The Oil Searching Problem

Given n potential oil locations, where each has oil at a certain depth, we seek good trade-offs between the number of oil sources found and the total amount of drilling performed. The cost of exploring a location is proportional to the depth to which it is drilled. The algorithm has no clue about the depths of the oil sources at the different locations or even if there are any. Abstraction of the oil searching problem applies naturally to several life contexts. Consider a researcher who wants to decide which research problems to invest time into. A natural dilemma whether to invest all the time into a few problems, or share time across many problems. When you have spent a lot of time on one problem with no success, should you continue or move to another problem?