LargeScale Modeling of MultiPhase 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 longrange 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 
Project w09, d59 Facilities Used VPP, PC, SC 
CoInvestigators Christoph ArnsStephen Hyde Viet Nguyen Mohammad Saadatfar Arthur Sakalleriou Adrian Sheppard Applied Mathematics RSPhysSE ANU
JiYoun Lee
Lincoln Paterson
Brent Lindquist

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 powerlaw growth of the mean square displacements with the time in which the effective exponents characterizing the power law oscillates logperiodically with the time. This result cannot be predicted by any of the currentlyavailable 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 23 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 LongRange Correlations: Anomalous, Fickian and Superdiffusive Transport and LogPeriodic Oscillations, Phys. Rev. E 65, 2002, 0261XX1