Permeability of fractured rock: Effect of fracture size and data uncertainties

Abstract

Equations are derived for equivalent Darcy permeability of a fracture system that consists of nonextensive fractures having arbitrary orientations with respect to the rock block considered. It is shown that in the case of nonextensive fractures, one cannot proceed to the limiting process where the volume of the bounding rock block can be reduced to an arbitrarily small value. Therefore, the dimensions of the block along with the dimensions of the fractures appear in the equations for the nine components of permeability. This leads to a situation where the equivalent permeability matrix is not only nonsymmetric, but also it does not follow the tensorial rules of axis rotation. This leads to the conclusion that, in the case of nonextensive fractures, the equivalent permeability cannot be defined as a characteristic intrinsic property of the medium as is possible in the case of porous medium or in a fractured medium that has extensive fractures. To analyze the effect of uncertainties in the field data on the estimation of equivalent permeability, a ‘second‐order’ statistical analysis is proposed. The method is quite general and can be used to analyze the propagation of parameter uncertainties in models. A numerical example using preliminary fracture data from the Columbia River basalt in Washington State is presented. The mean and standard deviation of the equivalent permeability of a 5×5 m block is estimated. Assuming that each component of the equivalent permeability has log normal distribution, the probability of their assuming values different from the mean is found to be significant. This points out the uncertainty that would be inherent in the flow simulation, if the mean value of permeability is used.