Solving the "enigma" of reser...

As a good rule of thumb, when you have a promising model describing some physical process, you might as well put it through its paces. Not only do you shake out some stubborn corner cases, but you often find something new and revealing. We did that in the last post, The Derivation of "Logistic-shaped" Discovery, grinding out the derivation of the classic Logistic/Sigmoid-shaped Hubbert curve based on the generalized Dispersive Discovery model. With this post, I use the same discovery model to derive the upward climb of the cumulative reserve growth curve which we empirically observe on many oil reservoirs and oil-bearing regions.

Lots of analysts have found this reserve growth behavior both curious and, ultimately, very important. I know that Khebab and Rembrandt have spent much time on TOD tracking this behavior as it plays an important role in how the peak will play out. Furthermore, I believe that the practice of "back-dating" discoveries based on reserve growth updates has muddied the waters and stalled progress in the basic understanding of the fundamental growth process. Unless we have a good model for the reserve growth dynamics we have to resort to using the heuristics supplied by USGS geologists, including the modified Arrington equation that Khebab has successfully used in the past. I find nothing wrong with using a heuristic and Khebab has really kick-started the "un-back-dating" approach with some excellent results. Still, a heuristic lacks some of the predictive power and room for insight that a fundamental model can provide.

The USGS crew have an interesting take on the reserve growth issue. In turns, the geologists working there have labeled fossil fuel (both oil and NG) reserve growth an "enigma" and a "puzzle".

For that reason the United States Geological Survey (USGS) considers [this] analysis "arguably the most significant research problem in the field of hydrocarbon resources assessment."

I have to admit that it has puzzled me for a while, but then again most of us here on TOD don't work at the USGS for 40+ hours a week. This post runs though a stochastic analysis that essentially explains how reserve growth can happen.

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One of the technical issues that the USGS have struggled with respect to their reserve growth analysis involves the use of "censored data". This essentially says that you should take special care of extrapolating data backwards considering you have only a truncated time-series data set of recent vintage. The "sweet spot" for good data basically doesn't exist, with very few values for old data and also relatively few values for the latest data (mainly due to diminishing finds). Working with such a limited data set, they massaged it the best they could and adequately normalized the fractional yearly growth, but they basically punted after this point and came up with only a heuristic to "explain" the trend. Pragmatically, I've decided to sweep the censoring problems under the rug, since in the long term, complaining about the possibility of statistical shortcomings still won't explain most of the trend, which rises steeply enough to make the cornucopians hopeful for great prospects ahead (listen to any right-wing radio program today for examples of this kind of unbridled optimism).

"We must accept finite disappointment, but we must never lose infinite hope." -- Martin Luther King Jr.

"Reserving judgments is a matter of infinite hope." -- F. Scott Fitzgerald in The Great Gatsby

Yet we still have left a riddle wrapped in a mystery inside an enigma. The actual problem with the half-baked reserve growth analysis has become obscured by the trickiness with using censored data. Inside, the issue really stems from a lack of a good value for the initial discovery estimate. Stating it bluntly, pick this number incorrectly and you can get numbers all over the map, with the possibility for some hugely absurd values.

To provide a path forward in unwrapping the riddle, I used the generalized Dispersive Discovery Model. In terms of modeling reserve growth, the dispersion generates a tail for accumulating further discoveries after the initial estimate occurs. For constant average growth, the model looks like this:

DD(t) = 1 / (1/L + 1/kt)

For the purposes of this analysis, I convert it to a reserve growth value U(t) and set T=L/k to make the math easier to handle later on:

U(t) = tT / (t+T)

Note that at time t=0, the discovered amount starts at zero and then the accumulation reaches some value proportional to L -- what one should consider as the characteristic depth or volume of the reservoir. It takes time=T for most of the search to reach this point. The basic premise of reserve growth and what USGS geologists such as Attanasi & Root¹ and Verma² frame their arguments on, has to do with the reserve growth considered as a multiplicative factor of the initial estimate. They see numbers that reach a value of nearly 10x after 100 years and claim (perhaps implicitly) that this has some real physical significance, almost offering up hope for still-to-come huge reserve benefits. Figure 1 exaggerates the claim for effect, as I want to make you aware that a finite asymptote certainly exists, but it gets obscured by the data trends commonly reported.

Figure 1: Potential for huge reserve growth

The contrived swindle of the USGS explanation has to do with exactly when the original estimate becomes available. Conceivably you can make estimates that occur very early in the lifespan of a reservoir, and you will get very low estimates for estimated discovery size. To take it to one extreme, you might find the initial estimate to fill a sewing thimble. Now, if that estimate grows at all, you can get huge apparent reserve growth factors, some fraction approaching infinite in fact. In contrast, you can wait a couple of years and then report the data. The later years' growth factor will proportionately account for much less of an increase. Now if you consider that in other parts of the world, countries report reserves less conservatively then the USA, then the reserve growth factors can vary even more wildly. I used USA oil data from an Attanasi & Root paper¹ which you can find a data dump of here. Initially, I plotted the data as a fractional yearly growth curve, basically reproducing the trend that A&R report:

Figure 2: Trend duplicated from the A&R data. The blue '+' symbols represent the data and the red curve represents a small-window moving average. To generate the curve correctly, you must normalize the yearly growth per reservoir, as the reservoir sizes vary widely.

The key insight to understand the growth factor in terms of the DD model has to do with averaging the initial discovery point over a relatively small window of time starting from t=0. This effectively samples the infinite values of growth against other finite values. The use of the sampling/integration window brings down the potentially infinite (or at least very large) growth factor to something more realistic. The math on this derives easily into an analytic form, and we end up with this function, where A indicates the time integration window:

U_bar (t) = (A - T ln((t+A+T)/(t+T))) / (A - T ln((A+T)/T))

The term on the right of the main divisor assures that the average U_bar starts at 1 for time t=0. Alternatively we can set A to some arbitrary value and skip the integration, which assumes that all initial discovery estimates start at t=A/2. This results in the conceptually simpler:

U_delta (t) = 2*(t+A/2)*(A/2+T)/A/(t+A/2+T)

The latter equation obviously starts at 1 and reaches an asymptotic value of 1+2T/A. Both the values of A and T describe the ultimate asymptote, but the non-zero A serves to avoid generating a singularity at the origin. For T= 24.6 and A=6.6, the figure to the right shows the negligible differences between using an integration window versus assuming a delta shift in the first estimate. The two curves essentially sit very close to one another.

So what do these numbers mean? Essentially, A=6.6 means that the first discovery estimate on occurs on average 3.3 years after they first made the discovery. This makes intuitive sense because if we make the estimate too early, we end up with the equivalent of a thimble-full of oil. For the T=24.6 number, this means that it takes about 25 years for the majority (i.e. the fast part) of the dispersive search to take place. The rest of the long tail results from the slower dispersion. The curve does eventually reach the asymptote for a cumulative growth factor of 8.5.

In terms of a spreadsheet, you can turn the CGF formula into a discrete generating function, with the yearly estimates based on the growth factor of the preceding year. I plotted the DD curve directly against the A&R data in Figure 3 and Figure 4 (see footnote). After the hairs on the back of my neck settled down, I realized that this model has some nice understandable properties. It essentially generates growth factors based solely on the maximum entropy dispersion in the underlying model. In other words, the "enigmatic" reserve growth has turned from a puzzle into a mathematical result resulting solely from basic stochastic effects. Interestingly, for the curve shown the characteristic time in the figure (T=24 years) indicates the point at which the reserve growth factor reaches around 50% of its ultimate value.

Figure 3: Fit to fractional yearly growth of the A&R data. Characteristic time T=24 years.

Figure 4: Fit to cumulative reserve growth of the A&R data. The data was calculated from the generating function in Figure 3. This correlates well with the direct use of the reserve growth equation, with value of T=24. The asymptotic value runs off the chart and reaches a value of 8.57 for the model.

Plotting the same data against an Attanasi & Root chart in Figure 5, it lays cleanly on top of it, showing discrepancies only on some very old outlier data. A&R went through the rationale of discounting the outliers, but I consider all the data valuable, if it gets used in the context of a decent model.

Figure 5: Fit on top of the original A&R chart duplicated from Reference 1. A&R forced a monotonic increase on their data to artificially generate a monotonically increasing curve. The blue line comes from the Dispersive Discovery reserve growth model.

Looking at the A&R data naively and with the limited data set available, the reserve growth looks like it will continue on and eventually reach infinite values -- but this becomes a mathematical impossibility if we refer to the model and integrate out the "thimble"-sized initial estimates. As a bottom-line, if we continue to make poor initial estimates for discoveries, we will continue to pay the price for acting surprised at the "huge" reserve growth we come up with (by "we" I mean the oil industry and oil observers). In other words, the USGS, condoned by the oil industry, has pawned off a contrivance on us by making believe that their trend lines venture beyond mere empiricism. If you look back at the previous "model" that the USGS's Verma postulated² based on the "modified Arrington" approach, you will realize that the trend line comes about purely from heuristic considerations. Their equation shows a fractional order power-law growth:

CGF=1.7378(YSD)^0.3152

To top it off, this heuristic shows unlimited growth! (and an infinite upward slope at t=YSD=0!) Do you suppose that the USGS geologists figured out that by reserving judgment on the possibility of an asymptote, that they could pawn off a fast one on the public, and defer the reality for years to come?

We have to ask ourselves how these professionals get away with publishing stuff based on hacking and punting, that when you get right to the bottom of it, really has such a simple statistical and mathematical underlying foundation. I honestly find nothing complicated about the mathematics (even though it has taken me some time to arrive at my current state of understanding). I call this combination of using simple models and using straightforward calculus and probabilities a form of pragmathematics -- just something you do to understand the physical foundation for the data we observe.

Referring again to the literature, you find hints that support the dispersive effects of accumulated reservoir estimates. I am not sure that they use it exactly the way I do (I highlighted the word dispersion in the following passage).

A graphic illustration of the very broad URA data dispersion that occurs when grouping fields across geologic types and geographic areas was provided by the National Petroleum Council (NPC) and is reproduced with minor modification in Figure FE5.

I claim that this new dispersive discovery reserve growth model has the potential for filling in the back-dated discovery curves and providing better estimates of future production levels. I would recommend starting with the USA data and using Khebab's approach for regenerating a profile of un-back-dated discoveries and then applying the Oil Shock Model to estimate extrapolated production levels. It may take some time to sort out the mess, but we likely have all the pieces necessary to formulate a complete model-based projection.

The actual enigma of reserve growth I think has to do with the cluelessness of the USGS and the secrecy and inscrutability of the oil industry. You would think they would have figured out the reserve growth puzzle long ago. I guess they thought that deferring the reality would surely provide us with infinite hope.

The "How Did We Get Ourselves Into This Mess?" Epilogue

A significant fraction of the population thinks that oil production remains a simple matter of turning the spigot clockwise to get more oil, and don't even see reserve growth on their radar. More than likely, they also believe that raising the current reserve inventory becomes a measured, conservative decision made entirely under the control of humans.

I am not an expert of the socio-economics of oil but have picked up a few nuggets from the tribal knowledge vault here on TOD. The number 1 bit of conventional wisdom: Overestimating the size of a reservoir has significant financial advantages. I agree that you can potentially attract huge initial investment capital if you exaggerate a claim. But this practice can also attract charlatans eager to make a quick fortune. So I agree with the SEC's decision to put the stops on that practice, with big fines for fraudulent claims. And thus the oil companies tried to make their reserve estimates as high as they could without overdoing it. But they still had no good ideas on how best to characterize their estimates (not smart? who knows?). Some people suggest that the industry just makes "conservative" decisions. Yet, I don't think that ambiguously-framed "conservative" decisions matches with well with what you can accomplish with a good reserve growth model. I would describe the industry's approach not "conservative" but "non-predictive" and "safe". I can point to lots of other engineering areas where through the routine process of characterization one can make equally conservative estimates, but they back that up by doing an excellent job of predicting the asymptotic behavior with safety margins for any errors or noise that might creep in.

As an example Andy Grove was the first to characterize silicon dioxide growth by evaluating simple diffusion-based models, and thus began Intel (in a nutshell :). Getting oxide thickness correct could follow a process where someone checks the thickness every two minutes like they were watching a loaf of bread rise, but the technology and modeling has gone way beyond that now in Silicon Valley and China. Not so with the oil industry however; to the untrained eye it looks like they use the primitive trick of plunging a thermometer into the turkey roasting in the oven. This is safe but, jeez, how primitive!

Bottom line, I believe the current industry estimates for reserve growth remain just as safe and non-predictive as a turkey thermometer, but that the underlying reality (having to do with the dispersion of searches through the volume of a reservoir) can vastly improve our educated guesses. This makes it potentially predictive if the industry had the balls to use a good validated model. The simple model I use for reserve growth is complementary to what Andy Grove did in the 60's. Unfortunately, I also don't think that petroleum engineers have any control over marketing decisions. It's possible that insider engineers knew about the characterization outlined here all along but were superceded by management's decisions. Perhaps, they used it in their own internal book-keeping, but presented only a "safe" facade for the outside world to view.

Unfortunately, we turned the clock to a new century, have maybe hit the plateau of peak oil, and only now have some nobodies on TOD finally figured out the obvious that had stared at them and the USGS in the face for years. At that slow a rate we should still be using vacuum tubes in all of our electronics.

Footnote:
As another approach to time window averaging, I selected the incremental fractional curve to integrate for Figure 3 and 4. The math gets a bit more complex. This results in a perhaps closer fit to the reserve growth data as the A&R approach uses the yearly fractional as a generating function as well. You can detect subtle differences only in the initial growth burst around t=0 as the integration tends to favor delta functions arising from the 1/t behavior close to the origin.

Journal References: ¹ Attanasi, E.D., and Root, D.H., 1994, The enigma of oil and gas field growth: American Association of Petroleum Geologists Bulletin, v. 78, no. 3, p. 321-332. ² Verma, M.K., 2003, Modified Arrington Method for Calculating Reserve Growth — A New Model for United States Oil and Gas Fields, U.S. Department of the Interior, U.S. Geological Survey

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