Geological structures in the subsurface ranging from fractures to reservoirs can be simplified as ellipsoidal inhomogeneities. For instance, one can model a reservoir as an inclusion by considering possibly different material properties and different fluid pressure in comparison to the surrounding rock. Hence, the stresses and displacements associated with the fluid withdrawal from or fluid injection into the formations can be determined by assuming no hydraulic communication between the inclusion and the surrounding medium. The lack of hydraulic communication could be the result of a cap rock or an impermeable seal/fault. On other hand, in the case of fractures, this assumption in not valid anymore and the hydraulic communication between the inclusion and medium should be considered in the calculations. This paper provides analytical solutions for deformation and stress distribution inside and outside of poroelastic ellipsoidal inclusions. Eshelby theory with the Biot theory of poroelasticity are combined to model the change in stress caused by changes in pore pressure or temperature inside the inclusions. Using the provided analytical solutions, we explore the effect of inclusion size, material properties and pressure/temperature condition. The results confirm that neglecting hydraulic communication between the inclusion and the sounding matrix may result in lower inclusion volume change ratio associated with mode (1) and higher inclusion volume change ratio associated with mode (2).
Despite considerable advances in numerical modeling of subsurface engineering problems, there is still a need for analytical solutions as a benchmark for numerical models as well as a quick resort to approximate solutions. Due to the nature of numerical techniques, boundary effects and discretization error may easily occur in any basic modelling efforts. Eshelby’s equivalent inclusion theory is recognized as one of the most powerful techniques for deriving analytical solutions for many elasticity problems, however, this techniques has not yet widely utilized for poroelastic problems. Presence of pore fluid in the porous materials and its coupling with material deformations leads to different class of material behaviors known as poroelastic behavior.