Similarity Solutions for Shallow Hydraulic Fracture in Impermeable Rock

The paper deals with the plane strain problem of a shallow fluid-driven fracture propagating parallel to a free surface in an impermeable elastic rock. For a shallow fracture, the aperture corresponds essentially to the deflection of the layer defined by the free-surface and the crack. This problem is thus governed by the Euler-Bernoulli beam equation and by the lubrication theory. We show that the solution generally depends on two independent time scales and that there exist 4 similarity solutions denoted as Ō, S̄, M̄ and K̄. Furthermore, we demonstrate that the parametric space of solution is a rectangle with the vertices corresponding to the similarity solutions. The Ō- and S̄-solutions are two different early time solutions characterized by a small length of the fluid-infiltrated region compared to the fracture length (i.e., the fluid lag occupies nearly the whole fracture), while the M̄ and A solutions are two large-time solutions characterized by a vanishingly small lag. The M̄- and the K̄- vertices are similarity solutions for the particular case when Gc = 0, while Ō- and S̄- are similarity solutions for the case σo = 0. Thus the S̄M̄-edge and the S̄M̄-edge represent two particular fracture evolutions, the first one when Gc = 0 and the second one when σo = 0. However, the large time solution for the general case Gc > 0 and σo > 0 has not yet been identified, as the numerical solutions conflict with the large time matched asymptotic solution.