Migration of clay fines can be a concern when less saline fluids are injected into brine-saturated sandstone formations containing clays. If the salinity near fluid injection wells decreases below a critical value, the clay fines near the injection may detach, start migrating, and finally clog the pores. This effect can cause permeability decline near the well and may rapidly reduce the well injectivity. The focus of this work is on evaluating the impacts of clay fines migration on permeability decline in the field, using a numerical model and pressure buildup data collected during successive variable-rate water injections in a deep sandstone reservoir. The numerical model accounts for the mixing of low-salinity water with native brine and the migration of clay fines with the detachment and pore-clogging processes. The model interpretation of the pressure buildup data implies that the observed reduction in well injectivity is mainly associated with the clay fines migration and related pore clogging near the well. The model reasonably well represents the pressure buildup data during the injections. Our simulations demonstrate that the permeability near the well can rapidly decline within the first hour of injection. The measured pressure buildup in post-injection periods appears to decay more rapidly, compared to the simulation results of the model that assume irreversible permeability damage. This raises the question whether the permeability damage may be partly reversible near the well by backflow of brine after the injection of low-salinity water.