Reliable simulation of enhanced oil recovery processes depends on an accurate description of fluid transport in the subsurface. Current empirical transport models of rock-fluid interactions are fit to limited experimental data for specific rock types, fluids, and boundary conditions. In this paper, a general equation-of-state (EoS) approach is developed for relative permeability (kr) based on a set of geometric state parameters: normalized Euler characteristic (connectivity) and saturation. Literature data and pore-network modeling (PNM) simulations are used to examine the functional form of the EoS. Our results show that the new kr-EoS matches experimental data better than the conventional Corey form, especially for highly nonlinear relative permeabilities at low saturations. Using hundreds of PNM simulations, relative permeability scanning curves show a locus of residual saturation and connectivity which defines an important limit for the physical kr region. The change of this locus is also considered for two contact angles. PNM data further allows for the estimation of the relative permeability partial derivatives which are used as inputs in the EoS. Linear functions of these partials in the connectivity-saturation space renders a quadratic response of kr, which shows excellent predictions. Unlike current empirical models that are based on only one residual saturation, the state function approach allows for dynamic residual conditions critical for capturing hysteresis in relative permeability.