Three-phase relative permeabilities are typically measured in cores using either steady-state or Johnson Bossier, and Naumann (JBN) methods. These methods require assumptions that can lead to erroneous relative permeability data. One alternative approach is a gravity drainage method, which has previously been used in sandpacks and recently extended to cores at atmospheric conditions. Here, we test a gravity drainage method that can be used to measure relative permeability in cores at elevated pressures. To achieve this, nitrogen gas is injected to the core at a low flow rate to overcome capillary pressure. We test the method by measuring two-phase water relative permeability in a Berea sandstone core using two gas flow rates: one that is low enough that gravity is a significant driving force for the flow, and a higher flow rate for comparison. During drainage, water saturation is measured along the length of the core at different times using a CT scanner, and pressure drops are measured across five sections of the core. The relative permeability of water is calculated using data points in regions of the core where the saturation is changing in time but not space, allowing capillary end effects and capillary pressure gradients to be ignored. Relative permeability data from the low flow rate experiment are scattered widely; the low gas flow rate likely hindered the free drainage of the water. Relative permeability data from the higher flow rate experiment formed a distinct curve. More flow rates will need to be tested to determine an optimum flow rate for gravity drainage experiments at reservoir pressures.