Page 173 - Reservoir Geomechanics

156 Reservoir geomechanics

Alternatively, one can determine the shear and normal stress via tensor transforma-
tion. If principal stresses at depth are represented by
 
0 0
S 1
S =  0 S 2 0 


0 0 S 3
we can express stress in a geographical coordinate system with the transform

S g = R SR 1 (5.10)
1
where
 
cos a cos b sin a cos b −sin b
 cos a sin b sin c − sin a cos c
R 1 = sin a sin b sin c + cos a cos c cos b sin c 
cos a sin b cos c + sin a sin c sin a sin b cos c − cos a sin c cos b cos c
(5.11)
and the Euler (rotation) angles that define the stress coordinate system in terms of
geographic coordinates are as follows:

a = trend of S 1
b =−plunge of S 1
c = rake S 2 .
However, if S 1 is vertical (normal faulting), these angles are defined as

a = trend of S Hmax − π/2
b =−trend of S 1
c = 0.

Using the geographical coordinate system, it is possible to project the stress tensor on
to an arbitrarily oriented fault plane. To calculate the stress tensor in a fault plane coor-
dinate system, S f ,we once again use the principles of tensor transformation such that

S f = R 2 S g R (5.12)
2
where
 
cos(str) sin(str) 0
R 2 =  sin(str) cos(dip) −cos(str) cos(dip) −sin(dip)  (5.13)


−sin(str) sin(dip) cos(str) sin(dip) −cos(dip)
where str is the fault strike and dip is the fault dip (positive dip if fault dips to the right
when the fault is viewed in the direction of the strike). The shear stress, τ, which acts
in the direction of fault slip in the fault plane, and normal stress, S n , are given by

τ = S r (3, 1) (5.14)
S n = S f (3, 3) (5.15)