Propagation of a fluid-driven crack in an impermeable linear elastic medium under axis-symmetric conditions is investigated in the present work. The fluid exerting the pressure inside the crack is an incompressible Newtonian one and its front is allowed to lag behind the propagating fracture tip. The tip cavity is considered as filled by fluid vapors under constant pressure having a negligible value with respect to the far field confining stress. A novel algorithm is here presented, which is capable of tracking the evolution of both the fluid and the fracture fronts. Particularly, the fracture tracking is grounded on a recent viscous regularization of the quasi-static crack propagation problem as a standard dissipative system. It allows a simple and effective approximation of the fracture front velocity by imposing Griffith's criterion at every propagation step. Furthermore, for each new fracture configuration, a non linear system of integro-differential equations has to be solved. It arises from the non local elastic relationship existing between the crack opening and the fluid pressure, together with the non linear lubrication equation governing the flow of the fluid inside the fracture.

中文翻译：

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一种新型水力压裂生长配方

在当前的工作中研究了在轴对称条件下不可渗透的线性弹性介质中流体驱动裂纹的传播。在裂缝内部施加压力的流体是不可压缩的牛顿流体，其前端允许滞后于扩展的裂缝尖端。尖端腔被认为是由恒压下的流体蒸汽填充的，其值相对于远场限制应力可以忽略不计。这里提出了一种新算法，该算法能够跟踪流体和裂缝前沿的演化。特别是，断裂跟踪基于最近作为标准耗散系统的准静态裂纹扩展问题的粘性正则化。它允许通过施加 Griffith' 来简单有效地逼近裂缝前沿速度 s 标准在每个传播步骤。此外，对于每个新的裂缝配置，必须求解非线性积分微分方程组。它源于裂缝张开与流体压力之间存在的非局部弹性关系，以及控制裂缝内流体流动的非线性润滑方程。