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.