration and reliability in groundwater modelling, Proceedings of the Model Care Conference, IAHS publication No. 195, pp 241–250 (1990)
Hassanizadeh S.M., Leijnse A.: On the modeling of brine transport in porous media. Water Resour. Res. 24(3), 321–330 (1988)
Hassanizadeh S.M., Leijnse A.: A non-linear theory of high-concentration gradient dispersion in porous media. Adv. Water Resour. 18(4), 203–215 (1995)
Hassanizadeh, S.M., Leijnse, A., de Vries, W.J., Stapper, R.A.M.: Experimental study of brine transport in porous media. RIVM Report 728514005, Bilthoven, The Netherlands (1990)
Jiao C.-Y., Hotzl H.: An experimental study of miscible displacements in porous media with variation of fluid density and viscosity. Transp. Porous Media 54, 125–144 (2004)
Johannsen K.: On the validity of the Boussinesq approximation for the Elder problem. Computat. Geosci. 7(3), 169–182 (2003)
Johannsen K.: Numerische Aspekte dichtegetriebener Stroemung in poroesen Medien. Heidelberg University, Habilitationsschrift (2004)
Johannsen K., Kinzelbach W., Oswald S., Wittum G.: The saltpool benchmark problem—numerical simulation of saltwater upconing in a porous medium. Adv. Water Resour. 25(3), 335–345 (2002)
Johannsen K., Oswald S., Held R., Kinzelbach W.: Numerical simulation of three-dimensional saltwater-freshwater fingering instabilities observed in a porous medium. Adv. Water Resour. 29(11), 1690–1704 (2006)
Landman, A. J.: Analysis of physical mechanisms underlying density-dependent transport in porous media. Utrecht University (2005)
Landman A.J., Johannsen K., Schotting R.J.: Density-dependent dispersion in heterogeneous porous media. Part I: A numerical study. Water Resour. Res. 30(12), 2467–2480 (2007a)
Landman A.J., Schotting R.J., Egorov A., Demidov D.: Density-dependent dispersion in heterogeneous porous media Part II: Comparison with nonlinear models. Water Resour. Res. 30(12), 2481–2498 (2007b)
Leroy C., Hulin J.P., Lenormand R.: Tracer dispersion in stratified porous media: influence of transverse dispersion and gravity. J. Contaminant Hydrol. 11, 51–68 (1992)
McLaughlin D., Ruan F.: Macrodispersivity and large-scale hydrogeologic variability. Transp. Porous Media 42(1–2), 133–154 (2001)
Moser, H.: Einfluß der Salzkonzentration auf die hydrodynamische Dispersion im porösen Medium. Mitteilungen des Institutes für Wasserbau u. Wasserwirtschaft. Mitteilung nr. 128, TU Berlin
Oberkampf W.L., Trucano T.G.: Verification and validation in computational fluid dynamics. Prog. Aerospace Sci. 38, 209–272 (2002)
Pickens J.F., Grisak G.E.: Scale-dependent dispersion in a stratified granular aquifer. Water Resour. Res. 17(4), 1191–1211 (1981)
Robin M.J.L., Gutjar A.L., Sudicky E.A., Wilson J.L.: Cross-correlated random field generation with the direct Fourier transform method. Water Resour. Res. 29(7), 2385–2397 (1993)
Roy C.J.: Review of code and solution verification procedures for computational simulation. J. Comput. Phys. 205(1), 131–156 (2005)
Schotting, R., Landman, A.L.: Towards a physically based theory of high-concentration-gradient dispersion in porous media. Experimental, theoretical and numerical studies. In: Ingham, D.B., et al. (eds.) Emerging Technologies and Techniques in Porous Media, NATO Science Series II, vol. 134, pp. 321–336. Kluwer Academic. Publishers, Dordrecht (2004)
Schotting R.J., Moser H., Hassanizadeh S.M.: High-concentration-gradient dispersion in porous media: experiments, analysis and approximations. Adv. Water Resour. 22(7), 665–680 (1999)
Scheidegger A.E.: General theory of dispersion in porous media. J. Geophys. Res. 66, 3273–3278 (1961)
Simmons C.T.: Variable density groundwater flow: From current challenges to future possibilities. Hydrogeol. J. 13, 116–119 (2005)
Starke, B.: Experimental and numerical investigations of macrodispersion of density-dependent flow and transport in stochastic porous media. PhD- Dissertation (in German), University Kassel, 199 pp (2005)
Starke, B., Koch, M.: Laboratory experiments and Monte Carlo simulations to validate a stochastic theory of density-dependent macrodispersion, CMWRXVI (2006)
Thiele M.: Gravity affected lateral dispersion and diffusion in a stationary horizontal porous medium shear flow. Transp. Porous Media 26, 553 (1997)
Thiele M.: Stationary transverse diffusion in a horizontal porous medium boundary-layer flow with concentration-dependent fluid density and viscosity. Transp. Porous Media 36, 341–355 (1999)
Tompson A.F.B., Gelhar L.W.: Numerical simulation of solute transport in three-dimensional, randomly heterogeneous porous media. Water Resour. Res. 26(10), 2541–2562 (1990)
Watson S.J., Barry D.A., Schotting R.J., Hassanizadeh S.M.: On the validity of Darcy’s law for stable high-concentration displacements in granular porous media. Transp. Porous Media 47, 149–167 (2002a)
Watson S.J., Barry D.A., Schotting R.J., Hassanizadeh S.M.: Validation of classical density-dependent solute transport theory for stable, high-concentration gradient brine displacements in coarse and medium sands. Adv. Water Resour. 25, 611–635 (2002b)
Welty C., Gelhar L.: Stochastic analysis of the effect of fluid density and viscosity variability on macrodispersion in heterogeneous porous media. Water Resour. Res. 27(8), 2061–2075 (1991)
Welty C., Kane A.C. III, Kaufman L.J.: Stochastic analysis of transverse dispersion in density-coupled transport in aquifers. Water Resour. Res. 39(6), 1150 (2003)
Yates S.R.: An analytical solution for one-dimensional transport in heterogeneous porous media. Water Resour. Res. 26(10), 2331–2338 (1990)
Acknowledgements
We wish to thank Majid Hassanizadeh, Rainer Helmig, and Robert Zimmerman for their scientific feedback and comments. We thank IWR in Heidelberg for using their cluster.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Nick, H.M., Schotting, R., Gutierrez-Neri, M. et al. Modeling Transverse Dispersion and Variable Density Flow in Porous Media. Transp Porous Med 78, 11–35 (2009). https://doi.org/10.1007/s11242-008-9277-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-008-9277-x