The equations governing the motion of the paper sheet in a press nip and the anisotropic percolation of the water in the sheet, are derived in invariant form. In order to avoid restrictive assumptions on the configuration of the paper sheet, the system of coordinates is generated by the flow of fibrous material through the nip region. With the corresponding constitutive equations, the formulation consists of a system of partial differential equations for the metric tensor of the coordinate system, and for the water velocity. For practical use, the solution is then mapped back into a cartesian frame of reference. Quantities of industrial interest, such as the residual water content and stresses, as well as the press-induced anisotropy, can be calculated in principle. A Galerkin finite-element approximation is implemented using rectangular linear elements. Two case studies are presented, for large and small permeabilities, and the corresponding differences in water pressure and velocity are in general qualitative agreement with the observations. Finally, the predictive value of the model is demonstrated by the dependence of the solution on the imposed shear stress and its gradient across the sheet.