Sublinear scaling of fracture aperture versus length: An exception or the rule?

Abstract

Observations of natural fracture dimensions have sparked a continuing debate as to the nature of the fundamental relationship between fracture aperture (maximum opening) and length. On the basis of theoretical fracture mechanics, some have argued aperture‐to‐length scaling should be linear. This relationship implies that all fractures in a given population have the same driving stress regardless of fracture length, arguably a state that is difficult to reconcile with fracture propagation criteria. Also, some field observations indicate sublinear aperture‐to‐length scaling that is apparently inconsistent with the linear elastic fracture mechanics theory. In this work, a nonlinear aperture‐to‐length relationship is derived, still based on linear elastic fracture mechanics in a homogeneous body, but incorporating subcritical and critical (equilibrium law) fracture propagation criteria. The new hypothesis postulates that fractures of different lengths preserved in a body of rock are all in the same condition with respect to propagation (i.e., they all have the same stress intensity factor). This requires that fractures have driving stresses that vary inversely with the square root of fracture length, producing fracture apertures that scale with length to the 1/2 power. Under these conditions, fracture aspect ratio (aperture/length) decreases with increasing fracture length to the negative 1/2 power. Linear aperture‐to‐length scaling is still considered a possibility but is attributed to a relaxed, postpropagation mechanical state. Deviations in fracture aperture‐to‐length relationships from these idealized models can result from mechanical fracture interaction, fracture segmentation into en echelon arrays, and three dimensional effects in stratabound fractures.