  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.
