Plane-Strain Propagation of a Hydraulic Fracture in Porous Media

Abstract

Hydraulic fracturing (HF) in a plane-strain environment is addressed by using continuum mechanics. Fracture propagation is expected in quasi-stationary mode (i.e., not account for inertial effects and the fracture width is smaller than its length). Irwin's fracture criterion is used to govern the growth of fracture length. The internal problem of incompressible fluid motion in the fracture is described by Poiseuille's law. Fracture and fluid are propagated independently, so a fluid-free domain (lag) might exist near the fracture tip. The external problem of stress state in the unbounded medium with the fracture is addressed by the linear elastic law. The solutions to the internal and external problems are fully coupled (pressure in fluid is equal to normal stress on the fracture boundary; the half width of the fracture in the internal problem is equal to the displacement of the fracture boundary in the external problem). Fluid leakoff to the media through the boundary of the fracture is addressed by local Darcy's law. Under these assumptions, a system of integral-differential equations is built to address the problem. The equations of the system are analyzed by nondimensionalization and application of the method of similarity. Three modes of fracture propagation are examined, depending on the relative magnitude of the fluid leakoff to the media—no leaks, with predominant leaks, and general. The no leaks and predominant leaks modes are revealed to have asymptotically self-similar behavior to the solution in all physically meaningful cases. In a general case of nonzero in-situ stress, a new self-similar solution exists for the power law fluid injection rate with an exponent 1/3. Symmetry of the system of equations under the group of scaling transformations enables the time dependence of self-similar solutions in explicit form. An implicit conservative difference scheme is used to construct the effective algorithm for a numerical solution to fracturing in a complex plane-strain environment. Verification of obtained results is determined by a qualitative comparison with experimental data and known self-similar solutions.