Soft processes at the LHC I: Multi-component model

We emphasise the sizeable effects of absorption on high-energy 'soft' processes, and, hence, the necessity to include multi-pomeron-pomeron interactions in the usual multi-channel eikonal description. We present a model which includes a complete set of the multi-pomeron vertices and which accounts for the diffusion in both the impact parameter and In(k(t)) of the parton during its evolution in rapidity. We tune the model to the available data for soft processes in the CERN-ISR to the Tevatron energy range. We make predictions for 'soft' observables at the LHC.