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

In the first part of this paper a universal fluid velocity based algorithm for simulating hydraulic fracture with leak-off was created for a penny-shaped crack. The power-law rheological model of fluid was assumed and the final scheme was capable of tackling both the viscosity and toughness dominated regimes of crack propagation. The obtained solutions were shown to achieve a high level of accuracy. In this paper simple, accurate, semi-analytical approximations of the solution are provided for the zero leak-off case, for a wide range of values of the material toughness and parameters defining the fluid rheology. A comparison with other results available in the literature is undertaken. Electronic supplementary material The online version of this article (10.1007/s11012-018-0903-6) contains supplementary material, which is available to authorized users.


Coefficients of the approximate solutions
For any value of the fluid behaviour index n and self-similar material toughnessK I , the selfsimilar crack propagation speedv 0 is given in the form (24) 1 . The values of respective coefficients C i are provided in Table S1 forK I = {0, 1, 10}.  Table S1: Values of the coefficients C i used to approximatev 0 (24) for different values of the fracture toughness.
Meanwhile, the coefficients of the constantĈ p (n), which takes the form (24) 2 in the viscosity dominated case (K Ic = 0), are provided in Table S2.
The remaining coefficients for the fracture aperture, pressure and fluid velocity approximations are outlined for different values of the fracture toughness below.
1.1 Viscosity dominated regime (K Ic = 0) In the general case 0 < n < 1 the coefficients of approximation for the aperture (21), fluid velocity (22) and pressure (23) are given as: with the values of r k , s k and κ for the case 0 < n < 1 being listed in Tables S3 and S4.
In the case of a Newtonian fluid n = 1 the coefficients used to approximate the aperture (21) and the fluid velocity (22) remain the same as in the general case. The coefficients of the pressure approximation (42) are now given by:

Toughness dominated regime withK I = 1
In the general case 0 < n < 1 the approximation coefficients for the aperture (29), fluid velocity (30) and pressure (31) are given in the form: with the values of r k , s k for the case 0 < n < 1 being listed in Tables S5 and S6. For the Newtonian fluid n = 1 the coefficients for the approximate pressure function (43) are now given by: while those for the aperture (29) and fluid velocity (30) remain the same as in the general case.
S2  Table S3: The values of coefficients r i used in approximation (S1) in the general case 0 < n < 1 withK  Table S4: The values of coefficients s i and κ used in approximation (S1) in the general case 0 < n < 1 withK I = 0.

S4
In the case of a perfectly plastic fluid n = 0 the coefficients used to approximate the aperture (48) and fluid velocity (46) are as follows: while those for the pressure function (31) remain the same as in the general case.

Toughness dominated regime withK I = 10
In the general case 0 < n < 1 the approximation coefficients for the aperture (29), fluid velocity (30) and pressure (31) are given in the form: with the values of r k , s k for the case 0 < n < 1 being listed in Tables S7 and S8. For a Newtonian fluid n = 1 the coefficients for the approximate pressure function (43) are now given by: p 1 = 8.86228, p 2 = 9.23151 × 10 −6 , p 3 = −1.384716 × 10 −5 , p 4 = −8.6771 × 10 −11 , (S8) while those for the aperture (29) and fluid velocity (30) remain the same as in the general case.
Finally, in the case of a perfectly plastic fluid n = 0 the coefficients used to approximate the aperture (48) and fluid velocity (46) are as follows: