Abstract This paper presents an extension of the variational interface fracture model proposed in Khisamitov and Meschke (2018) to model fluid driven fracture in porous materials. The fluid saturated material is described by the theory of poroelasticity and the effective stress concept, while the coupling between fluid flow and fracture initiation and propagation is accomplished via a variational interface element formulation, using a damage variable c as an additional degree of freedom in conjunction with linear momentum and mass balance equations along the interface zones. Introducing the damage variable c , fracture propagates according to the values of the minimizers of the total potential energy expressed in terms of effective stresses. Hence, neither local crack propagation criteria nor tracking algorithms are required. Taking into account that a fracture grows quasi-statically, the discretized system of PDEs is solved by the backward Euler-scheme, ignoring the contribution from inertia forces. In the Newton–Raphson iterative solution procedure, an operator splitting algorithm is employed to solve first the poroelastic equations, and updating subsequently the damage variable c . The proposed model for hydraulic fracturing is validated first by means of one-dimensional analytical solutions in toughness as well as in viscosity dominated regimes. The performance and the applicability of the model to simulate interactions between propagating and existing fractures is demonstrated by means of 2D and 3D numerical examples.

中文翻译：

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用于 2D 和 3D 水力压裂模拟的变分界面元素模型

摘要 本文介绍了 Khisamitov 和 Meschke (2018) 提出的变分界面裂缝模型的扩展，以模拟多孔材料中的流体驱动裂缝。流体饱和材料由多孔弹性理论和有效应力概念描述，而流体流动与裂缝起裂和扩展之间的耦合通过变分界面单元公式完成，使用损伤变量 c 作为附加自由度结合具有沿界面区域的线性动量和质量平衡方程。引入损伤变量 c ，裂缝根据以有效应力表示的总势能的最小值的值扩展。因此，既不需要局部裂纹扩展标准，也不需要跟踪算法。考虑到裂缝准静态增长，偏微分方程的离散系统由后向欧拉方案求解，忽略惯性力的贡献。在 Newton-Raphson 迭代求解过程中，使用算子分裂算法首先求解多孔弹性方程，然后更新损伤变量 c。提出的水力压裂模型首先通过韧性和粘度主导状态的一维解析解进行验证。通过 2D 和 3D 数值示例证明了该模型在模拟扩展裂缝和现有裂缝之间的相互作用方面的性能和适用性。忽略惯性力的贡献。在 Newton-Raphson 迭代求解过程中，使用算子分裂算法首先求解多孔弹性方程，然后更新损伤变量 c。提出的水力压裂模型首先通过韧性和粘度主导状态的一维解析解进行验证。通过 2D 和 3D 数值示例证明了该模型在模拟扩展裂缝和现有裂缝之间的相互作用方面的性能和适用性。忽略惯性力的贡献。在 Newton-Raphson 迭代求解过程中，使用算子分裂算法首先求解多孔弹性方程，然后更新损伤变量 c。提出的水力压裂模型首先通过韧性和粘度主导状态的一维解析解进行验证。通过 2D 和 3D 数值示例证明了该模型在模拟扩展裂缝和现有裂缝之间的相互作用方面的性能和适用性。提出的水力压裂模型首先通过韧性和粘度主导状态的一维解析解进行验证。通过 2D 和 3D 数值示例证明了该模型在模拟扩展裂缝和现有裂缝之间的相互作用方面的性能和适用性。提出的水力压裂模型首先通过韧性和粘度主导状态的一维解析解进行验证。通过 2D 和 3D 数值示例证明了该模型在模拟扩展裂缝和现有裂缝之间的相互作用方面的性能和适用性。