Page 208 - Introduction to Petroleum Engineering

HYDRAULIC FRACTURING 195

hmin

hmin
hmax hmax hmax

v
hmin
Stress orientation Plan view
FIGURE 10.3 Orientation of three principal stresses and plan view of borehole breakout.

high‐permeability rock. (This Biot refers to Maurice Anthony Biot who died in 1985
and is not the same as the namesake of the Biot number in heat transfer—that was
Jean‐Baptiste Biot who died in 1862.)
In addition to vertical stress, we need to describe stresses in the horizontal direc-
tions. In general, there are two horizontal stresses: the smaller horizontal stress is
minimum horizontal stress σ hmin , and the larger horizontal stress is maximum
horizontal stress σ hmax . Figure 10.3 shows the orientation of the three principal stresses
and a plan view of wellbore failure known as borehole breakout for a vertical well.
Borehole breakout occurs in the direction of minimum principal stress and can be
detected using a caliper log. The orientation of borehole breakout can be used to
indicate the orientation of principal horizontal stresses.
The effective horizontal stress can be found from the effective vertical stress using
Poisson’s ratio ν:
νσ
σ = ve (10.14)
he 1 −ν

Poisson’s ratio is the ratio of radial to axial strain for a cylindrical sample in tension.
For an incompressible material, Poisson’s ratio is 0.5. Poisson’s ratio is about 0.3 for
carbonates, about 0.2 for sandstones, and greater than 0.3 for shales. The actual
horizontal stress can be expressed using Biot’s constant:
α
σ = σ + p (10.15)
he
h
This horizontal stress is the minimum horizontal stress, σ hmin . To find the maximum
horizontal stress, we need to include the additional component of tectonic stress:

σ hmax = σ hmin + σ tectonic (10.16)
The “plane” of a hydraulic fracture will be roughly perpendicular to the direction
of the smallest of these three stresses (σ , σ hmin , and σ hmax ), which is usually σ hmin .
v
The above relationships for stress provide first‐order estimates, but they are usually
on the low side. Applications of these relationships are demonstrated using exam-
ples. The first two examples show how to calculate the overburden stress gradient.