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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 K and remote stress T (monotonically increasing with time). At each point in the trajectory P ( t ) = ( K , T ) ( t ) , the configuration of the fracture is essentially described by several non-dimensional variables, in this case the opening Ω 0 and pressure Π 0 at the inlet, and the length γ . When fluid lag is considered, as in this case, a fourth variable (e.g., the fluid fraction ξ f ) can be appended to build the descriptive set F 0 = { Ω 0 , Π 0 , γ , ξ f } . Various propagation regimes are observed across the MKO triangle. As the main results, we: (1) provide specific, 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 ( K ≳ 0 . 3 ), from the OK edge to the MK edge, are contained in a thin bundle about a universal curve in the 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. |
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