During its lifetime, a hydraulic fracture is known to traverse a trajectory in a region of a parametric space of non-dimensional evolutionary parameters. The topology of this diagram depends upon the phenomena considered. For the specific case of a 2D-plane strain fracture propagating in an elastic solid on a straight path normal to the minimum compressive stress, with a constant rate of injection of an incompressible newtonian fluid, and without leak-off, the diagram is a triangle whose vertices are typically called O, M, and K. The non-dimensional parameters are the toughness $\mathcal{K}$ and remote stress $\mathcal{T}$ (monotonically increasing with time). At each point in the trajectory $\mathcal{P}\left(t\right)=(\mathcal{K},\mathcal{T})\left(t\right)$, the configuration of the fracture is essentially described by several non-dimensional variables, in this case the opening ${\Omega }_0$ and pressure ${\Pi }_0$ at the inlet, and the length $\gamma$. When fluid lag is considered, as in this case, a fourth variable (e.g., the fluid fraction ${\xi }_f$) can be appended to build the descriptive set ${\mathcal{F}}_0=\{{\Omega }_0,{\Pi }_0,\gamma ,{\xi }_f\}$. Various propagation regimes are observed across the MKO triangle.

As the main results, we: (1) provide specific, $\mathcal{K}$-dependent transition times among the propagation regimes; and (2) found that the transient evolutions of all propagating cracks with moderate values of the non-dimensional toughness ($\mathcal{K}\gtrsim 0.3$), from the OK edge to the MK edge, are contained in a thin bundle about a universal curve in the ${\mathcal{F}}_0$-space. This result can be applied, e.g., to readily setup approximate initial conditions for more detailed hydraulic fracture propagation simulations. In addition, we developed a four-parameter family of parametrizations of the MKO triangle suitable for plotting trajectories and other loci on the triangle.

在其生命周期中，已知水力压裂在无量纲演化参数的参数空间区域中穿过轨迹。该图的拓扑取决于所考虑的现象。对于二维平面应变裂缝在弹性固体中沿垂直于最小压应力的直线路径传播的特定情况，以恒定速率注入不可压缩的牛顿流体，并且没有泄漏，该图是一个三角形其顶点通常称为 O、M 和 K。无量纲参数是韧性$\mathcal{钾}$ 和远程压力 $\mathcal{吨}$（随时间单调增加）。在轨迹中的每个点$\mathcal{磷}\left(吨\right)=(\mathcal{钾},\mathcal{吨})\left(吨\right)$，裂缝的配置基本上由几个无量纲变量描述，在这种情况下，开口 ${\Omega }_0$ 和压力 ${\Pi }_0$ 在入口处，长度 $\gamma$. 当考虑流体滞后时，如在这种情况下，第四个变量（例如，流体分数${\xi }_F$) 可以附加到构建描述集 ${\mathcal{F}}_0=\{{\Omega }_0,{\Pi }_0,\gamma ,{\xi }_F\}$. 在 MKO 三角形中观察到各种传播方式。

作为主要结果，我们：（1）提供具体的， $\mathcal{钾}$- 传播机制之间的依赖过渡时间；(2) 发现所有具有中等无量纲韧性值的扩展裂纹的瞬态演化 ($\mathcal{钾}\gtrsim 0.3$)，从 OK 边到 MK 边，包含在一个关于通用曲线的细丛中 ${\mathcal{F}}_0$-空间。该结果可用于，例如，为更详细的水力压裂扩展模拟容易地设置近似初始条件。此外，我们开发了 MKO 三角形的四参数系列参数化，适用于绘制三角形上的轨迹和其他轨迹。