The relative roles of parameters governing relative permeability, a crucial property for two-phase fluid flows, are imperfectly known. To characterize the influence of viscosity ratio (M) and capillary number (Ca), we calculated relative permeabilities of nonwetting fluids (knw) and wetting fluids (kw) in a 3D model of Berea sandstone using the lattice Boltzmann method. We applied the Euler–Poincaré characteristic to quantify the morphology of both fluids. We show that knw increases and kw decreases as M increases due to the lubricating effect and instability at fluid interfaces resulting from viscosity contrast. We also show that knw decreases markedly at low Ca (log Ca < −1.25), whereas kw undergoes negligible change with changing Ca. An M–Ca–knw correlation diagram, displaying the simultaneous effects of M and Ca, shows that they cause knw to vary by an order of magnitude, an effect not incorporated in current estimation techniques.