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. Poiseuillc flow is assumed in the fracture. We consider both Newtonian and Poiseuillc flow is assumed in the fracture. We consider both Newtonian and non-Newtonian fluids with and without wall building. For non-Newtonian fluid, we assume the usual power-law relation between shear stress and shear rate. The Lagrangian formulation yields a pair of nonlinear equations in and, 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 phenomena, such as plasticity, to be included. A practical computer program developed from this theory has been used for more than 10 years to 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. Introduction We present here a new approach to the 2D problem of fracture propagation based on Lagrangian methods. The Lagrangian formulation has been applied to a variety of problems in physics and chemistry. To the best of our knowledge, however, this is the first application to fracture mechanics. The Lagrangian formulation is based on the classical form of Lagrange's equations. As applied here, it produces a basic equation that expresses the balance between work expended and work done in propagating a 2D crack. Existing theories of crack propagation have all been developed by the application of equations from classical elasticity theory. This approach assumes linear elastic behavior of the reservoir rock and ignores surface energy considerations at the crack tip and plastic deformation effects. Leakoff, if it is included, is treated as an independent process and merged with the crack propagation problem by iterative methods that assume self-consistency. Some well-known examples of this approach have been presented by Zheltov and Khristianovich, Perkins and Kern, Nordgren presented by Zheltov and Khristianovich, Perkins and Kern, Nordgren Geertsma and de Klerk, Daneshy, Le Tirant and Dupuis, and Cleary. Geertsma and Haafkens have compared many of the results of these theories. A more general approach, the Lagrangian method is not restricted to elastic behavior, and leakoff can be included as an integral part of the formulation. We include leakoff by assuming a piston-like displacement of compressive reservoir fluid by an incompressible fracture fluid filtrate with a moving boundary between the two. The Lagrangian formulation yields a pair of nonlinear differential equations in fracture length and fracture half-width, which are reduced to a single equation in by introduction of a virtual work analysis. This equation can be solved numerically and can be used with other relations to obtain fracture dimensions and injection pressure as a function of time at constant injection rate. Experimental laboratory measurements reported previously confirm basic results obtained from such computations. The Lagrangian formulation presented here has been used for many years in our field operations to predict fracture dimensions. It has provided a means to plan and to optimize fracture treatments in a variety of field operations. Some specific field applications will be discussed.