Solving the "Enigma" of Reserve Growth

Khebab posted my solution to the "enigma" on The Oil Drum last week, which you can find here. I also left the working manuscript on Google Docs here which contains a few edits which missed getting into the final submission.

I didn't go into much detail on another aspect of reserve growth (or what looks a lot like reserve growth), which came up in a subsequent post by Phil Hart of TOD/DownUnder. Phil detailed some charts of technology boosted production on a field in Indonesia, and in the comments, Elwoodelmore turned the discussion to the effects of gravity drainage on lengthening the lifespan on many fields. Professional geologist Rockman mentioned that gravity drainage plays a real effect, and so I asked indirectly whether diffusion plays any kind of role in this phenomenon. Elwoodelmore provided the fundamental equations, which he said forms the basis of Darcy's Law:

the basic's of gravity drainage are contained in d'arcy's law which stated mathmatecally is:

v = c(k/u)(dp/ds)

with a substitution v=q/A then c becomes 1.127 (usually presented with a minus sign, because dp/ds is negative)

where q is in bbls/day
and A is the x-sectional area of flow, sf

k, permeability, is in darcies

u, viscosity is in centipoise

dp/ds is pressure drop, psi/ft

for the case of gravity drainage dp/ds can be taken as (dp/dh)*(dh/ds)

dp/dh is the bouyancy of the oil or difference between the fluid gradient for oil and water, psi/ft.
= 0.433 (rhowater-rhooil), at reservoir conditions.

and finally dh/ds is the change in elevation over a distance. or simply sin dipangle.

and strictly speaking d'arcy's law only applies to single phase isothermal flow under steady state conditions. and for cases where the capilary pressure is zero(well above the oil/water contact)petroleum engineering usually makes lots of assumptions, so these restrictions are usually just ignored.

The v term above essentially provides the flow of oil into the region. Right away from the elements of the equation, one can tell Darcy's Law acts pretty much the same as the ordinary Fick's Law in diffusion problems, which I have written about previously here and here. Wikipedia also makes the connection (which makes it a solid case :) :).

I find this intriguing in light of the "enigma" post because diffusion and dispersion have a close association in terms of outcome. The phenomena of diffusion provides a physical mechanism for material to move into a region of lower concentration. In this case, dispersion occurs because the material does indeed spread out over the volume, and the greater the volume or distance, the slower the relative dispersion proceeds. By the same token, I use dispersion in the Dispersive Discovery model to indicate that a search through a volume proceeds at many different effective rates, with the spread in rates always proportional to the cumulative volume searched. So, in the physical diffusion case, dispersion occurs as material fills up the new volume, and in the Dispersive Discovery case, humans do the work of randomly filling up the volume. In other words, diffusion implies the material comes to us and in dispersive discovery, we (as prospectors) go to the materials. And both of these processes occur randomly.

So the basic premises sound similar and you would expect that the Fick's Law solution would have the same diminishing rate of return as the Dispersive Discovery case.

With that, let us turn to how to cast Darcy's Law into Fick's Law of diffusion -- which has a a rather simple temporal behavior in the first-order case.

First take the equation for Darcy's law, that Elwoodelmore stated:

v=c(k/u)(dp/ds)

and then substituting (dp/dh)*(dh/ds)

v=c(k/u)(dp/dh)(dh/ds)

The key involves the dh/ds term, the "dip angle", which provides a driving gradient at the heart of any diffusion process.

The ds term expresses the displacement in volume as the gravity drainage starts to move material from one volume to the other. So whatever goes from one side of "s" goes to the other side, the "v" side. This means that the length of the partially drained volume gets bigger and bigger with time.

For a more three-dimensional view, look at the width in the following figure where x corresponds to the s term with a diffusive flow in the opposite direction as above:

So as x or s will get longer and longer over time, with a cumulative increase proportional to the integral of v. With the trigonometric small angle approximation for sin (dip angle) you get :

dh/ds = h/s

so rewriting this, replacing U with s to denote cumulative displaced volume

dU/dt = k / U

this solves simply as

U(t) = k * sqrt(t)

which represents the time dependence of the Fick's First Law of Diffusion. As a bottom-line you get progressively diminishing return in oil production over time with this law. You can also see this by taking the first derivative.

dU(t)/dt = k /(2*sqrt(t))

Note that the rate of production slows down inversely as the square root of time. As the long lifetime of a stripper well attests, the gravity drainage does exist but it also does have physical limits, mainly because of the diminishing rate of return.

But notice how similar the curves look for diffusive discovery and gravity diffusion:

This means that DD reserve growth and gravity drainage likely reinforce each other in terms of temporal behavior. But as Rockman says:

Gravity drainage fields can really confuse the villagers. I have one such fld that has produced for 50 years (20 mmbo so far) and will produce for another 100 years (maybe another 20 mmbo). When the angry villagers hear such tales they begin to think there really is help out there for them. The wells in this field make about 1 bb/ per day. It obvious has no bearing on PO. But the little ma and pa operators are slowly becoming millionaires.

The Role of Dispersive Discovery in Reserve Growth

Shorter post: Enigma Solved

"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

We have on our hands a huge swindle pertaining to the reserve growth analysis promulgated by USGS geologists. In turns, the establishment has labeled the fossil fuel reserve growth issue an "enigma" or 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 believe that the big technical issue that the USGS have historically had with their analysis has to do with using "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. But after more consideration, the problem has become even more painfully obvious. It really stems from a lack of a good value for the initial discovery estimate.

To lead me down this path, I used my generalized Dispersive Discovery Model. In terms of modeling reserve growth, the dispersion generates a tail for accumulating further discoveries after the initial estimate. For constant average growth, the model looks like this:

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

Note that at time t=0, the discovered amount starts at zero and then the accumulated reaches some value proportional to L -- what one should consider as the characteristic depth of the reservoir. 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 10x after 90 years and claim that this has some real physical significance, almost offering up hope for still-to-come huge reserve benefits.


The swindle with all this has to do with when the original estimate is made. 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. 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 be proportionately much less. 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 data from an Attanasi & Root paper¹ which you can find a dump of here. Initially, I plotted the data as a fractional yearly growth curve:

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 strange-looking function, where A indicates the integration window:

U(t) = exp( ((A+t)*ln(A+t) - t*ln(t) - A) / A )

It turns out that the value for characteristic depth L does not even play into the result, as long as it gets set to some relatively large number. The value for the integration window A does not play a big role either, as it serves mainly to avoid generating a singularity and turning it back into the original DD formula.

In terms of a spreadsheet, this turns into a discrete generating function, with the yearly estimates based on the growth factor of the preceding year. With absolutely no fudge factors, I plotted the curve directly against the A&R data below. After the hairs on the back of my neck settled down, I realized that this function has some type of fundamental golden ratio property. 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 curiosity resulting solely from simple stochastic effects.



Plotting against an Attanasi & Root figure, it lays cleanly on top of it, showing discrepancies only on some very old outlier data.

Interestingly, the reserve growth looks like it will continue to reach infinite values, but this turns out to stem solely from the possibility of "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 have. In other words, the swindle has played out in our heads.

If you look back in the literature, you find hints that support the dispersive effects of accumulated reservoir estimates (note that they use the term dispersion).

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 peaked back at the previous "model" that the USGS's Verma postulated² based on the "modified Arrington" approach and realize that this comes about purely from heuristic considerations.

CGF=1.7378(YSD)^0.3152

You have to ask yourself how these professionals get away with publishing stuff based on hacking and speculation that instead has such a simple statistical and mathematical foundation. I really 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.

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. As King would likely offer, we just have to look elsewhere; certainly BigMrBossman won't provide any guidance.

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References:
¹Attanasi, 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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Update:
Google Spreadsheet featuring full formula here: http://spreadsheets.google.com/pub?key=prVeQf4uJHD1AJVF_pGg9Jw

Update 2:
As I find it virtually impossible to show math derivations with the Blogger editor, I pasted my derivation from a math markup tool below. Equation 3 shows the technique for averaging initial estimates over the time window A, transforming U into U-bar. Equation 5 describes the final result, where the constant C scales the reserve growth factor to start at 1. The term L/k essentially gives the shape of the curve, with smaller values relative to A giving a more horizontal asymptote.

Continuation of TOD commentary

TheOilDrum.com doesn't allow comments on posts after a certain amount of time, so I will use this post to continue the discussion. For this subthread:

Vitalis on July 4, 2008 - 3:36am

Yes, if L is a sum of independent exponentials with means Lo and L1, then L has density

(*) 1/(L1-L0)*( exp(-x/L1)-exp(-x/L0) ), for x >= 0.

(Of course, when L0->L1 this will collapse to a Gamma distribution with 2 degrees of freedom).

In the formulas above, I assumed that S is also Gamma(2,s) distributed, which perhaps is harder to give a physical interpretation than for L. If we assume S ~ Exp(lambda) and L density (*), then

E[D(S,L)] = lambda*(1 - 1/(L1-Lo)*(1/(1/lambda + 1/L1) - 1/(1/lambda + 1/Lo)) ).

But perhaps all this fiddling around with different distibutions is of moderate interest: it is probably most important to recognice that the most important thing is to have some kind of uncertainty ("dispersion") in lambda and L. Whether this is the case, I would say, comes down to how sensitive the "output" is to the various assumptions on the distributions, the "output" being whatever the model help us to say about the real world. (In this case, perhaps the expected volume of discoveries in the next few years.)

Another thing we could try to adress is the sample variability: we have focused on the mean of D(S,L), but it should be straight forward to compute condfidence intervals for future discoveries, given assumptions on the distributions of S and L. (Maybe you already did this).

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Another (minor) comment: i don't think the single dispersive model corresponds to a uniformly distributed L --- in that model, L = Lo deterministically (i.e. point mass 1 at Lo in probability lingo).

In fact, L uniformly distributed on [0,2*L0] (to make E[L] = L0) gives

E[D(S,L)] = labmda*( 1 - 1/(2*Lo) + 1/(2*Lo)*exp(-2*Lo/lambda) ).

I agree with Vitalis about the statement "it is probably most important to recognice that the most important thing is to have some kind of uncertainty ("dispersion") in lambda and L."

The following figure shows how incremental dicoveries basically trend to the same asymptote, independent of the volume distribution.


I would also note that, contrary to Vitalis' suggestion, the Delta "thin seam" volume situated at L₀ would have a cumulative discovery profile that looks like this:

DeltaDiscovery(x) = L₀*exp(-L₀/x)

Striking Paydirt

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)

Let

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?

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.

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.