Measurement of nuclear magnetic resonance (NMR) relaxation is a well-established laboratory/borehole method to characterize the storage and transport properties of rocks due to its direct sensitivity to the corresponding pore-fluid content (water/oil) and pore sizes. Using NMR, the correct estimation of, e.g., permeability strongly depends on the underlying pore model. Usually, one assumes spherical or cylindrical pores for interpreting NMR relaxation data. To obtain surface relaxivity and thus, the pore-size distribution, a calibration procedure by, e.g., mercury intrusion porosimetry or gas adsorption has to be used. Recently, a joint inversion approach was introduced that used NMR measurements at different capillary pressures/saturations (CPS) to derive surface relaxivity and pore-size distribution (PSD) simultaneously. We further extend this approach from a bundle of parallel cylindrical capillaries to capillaries with triangular cross sections. With this approach, it is possible to account for residual or trapped water within the pore corners/crevices of partially saturated pores. In addition, we have developed a method that allows determining the shape of these triangular capillaries by using NMR measurements at different levels of drainage and imbibition. We show the applicability of our approach on synthetic and measured data sets and determine how the combination of NMR and CPS significantly improves the interpretation of NMR relaxation data on fully and partially saturated porous media.