Numerical and experimental study of fluid flow in a rough-walled rock fracture

Abstract

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 /i^/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 flow-rates are suitably low. Recent previous investigations 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 estimated that the cubic law will also break down if the Reynolds numbers reach some critical value, variously estimated to be between 1-10. The implication of these results has been that the full three-dimensional, nonlinear Navier-Stokes equations are actually needed to accurately simulate fluid flow in a rock fracture. This conjecture, however, has never been verified, and its verification is the main objective of this research. A surface profilometer was used to measure fracture profiles every 20 nm over the surface of a replica of a fracture in a red Permian sandstone of size 4 cm^. 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 then carried out at different values of the mean aperture, which corresponds to different values of the relative roughness (which is defined as the ratio of the roughness to the mean aperture). At low Reynolds numbers, the Navier-Stokes simulations yielded transmissivities that were 10-100% lower (depending on the relative roughness) than those predicted by the Reynolds simulations, in close agreement with the range of discrepancy between the Reynolds equation and experimental flow measurements reported by previous investigators. At Reynolds numbers <<1, the computed transmissivity is constant. Appreciable deviations from Darcy's law began to be observed when the Reynolds number (defined using the mean aperture as the length scale) exceeded unity. In the regime Re > 20, the computed transmissivities could be fit very well to a Forchheimer-type equation, in which the additional pressure drop varies quadratically with the Reynolds number. The initial deviations from linearity, for Reynolds number around 1, are consistent with the "weak inertia" model of Mei and Auriault (1991). Experimental flow measurements were conducted, on the same fracture replica for which the surface measurements were taken and on which the simulations were conducted. The experiments showed that the measured transmissivities were in close agreement to the values predicted by the Navier-Stokes equations, whereas the Reynolds equation over-estimated fracture transmissivity by as much as 30%. Appreciable deviations from Darcy's law began to be observed when the Reynolds number exceeded unity. When the Reynolds number exceeds about 15, measured transmissivities could be described by the Forchheimer equation, which is consistent with the numerical solutions of the Navier-Stokes equations.