Stress around circular cavity

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.1, pp. 169)

Kirsch solution

\[\begin{align} \sigma_{rr} &= \frac{ \sigma_{Hmax} + \sigma_{hmin} }{2}\left( 1 - \frac{a^2}{r^2} \right) + \frac{ \sigma_{Hmax} - \sigma_{hmin} }{2}\left( 1 - 4 \frac{a^2}{r^2} + 3 \frac{a^4}{r^4}\right)\cos 2\theta + (P_w-P_p)\left(\frac{a^2}{r^2}\right) \\ \sigma_{\theta\theta} &= \frac{ \sigma_{Hmax} + \sigma_{hmin} }{2}\left( 1 + \frac{a^2}{r^2} \right) - \frac{ \sigma_{Hmax} - \sigma_{hmin} }{2}\left( 1 + 3 \frac{a^4}{r^4}\right)\cos 2\theta - (P_w-P_p)\left(\frac{a^2}{r^2}\right) \\ \sigma_{r\theta} &= \frac{ \sigma_{Hmax} - \sigma_{hmin} }{2}\left( 1 + 2\frac{a^2}{r^2} - 3 \frac{a^4}{r^4} \right) - \sin 2\theta \\ \sigma_{zz} &= \sigma_v - 2 \nu (\sigma_{Hmax} -\sigma_{hmin})\left(\frac{a^2}{r^2}\right)\cos 2 \theta \end{align}\]

Example 1

\[\begin{align} S_{Hmax} &= 90 \mbox{MPa} \\ S_{v} &= 88.2 \mbox{MPa} \\ S_{hmin} &= 51.5 \mbox{MPa} \\ P_{p} &= 31.5 \mbox{MPa} \\ P_{w} &= 31.5 \mbox{MPa} \end{align}\]

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.2a, pp. 171)

Example 2

	\[\quad\]
Along azimuth of \(S_{Hmax}\)	\[\quad\]	Along azimuth of \(S_{hmin}\)

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.2b,c, pp. 171)

Variation of wellbore stresses

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.3a, pp. 173)

Wellbore breakout region

\[\quad\]

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.3b,c, pp. 173)

Mudweight stabilization

As \(\Delta P\) increases, \(\sigma_{\theta\theta}\) decreases and \(\sigma_{rr}\) increases.

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.3b, pp. 173 and Fig. 6.5a, pp. 177)

Breakouts as indicators of far-field stresses

Simplify Kirsch equations at wellbore wall \(a = r\), so

\[\begin{align} \sigma_{rr} &= (P_w-P_p) = \Delta P \\ \sigma_{\theta\theta} &= \sigma_{Hmax} + \sigma_{hmin} - 2 (\sigma_{Hmax} - \sigma_{hmin} ) \cos 2 \theta - \Delta P \\ \sigma_{zz} &= \sigma_v - 2 \nu (\sigma_{Hmax} -\sigma_{hmin})\cos 2 \theta \end{align}\]

\(\sigma_{\theta\theta}\) has min at 0\(^\circ\) and 180\(^\circ\)

\[\begin{align} \sigma_{\theta\theta}^{min} &= 3\sigma_{Hmin} - \sigma_{Hmax} - \Delta P \\ \end{align}\]

\(\sigma_{\theta\theta}\) has max at 90\(^\circ\) and 270\(^\circ\), so

\[\begin{align} \sigma_{\theta\theta}^{max} &= 3\sigma_{Hmax} - \sigma_{hmin} - \Delta P \\ \end{align}\]

so

\[\begin{align} \sigma_{\theta\theta}^{max} - \sigma_{\theta\theta}^{min} &= 4(\sigma_{Hmax} - \sigma_{hmin}) \end{align}\]

Tensile induced fractures

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.5a, pp. 177)

Safe drilling mud window

• Mud weight too low
  • Breakouts
• Mud weight too high
  • Tensile induced fractures leading to lost circulation

Imaging breakouts

	\[\quad\]		\[\quad\]
Ultrasonic \(P\)-wave	\[\quad\]	Electrical resistivity	\[\quad\]	Breakout cross-section

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.4a,b,c, pp. 176)

Four-arm caliper data

	\[\quad\]		\[\quad\]
Caliper data	\[\quad\]	Breakout indication	\[\quad\]	Examples of variations

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.9a,b,c, pp. 183)

Thermal effects on wellbore stress

Strongly time dependent

\[ \frac{\partial T}{\partial t} = \alpha_T \nabla^2 T \]

\(\alpha \to\) strongly depenendent of the silica content of the rock.

Under steady-state conditions,

\[ \Delta \sigma_{\theta\theta}^T = \frac{\alpha_T E \Delta T}{1-\nu} \]

Time-temperature effects

	\[\begin{align}\Delta T =& 25^{\circ} \mbox{ C} \\ \Delta P =& 6 \mbox{ MPa}\end{align}\]

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.14a,b pp. 194)

Stability through cooling?

Cooling	Reference

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.14c pp. 194 and Fig. 6.3 pp. 173)

Rock strength anisotropy

\[ \sigma_1 = \sigma_3 \frac{2 (S_w + \mu_w \sigma_3)}{(1-\mu_w \cot\beta_w)\sin 2\beta} \]

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 4.12, pp. 106)

Rock strength anisotropy effects on breakouts

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.16a,b pp. 199)

Two mechanisms

• Stresses exceed intact rock strength
• Stresses activate slip on weak bedding planes

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 6.16c pp. 199)

Chemical effects

• Water Activity (\(A_w \sim \frac{1}{\mbox{salinity}}\)) can to increased pore pressure

\(S_{Hmax}\) from breakout data

\[ S_{Hmax} = \frac{(C_0 + 2 P_p + \Delta P + \Delta \sigma^T) - S_{hmin}(1 + 2 \cos(\pi - w_{bo}))}{1 - 2 \cos(\pi - w_{bo})} \]

Example

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 7.7 pp. 223)

Wellbore stability

Defining a “stable” wellbore

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 10.1a,b pp. 304)

Emperical model: Maximum 90\(^{\circ}\) breakouts

\(\quad \quad \quad \quad\)

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 10.2a,b pp. 305)

Comprehensive model

Design based on pore pressure and frac gradient	Model considering collapse pressure	Final design

© Cambridge University Press Zoback, Reservoir Geomechanics (Fig. 10.3a,b,c pp. 307)

Occurance of both drilling induced tensile fractures and breakouts

Allows for estimate of rock strength in-situ

Recall: \(S_{Hmax}\) from breakout data

\[ S_{Hmax} = \frac{(C_0 + 2 P_p + \Delta P + \Delta \sigma^T) - S_{hmin}(1 + 2 \cos(\pi - w_{bo}))}{1 - 2 \cos(\pi - w_{bo})} \]

Recall: From Kirsch solution \[ S_{Hmax} = 3 S_{hmin} - 2 P_p - \Delta P - T_0 - \sigma^{\Delta T} \]