Joel Koplik, City College of New York

Surface elevation plot (top) and contour plot (left) of a typical self-affine fractal surface with Hurst exponent 0.8. The velocity field in a simulated fracture consisting of two such surfaces separated vertically by one-eighth of the lateral size is shown at right; the arrows represent the deviation of the local velocity from the mean.

Research Objectives
(1) Complete our numerical simulations and analytic estimates which aim to relate permeability, dispersivity, and more detailed characterizations of fluid transport to the fractal parameters of the three-dimensional fractured medium. (2) Consider the evolution of the pore space and fracture walls when small colloidal suspended particles in the flow are allowed to deposit along the walls, gradually shrinking the pore space. (3) Consider the dynamics of finite-sized suspended particles, large enough to alter the flow field but still small enough to be sensitive to the surface roughness. (4) Consider flow of fluids containing either passive tracers or suspended particles in fracture junctions, with the aim of understanding how fluids and particulates choose among flow channels, and studying the persistence of correlations between different parts of a fracture network.

Computational Approach
The computations are based on the lattice Boltzmann method for fluid flows. In this algorithm, fictitious particles move from node to node on a regular lattice with certain rules for collisions, which are designed so that the average motion of the lattice particles reproduce the appropriate solutions of the continuum Navier-Stokes equations. Solid surfaces correspond to nodes from which the particles are reflected, so that walls and particulates may be included. More elaborate rules allow one to simulate non-Newtonian fluids as well.

Accomplishments
We considered the fluid permeability and the dispersion of a passive tracer in a rough-walled fracture, modeled as the gap between two complementary self-affine surfaces rigidly translated with respect to each other. When the mean gap is large compared to the range of the surface height fluctuations, the effect of roughness on permeability is given by a systematic perturbation expansion, whose leading correction is readily expressed in terms of the Hurst exponent characterizing the fracture surface, and which agrees very well with the results of numerical simulations using the lattice Boltzmann method. These results go beyond the common lubrication approximation, because we find that an important ingredient in permeability reduction is the presence of stagnant zones of fluid in the roughness interstices, an effect not allowed for in lubrication.

In the opposite limit of a narrow gap, we began with the two-dimensional case first, where tortuosity effects dominate. Straightforward arguments based on dividing the flow path into decorrelated segments, and supported by numerical simulations, provide a relation between the permeability, the mean aperture, and the fractal exponent of the surface. The ensemble averaged results agree with theoretical scaling predictions for the variation of the effective dispersivity with fracture geometry and transit distance. The dispersivity shows strong anisotropic effects, and furthermore the dispersion front progressively wrinkles into a self-affine curve with a predictable dimension. We then revisited the two-dimensional case and used tortuosity-based arguments to obtain a relation between the surface exponent and the resulting change in dispersivity. Again, lattice-Boltzmann simulations were in agreement with the analytic arguments, but in this case the method required the development of a new concentration boundary condition.

Significance
The general goal of this work is to enhance our understanding of the motions of fluids and particles in geological formations, with application to water supply, hydrocarbon production, and waste disposal.

Publications
G. Drazer and J. Koplik, "Tracer dispersion in two dimensional rough fractures," Phys. Rev. E 63, 056104 (2001).

G. Drazer and J. Koplik, "Permeability of self-affine rough fractures," Phys. Rev. E 62, 8076 (2000).

J. Lee and J. Koplik, "Network models of deep bed filtration," Phys. Fluids 13, 1076 (2001).