| This research is oriented to the analysis of a fluid-driven, plane-strain fracture propagating in a permeable, elastic medium. The fluid flow within the fracture is modeled using the lubrication theory, and the fluid losses are described using Carter's law. The aim is to understand the propagation regimes that occur during hydraulic fracturing (HF) operations that are carried out for enhancement of hydrocarbon reservoir recoveries.; Scaling laws are presented for each of three energy dissipative processes: viscous flow of the fracturing fluid (M), fracturing of the rock (K), and diffusion of fluid into the formation (C). The scaling laws are represented as a triangular parametric space (the “MKC triangle”), with each vertex corresponding to an ideal situation with only one dissipative process. A hydraulic fracture evolves in this space, moving with time from one propagation regime to another. The changing behavior of the fracture tip is analyzed by studying a semi-infinite fluid-driven fracture in steady propagation.; Semi-analytical solutions are constructed for the MKC triangle vertices: the M-vertex solution (impermeable medium, zero toughness) is obtained by expanding the crack opening in a series of Gegenbauer polynomials, with the series coefficients calculated using a minimization procedure. The C-vertex (permeable medium, infinite time) represents the paradoxical case of a propagating fracture with no volume. The K-vertex (impermeable medium, inviscid fluid) corresponds to the classical problem of a uniformly pressurized crack. A regular asymptotic expansion is used to find a solution in the vicinity of the C-vertex.; Solutions along two edges of the triangle are also presented. The MK-edge solution (impermeable medium, finite toughness) is found by expanding the crack opening in a series of Chebyshev polynomials. The MC-edge solution (permeable medium, zero toughness) is obtained using a numerical algorithm that combines an explicit finite-difference scheme with the displacement discontinuity method.; Results of the numerical simulations indicate that the “leak-off-dominated” propagation regime (in which the fracture length evolves as a square-root of time) may never be reached in actual HF treatments. Also, the M-vertex solution approximates well the MK-edge solution for low values of dimensionless toughness. |
