Large-Scale Modeling of Multi-Phase Flows in Heterogeneous Porous Media

The purpose of this work is to study diffusion in a disordered medium which is characterized by a distribution of local conductances that contains long-range correlations. Our main interest in this problem is to understand whether diffusion of pollutants in soil can be modelled by the classical diffusion equation with a constant diffusivity, or whether the presence of the correlations gives rise to a nonlocal transport process at the macroscopic level that cannot be represented by the diffusion equation. In the latter case, one must develop the appropriate transport equation.

Principal Investigator

Mark Knackstedt
Applied Mathematics


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Facilities Used



Christoph Arns
Stephen Hyde
Viet Nguyen
Mohammad Saadatfar
Arthur Sakalleriou
Adrian Sheppard
Applied Mathematics

Ji-Youn Lee
Walid Mahmud
Val Pinczewski
Rob Sok
Petroleum Engineering
University of NSW

Lincoln Paterson
CSIRO Petroleum Resources
CSIRO and Bureau of Meteorology

Brent Lindquist
New York, USA

RFCD Codes

260204, 240205, 240503, 290703, 240202, 291406

Significant Achievements, Anticipated Outcomes and Future Work

The correlations are generated by a fractional Brownian motion characterized by a Hurst exponent H. For H>1/2 the correlations appear to have no effect, and the transport process is diffusive. However, for H<1/2 and depending on the morphology of the medium, three distinct types of transport processes, namely, anomalous, Fickian, and superdiffusive transport may emerge. Moreover, if the medium is anisotropic and stratified, biased diffusion in it is characterized by power-law growth of the mean square displacements with the time in which the effective exponents characterizing the power law oscillates log-periodically with the time. This result cannot be predicted by any of the currently-available continuum theories of transport in disordered media.


Computational Techniques Used

We used the power spectrum method to generate the FBM distribution. All of the results presented in this paper were obtained with 128 x 128 x 128 lattices. This size of the lattice gives rise to a conductance distribution the broadness of which is about 2-3 orders of magnitude variations in the bonds' conductances. Periodic boundary conditions were used in all the directions. The diffusion process was simulated by the random walk of a particle which is initially (at time t=0) inserted into the lattice at a randomly selected site. The particle executes a random walk between the nearest neighbor sites of the lattice. Each step of the walk from one site to another is taken with a transition probability proportional to the conductance of the bond between the two sites. After each step, the time t is increased by one unit. The mean square displacements (MSD) R^2(t) of the walkers at time t are computed, where the averaging is taken over the initial positions of the walkers, and the different realizations of the lattice. Typically, we used 4,000 walkers (i.e., 4,000 initial positions) and 40 realizations of each of the conductance distributions. All the random walkers took 2 x 10^6 steps, (i.e., the MSD were computed up to time t=2 x 10^6).


Publications, Awards and External Funding

M. Saadatfar M. Sahimi , Diffusion in Disordered Media with Long-Range Correlations: Anomalous, Fickian and Superdiffusive Transport and Log-Periodic Oscillations, Phys. Rev. E 65, 2002, 0261XX-1