| The problem of a fluid-driven, plane-strain fracture propagating in a permeable, elastic rock of arbitrary toughness is investigated in this work. The fluid flow within the fracture is modeled using the lubrication theory, and the fluid losses to the permeable rock are described using the Carter's law. The main objective of this research is to obtain a general solution which prescribes evolution of the fluid pressure, the fracture opening and the fracture extent given rock properties, fluid rheology and injection rate. This general solution provides fracture evolution/transition between appropriate asymptotic regimes.; The physics of the problem is determined by the two competing dissipation mechanisms corresponding to the viscous dissipation in the fluid flow in the fracture and to the dissipation in fracturing the rock, and by the two competing fluid storage mechanisms corresponding to storage in the fracture and in the rock, respectively.; Four primary asymptotic regimes of hydraulic fracture propagation, where one of the two dissipative mechanisms and one of the two fluid storage mechanisms are vanishing, can then be identified: storage-viscosity (M), storage-toughness (K), leak-off-viscosity (M˜ ), and leak-off-toughness (K˜) dominated regimes. Scaling laws can then be presented (Detournay and Garagash, 2005) in a rectangular parametric space (the "MM˜K˜K rectangle"), with each vertex corresponding to one of the four primary asymptotic regimes.; General solutions (inside the MM˜K˜K parametric rectangle) of a fracture driven by Newtonian and shear-thinning non-Newtonian fluids for the crack length evolution, the fracture opening, the fluid pressure and the fluid flow rate inside the fracture as function of position and two dimensionless parameters, are obtained numerically. Numerical solutions are guided by existing and new asymptotic solutions, corresponding to various vertices and edges of the parametric space. In particular, new asymptotic solutions for the MM˜-edge ("viscosity" edge of the parametric space corresponding to fracture propagation along pre-existing discontinuity with zero toughness) and for the M˜-vertex (viscosity/leak-off dominated regime) are obtained in the case of a non-Newtonian (power law) fluid. The solutions are obtained using numerical method of lines which utilizes a dominant fracture tip asymptote. Obtained mapping of the solution in the parametric space allows for in-flight HF data inversion and, possibly, real-time control of the treatment. |
