紊流驱动下的不透水岩石叶片形（PKN）水力压裂
Blade-shaped (PKN) Hydraulic Fracture Driven By A Turbulent Fluid In An Impermeable Rock
摘要：
在石油和天然气工业中，高流速水驱动水力压裂比以往任何时候都更常见。虽然裂缝很小，但水的高注入速率和低粘度导致裂缝中的高雷诺数和潜在湍流。在这里，我们提出了一个半解析解的叶片形（PKN）几何水力压裂驱动的湍流流体在零流体泄漏到地层的限制。我们使用Gaukler-Manning-Strickler参数化来模拟PKN裂缝中的湍流，该参数化将水的流速与沿着裂缝的压力梯度联系起来。在这个关系中的关键参数是达西-韦斯巴赫摩擦系数的粗糙度的裂纹壁。耦合这种湍流参数化与质量守恒允许我们写一个非线性偏微分方程的裂纹宽度作为空间和时间的函数。通过相似性分析，我们得到了一个半解析解使用正交多项式级数。将裂缝尖端附近的渐近行为嵌入到多项式级数中，我们发现非常快速的收敛：用级数的两个项得到适当精确的解。这种封闭形式的解决方案有利于层流和湍流水力压裂的结果和参数之间的清晰比较。特别是，它解决了一个众所周知的问题，即校准模型的数据难以同时匹配的水力压裂长度和井筒压力。
英文摘要：
High flow rate, water-driven hydraulic fractures are more common now than ever in the oil and gas industry. Although the fractures are small, the high injection rate and low viscosity of the water, lead to high Reynolds numbers and potentially turbulence in the fracture. Here we present a semi-analytical solution for a blade-shaped (PKN) geometry hydraulic fracture driven by a turbulent fluid in the limit of zero fluid leak-off to the formation. We model the turbulence in the PKN fracture using the Gaukler-Manning-Strickler parametrization, which relates the the flow rate of the water to the pressure gradient along the fracture. The key parameter in this relation is the Darcy-Weisbach friction factor for the roughness of the crack wall. Coupling this turbulence parametrization with conservation of mass allows us to write a nonlinear pde for the crack width as a function of space and time. By way of a similarity ansatz, we obtain a semi-analytical solution using an orthogonal polynomial series. Embedding the asymptotic behavior near the fracture tip into the polynomial series, we find very rapid convergence: a suitably accurate solution is obtained with two terms of the series. This closed-form solution facilitates clear comparisons between the results and parameters for laminar and turbulent hydraulic fractures. In particular, it resolves one of the well known problems whereby calibration of models to data has difficulty simultaneously matching the hydraulic fracture length and wellbore pressure.
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