In the first part of this paper, a universal fluid velocity based algorithm for simulating hydraulic fracture with leak-off, previously demonstrated for the PKN and KGD models, is extended to obtain solutions for a penny-shaped crack. The numerical scheme is capable of dealing with both the viscosity and toughness dominated regimes, with the fracture being driven by a power-law fluid. The computational approach utilizes two dependent variables; the fracture aperture and the reduced fluid velocity. The latter allows for the application of a local condition of the Stefan type (the speed equation) to trace the fracture front. The obtained numerical solutions are carefully tested using various methods, and are shown to achieve a high level of accuracy.

中文翻译：

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基于流体速度的水力压裂模拟：硬币形模型——第一部分：数值算法

在本文的第一部分，基于通用流体速度模拟渗漏水力压裂的算法（之前针对 PKN 和 KGD 模型进行了演示）进行了扩展，以获得便士形裂缝的解决方案。数值方案能够处理粘度和韧性主导的状态，裂缝由幂律流体驱动。计算方法利用两个因变量；裂缝孔径和降低的流体速度。后者允许应用 Stefan 类型的局部条件（速度方程）来追踪裂缝前沿。使用各种方法对获得的数值解进行了仔细测试，并证明可以达到很高的精度。