Deviated Wells

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

Learning objectives

Rotate the far-field stress tensor into the borehole frame of a deviated well
Explain why wellbore stability depends on trajectory, not just depth
Show that the safest and worst trajectories are set by the stress regime
Read a stability map: required mud weight as a function of well azimuth and inclination

The Hole Chooses Which Stresses It Feels

The Kirsch equations of this part assumed a vertical well, feeling the two horizontal stresses. A deviated well, drilled at some inclination and azimuth, feels a different combination, because the stresses acting across a tilted borehole are the far-field tensor rotated into the borehole's own frame. A horizontal well drilled along $S_{hmin}$ hmin has the two large stresses, $S_v$ v and $S_{Hmax}$ Hmax, acting across it; a horizontal well along $S_{Hmax}$ Hmax has $S_v$ v and $S_{hmin}$ hmin across it, a very different and generally gentler loading. The wellbore-stability problem, in full, is the Kirsch calculation run on the rotated stress state at every point around a hole of arbitrary orientation, and its answer, the required mud weight, depends strongly on where you point the well.

The lower-hemisphere map in the figure colors every well trajectory by the mud weight it requires to stay stable: cool where the well is easy, hot where it is hard. The pattern is dictated by the regime. In our canon normal-faulting state, where $S_v$ v is the largest stress, a vertical well is relatively stable, because $S_v$ v acts along the hole rather than across it, so the wall feels only the two smaller horizontal stresses. The worst trajectory is a horizontal well drilled along the $S_{Hmax}$ Hmax direction, because it then feels $S_v$ v and $S_{hmin}$ hmin across the hole, the full span from 67.7 down to 46 MPa, the largest stress difference available, and needs the heaviest mud. A horizontal well along $S_{hmin}$ hmin, by contrast, feels $S_v$ v and $S_{Hmax}$ Hmax across it, a much smaller span and a gentler well. Drag the well point and watch the required mud weight swing by pounds per gallon between the best and worst headings.

Trajectory as a Design Variable

This turns wellbore stability from a fixed constraint into a design choice. When a well must be deviated, to reach a distant target, to land horizontally in a reservoir, the geomechanicist advises the azimuth and inclination that keep it in the easy part of the map, and warns against the headings that would demand a mud weight outside the window of the last section. In a strike-slip or reverse regime the map rotates and the safe directions change, which is why the regime of Part 5 is the first thing established: it tells you, before any well is planned, which way to point. The stability map is thus the meeting point of the whole course, the stress field from Part 5, the strength from Part 3, the pore pressure from Part 4, and the Kirsch machinery of this part, all resolved onto the geometry of a real, crooked hole. One task remains in this part: to read, from a well already drilled, the breakouts and fractures that measure the stresses, which is the next section, and then Part 7 turns to fracturing the rock on purpose.

References

Peska, P., & Zoback, M. D. (1995). Compressive and tensile failure of inclined well bores and determination of in-situ stress and rock strength. Journal of Geophysical Research, 100(B7), 12791-12811.
Zoback, M. D. (2007). Reservoir Geomechanics. Cambridge University Press.
Fjaer, E., Holt, R. M., Horsrud, P., Raaen, A. M., & Risnes, R. (2008). Petroleum Related Rock Mechanics (2nd ed.). Elsevier.