Compaction of sedimentary porous rock by dissolution and precipitation is a complex deformation mechanism, that is often localized on stylolites and pressure solution seams. We consider a one-dimensional model of compaction near a thin clay-rich stylolite embedded in a porous rock. Under the assumption that the clay enhances solubility, the model predicts a reactive transport away from the clay layer followed by pore cementation. The evolution of the porosity, reactant transport, and compaction rate are studied as functions of model parameters and shown to reach a stationary state. We find good agreement between the porosity distribution predicted by the model and previously reported field measurements. The model provides quantitative estimates for compaction rates on stylolitic surfaces.