In this study we present a numerical analysis for the self-averaging of the longitudinal dispersion coefficient for transport in heterogeneous media. This is done by investigating the mean-square sample-to-sample fluctuations of the dispersion for finite times and finite numbers of modes for a random field using analytical arguments as well as numerical simulations. We consider transport of point-like injections in a quasi-periodic random field with a Gaussian correlation function. In particular, we focus on the asymptotic and pre-asymptotic behaviour of the fluctuations with the aid of a probability density function for the dispersion, and we verify the logarithmic growth of the sample-to-sample fluctuations as earlier reported in Eberhard (2004 J. Phys. A: Math. Gen. 37 2549–71). We also comment on the choice of the relevant parameters to generate quasi-periodic realizations with respect to the self-averaging of transport in statistically homogeneous Gaussian velocity fields.
Simulation in Technology, University of Heidelberg, Germany
Institute of Applied Mathematics, University of Erlangen-Nuremberg, Germany
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
J. Eberhard, N. Suciu, C. Vamoş (2007), On the self-averaging of dispersion for transport in quasi-periodic random media, J. Phys. A: Math. Theor., 40, 597-610, doi: 10.1088/1751-8113/40/4/002
see the expanding block below
J. Phys. A: Math. Theor.
Not available yet.
Not available yet.
Google Scholar Profile
google scholar profile
 M. Clincy and H. Kinzelbach. Stratified disordered media: exact solutions for transport parameters and their self-averaging properties. J. Phys. A: Math. Gen., 34:7141–7152, 2001.
 G. Dagan. Theory of solute transport by groundwater. Water Resour. Res., 19:183–215, 1987.
 G. Dagan. Flow and Transport in Porous Formations. Springer Verlag, Berlin, 1989.
 M. Dentz, H. Kinzelbach, S. Attinger, and W. Kinzelbach. Temporal behavior of a solute cloud in a heterogeneous porous medium 1. Point-like injection. Water Resour. Res., 36(12):3591–3604, 2000.
 M. Dentz, H. Kinzelbach, S. Attinger, and W. Kinzelbach. Temporal behavior of a solute cloud in a heterogeneous porous medium 3. Numerical simulations. Water Resour. Res.,
 M. Dentz, H. Kinzelbach, S. Attinger, and W. Kinzelbach. Numerical studies of the transport behavior of a passive solute in a two-dimensional incompressible random flow field. Phys.
Rev. E, 67(046306):1–10, 2003.
 J. Eberhard. Approximations for transport parameters and self-averaging properties for pointlike injections in heterogeneous media. J. Phys. A: Math. Gen, 37:2549–2571, 2004.
 C. W. Gardiner. Handbook of Stochastic Methods (for Physics, Chemistry and Natural Science). Springer, New York, 1985.
 L. W. Gelhar. Stochastic Subsurface Hydrology. Prentice Hall, New Jersery, 1993.
 E. Guyon, C. D. Mitescu, J. Hulin, and S. Roux. Fractals and percolution in porous media and flows? Physica D, 38:172–178, 1989.
 U. Jaekel and H. Vereecken. Renormalization group analysis of macrodispersion in a directed random flow. Water Resour. Res., 33(10):2287–2299, 1997.
 I. Jankovic, A. Fiori, and G. Dagan. Flow and transport in highly heterogeneous formations: 3. Numerical simulations and comparison with theoretical results. Water Resour. Res., 39(9):16.1–16.13, 2003.
 H. Kesten and G. C. Papanicolaou. A limit theorem for turbulent diffusion. Commun. Math. Phys., 65:97–128, 1979.
 R. Lenormand and C. Zarcone. Invasion percolation in an etched network: measurement of a fractal dimension. Physical Review Letters, 54(20):2226–2229, 1985.
 W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C: the art of scientific computing. Cambridge University Press, Cambridge, UK, 1992.
 H. Schwarze, U. Jaekel, and H. Vereecken. Estimation of macrodispersion by different approximation methods for flow and transport in randomly heterogeneous media. Transport
in Porous Media, 43(2):265–287, 2001.
 N. Suciu, C. Vamos, and J. Eberhard. Evaluation of the first-order approximations for transport in heterogeneous media. Water Resour. Res., 2006.
CrossRef (DOI) (in press).
 N. Suciu, C. Vamo¸s, J. Vanderborght, H. Hardelauf, and H. Vereecken. Numerical investigations on ergodicity of solute transport in heterogeneous aquifers. Water Resour. Res., 2006. 42,W04409,
 M. G. Trefry, F. P. Ruan, and D. McLaughlin. Numerical simulations of preasymptotic transport in heterogeneous porous media: Departures from the Gaussian limit. Water Resour. Res., 39(3):10:1–12, 2003.
 N. G. van Kampen. Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam, 1981.
 C. L. Winter, C. M. Newman, and S. P. Neuman. A perturbation expansion for diffusion in random velocity field. SIAM J. Appl. Math., 44(2):411–424, 1984.