Page 257 - Reservoir Geomechanics

238 Reservoir geomechanics

are given by
 
0 0
S 1
S s =  0 S 2 0  (8.1)


0 0 S 3
To rotate these stresses into a wellbore coordinate system we first need to know how to
transform the stress field first into a geographic coordinate system using the angles α,
β, γ (Figure 8.1c). This is done using
   
x s X
(8.2)
   
y s = R s Y 
z s Z
where,
 
cos α cos β sin α cos β −sin β
 
R s =  cos α sin β sin γ − sin α cos γ sin α sin βsin γ + cos α cos γ cos β sin γ 
cos α sin β sin γ + sin α sin γ sin α sin βcos γ − cos α sin γ cos βcos γ
(8.3)
To transform the stress field from the geographic coordinate system to the borehole
system, we use
   
x b X
(8.4)
   
y b = R b Y 
Z
z b
where,
 
−cos δ cos φ −sin δ cos φ sin φ
sin δ −cos δ 0  (8.5)
 
R b = 
cos δ sin φ sin δ sin φ cos φ
With R s and R b defined, we can define the stress first in a geographic, S g , and then in
a wellbore, S g , coordinate system using the following transformations
T
S g = R S s R s
s (8.6)
T
S b = R b R S s R s R T
s b
where we define effective stress using the generalized form of the effective stress
law described above (equation 3.10). We go on to define individual effective stress
components around the well (simplified here for the wellbore wall) as

σ zz = σ 33 − 2ν (σ 11 − σ 22 ) cos 2θ − 4νσ 12 sin 2θ
σ θθ = σ 11 + σ 22 − 2 (σ 11 − σ 22 ) cos 2θ − 4σ 12 sin 2θ −
P
(8.7)
τ θz = 2 (σ 23 cos θ − σ 13 sin θ)
σ rr =
P