Page 157 - Applied Petroleum Geomechanics

150 Applied Petroleum Geomechanics

4.3 Sneddon solutions of fracture widths
4.3.1 2-D plane strain solution of the Griffith fracture
Numerous analytical or semi-analytical solutions for estimating fracture
width have been proposed. The line crack solution proposed by Sneddon
(1946) and Sneddon and Elliott (1946) can be used to determine the width
of a 2-D plane strain crack (fracture) in an isotropic stress environment
without the presence of a borehole. Sneddon and Elliott (1946) considered
the distribution of stresses in the interior of an infinite two-dimensional
elastic medium when a very thin internal crack is opened under the ac-
tion of a pressure, which may be considered to vary in magnitude along the
length of the crack. Sneddon (1946) derived stress distributions in
the interior of an infinite two-dimensional elastic medium produced by the
opening of an internal crack (the length was 2c in his paper, and here 2L is
used as shown in Fig. 4.9) under the action of a pressure in the crack in the
plane strain condition. For the case of a uniform pressure (p 0 ), Sneddon and
Elliott (1946) proposed the following solution for the fracture width (two-
side displacements in y-direction):
2
4ð1 n Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
wðxÞ¼ p o L x 2 (4.45)
E
where w(x) is the fracture width; E is Young’s modulus of the rock; n is
Poisson’s ratio; p 0 is the internal pressure in the fracture; L is the fracture
half length; x is the distance from the fracture center, as shown in Fig. 4.9.
The maximum fracture width appears at the center of the fracture,
which can be obtained from Eq. (4.45) when x ¼ 0.
Sneddon (1946) recognized that an adjustment might be required to the
analytical solution for the effects of shorter height fractures when he stated
that “the most striking feature of the analysis in the three-dimensional case
is that the expressions for the components of stress in the neighborhood of

y

L
p(x) x
p

Figure 4.9 2-D plane strain fracture model.