The pore-scale flow mechanisms and the relative permeabilities during steady-state two-phase flow in a glass model pore network were studied experimentally for the case of strong wettability (theta(e) < 10 degrees). The capillary number, the fluid flow rate ratio, and the viscosity ratio were changed systematically, while all other parameters were kept constant. The flow mechanisms at the microscopic and macroscopic scales were examined visually and videorecorded. As in the case of intermediate wettability, we observed that over a broad range of values of the system parameters the pore-scale flow mechanisms include many strongly nonlinear phenomena, specifically, breakup, coalescence, stranding, mobilization, etc. Such microscopically irreversible phenomena cause macroscopic nonlinearity and irreversibility, which make an Onsager-type theory inappropriate for this class of flows. The main effects of strong wettability are that it changes the domains of the system parameter values where the various flow regimes are observed and increases the relative permeability values, whereas the qualitative aspects of the flow remain the same. Currently, a new true-to-mechanism model is being developed for this class of flows.