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Abstract
Hydraulic fracturing stimulation is one of the key technologies for shale gas development. The recent advances in microseismic data acquisition and processing suggest that hydraulic fracturing stimulation has often resulted in complex fracture network due to the pre-existing natural fractures. Modeling hydraulic fracturing processes needs to couple in-situ stress response and flow of engineered fluid that includes water, proppant and other chemicals. Moreover, the high Reynolds number indicates that the flow in the hydraulic fracturing processes is either in transition or turbulent flow regime. Consequently, the resulting mathematical model is complex and needs to be numerically solved.

In this paper, we have developed a hydraulic fracturing model considering the in-situ stress response to turbulent flow process. The mixed finite element method is employed for numerical solution of the resulting system of coupled nonlinear partial different equations. The proposed model has been validated with bi-wing hydraulic fracture model through regression tests. The preliminary numerical results show the significant differences in hydraulic fracture growth in comparison with the models that assume laminar flow in hydraulic fracturing processes. We have also integrated proposed hydraulic fracturing model into a numerical reservoir simulator and are currently conducting field-scale numerical simulation studies. The preliminary results also suggest that the proposed model is also capable of modeling the interactions between the hydraulic fracture and pre-existing natural fractures based on initial fracture mapping.

The proposed model provides an opportunity to optimize hydraulic fracturing stimulation design through numerical simulations, which is vital in unconventional reservoir production.

Introduction
Shale formations usually exhibit very complex geological structures with ultra-low permeability that can go as low as nano-darcy. Hydraulic fracturing stimulation is one of the key technologies to produce shale gas in an economically viable fashion (Himanshu Yadav. 2011). It is thus vitally important to accurately model hydraulically induced fracture network and, consequently, predict the production performance and optimize the development plan.

The traditional hydraulic fracturing model for conventional reservoirs is the bi-wing model that is an analytical solution on the induced fracture propagation in homogeneous and isotropic porous media. In the bi-wing model, the hydraulically induced fracture propagates in the direction that is perpendicular to the least principal stress (Perkins and Kern. 1961, Geertsma and Klerk. 1969, Nordgren. 1972). Although bi-wing model generates simple planar fracture geometries in homogeneous and isotropic formations, they provides a simple and efficient prediction of hydraulic fracture propagation in conventional reservoirs and have been widely used in oil and gas industry for many years. Hydraulic fracturing stimulation in shale gas usually introduces a complex induced fracture network due to the pre-existing natural fractures and, the heterogeneity and anisotropy of the rock properties. Micro-seismic data indicate the complexity of induced fracture networks (Fisher et al. 2002; Maxwell et al. 2002; Daniels et al. 2007; Le Calvez et al. 2007; X. Wang et al. 2011). Consequently, a complex fracture network model is required to adequately model the hydraulic fracture network propagation in shale formation.