A Simple Model for Deviations from the Cubic Law for a Fracture Undergoing Dilation or Closure

Abstract

— Experimental observations show that flow through a fracture decreases more rapidly than the cube of the mean aperture (Cook, 1992). In order to provide a possible explanation of these experimental findings, we study creeping flow through a fracture of varying aperture that is symmetric about its midplane, using the power series of the stream function obtained by Van Dyke (1987) for low Reynolds numbers. For the case of sinusoidally-varying walls, a simple expression relating the effective hydraulic aperture of the channel to the mean aperture and to the amplitude and wavelength of the sinusoidal wall profiles is obtained. Comparison is made to previous studies (Kitanidis and Dykaar, 1997) and to finite element calculations, and good agreement is obtained. The effect of fracture closure is then modelled as a decrease of the mean aperture without a change in the roughness. A power law relationship can be obtained between the flowrate and the mean aperture, with an exponent as high as 10, thus providing a potential mechanistic explanation of the experimental findings of Pyrak-Nolte et al. (1987).