This paper analyses the problem of a semi-infinite fluid-driven fracture propagating through multiple stress layers in a permeable elastic medium. Such a problem represents the tip region of a planar hydraulic fracture. When the hydraulic fracture crosses a stress layer, the use of a standard tip asymptotic solution may lead to a considerable reduction of accuracy, even for the simplest case of a height-contained fracture. In this study, we propose three approaches to incorporate the effect of stress layers into the tip asymptote: non-singular integral formulation, toughness-corrected asymptote, and an ordinary differential equation approximation of the non-singular integral formulation mentioned above. As illustrated in the paper, these approaches for stress-corrected asymptotes differ in computational complexity, the complexity of implementation, and the accuracy of the approximation. In addition, the size of the validity region of the stress-corrected asymptote is evaluated, and it is shown to be greatly reduced relative to the case without layers. In order to address the issue, the stress relaxation factor is introduced. This, in turn, allows for enhancing the accuracy of the layer-crossing computation on a relatively coarse mesh to utilize the stress-corrected asymptote in hydraulic fracturing simulators for the purpose of front tracking.

翻译：本文分析了半无限流体驱动裂缝在渗透性弹性介质中穿过多应力层扩展的问题。该问题代表了平面水力裂缝的尖端区域。当水力裂缝穿过应力层时，即使对于最简单的高度受限裂缝情况，使用标准尖端渐近解也可能导致精度显著下降。本研究提出了三种将应力层效应纳入尖端渐近解的方法：非奇异积分公式、韧性修正渐近解以及上述非奇异积分公式的常微分方程近似。如文中所示，这些应力修正渐近解方法在计算复杂度、实现难度和近似精度方面存在差异。此外，本文评估了应力修正渐近解的有效区域尺寸，结果表明该区域相对于无层状工况显著减小。为解决该问题，引入了应力松弛因子，从而能够在相对粗糙的网格上增强穿层计算的精度，以便在水力压裂模拟器中利用应力修正渐近解进行前沿追踪。