Analysis of Hydraulic Fracture Propagation in Toughness Dominant with Considering Fluid Inertia and Viscosity Parameters Interaction: Higher order terms

In the process of hydraulic fracture, various physical parameters such as; viscosity, inertia of fluid and toughness of rock do not influence the fracture propagation identically, and it is probable that one or more of the parameters be more pronounced. Therefore, it may persuade one special regime which is named base on dissipation of energy. In an impermeable rock, the two limiting regimes can be identified with the dominance of one or the other of the two energy dissipation mechanisms corresponding to extending the fracture in the rock and to flow of viscous fluid in the fracture, respectively. In the viscosity-dominated regime, dissipation in extending the fracture in the rock is negligible compared to the dissipation in the viscous fluid flow, and in the toughness-dominated regime, the opposite holds. It is supposed that the flow of incompressible fluid in the fracture is unidirectional and laminar. The contribution of this research is the evaluation of parameters effects on the propagation of hydraulic fracture an impermeable brittle rock. Here, the modified perturbation method suggested for evaluating fluid viscosity and inertia parameters interaction (FVII). The proposed method provides a good estimate of the solution in the wide range of the viscosity/inertia parameters because of the coexistence of both small parameters in the governing equations. The result shows, neglecting FVII is reasonable for very small fluid viscosity as well as small inertia. However, the influence of FVII become considerable for slightly higher values of these parameters.