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.