Analytical stress functions applied to hydraulic fracturing: Scaling the interaction of tectonic stress and unbalanced borehole pressures

Some 19,000 new oil and gas wells were drilled in the US alone in 2010, and many need hydraulic fracturing aimed at improving well productivity. The Kirsch equations have been widely applied to model borehole stresses; in the basic formulation these equations are valid only for isotropic elastic media with a non-pressurized hole. A stress function, from which the Kirsch equations can be derived, is developed here to account for the effect of hydraulic pressure on the cylindrical surface of the wellbore. The concise solution via the stress function superposition is simpler than any previous analytical derivation. The function can be scaled to map the typical stress patterns for unbalanced hydraulic well bore pressures and provides a more comprehensive solution than the early Kirsch equations. The stress fields visualized can account for both over-pressurized and under-pressurized boreholes - with and without a tectonic background stress - and are valid for horizontal and vertical borehole sections. These analytical solutions allow fast mapping of stress trajectories around wellbores, which is useful for applications that aim to optimize hydraulic fracturing and thereby improve oil and gas well productivity. The analytical insight developed here may also contribute to develop better visual images (using stress trajectories) to prevent blow-outs and loss of drilling fluid in damaged formations.