Using the observed submillimetre source counts, from 250 to 1200 μm[including the most recent 250, 350 and 500 μm counts from Balloon-borne Large-Aperture Submillimetre Telescope (BLAST)], we present a model capable of reproducing these results, which is used as a basis to make predictions for upcoming surveys with the Spectral and Photometric Imaging Receive (SPIRE) instrument aboard the Herschel Space Observatory. The model successfully fits both the integral and differential source counts of submillimetre galaxies in all wavebands, predicting that while ultra-luminous infrared (IR) galaxies dominate at the brightest flux densities, the bulk of the IR background is due to the less luminous IR galaxy population. The model also predicts confusion limits and contributions to the cosmic IR background that are consistent with the BLAST results. Applying this to SPIRE gives predicted source confusion limits of 19.4, 20.5 and 16.1 mJy in the 250, 350 and 500 μm bands, respectively. This means the SPIRE surveys should achieve sensitivities 1.5 times deeper than the BLAST, revealing a fainter population of IR-luminous galaxies and detecting approximately 2600, 1300 and 700 sources per deg2 in the SPIRE bands (with one in three sources expected to be a high-redshift ultra-luminous source at 500 μm). The model number redshift distributions predict a bimodal distribution of local quiescent galaxies and a high-redshift peak corresponding to strongly evolving star-forming galaxies. It suggests the very deepest surveys with Herschel–SPIRE ought to sample the source population responsible for the bulk of the IR background.