Initiation Pressure and Location of Fracture Initiation in Elliptical Wellbores

Abstract

Due to the shear failure caused by the compressive stress concentration in the wellbore wall and the non-isotropy of the in situ stresses, the actual shape of the wellbore cross-section is oval instead of what is usually considered a circle. The purpose of this paper is to find the fracture initiation pressure and the initiation location in elliptical wellbores. For this purpose, a simple analytical model is used. This model is obtained by combining analytical relations of stress distribution around elliptical wellbores and material tensile strength criterion to predict crack initiation. According to the results, if the in situ stresses are isotropic, the fracture initiation location is in the direction of the longest diameter of the wellbore, and the initiation pressure decreases with the increase of the shape parameter. In the case that the in situ stresses are non-isotropic, a critical shape parameter is defined. If the shape parameter is less than the critical value, the fracture initiation in the wellbore wall is along the maximum in situ stress and the initiation pressure increases with the increase of the shape parameter, and if the shape parameter is greater than the critical value, up to 0.33, the fracture initiation in the wellbore wall occurs in the direction of the minimum in situ stress and the initiation pressure decreases with the increase of the shape parameter. This model can be used in estimating the fracture initiation pressure and determining the fracture initiation position in elliptical cavities in the field and laboratory.

Appendices

Appendix

Appendix: Stress Distribution Around Elliptical Wellbores

The stress distribution around the elliptical wellbore have been obtained using the conformal mapping technique. Conformal mapping/transformation is a technique used in complex analysis to transform one complex plane onto another while preserving the angles between intersecting curves. First, Muskhelishvili (Muskhelishvili 1963) provided the following mapping function to map an ellipse to a unit circle:

$$ f\left( \varsigma \right) = R\left( {\frac{1}{\varsigma } + m\,\varsigma } \right) $$

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where

$$ R = \frac{a + b}{2} $$

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and

$$ m = \frac{a - b}{{a + b}} $$

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$$ \varsigma = \rho \left( {\cos \theta + i\sin \theta } \right) $$

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$$ \rho = \frac{R}{r}\,\,\,\,\,;\,\,\,\,\left( {R \le r} \right) $$

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where a, b are the semi-axes of the ellipse (Fig. 

Fig. 22

Stress around the elliptical wellbore

22). Indeed, the distance from the center of the unit circle is defined by $\rho = \frac{R}{r}$, where $R = \frac{a + b}{2}$ is a constant and r is variable. Any point with $\rho = 1$ is on the ellipse wall or on the circumference of the unit circle.

By using this mapping function and converting the ellipse into a circle, the stress distribution at any point around the elliptical hole is obtained as follows (Liu et al. 2021):

$$ \begin{gathered} \sigma_{\theta \theta } = \frac{1}{{2\,\left( {m^{2} - 2m\rho^{ - 2} \cos 2\theta + \rho^{ - 4} } \right)}}\, \times \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\left[ {\left( { - \rho^{ - 2} + m\,\cos 2\theta } \right)\,\frac{{\rho^{ - 2} \left( { - \sigma_{1} - \sigma_{3} } \right)\,\left( {\rho^{ - 4} - m^{2} \cos 4\theta } \right)}}{{m^{2} - 2m\rho^{ - 2} \cos 2\theta + \rho^{ - 4} }}} \right. - \frac{{m^{3} \rho^{ - 2} \left( { - \sigma_{1} - \sigma_{3} } \right)\,\sin 4\theta \,\sin 2\theta }}{{m^{2} - 2m\rho^{ - 2} \cos 2\theta + \rho^{ - 4} }} - \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,4\,\rho^{ - 2} \,\left( { - \rho^{ - 2} + m\,\cos 2\theta } \right)\,\frac{{\left( {\rho^{ - 2} \cos 2\theta + m\cos 4\theta } \right)\,\left( {GE + HF} \right) + \,\left( {HE - GF} \right)\,\left( {m\,\sin 4\theta + \rho^{ - 2} \sin 2\theta } \right)}}{{E^{2} + F^{2} }} + \hfill \\ \,\,\,\,\,\,\,\,\,\,\,4m\,\sin 2\theta \,\frac{{\left( {\rho^{ - 4} \cos 2\theta + m\,\rho^{ - 2} \cos 4\theta } \right)\,\left( {HE - GF} \right) - \,\left( {GE + HF} \right)\,\left( {m\,\rho^{ - 2} \sin 4\theta + \rho^{ - 4} \sin 2\theta } \right)}}{{E^{2} + F^{2} }} + \hfill \\ \,\,\,\,\,\,\,\,\,\,\,2\,\left( { - \rho^{ - 2} + m\,\cos 2\theta } \right)\,\left( {\frac{{ - \sigma_{1} - \sigma_{3} }}{2} + P_{w} } \right) + \hfill \\ \,\,\,\,\,\,\,\,\,\,\rho^{ - 2} \left( {\sigma_{1} - \sigma_{3} } \right)\,\left( {\left( { - \rho^{ - 2} + m\,\cos 2\theta } \right)\cos \left( {2\alpha + 2\theta } \right)\, + m\,\sin 2\theta \sin \left( {2\alpha + 2\theta } \right)} \right)\, - \hfill \\ \,\,\,\,\,\,\,\,\,\frac{{2N\left( {\left( { - \rho^{ - 2} + m\,\cos 2\theta } \right) - \,\frac{{\sigma_{1} - \sigma_{3} }}{2}m\sin 2\theta \sin 2\alpha } \right)\,\left( {LJ + MK} \right)}}{{J^{2} + K^{2} }} - \hfill \\ \,\,\,\,\,\,\,\,\,\left. {\frac{{2\left( {MJ - LK} \right)\,\left( {\frac{{\sigma_{1} - \sigma_{3} }}{2}\sin 2\alpha \left( { - \rho^{ - 2} + m\,\cos 2\theta } \right) + Nm\,\sin 2\theta } \right)}}{{J^{2} + K^{2} }}} \right]\, + \hfill \\ \,\,\,\,\,\,\,\,\frac{{2\,\left( {m\,\left( {\frac{{ - \sigma_{1} - \sigma_{3} }}{2} + P_{w} } \right) + \frac{{\sigma_{1} - \sigma_{3} }}{2}\cos 2\alpha + \frac{{ - \sigma_{1} - \sigma_{3} }}{{4\rho^{2} }}\cos 2\theta } \right)\,\left( {m\, - \rho^{ - 2} \cos 2\theta } \right)}}{{m^{2} - 2m\rho^{ - 2} \,\cos 2\theta + \rho^{ - 4} }} + \hfill \\ \,\,\,\,\,\,\,\,\frac{{2\rho^{ - 2} \,\left( {\frac{{\sigma_{1} - \sigma_{3} }}{2}\sin 2\alpha - \frac{{ - \sigma_{1} - \sigma_{3} }}{{4\rho^{2} }}\sin 2\theta } \right)\,\sin 2\theta \,}}{{m^{2} - 2m\rho^{ - 2} \,\cos 2\theta + \rho^{ - 4} }} \hfill \\ \end{gathered} $$

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$$ \begin{gathered} \sigma_{\rho \rho } = \frac{ - 1}{{2\,\left( {m^{2} - 2m\rho^{ - 2} \cos 2\theta + \rho^{ - 4} } \right)}}\, \times \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\left[ {\left( { - \rho^{ - 2} + m\,\cos 2\theta } \right)\,\frac{{\rho^{ - 2} \left( { - \sigma_{1} - \sigma_{3} } \right)\,\left( {\rho^{ - 4} - m^{2} \cos 4\theta } \right)}}{{m^{2} - 2m\rho^{ - 2} \cos 2\theta + \rho^{ - 4} }}} \right. - \frac{{m^{3} \rho^{ - 2} \left( { - \sigma_{1} - \sigma_{3} } \right)\,\sin 4\theta \,\sin 2\theta }}{{m^{2} - 2m\rho^{ - 2} \cos 2\theta + \rho^{ - 4} }} - \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,4\,\rho^{ - 2} \,\left( { - \rho^{ - 2} + m\,\cos 2\theta } \right)\,\frac{{\left( {\rho^{ - 2} \cos 2\theta + m\cos 4\theta } \right)\,\left( {GE + HF} \right) + \,\left( {HE - GF} \right)\,\left( {m\,\sin 4\theta + \rho^{ - 2} \sin 2\theta } \right)}}{{E^{2} + F^{2} }} + \hfill \\ \,\,\,\,\,\,\,\,\,\,\,4m\,\sin 2\theta \,\frac{{\left( {\rho^{ - 4} \cos 2\theta + m\,\rho^{ - 2} \cos 4\theta } \right)\,\left( {HE - GF} \right) - \,\left( {GE + HE} \right)\,\left( {m\,\rho^{ - 2} \sin 4\theta + \rho^{ - 4} \sin 2\theta } \right)}}{{E^{2} + F^{2} }} + \hfill \\ \,\,\,\,\,\,\,\,\,\,\,2\,\left( { - \rho^{ - 2} + m\,\cos 2\theta } \right)\,\left( {\frac{{ - \sigma_{1} - \sigma_{3} }}{2} + P_{w} } \right) + \hfill \\ \,\,\,\,\,\,\,\,\,\,\rho^{ - 2} \left( {\sigma_{1} - \sigma_{3} } \right)\,\left( {\left( {m\,\cos 2\theta - \rho^{ - 2} } \right)\cos \left( {2\alpha + 2\theta } \right)\, + m\,\sin 2\theta \sin \left( {2\alpha + 2\theta } \right)} \right)\, - \hfill \\ \,\,\,\,\,\,\,\,\,\frac{{2N\left( {\left( { - \rho^{ - 2} + m\,\cos 2\theta } \right) - \,\frac{{\sigma_{1} - \sigma_{3} }}{2}m\sin 2\theta \sin 2\alpha } \right)\,\left( {LJ + MK} \right)}}{{J^{2} + K^{2} }} - \hfill \\ \,\,\,\,\,\,\,\,\,\left. {\frac{{2\left( {MJ - LK} \right)\,\left( {\frac{{\sigma_{1} - \sigma_{3} }}{2}\sin 2\alpha \left( { - \rho^{ - 2} + m\,\cos 2\theta } \right) + Nm\,\sin 2\theta } \right)}}{{J^{2} + K^{2} }}} \right]\, + \hfill \\ \,\,\,\,\,\,\,\,\frac{{2\,\left( {m\,\left( {\frac{{ - \sigma_{1} - \sigma_{3} }}{2} + P_{w} } \right) + \frac{{\sigma_{1} - \sigma_{3} }}{2}\cos 2\alpha + \frac{{ - \sigma_{1} - \sigma_{3} }}{{4\rho^{2} }}\cos 2\theta } \right)\,\left( {m\, - \rho^{ - 2} \cos 2\theta } \right)}}{{m^{2} - 2m\rho^{ - 2} \,\cos 2\theta + \rho^{ - 4} }} + \hfill \\ \,\,\,\,\,\,\,\,\frac{{2\rho^{ - 2} \,\left( {\frac{{\sigma_{1} - \sigma_{3} }}{2}\sin 2\alpha - \frac{{ - \sigma_{1} - \sigma_{3} }}{{4\rho^{2} }}\sin 2\theta } \right)\,\sin 2\theta \,}}{{m^{2} - 2m\rho^{ - 2} \,\cos 2\theta + \rho^{ - 4} }} \hfill \\ \end{gathered} $$

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$$ \begin{gathered} \tau_{\rho \theta } = \frac{1}{{2\,\left( {m^{2} - 2m\rho^{ - 2} \cos 2\theta + \rho^{ - 4} } \right)}}\, \times \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\left[ {\,\frac{{\left( { - \sigma_{1} - \sigma_{3} } \right)\,\left( {\rho^{ - 4} - m^{2} \cos 4\theta } \right)\rho^{ - 2} m\,\sin 2\theta }}{{m^{2} - 2m\rho^{ - 2} \cos 2\theta + \rho^{ - 4} }}} \right. + \frac{{\left( { - \sigma_{1} - \sigma_{3} } \right)\,\left( {m\cos 2\theta - \rho^{ - 2} } \right)m^{2} \rho^{ - 2} \sin 4\theta \,}}{{m^{2} - 2m\rho^{ - 2} \cos 2\theta + \rho^{ - 4} }} - \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,4m\,\sin 2\theta \,\,\frac{{\left( {\rho^{ - 4} \cos 2\theta + m\rho^{ - 2} \cos 4\theta } \right)\,\left( {GE + HF} \right) + \,\left( {HE - GF} \right)\,\left( {m\rho^{ - 2} \sin 4\theta + \rho^{ - 4} \sin 2\theta } \right)}}{{E^{2} + F^{2} }} - \hfill \\ \,\,\,\,\,\,\,\,\,\,\,4\left( {m\cos 2\theta - \rho^{ - 2} } \right)\,\frac{{\left( {\rho^{ - 4} \cos 2\theta + m\,\rho^{ - 2} \cos 4\theta } \right)\,\left( {HE - GF} \right) - \,\left( {GE + HF} \right)\,\left( {m\,\rho^{ - 2} \sin 4\theta + \rho^{ - 4} \sin 2\theta } \right)}}{{E^{2} + F^{2} }} + \hfill \\ \,\,\,\,\,\,\,\,\,\,\,2\,m\,\left( {\frac{{ - \sigma_{1} - \sigma_{3} }}{2} + P_{w} } \right)\sin 2\theta + \hfill \\ \,\,\,\,\,\,\,\,\,\,2\frac{{\sigma_{1} - \sigma_{3} }}{2}\rho^{ - 2} m\,\sin 2\theta \,\cos \left( {2\alpha + 2\theta } \right) - 2\left( {m\,\cos 2\theta - \rho^{ - 2} } \right)\frac{{\sigma_{H} - \sigma_{h} }}{2}\rho^{ - 2} \sin \left( {2\alpha + 2\theta } \right)\, - \hfill \\ \,\,\,\,\,\,\,\,\,\frac{{2\left( {\frac{{\sigma_{1} - \sigma_{3} }}{2}\left( {m\,\cos 2\theta - \rho^{ - 2} } \right)\sin 2\alpha + Nm\,\sin 2\theta } \right)\,\left( {LJ + MK} \right)}}{{J^{2} + K^{2} }} - \hfill \\ \,\,\,\,\,\,\,\,\left. {\frac{{2\left( {MJ - LK} \right)\left( {N\left( {m\,\cos 2\theta - \rho^{ - 2} } \right) - \frac{{\sigma_{H} - \sigma_{h} }}{2}m\,\sin 2\theta \sin 2\alpha } \right)\,}}{{J^{2} + K^{2} }}} \right] \hfill \\ \,\,\,\,\,\,\, \hfill \\ \end{gathered} $$

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where

$$ E = m^{2} - 2m\rho^{ - 2} \cos 2\theta + \rho^{ - 4} \cos 4\theta $$

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$$ F = 2m\rho^{ - 2} \sin 2\theta - \rho^{ - 4} \sin 4\theta $$

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$$ G = m\,\left( {\frac{{ - \sigma_{1} - \sigma_{3} }}{4} + P_{w} } \right) + \frac{{\sigma_{1} - \sigma_{3} }}{2}\,\cos 2\alpha + \frac{{ - \sigma_{1} - \sigma_{3} }}{2}\,\rho^{ - 2} \cos 2\theta $$

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$$ H = \frac{{\sigma_{1} - \sigma_{3} }}{2}\,\sin 2\alpha - \frac{{ - \sigma_{1} - \sigma_{3} }}{2}\,\rho^{ - 2} \sin 2\theta $$

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$$ J = m^{2} \rho^{4} \cos 4\theta - 2m\rho^{2} \cos 2\theta + 1 $$

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$$ K = m^{2} \rho^{4} \sin 4\theta - 2m\rho^{2} \sin 2\theta $$

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$$ L = m\rho^{4} \cos 4\theta - \left( {m^{2} + 3} \right)\rho^{2} \cos 2\theta - m $$

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$$ M = m\rho^{4} \sin 4\theta - \left( {m^{2} + 3} \right)\rho^{2} \sin 2\theta $$

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$$ N = m\,\left( {\frac{{ - \sigma_{1} - \sigma_{3} }}{4} + P_{w} } \right) + \frac{{\sigma_{1} - \sigma_{3} }}{2}\,\cos 2\alpha $$

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where $\alpha$ is the angle between the major axis of the ellipse and the maximum principal stress direction (Fig. 22). In this article, to find the fracture initiation pressure, only the tangential stress in the borehole wall is needed. In the borehole wall, $\rho = 1$ and according to Fig. 22, the major axis of the ellipse is considered to be perpendicular to the maximum principal stress direction, i.e. $\alpha = 90^\circ$; By applying these two conditions in relation 19, this relation is simplified as follows, which expresses the tangential stress in the borehole wall:

$$ \sigma_{\theta \theta } = \frac{{\left( { - \sigma_{1} - \sigma_{3} } \right)\,\left( {m^{2} - 1} \right) + 2\,\left( {\sigma_{1} - \sigma_{3} } \right)\,\left[ { - m + \cos 2\theta } \right] + 4mP_{w} \left( {m - \cos 2\theta } \right)}}{{\left( {m - \cos 2\theta } \right)^{2} + \sin^{2} 2\theta }} - P_{w} $$

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This relationship can be written as follows

$$ \sigma_{\theta \theta } = A\,\sigma_{1} \, + \,B\,\sigma_{3} \, - \,C\,P_{w} $$

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where

$$ A = \frac{{1 - m^{2} - 2m + 2\cos 2\theta }}{{1 + m^{2} - 2m\,\cos 2\theta }} $$

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$$ B = \frac{{1 - m^{2} + 2m - 2\cos 2\theta }}{{1 + m^{2} - 2m\,\cos 2\theta }} $$

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$$ C = \frac{{1 - 3m^{2} + 2m\,\cos 2\theta }}{{1 + m^{2} - 2m\,\cos 2\theta }} $$

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