Data from: An approximate solution for a penny-shaped hydraulic fracture that accounts for fracture toughness, fluid viscosity, and leak-off

This paper develops a closed form approximate solution for a penny-shaped hydraulic fracture, whose behavior is determined by an interplay of three competing physical processes that are associated with fluid viscosity, fracture toughness, and fluid leak-off. The primary assumption that permits one to construct the solution is that the fracture behavior is mainly determined by the three-process multiscale tip asymptotics and the global fluid volume balance. First, the developed approximation is compared to the existing solutions for all limiting regimes of propagation. Then, a solution map, which indicates applicability regions of the limiting solutions, is constructed. It is also shown that the constructed approximation accurately captures the scaling that is associated with the transition from any one limiting solution to another. The developed approximation is tested against a reference numerical solution, showing that accuracy of the fracture width and radius predictions lie within a fraction of a percent for a wide range of parameters. As a result, the constructed approximation provides a rapid solution for a penny-shaped hydraulic fracture, which can be used for a quick fracture design calculations or as a reference solution to evaluate accuracy of various hydraulic fracture simulators.本研究针对币形水力压裂裂缝（penny-shaped hydraulic fracture）推导得到闭式近似解。该类裂缝的演化行为由三类相互竞争的物理过程共同调控，三类过程分别关联流体黏度、断裂韧性与流体滤失。构建该近似解的核心前提为：裂缝的整体行为主要由三类过程的多尺度尖端渐近行为与全局流体体积平衡所主导。首先，将本文推导的近似解与现有覆盖所有极限传播工况的解析解进行对标验证。随后，构建用以标注各类极限解适用范围的解映射图（solution map）。研究同时证实，该近似解可精准捕捉从任意一类极限解过渡至另一类极限解时对应的尺度律变化。将所提近似解与参考数值解（reference numerical solution）开展对比验证，结果表明：在宽参数区间内，裂缝宽度与半径的预测误差仅在百分之一以内。综上，本文构建的近似解可为币形水力压裂裂缝提供高效求解方案，可快速应用于裂缝设计计算，亦可作为基准解用于评估各类水力压裂模拟器的求解精度。