Page 255 - Reservoir Geomechanics

236 Reservoir geomechanics

the principal stresses are not in horizontal and vertical planes. The generalized theory
presented in this chapter allows us to do so.
Finally, I address the issue of determination of stress orientation from cross-dipole
sonic logs. I begin by reviewing the basic idea of deriving stress orientations from this
data in vertical wells when bedding is sub-horizontal and go on to illustrate how the
technique can work in deviated wells when the bedding (or aligned fractures) is likely
to be at an oblique angle to the well trajectory.

State of stress surrounding an arbitrarily deviated well

Various authors have addressed different aspects of wellbore failure in deviated wells.
Bradley (1979)was the first to model for compressive well failure of a deviated well
for the purpose of recommending proper mud weights to prevent borehole failure.
However, he did all of his analyses for the rare case where the two horizontal stresses are
equal and less than the vertical stress. Daneshy 1973; Richardson (1981), Roegiers and
Detournay (1988), Yew and Li (1988) and Baumg¨artner, Carvalho et al.(1989)have
done numerical and experimental analyses of hydraulic fracture formation in wells
at various orientations to principal stresses, although only several specific borehole
orientations and stress states were considered. In this chapter, we present a systematic
analysis of wellbore stability (including both compressive and tensile failures) for
arbitrarily inclined boreholes in a wide variety of stress states ranging from normal
faulting, to strike-slip to reverse faulting environments. We also consider the likelihood
of compressive and tensile borehole failure as a function of rock strength and borehole
fluid pressure over a wide range of conditions.
In a deviated well, the principal stresses acting in the vicinity of the wellbore wall are
generally not aligned with the wellbore axis (Figure 8.1a). To consider failure in a well
of arbitrary orientation, we must define three coordinate systems (Figure 8.1b): (1) a
geographic coordinate system, X, Y and Z oriented north, east and vertical (down); (2)
a stress coordinate system, x s , y s and z s (corresponding to the orientations S 1 , S 2 , and
S 3 ) and (3) the wellbore coordinate system x b , y b and z b where x b is radial, pointing
to the bottom of the well, z b is down along the wellbore axis and y b is orthogonal in a
right-hand coordinate system. To most easily visualize wellbore failure we will always
look down deviated wells and evaluate wellbore failure as a function of angle, θ, from
the bottom of the well in a clockwise direction. Despite the complexities associated with
such cases, to analyze whether (and how) failure might initiate at the wellbore wall,
we simply need to consider whether the principal stresses acting in a plane tangential
to the wellbore wall, σ tmax and σ tmin (and σ rr acting normal to the wellbore wall) are
such that that they exceed the strength of the rock. I define the angle between the axis
of the wellbore and the plane normal to σ tmin as ω (Figure 8.1a), and consider stress