Analysis of the classical pseudo-3D model for hydraulic fracture with equilibrium height growth across stress barriers

This paper deals with the so-called “pseudo three-dimensional” (P3D) model for a hydraulic fracture with equilibrium height growth across two symmetric stress barriers. The key simplifying assumptions behind the P3D model are that (i) each cross section perpendicular to the main propagation direction is in a condition of plane-strain, and (ii) the local fracture height is determined by a balance between the effect of the stress jump across the barriers and that of the rock toughness. Furthermore, in the equilibrium height growth P3D models, the pressure is assumed to be uniform in each vertical cross-section. We revisit this particular model by first formulating the non-linear differential equations governing the evolution of the length, height, and aperture of the hydraulic fracture, in contrast to the numerical formulations adopted in many previous studies. Scaling of these equations shows that the solution depends, besides the dimensionless space and time coordinates, on only two numbers representing a scaled toughness and a scaled leak-off coefficient. Analysis of the governing equations enables us to determine explicitly the conditions under which breakthrough takes place (i.e., the onset of growth into the bounding layers), as well as the conditions of unstable height growth (i.e., the conditions of “runaway height” when the main assumptions of the equilibrium height model become invalid). The mathematical model is solved numerically using a novel implicit fourth order collocation scheme on a moving mesh, which makes explicit use of the fracture tip asymptotics. We then report the results of several numerical simulations conducted for different values of the dimensionless toughness and the dimensionless leak-off coefficients, as well as a comparison with closed-form small and large time similarity solutions that are valid under conditions where the fracture remains contained within the reservoir layer.