A remark on the poroelastic center of dilation

In the original derivation due to Dougall (Proc. Edinb. Math. Soc. 16(1):82-98, [22]), the center of dilation is constructed from the superposition of orthogonal force dipoles. However, it can also be built by combining dislocation dipoles. While the two representations are equivalent in elasticity, the different nature of the force and dislocation dipoles leads to different transient behaviors in poroelasticity, when the response evolves from undrained to drained. Indeed, we show that the poroelastic solution of a center of dilation is obtained by superposing the corresponding elastic solution and either an instantaneous fluid sink or an instantaneous fluid source, depending on whether the center of dilation is treated as a singular stress or displacement discontinuity. A simple explanation to the instantaneous injection or withdrawal of a finite volume of fluid from/to the singularity is given by adopting an exploded view of the poroelastic center of dilation. The exploration of the drained, undrained and transient responses of this singularity also requires to clarify the close relationship between an instantaneous fluid source and a continuous fluid dodecapole, an object constructed by collapsing onto a point three orthogonal pairs of fluid dipoles.