The Hubbert linearization is a way to plot production data to estimate two important parameters of a

Hubbert curve The Hubbert curve is an approximation of the production rate of a resource over time. It is a symmetric logistic distribution curve, often confused with the "normal" gaussian function. It first appeared in "Nuclear Energy and the Fossil Fuels ...

, the approximated production rate of a nonrenewable resource following a

logistic distribution Logistic may refer to: Mathematics * Logistic function, a sigmoid function used in many fields ** Logistic map, a recurrence relation that sometimes exhibits chaos ** Logistic regression, a statistical model using the logistic function ** Logit, ...

: * the logistic growth rate and * the quantity of the resource that will be ultimately recovered. The linearization technique was introduced by

Marion King Hubbert Marion King Hubbert (October 5, 1903 – October 11, 1989) was an American geologist and geophysicist. He worked at the Shell research lab in Houston, Texas. He made several important contributions to geology, geophysics, and petroleum geolo ...

in his 1982 review paper. The Hubbert curve is the first derivative of a logistic function, which has been used for modeling the depletion of crude oil in particular, the depletion of finite mineral resources in general and also

population growth Population growth is the increase in the number of people in a population or dispersed group. Actual global human population growth amounts to around 83 million annually, or 1.1% per year. The global population has grown from 1 billion in 1800 to ...

patterns.

Principle

The first step of the Hubbert linearization consists of plotting the yearly production data (''P'' in bbl/y) as a fraction of the cumulative production (''Q'' in bbl) on the vertical axis and the cumulative production on the horizontal axis. This representation exploits the linear property of the logistic differential equation: with *''k'' as logistic growth rate and *''URR'' as the ultimately recoverable resource. We can rewrite (1) as the following: The above relation is a line equation in the ''P/Q'' versus ''Q'' plane. Consequently, a

linear regression In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is call ...

on the data points gives us an estimate of the line slope calculated by ''-k/URR'' and intercept from which we can derive the Hubbert curve parameters: * The ''k'' parameter is the intercept of the vertical axis. * The ''URR'' value is the intercept of the horizontal axis.

Examples

Global oil production

The geologist

Kenneth S. Deffeyes Kenneth S. Deffeyes was a geologist who worked with M. King Hubbert, the creator of the Hubbert peak theory, at the Shell Oil Company research laboratory in Houston, Texas. He claimed Chickasaw ancestry. He did not make a distinction between "conventional" and "non-conventional" oil produced by fracturing, aka tight oil, which has continued further growth in oil production. However, since 2005 conventional oil production has not grown anymore.

US oil production

The charts below gives an example of the application of the Hubbert Linearization technique in the case of the US Lower-48 oil production. The fit of a line using the data points from 1956 to 2005 (in green) gives a URR of 199 Gb and a logistic growth rate of 6%. Image:HubbertLin_US_Lower48.svg, Hubbert Linearization on US's oil production Image:Hubbert_US_Lower48.svg, Hubbert curve on US's oil production

Norway oil production

The Norwegian Hubbert linearization estimates an URR = 30 Gb and a logistic growth rate of k = 17%. Image:HubbertLin_Norway.svg, Hubbert Linearization on

Norway Norway, officially the Kingdom of Norway, is a Nordic country in Northern Europe, the mainland territory of which comprises the western and northernmost portion of the Scandinavian Peninsula. The remote Arctic island of Jan Mayen and t ...

's oil production Image:Hubbert_Norway.svg, Hubbert curve on

's oil production

Alternative techniques

Second Hubbert linearization

The Hubbert linearization principle can be extended to the first derivatives of the production rate by computing the derivative of (2): The left term, the rate of change of production per current production, is often called the decline rate. The decline curve is a line that starts at +k, crosses zero at URR/2 and ends at −k. Thus, we can derive the Hubbert curve parameters: * The growth parameter k is the intercept of the vertical axis. * The URR value is twice the intercept of the horizontal axis.

Hubbert parabola

This representation was proposed by Roberto Canogar and applied to the oil depletion problem. It is equation (2) multiplied by Q. The parabola starts from the origin (0,0) and passes through (URR,0). Data points until t are used by the least squares fitting method to find an estimate for URR.

Logit transform

David Rutledge applied the logit transform for the analysis of coal production data, which often has a worse signal-to-noise ratio than the production data for hydrocarbons. The integrative nature of cumulation serves as a low pass, filtering noise effects. The production data is fitted to the logistic curve after transformation using ''e''(''t'') as normalized exhaustion parameter going from 0 to 1. The value of URR is varied so that the linearized logit gives a best fit with a maximal

coefficient of determination In statistics, the coefficient of determination, denoted ''R''2 or ''r''2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s). It is a statistic used i ...

R^2

External links

* Robert Rapier
Does the Hubbert Linearization Ever Work?
The Oil Drum, 2007-03-22 * David Rutledge
Energy Supplies and Climate

Caltech The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...

, 2019 - on curve fits to the production history (includin
excel data
on historic coal production and logit fits)

References

{{Reflist Peak oil