Stress around a borehole

If a circular hole is made in a homogeneous body experiencing a homogeneous stress field, stress will concentrate around the hole since no force can be carried through the interior void. Figure 3A shows the stress concentration around the hole in a body under uniaxial compression, of magnitude $\sigma _1^R$ in the far-field, in the x-direction. Since the hole boundary is a free surface, the stresses acting normal to the boundary must decrease to zero at the boundary. As I will show below, at the point x = 0, y = $\pm$R, we have the largest compressive stress, $\sigma_{xx}$ = 3 $\sigma _1^R$ and $\sigma_{yy}$ = 0. At x = $\pm$R and y = 0, we have the largest tensile stress, $\sigma_{xx}$ = 0 and $\sigma_{yy}$ = - $\sigma _1^R$.

Figure: A) Circular hole in a material under uniaxial, compressive stress of magnitude $\sigma _1^R$ at infinity. B) Coordinate system and orientations of the components of the stress tensor in cylindrical coordinates.

Even if the hole is filled with other material, of differing elastic moduli, there would be a perturbation in the stress field around the inclusion. The equations governing the stresses around a hole are best represented in polar coordinates, for the 3D case cylindrical coordinates will be used, see Figure 3B. The stress equations are obtained by considering the equilibrium equations in the three coordinate directions, e.g. in the case of the tangential (or hoop) stress, $\sigma_{\theta\theta}$, we require the tangential forces to sum to zero, $\sum F_{\theta} = 0$, which yields

$$\frac{1}{r}\frac{\partial\ensuremath{\sigma_{\theta\theta}}}{\par... ...\sigma_{r\theta}}}{\partial r} + \frac{2}{r}(\ensuremath{\sigma_{r\theta}}) = 0$$ (24)

If we assume that the rock is elastic and isotropic and that the borehole is parallel to one of the principal stresses, an example is a vertical borehole in an area where the vertical stress, $\sigma_V$, is a principal stress, we obtain the analytical solutions described already by Kirsch [1898].

$$\ensuremath{\sigma_{rr}} = \frac{1}{2}(\sigma_H + \sigma_h)\left ( 1 - \frac{R^2}{r^2}\right... ...ac{R^2}{r^2} + 3\frac{R^4}{r^4}\right )\cos 2\theta + \frac{\Delta P R^2} {r^2}$$

$$\ensuremath{\sigma_{\theta\theta}} = \frac{1}{2}(\sigma_H + \sigma_h)\left ( 1 + \frac{R^2} {r^2}\righ... ...a_h)\left ( 1 + 3\frac{R^4}{r^4}\right )\cos 2\theta - \frac{\Delta P R^2}{r^2}$$

$$\ensuremath{\sigma_{r\theta}} = -\frac{1}{2}(\sigma_H + \sigma_h)\left ( 1 + 2\frac{R^2}{r^2} - 3\frac{R^4}{r^4}\right )\sin 2\theta$$

$\sigma_H$ and $\sigma_h$ are the maximum and minimum horizontal stresses, $\theta$ is the angle around the borehole from the direction of the maximum horizontal stress, see Figure 3B, $R$ is the borehole radius and $r$ the radial distance to the point of measurement ($r \geq R$) and $\Delta P$ is the difference between the fluid pressure in the borehole, $P_b$, and the formation pore pressure, $P_0$. At the borehole wall, $r = R$, the equations simplify to

$$\ensuremath{\sigma_{rr}} = \Delta P$$ (25)

$$\ensuremath{\sigma_{\theta\theta}} = \sigma_H + \sigma_h - 2(\sigma_H - \sigma_h) \cos 2\theta -\Delta P$$ (26)

$$\ensuremath{\sigma_{r\theta}} =$$ 0 (27)

Comparing these equations with the simple case considered in Figure 3A, we see that the equations further simplify to the results stated above, at $\theta = 0$ and $\theta = \ensuremath{{90}^{\circ}}$ with $\sigma_H = \ensuremath{\sigma_{1}}$, $\sigma_h = 0$ and $\Delta P = 0$.

The general case with a borehole arbitrarily inclined in the stress field was considered by Hiramatsu and Oka [1962] and Fairhurst [1968], here I will only state the equations evaluated at the borehole wall, $r = R$. The stresses referred to below, $\ensuremath{\sigma_{ij}}\hspace{0.3cm}i,j = 1,2,3$, are stresses in a borehole local Cartesian coordinate system where the z-axis lies along the borehole axis, the x-axis is in the plane perpendicular to the borehole axis directed towards the bottom side of the borehole and the y-axis is in the same plane but perpendicular to x. This is the coordinate system utilized by Peška and Zoback [1995]. The cylindrical coordinate system is also local to the borehole with the z-axis parallel to the borehole axis.

$$\begin{array}{ccl} \ensuremath{\sigma_{rr}}& = & \Delta P \\ ... ...3}}\cos\theta - \ensuremath{\sigma_{13}}\sin\theta) \end{array}$$ (28)

In this Cartesian coordinate system, $\theta$ is the angle from the x-axis around the borehole wall towards the y-axis. $\nu$ is Poisson's ratio. These general equations will be used for the borehole stress analysis in Paper I.