We propose a model to reconcile the theory of inter-temporal non-renewable resource depletion with well-known stylized facts concerning the exploitation of exhaustible resources such as oil. Our approach introduces geological constraints into a Hotelling type extraction-exploration model. We show that such constraints, in combination with initially small reserves and strictly convex exploration costs, can coherently explain bell-shaped peaks in natural resource extraction and hence U-shapes in prices. As production increases, marginal profits (marginal revenues less marginal extraction cost) are observed to decline, while as production decreases, marginal profits rise at a positive rate that is not necessarily the rate of discount.A numerical calibration to the global oil market predicts substantially higher future oil prices and considerably lower global oil production with the more realistic geological constraints set-up than with the Hotelling simulation. While mainly (small) non-OPEC producers increase production in response to higher oil prices induced by the geological constraints, most (large) producers' production declines, leading to a lower peak level for global oil production. High future oil prices therefore, do not necessarily translate to increased oil supplies on global markets.