The fracture equation used in the oil industry is derived from the Kirsch equation for the hoop stress. Due to its simplicity it is almost exclusively used for 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 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 Poissons 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 Poissons 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.