L. David Roper
http://www.roperld.com/personal/RoperLDavid.htm
6 March, 2018
World Fossil Fuels Depletion
The U.S. state of Arkansas is having a boom in extracting natural gas by the technique of fractionating ("fracking") shale/dolomite formations. A "boom" in nonrenewable resource extraction from the Earth is defined as a time period in which extraction is occurring very fast in a given area; thus, many workers come in from outside the area to man the drilling rigs, to build housing for the oil workers and to provide other services for the increased population.
This article shows mathematically that the Arkansas natural-gas boom will become a bust within a decade. A "bust" in nonrenewable resource extraction from the Earth is defined to begin at the time when extraction of the resource peaks and then falls to negligible amounts over a time period.
The U.S. Energy Information Administration gives monthly and annual natural-gas extraction data for Arkansas since 1981.
Those data are fitted by a depletion function, the Verhulst function, in this study to determine when the extraction will peak.
The data and the fits to the data are given in a later section.
A reliable estimate of reserves is needed to fit extraction data by a function for projecting into the future for a nonrenewable resource Here is a good definition of reserves of a nonrenewable resource.
The U.S. Energy Information Administration gives reserves estimates from 1977 to 2016 for natural-gas extraction in Arkansas, which are shown here by black dots for years 2005 to 2016:
The curve is a fit to the 2005-2016 data using the Verhulst function described in the next section, assuming that the curve will be symmetrical. The fit is done to get a rough estimate of the peak value of the reserves estimate in the future, which is ~15 x 1012 ft3.
The depletion function that is used in this article is the Verhulst function:
The asymmetry parameter, n, must be greater than 0.
where 
The maximum of P(t) occurs at 
, which yields the peak value 
.
For the symmetric case (n=1): 
and 
.
For a depletion situation for which there are N peaks the depletion function is:
One needs an estimate of the amount of asymmetry, described by the parameter n, for the future peak due to fracking for shale natural gas, which can be obtained from the macro-analysis of the Fayetteville play by J. David Hughes (Drilling Deeper).
J. David Hughes (Drilling Deeper) has done a micro-analysis of the Haynesville play, which is totally in Arkansas.:
I fitted a Verhulst function to the Hughes Arkansas curve to get the asymmetric parameter n:
The asymmetric parameter is 36. (The Y-axis has been changed from 10^9 ft^3/day to 10^12 ft^3/yr.)
Then a Verhulst fit was done to the Arkansas natural-gas extraction data varying the asymmetry parameter n:
All Verhulst parameters were varied, the asymmetry parameter for the final peak searched to 3.69. The calculated 2015 reserves for this curve is ~4.27 x 1012 ft3, much less than the estimated reserves value (~15 x 1012 ft3) given above. So, there may be another extraction peak in the future.
The fit to the data for early years:
Even for very high estimates of natural-gas reserves for its extraction in Arkansas, the current boom will turn into a bust in less than two decades.
It would be wise for Arkansas to use the current natural-gas boom to build the policies and infrastructure for collecting energy from wind and solar, for encouraging drivers to drive electric vehicles and for fast charging stations for electric vehicles in personal and parking garages. Wind energy in Arkansas is off to a very slow start.
See http://www.roperld.com/science/RenewableEnergyElectricityDemand.pdf.
It would be wise for the government of Arkansas to do some decade-long planning about how to best manage the coming natural-gas-extraction bust. A tax on natural-gas extraction to put in a fund to help manage the bust and to clean up the mess made by the extraction would be wise. Such tax might have an added benefit of slowing down the extraction so that the bust will not occur so soon, giving more time to prepare for it.
L. David Roper interdisciplinary studies
World Fossil Fuels Depletion
L. David Roper, http://arts.bev.net/RoperLDavid/
6 March, 2018