Abstract
A stochastic numerical method is developed for simulation of flows and particle transport in a 2D layer of porous medium. The hydraulic conductivity is assumed to be a random field of a given statistical structure, the flow is modeled in the layer with prescribed boundary conditions. Numerical experiments are carried out by solving the Darcy equation for each sample of the hydraulic conductivity by a direct solver for sparse matrices, and tracking Lagrangian trajectories in the simulated flow. We present and analyze different Eulerian and Lagrangian statistical characteristics of the flow such as transverse and longitudinal velocity correlation functions, longitudinal dispersion coefficient, and the mean displacement of Lagrangian trajectories. We discuss the effect of long-range correlations of the longitudinal velocities which we have found in our numerical simulations. The related anomalous diffusion is also analyzed.
Similar content being viewed by others
References
Ababou R., McLaughlin D., Gelhar L.W., Tompson A.F.B.: Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media. Transp. Porous Media 4(6), 549–565 (1989)
Dagan G.: Flow and Transport in Porous Formations. Springer-Verlag, Berlin, Heidelberg, Germany (1989)
Dagan G.: Spatial moments, Ergodicity, and Effective Dispersion. Water Resour. Res. 26(6), 1281–1290 (1990)
Dentz M., Kinzelbach H., Attinger S., Kinzelbach W.: Temporal behaviour of a solute cloud in a heterogeneous porous medium. 3. Numerical simulations. Water Resour. Res. 38(7), 1118–1130 (2002)
Dreuzy J.-R., Beaudoin A., Erhel J.: Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations. Water Resour. Res. 43, W10439 (2007)
Freeze R.A.: A stochastic-conceptual analysis in groundwater flow in non-uniform, homogeneous media. Water Resour. Res. 11(5), 725–741 (1975)
Gelhar L.W.: Stochastic Subsurface Hydrology. Prentice-Hall, Englewood Cliffs, N.J (1993)
Ghanem R.G., Spanos P.D.: Stochastic Finite Elements. A Spectral Approach. Courier Dover Publications, Chemsford (2003)
Glimm J., Sharp D.H.: A random field model for anomalous diffusion in heterogeneous porous media. J. Stat. Phys. 62(N1/2), 415–424 (1991)
Kraichnan R.H.: Diffusion by a random velocity field. Phys.Fluids 13(N1), 22–31 (1970)
Kramer P., Kurbanmuradov O., Sabelfeld K.: Comparative analysis of multiscale Gaussian random field simulation algorithms. J. Comput. Phys. 226, 897–924 (2007)
Kurbanmuradov O., Sabelfeld K., Koluhin D.: Stochastic Lagrangian models for two-particle motion in turbulent flows. Numerical results. Monte Carlo Methods Appl 3(3), 199–223 (1997)
Mikhailov G.A.: Approximate models of random processes and fields. Russ. J. Comput. Math. Math. Phys. 23(3), 558–566 (1983) (in Russian)
Monin A.S., Yaglom A.M.: Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 1. The MIT Press, Cambridge (1971)
Pollock D.W.: Semianalytical computation of path lines for finite-difference models. Ground Water Res. 28, 743–750 (1988)
Sabelfeld K.K.: Monte Carlo Methods in Boundary Value Problems. Springer-Verlag, Berlin – Heidelberg – New York (1991)
Sabelfeld K.: Stokes flows under random boundary velocity excitations. J. Stat. Phys. 132(6), 1071–1095 (2008)
Sabelfeld K., Kolyukhin D.: Stochastic Eulerian model for the flow simulation in porous media. Monte Carlo Methods and Applications. 9(3), 271–290 (2003)
Salandin P., Fiorotto V.: Solute transport in highly heterogeneous aquifers. Water Resour. Res 34, 949–961 (1998)
Samarskii A.A., Nikolaev E.S.: Numerical Methods for Grid Equations. Birkhauser, Basel (1989)
Schwarze H., Jaekel U., Vereecken H.: Estimation of macrodispersion by different approximation methods for flow and transport in randomly heterogeneous media. Transp. Porous Media 49(2), 267–287 (2001)
Smith L., Freeze R.A.: Stochastic analysis of steady state groundwater flow in a bounded domain, 2. Two-dimensional simulation. Water Resour. Res. 15(6), 1543–1559 (1979)
Suciu N., Vamos C., Vereecken H., Sabelfeld K., Knabner P.: Memory effects induced by dependence on initial conditions and ergodicity of transport in heterogeneous media. Water Resour. Res. 44, W08501 (2008). doi:10.1029/2007WR006740
Sun L., Peng C., Liu H., Hu Y.: Analogy in the adsorption of random copolymers and homopolymers at solid-liquid interface: a Monte Carlo simulation study. J. Chem. Phys 126, 094905 (2007). doi:10.1063/1.2567301
Tompson A.F.B., Gelhar L.W.: Numerical simulation of solute transport in tree-dimensional randomly heterogeneous porous media. Water Resour. Res. 26(10), 2541–2562 (1990)
Trefry M.G., Ruan F.P., McLaughlin D.: Numerical simulations of preasimptotic transport in hetoregenous porous media: departures from the Gaussian limit. Water Resour. Res. 39(3), 1063–1077 (2003)
Xiu D., Karniadakis G.E.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)
Yang J., Zhang D., Lu Z.: Stochastic analysis of saturated-unsaturated flow in heterogeneous media by combining Karhunen-Loeve expansion and perturbation method. J. Hydrol. 294, 18–38 (2004)
Zhang D., Lu Z.: An efficient, high-order perturbation approach for flow in random porous media via Karhunen–Loeve and polynomial expansions. J. Comput. Phys. 194, 773–794 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kurbanmuradov, O.A., Sabelfeld, K.K. Stochastic Flow Simulation and Particle Transport in a 2D Layer of Random Porous Medium. Transp Porous Med 85, 347–373 (2010). https://doi.org/10.1007/s11242-010-9567-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-010-9567-y