Numerical models of hydraulic fracturing and the interpretation of syntectonic veins

Abstract

The hydraulic fracture is modelled as an ellipse in an infinite elastic medium with an internal fluid pressure and loaded under biaxial stresses at infinity. The available stress function for this model has been evaluated numerically, and the magnitudes of the stresses generated around the crack calculated for a variety of loading conditions and crack orientations. Fracture initiation is predicted from the Griffith maximum tensile stress criterion. The location of the maximum tensile stress around the crack is recorded and it is found that for many conditions of applied stresses and crack fluid pressure, the hydraulic shear fracture has a symmetrically developed maximum tensile stress and fracture initiation will occur by growth along the direction of the crack. It is also predicted that fracture initiation will occur when the ratio of fluid pressure to applied least principal stress is considerably less than one. The elastic strain energy fields around elliptical hydraulic flaws have been calculated, and in particular, the change in strain energy upon introduction of a small flaw, and the change in strain energy upon growth of this flaw, have been investigated. The results allow an evaluation of the second part of the Griffith criterion-that fracture growth is accompanied by a decrease in strain energy-for hydraulic fractures. Changes in strain energy with small increases in fluid pressure provide a physical basis for dilatancy hardening and fracture instability. Quasi-static growth from a flaw is modelled by calculating changes in strain energy for unit increases in half length. The distinction between fractures which show an increasing and a decreasing rate of change in strain energy with increasing length, and between fractures which may only extend spontaneously for short distances and those which may show extensive spontaneous growth on the basis of the rate of change of strain energy with length, is made. A gradual drop in crack fluid pressure once the threshold for fracture initiation has been passed may promote the extent to which spontaneous crack growth occurs. The formation of syntectonic veins, particularly in rocks being deformed under low grade metamorphic conditions, is often the most abundant evidence of natural hydraulic fracturing in rocks. Commonly observed geometric features of syntectonic veins-length, simple tapering, symmetric and asymmetric forking, branching, irregular zig-zag traces, en échelon patterns—are discussed primarily with reference to the strain energy model for growth established, and the geometric variation is interpreted in terms of variation in applied stress and fluid pressure conditions and the rate of change of stored strain energy with crack growth. In particular, terminal branching arises when the minimum stress changes from a symmetric to an asymmetric location at the tip of a growing shear fracture, and terminal forking results when there is an increase in the energy release rate during crack growth, and may be symmetric or asymmetric depending on the location of the minimum stress at the crack tip at the time of forking.