Fluid velocity based simulation of hydraulic fracture: a penny shaped model—part I: the numerical algorithm

Abstract

In the first part of this paper, a universal fluid velocity based algorithm for simulating hydraulic fracture with leak-off, previously demonstrated for the PKN and KGD models, is extended to obtain solutions for a penny-shaped crack. The numerical scheme is capable of dealing with both the viscosity and toughness dominated regimes, with the fracture being driven by a power-law fluid. The computational approach utilizes two dependent variables; the fracture aperture and the reduced fluid velocity. The latter allows for the application of a local condition of the Stefan type (the speed equation) to trace the fracture front. The obtained numerical solutions are carefully tested using various methods, and are shown to achieve a high level of accuracy. Electronic supplementary material The online version of this article (10.1007/s11012-018-0899-y) contains supplementary material, which is available to authorized users.

Fluid velocity based simulation of hydraulic fracture—a penny shaped model. Part II: new, accurate semi-analytical benchmarks for an impermeable solid

“…In this section, the algorithm described in part I of this paper [ 21 ] is used to deliver highly accurate numerical benchmark solutions. A comparative analysis with other data available in the literature is given.…”

“…While the numerical simulator constructed in the first part of the paper [ 21 ] is capable of providing high quality reference data, it is not necessarily in a form which can be easily utilized when testing various computational algorithms. Following the idea from [ 12 ], we shall also deliver simple and accurate semi-analytical approximations of the numerical solutions, which can easily be used as benchmark examples without the need for advanced computational programs.…”

“…In the viscosity dominated regime the crack tip asymptotics of the aperture and pressure derivative can be expressed as follows: The asymptotic behaviour of the pressure function can be derived from the above, however, this form is given due to its use in computations (see the first part of this paper [ 21 ] for more details).…”

“…In the previous paper [ 21 ], the behaviour of the solution as was shown to take the form: where , for the case when is defined by ( 8 ), is as stated in ( 9 ).…”

“…The separation of variables used in the self-similar formulation, description of the crack tip asymptotics and limiting behaviour in the case of infinite toughness (given above) provide all of the details we need to define the semi-analytical approximations of numerical solutions and perform comparisons with other benchmarks available in the literature. The full set of governing equations (both the standard and self-similar forms), alongside a complete description of the computational algorithm used to obtain the numerical reference data, are provided in part I of this paper [ 21 ] and will not be repeated here.…”