This paper investigates the dependence of breakdown pressure, the critical pressure at which tensile failure is initiated at the borehole wall by fluid injection, on the rate of pressurization. The mathematical model explicitly accounts for the existence of micro-cracks at the borehole wall that trigger the failure process. Breakdown, in this context, occurs when the stress intensity factor of a critically oriented micro-crack reaches the rock toughness. The model is presently restricted to low-permeability/low-porosity rocks. By considering one-dimensional lubrication flow in the crack coupled with the non-local elastic response of the crack, the evolution of the net pressure, crack opening and stress intensity factor is obtained as functions of the pressurization rate. The relation between breakdown pressure and pressurization rate in the case of zero initial net pressure is shown to be controlled by only one dimensionless number: the ratio between the initial width of the unstressed micro-crack and the induced elastic opening at failure. It is shown that (i) the fluid pressure in the early stages of the pressurization history drops in the crack and that cavitation can occur, and (ii) local back-flow in the crack occurs. The dependence of breakdown pressure, pb, on the pressurization rate, A, is determined as well as the range of A, where pb varies significantly. The lower and upper bounds of this range of pressurization rate correspond to limiting regimes of slow and fast pressurization.
The breakdown pressure, pb, the critical pressure at which tensile failure (breakdown) is initiated at the borehole wall by fluid injection, is a crucial parameter in hydraulic fracturing. Several breakdown criteria (Hubbert and Willis, 1957; Haimson and Fairhurst, 1967) have been proposed to estimate the value of pb in terms of the far-field stress and rock properties. These criteria predict pb to be independent of the pressurization rate A. However, there is strong experimental evidence that the breakdown pressure does vary with the pressurization rate, pb = pb (A) (e.g., Haimson and Zhao, 1991; Schmitt and Zoback, 1992, 1993). Recent studies (Detournay and Cheng, 1992; Detournay and Carbonell, 1994) actually suggest that the classical breakdown criteria correspond to particular asymptotic regimes of pressurization rate, under the limiting condition that the length. An attempt to model the dependence of the breakdown pressure on the pressurization rate in permeable rock was previously made by introducing a length scale, ë, to characterize the failure process and by assuming that breakdown takes place when the Terzaghi effective stress, averaged over ë, reaches a critical value (Detournay and Cheng, 1992). Threshold values of the pressurization rate, AI and Au, for the slow and fast asymptotic regimes, respectively, were then obtained by considering the evolution of the pore- pressure perturbation near the permeable boundary of the borehole. However the values AI and Au, calculated for typical sets of material parameters, were much higher than those observed in experiments, and their ratio is also two to three orders of magnitude higher than in the experiment.
