Page 89 - Physical Principles of Sedimentary Basin Analysis

3.20 Momentum balance (Newton’s second law) 71

where the “body” is defined as all particles contained by the volume V (t).Thevolume
may change shape and size, but it always contains the same particles. The left-hand side
of equation (3.170) is the rate of change of momentum, and the right-hand side has as the
first term the body force (gravity) and as the second term the surface force acting on the
body. The Kronecker delta δ i,z makes sure that gravity acts only in the vertical direction.
We have seen that the force acting on a surface with area dA and outward unit normal
vector n j is σ ij n j dA. Notice that ∂V (t) is used as a notation for the entire surface of the
volume V (t). The integral over the surface is converted to a volume integral by means of
Gauss’s theorem,

∂ f j
f j n j dA = dV. (3.171)
∂V (t) V (t) ∂x j
Furthermore, the rate of change of momentum in direction i is rewritten as

d Dv i
v i dV = dV (3.172)
dt V (t) V (t) dt
by means of equation (3.168) when f = v i . Newton’s second law as expressed by
equation (3.170) is therefore

Dv i ∂σ ij
= g δ i,z + . (3.173)
dt ∂x j
The acceleration Dv i /dt is often small compared with the constant of gravity, and the
left-hand side of equation (3.173) can be approximated by zero. Newton’s second law then
reduces to a force balance
∂σ ij
=− g δ i,z (3.174)
∂x j
and it is in this last form that we will be using Newton’s second law. But we have to change
the sign of the right-hand side before we can use equation (3.174) with a z-axis pointing
downwards. A 1D model along the z-axis gives after sign reversal

∂σ zz
= g (3.175)
∂z
and an integration then gives σ zz = gz as wanted. The force balance (3.174)isnow
written once more, adapted for a z-axis pointing downwards,

∂σ ij
= g δ i,z . (3.176)
∂x j
We have already seen that effective stress is the cause for deformations in a fluid-saturated
porous medium. The fluid pressure acts to expand the medium while (positive) normal
stress will compress it. The force balance (3.176) becomes
∂σ ij ∂ p f
+ α = g δ i,z (3.177)
∂x j ∂x i