**Acknowledgements**

*<sup>v</sup>* <sup>=</sup> *<sup>k</sup> x*\*

*v*\* (*t*)= *<sup>d</sup> <sup>x</sup>*\* *dt* <sup>=</sup> *<sup>k</sup> x*\* 1/*<sup>n</sup>* (- ∂ *y* <sup>∂</sup> *<sup>ς</sup>* )1/*<sup>n</sup>*

where

sence in the SE (48).

654 Effective and Sustainable Hydraulic Fracturing

ing. Since *dp* =( *dp*

**6. Conclusions**

ous section, are of use for the P3D models.

of the modified pressure *P* = *p* 1/*α* is obvious.

1/*<sup>n</sup> H* (*y*)(-

*y*(*ς*, *t*0) = *y*<sup>0</sup>

*<sup>y</sup> <sup>α</sup><sup>v</sup> <sup>ς</sup>*=0 <sup>=</sup>*q*<sup>0</sup>

( ) ( ( )) ( )

*p*

*d wG w H y <sup>w</sup> dw* é ù ê ú <sup>=</sup> ê ú ë û

1/ G *n*

is a function to be found in advance through the functions *Gp*(*w*) and G*v*(*w*) of the argu‐ ment *<sup>w</sup>* <sup>=</sup> *<sup>y</sup> <sup>α</sup>*. By the properties of *Gp*(*w*) and G*v*(*w*), we have *<sup>H</sup>* (0 )=1 which explains its ab‐

The problem (43)-(49) differs from the problem (27)-(33) in the only detail: equation (44) for the velocity contains the function *H* (*y*) defined by (50). The latter function, being smooth and tending to the unity at the front, the efficient numerical schemes, discussed in the previ‐

*Comment*. In some cases, it may be convenient to use the net-pressure rather than the open‐

The discussion above demonstrates the analytical and computational advantages of using the modified formulation. The analytical advantages are evident from the obtained simple analytical solutions for the PKN and KGD models, which otherwise require involved calcu‐ lations. The computational advantages include: (i) the possibility to use the well-established theory of propagating surfaces, (ii) avoiding deterioration of numerical solution caused by ill-posedness of the problem when neglecting the lag and fixing the fracture contour at a

*dw* )*dw*, reformulation of the equations and computational schemes in terms

*v*

∂ *y*

<sup>∂</sup> *<sup>ς</sup>* )1/*<sup>n</sup>* (44)

(*ς*) (45)

(*t*), (46)

*<sup>ς</sup>*=1 (48)

(50)

*y*(1, *t*)=0 (47)

*x\**(*t*0) = *x\**<sup>0</sup> (49)

The authors appreciate support of the EU Marie Curie IAPP program (Grant # 251475). The first author (AL) also thankfully acknowledges support of the Russian Fund of Fundamental Investigations as concerns with the results on non-Newtonian fluids (Grant # 12-05-00140). Both authors are grateful to Piotr Kusmierczyk and Michal Wrobel for the help in performing many numerical experiments for the Nordgren's problem with Carter's leak-off.
