# Peak Oil: Testing Hubbert’s Curve via Theoretical Modeling

## Abstract

A theoretical model of conventional oil production has been developed. The model does not assume Hubbert’s bell curve, an asymmetric bell curve, or a reserve-to-production ratio method is correct, and does not use oil production data as an input. The theoretical model is in close agreement with actual production data until the 1979 oil crisis, with an *R*^{2} value of greater than 0.98. Whilst the theoretical model indicates that an ideal production curve is slightly asymmetric, which differs from Hubbert’s curve, the ideal model compares well with the Hubbert model, with *R*^{2} values in excess of 0.95. Amending the theoretical model to take into account the 1979 oil crisis, and assuming the ultimately recoverable resources are in the range of 2–3 trillion barrels, the amended model predicts conventional oil production to peak between 2010 and 2025. The amended model, for the case when the ultimately recoverable resources is 2.2 trillion barrels, indicates that oil production peaks in 2013.

### Keywords

Peak oil modeling Hubbert’s curve### Nomenclature

- [
*C*_{d}(*t*)] The cumulative discoveries for the world as a function of time (TL)

- [
*C*_{p}(*t*)] The cumulative production for the world as a function of time (TL)

- [\(C_{{\rm p}_{l}}(t)\)]
The cumulative production for the reservoir

*l*as a function of time (TL)- [\(C_{{\rm p}_{l_{i}}}(t)\)]
The cumulative production for the

*i*th well in reservoir*l*(TL)- [
*k*(*t*)] The technology function

- [
*p*(*t*)] The expected discovery percentage function

- [
*P*′(*t*)] The production function as used in Brandt (2007) (b/year)

- [\( P_{l_i}(t)\)]
The production in the

*i*th well of reservoir*l*(TL/year)- [\( Pr_{l_i}(t)\)]
The pressure in the

*i*th well of reservoir*l*as a function of time (Pa)- [
*R*^{2}] The coefficent of determination

- [
*w*_{l}(*t*)] The number of wells in operation for the reservoir

*l*as a function of time- [
*y*_{d}(*t*)] The yearly discoveries function (TL/year)

- [
*b*_{t}] The slope constant for the technology function(year

^{−1})- [\( k_{1_{l_i}}\)]
The proportionality constant relating production to pressure, in the

*i*th well (TL/Pa.year)- [\( k_{2_{l_i}} \)]
The proportionality constant relating pressure to remaining reserves (Pa/TL)

- [\( k_{{\rm p}_{l_i}} \)]
The proportionality constant relating the production to the remaining reserves (year

^{−1})- [
*k*_{w}] The proportionality constant in the wells model

- [\( k_{{\rm w}_{l}} \)]
The proportionality constant for reservoir

*l*in the wells model- [
*P*_{0}] The initial production of the wells in all reservoirs (TL/year)

- [\( P_{0_{l}} \)]
The initial production of the wells in reservoir

*l*(TL/year)- [\( P_{0_{l_i}} \)]
The initial production from the

*i*th well in reservoir*l*(TL/year)- [
*r*_{dec}] The rate of decrease, as used by Brandt (2007) (year

^{−1})- [
*r*_{inc}] The rate of increase, as used by Brandt (2007) (year

^{−1})- [Δ
*r*] The difference between the rate of increase and rate of decrease, as used by Brandt (2007) (year

^{−1})- [
*t*] Time (year)

- [
*t*_{l}] The year the

*l*th reservoir is found (year)- [\( t_{l_i} \)]
The year the

*i*th well comes on-line in reservoir*l*(year)- [
*T*_{peak}] The Peak year for the production curve as used in Brandt (2007) (year)

- [
*T*_{start}] The start year for the production curve as used in Brandt (2007) (year)

- [
*t*_{t}] The year the technology function reaches 0.5 (year)

- [
*URR*] The Ultimate Recoverable Resources (TL)

- [
*URR*_{l}] The Ultimately Recoverable Resources for the reservoir

*l*(TL)- [\( w_{l_{\rm T}} \)]
The total number of wells for reservoir

*l*, if cumulative production were infinite- [\( w_{l_{{\rm T}_{\rm act}}} \)]
The total number of wells for reservoir

*l*given the cumulative production is finite