Page 80 - Physical Principles of Sedimentary Basin Analysis

62 Linear elasticity and continuum mechanics

We see that the gravitational force on the body is counteracted by the weight of the fluid
displaced by the body, which implies that the body will sink (F points downwards) when
b > f , or otherwise float.
The reason for deriving Archimedes’ principle is that sediments are normally immersed
in fluid and therefore experience buoyancy. Let us consider a column of sediments of height
dz, porosity φ, matrix density s and a horizontal surface area A. The net gravitational
force of the sediments on the base of the column is then
F = ( s − f )gA(1 − φ)dz, (3.121)

because the volume of fluid displaced by the solid (sediments) is A (1 − φ) dz. The stress
on the base of the column becomes

dσ = F/A = ( s − f )(1 − φ)gdz (3.122)

where the vertical stress σ is called the effective stress. It is written properly as an integral
z

σ = ( s − f ) 1 − φ(z) gdz (3.123)
0
because the porosity is a function of depth. The effective stress plays an important role
in controlling the compaction of the sediments. The porosity (or the void) ratio of loose
(unlithified) sediments is shown with laboratory experiments to be a function of the
effective stress.
We will now introduce two more vertical stress definitions. The first one is the
hydrostatic fluid pressure,
z
p h = f gdz (3.124)
0
which is simply the stress from the weight of the water column. The second one is the
vertical lithostatic stress

z
p b = φ f + (1 − φ) s gdz (3.125)
0
which is the stress from the weight of both the fluid and the solid sediment matrix. The
difference between the lithostatic stress and the hydrostatic fluid pressure is the effective

vertical stress, σ = p b − p h .
It often turns out that sediments are overpressured, which means that the fluid pressure
p f is larger than the hydrostatic pressure. The effective stress is therefore generally stated
as the lithostatic stress minus the fluid pressure,

σ = p b − p f . (3.126)
It is important to note that the effective stress is not the same as the average stress on
grain–grain contacts. The stress at grain contacts may be quite large because the weight
of a sedimentary column can be distributed over small contact areas. The vertical effective