A Simple Model for the Variation of the Transmissivity of a Rock Fracture under Normal Stress

A simple model is described that relates the normal stiffness of a fracture to the variation in transmissivity that occurs when the fracture is subjected to a normal load. The fracture is assumed to be represented by a row of evenly spaced, open elliptical channels (Myer, Int. J. Rock Mech., 2000). The normal stiffness of such a fracture is given exactly by the elasticity solution derived by Sneddon and Lowengrub ( Crack Problems in the Classical Theory of Elasticity, 1969). At zero load, these channels may have an arbitrary distribution of aspect ratios. As the normal load is increased, the channels close up sequentially, with those having smaller initial aspect ratios closing first, etc. As the stiffness depends on the number of open channels, and the normal stress required to completely close a channel depends on the initial aspect ratio, in principle the aspect ratio distribution determines the nonlinear stress-strain curve of the fracture. We can derive an explicit relation between the stress-strain curve and the aspect ratio distribution by considering the behavior of Sneddon and Lowengrub's equation in the limit of large contact area (in practice, greater than about 0.2). The transmissivity of such a fracture is given by the sum of the transmissivities of the various elliptical channels, as given by the well-known expression for flow through an elliptical tube (Landau and Lifschitz, Fluid Mechanics, 1959). The instantaneous aspect ratio of each tube can be expressed as a function of the normal stress, and so the overall transmissivity can be computed as a function of stress. This simple model has been applied to the classic data of Iwai ( PhD, UC Berkeley, 1976) on a granite fracture, with reasonable results. Obvious future extensions and refinements of the model include (a) use of Sneddon and Lowengrub's exact expression for the fracture compliance, rather than the approximation that assumes a high contact fraction, and (b) allowing the various flow channels to be connected in a two- dimensional network, rather than assuming that they are in parallel with each other. Nevertheless, the present simple version of this model is capable of providing a rationally based connection between fracture stiffness and transmissivity.