A new unsteady state laboratory displacement experiment has been developed to measure two-phase relative permeability, fluid saturation, and capillary head (kr-S-hc) relationships under moderate to low displacement rates and fluid pressures. The new technique maintains approximately uniform fluid saturation along the length of a core sample through the innovative use of a porous top disc to control internal pressure of the displacing fluid. Internal fluid pressure heads are measured in both phases at each end of the core, and permeabilities are calculated directly from Darcy's law. This approach avoids the bias inherent in other techniques that rely on assumptions related to inverse solution of unsteady state flow equations or a priori parameterization of kr-S-hc relationships. ¿ 1999 American Geophysical Union