Fluid injection via a vertical well into a high-temperature formation may induce multiple longitudinal thermal fractures, which may eventually transition to two-wing fractures during fracture propagation, depending on horizontal stress ratio κ. In this study, we develop a plane strain model with radial heat conduction to investigate either two-wing fractures under highly anisotropic stresses κha≫1 or multiple fractures under isotropic stresses κ=1. The coupled dimensionless elasticity equation and criteria of fracture propagation (and arrest) are formulated, discretized, and solved iteratively (with two special algorithms). Two additional critical model parameters are dimensionless effective confining stress T and wellbore radius A. The multiple-fracture solution of dimensionless fracture length L, angular spacing D, and aperture consists of solutions for competitive propagation of fractures with arrests in the near-wellbore region and the subsequent stable propagation of fractures away from the wellbore. Both the multiple-fracture and two-wing-fracture solutions accurately capture the early-time transient and late-time power-law changes with dimensionless time τ, as verified numerically. The late-time fracture propagation follows scaling law L=fT,A,Dτ1−2T/2. These solutions and scaling laws can be used to well bound a general solution with 1<κ<κha as demonstrated numerically for a geothermal site, for which the maximum fracture length reaches 0.45, 2.71, and 16.42 m in 1, 100 and 10,000 days of cooling, respectively. The applicability of the assumptions used in the theoretical and numerical analysis is discussed.

中文翻译：

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沿垂直井的热驱动纵向裂缝建模


通过垂直井注入高温地层的流体可能会诱发多个纵向热裂缝，最终可能在裂缝扩展过程中转变为双翼裂缝，具体取决于水平应力比 κ。在这项研究中，我们开发了一个具有径向热传导的平面应变模型，以研究高各向异性应力 κha≫1 下的双翼断裂或各向同性应力 κ=1 下的多条断裂。耦合的无量纲弹性方程和裂缝扩展（和停滞）准则被制定、离散化和迭代求解（使用两种特殊算法）。另外两个关键模型参数是无量纲有效围压 T 和井筒半径 A。无量纲裂缝长度 L、角间距 D 和孔径的多裂缝解包括在近井筒区域停滞的裂缝竞争扩展以及随后裂缝从井筒外稳定扩展的解。多裂缝和双翼裂缝解都准确地捕获了无量纲时间 τ 的早期瞬态和晚期幂律变化，如数值验证的那样。晚期裂缝扩展遵循缩放定律 L=fT，A，Dτ1−2T/2。这些解和缩放定律可用于将 1<κ<κha 的一般解很好地结合，如地热场地的数值所示，其最大裂缝长度在 1、100 和 10,000 天的冷却中分别达到 0.45、2.71 和 16.42 m。讨论了理论和数值分析中使用的假设的适用性。