The Kirsch Equations

Part 6, Part 6: The Wellbore, Kirsch and the Window

Learning objectives

State the Kirsch solution: how a circular hole concentrates the far-field stress around its wall
Read the hoop stress around the wall, maximum at the minimum-stress azimuth and minimum at the maximum-stress azimuth
Compute the canon maximum effective hoop stress, 68.7 MPa, and see it exceed the rock strength
Confirm the concentration decays as one over radius squared, back to the far field within a few hole radii

A Hole Rearranges the Stress

Drill a hole and you do not simply remove rock; you force the stress that the rock was carrying to flow around the opening, and it piles up at the wall. Gustav Kirsch solved this in 1898 for a circular hole in a stressed plate, and his equations are the single most important tool in this book, the wellbore's equivalent of Gassmann. They give the full stress field $(\sigma_{rr}, \sigma_{\theta\theta}, \sigma_{r\theta})$ rr,sigmathetatheta,sigmartheta) at every radius and angle around the hole. Two features matter most. At the wall the radial stress is just the mud pressure $P_w$ w pushing back, but the hoop stress $\sigma_{\theta\theta}$ thetatheta, the stress running around the wall, is dramatically concentrated. And the concentration is local: it dies off as $1/r^2$ , so within three or four hole radii the stress is back to its undisturbed far-field value. A borehole disturbs the stress only in its immediate neighborhood, but there, intensely.

The polar map in the figure shows the hoop stress around the hole; the sliders set the far-field stresses. The wall hoop stress swings enormously with azimuth. It is largest at the azimuth of the minimum horizontal stress, where the equation gives, in effective terms, $\sigma_{\theta\theta}^{max} = 3S_{Hmax} - S_{hmin} - P_w - P_p$ thetathetamax=3SHmax−Shmin−Pw−Pp, and smallest at the azimuth of the maximum horizontal stress, $\sigma_{\theta\theta}^{min} = 3S_{hmin} - S_{Hmax} - P_w - P_p$ thetathetamin=3Shmin−SHmax−Pw−Pp. The factor of three on the near stress is the stress-concentration signature. For the canon well, with $S_{Hmax} = 62$ Hmax=62, $S_{hmin} = 46$ hmin=46, mud $P_w = 36$ w=36, and $P_p = 35.3$ p=35.3, the maximum effective hoop stress is $3(62) - 46 - 36 - 35.3 = 68.7$ MPa, which exceeds the rock's UCS of 65: the wall is being crushed at that azimuth, and the breakouts of the next section are the result.

Why This Is the Workhorse

Everything the rest of this part does runs on these equations. The wall is compressively overstressed where the hoop stress is highest, at the minimum-stress azimuth, producing breakouts. The wall can go into tension where the hoop stress is lowest, at the maximum-stress azimuth, producing tensile fractures. The mud weight $P_w$ w appears in every hoop-stress expression, which is exactly why choosing it correctly, the mud-weight window, controls both failures. And because the failures point in known directions relative to the stress field, reading them backward measures the stress, the inversion Part 5 was already invoking. One check confirms the machinery: set $S_{Hmax} = S_{hmin}$ , an isotropic horizontal field, and the azimuthal swing vanishes, leaving a uniform wall hoop stress of $2S - P_w - P_p$ at every angle, the classic thick-walled-cylinder result. With the Kirsch field in hand, the next section watches the wall break.

References

Kirsch, G. (1898). Die Theorie der Elastizitaet und die Beduerfnisse der Festigkeitslehre. Zeitschrift des Vereines deutscher Ingenieure, 42, 797-807.
Zoback, M. D. (2007). Reservoir Geomechanics. Cambridge University Press.
Jaeger, J. C., Cook, N. G. W., & Zimmerman, R. W. (2007). Fundamentals of Rock Mechanics (4th ed.). Blackwell.