Coupling of Fracture/Non-Newtonian Flow for Simulating Nonplanar Evolution of Hydraulic Fractures

Abstract

Abstract A computational technique has been developed for simulating non-planar evolution of hydraulic fractures in three-dimensional elastic media. The technique utilizes a symmetric Galerkin boundary element method to treat the elasticity problem associated with a fracture in an unbounded domain, and a Galerkin finite element method to model the (non-Newtonian) fluid flow within the fracture. The boundary element method is based upon a weakly-singular, weak-form traction boundary integral equation which is applicable to non-planar fractures in generally anisotropic media. Since the integral equation is only weakly-singular, standard C[o] elements can be employed, and such elements are adopted everywhere except along the fracture front where a special crack-tip element is used. The fluid flow within the (non-planar) fracture is described in terms of a "channel flow relation" cast in a weak form well-suited for numerical treatment. The final system of non-linear algebraic equations (for the relative crack-face displacement and fluid pressure) is then solved using the Newton-Raphson method. To illustrate the capabilities of the method, example simulations are presented including ones involving complex non-planar crack growth. Introduction In order to design and control the hydraulic fracturing process, it is necessary to have predictive capabilities which adequately capture the complexities of the event. When the fracture is planar and well contained, two-dimensional (or "pseudo three-dimensional") simulations typically suffice (e.g. Yew, 1997), but in more complicated situations a truly three-dimensional simulation may be required to properly predict/understand the fracturing event. Various numerical strategies have been developed for three-dimensional hydraulic fracturing (e.g. Clifton, 1989; Ouyang et al., 1997; Yew, 1997; Carter et al, 2000), but these strategies are either restricted to planar crack growth or to cracks which evolve according to a predetermined shape (Hsu, 1994). Under certain conditions the crack may evolve in a complex non-planar fashion, and the capability to model this phenomenon is necessary in order to understand its impact on the completion process and/or to ensure that such situations do not arise. An example is fracturing from inclined or horizontal wellbores where, if the fracture initiates at a non-preferred orientation, a tortuous final crack shape may result. This in turn may significantly alter the relationship between the injection history and the crack size, and it may lead to complications associated with proppant transport due to a restriction of the crack width near the wellbore. In this paper, a computational procedure is developed to treat general non-planar evolution of hydraulic fractures. The elasticity problem associated with the fracture is modeled by means of a weak-form traction integral equation applicable to both isotropic and anisotropic media (Li and Mear, 1998; Rungamornrat and Mear, 2005). The fluid flow within the fracture is modeled by a two-dimensional flow relation that applies for an arbitrary curved fracture surface and for power-law non-Newtonian fluid behavior. The resulting coupled fracture/flow equations are fully nonlinear and solved using a Newton iterative procedure.