Page 81 - Physical Principles of Sedimentary Basin Analysis

3.15 Effective stress in 3D 63

stress is simply the weight of the sedimentary column minus the buoyancy on a macro-
scopic xy-surface. The pores can be thought of as so small that they become an unimportant
perforation of planes that carry stress or force. The effective stress concept dates back to
Terzaghi (Terzaghi, 1925) and his studies of compaction of soils in civil engineering.
Exercise 3.19 Let the porosity by related to depth by Athy’s function (2.4). Show that the
lithostatic pressure becomes
p b (z) = s gz + ( s − w ) gφ 0 z 0 (1 − e −z/z 0 ) (3.127)

as a function of depth z.

3.15 Effective stress in 3D
We have so far only considered effective stress in the vertical direction. It has already been
mentioned that the vertical effective stress controls how loose sediments compact. The
generalization to 3D of vertical effective stress (3.126) is the effective stress tensor

σ = σ ij − α p f δ ij (3.128)
ij
where α is the Biot coefficient. The Biot coefficient is 1 in the vertical effective
stress (3.126), and it is therefore often approximated by 1. The effective stress tensor con-
trols deformations (strain) of porous media just like the effective vertical stress controls
deformation (compaction) of porous sediments. In the case of linear elasticity the Lamé
equation becomes

− σ =−(σ ij − α p f δ ij ) = 2Gε ij + λε kk δ ij . (3.129)
ij
Our sign convention implies that a positive fluid pressure leads to positive strain, which
causes expansion of the porous media. The inverse of the Lamé equation is just like
equation (3.96) except that strain is now produced by effective stress instead of stress:
1 ν

− ε ij = (1 + ν) σ − σ δ ij . (3.130)
ij
kk
E E
One way to get some feeling of the effective stress is to compute the volume strain ε kk
from equation (3.130), which gives
V (α p −¯σ)
= (3.131)
V K
1
where ¯σ = σ kk is the average stress in the solid porous matrix. (See Exercise 3.15 and
3
Section 3.2.) The volume strain (3.131) shows that
V > 0, when α p > ¯σ (3.132)
V < 0, when α p < ¯σ. (3.133)

Our sign convention now tells us that the porous matrix expands when the pore fluid pres-
sure is larger than the average normal stress. We assume for simplicity that α = 1. The