Fluid flow in rock fractures: From the Navier-Stokes equations to the cubic law

The mathematical analysis of the flow of a single-phase Newtonian fluid through a rough-walled rock fracture is reviewed, starting with the Navier-Stokes equations. By a combination of order-of-magnitude analysis, appeal to available analytical solutions, and reanalysis of some data from the literature, it is shown that the Navier-Stokes equations can be linearized if the Reynolds number is less than about 10. Further analysis shows that the linear Stokes equations can be replaced by the simpler Reynolds lubrication equation if the wavelength of the dominant aperture variations is about three times greater than the mean aperture. However, this criterion does not seem to be strongly obeyed by all fractures. The Reynolds equation (i.e., the local cubic law) may therefore suffice in estimating fracture permeabilities to within a factor of about 2, but more accurate estimates will require solution of the Stokes equations. Similarly, estimates of mean aperture based on inverting transmissivity data may have errors of a factor of two if any version of the local cubic law is used to relate transmissivity to mean aperture.

[1]  Pierre M. Adler,et al.  Permeability of a Single Fracture; Validity of the Reynolds Equation , 1995 .

[2]  Robert W. Zimmerman,et al.  Effect of shear displacement on the aperture and permeability of a rock fracture , 1998 .

[3]  Robert W. Zimmerman,et al.  The effect of contact area on the permeability of fractures , 1989 .

[4]  George Keith Batchelor,et al.  An Introduction to Fluid Dynamics. , 1969 .

[5]  J. Gudmundsson,et al.  High-velocity flow in a rough fracture , 1999, Journal of Fluid Mechanics.

[6]  J. Wan,et al.  Water film flow along fracture surfaces of porous rock , 1997 .

[7]  J. S. Y. Wang,et al.  Validity of cubic law for fluid flow in a deformable rock fracture. Technical information report No. 23 , 1979 .

[8]  Gudmundur S. Bodvarsson,et al.  Lubrication theory analysis of the permeability of rough-walled fractures , 1991 .

[9]  L. J. Pyrak-Nolte,et al.  The fractal geometry of flow paths in natural fractures in rock and the approach to percolation , 1989 .

[10]  J. Maxwell.  A Treatise on Electricity and Magnetism , 1873, Nature.

[11]  Chiang C. Mei,et al.  The effect of weak inertia on flow through a porous medium , 1991, Journal of Fluid Mechanics.

[12]  Gudmundur S. Bodvarsson,et al.  Hydraulic conductivity of rock fractures , 1996 .

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

[14]  J. Aitchison,et al.  The Lognormal Distribution. , 1958 .

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

[16]  Stephen R. Brown,et al.  Applicability of the Reynolds Equation for modeling fluid flow between rough surfaces , 1995 .

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

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

[19]  Y. Tsang,et al.  On Two-Phase Relative Permeability and Capillary Pressure of Rough-Walled Rock Fractures , 1989 .

[20]  Peter K. Kitanidis,et al.  Stokes Flow in a Slowly Varying Two-Dimensional Periodic Pore , 1997 .

[21]  Brian P. Bonner,et al.  Self‐propping and fluid flow in slightly offset joints at high effective pressures , 1994 .

[22]  Katsuhiko Iwai,et al.  Fundamental studies of fluid flow through a single fracture , 1976 .

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

[24]  Gedeon Dagan,et al.  Higher-order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution , 1993 .

[25]  Derek Elsworth,et al.  Laboratory assessment of the equivalent apertures of a rock fracture , 1993 .

[26]  P. A. Witherspoon,et al.  Hydromechanical behavior of a deformable rock fracture subject to normal stress , 1981 .

[27]  Peter K. Kitanidis,et al.  Adaptive finite element simulation of Stokes flow in porous media , 1998 .