Stresses on deviated wellbores

Let us define a coordinate system for a point along the trajectory of a deviated wellbore. The first element $x_b$ of the cartesian base goes from the center of a cross-section of the wellbore at a given depth to the deepest point around the cross-section (perpendicular to the axis). The second element of the base $y_b$ goes from the center to the side on a horizontal plane. The third element of the base $z_b$ goes along the direction of the wellbore.

Figure 6.20: The wellbore coordinate system.

Based on the previous definition, it is possible to construct a transformation matrix $R_{GW}=f(\delta,\varphi)$ that links the geographical coordinate system and the wellbore coordinate system.

$$\uuline{S}{}_W = R_{GW} \uuline{S}{}_G R_{GW}^T$$ (6.19)

Furthermore, the wellbore stresses can be calculated from the principal stress tensor according with:

$$\uuline{S}{}_W = R_{GW} R_{PG}^T \uuline{S}{}_P R_{PG} R_{GW}^T$$ (6.20)

Where $\uuline{S}{}_P$ and $R_{PG}$ are the principal stress tensor and the corresponding change of coordinate matrix to the geographical coordinate system. The tensor $\uuline{S}{}_W$ is composed by the following stresses:

$$\uuline{S}{}_W = \left[ \begin{array}{ccc} S_{11} & S_{12} & ... ...2} & S_{23} \\ S_{31} & S_{32} & S_{33} \\ \end{array}\right]$$ (6.21)

Stresses on the plane of the cross-section of the deviated wellbore at the wellbore wall $(\sigma_{rr}, \sigma_{\theta \theta}, \sigma_{zz}, \sigma_{\theta z})$ depend on far-field stresses $\sigma_{11} = S_{11} - P_p$, $\sigma_{22} = S_{22} - P_p$, $\sigma_{33} = S_{33} - P_p$, $\sigma_{12} = S_{12}$, $\sigma_{13} = S_{13}$, and $\sigma_{23} = S_{23}$. The Kirsch equations require additional far field shear terms $\sigma_{12}$, $\sigma_{13}$, and $\sigma_{23}$ in order to account for principal stresses not coinciding with the wellbore orientation. The solution of Kirsch equation with far-field shear stresses is provided in Fig. 6.21.

Figure 6.21: Stresses around the wall of a deviated wellbore.

Solving for principal stresses $(\sigma_{tmax}, \sigma_{rr}, \sigma_{tmin})$ on the wellbore wall permits checking for rock failure (tensile of shear). The principal stresses may not be necessarily aligned with the wellbore axis leading to an angle $\omega$ (see Fig. 6.22). Because of such angle, tensile fractures in deviated wellbores can occur at an angle $\omega$ from the axis of the wellbore and appear as a series of short inclined (en-échelon) fractures instead of a long tensile fracture parallel to the wellbore axis as in Fig. 6.16.

Figure 6.22: Principal stresses around the wall of a deviated wellbore. ( $\Delta P = P_W - P_p$).