Lubrication theory analysis of the permeability of rough-walled fractures

Abstract Lubrication theory is used to study the permeability of rough-walled rock fractures. In this approximation, which is valid for low Reynolds numbers and under certain restrictions on the magnitude of the roughness, the Navier-Stokes equations that govern fluid flow are reduced to the more tractable Reynolds equation. An idealized model of a fracture, in which the roughness follows a sinusoidal variation, is studied in detail. This fracture is considered to consist of a random mixture of elements in which the fluid flows either parallel or transverse to the sinusoidal bumps. The overall permeability is then found by a suitable averaging procedure. The results are similar to those found by other researchers from numerical analysis of the Reynolds equation, in that the ratio of the hydraulic aperture to the mean aperture correlates well with the ratio of the mean aperture to the standard deviation of the aperture. Higher-order approximations to the Navier-Stokes equations for flow between sinusoidal walls are then studied, and it is concluded that in order for the lubrication approximation to be valid, the fracture walls must be smooth over lengths on the order of one standard deviation of the aperture, which is much less restrictive a condition than had previously been thought to apply.

[1]  Gedeon Dagan,et al.  Models of groundwater flow in statistically homogeneous porous formations , 1979 .

[2]  J. B. Walsh,et al.  EFFECT OF PORE PRESSURE AND CONFINING PRESSURE ON FRACTURE PERMEABILITY , 1981 .

[3]  Gudmundur S. Bodvarsson,et al.  Fractal study and simulation of fracture roughness , 1990 .

[4]  Stephen R. Brown,et al.  Broad bandwidth study of the topography of natural rock surfaces , 1985 .

[5]  H. Cheng,et al.  An Average Flow Model for Determining Effects of Three-Dimensional Roughness on Partial Hydrodynamic Lubrication , 1978 .

[6]  C. Pozrikidis.  Creeping flow in two-dimensional channels , 1987, Journal of Fluid Mechanics.

[7]  K. Raven,et al.  Interpretation of field tracer tests of a single fracture using a transient solute storage model , 1988 .

[8]  W. E. Hughes,et al.  An Introduction to Viscous Flow , 1980 .

[9]  E. Hasegawa,et al.  On Steady Flow through a Channel Consisting of an Uneven Wall and a Plane Wall : Part 1. Case of No Relative Motion in Two Walls , 1983 .

[10]  Michel Deville,et al.  Slow viscous flow , 1964 .

[11]  I. S. Gradshteyn.  Table of Integrals, Series and Products, Corrected and Enlarged Edition , 1980 .

[12]  T. Madden.  Random Networks and Mixing Laws , 1976 .

[13]  D. T. Snow.  ROCK FRACTURE SPACINGS, OPENINGS, AND POROSITIES , 1968 .

[14]  M. V. Dyke,et al.  Slow variations in continuum mechanics , 1987 .

[15]  Stephen R. Brown,et al.  Transport of fluid and electric current through a single fracture , 1989 .

[16]  Stephen R. Brown,et al.  Fluid flow through rock joints: The effect of surface roughness , 1987 .

[17]  R. E. Goodman,et al.  Characterization of rock fissure hydraulic conductivity using idealized wall roughness profiles , 1986 .

[18]  Stephen E. Silliman,et al.  An interpretation of the difference between aperture estimates derived from hydraulic and tracer tests in a single fracture , 1989 .

[19]  J. V. Tracy,et al.  Flow through fractures , 1981 .

[20]  S. Shtrikman,et al.  A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials , 1962 .