This paper is concerned with the initiation of plane strain hydraulic fractures (HF) from a wellbore in an impermeable homogeneous elastic rock. The fractures are driven by Newtonian fluids injected into the wellbore. The solid deformation is modeled according to linear elasticity, and the viscous fluid flow within the fracture is modeled using lubrication theory. Compressibility effects are introduced though the addition of a fluid pressure-dependent wellbore storage term in the condition on the fluid flow at the fracture inlet. A solution is obtained in terms of fluid net pressure, fracture length, and opening. The problem depends on a dimensionless viscosity and two additional dimensionless parameters, one that is related to the compressibility and the other that is related to the deviatoric in-situ stress. We compute the solution for initiation and the early stages of propagation of a HF using an implicit finite difference scheme with a fixed spatial grid coupled with the displacement discontinuity method. An instability is identified in the problem after breakdown for an inviscid fluid, and results of the numerical simulation indicate that viscosity effects mitigate the initial instability. We also show that the difference between the breakdown (peak) pressure and the fracture initiation pressure increases with the viscosity of the fracturing fluid.
In the beginning stages of a hydraulic fracturing treatment, prior to fracture initiation, fluid that is injected is stored in the wellbore due to the compressibility of the fluid/wellbore system. The release of this compressed volume during the initiation process can lead to a sudden instability that includes a jump in fracture length and a sudden decrease of the injection pressure at breakdown. However, this sudden instability is mitigated by dissipation of the stored energy through flow of the viscous fluid. Experiments have shown that the combined effect of compressibility and viscous flow can cause the injection pressure to continue to rise for some period of time after initial crack growth [1, 2]. This effect also could make it difficult to interpret the observed breakdown pressures in microfracturing experiments [3]. In this paper, we consider the plane strain propagation of a fracture that is growing away from a wellbore in an impermeable linear elastic rock characterized by Young's modulus E, Poisson's ratio ν, and fracture toughness K Ic, see Figure 1. The fracture is driven by a Newtonian fluid of viscosity µ, injected at the wellbore at a constant rate Q 0, so that the volume of injected fluid is given at any time t, by V(t) = Q 0 t. The crack surfaces are loaded by the internal fluid pressure p ƒ (x,t) (where x is the position along the crack) and by the far-field stresses δ1, δ3. The fracture is assumed to be in mobile equilibrium at all times so that K I= K Ic and its propagation is quasi-static under conditions of negligible fluid lag.
