Page 79 - Physical Principles of Sedimentary Basin Analysis

3.14 Buoyancy and effective stress 61

The two terms in the right-hand side of the equation are the contributions to the horizontal
stress from the lithostatic stress and the horizontal thermal stress, respectively.
These contributions can be compared when the horizontal stress (3.117) is rewritten as

ν Eβc
σ h = 1 + b gz (3.118)
1 − ν 3ν b g
where the lithostatic stress is σ b = b gz and the temperature is given by a constant thermal
gradient c ( T = cz). One sees that the number
Eβc
N T = (3.119)
3ν b g
measures the size of the thermal stress relative to the lithostatic stress as a cause for hor-
izontal stress. We get N T = 0.96 using typical values for the parameters like ν = 0.25,
4
E = 60 GPa, β = 1 · 10 −5 K −1 , c = 0.03 K/m and b g = 2.5 · 10 Pa/m, which means
that the thermal expansion of the rock is almost equally important as compression caused
by the lithostatic stress.

3.14 Buoyancy and effective stress
We will first examine a body immersed in a fluid, and show the renowned Archimedes’
principle. It states that the upward vertical buoyancy force is equal to the weight of the
fluid displaced by the body. Archimedes’ principle is shown for a solid box of bulk den-
sity b and volume V = Ah, where A is the surface area and h is the thickness (see
Figure 3.15). The surface forces from the fluid on the vertical sides balance each other, but
the lower and upper horizontal surfaces are acted on by the forces F 1 =− f gA(z + h)e z
and F 2 = f gA e z , respectively, where z is the depth down to the upper surface. The
gravitational force on the body is F 3 = b Vg e z , and the sum of these three forces F i
(i = 1, 2, 3) is
F = ( b − f )Vg e z . (3.120)

F (2) = ρ gAz
f
z

F (3) = ρ gV
s
z + h
F (1) = ρ gA(z + h)
f
Figure 3.15. The gravitational force on the body is counteracted by the surface forces from the fluid.
(The surface forces on the vertical sides balance each other and they are therefore not shown.)