**5. Extension to P3D models**

In the previous section we have stated that the modified formulation provides prerequisites for efficient solving the Nordgren's problem. The latter, being the basis for the P3D models, we may extend the efficient schemes to these models.

The detailed description of the P3D models is given by Mack and Warpinski [1]. Thus it is sufficient to list only those their features, which distinguish them from the PKN model (Fig. 1) and which are significant for the extension.

(i) In P3D models, the in-situ rock stresses are different in various layers while the fluid pressure is assumed constant in any vertical cross-section. This implies that, in contrast with the PKN model, the net-pressure is now not constant in a vertical cross-section. Thus the P3D models employ the pressure itself rather than the net-pressure. Alternatively, one may employ the difference of the fluid pressure with a *fixed* value of in-situ rock pressure, say the pressure in the pay-layer. Below to keep clear connection with the PKN model, we shall use this option and conditionally call the difference the 'net-pressure'.

(ii) The linear dependence (13) between the net-pressure *p* and the average opening *wav* of a cross-section is changed to a non–linear dependence *p* = *p*(*wav*). To have clear connection with the initial PKN model, we re-write it as

$$\mathbf{w} = k\_r \mathbf{w}\_{av} F\_p(\mathbf{w}\_{av}) \tag{38}$$

where for sufficiently small *wav*, in particular near the fracture front, *F <sup>p</sup>*(*wav*) =1. A specific form of the dependence (38) is found from solving plane-strain elasticity equation for a straight vertical crack under the fracture conditions of the form *KI* ≤ *K Ic* at the upper and bottom tips. Although looking for this dependence may be involved, it is found in advance for a prescribed geometry of layered stratum, in-situ stresses in layers and critical SIFs. These preliminary calculations also give the positions *zu*(*p*) and *zl*(*p*) of the upper and bot‐ tom tips, respectively. Consequently, the height *h <sup>f</sup>* (*p*) of the fracture in a cross section is a known function of the pressure:

$$h\_f(p) = z\_u(p) \cdot z\_l(p) \tag{39}$$

(iii) The dependence (12) between the particle velocity and the pressure, after averaging the velocity over the height of a cross-section, obtains a factor *Fv*(*wav*):

A similarly efficient computational scheme consists of solving the PDE (27), (28) with *ε*-

*Comment.* In view of the accepted linear dependence (13) between the net-pressure and the opening, the pressure may replace the opening as an unknown variable. Actually, under the

In the previous section we have stated that the modified formulation provides prerequisites for efficient solving the Nordgren's problem. The latter, being the basis for the P3D models,

The detailed description of the P3D models is given by Mack and Warpinski [1]. Thus it is sufficient to list only those their features, which distinguish them from the PKN model (Fig.

(i) In P3D models, the in-situ rock stresses are different in various layers while the fluid pressure is assumed constant in any vertical cross-section. This implies that, in contrast with the PKN model, the net-pressure is now not constant in a vertical cross-section. Thus the P3D models employ the pressure itself rather than the net-pressure. Alternatively, one may employ the difference of the fluid pressure with a *fixed* value of in-situ rock pressure, say the pressure in the pay-layer. Below to keep clear connection with the PKN model, we shall use

(ii) The linear dependence (13) between the net-pressure *p* and the average opening *wav* of a cross-section is changed to a non–linear dependence *p* = *p*(*wav*). To have clear connection

where for sufficiently small *wav*, in particular near the fracture front, *F <sup>p</sup>*(*wav*) =1. A specific form of the dependence (38) is found from solving plane-strain elasticity equation for a straight vertical crack under the fracture conditions of the form *KI* ≤ *K Ic* at the upper and bottom tips. Although looking for this dependence may be involved, it is found in advance for a prescribed geometry of layered stratum, in-situ stresses in layers and critical SIFs. These preliminary calculations also give the positions *zu*(*p*) and *zl*(*p*) of the upper and bot‐ tom tips, respectively. Consequently, the height *h <sup>f</sup>* (*p*) of the fracture in a cross section is a

*p* =*krwavF <sup>p</sup>*(*wav*) (38)

*h <sup>f</sup>* (*p*)= *zu*(*p*) - *zl*(*p*) (39)

regularized BC and SE by the Crank-Nicolson method [3,6].

we may extend the efficient schemes to these models.

this option and conditionally call the difference the 'net-pressure'.

1) and which are significant for the extension.

with the initial PKN model, we re-write it as

known function of the pressure:

**5. Extension to P3D models**

652 Effective and Sustainable Hydraulic Fracturing

normalization, yielding *kr* =1, there is no difference between *p* and *w*.

$$
\omega\_{av} = \left(\therefore k\_f \, w\_{av}^{n+1} \frac{\partial \, p}{\partial x}\right)^{1/n} F\_v \left(\!\!/ w\_{av}\right) \tag{40}
$$

where for sufficiently small *wav*, we have *Fv*(*wav*) =1. Since *wav* →0 when approaching the fracture front, the SE becomes:

$$
\omega\_{\upsilon\ast} = \left( -k\_f \, \text{tr}\_{av} \, w\_{av}^{n+1} \, \frac{\partial \, p}{\partial \, x} \right)^{1/n} \, \text{x} \, \text{w} \, \text{s} \tag{41}
$$

(iv) The continuity equation is integrated over the cross-sectional height. As a result, the to‐ tal flux through a cross-section and the total leak-off replace, respectively, the flux and leakoff per unit height. The average velocity *vav*, being defined as the ratio of the total flux to the area *A* of the cross-section, the area *A* replaces the opening present in the PKN model. To preserve connection with the PKN model, we may use the flux, leak-off losses and area div‐ ided by a fixed reference height *hr*, say the height of the pay-layer. Then denoting

$$
\varpi \upsilon = \frac{A}{h\_r} \, \, \, q = \frac{A \upsilon\_{av}}{h\_r} = \varpi \upsilon \upsilon\_{av} \, \, \, q\_l = \frac{Q\_l}{h\_r} \, \, \, \, \tag{42}
$$

we have the modified continuity equation (15) in the unchanged form. The BC (18) at the inlet is also the same when denoting *q*0(*t*) =*Q*0(*t*) / *hr*, where *Q*0(*t*) is the prescribed total in‐ flux.

Take into account that by the definition of the average opening, we have *A*=*wavh <sup>f</sup>* , where *h <sup>f</sup>* is the fracture height in a considered cross-section. Thus, by the first of (42), *w* = *Ah <sup>f</sup>* / *hr*. Then, in view of (39) and (38), we may use the argument *w* instead of *wav* in the equations (38), (40) and(41) writing them, respectively, as

$$p = k\_r \boldsymbol{\omega} \upsilon \, G\_p(\boldsymbol{\omega} \upsilon), \upsilon = \left( \cdot k\_f \, \upsilon \, \upsilon^{n+1} \frac{\partial \, p}{\partial \, \boldsymbol{x}} \right)^{1/n} \, G\_v(\boldsymbol{\omega} \upsilon), \text{ and } \upsilon \ast = \left( \cdot k\_f \, \upsilon \, \upsilon^{n+1} \frac{\partial \, p}{\partial \, \boldsymbol{x}} \right)^{1/n} \, \upsilon \ast \upsilon.$$

where to simplify notation, we have omitted the subscript in the averaged velocity *vav*. The functions *Gp*(*w*) and *Gv*(*w*) are found in advance through the functions *F <sup>p</sup>*(*wav*) and *Fv*(*wav*). They are such that *Gp*(*w*)=1 and *Gv*(*w*)=1 for sufficiently small *w*.

Finally, by introducing the variables *<sup>y</sup>* <sup>=</sup>*<sup>w</sup>* 1/*α* and *<sup>ς</sup>* <sup>=</sup> *<sup>x</sup>* / *x\**, we arrive at the system similar to (27)-(33):

$$\frac{\partial \underline{y}}{\partial t} = -\frac{y}{\alpha \text{x}\_\*} \frac{\partial \underline{v}}{\partial \underline{\varphi}} + \frac{\varphi \upsilon\_\* - \upsilon}{\text{x}\_\*} \frac{\partial \underline{y}}{\partial \underline{\varphi}} - \underline{y^{1-\alpha}\_\*} q\_{1'} \tag{43}$$

$$\sigma = \frac{k}{x\_\*^{1/n}} H\left(y\right) \Big(\cdot, \frac{\partial \cdot y}{\partial \cdot \zeta}\Big)^{1/n} \tag{44}$$

$$y\_0(\varepsilon\_\prime \ t\_0) = y\_0(\varepsilon) \tag{45}$$

time step, (iii) avoiding singularities on the fluid front, (iv) the possibility to use highly effi‐ cient numerical schemes for the PKN and P3D models. These beneficial features are of sig‐ nificance when developing simulators able to efficiently solve truly 3D and pseudo-threedimensional problems in real time. The work on simulators of both types is in progress.

Modified Formulation, ε-Regularization and the Efficient Solution of Hydraulic Fracture Problems

http://dx.doi.org/10.5772/56218

655

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.

1 Department of Mathematics, Rzeszow University of Technology, Rzeszow, Poland

3 Institute of Mathematical and Physical Sciences, Aberystwyth University, Aberystwyth, UK

[1] Mack MG., Warpinski NR. Mechanics of hydraulic fracturing. In: Economides MJ., Nolte KG. (eds) Reservoir simulation, 3-rd edn. John Willey & Sons; 2000. p6.1-6.49.

[2] Linkov AM. Speed equation and its application for solving ill-posed problem of hy‐

[3] Linkov AM. Use of speed equation for numerical simulation of hydraulic fractures.

[4] Linkov AM. On efficient simulation of hydraulic fracturing in terms of particle veloc‐

[5] Linkov AM. Numerical modeling of hydraulic fractures: State of art and new results. In: Advanced Problems in Mechanics, APM 2012: proceedings of International XL

2 Institute for Problems of Mechanical Engineering, Saint-Petersburg, Russia

draulic fracturing. Doklady Physics 2011; 56 (8) 436-438.

Available at: http://arxiv.org/abs/1108.6146.

ity. Int. J. Engineering Sci. 2012; 52, 77-88.

**Acknowledgements**

**Author details**

**References**

Alexander M. Linkov1,2\* and Gennady Mishuris3

\*Address all correspondence to: voknilal@hotmail.com

$$\left[y^{\;\;\alpha}v\right]\_{\prec^{\alpha}0} = q\_0(t),\tag{46}$$

$$(y(1,t) = 0)\tag{47}$$

$$\upsilon\_{\ast}(t) = \frac{d\,\,\mathbf{x}\_{\ast}}{dt} = \frac{k}{\chi\_{\ast}^{1/n}} \mathbb{E}\left[ \left( -\frac{\partial\_{\ast}\,y}{\partial\_{\ast}\zeta} \right)^{1/n} \right]\_{\zeta=1} \tag{48}$$

$$\mathbf{x}\_{\ast}(t\_0) = \mathbf{x}\_{\ast 0} \tag{49}$$

where

$$H(y) = \left[\frac{d\left(w\mathcal{G}\_p(w)\right)}{dw}\right]^{1/n}\mathcal{G}\_v(w) \tag{50}$$

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‐ sence in the SE (48).

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‐ ous section, are of use for the P3D models.

*Comment*. In some cases, it may be convenient to use the net-pressure rather than the open‐ ing. Since *dp* =( *dp dw* )*dw*, reformulation of the equations and computational schemes in terms of the modified pressure *P* = *p* 1/*α* is obvious.

#### **6. Conclusions**

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 time step, (iii) avoiding singularities on the fluid front, (iv) the possibility to use highly effi‐ cient numerical schemes for the PKN and P3D models. These beneficial features are of sig‐ nificance when developing simulators able to efficiently solve truly 3D and pseudo-threedimensional problems in real time. The work on simulators of both types is in progress.
