HR: 08:15h
AN: H51B-02 INVITED     [PDF]
TI: Fluid Flow in Rock Fractures
AU: * Zimmerman, R W
EM: r.w.zimmerman@ic.ac.uk
AF: Department of Earth Science and Engineering, Imperial College, London, SW7 2AZ United Kingdom
AB: Understanding the hydraulic properties of rock fractures is an issue of great importance in fields such as petroleum engineering, groundwater hydrology, and underground waste isolation. Traditionally, models of fluid flow through rock fractures have been based on the cubic law, which asserts that the local value of the fracture transmissivity is equal to h$^{3}$/12, where h is the fracture aperture. The local cubic law is mathematically equivalent to assuming that flow in the fracture is governed by the Reynolds lubrication equation. However, this equation is only applicable if the aperture does not change too abruptly, and flowrates are suitably low. These two limitations have led, over the previous decade or so, to a series of computational and experimental investigations aiming to better understand the applicability of the lubrication approximation. These investigations (Mourzenko et al., J. Phys., 1995; Brown et al., GRL, 1995; Yeo et al., Int. J. Rock Mech., 1998; Nicholl et al., WRR, 1999) have showed that the Reynolds equation may over-predict the transmissivity of a fracture by as much as 100%. Other analyses, both theoretical and computational, have concluded that the linear relationship between pressure drop and flowrate will break down if the Reynolds numbers reach some critical value, variously estimated to be between 1-100 (Oron and Berkowitz, WRR, 1998; Brush and Thomson, WRR, 2003). The implication of these results has been that the full three-dimensional, nonlinear Navier-Stokes equations are needed to accurately simulate fluid flow in a rock fracture. In a recently completed project at Imperial College, a surface profilometer was used to measure fracture profiles every 10 microns over the surface of a replica of a fracture in a red Permian sandstone, to within an accuracy of a few microns. These surface data were used as input to two finite element codes that solve the Navier-Stokes equations and the Reynolds equation, respectively. Numerical simulations of flow through these measured aperture fields were carried out at different values of the mean aperture, corresponding to different values of the relative roughness. Flow experiments were also conducted in molds of two regions of the fracture. At low Reynolds numbers, the Navier-Stokes simulations yielded transmissivities for the two fracture regions that were within 6% and 11% of the measured values, whereas the value predicted by the Reynolds simulations were too high by 21% and 43%. We interpret these results as verifying, for apparently the first time in a real rock fracture, that (a) that the Reynolds equation over-predicts the transmissivities, and (b) the Navier-Stokes equation, with the no-slip boundary condition on the fracture walls, is indeed the correct model for fluid flow in a fracture. The initial deviations from linearity, for Reynolds number around 1, are consistent with the "weak inertia" model developed by Mei and Auriault (J. Fluid Mech., 1991) for porous media, and with the results obtained computationally by Skjetne et al. (J. Fluid Mech., 1999) on a two-dimensional self-affine fracture. In the regime 15$<$Re$<$100, both the computed and measured transmissivities could be fit very well to a Forchheimer-type equation, in which the additional pressure drop varies quadratically with the Reynolds number. The computed and "measured" Forchheimer coefficients differed by about 30%, however. Examination of computed velocity profiles indicates that, in some regions of the fracture, there are strong deviations from a parabolic velocity profile. This should have implications for processes such as solute transport, and dissolution/precipitation of minerals.
DE: 1831 Groundwater quality
DE: 5104 Fracture and flow
DE: 5114 Permeability and porosity
SC: Hydrology [H]
MN: 2003 Fall Meeting