A radial hydraulic fracture driven by a Herschel–Bulkley fluid

Abstract

We analyse the influence of fluid yield stress on propagation of a radial (penny-shaped) hydraulic fracture in a permeable reservoir. In particular, the Herschel-Bulkley rheological model is adopted that includes yield stress and non-linearity of the shear stress. The rock is assumed to be linear elastic, and the fracture is driven by the point source fluid injection with a constant volumetric rate. The fracture propagation condition follows the theory of linear elastic fracture mechanics, and Carter's leak-off law is selected to govern the fluid exchange process between the fracture and formation. The numerical solution for the problem is found using the algorithm based on Gauss-Chebyshev quadrature and Barycentric Lagrange interpolation techniques. We also construct an approximate solution with the help of the global fluid balance equation and the near-tip region asymptote. The latter approximation is computationally efficient, and we estimate its accuracy by comparing the primary crack characteristics such as opening, pressure, and radius with that provided by the full numerical solution. We present examples corresponding to typical field cases and demonstrate that the addition of yield stress can lead to shorter radius and wider opening compared to the corresponding case with a simpler power-law fluid rheology. Further, we quantify the limiting propagation regimes (or vertex solutions) characterised by dominance of a particular physical phenomenon. Relative to the power-law results, there are two new vertices that are associated with domination of yield stress: storage-yield-stress and leak-off-yield-stress. To understand the influence of various problem parameters, we utilise the constructed approximate solution to investigate the dimensionless parametric space for the problem, in which the applicability domains of the limiting solutions are quantified. This enables one to quickly determine whether the yield stress provides a strong influence for a given problem parameters.