Propagation regimes, transition times, and approximate universality in 2D hydraulic fracture propagation with fluid lag

Abstract

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 Ω0 and pressure Π0 at the inlet, and the length γ. When fluid lag is considered, as in this case, a fourth variable (e.g., the fluid fraction ξf) can be appended to build the descriptive set F0={Ω0,Π0,γ,ξf}. 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 F0-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.