Coupled SGBEM-FEM for efficient simulation of height-contained hydraulic fractures

An efficient computational model is developed to simulate the growth of vertically oriented, height-contained hydraulic fractures. A symmetric Galerkin boundary element method, used to model the behavior of the fracture, is specialized by exploiting knowledge of the fracture surface geometry and an assumption on the approximately elliptical, vertical cross section of the fracture. This geometric knowledge is used to reduce the governing weakly singular, weak-form traction integral equation from an integral over the fracture surface to an integral along the centerline of the fracture through an analytical integration with respect to the fracture height. The fluid flow within the fracture is treated using a Galerkin finite element method to model one-dimensional flow through an arbitrarily curved channel. Under the assumption that the fluid pressure is uniform over the fracture height (as in the case of a tunnel crack) and using the cross sectional form assumed by the fracture model, a specialized, weak-form, fluid flow equation is developed and integrated analytically with respect to the fracture height. The symmetric Galerkin boundary element method and Galerkin finite element method are coupled and the resulting system is solved using a Newton-Raphson method. The fracture propagation is governed by a mixed mode-I/II growth law based on linear elastic fracture mechanics, with stress intensity factors computed directly from the degrees of freedom associated with special crack tip elements designed to capture the square root behavior near the fracture tip. This new computational model is compared to an existing coupled SGBEM-FEM model designed for general, three-dimensional, non-planar fractures to illustrate the efficiency of the new model and the dramatic speedup it offers for modeling height-contained hydraulic fracturing scenarios. The model is extended to treat various conditions typical of hydraulic fracturing design including fracture growth near completed hydraulic fractures, staged fluid injection scenarios, fracture growth in multiple, vertically stacked pay zones, and the distribution of fluid injection to a set of fractures growing from a shared wellbore.