Due to the complexity of the interaction between hydraulic fractures (HFs) and natural fractures (NFs), several different interaction behaviours (arrest/cross/offset) may occur, according to hydraulic fracturing experiments. The existing analytical criteria cannot accurately describe this interaction process and predict the interaction behaviours by adopting several simple assumptions. An innovative numerical method combining strength and energy is proposed to determine the critical average stress at the most likely re-initiation position along the NF. The critical average stress for the HF crossing can be calculated by averaging the stresses around the fracture re-initiation point when the NF does not open and slip and simultaneously satisfying the energy criterion. Then, the proposed numerical method is incorporated in an extended finite element method (XFEM) scheme, in which the friction and contact of the NF are solved by a penalty function method. The numerical results suggest that the average stress can be numerically calculated and used to predict the HF crossing behaviour. The varying trends of the required maximum confining stress with respect to the coefficient of friction obtained by the proposed numerical method agree well with those predicted by the R-P and ER-P criteria. In addition, the proposed average stress method can quantitatively predict the crossing behaviour in the presence of slippage and opening after the HF terminates at the NF.

中文翻译：

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基于平均应力水平的裂缝交叉数值方法

根据水力压裂实验，由于水力压裂 (HF) 和天然裂缝 (NF) 之间相互作用的复杂性，可能会发生几种不同的相互作用行为（截断/交叉/偏移）。现有的分析标准不能准确地描述这种交互过程，并不能通过采用几个简单的假设来预测交互行为。提出了一种结合强度和能量的创新数值方法来确定沿 NF 最可能重新起始位置的临界平均应力。HF 交叉的临界平均应力可以通过在 NF 不张开和滑移并同时满足能量准则时对裂缝重新起始点周围的应力求平均值来计算。然后，所提出的数值方法包含在扩展有限元方法 (XFEM) 方案中，其中 NF 的摩擦和接触通过惩罚函数方法求解。数值结果表明平均应力可以通过数值计算并用于预测 HF 交叉行为。所提出的数值方法获得的摩擦系数所需的最大围压的变化趋势与 RP 和 ER-P 标准预测的趋势非常吻合。此外，所提出的平均应力方法可以定量预测在 HF 终止于 NF 后存在滑移和打开的情况下的交叉行为。数值结果表明平均应力可以通过数值计算并用于预测 HF 交叉行为。所提出的数值方法获得的摩擦系数所需的最大围压的变化趋势与 RP 和 ER-P 标准预测的趋势非常吻合。此外，所提出的平均应力方法可以定量预测在 HF 终止于 NF 后存在滑移和打开的情况下的交叉行为。数值结果表明平均应力可以通过数值计算并用于预测 HF 交叉行为。所提出的数值方法获得的摩擦系数所需的最大围压的变化趋势与 RP 和 ER-P 标准预测的趋势非常吻合。此外，所提出的平均应力方法可以定量预测在 HF 终止于 NF 后存在滑移和打开的情况下的交叉行为。