Complete Kirsch set of equations

Consider a vertical wellbore subjected to horizontal stresses $S_{Hmax}$ and $S_{hmin}$, both principal stresses, vertical stress $S_v$, pore pressure $P_p$, and wellbore pressure $P_W$. The corresponding effective stresses are $\sigma_{Hmax}$, $\sigma_{hmin}$, and $\sigma _v$. The Kirsch solution for a linear elastic and isotropic solid yields:

$$\left\lbrace \begin{array}{rcl} \sigma_{rr} & = & (P_W - P_p)... ...left( \frac{a^2}{r^2} \right) \cos (2\theta) \end{array}\right.$$ (6.2)

where $\sigma _{rr}$ is the radial effective stress, $\sigma _{\theta \theta }$ is the tangential (hoop) effective stress, $\sigma _{r\theta }$ is the shear stress in a plane perpendicular to $r$ in tangential direction $\theta$, and $\sigma_{zz}$ is the vertical effective stress in direction $z$.

The solution corresponds to a vertical wellbore with radius $a$, subjected to an internal pressure $P_W$, with far-field principal stresses $S_v$, $S_{Hmax}$, and $S_{hmin}$, and pore pressure $P_p$. The angle $\theta$ is the angle between the direction of $S_{Hmax}$ and the point at which stress is considered. The distance $r$ is measured from the center of the wellbore. For example, at the wellbore wall $r=a$.

Figure 6.7 shows an example of the solution of Kirsch equations for $S_{Hmax}=22$ MPa, $S_{hmin}=13$ MPa, and $P_W=P_p=10$ MPa. The plots show radial $\sigma _{rr}$ and tangential $\sigma _{\theta \theta }$ effective stresses, as well as the calculated principal stresses $\sigma _3$ and $\sigma_{1}$.

Figure 6.7: Example of solution of Kirsch equations. The effective radial stress at the wellbore wall is $P_W-P_p = 0$. Top: solution for radial $\sigma _{rr}$ and hoop $\sigma _{\theta \theta }$ stresses. Bottom: solution for principal stresses (eigenvalues from $\sigma _{\theta \theta }$, $\sigma _{rr}$, and $\sigma _{r\theta }$ ). Notice that $\sigma _1$ is the highest at top and bottom and $\sigma _3$ is the lowest at the sides. The influence of the cavity extend to a few wellbore radii.

Let us obtain $\sigma _{rr}$ and $\sigma _{\theta \theta }$ at the wellbore wall $r=a$. The radial stress for all $\theta$ is

$$\sigma_{rr}(r=a) = P_W - P_p$$ (6.3)

The hoop stress depends on $\theta$,

$$\sigma_{\theta \theta} (r=a) = -(P_W - P_p) +(\sigma_{Hmax}+\sigma_{hmin}) -2(\sigma_{Hmax}-\sigma_{hmin}) \cos (2\theta)$$ (6.4)

and it is the minimum at $\theta = 0$ and $\pi$ (azimuth of $S_{Hmax}$) and the maximum at $\theta = \pi/2$ and $3\pi/2$ (azimuth of $S_{hmin}$):

$$\left\lbrace \begin{array}{rcl} \sigma_{\theta \theta} (r=a,\... ... -(P_W - P_p) +3\sigma_{Hmax} -\sigma_{hmin} \end{array}\right.$$ (6.5)

These locations will be prone to develop tensile fractures ( $\theta = 0$ and $\pi$) and shear fractures ( $\theta = \pi/2$ and $3\pi/2$). The shear stress around the wellbore wall is $\sigma_{r \theta} = 0$. This makes sense because fluids (drilling mud) cannot apply steady shear stresses on the surface of a solid. Finally, the effective vertical stress is

$$\sigma_{zz} (r=a) = \sigma_v - 2\nu (\sigma_{Hmax}-\sigma_{hmin}) \cos (2\theta)$$ (6.6)