I find it so rare when someone else checks your math, that I like to point it out. From TOD:
Ok, I think I understand now. Would this be a correct interpretation of your model above? (I define some new notations here, but I feel I have to in order to be unambiguous)I appreciate the way Vitalis framed it as an expected value. I do this calculation so often, that I tend to forget to qualify with the E[ ] notation.
L = total, finite volume where it is possible to find oil,
alpha = density of oil = (total volume of oil)/L,
S = total volume we have searched.
Define as the volume of discovered oil:
D(S,L) = alpha*S if S <= L, and D(S,L) = alpha*L if S > L.
Now, model S and L as exponentially distributed stochastic variables with means lambda and L_0 respectively. Thus D(S,L) is random too. Compute the mean (by evaluating integrals like in the derivation above) gives
E[D(S,L)] = alpha*/(1/lambda + 1/L_0).
Now, we may let lambda depend on time, for instance lambda = lambda(t) = k*t^N or lambda(t) = A*exp(B*t) as above.
With this in mind, perhaps one should try to use a more realistic distribution for the true depth L than the exponential distribution (it should clearly not have so much mass close to zero)? Any ideas?
To answer his question, we have 3 candidate distributions: the Singly dispersive discovery describing a rectangular L0, where you have a finite constant depth "box" with uniform distribution; and then the damped exponential, which forms the Double dispersed discovery that Vitalis confirmed. The third involves a suggestion by Khebab upthread, where he thinks a sweet spot exists between 7,000 to 15,000 feet. This last one I don't believe differs much from the uniform model. I would not think it too difficult to mathematically derive but intuition and history says that we find enough oil near the surface (think Texas and Pennsylvania) that it could prove harder to rationalize without unduly complicating the model.
So if we use the single and doubly dispersive model, it can give us some good bounds, with the third one probably being more sharply defined than the single.