X-FEM Modeling of Multizone Hydraulic Fracturing Treatments Within Saturated Porous Media

Abstract

In this paper, a fully coupled hydro-mechanical model is presented for the study of multizone hydraulic fracturing. The momentum balance equation of the bulk together with the mass balance and momentum balance equation of the fluid phase are employed in order to derive the hydro-mechanical coupled system of governing equations of the porous media known as the $$({\mathbf{u}} - p)$$ formulation. The hydro-fracture inflow is modeled based on the Darcy law, where the fracture permeability is determined by using the cubic law. Provisions are made for the plausible closure as well as the frictional resistance of the fracture edges in the solid phase by means of Kuhn–Tucker inequalities embedded in an X-FEM penalty method. In addition, for the fluid phase, the zero leak-off constrain is imposed through the application of the large time increment-based contact algorithm in the case of crack closure. The cohesive crack model is employed to account for the nonlinear fracturing process at the hydro-fracture tips. Multiple crack growth patterns are determined by means of energy based cohesive stress functions. Based on the X-FEM, the strong discontinuities in the displacement field due to fracture opening as well as the weak discontinuities within the pressure field due to leak-off flow are incorporated by using the Heaviside and modified level-set enrichment functions, respectively. A consistent computational algorithm is proposed for the determination of the fracturing fluid flow distribution across the existing perforations. Finally, several numerical examples are presented to demonstrate the robustness of the proposed X-FEM framework in the study of multizone hydraulic fracturing treatments through saturated porous media. The results appear to accord with the field observations reporting numerous failed attempts of multistage multizone fracturing treatments, which provide a great insight into the complexities encountered in practice.