We analyse the influence of fluid yield stress on the propagation of a radial (penny-shaped) hydraulic fracture in a permeable reservoir. In particular, the Herschel–Bulkley rheological model is adopted that includes yield stress and non-linearity of the shear stress. The rock is assumed to be linear elastic, and the fracture is driven by the point source fluid injection with a constant volumetric rate. The fracture propagation condition follows the theory of linear elastic fracture mechanics, and Carter’s leak-off law is selected to govern the fluid exchange process between the fracture and formation. The numerical solution of the problem is found using the algorithm based on Gauss–Chebyshev quadrature and Barycentric Lagrange interpolation techniques. We also construct an approximate solution with the help of the global fluid balance equation and the near-tip region asymptote. The latter approximation is computationally efficient, and we estimate its accuracy by comparing the primary crack characteristics such as opening, pressure, and radius with those provided by the full numerical solution. We present examples corresponding to typical field cases and demonstrate that the addition of yield stress can lead to a shorter radius and wider opening compared to the corresponding case with simpler power-law fluid rheology. Further, we quantify the limiting propagation regimes (or vertex solutions) characterised by the dominance of a particular physical phenomenon. Relative to the power-law results, there are two new vertices that are associated with the domination of yield stress: storage-yield-stress and leak-off-yield-stress. To understand the influence of various problem parameters, we utilise the constructed approximate solution to investigate the dimensionless parametric space of the problem, in which the applicability domains of the limiting solutions are quantified. This enables one to quickly determine whether yield stress provides a strong influence for given problem parameters.

我们分析了流体屈服应力对渗透性储层中径向（硬币形）水力压裂扩展的影响。特别是采用了 Herschel-Bulkley 流变模型，其中包括屈服应力和剪切应力的非线性。假设岩石是线弹性的，裂缝是由恒定体积速率的点源流体注入驱动的。裂缝扩展条件遵循线弹性断裂力学理论，选择卡特漏失定律来控制裂缝与地层之间的流体交换过程。该问题的数值解是使用基于高斯-切比雪夫正交和重心拉格朗日插值技术的算法找到的。我们还在全局流体平衡方程和近尖端区域渐近线的帮助下构建了一个近似解。后一种近似在计算上是有效的，我们通过将主要裂纹特征（如开口、压力和半径）与完整数值解提供的特征进行比较来估计其精度。我们提供了对应于典型现场案例的示例，并证明与具有更简单幂律流体流变学的相应案例相比，增加屈服应力可以导致更短的半径和更宽的开口。此外，我们量化了以特定物理现象的优势为特征的限制传播机制（或顶点解）。相对于幂律结果，有两个与屈服应力支配相关的新顶点：储存屈服应力和泄漏屈服应力。为了理解各种问题参数的影响，我们利用构建的近似解来研究问题的无量纲参数空间，其中量化了限制解的适用范围。这使人们能够快速确定屈服应力是否对给定的问题参数有很大的影响。