Aggregate properties of fracture populations

Empirical studies indicate that the individual attributes of both faults and extension fractures follow power-law scaling. Aggregate properties of fracture populations are important in a variety of problems and can be specified in terms of the scaling parameters of individual fracture attributes. Development of an expression for an aggregate property requires consideration of a number of independent factors, including the topologic dimension of the aggregate property, the topologic dimension of sampling and the possibility of scaling changes for fractures that span some mechanically significant layer. The Riemann zeta function provides an alternative to integration for the analytical and numerical solution of aggregate problems. Previous work regarding aggregate properties of fracture populations has focused on the strain due to faulting. New expressions are developed here for other aggregate properties of interest: fracture surface area, fracture porosity, fracture permeability and shear-wave anisotropy. A general characteristic of these aggregate properties is that, for most values of scaling exponents, the aggregate properties are dependent on the size of the sampling domain. This implies that the aggregate properties are scale-dependent. Additionally, it appears that fracture surface area is concentrated in the smallest fractures of many populations. Fracture porosity is concentrated in the largest fractures of most populations but not as strongly as fracture permeability, which probably derives almost entirely from the largest fractures in populations.