An analytical solution for displacements due to reservoir compaction under arbitrary pressure changes

Production of fluids in reservoirs, such as oil, gas and water, leads to changes in the stress and strain fields, generating compaction in the reservoir and subsidence of the ground surface. Some of the severe consequences of these displacements are rock fracture, induced seismicity, fault reactivation, wellbore failure, and reduction in the storage capacity of aquifer systems. Analytic methods based on the nucleus of strain approach with simplifying assumptions can be useful to estimate compaction and subsidence, and to indicate when more detailed analyses are required. However, these methods may present problems by the evaluation of the displacement response inside and in the vicinity of the reservoir due to the presence of singularities in the solution. In this work, an analytical displacement solution is obtained for reservoirs of arbitrary shape in a linear elastic semi-infinite medium under arbitrary distribution of pressure changes. For this, the reservoir is discretized in parallelepiped cells. A three-dimensional integration of the solution is carried out over each cell for an infinitesimal reservoir based on the nucleus of strain approach. The final displacement field is given by the superposition of the solutions of individual cells. The three-dimensional integration eliminates the singularities and allows the evaluation of displacements throughout the entire model, including the reservoir. The solutions presented can be easily implemented in a computational procedure, maintaining the low effort of similar methods. Finite element models are employed to verify the accuracy of the proposed solution.