Summary
The fracture equation used in the oil industry is derived from the Kirsch
equation for the hoop stress. Because of its simplicity, it is almost
exclusively used for the prediction of fracture initiation pressures. However,
it is not useful for analysis of load history.
An analytic study was undertaken to model load history leading to the
fracturing of the borehole. To use the model, initial conditions must be
established, given by the virgin in-situ stress state and the pore pressure,
followed by the load history and the temperature history. Imposing a volumetric
strain balance, a new fracturing equation is developed. Because the borehole is
loaded in the radial direction, causing tension in the tangential direction, a
Poisson's effect arises. In addition, the general solution includes effects of
temperature history.
Example cases will show the improvement with the new model. The first case
compares the new load-history fracture model with the Kirsch solution. The
Poisson's scaling factor in the new solution leads to a higher fracture
pressure than the conventional solution. This may explain some of the
discrepancy between models and field data. The second case investigates the
thermal effects by comparing the fracture pressure for the drilling phase with
a hot-production phase and a cold-water-injection phase.
It is believed that by including the pressure and temperature load history,
a better assessment of the fracture strength is obtained, leading to better
predictions.
Introduction
Basis for the Model. Although the Kirsch solution for stresses in a circular
hole was published more than 100 years ago, it was not untill 1980 that
borehole mechanics started being applied to petroleum drilling. At that time,
deviated wells were evolving, and because of the complexity, high inclination,
and increasing length of these, borehole stability was identified as a critical
factor. Bradley (1979) is considered the person that introduced application of
classical mechanics into the petroleum industry by analyzing borehole
fracturing and collapse in deviated boreholes. Later, Aadnoy and Chenevert
(1987) presented the mathematical framework and elaborated on the applications.
These early works still form the basis for modern wellbore-stability
analyses.
Continuing work, the past two decades have of course led to many
contributions to the classical solutions. A full review will not be given here,
but Fjaer et al. (1992) serves as a good general reference. More recently,
Aadnoy and Belayneh (2004) have shown that the boundary condition given by the
drilling fluid can be better represented as an elastoplastic barrier. The
temperature effects are identified as having effect on the fracture pressure.
Examples are given by Maury and Sauzay (1987) and Maury and Guenot (1995). A
more recent paper by Gil et al. (2006) considers the temperature issues from a
poroelastic and "stress-cage" perspective.
The variable in the solution is the borehole pressure, which causes the
borehole to be loaded in the radial direction. However, the solution used today
does not include the full Poisson's effect (i.e., the effects in the tangential
and axial direction of a radial loading). The objective of this paper is to
include this effect. In the following, we will present the resulting expression
for the new model, which is derived in the Appendix.
© 2009. Society of Petroleum Engineers
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History
- Original manuscript received:
7 May 2008
- Meeting paper published:
25 August 2008
- Revised manuscript received:
12 September 2008
- Manuscript approved:
3 October 2008
- Published online:
16 July 2009
- Version of record:
28 September 2009
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