General Dispersive Discovery : The Laplace Transform Technique
It turns out that the generalized Dispersive Discovery model fits into a canonical mathematical form that makes it very accessible to all sorts of additional analysis. Much of the basis of this formulation came from a comment discussion started by Vitalis. I said in the comments that the canonical end result turns into the Laplace transform of the underling container volume density. The various densities include an exponential damping (e.e. more finds near the surface), a point value (corresponding to a seam at a finite depth), a uniform density abruptly ending at a fixed depth, and combinations of the above.
The following derivation goes through the steps in casting the dispersive discovery equations into a Laplace transform. The s variable in Laplace parlance takes the form of the reciprocal of the dispersed depth, 1/lambda. The basic idea again assumes that we search through the probability space of container densities, and accumulate discoveries proportional to the total size searched. The search depths themselves get dispersed so that values exceeding the cross-section of the container density random variable x with the largest of the search variables h get weighted as a potential find. In terms of the math, this shows up as a conditional probability in the 3rd equation, and due to the simplification of the inner integral, it turns into a Laplace transform as shown in the 4th equation.
The fun starts when we realize that the container function f(x) becomes the target of the Laplace transform. Hence, for any f(x) that we can dream up, we can short-circuit much of the additional heavy-duty math derivation by checking first to see if we can find an entry in any of the commonly available Laplace transform tables.
In the square bracketed terms shown after the derivation, I provide a few selected transforms giving a range of shapes for the cumulative discovery function, U-bar. Remember that we still need to substitute the lambda term with a realistic time dependent form. In the case of substituting an exponential growth term for an exponentially distributed container,
lambda ~ exp(kt), the first example turns directly into the legendary Logistic sigmoid function that we demonstrated previously.
The second example provides some needed intuition how this all works out. A point container describes something akin to a seam of oil found at a finite depth L0 below the surface. Note that it takes much longer for the dispersive search to probabilistically "reach" this quantity of oil as shown in the following figure. Only an infinitesimal fraction of the fast dispersive searches will reach this point initially as it takes a while for the bulk of the searches to approach the average depth of the seam. I find it fascinating how the math reveals the probability aspects so clearly while we need much hand-waving and subjective reasoning to convince a lay-person that this type of behavior could actually occur.
The 3rd example describes the original motivator for the Dispersive Discovery model, that of a rectangular or uniform density. I used the classical engineering unit-step impulse function u(x) to describe the rectangular density. As a a sanity check, the lookup in the Laplace transform table matches exactly what I derived previously in a non-generalized form, i.e. without the benefit of the transform.
Khebab also suggests that an oil window "sweet spot" likely exists in the real world, which would correspond to a container density function somewhere in between the "seam" container and the other two examples. I suggest two alternatives that would work (and would conveniently provide straightforward analytical Laplace transforms). The first would involve a more narrow uniform distribution that would look similar to the 3rd transform. The second would use a higher order exponential, such as a gamma density that would appear similar to the 1st transform example (see table entry 2d in the Wikipedia Laplace transform table):
Interestingly, this function, under an exponentially increasing search rate will look like a Logistic sigmoid cumulative raised to the nth power, where n is the order of the gamma density! I do wonder if any oil depletion analysts have ever empirically observed such a shape ? (hint, hint, this may require a TOD re-posting)