What is the denser liquid water or the oil? Normally it is the water. In a transparent glass beaker (with flat side walls) you should be able to image a drop of less dense liquid floating on the denser liquid. From these side view images, you can calculate the contact angle. Take care to take into account that the drop partially sinks in the denser liquid. In the inverse case (denser liquid sitting on top of the less dense liquid), thing are more complex, because the drop will completely sink into the less dense liquid.
You need to measure two angles: water/oil/air and oil/water/air. They should be both independant of geometry (no matter if it is water lense or oil lense), in the ideal case. Normally, the impurities are making things more complex, and since they are partitioning differently for oil lense in water and for water lense in oil, you will get different results. If the denser fluid is forming the lense, it may sink or it may not sink depending on the three interfacial tensions. The condition for a small lense not to sink is sigma12 < sigma23+sigma13 (small compared to capillary length; 12 is the lense-air interface). A large lense can also be stable