A porous plate technique was developed to measure capillary pressure during both primary drainage and waterflooding in a reservoir carbonate rock sample. During primary drainage, a water-wet ceramic disc at the end of the sample allowed brine to be displaced but prevented the escape of oil while oil was injected at a sequence of increasing pressures. Saturation was measured using high-resolution three-dimensional X-ray imaging from the differences in greyscale (X-ray adsorption) between dry, partially-saturated and completely-saturated images. A two-step displacement process was observed, with the resolvable macro-pores displaced by oil followed by invasion into unresolved micro-porosity with a variation of two orders of magnitude in capillary pressure. The sample was then exposed to crude oil to render some of the solid surfaces oil-wet. A small amount of spontaneous imbibition (displacement at a positive capillary pressure) was observed in micro-porosity. Then an oil-wet porous plate was added to the outlet, so that water could be injected at a sequence of increasing pressures while only oil could escape. As expected, the oil-wet macro-porosity was displaced by brine at a low capillary pressure with a magnitude similar to that seen during drainage (but of opposite sign). Remarkably though the displacement of oil from micro-porosity occurred at a capillary pressure approximately an order of magnitude lower than in drainage, implying the existence of mixed-wettability with fluid menisci that are approximately minimal surfaces. The work demonstrates that in mixed-wet media displacement of oil from micro-porosity can occur at much lower capillary pressures that would be estimated from primary drainage results using the calculated pore size distribution.