The classical constitutive relations for multiphase flows in porous media assume instantaneous and local phase-equilibrium. Several alternative nonequilibrium/dynamic constitutive relations have been proposed in the literature including the works of Barenblatt, and Hassanizadeh and Gray. This work applies a Bayesian model selection framework in order to examine the relative efficacy of these three models to represent experimental observations. Experimental observations of multiphase displacement processes in natural porous media are often sparse and indirect, leading to considerable uncertainty in control conditions. Data from three core-scale drainage experiments are considered. Gaussian prior probability models are assumed for key multiphase flow parameters and measurements. Accurate numerical simulation approximations using the three constitutive relation models are implemented. The model selection analysis comprises a data-assimilation stage that calibrates the assumed model to the data while quantifying uncertainty. The second stage is the computation of the maximum likelihood estimate and its application to compute the Bayesian Information Criterion. It is observed that Barenblatt's nonequilibrium model is more likely to match data from unstable displacements that involve higher viscosity ratios of the invading phase to the resident fluid. At the lowest viscosity ratio, there is no delineation between the goodness of fit obtained using the classical model and the model proposed by Hassanizadeh and Gray, and both outperform Barenblatt's nonequilibrium model. (C) 2016 Elsevier Ltd. All rights reserved.