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Rock physical and mechanical properties 59

smaller than that in unpropped fractures, as shown in Fig. 2.17. This implies
that a higher proppant concentration, larger proppant size, and larger
propped fracture area can enhance fracture conductivity and increase oil
and gas production.

2.4.6 Stress and permeability relation in porous rocks
For porous media, fluids flow mainly through pore spaces. The variation in
grain sizes or pore spaces due to the effect of an applied stress causes a
change of permeability. For a simple cubical grain packing structure, the
permeability k can be expressed as follows (Bai and Elsworth, 1994):

2R 2
k ¼ (2.44)
p 2
where R is the grain radius of the porous rock.
When the stresses exerted on the grains change, it will cause the grain
size and pore space to change and result in permeability change. From
Eq. (2.44), the following equation can be used to describe the relation
between the grain size and permeability:
R 2
k p ¼ k 0 2 (2.45)
R 0
where k p is the permeability after the change of the grain radius (R); k 0 is
the initial permeability; R 0 is the initial grain radius.
Under a three-dimensional effective stress condition, the change in grain
sizes within a cubical packing (Fig. 2.18) can be determined by analyzing the
elastic contact of spheres. Applying the theory of the Hertzian contact of

Δσ z Δσ z

Δσ x Δσ x
Δσ y Δσ y

Δσ x Δσ x
Δσ y Δσ
y
z
y
Δσ z Δσ z x
Figure 2.18 Spherical contact of grains under 3-D stresses.