Skip to main content

Descriptive and Predictive Growth Curves in Energy System Analysis

Abstract

This study reviews a variety of growth curve models and the theoretical frameworks that lay behind them. In many systems, growth patterns are, or must, ultimately be subjected to some form of limitation. A number of curve models have been developed to describe and predict such behaviours. Symmetric growth curves have frequently been used for forecasting fossil fuel production, but others have expressed a need for more flexible and asymmetric models. A number of examples show differences and applications of various growth curve models. It is concluded that these growth curve models can be utilised as forecasting tools, but they do not necessarily provide better predictions than any other method. Consequently, growth curve models and other forecasting methods should be used together to provide a triangulated forecast. Furthermore, the growth curve methodology offers a simple tool for resource management to determine what might happen to future production if resource availability poses a problem. In the light of peak oil and the awareness of natural resources being considered as a basis for the continued well-being of the society and the mankind, resource management should be treated as an important factor in future social planning.

This is a preview of subscription content, access via your institution.

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5

References

  • Aleklett, K., and Campbell, C., 2003, The peak and decline of world oil and gas production: Miner. Energy, v. 18, no. 1, p. 5–20.

    Article  Google Scholar 

  • Ang, B. W., and Ng, T. T., 1992, The use of growth curves in energy studies: Energy, v. 17, no. 1, p. 25–36.

    Article  Google Scholar 

  • Ayres, R. U., 2006, Turning point: The end of exponential growth?: Technol. Forecast. Soc. Change, v. 73, no. 9, p. 1188–1203.

    Article  Google Scholar 

  • Bardi, U., 2005, The mineral economy: a model for the shape of oil production curves: Energy Policy, v. 33, no. 1, p. 53–61.

    Article  Google Scholar 

  • Bardi, U., 2007, Energy prices and resource depletion: lessons from the case of whaling in the nineteenth century: Energy Sources B Econ. Plann. Policy, v. 2, no. 3, p. 297–304.

    Article  Google Scholar 

  • Bartlett, A. A., 1993, Arithmetic of growth: methods of calculation: Popul. Environ., v. 14, no. 4, p. 359–387.

    Article  Google Scholar 

  • Bartlett, A. A., 1999, Arithmetic of growth: methods of calculation II: Popul. Environ., v. 20, no. 3, p. 215–246.

    Article  Google Scholar 

  • Bartlett, A. A., 2000, An analysis of U.S. and world oil production patterns using Hubbert-style curves: Math. Geol., v. 32, no. 1, p. 1–17.

    Article  Google Scholar 

  • Bartlett, A. A., 2004, The essential exponential! For the future of our planet: Center for Science, Mathematics and Computer Education, University of Nebraska-Lincoln, 302 p.

  • Bass, F., 1969, A new product growth model for consumer durables: Manage. Sci., v. 15, p. 215–227.

    Article  Google Scholar 

  • Bentley, R., and Boyle, G., 2007, Global oil production: forecasts and methodologies: Environ. Plann. B: Plann. Des., v. 35, no. 4, p. 609–626.

    Article  Google Scholar 

  • Bentley, R., Mannan, S. A., and Wheeler, S. J., 2007, Assessing the date of the global oil peak: the need to use 2P reserves: Energy Policy, v. 35, no. 12, p. 6364–6382.

    Article  Google Scholar 

  • Berndes, G., Hoogwijk, M., and van den Broek, R., 2003, The contribution of biomass in the future global energy supply: a review of 17 studies: Biomass Bioenergy, v. 25, no. 1, p. 1–28.

    Article  Google Scholar 

  • Bertalanffy, L. V., 1957, Wachstum, in Helmcke, J. G., Len-Gerken, H. V., and Starck, D., eds., Handbuch der Zoologie: Walter de Gruyter, Berlin, 68 p.

    Google Scholar 

  • Bickel, P. J., Li, B., Tsybakov, A. B., van de Geer, S. A., Yu, B., Valdes, T., Rivero, C., Fan, J., and van der Aart, A., 2006, Regularization in statistics: TEST, v. 15, no. 2, p. 271–344.

    Article  Google Scholar 

  • Billo, E. J., 2007, Excel for scientists and engineers: numerical methods: Wiley Interscience, New Jersey, 454 p.

    Book  Google Scholar 

  • Birch, C. P. D., 1999, A new generalized logistic sigmoid growth equation compared with the Richards growth equation: Ann. Bot., v. 83, p. 713–723.

    Article  Google Scholar 

  • Brandt, A. R., 2007, Testing Hubbert: Energy Policy, v. 35, no. 5, p. 3074–3088.

    Article  Google Scholar 

  • Brody, S., 1945, Bioenergetics and growth: Reinhold Publishing, New York, 1033 p.

    Google Scholar 

  • Brown, D., 2007, World fields study shows trends: giants like stable environments. AAPG Explorer (March 2007), p 36–38.

  • Browne, M. W., 2000, Cross-validation methods: J. Math. Psychol., v. 44, no. 1, p. 108–132.

    Article  Google Scholar 

  • Caithamer, P., 2008, Regression and time series analysis of the world oil peak of production: another look: Math. Geosci., v. 40, no. 6, p. 653–670.

    Article  Google Scholar 

  • Carlson, W. B., 2007, Analysis of world oil production based on the fitting of the logistic function and its derivatives: Energy Sources B Econ. Plann. Policy, v. 2, no. 4, p. 421–428.

    Article  Google Scholar 

  • Cavallo, A. J., 2004, Hubbert’s petroleum production model: an evaluation and implications for world oil production forecasts: Nat. Resour. Res., v. 13, no. 4, p. 211–221.

    Article  Google Scholar 

  • Chatfield, C., 2004, The analysis of time series: an introduction (6th edn.): CRC Press, Boca Raton, 333 p.

    Google Scholar 

  • Clark, T. E., 2004, Can out-of-sample forecast comparisons help prevent overfitting?: J. Forecast., v. 23, no. 2, p. 115–139.

    Article  Google Scholar 

  • Cleveland, C. J., and Kaufmann, R. K., 1991, Forecasting ultimate oil recovery and its rate of production: incorporating economic forces into the models of M. King Hubbert: Energy J., v. 12, no. 2, p. 17–46.

    Google Scholar 

  • de Levie, R., 2001, How to use Excel in analytical chemistry and in general scientific data analysis: Cambridge University Press, New York, 487 p.

    Google Scholar 

  • EIA Monthly Energy Review, 2010, Data taken from http://www.eia.doe.gov/emeu/mer/nuclear.html.

  • Energywatch Group, 2007, Coal: resources and future production. See also http://www.energywatchgroup.org/.

  • Feng, L., Junchen, L., and Pang, X., 2008, China’s oil reserve forecast and analysis based on peak oil models: Energy Policy, v. 36, no. 11, p. 4149–4153.

    Article  Google Scholar 

  • Fitzpatrick, A., Hitchon, B., and McGregor, J. R., 1973, Long-term growth of the oil industry in the United States: Math. Geol., v. 5, no. 3, p. 237–267.

    Article  Google Scholar 

  • Fylstra, D., Lasdon, L., Warren, A., and Watson, J., 1998, Design and use of the Microsoft Excel Solvers: Interfaces, v. 28, no. 5, p. 29–55.

    Article  Google Scholar 

  • Gompertz, B., 1825, On nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies: Philos. Trans. R. Soc. Lond., v. 115, p. 513–585.

    Article  Google Scholar 

  • Guyon, X., and Yao, J. F., 1999, On the underfitting and overfitting sets of models chosen by order selection criteria: J. Multivar. Anal., v. 70, no. 2, p. 221–249.

    Article  Google Scholar 

  • Hamilton, J. D., 1994, Time series analysis (1st edn.): Princeton University Press, New Jersey, 820 p.

    Google Scholar 

  • Höök, M., and Aleklett, K., 2008, A decline rate study of Norwegian oil production: Energy Policy, v. 36, no. 11, p. 4262–4271.

    Article  Google Scholar 

  • Höök, M., and Aleklett, K., 2009, Historical trends in American coal production and a possible future outlook: Int. J. Coal Geol., v. 78, no. 3, p. 201–216.

    Article  Google Scholar 

  • Höök, M., and Aleklett, K., 2010, Trends in U.S. recoverable coal supply estimates and future production outlooks: Nat. Resour. Res., v. 19, no. 3, p. 189–208.

    Article  Google Scholar 

  • Höök, M., Söderbergh, B., Jakobsson, K., and Aleklett, K., 2009, The evolution of giant oil field production behaviour: Nat. Resour. Res., v. 18, no. 1, p. 39–56.

    Article  Google Scholar 

  • Höök, M., Zittel, W., Schindler, J., and Aleklett, K., 2010a, Global coal production outlooks based on a logistic model: Fuel, v. 89, no. 11, p. 3546–3558.

    Article  Google Scholar 

  • Höök, M., Bardi, U., Feng, L., and Pang, X., 2010b, Development of oil formation theories and their importance for peak oil: Mar. Petrol. Geol., v. 27, no. 9, p. 1995–2004.

    Article  Google Scholar 

  • Hotard, D. G., and Ristroph, J. H., 1984, A regional logistic function model for crude oil production: Energy, v. 9, no. 7, p. 565–570.

    Article  Google Scholar 

  • Hu, J., Chen, Y., and Zhang, S., 1995, A new model for predicting production and reserves of oil and gas fields: Acta Petrolei Sin., v. 16, no. 1, p. 79–86.

    Google Scholar 

  • Hubbert, M. K., 1956, Nuclear energy and the fossil fuels, Presented before the Spring Meeting of the Southern District, American Petroleum Institute, Plaza Hotel, San Antonio, TX, March 7–9, http://www.hubbertpeak.com/Hubbert/1956/1956.pdf.

  • Hubbert, M. K., 1959, Techniques of prediction with application to the petroleum industry, Published in 44th Annual Meeting of the American Association of Petroleum Geologists, Shell Development Company, Dallas, TX, 43 p.

  • Imam, A., Startzman, R. A., and Barrufet, M., 2004, Multicyclic Hubbert model shows global conventional gas output peaking in 2019: Oil Gas J., v. 102, p. 3131–3139.

    Google Scholar 

  • Janoschek, A., 1957, Das reaktionskinetische Grundgesetz und seine Beziehungen zum Wachstums- und Ertragsgesetz: Statistische Vierteljahresschrift, v. 10, p. 25–37.

    Google Scholar 

  • Kaufmann, R. K., 1991, Oil production in the lower 48 states: reconciling curve fitting and econometric models: Resour. Energy, v. 13, no. 1, p. 111–127.

    Article  Google Scholar 

  • Laherrère, J., 1997, Multi-Hubbert modeling, http://www.oilcrisis.com/laherrere/multihub.htm, Accessed 26 May 2010.

  • Laherrère, J., 2000, Distribution of field sizes in a petroleum system: parabolic fractal, lognormal or stretched exponential?: Mar. Petrol. Geol., v. 17, no. 4, p. 539–546.

    Article  Google Scholar 

  • Laherrère, J., 2004, Natural gas future supply, Paper presented at International Energy Workshop jointly organized by the Energy Modeling Forum, International Energy Agency (IEA including ETSAP) and International Institute for Applied Systems Analysis (IIASA), 22–24 June 2004 at the IEA, Paris, France, see also http://www.iiasa.ac.at/Research/ECS/IEW2004/docs/2004P_Laherrere.pdf.

  • Levenberg, K., 1944, A method for the solution of certain non-linear problems in least squares: Q. Appl. Math., v. 2, no. 2, p. 164–168.

    Google Scholar 

  • Lynch, M. C., 2002, Forecasting oil supply: theory and practice: Q. Rev. Econ. Finance, v. 42, no. 2, p. 373–389.

    Article  Google Scholar 

  • Mabel, M. C., and Fernandez, E., 2008, Growth and future trends of wind energy in India: Renew. Sustain. Energy Rev., v. 12, no. 6, p. 1745–1757.

    Article  Google Scholar 

  • Mann, P., Gahagan, L., and Gordon, M. B., 2003, Tectonic setting of the world’s giant oil and gas fields, in Halbouty, M. T., ed., AAPG Memoir 78: Giant Oil and Gas Fields of the Decade 1990–1999, p. 15–105.

  • Marquardt, D., 1963, An algorithm for least squares estimation of nonlinear parameters: SIAM J. Appl. Math., v. 11, no. 2, p. 431–441.

    Article  Google Scholar 

  • McArdle, J. J., 2001, Growth curve analysis, in Smelser, N. J., and Baltes, P. B., eds., International Encyclopedia of the Social & Behavioral Sciences: Pergamon, Oxford, p. 6439–6445.

    Google Scholar 

  • Meng, Q. Y., and Bentley, R. W., 2008, Global oil peaking: responding to the case for ‘abundant supplies of oil’: Energy, v. 33, no. 8, p. 1179–1184.

    Article  Google Scholar 

  • Meyer, P. S., and Ausubel, J. H., 1999, Carrying capacity: a model with logistically varying limits: Technol. Forecast. Soc. Change, v. 61, no. 3, p. 209–214.

    Article  Google Scholar 

  • Milici, R. C., and Campbell, E. V. M., 1997, A predictive production rate life-cycle model for southwestern Virginia coalfields. Geological Survey Circular 1147, http://pubs.usgs.gov/circular/c1147/.

  • Miller, R. G., 1996, Estimating global oil resources and their duration, Norwegian Petroleum Society Special Publications 6, p. 43–56.

    Google Scholar 

  • Modis, T., 2007, Strengths and weaknesses of S-curves: Technol. Forecast. Soc. Change, v. 74, no. 6, p. 866–872.

    Article  Google Scholar 

  • Modis, T., and Debecker, A., 1992, Chaoslike states can be expected before and after logistic growth: Technol. Forecast. Soc. Change, v. 41, no. 2, p. 111–120.

    Article  Google Scholar 

  • Mohamed, Z., and Bodger, P., 2004, Forecasting electricity consumption in New Zealand using economic and demographic variables: Energy, v. 30, no. 10, p. 1833–1843.

    Article  Google Scholar 

  • Mohr, S. H., and Evans, G. M., 2009, Forecasting coal production until 2100: Fuel, v. 88, no. 11, p. 2059–2067.

    Article  Google Scholar 

  • Moore, C. L., 1966, Projections of U.S. petroleum supply to 1980, with annex entitled: the Gompertz curve for analyzing and projecting the historic supply patterns of exhaustible natural resources, Technical Report: Office of Oil and Gas, Washington, DC, 47 p.

  • Moriarty, P., and Honnery, D., 2009, What energy levels can the Earth sustain?: Energy Policy, v. 37, no. 7, p. 2469–2474.

    Article  Google Scholar 

  • Nashawi, I. S., Malallah, A., and Al-Bisharah, M., 2010, Forecasting world crude oil production using multicyclic Hubbert model: Energy Fuels, v. 24, no. 3, p. 1788–1800.

    Article  Google Scholar 

  • Nehring, R., 2006a, Two basins show Hubbert’s method underestimates future oil production: Oil Gas J., v. 104, no. 13, p. 37–42.

    Google Scholar 

  • Nehring, R., 2006b, How Hubbert method fails to predict oil production in the Permian Basin: Oil Gas J., v. 104, no. 15, p. 30–35.

    Google Scholar 

  • Nehring, R., 2006c, Post-Hubbert challenge is to find new methods to predict production: Oil Gas J., v. 104, no. 16, p. 43–46.

    Google Scholar 

  • Newton, I., 1726, Philosophiae naturalis principia mathematica, general scholium (3rd edn.), page 943 of I. Bernard Cohen and Anne Whitman’s 1999 translation: University of California Press, 974 p.

  • NIST/SEMATECH, 2010, NIST/SEMATECH e-handbook of statistical methods, See also http://www.itl.nist.gov/div898/handbook/.

  • Owen, N. A., Inderwildi, O. R., and King, D. A., 2010, The status of conventional world oil reserves—Hype or cause for concern?: Energy Policy, v. 38, no. 8, p. 4743–4749.

    Article  Google Scholar 

  • Patzek, T. W., 2008, Exponential growth, energetic Hubbert cycles, and the advancement of technology: Arch. Min. Sci., v. 53, no. 2, p. 131–159.

    Google Scholar 

  • Patzek, T. W., and Croft, G. D., 2010, A global coal production forecast with multi-Hubbert cycle analysis: Energy, v. 35, no. 8, p. 3109–3122.

    Article  Google Scholar 

  • Radetzki M, 2007. Råvarumarknaden (in Swedish): SNS Förlag, 312 p.

  • Richards, F. J., 1959, A flexible growth curve for empirical use: J. Exp. Bot., v. 10, no. 2, p. 290–301.

    Article  Google Scholar 

  • Rotty, R. M., 1979, Growth in global energy demand and contribution of alternative supply systems: Energy, v. 4, no. 5, p. 881–890.

    Article  Google Scholar 

  • Silvennoinen, P., and Väänänen, J., 1987, Forecasting technological substitution: the logistic model of energy systems revisited: Technol. Forecast. Soc. Change, v. 32, no. 3, p. 273–280.

    Article  Google Scholar 

  • Simon, J., 1998, The ultimate resource 2. Revised version: Princeton University Press, 778 p.

  • Solow, R., 1974, The economics of resources or the resources of economics: Am. Econ. Rev., v. 64, no. 2, p. 1–14.

    Google Scholar 

  • Sorrell, S., and Speirs, J., 2009, Methods of estimating ultimately recoverable resources, Technical report 5 of the UK Energy Research Centre report on Global Oil Depletion, See also: http://www.ukerc.ac.uk/support/Global%20Oil%20Depletion.

  • Sorrell, S., and Speirs, J., 2010, Hubbert’s legacy: a review of curve-fitting methods to estimate ultimately recoverable resources: Nat. Resour. Res., v. 19, no. 3, p. 209–230.

    Article  Google Scholar 

  • Sorrell, S., Speirs, J., Bentley, R. W., Brandt, A. R., and Miller, R. G., 2009, Global oil depletion: an assessment of the evidence for a near-term peak in global oil production: UK Energy Research Centre, London.

    Google Scholar 

  • Sorrell, S., Speirs, J., Bentley, R. W., Brandt, A. R., and Miller, R. G., 2010a, Global oil depletion: a review of the evidence: Energy Policy, v. 38, no. 9, p. 5290–5295.

    Article  Google Scholar 

  • Sorrell, S., Miller, R., Bentley, R., and Speirs, J., 2010b, Oil futures: a comparison of global supply forecasts: Energy Policy, v. 38, no. 9, p. 4990–5003.

    Article  Google Scholar 

  • Sprott, J. C., 2003, Chaos and time-series analysis (1st edn.): Oxford University Press, New York, 507 p.

    Google Scholar 

  • Stone, M., 1978, Cross-validation: a review: Math. Oper. Res. Stat., v. 9, no. 1, p. 127–139.

    Google Scholar 

  • Szklo, A., Machado, G., and Schaeffer, R., 2007, Future oil production in Brazil—estimates based on a Hubbert model: Energy Policy, v. 35, no. 4, p. 2360–2367.

    Article  Google Scholar 

  • Tao, Z., and Li, M., 2007, System dynamics model of Hubbert Peak for China’s oil: Energy Policy, v. 35, no. 4, p. 2281–2286.

    Article  Google Scholar 

  • Tsoularis, A., and Wallace, J., 2002, Analysis of logistic growth models: Math. Biosci., v. 179, no. 1, p. 21–55.

    Article  Google Scholar 

  • United States Geological Survey, 2000, World Petroleum Assessment 2000: USGS World Energy Assessment Team, http://pubs.usgs.gov/dds/dds-060/.

  • van Rensburg, W. C. J., 1975, ‘Reserves’ as a leading indicator to future mineral production: Resour. Policy, v. 1, no. 6, p. 343–356.

    Article  Google Scholar 

  • Verhulst, P. F., 1838, Notice sur la loi que la population suit dans son accroissement: Correspondence Mathematique et Physique, v. 10, p. 113–121.

    Google Scholar 

  • Watkins, G. C., 2006, Oil scarcity: What have the past three decades revealed?: Energy Policy, v. 34, no. 5, p. 508–514.

    Article  Google Scholar 

  • Weibull, W., 1951, A statistical distribution function of wide applicability: J. Appl. Mech., v. 18, no. 3, p. 293–297.

    Google Scholar 

  • Wikipedia, 2009, Wheat and chessboard problem, Available from http://en.wikipedia.org/wiki/Wheat_and_chessboard_problem.

Download references

Acknowledgments

The authors would like to thank Dr. Herbert West for providing valuable inspiration. Two anonymous reviewers also merit our most sincere gratitude for presenting comments that greatly improved this manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mikael Höök.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Höök, M., Li, J., Oba, N. et al. Descriptive and Predictive Growth Curves in Energy System Analysis. Nat Resour Res 20, 103–116 (2011). https://doi.org/10.1007/s11053-011-9139-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11053-011-9139-z

Keywords

  • Growth curve models
  • curve fitting
  • logistic model
  • resource management