In this study, we consider a one-dimensional three-phase model describing wet pressing of paper. Part I is devoted to the simplified case in which air is assumed incompressible. In Part II we drop this assumption. The model is formulated in terms of water saturation and void ratio and it uses a material coordinate to describe spatial dependence. It also involves cross or matching conditions between the wet paper and the felt. In mathematical terms, we end up with a coupled system of equations: a nonlinear diffusion equation and a first order hyperbolic equation. We present some analytical observations to explain the essential behaviour of the model and we carry out numerical experiments using an upwind and a front tracking method.