# The Bakken Dispersive Diffusion Oil Production Model

This post continues from Bakken Growth.

The Model
Intuition holds that oil production from a typical Bakken well is driven by diffusion.  The premise is that a volume of trapped oil diffuses outward along the fractures.  After the initial fracturing, oil close by the collection point will quickly diffuse through the new paths. This does not last long, however, as this oil is then replenished by oil from further away and since it takes longer to diffuse, the flow becomes correspondingly reduced. Eventually, the oil flow is based entirely on diffusing oil from the furthest points in the effective volume influenced by the original fractured zone. This shows the classic law of diminishing returns, characteristic of Fickian diffusion.

This class of problems is very straightforward to model.  The bookkeeping is that the diffusing oil has to travel various distances to reach the collection point. One integrates all of these paths and gets the production profile. I call it dispersive because the diffusion coefficient is actually smeared around a value.

One can start from the master diffusion equation, also known as the Fokker-Planck equation.
$$\frac{\partial f(x,t)}{\partial t} = \frac{D_0}{2} \frac{\partial^2 f(x,t)}{\partial x^2}$$

Consider that a plane of oil will diffuse outward from a depth at position x. The symmetric kernel solution is given by:
$$f(x,t) = {1\over{2\sqrt{D_0 t}}}e^{-x/\sqrt{D_0 t}}$$
If we assume that the diffusion coefficient is smeared around the value D0 with maximum entropy uncertainty, integrate from all reasonable distances from the collection point, the cumulative solution becomes

$$P(t) = \frac{P_0}{1+ \frac{1}{\sqrt{D t}}}$$

The reasonable distances are defined as a mean distance from the collection point and with a distribution around the mean with maximum entropy uncertainty. P0 is the effective asymptotic volume of oil collected and the diffusion coefficient turns into a spatially dimensionless effective value D. The details of the derivation are found in the text The Oil Conundrum and is what I refer to as a general dispersive growth solution; in this case the dispersive growth follows a fundamental Fickian diffusive behavior proportional to the square root of time.  This is all very basic statistical mechanics applied to a macroscopic phenomena, and the only fancy moves are in simplifying the representation through the use of maximum entropy quantification.



Some Data and a Model Fit
More recent data on oil production is available from an article Oil Production Potential of the North Dakota Bakken  in the Oil&Gas Journal written by James Mason.

Figure 1 below shows the averaged monthly production values from Bakken compiled by Mason. The first panel shows in blue his yearly production and his cumulative. I also plotted the dispersive diffusion model in red with two parameters, an effective diffusion coefficient and an equivalent scaled swept oil volume. Note that the model is shifted to the left compared to the blue line, indicating that the fit may be bad. But after staring at this for awhile, I discovered that Mason did not transcribe his early year numbers correctly. The panel on the bottom is the production data for the first 12 months and I moved those over as black markers on the first panel, which greatly improved the fit. The dashed cumulative is the verification as the diffusive model fits very well over the entire range.

 Figure 1: Model adapted to Mason Bakken Data. Top is yearly and bottom is monthly data.

For this model, the asymptotic cumulative is set to 2.6 million barrels. This is a deceptive number since the fat-tail is largely responsible for reaching this value asymptotically. In other words, we would have to wait for an infinite amount of time to collect all the diffused oil — such is the nature of a random walk. Even to collect 800K barrels will take 100 years from extrapolating the curve. After 30 years, the data says 550K barrels, so one can see that another 70 years will lead to only 250K barrels, should the well not get shut-in for other reasons.

If these numbers that Mason has produced are high quality, and that is a big if (considering how he screwed up the most important chart) this may become a de facto physical model describing oil production for fractured wells. I can guarantee that you won’t find a better fit than this considering it is only two parameters, essentially describing a rate and a volume. This is likely the actual physical mechanism as diffusional laws are as universal as entropy and the second law.

The connection to the previous post is that the substantial production increase is simply a result of gold-rush dynamics and the acceleration of the number of new wells. Wait until these new starts stop accelerating. All the declines will start to take into effect, as one can see from the steep decline in the dispersive diffusion profiles.  We may still get returns from these wells for many years, but like the 5 barrel/day stripper wells that dot the landscape in Texas and California, they don’t amount to much more than a hill of beans.  The peak oil problem has transformed into a flow problem, and unless thousands of new wells are added so that we can transiently pull from the initial production spike or to continuously pull from the lower diminishing returns, this is what Bakken has in store — a few states with thousands and thousands of wells cranking away, providing only a fraction of the oil that we demand to keep the economy running.

If someone comes up with a way to increase diffusion, it might help increase flow, but diffusion is a sticky problem. That is what nature has laid out for us, and we may have gotten as far as we can by applying hydraulic-fracturing to lubricate the diffusion paths.

This analysis fits in perfectly with the mathematical analysis laid out in The Oil Conundrum book, and will likely get added in the next edition.