Large-Scale Geological Carbon Sequestration - Introduction to Carbon Capture and Sequestration - page 489

Box 10.4.1 Mohr diagrams

Mohr circles provide a convenient and illustrative way to understand stress and brittle
failure in rocks. The stress state of rock in the subsurface can be resolved into three
principal mutually perpendicular vectors:

σ 3 . These vectors are referred to
as the maximum principal stress, intermediate stress, and minimum principal stress
directions, respectively (

σ 1 , σ 2 , and

σ 3 ). If we consider any plane within the earth whose
strike (intersection of horizontal plane with this plane of interest) is parallel to

σ 1 ≥ σ 2 ≥

σ 2 , then

(a) shows that the two-dimensional plane stress in the plane containing σ 1 and σ 3 can

be resolved using the appropriate tensor operations into a shear stress ( σ s , often

denoted by τ ) and normal stress ( σ n ).

θ

θ

(a)

σ 2
can be found using a Mohr diagram. Figure (b) shows a Mohr diagram, in which the
shear stress (

The shear and normal stresses on any plane of interest with strike parallel to

σ s ) is plotted on the y-axis and normal stress ( σ n ) on the x-axis. The Mohr

circle has diameter σ 1 - σ 3 = σ d (differential stress), and is centered at ½( σ 1 + σ 3 ) on the

normal-stress axis.
Let us now determine the shear and normal stress of our plane in (a).
Mathematically, these stresses are given by:

1

( )

()

σ= σ−σ sin 2 θ

s 13

2

1 1

( ) ( ) ()

σ= σ+σ + σ−σ cos 2 θ

n 13 13

2 2

Figure (b) shows that these components can be obtained from a simple graphical
construction of the Mohr diagram.

( Continued )

starting from a stress state in which a fault is stuck because shear stress
is not large enough to overcome the normal stress on it, pore pressure
increases can reduce the effective normal stress to the point that the fault
will fail with no increase in shear stress.