Propagation of a penny-shape hydraulic fracture in an impermeable rock

This paper deals with the self-similar solution of a penny-shape hydraulic fracture propagating in an impermeable elastic rock. Growth of the fracture is driven by injection of an incompressible Newtonian fluid at the center of the fracture, at a flow rate varying according to a power law of time (which includes the practically important case of a constant injection rate). The solution is restricted to the so-called viscosity-dominated regime where it can be assumed that the rock has zero toughness. In this regime, the fracture tip is characterized by a singularity which is weaker than the classical square root singularity of linear elastic fracture mechanics. The paper describes the construction of a semi-analytical similarity solution, which incorporates the known singularity of the fluid pressure at the center of the fracture and at the tip and which is based on series expansions of the fracture opening and fluid pressure in terms of Jacobi polynomials. It is shown that very few terms in the expansions are needed to capture the solution accurately.