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Monday, April 30, 2007

Apparent Reserve Growth

In several prior posts, I have tried to explain the enigma of reserve growth via some physical mechanisms, concentrating primarily on diffusion. The process of diffusion makes some physical sense, particularly if you consider that oil actually can diffuse through porous material, and that mathematical relationships such as Fick's first law map fairly well to a reservoir depth acting as a permeable membrane.

However, we should also consider the psycho-business aspects of making reserve growth predictions. These often have little to do with the physical processes, but have much more to do with following conservative rules which guard against overly speculative forecasts. Fortunately, we can just as easily make a model of econometric predictions as we can of a real physical process. Watch this unfold, as the math points to an almost counter-intuitive result.

First, consider that any kind of prediction has to deduce from a probability model. To keep things simple, say that a reservoir has a finite depth L0, which corresponds to a given extractable oil volume. Importantly, we do not know the exact value of L0, but can make educated guesses based on a depth that we do have confidence in. We call this the "depth of confidence" and assign it to the random variable Lambda. This has the property that as we go beyond this depth, our confidence in our prediction becomes less and less concrete, or alternatively, more and more fuzzy. With this simple premise providing a foundation, we can use basic probability arguments to estimate a value for the unknown L0 which we call L-bar.

A line by line analysis of the derivation:
1. Probability density function for the depth of confidence.
2. Estimator for reservoir depth, a mean value.
3. Estimator shows a piecewise integration; the first part integrating to the actual depth, and the second part adding in a higher confidence factor as we probe blow the actual depth.
4. Solution to the integration, giving the enigmatic reserve growth dynamics as Lambda increases with time.
5. If Lambda increases linearly with time
6. If Lambda increases with a parabolic dependence, which matches a diffusional process.
On the face of it, the estimator for reservoir depth looks like a classical exponential damping formula. But if you look closely, you see that the random variable lands in the denominator of the exponent, leading to a sharper initial rise but much more gradual dampening than the traditional form. The conservative nature of the estimation comes about because Lambda rises only gradually with time.

This function does reach a clear asymptote, given by L0, however much it superficially looks as if it may grow indefinitely. I consider this analysis a huge breakthrough in understanding the "enigma" of reserve growth. The fundamental form basically provides a simple explanation to what we observe, and dissipates the smoke and mirrors that the peak oil deniers trot out when they talk about reserve growth as some magical phenomenon. Clearly, we can have reserve growth as a pure book-keeping exercise, which basically deflates the preposterous Lynch and Attanasi & Root arguments.

As a bottom-line, I suggest that we start using the "apparent reserve growth" equation on observed profiles. A single variable essentially guides the growth, from which we can infer asymptotic behavior simply and quickly.

Thank you for participating in Part 72 of the series, "Why big oil refuses to admit to the principles of basic math."

As I finish this post up, I hear the voice of Jerome a Paris from DKos as he makes an appearance on the Thom Hartmann radio show.