The Cubic Growth Discovery Model

A few months ago, I posited an oil discovery growth model which I consider follows reality better than any currently available model (which basically consists of none, since everyone bar a few contrarians seem to consider the Logistic model the end-game, and no one apparently seems to consider the discovery phase interesting from a modelling perspective). I used a model that features a time dependence which increases quadratically. This has worked in the past for human-effected discovery phenomena such as Wiki word growth. After thinking about a solid first-principle physical basis for such a growth regime, it finally dawned on me to re-think the geometry of the problem domain. Also, even though the quadratic solution showed a nice simplicity, featuring only an amplitude and a time-scale factor, the quality of the fit I thought could have used some more work. (Note that the profile could narrow a bit for USA discoveries below)

The insight came to me as I thought how quadratic growth could occur in various geometric contexts. Of course, quadratic growth occurs in two-dimensions in the case of a circle which has a radius that increases linearly with time:

Area(Time) ~ (a*Time)²

Now, this makes sense from a discovery perspective in one key respect. If you consider that technology and manpower increases at least linearly with time (Moore's law notwithstanding), then the size of the discovery cross-section should at least track a radial increase. So the size of the prospected regions would follow a quadratic (n=2 power law). Each year a progressively larger geographical area gets sampled for oil until it hits a real physical constraint, i.e. the cumulative area sampled. This gives rise to the differential equation I presented last time:

d²Discovery(t)/dt² = c - a³*Integral(Discovery(t))

Note that I make the assumption that a sampled region generates a discovery proportional to the size of the region, or Discovery ~ SampledArea. So that makes sense from a first-principles perspective and I didn't have to drag in other productivity factors to get the squared-law to work.

But then I realized that sampling actually works from a volumetric perspective as well:

Volume(Time) ~ k*Time³

In this case, not only does the radius of a sampled area increase with time, but so does the depth. Again this makes sense, as oil prospecting has likely shown a monotonic increase in sampled depths over the years. I tried to describe this phenomena in the following illustration.

Figure 1: Cubic model showing the two regimes, A and B

If you look at the part of the figure labelled Regime A, a time sequence of progressively increasing sampling volumes probe the "easy access" volume of size V_d. Every year, a new volume probed gets used up and contributes to a declining pool of remaining volume. In computer science terms, this looks an awful lot like a bin-packing problem. As the remaining bins become progressively more sparse and small, a negative feedback gets applied to the linear rate term. This effect leads to the following linear differential equation and surprisingly simple solution.

The differential equation transitions from a 3^rd order solution in the case of the quadratic model to a 4^th order solution in the case of this cubic model. The second term on the RHS provides the negative feedback in terms of the cumulative discovery volume (D(t)) probed.

The following figure shows the qualitative difference between the quadratic and cubic growth models:

The cubic growth model shows the expected clear narrowing of the distribution around the peak, which also seems to fit the discover data better.

We should worry about the region to the right of the figure, where the discovery model goes to zero while the real data indicates substantial finite discoveries. But this has a fairly obvious interpretation when we refer back to Figure 1. Consider that the cubic discovery model needs a constraint related to a finite pool of easily discoverable oil -- i.e. Regime A in the figure. Yet we don't really know the exact extend of this finite volume and especially what lies beyond the easily accessible volume. In particular we don't know how deep the volume goes. To model this, we add an extra regime, Regime B, which lies underneath the "easy access" volume (or in more inhospitable offshore or polar regions).

Unfortunately, the remaining "difficult access" area gets progressively harder to probe, likely proportional to 1/Depth of the unknown area (i.e. we have to work harder for region at time T6 than for the shallower region at time T5). We also no longer can make increasingly large volume projections as in the cubic Regime A, as we have pretty much exhausted the entire planet in terms of areal coverage. So at best, any technological increases that track linearly with time gets offset by a factor proportional to the inverse of the depth, leading to a discovery rate proportional to that remaining. This leads to the 1^st-order differential equation:

Discovery(t) = c0 - a0*Integral(Discovery(t))

leading to a solution described by a simple decaying exponential -- in other words, the only part of the Logistic model that makes sense.

I didn't spend too much time fitting the declining exponential to real world data as we have limited data here and clearly it won't make much difference to the overall future production of quality crude.

If this model indeed effectively describes discovery, why should it work? My thinking goes like this: we collectively have the one of the most highly sampled data source in human history. Every region of the globe has gotten probed, either randomly or systematically, such that we have few remaining unexplored regions on Earth. The cubic discovery model does basically puts a mathematical basis to a blind-man's dart game. I contend once again that we do not have to understand much about the geology of oil deposits. This model would probably work just effectively if we happened to start looking for expensive buried coins on a hypothetically previously unexplored beach and we could start by digging by hand, then using a metal detector, and finally using earth moving equipment and a sifter (you get the idea). This essentially explains why all the unique elements of petroleum geology get washed out; in the end if we deal with a sparsely populated sampled system, this kind of math should work out fine.

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I combined the cubic growth discovery model with the oil shock depletion model via a convolution and came up with this model fit for the global oil production curve. I used 12.5 years for each of the shock model lags, which you can see in the progressively deeply shaded curves from discovery to production. (caveat, I only used Regime A of the cubic discovery so the discovery decay dies out rapidly around the year 2000)

Is this the only first-principles model for global oil discovery and production?

Note that at the time origin at year 1858, I have no choice but to place the original discovery stimulus for Titusville, PA. If this deterministic stimulus did not exist the oil production profile would not match the model, which naively would suggest a starting value of 0 barrels/year. This adds an extra variable to the model, but the residual error only effects the early years, and nothing much past 1900.