In this report, the use of an asymptotic method, based on the time scaling of the ratio of produced fluids, to infer the relative permeability exponent of the displaced phase near its residual saturation, for immiscible displacements in laboratory cores was proposed. Sufficiently large injection rates, the existence of a power law can be detected, and its exponent inferred, by plotting in an appropriate plot the ratio of the flow rates of the two fluids at the effluent for some time after breakthrough.