**5. Results and discussion**

The problem of the initiation of an axisymmetric hydraulic fracture from a wellbore depends only on three dimensionless parameters i.e. the timescale ratio *χ*, dimensionless wellbore radius and the ratio of the initial defect length and the wellbore radius l<sup>o</sup> *<sup>a</sup>* <sup>=</sup> <sup>γ</sup><sup>o</sup> . The parameter space for these three dimensionless quantities is now explored. We aim to see their effect on the transition to the analytical solution of a penny-shaped hydraulic fracture propagating in an infinite medium [15,17] as well as on the breakdown pressure (i.e. the maximum pressure recorded).

In Figure. 2, the evolution of fracture length *γ*, inlet opening Ω(0) and wellbore pressure Π*wb*

are plotted for different values of the timescale ratio *<sup>χ</sup>* for values of =0.1 and *γ<sup>o</sup>* =0.3. This is done in order to investigate the transition to the infinite medium solution. These results are presented in the Viscosity-Toughness (*MK*) scaling. The infinite medium solution goes from the viscosity dominated dissipation (*M* vertex solution) to the toughness dominated dissipation (*K* vertex solution). This transition is called the *MK* edge of the semi-infinite me‐ dium solution. It can be observed in Figure. 2 that fracture length, opening and wellbore pressure of our numerical solution accounting for the wellbore effects converges at large time to the *MK* edge solution in an infinite medium. The timescales ratio *χ* governs where the transient effects end along the MK edge of the infinite medium solution. For larger value of *χ*, the transient effects end on the viscosity asymptote of the infinite medium solution, whereas for small value of *χ*, the transient effect ends on the toughness asymptote of the in‐ finite medium solution. The fact that the infinite medium solution is recovered at large time ultimately validates the large time behavior obtained by our numerical algorithm.

similar while higher breakdown pressures are obtained for higher values of *χ* and lower breakdown pressures are obtained for lower values of *χ*. The breakdown is also much more abrupt for the case of low values of *χ*. Such an abrupt breakdown corresponds to the unstable

Initiation and Breakdown of an Axisymmetric Hydraulic Fracture Transverse to a Horizontal Wellbore

along the inviscid solution of [6] in Figure. 4, where we can see that the numerical solution converges to the inviscid fluid solution for small *χ*. Similar results for the case of a plane-strain

**Figure 3.** Effect of χ on the wellbore pressure Π*wb*, fracture length γ and effective flux entering the fracture for

lines represent γ*<sup>o</sup>* =0.005, magenta colour lines represent γ*<sup>o</sup>* =0.02 and the golden colour lines represent γ*<sup>o</sup>* =0.32. The dashed magenta line represents the *K*-vertex solution whereas the dashed green line represents the inviscid fluid sol‐

). Our numerical results are plotted

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413

, magenta colour lines represent χ =10<sup>3</sup>

, =0.1 in *UK* scaling. The blue coloured

and

crack growth in the limiting case of a inviscid fluid (*χ* <10-4

<sup>γ</sup>*<sup>o</sup>* =0.003, =0.1 in *UK* scaling. The blue coloured lines represent <sup>χ</sup> =10-4

**Figure 4.** Unstable crack growth in the case of inviscid fluid for χ =1.5×10-4

the golden colour lines represent χ =10<sup>4</sup>

ution of [6].

hydraulic fracture are reported in [5].

**Figure 2.** Effect of χ on the convergence of fracture length γ, inlet opening Ω and wellbore pressure Π*wb* to the infinite medium propagation solution for γ*<sup>o</sup>* =0.003, =0.1. The results are displayed in the *MK* scaling. The solid blue lines correspond to the simulation results, the dashed blue line is the M vertex solution, the dashed red line is the K vertex solution and the solid magenta line is the ∞ - medium solution.

It is also observed that the pressure and opening do not converge monotonically to the asymptotic solution. This is a characteristic feature of the near wellbore solution also obvious from Figure. 3 where the results are displayed in the Compressibility-Toughness (*UK*) scaling. Such a non-monotonic behavior is associated with fracture breakdown and is more pro‐ nounced for larger injection system compressibility, i.e. when the release of the stored fluid in the newly created fracture is more sudden.

Theeffectsof*χ*onthebreakdownpressure(definedasthemaximumwellborepressure),fracture length and effective flux entering the fracture are shown in Figure. 3 in the Compressibility-Toughness (*UK*) scaling. It has been observed that due to the strong fluid-solid coupling, the pressure in the wellbore keeps rising even after the fracture has already initiated. The pressure at which a fracture starts to propagate is called the initiation pressure and the highest pressure recorded is called the breakdown pressure. This difference in the initiation and the break‐ downpressurehasbeenobservedtheoretically[5,4]aswellasexperimentally[23]anditdepends upon the fluid viscosity, injection rate and the system compressibility. It is observed from Equation (20) that low value of the timescale ratio *χ* corresponds to low viscosity and high injection system compressibility. It can be seen in Figure. 3 that the initiation pressure remains similar while higher breakdown pressures are obtained for higher values of *χ* and lower breakdown pressures are obtained for lower values of *χ*. The breakdown is also much more abrupt for the case of low values of *χ*. Such an abrupt breakdown corresponds to the unstable crack growth in the limiting case of a inviscid fluid (*χ* <10-4 ). Our numerical results are plotted along the inviscid solution of [6] in Figure. 4, where we can see that the numerical solution converges to the inviscid fluid solution for small *χ*. Similar results for the case of a plane-strain hydraulic fracture are reported in [5].

of *χ*, the transient effects end on the viscosity asymptote of the infinite medium solution, whereas for small value of *χ*, the transient effect ends on the toughness asymptote of the in‐ finite medium solution. The fact that the infinite medium solution is recovered at large time

**Figure 2.** Effect of χ on the convergence of fracture length γ, inlet opening Ω and wellbore pressure Π*wb* to the infinite medium propagation solution for γ*<sup>o</sup>* =0.003, =0.1. The results are displayed in the *MK* scaling. The solid blue lines correspond to the simulation results, the dashed blue line is the M vertex solution, the dashed red line is the K vertex

It is also observed that the pressure and opening do not converge monotonically to the asymptotic solution. This is a characteristic feature of the near wellbore solution also obvious from Figure. 3 where the results are displayed in the Compressibility-Toughness (*UK*) scaling. Such a non-monotonic behavior is associated with fracture breakdown and is more pro‐ nounced for larger injection system compressibility, i.e. when the release of the stored fluid in

Theeffectsof*χ*onthebreakdownpressure(definedasthemaximumwellborepressure),fracture length and effective flux entering the fracture are shown in Figure. 3 in the Compressibility-Toughness (*UK*) scaling. It has been observed that due to the strong fluid-solid coupling, the pressure in the wellbore keeps rising even after the fracture has already initiated. The pressure at which a fracture starts to propagate is called the initiation pressure and the highest pressure recorded is called the breakdown pressure. This difference in the initiation and the break‐ downpressurehasbeenobservedtheoretically[5,4]aswellasexperimentally[23]anditdepends upon the fluid viscosity, injection rate and the system compressibility. It is observed from Equation (20) that low value of the timescale ratio *χ* corresponds to low viscosity and high injection system compressibility. It can be seen in Figure. 3 that the initiation pressure remains

solution and the solid magenta line is the ∞ - medium solution.

412 Effective and Sustainable Hydraulic Fracturing

the newly created fracture is more sudden.

ultimately validates the large time behavior obtained by our numerical algorithm.

**Figure 3.** Effect of χ on the wellbore pressure Π*wb*, fracture length γ and effective flux entering the fracture for <sup>γ</sup>*<sup>o</sup>* =0.003, =0.1 in *UK* scaling. The blue coloured lines represent <sup>χ</sup> =10-4 , magenta colour lines represent χ =10<sup>3</sup> and the golden colour lines represent χ =10<sup>4</sup>

**Figure 4.** Unstable crack growth in the case of inviscid fluid for χ =1.5×10-4 , =0.1 in *UK* scaling. The blue coloured lines represent γ*<sup>o</sup>* =0.005, magenta colour lines represent γ*<sup>o</sup>* =0.02 and the golden colour lines represent γ*<sup>o</sup>* =0.32. The dashed magenta line represents the *K*-vertex solution whereas the dashed green line represents the inviscid fluid sol‐ ution of [6].

Lastly, the value of *<sup>γ</sup>*ϵ is plotted against the values of *χ* in Figure. 5, where *γ*ϵ is defined as the frature length *γ* for which the infinite medium solution has been reached within a given tolerence ϵ*<sup>∞</sup>* where

$$\mathfrak{c}\_{\simeq} = \frac{\|\boldsymbol{\gamma}\_{-} \cdot \boldsymbol{\gamma}^{\prime}\|}{\|\boldsymbol{\gamma}\_{-}\|},\tag{25}$$

**Figure 6.** Effect of γ*<sup>o</sup>*

<sup>χ</sup> =1, <sup>γ</sup><sup>0</sup>

golden colour lines represent =0.05

and the magenta colour lines represent γ*<sup>o</sup>*

fracture for <sup>χ</sup> =1, =0.1 in *UK* scaling. The blue coloured lines represent γ*<sup>o</sup>*

=0.01.

on the dimensionless wellbore pressure Π*wb*, fracture length γ and effective flux entering the

Initiation and Breakdown of an Axisymmetric Hydraulic Fracture Transverse to a Horizontal Wellbore

**Figure 7.** Effect of on the wellbore pressure Π*wb*, fracture length γ and effective flux entering the fracture for

=0.1 in *UK* scaling. The blue coloured lines represent =1.2, magenta colour lines represent =0.4 and the

=1, golden colour lines represent γ*<sup>o</sup>*

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415

=0.05

here *γ∞* is the fracture radius for the penny shaped fracture propagation in an infinite medium. It can be seen from Figure. 5, that *γ*ϵ varies, for that case, from 11 to 13 times the wellbore radius for different values of *χ*. This variation shows that there is minimal effect of *χ* on the length to wellbore ratio at which the hydraulic fracture is no longer affected by near-wellbore effects.

Let us now compare the effect of different ratios of *γ<sup>o</sup>* <sup>=</sup> <sup>l</sup>*<sup>o</sup> <sup>a</sup>* i.e. the ratio of the initial defect length to the wellbore radius. In Figure. 6, the dimensionless wellbore pressure, fracture length and effective flux entering the fracture are plotted for various ratios *γ<sup>o</sup>* for *χ* =1 and =0.1 in the Compressibility-Toughness (*UK*) scaling. It can be seen that higher breakdown pressures are obtained for lower ratios of *γ<sup>o</sup>* and lower breakdown pressures are obtained for higher values of *γ<sup>o</sup>* . For lower values of *γ<sup>o</sup>* , there is a sudden drop in pressure after the breakdown similar to the behavior observed for a low viscosity fluid/ highly compressible injection system.

**Figure 5.** Effect of χ on the convergence to infinite space solution for γ*<sup>o</sup>* =0.003 and =0.1.

It can be seen from Figure. 6, that there is no significant difference in convergence to the infinite space solution for different ratios of *γ<sup>o</sup>* .

Finally,inFigure. 7,the effectofthedimensionlesswellbore radius is consideredfor a fixedratio *γ*0 =0.1 and *γ*<sup>0</sup> =0.01. The results are displayed in the Compressibility-Toughness (*UK*) scal‐ ing. It is observed that the breakdown pressure is higher for a smaller dimensionless wellbore radius.Itistobenoticedthattheconvergenceofonlyonesolutiontotheinfinitemediumsolution is shown in the figure. All other solutions also converge to the infinite medium solution but at a

Initiation and Breakdown of an Axisymmetric Hydraulic Fracture Transverse to a Horizontal Wellbore http://dx.doi.org/10.5772/56262 415

Lastly, the value of *<sup>γ</sup>*

414 Effective and Sustainable Hydraulic Fracturing

ϵ*<sup>∞</sup>* where

tolerence

of *γ<sup>o</sup>*

*γ*0

=0.1 and *γ*<sup>0</sup>

ϵ

It can be seen from Figure. 5, that *γ*

obtained for lower ratios of *γ<sup>o</sup>*

. For lower values of *γ<sup>o</sup>*

is plotted against the values of *χ* in Figure. 5, where *γ*

frature length *γ* for which the infinite medium solution has been reached within a given

here *γ∞* is the fracture radius for the penny shaped fracture propagation in an infinite medium.

for different values of *χ*. This variation shows that there is minimal effect of *χ* on the length to wellbore ratio at which the hydraulic fracture is no longer affected by near-wellbore effects.

to the wellbore radius. In Figure. 6, the dimensionless wellbore pressure, fracture length and

Compressibility-Toughness (*UK*) scaling. It can be seen that higher breakdown pressures are

to the behavior observed for a low viscosity fluid/ highly compressible injection system.

It can be seen from Figure. 6, that there is no significant difference in convergence to the infinite

Finally,inFigure. 7,the effectofthedimensionlesswellbore radius is consideredfor a fixedratio

ing. It is observed that the breakdown pressure is higher for a smaller dimensionless wellbore radius.Itistobenoticedthattheconvergenceofonlyonesolutiontotheinfinitemediumsolution is shown in the figure. All other solutions also converge to the infinite medium solution but at a

=0.01. The results are displayed in the Compressibility-Toughness (*UK*) scal‐

.

<sup>=</sup> <sup>l</sup>*<sup>o</sup>*

ϵ

ϵ

effective flux entering the fracture are plotted for various ratios *γ<sup>o</sup>*

Let us now compare the effect of different ratios of *γ<sup>o</sup>*

**Figure 5.** Effect of χ on the convergence to infinite space solution for γ*<sup>o</sup>*

space solution for different ratios of *γ<sup>o</sup>*

<sup>∞</sup> <sup>=</sup> <sup>|</sup>*γ*<sup>∞</sup> - *<sup>γ</sup>*<sup>|</sup>

ϵ

*<sup>a</sup>* i.e. the ratio of the initial defect length

for *χ* =1 and =0.1 in the

<sup>|</sup>*γ*∞<sup>|</sup> , (25)

varies, for that case, from 11 to 13 times the wellbore radius

and lower breakdown pressures are obtained for higher values

, there is a sudden drop in pressure after the breakdown similar

=0.003 and =0.1.

is defined as the

**Figure 6.** Effect of γ*<sup>o</sup>* on the dimensionless wellbore pressure Π*wb*, fracture length γ and effective flux entering the fracture for <sup>χ</sup> =1, =0.1 in *UK* scaling. The blue coloured lines represent γ*<sup>o</sup>* =1, golden colour lines represent γ*<sup>o</sup>* =0.05 and the magenta colour lines represent γ*<sup>o</sup>* =0.01.

**Figure 7.** Effect of on the wellbore pressure Π*wb*, fracture length γ and effective flux entering the fracture for <sup>χ</sup> =1, <sup>γ</sup><sup>0</sup> =0.1 in *UK* scaling. The blue coloured lines represent =1.2, magenta colour lines represent =0.4 and the golden colour lines represent =0.05

larger time which is not shown here in order to focus on the breakdown phase. The effect of on the convergence to the infinite space solution is displayed on Figure. 8. It can be seen that for dimensionless wellbore radius greater than 0.2, *γ*ϵ is only about 6 to 8 times . For wellbore radius smaller than 0.2, an exponential increase is observed on the transitional fracture length. The case of small corresponds to conditions where a large amount of fluid can be stored in the

It was also found that the near wellbore effects are present up to a fracture length about 12 times the wellbore radius for different values of *χ*. This shows that the transitional length is not affected by the value of *χ*. The variation of the ratio of initial defect length to the wellbore

a small dimensionless wellbore radius has a profound effect on convergence to the infinite

transitional length is observed. This behavior is due to the injection system compressibility. The larger the compressibility, the larger the volume of fluid stored during pressurization and ultimately, the larger is the effective flux of fluid entering the fracture at breakdown compared to the nominal injection rate. This has the consequence of delaying the transition toward the solution of a hydraulic fracture propagating in an infinite medium under constant injection rate. It is important to note, however, that the presence of valves/perforations in the injection system will help dissipate the energy associated with such compressibility effects observed for

The edge dislocation kernel *J*(*r*, *r '*) mentioned in Equation (2) is given by [11] as follows

*<sup>r</sup>* ') - <sup>1</sup>

)*α*(*ξ*, *x*) + *Q*(*ξ*, *x* '

where *E* and *K* are complete elliptic integrals of first and second kind respectively and

) + *ξI*0*K*<sup>1</sup>

) + *S*(*r*, *r* '

*rr* ' *<sup>K</sup>*( *<sup>r</sup>*

*<sup>α</sup>*(*ξ*, *<sup>r</sup>*)= - 2(2 - *<sup>ν</sup>*) <sup>+</sup> *<sup>ξ</sup>* <sup>2</sup> *<sup>K</sup>*1*K*0(*ξr*) / *<sup>ξ</sup>* <sup>+</sup> *<sup>ξ</sup>xK*0*K*1(*ξr*) <sup>+</sup> *rK*1*K*1(*ξr*) - <sup>3</sup>*K*0*K*0(*ξr*), (30)

(*ξr* ' ) - *ξr* ' *I K*<sup>0</sup> (*ξr* ' ) + *ξ* <sup>2</sup> *r* ' *I K*0(*ξr* '

*β*(*ξ*, *r*)=2*K*1*K*0(*ξr*) + *ξrK*1*K*1(*ξr*) + *ξK*0*K*0(*ξr*), (32)

*<sup>r</sup>* '), *<sup>r</sup>* <sup>&</sup>lt;*<sup>r</sup>* '

*<sup>r</sup>* '), *<sup>r</sup>* <sup>&</sup>gt;*<sup>r</sup>* '

) =*r R*(*r*, *r* '

medium solution. For small dimensionless wellbore radius <sup>=</sup> *<sup>a</sup>*

*J*(*r*, *r* '

(*<sup>r</sup>* <sup>2</sup> - *<sup>r</sup>* '2)2 *<sup>E</sup>*( *<sup>r</sup>*

1 *<sup>r</sup>* <sup>2</sup> - *<sup>r</sup>* '2 *<sup>E</sup>*( *<sup>r</sup>*

) ={ <sup>1</sup>

*r r* '

*P*(*ξ*, *x* '

*P*(*ξ*, *r* '

) =*ξ* <sup>2</sup> *I*0*K*<sup>1</sup> (*ξr* ' ) - *ξ* <sup>2</sup> *r* ' *I K*<sup>0</sup> (*ξr* '

(*ξr* '

*R*(*r*, *r* '

) = *∫* 0 ∞

) <sup>=</sup> - 2(1 - *<sup>ν</sup>*) <sup>+</sup> *<sup>ξ</sup>* <sup>2</sup> *<sup>I</sup>*1*K*<sup>1</sup>

*S*(*r*, *r* '

*<sup>a</sup>* has a minimal effect on convergence to the infinite medium solution. In contrast,

Initiation and Breakdown of an Axisymmetric Hydraulic Fracture Transverse to a Horizontal Wellbore

(*E '*

*<sup>U</sup>* )1/3 , a large increase in the

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417

) (26)

)*β*(*ξ*, *x*) / Δ(*ξ*)*dξ*, (28)

), (29)

), (31)

(27)

radius *γ<sup>o</sup>*

small .

**Appendix A**

*Q*(*ξ*, *r* '

<sup>=</sup> <sup>l</sup>*<sup>o</sup>*

wellbore during the pressurization phase, i.e *a*≪(*E ' U* ) 1 3 . For those cases, the release of this stored volume of fluid in the fracture after its initiation results in an effective flow rate enter‐ ing the crack much larger the injected one. This is the causes of such a large transition length to the infinite medium propagation for small values of .

**Figure 8.** Effect of on the convergence to infinite space solution for γ*<sup>o</sup>* =0.1. Two convergence criteria are used i.e ϵ<sup>∞</sup> =0.1 and ϵ<sup>∞</sup> =0.01.

#### **6. Conclusions**

The initiation of axisymmetric hydraulic fractures from a horizontal wellbore has been investigated with an emphasis on near wellbore effects. Through a detailed dimensional analysis, two characteristic timescales were identified. The first characteristic time *tmk* defines the timescale of transition from viscosity dominated propagation to the toughness dominat‐ ed propagation and the second characteristic time *tuk* defines the timescale of transition from near wellbore effects to the infinite medium propagation. The ratio of these timescales *χ* (see Equation (20)) increases with increasing fluid viscosity and decreases with increasing injection system compressibility. The large time behavior of the numerical algorithm was verified by the convergence of the numerical solution to the propagation solution in an infinite medi‐ um for a sufficiently large fracture compared to the wellbore size. The effect of the time‐ scale ratio *χ* on the convergence to the infinite medium solution was investigated. It was found that the numerical solution converges to the infinite medium solution for each value of *χ*. The value of *χ* dictates on which infinite medium regime of propagation, the transi‐ ent solution converges to. The solution converges to the toughness dominated propagation regime for small values of *χ*, whereas it converges to the viscosity dominated regime for large values of *χ*.

It was also found that the near wellbore effects are present up to a fracture length about 12 times the wellbore radius for different values of *χ*. This shows that the transitional length is not affected by the value of *χ*. The variation of the ratio of initial defect length to the wellbore radius *γ<sup>o</sup>* <sup>=</sup> <sup>l</sup>*<sup>o</sup> <sup>a</sup>* has a minimal effect on convergence to the infinite medium solution. In contrast, a small dimensionless wellbore radius has a profound effect on convergence to the infinite medium solution. For small dimensionless wellbore radius <sup>=</sup> *<sup>a</sup>* (*E ' <sup>U</sup>* )1/3 , a large increase in the transitional length is observed. This behavior is due to the injection system compressibility. The larger the compressibility, the larger the volume of fluid stored during pressurization and ultimately, the larger is the effective flux of fluid entering the fracture at breakdown compared to the nominal injection rate. This has the consequence of delaying the transition toward the solution of a hydraulic fracture propagating in an infinite medium under constant injection rate. It is important to note, however, that the presence of valves/perforations in the injection system will help dissipate the energy associated with such compressibility effects observed for small .

### **Appendix A**

larger time which is not shown here in order to focus on the breakdown phase. The effect of on the convergence to the infinite space solution is displayed on Figure. 8. It can be seen that for

radius smaller than 0.2, an exponential increase is observed on the transitional fracture length. The case of small corresponds to conditions where a large amount of fluid can be stored in the

stored volume of fluid in the fracture after its initiation results in an effective flow rate enter‐ ing the crack much larger the injected one. This is the causes of such a large transition length to

The initiation of axisymmetric hydraulic fractures from a horizontal wellbore has been investigated with an emphasis on near wellbore effects. Through a detailed dimensional analysis, two characteristic timescales were identified. The first characteristic time *tmk* defines the timescale of transition from viscosity dominated propagation to the toughness dominat‐ ed propagation and the second characteristic time *tuk* defines the timescale of transition from near wellbore effects to the infinite medium propagation. The ratio of these timescales *χ* (see Equation (20)) increases with increasing fluid viscosity and decreases with increasing injection system compressibility. The large time behavior of the numerical algorithm was verified by the convergence of the numerical solution to the propagation solution in an infinite medi‐ um for a sufficiently large fracture compared to the wellbore size. The effect of the time‐ scale ratio *χ* on the convergence to the infinite medium solution was investigated. It was found that the numerical solution converges to the infinite medium solution for each value of *χ*. The value of *χ* dictates on which infinite medium regime of propagation, the transi‐ ent solution converges to. The solution converges to the toughness dominated propagation regime for small values of *χ*, whereas it converges to the viscosity dominated regime for

ϵ

*U* ) 1 3

is only about 6 to 8 times . For wellbore

. For those cases, the release of this

=0.1. Two convergence criteria are used i.e

dimensionless wellbore radius greater than 0.2, *γ*

416 Effective and Sustainable Hydraulic Fracturing

wellbore during the pressurization phase, i.e *a*≪(*E '*

the infinite medium propagation for small values of .

**Figure 8.** Effect of on the convergence to infinite space solution for γ*<sup>o</sup>*

ϵ<sup>∞</sup> =0.1 and

ϵ<sup>∞</sup> =0.01.

**6. Conclusions**

large values of *χ*.

The edge dislocation kernel *J*(*r*, *r '*) mentioned in Equation (2) is given by [11] as follows

$$\mathbf{J}\begin{pmatrix}r,r'\end{pmatrix} = r\mathbf{J}\begin{bmatrix}\mathcal{R}\begin{pmatrix}r,r'\end{bmatrix} + \mathcal{S}\begin{pmatrix}r,r'\end{pmatrix}\end{bmatrix} \tag{26}$$

$$R\left(r\_{r},r^{2}\right) = \begin{cases} \frac{1}{\left(r^{2}-r^{-2}\right)^{2}}E\left(\frac{r}{r}\right), & r \le r^{\circ} \\ \frac{r}{r^{\circ}}\frac{1}{\left(r^{2}-r^{-2}\right)^{2}}E\left(\frac{r}{r^{\circ}}\right)\cdot\frac{1}{rr}K\left(\frac{r}{r}\right), & r \ge r^{\circ} \end{cases} \tag{27}$$

$$S(r,r') = \coprod\_{0}^{a} \left[ P\{\xi\_{\cdot} \ge \} \alpha(\xi\_{\cdot}, \ge) + Q\{\xi\_{\cdot} \ge \} \beta(\xi\_{\cdot}, \ge) \right] / \Delta(\xi) d\xi\_{\cdot} \tag{28}$$

where *E* and *K* are complete elliptic integrals of first and second kind respectively and

$$P(\xi\_r | r') = \xi^{-2} I\_0 K\_1(\xi\_r r') \text{ - } \xi^2 r' I\_0(\xi\_r r') \text{ } \tag{29}$$

$$\ln a(\underline{\xi}, r) = -\left[2(2 - \nu) + \xi^2\right] \mathcal{K}\_1 \mathcal{K}\_0\{\underline{\xi}r\} / \xi + \xi \ge \mathcal{K}\_0 \mathcal{K}\_1\{\underline{\xi}r\} + r \mathcal{K}\_1 \mathcal{K}\_1\{\underline{\xi}r\} - 3 \mathcal{K}\_0 \mathcal{K}\_0\{\underline{\xi}r\},\tag{30}$$

$$Q(\xi, r') = -\left[\mathbb{1}(1 \cdot \nu) + \xi^2\right] I\_1 K\_1(\xi r') + \xi I\_0 K\_1(\xi r') - \xi r' I\_0 K\_0(\xi r') + \xi^2 r' I\_0(\xi r'),\tag{31}$$

$$\beta(\xi, r) = 2K\_1 K\_0(\xi r) + \xi r K\_1 K\_1(\xi r) + \xi' K\_0 K\_0(\xi r), \tag{32}$$

where *In* and *Kn* are the modified Bessel functions of first and second kind respectively where *n* =0,1. In Equation (26), *S* is a semi-infinite integral given by Equation (28) which has a very slow rate of convergence. In order to improve the convergence, we follow a method similar to [11]: *S* is taken out of the integral and evaluated separately in closed form as follows

$$\mathcal{S}\_{\{\max\}}(r,r') = \tilde{\int} \left[ \mathbb{L}P(\xi\_{\prime},r')\alpha(\xi\_{\prime},r) + \mathbb{Q}(\xi\_{\prime},r')\mathbb{\hat{\beta}}(\xi\_{\prime},r) \right] / \Delta(\xi) \text{ - A}^{\prime}(\xi\_{\prime},r\_{\prime},r') \right] d\xi + \tilde{\int} A^{\prime}(\xi\_{\prime},r\_{\prime},r\_{\prime}) d\xi\_{\prime} \tag{33}$$

where *A\**(*ξ*, *<sup>r</sup>*, *<sup>r</sup> '*) is the third order Taylor expansion of (*P<sup>α</sup>* <sup>+</sup> *<sup>Q</sup>β*) <sup>Δ</sup> at infinity. The integral of *A\**(*ξ*, *r*, *r '*) can be obtained analytically.

#### **Appendix B**

The time step is computed by imposing the LEFM tip asymptote (15) in a weak form in the tip element. In the case of negligible toughness, care should be taken as the governing equations degenerate in the tip region. An algorithm based on the LEFM asymptote then requires very fine mesh for good convergence (see [24] for discussion).

The asymptotic volume of the tip element corresponding to the LEFM crack opening asymptote Eq. (15) is given as

$$
\stackrel{\frown}{\mathbf{V}} = \mathcal{G}\_{\mathbf{k}} \stackrel{\scriptstyle \alpha}{\int} \stackrel{\scriptstyle \rho}{\rho}^{1/2} d\stackrel{\scriptstyle \rho}{\rho} = \frac{2}{3} \mathcal{G}\_{\mathbf{k}} \Delta \mathbf{q}^{3/2} \tag{34}
$$

**Appendix C**

where *ρ<sup>i</sup>*

The coupled problem of fluid-solid coupling inside the fracture is described as: for a given fracture length γ, and time τ + Δ*τ <sup>n</sup>*, find the opening Ω<sup>n</sup> + ΔΩ*n*. The continuity equation (12)

Initiation and Breakdown of an Axisymmetric Hydraulic Fracture Transverse to a Horizontal Wellbore

is the element midpoint coordinate and *ρi*-1/2 and *ρi*+1/2 are the element end point

Π *<sup>f</sup>* (*i*) - Π *<sup>f</sup>* (*i*-1)

Π *<sup>f</sup>* (*i*+1) - Π *<sup>f</sup>* (*i*)

∂ Π*<sup>b</sup>*

( <sup>Ω</sup>*<sup>i</sup>* <sup>+</sup> <sup>Ω</sup>*i*-1 <sup>2</sup> )<sup>3</sup>

The flux entering the fracture from the wellbore is given by the inlet boundary condition

<sup>2</sup>*<sup>π</sup>* (*q*Ψ*<sup>o</sup>* - *<sup>U</sup>*

The elasticity equation (11) is discretized by the Displacement Discontinuity Method (DDM)

the expression for Π *<sup>f</sup>* (*i*) from Equation (43) in (38) and (39) and then Ψ from Equations (38)

and (39) in Equation (37) we get the following expression after rearranging the terms

is the stiffness matrix which is dense and symmetric for a regular mesh. Now putting

coordinates. The flux entering an element is denoted by Ψ*i*-1/2 while the outgoing flux is

denoted by Ψ*i*+1/2. These fluxes are computed using the Poiseuille equation as follows:

Ψi-1/2 = - *Ki*-1/2

Ψi+1/2 = - *Ki*+1/2

Where the entrance hydraulic conductivity of an element *Ki*-1/2 is given by

*Ki*-1/2 <sup>=</sup> <sup>1</sup> *m*

<sup>Ψ</sup>1/2 <sup>=</sup> <sup>1</sup>

*Aij* (Ω *<sup>j</sup>* <sup>0</sup> <sup>+</sup> ΔΩ *<sup>j</sup>*

while no flow is assumed out of the fracture tip

to the following form

where *Aij*

<sup>Δ</sup>*<sup>ρ</sup>* (*ρi*-1/2Ψ*i*-1/2 - *ρi*+1/2Ψ*i*+1/2) (37)

<sup>Δ</sup>*<sup>ρ</sup>* (38)

http://dx.doi.org/10.5772/56262

419

<sup>Δ</sup>*<sup>ρ</sup>* (39)

. (40)

<sup>∂</sup> *<sup>τ</sup>* ) (41)

Ψ<sup>N</sup> =0. (42)

) + Π*<sup>h</sup>* =Π *<sup>f</sup>* (*i*) (43)

is discretized using the Finite Volume Method as follows:

ΔΩi<sup>=</sup> *r*Δ*<sup>τ</sup> ρi*

where *ρ* ^ =(*<sup>γ</sup>* <sup>+</sup> ) - *<sup>ρ</sup>* is the distance from the fracture tip and Δρ is the size of the tip element. The new time step Δ*τ <sup>n</sup>*+1 is found by finding the zero of the function *F* defined as the mismatch between the current tip volume and the volume consistent with the LEFM asymptote:

$$F = \mathbf{V}\_{\mathcal{N}} \cdot \stackrel{\wedge}{\mathbf{V}} \tag{35}$$

where VN =Δρ×ΩN is the volume of the last element of the fracture. For each trial value of the time step, the coupled fluid-solid equations are solved in order to obtain a new esti‐ mate of the opening in the tip element ΩN. A secant method is used to find the zero of F. The iterative loop is repeated until the time step converges and the zero of (35) is found. The convergence criteria is

$$\frac{\|\Lambda\tau^{\ast\ast 1} \cdot \Lambda \tau^{\ast}\|\_{2}}{\|\Lambda\tau^{\ast\ast}\|\_{2}} < \varepsilon \qquad\qquad\text{with}\quad \varepsilon = 10^{-4}.\tag{36}$$

## **Appendix C**

where *In* and *Kn* are the modified Bessel functions of first and second kind respectively where *n* =0,1. In Equation (26), *S* is a semi-infinite integral given by Equation (28) which has a very slow rate of convergence. In order to improve the convergence, we follow a method similar to

The time step is computed by imposing the LEFM tip asymptote (15) in a weak form in the tip element. In the case of negligible toughness, care should be taken as the governing equations degenerate in the tip region. An algorithm based on the LEFM asymptote then requires very

The asymptotic volume of the tip element corresponding to the LEFM crack opening asymptote

^ =(*<sup>γ</sup>* <sup>+</sup> ) - *<sup>ρ</sup>* is the distance from the fracture tip and Δρ is the size of the tip element.

between the current tip volume and the volume consistent with the LEFM asymptote:

*F* =VN - V

where VN =Δρ×ΩN is the volume of the last element of the fracture. For each trial value of the time step, the coupled fluid-solid equations are solved in order to obtain a new esti‐ mate of the opening in the tip element ΩN. A secant method is used to find the zero of F. The iterative loop is repeated until the time step converges and the zero of (35) is found. The

, with

ϵ=10-4

is found by finding the zero of the function *F* defined as the mismatch

)*β*(*ξ*, *r*) / Δ(*ξ*) - *A*\*(*ξ*, *r*, *r* '

)|*dξ* + *∫* 0 ∞

<sup>3</sup> kΔρ3/2 (34)

^ (35)

. (36)

*A*\*(*ξ*, *r*, *r* '

<sup>Δ</sup> at infinity. The integral of

)*dξ*, (33)

[11]: *S* is taken out of the integral and evaluated separately in closed form as follows

*S*{*new*} (*r*, *r* ' ) = *∫* 0 ∞

**Appendix B**

Eq. (15) is given as

The new time step Δ*τ <sup>n</sup>*+1

convergence criteria is

where *ρ*


418 Effective and Sustainable Hydraulic Fracturing

*A\**(*ξ*, *r*, *r '*) can be obtained analytically.

)*α*(*ξ*, *r*) + *Q*(*ξ*, *r* '

fine mesh for good convergence (see [24] for discussion).

V ^ <sup>=</sup><sup>k</sup> *<sup>∫</sup>* 0 Δ*ρ ρ* ^1/2 *dρ* ^ <sup>=</sup> <sup>2</sup>

Δ*τ <sup>n</sup>*+1 - Δ*τ <sup>n</sup>* <sup>2</sup> Δ*τ <sup>n</sup>* <sup>2</sup>

<ϵ

where *A\**(*ξ*, *<sup>r</sup>*, *<sup>r</sup> '*) is the third order Taylor expansion of (*P<sup>α</sup>* <sup>+</sup> *<sup>Q</sup>β*)

The coupled problem of fluid-solid coupling inside the fracture is described as: for a given fracture length γ, and time τ + Δ*τ <sup>n</sup>*, find the opening Ω<sup>n</sup> + ΔΩ*n*. The continuity equation (12) is discretized using the Finite Volume Method as follows:

$$
\Delta\Omega\_i = \frac{\mathcal{G}\_r \Lambda \tau}{\rho\_i \Delta \rho} (\rho\_{i \text{-1/2}} \Psi\_{i \text{-1/2}} \text{ - } \rho\_{i \text{+1/2}} \Psi\_{i \text{+1/2}}) \tag{37}
$$

where *ρ<sup>i</sup>* is the element midpoint coordinate and *ρi*-1/2 and *ρi*+1/2 are the element end point coordinates. The flux entering an element is denoted by Ψ*i*-1/2 while the outgoing flux is denoted by Ψ*i*+1/2. These fluxes are computed using the Poiseuille equation as follows:

$$\mathbf{W}\_{i\text{-}1/2} = \mathbf{-K}\_{i\text{-}1/2} \frac{\Pi\_{f\text{-}(i)} \cdot \Pi\_{f\text{-}(i\text{-}1)}}{\Delta \rho} \tag{38}$$

$$\Psi\_{i+1/2} = -K\_{i+1/2} \frac{\Pi\_{f'(i+1)} \cdot \Pi\_{f'(i)}}{\Delta \rho} \tag{39}$$

Where the entrance hydraulic conductivity of an element *Ki*-1/2 is given by

$$K\_{i-1/2} = \frac{1}{\mathcal{G}\_m} \left(\frac{\Omega\_i + \Omega\_{i-1}}{2}\right)^3. \tag{40}$$

The flux entering the fracture from the wellbore is given by the inlet boundary condition

$$\Psi\Psi\_{1/2} = \frac{1}{2\pi} \left( \mathcal{G}\_q \Psi\_o - \mathcal{G}\_{II} \frac{\partial \,\Pi\_b}{\partial \tau} \right) \tag{41}$$

while no flow is assumed out of the fracture tip

$$
\Psi\_N = 0.\tag{42}
$$

The elasticity equation (11) is discretized by the Displacement Discontinuity Method (DDM) to the following form

$$A\_{ij} \{ \Omega\_j^0 + \Lambda \Omega\_j \} + \Pi\_h = \Pi\_{f^{\prime}(i)} \tag{43}$$

where *Aij* is the stiffness matrix which is dense and symmetric for a regular mesh. Now putting the expression for Π *<sup>f</sup>* (*i*) from Equation (43) in (38) and (39) and then Ψ from Equations (38) and (39) in Equation (37) we get the following expression after rearranging the terms

$$
\Xi(\Delta\Omega) \cdot \Delta\Omega = \Gamma(\Delta\Omega) \tag{44}
$$

where Ξ is expressed as

$$
\Delta = \delta\_{ij} - \mathbf{B}\_{ij} \tag{45}
$$

**Author details**

and Brice Lecampion

\*Address all correspondence to: sabbas10@slb.com

[1] Hubbert, M K, & Willis, d D. G. Mechanics of hydraulic fracturing. Trans. Am. Inst.

Initiation and Breakdown of an Axisymmetric Hydraulic Fracture Transverse to a Horizontal Wellbore

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[2] Haimson, B, & Fairhust, C. In-situ stress determination at great depth by means of hydraulic fracturing. In The 11th U.S. Symposium on Rock Mechanics (USRMS); June

[3] Lecampion, B, Abbas, S, & Prioul, R. Competition between transverse and axial hy‐ draulic fractures in horizontal wells, SPE 163848. In SPE Hydraulic Fracturing Tech‐

[4] Garagash, D I, & Detournay, E. An analysis of the inuence of the pressurization rate on the borehole breakdown pressure. Int. J. Sol. and Struct. 1997; 34(24). (1997). ,

[5] Bunger, A, Larikouhani, A, & Detournay, E. Modelling the effect of injection system compressibility and viscous fluid flow on hydraulic fracture breakdown pressure. In Proceedings of the 5th International Symposium on In-Situ Rock Stress; (2010). Bei‐

[6] Lhomme, T, Detournay, E, & Jeffrey, R G. Effect of fluid compressibility and borehole on the initiation and propagation of a tranverse hydraulic fracture. Strength, Fracture

[7] Gordeliy, E, & Detournay, E. A fixed grid algorithm for simulating the propagation of a shallow hydraulic fracture with a fluid lag. International Journal for Numerical

[8] Lecampion, B. Hydraulic fracture initiation from an open-hole: wellbore size, pres‐ surization rate and fluid-solid coupling effects. In 46th US Rock Mechanics/Geome‐

[9] Kirsch, G. Die theorie der elastizität und die bedürfnisse der festigkeislehre. Zantral‐

Schlumberger Doll Research, Cambridge, USA

Min. Eng. (1957). , 153-158.

(1969). Berkeley, CA, USA. , 559-584.

nology Conference; (2013). The Woodlands, Texas, USA.

and Analytical Methods in Geomechanics. (2011). , 602-629.

Safdar Abbas\*

**References**

3099-3118.

jing: ISRSV. , 59-67.

and Complexity. (2005). , 149-162.

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blatt Verlin Deutscher Ingenieure. (1898).

where *δij* is the Kronecker delta and

$$B\_{\vec{\eta}} = \alpha \begin{cases} \frac{\mathcal{G}\_{01}\Lambda\rho}{2\pi\Lambda\tau\rho\_{\vec{\eta}}}A\_{(i,\cdot)} + K\_{\epsilon(j)}A\_{(i+1,\cdot)} \cdot K\_{\epsilon(j)}A\_{(i,\cdot)\rangle\prime} & i = 1 \quad j = 1,\; N \\ K\_{w(\cdot;1)}A\_{(i-1,\cdot)} \cdot \left(K\_{w(\cdot;1)} + K\_{\epsilon(j)}\right)A\_{(i,\cdot)} + K\_{\epsilon(j)}A\_{(i+1,\cdot)\prime} & i = 2,\; N \ -1,\; \quad j = 1,\; N \\ K\_{w(\cdot;1)}A\_{(i-1,\cdot)} \cdot K\_{w(\cdot;1)}A\_{(i,\cdot)\prime} & i = N,\; \quad j = 1,\; N \end{cases} \tag{46}$$

$$\Gamma \vdash \alpha \begin{cases} \frac{\mathcal{G}\_{\boldsymbol{\theta}} \Psi\_{\boldsymbol{\nu}} \Delta \boldsymbol{\rho}}{2 \pi \rho\_{i}} \mathcal{A}\_{(i,j)} + \mathcal{K}\_{\epsilon(j)} \{\mathcal{A}\_{(i+1,j)} \cdot \mathcal{A}\_{(i,j)}\} \cdot \boldsymbol{\Omega}\_{\boldsymbol{\nu}'}^{0} & i = 1, \quad j = 1, \; N \\ \boldsymbol{\mathcal{B}}\_{\boldsymbol{\mathcal{W}}} \cdot \boldsymbol{\Omega}\_{\boldsymbol{\mathcal{W}}}^{0} & i = 2, \; \mathcal{N} \cdot \mathbf{1}\_{\boldsymbol{\prime}} & j = 1, \; \mathcal{N} \\ \boldsymbol{\Psi}\_{\boldsymbol{m}} \Delta \boldsymbol{\rho} + \mathcal{B}\_{\mathcal{N}j} \cdot \boldsymbol{\Omega}\_{\boldsymbol{\mathcal{W}}}^{0} & i = \mathcal{N}, \quad j = 1, \; \mathcal{N} \end{cases} \tag{47}$$

where α<sup>=</sup> *r*Δ*<sup>τ</sup>* <sup>Δ</sup>*<sup>ρ</sup>* 2 and

$$\mathbf{K}\_{\mathbf{e}} = \frac{\rho\_{\text{i-1}/2}}{\rho\_{\text{i}}} \mathbf{K}\_{\mathbf{i}-1/2'} \quad \text{i} = \text{2, N} \tag{48}$$

$$\mathbf{K}\_{\mathbf{e}} = \frac{\rho\_{i+1/2}}{\rho\_i} \mathbf{K}\_{i+1/2'} \quad i = \mathbf{1}, \ N \ -1 \tag{49}$$

The nonlinear system (45) is solved by fixed point iteration

$$
\Xi \{ \Lambda \Omega^k \} \cdot \Lambda \Omega^{k+1} = \Gamma \{ \Lambda \Omega^k \} \tag{50}
$$

with under-relaxation

$$
\Delta\Omega^{\rm k+1} = \begin{pmatrix} 1 \ -\eta^{\gamma} \end{pmatrix} \Delta\Omega^{\rm k} + \eta^{\gamma} \Delta\Omega^{\rm k+1} \tag{51}
$$

where *η* is the relaxation parameter. The convergence criteria is

$$\frac{\|\Lambda\Omega^{k+1} \cdot \Lambda\Omega^{l}\|\_{2}}{\|\Lambda\Omega^{k+1}\|\_{2}} < \varepsilon, \qquad \text{with} \quad \varepsilon = 10^{-5}. \tag{52}$$
