[Above] Example of the geometry and velocity field in a 2D fracture with one planar and one self-affine surface of roughness exponent H = 0.8. The enlargements show the difference in the velocity decay near smooth and rough boundaries.
[Above] Flow field in a narrow self-affine fracture with a constant gap, and Hurst exponent H = 0.8. The vertical aperture is constant everywhere, but the effective local aperture for fluid flow, that is, the local width of the channel normal to the mean flow direction, strongly depends on the local angle between the surface and the mean plane. The enlargements illustrate the effect of the effective aperture on the flow field.

Joel Koplik, German Drazer, and Igor Baryshev, City College of the City University of New York

Research Objectives
We are concerned with flow, passive tracer dispersion, and depositional processes in the self-affine fractures often observed in naturally fractured rock. In such systems the rock surfaces have long-range correlations which have been shown to significantly enhance and modify the dynamics of passive tracers used as diagnostic tools. We further wish to examine the motion of solids suspended in flowing fluid in fractured geological formations, and in particular study the evolution of the fractured pore space as deposition occurs.

Computational Approach
The problems of flow, tracer motion and particle deposition in self-affine fractures require an efficient method for solving the Navier-Stokes and convection-diffusion equations in a highly irregular and evolving geometry. The lattice Boltzmann method is optimal for problems in complicated regions because the core of the calculation is the motion of particles in the region’s interior, with an adjustment to the motion when a boundary is reached. As the pore space evolves, only the geometrical specification of the solid region is needed. Other workers have used this method for both active and passive tracer dispersion studies, as well as the somewhat analogous problem of suspension dynamics.

Accomplishments
One of last year’s two projects, simulations of deep-bed filtration in a statistically homogeneous porous medium, was completed. The second project on fracture flows continued this year. To date, we have developed scaling relations for the permeability of 2D self-affine fractures and verified them by numerical simulations.

Significance
The efficient extraction of water and hydrocarbon resources from underground reservoirs, as well as the use of underground formations as waste disposal sites, requires a full understanding of the dynamics of the flow of fluids and various suspended matter in the disordered porous media which comprise geological formations and reservoirs. The proposed research looks at the transport and deposition of solid particulates which may clog or perhaps break open new flow channels in these systems, and the effects of subtle correlations resulting from fracture processes on tracer tests. In addition, some of the results are relevant to commercial filtration processes used in purification and manufacture.

Publications
J. Lee and J. Koplik, “Network model for deep-bed filtration,” Phys. Fluids (submitted, 2000).

G. Drazer and J. Koplik, “Permeability of self-affine rough fractures,” Phys. Rev. E (submitted, 2000). E-print cond-mat/0006287.

J. Lee and J. Koplik, “Microscopic motion of particles passing through a porous medium,” Phys. Fluids 11, 76 (1999).