Page 91 - Physical Principles of Sedimentary Basin Analysis

3.20 Momentum balance (Newton’s second law) 73

Exercise 3.29 Show that equation (3.177) for effective stress becomes
∂σ
ij
= ( − f ) g δ i,z (3.185)
∂x j
when the fluid pressure is hydrostatic p f = f gz and the Biot coefficient is α = 1.
Solution: We have
∂ p f ∂( f gz) ∂z
= = f g = f gδ i,z (3.186)
∂x i ∂x i ∂x i
(where it is assumed that both the fluid density f and the acceleration of gravity are
constants).
Exercise 3.30 Find the vertical displacement caused by gravity in a 1D fluid-saturated
linear-elastic sedimentary column. Assume that the fluid pressure is hydrostatic p f =
f gz, the horizontal strain is zero, the displacement at the top of column (z = 0) is zero,
the top of the column (z = 0) is unstressed, and the Biot coefficient is α = 1.
Solution: It is the effective stress that causes deformations in a saturated porous medium,
and the equilibrium equation in the case of a hydrostatic fluid is

∂σ
ij
= ( − f ) g δ i,z (3.187)
∂x j
as seen from Exercise 3.29. Using the Lamé equations we get
2
dσ zz =−(2G + λ) d u = ( − f ) g. (3.188)

dz dz 2
The solution of equation (3.188) is therefore the same as in Exercise 3.28, except that the
solution yields effective stress instead of stress, and that the density that enters the solution
is − f . The maximum effective stress is σ = ( − f ) gh at the base of the column

0
(z = h), and we get the stress at the same depth by adding the fluid pressure p h = f gh,
which is then σ 0 = f gh. We see that there will be no deformation of the sedimentary
column when the fluid density is equal to the bulk density, ( − f = 0).

Exercise 3.31 Show that the fluid density has to be equal to the matrix (solid) density if
the bulk density and the fluid density are equal.
Solution: The bulk density is = (1 − φ) s + φ f where φ is the porosity, s and f are
the matrix and fluid densities, respectively. We then get

− f = (1 − φ)( s − f ) (3.189)
which implies that if s = f then = f . (It is assumed that φ< 1.)

Exercise 3.32
(a) Show that the equilibrium equations (3.177) become

dσ
zz dp e
= ( − α f )g − α (3.190)
dz dz