We perform co-current spontaneous imbibition ambient-condition experiments in three carbonates with a wide range of permeability under strongly water-wet conditions. We measure water saturation profiles as a function of distance and time in air-filled rocks with no initial water saturation using X-ray CT scanning. We demonstrate that the saturation profiles are functions of distance divided by the square root of time. We also demonstrate that the profiles are consistent with analytical solutions for imbibition in one dimension, and using reasonable estimates of relative permeability and capillary pressure, we can match the experimental results. We discuss how, in combination with conventional measurements of relative permeability (steady-state or using Buckley-Leverett theory in an unsteady-state experiment), the capillary pressure can be determined, or how the relative permeability can be determined from the spontaneous imbibition experiment and the capillary pressure.