Page 291 - Physical Principles of Sedimentary Basin Analysis

8.7 Ductile flow and yield strength envelopes 273

which gives that the fluid pressure necessary for fracturing is
bf − a 2S 0

p f = σ 1 + . (8.37)
b − a b − a
The condition (8.35) for tension fracture is

− T 0 = σ 3 − p f (8.38)
which gives that the fluid pressure necessary for fracturing is

p f = f σ 1 + T 0 . (8.39)

An increasing fluid pressure leads to tension fracture if the necessary fluid pressure for
tension fracture is less than for shear fracture, and vice versa. A near isotropic stress state
leads to tension fracture, while a sufficient amount of anisotropy leads to shear fracture.
The condition (8.39) for tension fracture simplifies to
p f ≈ f σ 1 (8.40)

for depths where σ 1
T 0 . Observations from overpressured oil and gas reservoirs show
that the fluid pressure is normally bounded by 85% of the overburden, which might indicate
that the amount of anisotropy is f ∼ 0.85.

8.7 Ductile flow and yield strength envelopes
Rock behaves more ductile than brittle at higher temperatures, and its mode of deformation
becomes flow rather than fracturing and faulting. The ductile flow of rock is measured in
the laboratory by a strain rate, when the rock is subjected to differential stress. It could for
example be the strain rate of a cylindrical specimen compressed with a constant axial stress
σ 1 and a surrounding stress σ 2 = σ 3 . The stresses σ 1 and σ 3 are the largest and the least
principal stresses, respectively. Such an experiment resembles the triaxial test shown in
Figure 8.9, but the stresses σ 1 and σ 3 are now kept constant, while a strain rate is measured
in the direction of σ 1 . Experiments show that the strain rate ˙ for a rock sample under a
stress difference σ 1 − σ 3 is

n E
˙ = (σ 1 − σ 3 ) A exp − (8.41)
RT
where n is an empirical exponent. The strain rate is temperature dependent by means of an
Arrhenius factor A exp(−E/RT ), where A is the Arrhenius prefactor, E is the activation
energy, R is the gas constant and T is the temperature in kelvin. The stress–strain-rate
relationship (8.41)iscalled power law creep, because the strain rate is proportional to the
differential stress to the power of n. We notice that the strain rate becomes zero in the
case of zero differential stress, σ 1 − σ 3 = 0. A stress anisotropy is therefore necessary
for rocks to deform by ductile flow. The stress anisotropy is often given by means of the