A modified form of Biot's linear theory of poroelasticity is applied to shale swelling in contact with an aqueous electrolyte. The shale is assumed to behave as an isotropic, perfect ion exclusion membrane, and in this limit swelling depends only upon the total stress and on the chemical potential of water within the pores of the rock. An axisymmetric, plane-strain analysis of swelling around a wellbore is first presented, and this is subsequently extended to include swelling of a cylindrical hole in a finite, cylindrical shale sample. It is predicted that swelling is prevented if the chemical potential of water within the shale equals that within the wellbore. The predictions of the analysis are compared with experimental results obtained when drained outcrop shale swelled in contact with aqueous solutions of KCl or NaCl. The experimental swelling did indeed vary with water chemical potential, and could be prevented if the salt concentration within the wellbore fluid was sufficiently high. However, post-mortem chemical analysis of the shale showed that ion-exchange had taken place, with consequent modification of the shale's mechanical and chemical properties. Ion exclusion was therefore imperfect, and an analysis that incorporates the chemical potentials of components other than water is necessary.