Page 155 - Applied Petroleum Geomechanics

148 Applied Petroleum Geomechanics

σ σ σ

y
2a
σ(x)
= +
2a l a+l 2a l

σ σ
Figure 4.8 Schematic presentation of the DugdaleeBarenblatt model with a linear
closing stress (s (x)).

crack opening. As a first estimate of varying s (x), Labuz et al. (1985)
proposed a linear distributed s (x) over the length of the FPZ (l p ),
x a
sðxÞ¼ T 0 (4.38)
l p
where x is the distance from the center of the true crack tip along the x-axis.
Consider a crack of length 2a in an infinite plate (plane stress) acted on
by a uniform tensile stress, s, as shown in Fig. 4.8. A nonlinear zone of
length l p is formed (assuming a > l p ). Using a similar formulation of
Barenblatt (1962), Labuz et al. (1985) derived a solution of the length l p
of the FPZ zone as follows:
2
9p K IC
l p ¼ (4.39)
32 T 0
Therefore, with the assumption of a linear distribution of closing stress
and smooth crack closure (the singularity is zero), the nonlinear region is
over twice as long as with the constant distribution as derived by Dugdale
(1960), suggesting that the FPZ zone in rock is bigger than that in metal.

4.2.7 Fracture toughness of rock and its correlation to
tensile strength

The fracture toughness of rock is an important parameter for modeling
fracture failure and can be measured from laboratory tests. Gunsallus and