Multiphase flow simulations on imaged porous rock structures require numerical methods that are accurate and robust when applied on complex geometries. A key element in this context is to investigate how simulations behave under grid refinement. In this work, we couple an existing software for structured adaptive mesh refinement (AMR) and parallelism to a previously developed level set method for capillary-controlled displacement on the pore scale. The level set method accounts for wettability by using different evolution velocities in pore and solid space rather than implementing it as a boundary condition on the pore walls. We perform simulations with up to three nested refinement levels on idealized pore geometries to validate the AMR technology. Based on simulations, we identify suitable cell refinement criteria for our applications. We demonstrate effects of AMR in simulations relevant for drainage in porous media, such as in the computation of capillary entry pressures in pore throats with different shapes and wetting states, and obtain excellent agreement with analytic results. Finally, we investigate the effects of AMR during simulation of quasi-static drainage on sandstone. The comparison of capillary equilibrium fluid configurations, capillary pressure, and specific fluid/fluid interfacial area-saturation relationships for drainage with and without AMR shows differences that diminish for less water-wet states. The general behavior is that the capillary pressure and interfacial area for a given saturation increase in simulations with AMR. The largest deviation occurs for small water saturations, suggesting AMR can be an important component in simulation tools to describe more accurately capillary behavior in the low water saturation regime where interface curvature is high.