Approximate semi-analytical solution for a penny-shaped rough-walled hydraulic fracture driven by turbulent fluid in an impermeable rock

The popularity of high injection rate hydraulic fracturing treatments using low viscosity fluids is driving a need to consider the turbulent and laminar-turbulent transition regimes of fluid flow in hydraulic fracture simulators. The radial model is one of the most important geometries both for benchmarking and as a starter solution for 3D and Planar 3D models. Here we provide a semi-analytical, orthogonal polynomial series solution for a rough-walled radial (penny-shaped) hydraulic fracture driven by a fully turbulent fluid. Embedding the appropriate pressure singularities in a family of orthogonal polynomials used for derivation of the solution leads to very rapid convergence of the series, requiring just two terms for an accurate result. We conclude with an investigation of the occurrence of this limiting regime by comparison with numerical simulations, illustrating that the fully turbulent regime is typically not encountered for the radial geometry, although the present solution remains necessary as a starter solution and benchmark for the numerical simulators that are required to capture the laminar-turbulent transition. By comparison with numerical simulations that consider the laminar-turbulent transition, we find that such an estimate is expected to be sufficient for practical purposes when the inlet opening predicted by the turbulent solution exceeds the inlet opening predicted by the laminar solution.