We present a systematic comparison study of simulating two-phase flow (drainage) in single heterogeneous fractures by using two fundamentally different approaches, namely, a continuum-based two-phase flow model and an invasion-percolation (IP) model. We analyze both gravity neutral and gravity destabilized cases. In the continuum model, the two-phase mass conservation equations for the 2-D fracture plane are solved, based on modifications to TOUGH2, a numerical simulator for multiphase and multicomponent flow and transport in geological media. A specific capillary pressure-liquid saturation function is used to account for the sudden drainage of a local aperture location in the fracture once its local aperture-dependent nonwetting phase fluid entry pressure is exceeded. Results from the continuum model are compared to those from an IP model that includes trapping. We consider cases where the contribution of aperture-induced curvature in the capillary pressure term dominates over that of the in-plane curvature. The comparison shows that the presented continuum model can well reproduce the IP model results at low-capillary-number conditions and furthermore can also produce meaningful results in the high-capillary-number regimes where IP models are not valid. Taking into account the viscous forces in the fluid displacement process, the continuum model is used to examine the effect of capillary number (reflecting the injection rate) on the phase invasion. When the injection rate varies from low to high, simulations using the continuum model show that the invasion pattern changes from single dominant fingers to more homogeneous spreading and/or clusters with numerous tortuous fingers. This trend is comparable to results from previous experimental observations in the literature. The continuum model is also used to numerically construct the upscaled (fracture-scale) capillary pressure-saturation relationship. The upscaled relationship can be well fitted to the van Genuchten and the Brooks-Corey porous-medium-type models. Fracture capillary behavior depends on the aperture field heterogeneity. Simulation results indicate that increasing the aperture standard deviation leads to smaller entry pressure and larger residual water saturation.