A Two-Dimensional Theory of Fracture Propagation

Abstract

Summary A basic theory of two-dimensional (2D) fracture propagation has been developed with a Lagrangian formulation combined with a virtual work analysis. Fluid leakoff is included by the assumption that an incompressible filtrate produces a piston-like displacement of a compressible reservoir fluid with a moving boundary between the two. Poiseuille flow is assumed in the fracture. We consider both Newtonian and non-Newtonian fluids with and without wall building. For non-Newtonian fluids, we assume the usual power-law relation between shear stress and shear rate. The Lagrangian formulation yields a pair of nonlinear equations in Lf and bj, the fracture length and half-width. By introducing a virtual work analysis, we obtain a single equation that can be solved numerically. For non-wall-building fluids, it predicts much higher leakoff rates than existing methods. The Lagrangian method also allows nonelastic phenomena, such as plasticity, to be included. A practical computer program developed from this theory has been used for more than 10 years to design fracturing treatments in oil and gas reservoirs in Canada, California, the midcontinent and Rocky Mountain areas, the U.S. gulf coast, the North Sea, and in northern Germany. In most of these applications, it has predicted fracture dimensions that have been in line with production experience. Optimization methods based on this program led to very large fracturing treatments in low-permeability gas sands that were forerunners of massive fracturing treatments in tight gas sands. Specific examples in which this method was used to design fracturing programs in large gas fields in Kansas and Texas are discussed.