ABSTRACT: This paper deals with the self-similar solution of a penny-shape hydraulic fracture propa- gating 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.
INTRODUCTION Mathematical modeling of hydraulic fractures has attracted numerous contributions since the 1950's. While early efforts dealt mainly with (approxi- mate) analytical solutions (see e.g. Khristianovic and Zheltov, 1955; Barenblatt, 1962; Perkins and Kern, 1961; Nordgren, 1972; Geerstma and Haaf- kens, 1979), the focus of researchas shifted in re- cent years towards the development of numerical algorithms to model the three-dimensional propa- gation of hydraulic fractures in layered strata char- acterized by different mechanical properties and/or in-situ stresses (e.g. Clifton and Abou-Sayed, 1979; Advani et al., 1990, Sousa et al., 1993; Shah et al.,
Despite this trend towards the development of realistic models of hydraulic fractures, there is still interest in obtaining "exact" solutions for simpler models with rigorous consideration given to both the flow of fluid in the fracture (generally modeled according to the lubrication theory) and the elas- tic deformation and propagation of the fracture.
Such solutions can be used not only to benchmark numerical codes but also to explore the depen- dance of the solution to various parameters and to establish the existence of different regimes of propagation. These solutions are notoriously dif- ficult to construct, however, because of the strong non-linear coupling between the lubrication and elasticity equations and the non-local character of the elastic response of the fracture.
Within the realm of "simple" models for hy- draulic fracturing, the penny-shape fracture is guably the most relevant one. Yet it has been treated by an handful of authors only (e.g. Baren- blatt, 1959; Abe et al., 1976; Abe et al., 1979; Cleaxy and Wong, 1985; Nilson et al., 1985; Ad- vani et al., 1987; Barr, 1991; de Pater et al., 1996; Yuan, 1997). Furthermore, no rigorous analytical (or semi-analytical) solution of this problem ex- ists to our knowledge, as published analytical so- lutions are based on approximations involving ad hoc forms of the fluid pressure or the crac? aper- ture.
This paper deals with the construction of a semi-analytical solution for the problem of a penny- shape crack propagating in an unbounded imper- meable elastic medium, see Fig. 1. The fracture is driven by an incompressible Newtonlan fluid in- jected at the center of the fracture. As discussed in the paper, the injection flow rate is restricted to a power law of time, which includes, however, the practically important case of a constant flow rate.