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 behaviour 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 behaviour is mainly determined by the three-process multiscale tip asymptotics and the global fluid volume balance. First, the developed approximation is compared with 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 per cent 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 quick fracture design calculations or as a reference solution to evaluate accuracy of various hydraulic fracture simulators.

本文针对受流体黏度、断裂韧性与流体滤失三类竞争物理过程共同调控的币形水力压裂裂缝（penny-shaped hydraulic fracture），推导得到闭式近似解。构建该解的核心前提假设为：裂缝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，将所提出的近似解与现有各类极限传播工况下的解析解进行比对验证。随后，构建可表征极限解适用区间的解映射（solution map）。研究同时表明，所构建的近似解可精准捕捉从任意一种极限解向另一种极限解过渡时对应的尺度律。将所提出的近似解与参考数值解（reference numerical solution）开展对标测试，结果显示：在宽参数取值范围内，该近似解对裂缝宽度与裂缝半径的预测误差仅为百分之零点几，精度优异。综上，本文构建的闭式近似解可为币形水力压裂裂缝提供高效快速的求解方案，可用于快速开展裂缝设计计算，或作为基准参考解用于评估各类水力压裂模拟器的计算精度。