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

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 Rho(t) = (K,T)(t), 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 Tau(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, 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 greater than or similar to 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.

Automated summary

The paper discusses the trajectory and topology of hydraulic fractures in a non-dimensional evolutionary parameter space, as well as the description and parameterization of fracture configurations in specific cases, providing a method for setting approximate initial conditions for more detailed hydraulic fracture simulations.