عنوان مقاله [English]
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. Here, it is supposed that the flow of incompressible fluid in the fracture is unidirectional and laminar. Besides, the fracture is fully fluid-filled at all times and fracture propagation is described in the framework of linear elastic fracture mechanics (LEFM). In this paper, a new semi-analytical method has been introduced for solving the plane-strain fluid-driven fracture propagating in an impermeable medium in viscosity-toughness dominated (the MK-edge solution). Standard methods of analysis and improvement of diverging series have been applied on the expansion series method to gain the more convergence for the viscosity series diverge due to a nearest (non-physical) singularity on the negative real axis of the viscosity parameter for larger viscosity. For more explanation, Euler transformations have been suggested in terms of small parameter which is a function of viscosity parameter. Compared to the other analytical solution (e.g. Garagash, 2006), the new M-K edge solution represents a significant improvement in term of convergence. In addition to, the results have been compared to the numerical solution (e.g. Adachi, 2000) and it is shown good agreement in the light of quantity and quality. Contrary to numerical methods, the new proposed method can pragmatically be used for the range of M-K edge.