Page 187 - Reservoir Geomechanics

170 Reservoir geomechanics

trajectories to bend in such a way as to be parallel and perpendicular to the wellbore
wall because it is a free surface which cannot sustain shear traction. Moreover, as
the material removed is no longer available to support far-field stresses, there is a
stress concentration around the well. This is illustrated by the bunching up of stress
trajectories at the azimuth of S hmin , which indicates strongly amplified compressive
stress. In contrast, the spreading out of stress trajectories at the azimuth of S Hmax
indicates a decrease in compressive stress.
Mathematically, the effective stresses around a vertical wellbore of radius R are
described in terms of a cylindrical coordinate system by the following:

1
R 2 1
σ rr = (S Hmax + S hmin − 2P 0 ) 1 − + (S Hmax − S hmin )
2 r 2 2

2 4 2
4R 3R P 0 R
× 1 − + cos 2θ + (6.1)
r 2 r 4 r 2
1
R 2 1
σ θθ = (S Hmax + S hmin − 2P 0 ) 1 + − (S Hmax − S hmin )
2 r 2 2
3R P 0 R
T

4 2
× 1 + cos 2θ − − σ (6.2)
r 4 r 2
1
2R 2 3R 4
τ rθ = (S Hmax − S hmin ) 1 + − sin 2θ (6.3)
2 r 2 r 4

where θ is measured from the azimuth of S Hmax and
P is the difference between the
mud weight in the wellbore and the pore pressure, P 0 . σ
T represents thermal stresses
arising from the difference between the mud temperature and formation temperature
(
T). This will be ignored for the moment but is considered below. It can be shown
that for any reasonable amount of elastic anisotropy, the stress concentration around a
vertical well is not changed in any significant way (Lekhnitskii 1981). Hence, while
anisotropic rock strength induced by weak bedding planes can have an important effect
on wellbore failure (as described below), elastic anisotropy generally does not.
There are several important points about these equations that are illustrated in
Figure 6.2 for the following parameters:
S Hmax = 90 MPa

S Hmax orientation is N90 E (east–west)
◦
S v = 88.2 MPa (depth 3213m)

S hmin = 51.5 MPa

P p = P mud =31.5 MPa

First, the stress concentration varies strongly as a function of position around the
wellbore and distance from the wellbore wall. Also, the stress concentration is sym-
metric with respect to the direction of the horizontal principal stresses. For an east–west
direction of S Hmax , Figure 6.2a shows that σ θθ (the so-called effective hoop stress) is