Displacement field potentials for deformation in elastic Media: Theory and application to pressure-loaded boreholes

This study demonstrates how analytical solutions for displacement field potentials of deformation in elastic media can be obtained from known vector field solutions for analog fluid flow problems. The theoretical basis is outlined and a geomechanical application is elaborated. In particular, closed-form solutions for deformation gradients in elastic media are found by transforming velocity field potentials of fluid flow problems, using similarity principles. Once an appropriate displacement gradient potential is identified, solutions for the principal displacements, elastic strains, stress magnitudes and stress trajectories can be computed. An application is included using the displacement gradient due to the internal pressure-loading of single and multiple wellbores. The analytical results give perfect matches with results obtained with an independent discrete element modeling method.