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Domestic crude oil resource appraisal
Domestic crude oil resource appraisal N o e l D. Uri*
Department of Economics and Business, Catholic University of America, Washington DC, 20064, USA (Received December 1979)
This paper endeavours to estimate the cummulative level of discovery and production of crude oil in the United States where the importance of price and technological change is considered. Two separate functional specifications for the cumulative level are hypothesized and estimated. The results suggest that between 170 and 180 billion barrels will be ultimately recoverable of which 117 billion barrels have been produced through the end of 1989. Key words: crude oil, natural resources, production planning
Introduction The pervasive attitude prior to the events beginning in October 1973 were that oil resources in the United States were high and unbounded. To be sure, there were those who were aware of the falling rate of production of crude oil and increasing costs. These less optimistic assessments, however, were relegated to the background. The inherent riantness of the explorer with regard to potential discoveries was supported by past experience. The United States never seemed in danger of being anything less than the foremost producer of crude oil in the world. The reduction in exploratory activity was cause for little or no concern for those external to the industry. The prevailing opinion seemed to be that this voiced concern was simply another effort by the industry to elicit favourable treatment by Congress on taxation, incentives and protection from importers. The Arab oil embargo, however, eliminated the absence of public consideration. A group sponsored by the National Academy of Sciences, the Committee on Mineral Resources and the Environment, 1 in 1975 reviewed all of the then existing estimates of total producible domestic crude oil and concluded that of the 249 billion barrels initially existing, only 113 billion barrels remained to be discovered. This result severely tempered the prevailing optimistic view about the future of crude oil production in the United States. To have this assessment appear while the industry was lobbying for increased prices resulted in general public confusion and distrust of the industry. * The author was an economist with the Department of Energy when this paper was written. The views expressed are those of the author and do not necessarily represent the policies of the Department of Energy or the views of other Department of Energy staff members. 0307-904X/82/020119-05/$03.00 © 1982 Butterworth & Co. (Publishers) Ltd.
It is appropriate now to reassess the nature and extent of domestic crude oil resources that remain available for development. The issue is still as topical as it was a few years ago and while the accepted techniques for estimating resource availability are the same they can be combined in such a way as to shed additional light on the situation. In this paper a combination of two standard techniques are used (the third being geologic). One technique for crude oil resource estimation involves an engineering approach and uses projections of drilling, discovery, and production processes to infer the quantity of recoverable oil remaining in the ground. A second technique is an econometric approach and uses a structural model to suggest that future supply can be achieved by producers as they respond to price changes and technological improvements. These two approaches are combined to analyse the issue of interest. Before proceeding it is useful to note that there is little general agreement on the method appropriate for forecasting the availability of crude oil. One group contends ghat demand determines availability. Others hold that geologic criteria determine availability but within this group there is a significant disagreement on the appropriate method of forecasting and on the assumptions which are proper or necessary. The disagreement gap is large because the world has yet to witness its first exhaustion, on a global scale of any nonrenewable resource. Examples exist of the exhaustion of individual oil fields but there is no consensus as to the validity of extrapolating from the production histories of individual deposits and groups of deposits to the total quantity that may ultimately be available for use. There is even disagreement about the value, for planning purposes, of any long-range forecast of availability. The existence of such disagreements, however, should not preclude considering the issue of the estimation
Appl. Math. Modelling, 1982, Vol. 6, April
119
Domestic crude oil resource appraisal: N. D. Uri
resource availability. Whatever the ultimate value in this setting happens to be, it must necessarily be put within the perspective of the methodology employed.
Cumulativediscoveries, O ~>
Background The fact that crude oil production is a process in which production declines and costs increase became apparent soon after the first well was drilled. While there is considerable variability in production, extrapolation of past production of any well or field provides a picture of a downward trend in the absence of new discoveries and technological innovation. In contrast, projections made by individuals showing increasing future production are illustrations of how additional investment in exploration, drilling or applications of new technology can cause the aggregate production of oil to increase in the future, despite the fact that older wells are declining and future efforts will face a greater cost per barrel produced per foot drilled. Most projections of future production are not primarily designed to deal with the ultimate size of oil resources. Nevertheless, they may still foster a belief that resources are adequate in size to meet the projected goals. Additionally, there may only be minimal attention paid to the price required to elicit the necessary investments. Whatever the price may be, the accompanying change in demand for oil, given that price, may not be addressed at all. One specialized form of engineering projection that has been employed is to use a production-history prof'fle that follows the standard pattern observed when minerals or fuels are produced from a finite deposit. The classical configuration is a bell shaped distribution showing an upward trend, a peaking of production, followed by a decline to f'mal depletion. Not only does a specific deposit follow this pattern, but regions and the national aggregate often behave this way as well. Assuming that this is the general behaviour in production, it becomes possible to estimate when the peak will occur, the quantity that will be produced, and how it will be distributed over time. This is done by curve-fitting techniques using an appropriate functional specification. This specification is the subject of the analysis. Before turning to it, however, a brief review of the theoretical justification for the engineering projection methodology is in order. E s t i m a t i n g m a x i m u m r e c o v e r a b l e reserves From the record of annual production, dQp/dt, the cululatire production, Qp, can be obtained. From the value of cumulative production and proved reserves, Qr, for any given year, the cumulative proved discoveries, Qa, are defined by:
Oa = Op + Qr
(1)
That is, the oil whose discovery may be said to have been proved during any given year is the sum of the oil already produced plus the remaining proved reserves. During the complete cycle of production, the curve of the rate of production dQp/dt, begins at zero and then, after passing one or more maxima, ultimately returns to zero. In parallel, the cumulative production curve begins at zero and increases monotonically with time until it finally levels off asymptotically to the ultimate quantity, Q*, indicating total recoverable resources.
120' Appl. Math. Modelling, 1982, Vol. 6, April
/ t ~ /
d
~
/ Cumul producti otoiv0p en,
o
0
Time Variation with time of proved reserves,cumulative production, and cumulative proved discoveries Figure I
The relations between the curves of cumulative production, proved reserves, and cumulative proved discoveries for a single-cycle production lfistory are shown in Figure 1. For a small area, the production curve for the complete cycle may involve more than one major cycle. In a large area (e.g., the aggregate United States), the production-rate curve is the composite of the production from all its components, both old and new. In such an instance, production irregularities at the micro-level tend to cancel out so that for such an area the production history shows every promise of giving a comparatively smooth singlecycle curve. There is a close resemblance between the cumulative discovery curve and the cumulative production curve, except that in the mid-range the discovery curve precedes that of production by a nearly constant time interval At. Because of the similarity between the two curves, the discovery curves give an approximate preview of the behaviour of the production curve by the lead time interval At. That is, one may determine approximately how much oil will be produced At years later by examining what the discovery curve is currently doing. This is demonstrated in Figure 2. A third curve, that of the rate of increase of proved reserves, dQr/dt, is also of interest. There is a positive period representing the interval during which proved reserves are increasing and a negative period during which they are decreasing. The point at which the rate of increase is equal to zero coincides with the intersection between the rate-of-discovery and the rate-of-production curves. This result may be obtained from equation (1) by noting that:
Or = aa - ap the derivatives of which are:
dQr/dt = daa/dt -- dQp/dt
(2)
When proved reserves reach their maximum, the rate of increase of proved reserves is zero. That is:
dQr/dt = 0
and
dQa/dt = dQp/dt
(3)
The curves in Figure 2 give little information about the maghitude of the complete cycle until the maximum value of the rate of increase of the proved reserves is reached.
After that, the three curves taken together give an increasingly accurate estimate of the degree of advancement reached over the cycle at any given time. In particular, after the peak in the rate of discoveries has been reached, the
Appl. Math. Modelling, 1982, Vol. 6, April 123 the Gompertz curve which has a positive skew, will be used. Its cumulative distribution is given by: Qt = Q *ab(t - to)
(6)
The rate of discovery and production between successive periods will be given as: dat - - = gQt(log Q* - l o g at) dt
dOr/dt" ~ / / Figure2
Variation of rates of production, of proved discovery, and of rate of increase of proved reserves
peak in proved reserves may be expected to occur at about At/2, and that in the rate of production at about At later. In order to obtain analytical derivatives for these three curves, it is necessary to fit them to explicit functional rleations. Unfortunately, however, such relations are not theurgically supplied. There is simply no theoretical justification for selecting one specification over another. 2 Nevertheless, there is considerable support for the notion that when the production history of a nation is far enough advanced that it begins to fit something like a logistic curve (as is currently the case in the U.S.), one should abandon the other approaches in favour of the more rigorous exploration-history method (Cook, a p. 145). This is the tactic adopted here. The logistic curve, has been used effectively by others (see Schanz4 for a survey). The form of this equation is a t = a*/(1 + a e - b ( t - t ° ) )
(4)
where Qr denotes the cumulative quantity of crude oil discovered or produced up to period t; t - - t o is the time after a reference to; and Q*, a and b are constants to be estimated. (Note that if to is not chosen arbitrarily, a fourth parameter must be introduced.) As t increases, Q approaches the value Q* asymptotically. Hence, the curve of Q as a function of time begins at zero, rises initially approximately exponentially, then slows down in its growth rate, passes its inflection point, and eventually levels off asymptotically to an ultimate maximum value Q*. The derivative of equation (4) with respect to time, i.e., the rate of production from one year to the next, is just: dOt= cat(Q* - at) dt
(5)
where c = b/Q*. (This is easily demonstrated by taking the derivative of Qt in equation (4) and doing some simple algebraic manipulations.) One of the disadvantages of the logistic specification is that it is symmetric with respect to time. This assumption has been effectively criticized by Mayer et al. s and others. The contention is correctly made that there is no reason to believe that resource discovery and production should be the same on either side of the temporal midpoint. This suggests that other functional forms, in the hope of gaining a truly objective estimate, should be tried. While it is impossible to consider all of these, one that is assymetric is introduced with the goal of comparing the robustness (in the statistical sense) of the resulting estimates of Q*. Thus,
(7)
where g = -- log b (log denoting logarithmic transformation to the base e). With the Gompertz curve, growth in discovery or production rises rapidly to its maximum rate which occurs when actual discovery or production equals 37% of the maximum cumulative level. Thereafter, growth rate at any point above the maximum is greater than that equally distant point below the maximum. 6 (Note that the logistic curve reaches its maximum slope when actual discovery or production reaches 50% of the maximum level.) Either the logistic specification or the Gompertz specification can be used to estimate the maximum discovery or production of crude off in the United States. For comparative purposes, both will be used. Additionally, either the cumulative function or the first derivative of the cumulative function can be estimated. To simplify the estimation the latter form is selected. Before turning to the actual estimation, one additional factor should be considered. There is no reason to suppose that the asymptotic value Q* is static. It is reasonable to suppose that it is a function of price (both current and lagged), technological change, etc. In the case of price, it is a well established fact that price is causally related to the level of production (see e.g., UriC). Technological change is likewise a potentially important factor. One has to look only briefly at the history of crude oil production to note the significant impact technological change has had on secondary and tertiary recovery techniques, a This leads to the suggestion that equation (5) and equation (7) should be reformulated as: --=
1-Qt
~
Otliet_i+ot 2
(5a)
i=o and dt
gQt
~1i logPt_i +/32 log Tt--log Qt (7a) i
where P t - i denotes the well-head price of crude oil in period t -- i; Tt denotes a measure of technological change from period t -- 1 to period t; all ,/31i, (i = 0, 1 , . . . , k), a 2 and/32 are parameters to be estimated; and the other terms are as previously defined. What now becomes the objective is to estimate relations (5a) and (7a) and use the estimates together with asymptotic forecasts of price and technological change to yield estimates of the maximum discovery and production of crude oil in the United States. The only assumption imposed on this process is that the parameter estimates (beyond the Q*) do not change in the future. Estimation technique To estimate equation (7a), ordinary least squares with an adjustment for serial correlation could be employed. This is not feasible for equation (5a), however, because of its
Appl. Math. Modelling, 1982, Vol. 6, April 121
Domestic crude oil resource appraisal: N. D. Uri
nonlinear nature. Consequently, a maximum likelihood approach adjusting for serial correlation is used to estimate both equations to provide an element of consistency. Second, not all of the information contained in the theoretical considerations is used. Specifically, if both the discovery and production curves are estimated, then some efficiency will be gained if the asymptotic values are constrained to be equal subject, of course, to an appropriate hypothesis test. Preliminary results did not show this to be a realistic supposition. As a result, the estimation is done separately for each equation and each model. D a t a and s t r u c t u r e The data on discoveries and production used in the estimation were obtained from the American Gas Association. 9 This is the conventional source. Data on cumulative production are available back to 1920 but reliable data on cumulative proved discoveries only began in 1945. Consequently, for consistency reasons, the years 1945 through 1978 serve as the sample period. Constant dollar (1975) average well-head crude oil price data were obtained from the Department of Energy. z° In introducing price as a measure to induce increased discovery and production since there is not a full immediate response to price variation, various lengths of lag were tried. A two period lag (the current period price and two lagged values (i.e., k = 2 in equations (5a) and (7a))) was judged to be most appropriate on a significance and goodness-of-fit basis. Specification of a good measure of technological change proved elusive. There are no measures of efficiency improvement, drilling, enhanced recovery, etc., generally available. Therefore, it is felt that the impact of technology on discovery and production will eventually exhaust itself. After some preliminary analysis, a measure defined as 1/e ( t - to) was used where t and to are as previously defined and the complete term is just Tt. E m p i r i c a l results The logistic model characterized in equation (5a) and the Gompertz model characterized in equation (7a), were
Table I
estimated in the fashion previously indicated. The results are presented in Table I for the logistic model and in Table 2 for the Gompertz model. Standard errors of the estimates are given in parentheses. To be consistent with the earlier work of Hubbert, z~' ~2 to was taken to be 1900. All of the coefficient estimates for both model specifications are significantly different from zero at the 95% level. Additionally, serial correlation was adjusted for, being initially a problem as indicated by the statistical significance of p. The signs of the coefficient are what one would expect. For example, an increase in price would make exploration of potentially marginal reserves economically feasible, as well as increasing the degree of intensitiveness of production through the use of more expensive techniques. Further, the full impact of a price increase is such that the increase in discovery and production activity is greatest at a lag of one period (i.e., the coefficient is largest). (Note that no a priori restrictions were placed on the lag structures.) Additionally, the results indicate that, technologically, there is a maximum feasible level of reserves and production regardless of the price of crude. The quantity of crude oil available is indeed scarce and will become more so implying that the law of diminishing returns applies to it as well as to technology. Conclusions The present analysis was undertaken in an effort to obtain estimates of cumulative reserves and cumulative production of crude oil in the aggregate United States as a function of price and technological change. What do the results indicate in this regard? If the value of t is allowed to approach infinity (that is, consider the ultimate quantity of recoverable reserves), then the value of Q*, i.e., cumulative reserves and cumulative production in the limit, will approach the value oqoPt + t~,Pt_ l + a l 2 P t - 2 for the the logistic model and the value/31o logPt +/311 log Pt_ l + /3~2 log P r - 2 raised to the power e for the Gompertz model. If it is assumed that the price of crude oil will reach a maximum value of, say $45.43 per barrel, 13 then Q* will approach a value for the logistic model of 175.32 billion
Coefficient estimates for the logistic model (standard errors of estimates are given in parentheses)
2 Production
R2
0.6059
2.17
0.8765
0.6391 (0.2487)
2.30
0.9411
p (serial correlation coefficient)
DurbinWatson statistic
R2
~1o
CEIl
t'Yl2
(~2
1.3620 (0.6875)
1.4107 (0.3404)
1.0646" (0.4552)
0.1722 (0.0897)
(0.2576)
(0.4033)
1.3210
0.2890
(0.0754)
1.4040 (0.7172)
1.0719
(0.1497)
(0.6211 )
(0.1301 )
b 1 Discovery
DurbinWatson statistic
p (serial correlation coefficient)
0.2041 0.1003)
Table 2 Coefficient estimates for the Gompertz model (standard errors of estimates in parentheses)
1 Discovery
0.1882 (0.0899)
0.4717 (0.2171 )
0.5891 (0.2601)
0.3011 (0.1163)
0.1656 (0.0543)
0.6100 (0.2940)
1.92
0.9773
2 Production
0.0796 (0.0223)
0.4011 (0.1893)
0.6129
0.3445 (0.1243)
0.4207 (0.2014)
0.3179 (0.1121)
2.16
0.9~3
(0.3002)
12i
A p p l . Math'. Modelling, 1982, Vol. 6, April
Domestic crude oil resource appraisal: N. D. Uri
barrels for discovery and 172.49 billion barrels for production. Similarly for the Gompertz model the values for discovery and production are 180.73 billion barrels and 188.40 billion barrels, respectively. Within the arena of statistical inference, these estimates are remarkably close. These estimates, however, are somewhat larger than the 170 billion barrels assumed by Hubbert, the 154-178 billion barrels estimated of Mayer et al. s and the 159 billion barrel estimate obtained by Uri. 2 The difference of course, is accounted for by the methodology employed. In each of these previous situations no explicit account is taken of price or technology induced expansions of the resource base. The per capita consumption of crude oil is directly related to the standard of living in the U.S) 4 Energy consumption per capita varies in approximately a linear fashion with per capita income.IS Today we are intensely aware of these dependencies and to the extent that higher prices and technological change will induce an expanded resource base of crude oil, fluctuations in the standard of living due to supply interruptions can be minimized by substituting domestic production for supplies from uncertain foreign sources. The results supported the hypothesis that a higher real price of oil will result in increased aggregate production. The additional production resulting from an increase of the 1978 price o f crude oil of about $8 per barrel to $45 per barrel will ultimately be on the order of 20 to 30 billion barrels. Whether it is advantageous to encourage a price increase o f this order of magnitude, or whether a price of $45 per barrel will ever be reached as a backstop technologies take over, is outside the scope o f the present investigation. What is relevant is the observation that, based on historical performance, cumulative reserves and cumulative production can be expanded with an increase in the price of crude oil. It must be realized in making these projections that past experience is being projected into the future. Insofar as the past does not adequately represent the future as producer and governmental behaviour modulates, then the estimates are likely to be in error. This is particularly problematical in the current situation given the large degree of uncertainty surrounding energy related issues. This should not be an absolute deterrent, however, to making policy inferences. Indeed the implicit price elasticity coming from the logistic model is 0.063 and that coming from the Gompertz model is 0.090. That is, if the domestic price o f crude oil is allowed to expand to the level forecast,
then for each 1% increase in the domestic price, one will observe between a 0.06 and 0.09% increase in the amount of crude oil that can be feasibly produced. Thus, decontrol of the domestic price of crude oil will result in a smaller future dependence on foreign crude oil. Up to the end of 1978, the United States had produced 117 billion barrels from its resource base. Consequently, there are in the neighbourhood of 5 5 - 6 0 billion barrels still in place. If we continue at our current pace of production, this should last for 18 to 20 years. Hopefully, this will prove an adequate length of time to develop and refine alternative technologies.
References 1 Committee on Mineral Resources and the Environment, Mineral Resources and the Environment, National Academy of Science, Washington, D.C., 1975 2 Uri, N. D. 'Two models for estimating undiscovered oil reserves', Appl. Energy, (in press) 3 Cook, E. 'Forecasting depletion', in 'Background readings on energy policy', (Committee on Ways and Means, ed.) U.S. Government Printing Office, Washington, D.C., 1975 4 Schanz, J. J. 'Oil and gas resouxces: welcome to uncertainty', Resources, 1978, 58, 1 5 Mayer, L. S. et al. 'Modeling the rates of domestic crude oil discovery and production:, Department of Statistics, Princeton University, February 1979 6 I_;akhanJ,H. Technol. Forecasting and Social Change, 1975, 7, 33 7 Uri, N. D. 'Price, quantity and causality in the production of crude petroleum in the United States', Energy Sources, (in press)
8 9
Blair,J. M. 'The control of oil', Random House, New York, 1976 American Petroleum Institute, American Gas Association, Canadian Petroleum Association, Reserves o f Crude Oil, Natural Gas Liquids, and Natural Gas in the United States and Canada as of December 31, 1977, American Petroleum
II
Institute, Washington, 1978 Department of Energy, 'Historical review of domestic oil and gas exploration activity', Department of Energy, Washington, D.C., 1978 Hubbert, M. K. Energy Resources, A Report to the Committee
12
on Natural Resources: National Academy o f Science-National Research Council, Pub. 1000-D, Washington, D.C., 1962 Hubbert, M. K. 'U.S. Energy resources, A review as of 1972',
I0
U.S. Government Printing Office, Washington, D.C., 1974 Data Resources, Inc., 'Energy review: Summer 1979:, Data Resources, Inc., Lexington, 1979 14 Uri, N. D. 'Energy, GNP and causality: a statistical look at the issue', Energy Communication , (in press) 15 Daxmstadter, J. et al. 'How industrial societies use energy', Johns Hopkins University Press, Baltimore, MD, 1977 13
Appl. Math. Modelling, 1982, Vol. 6, April 123
Department of Economics and Business, Catholic University of America, Washington DC, 20064, USA (Received December 1979)
This paper endeavours to estimate the cummulative level of discovery and production of crude oil in the United States where the importance of price and technological change is considered. Two separate functional specifications for the cumulative level are hypothesized and estimated. The results suggest that between 170 and 180 billion barrels will be ultimately recoverable of which 117 billion barrels have been produced through the end of 1989. Key words: crude oil, natural resources, production planning
Introduction The pervasive attitude prior to the events beginning in October 1973 were that oil resources in the United States were high and unbounded. To be sure, there were those who were aware of the falling rate of production of crude oil and increasing costs. These less optimistic assessments, however, were relegated to the background. The inherent riantness of the explorer with regard to potential discoveries was supported by past experience. The United States never seemed in danger of being anything less than the foremost producer of crude oil in the world. The reduction in exploratory activity was cause for little or no concern for those external to the industry. The prevailing opinion seemed to be that this voiced concern was simply another effort by the industry to elicit favourable treatment by Congress on taxation, incentives and protection from importers. The Arab oil embargo, however, eliminated the absence of public consideration. A group sponsored by the National Academy of Sciences, the Committee on Mineral Resources and the Environment, 1 in 1975 reviewed all of the then existing estimates of total producible domestic crude oil and concluded that of the 249 billion barrels initially existing, only 113 billion barrels remained to be discovered. This result severely tempered the prevailing optimistic view about the future of crude oil production in the United States. To have this assessment appear while the industry was lobbying for increased prices resulted in general public confusion and distrust of the industry. * The author was an economist with the Department of Energy when this paper was written. The views expressed are those of the author and do not necessarily represent the policies of the Department of Energy or the views of other Department of Energy staff members. 0307-904X/82/020119-05/$03.00 © 1982 Butterworth & Co. (Publishers) Ltd.
It is appropriate now to reassess the nature and extent of domestic crude oil resources that remain available for development. The issue is still as topical as it was a few years ago and while the accepted techniques for estimating resource availability are the same they can be combined in such a way as to shed additional light on the situation. In this paper a combination of two standard techniques are used (the third being geologic). One technique for crude oil resource estimation involves an engineering approach and uses projections of drilling, discovery, and production processes to infer the quantity of recoverable oil remaining in the ground. A second technique is an econometric approach and uses a structural model to suggest that future supply can be achieved by producers as they respond to price changes and technological improvements. These two approaches are combined to analyse the issue of interest. Before proceeding it is useful to note that there is little general agreement on the method appropriate for forecasting the availability of crude oil. One group contends ghat demand determines availability. Others hold that geologic criteria determine availability but within this group there is a significant disagreement on the appropriate method of forecasting and on the assumptions which are proper or necessary. The disagreement gap is large because the world has yet to witness its first exhaustion, on a global scale of any nonrenewable resource. Examples exist of the exhaustion of individual oil fields but there is no consensus as to the validity of extrapolating from the production histories of individual deposits and groups of deposits to the total quantity that may ultimately be available for use. There is even disagreement about the value, for planning purposes, of any long-range forecast of availability. The existence of such disagreements, however, should not preclude considering the issue of the estimation
Appl. Math. Modelling, 1982, Vol. 6, April
119
Domestic crude oil resource appraisal: N. D. Uri
resource availability. Whatever the ultimate value in this setting happens to be, it must necessarily be put within the perspective of the methodology employed.
Cumulativediscoveries, O ~>
Background The fact that crude oil production is a process in which production declines and costs increase became apparent soon after the first well was drilled. While there is considerable variability in production, extrapolation of past production of any well or field provides a picture of a downward trend in the absence of new discoveries and technological innovation. In contrast, projections made by individuals showing increasing future production are illustrations of how additional investment in exploration, drilling or applications of new technology can cause the aggregate production of oil to increase in the future, despite the fact that older wells are declining and future efforts will face a greater cost per barrel produced per foot drilled. Most projections of future production are not primarily designed to deal with the ultimate size of oil resources. Nevertheless, they may still foster a belief that resources are adequate in size to meet the projected goals. Additionally, there may only be minimal attention paid to the price required to elicit the necessary investments. Whatever the price may be, the accompanying change in demand for oil, given that price, may not be addressed at all. One specialized form of engineering projection that has been employed is to use a production-history prof'fle that follows the standard pattern observed when minerals or fuels are produced from a finite deposit. The classical configuration is a bell shaped distribution showing an upward trend, a peaking of production, followed by a decline to f'mal depletion. Not only does a specific deposit follow this pattern, but regions and the national aggregate often behave this way as well. Assuming that this is the general behaviour in production, it becomes possible to estimate when the peak will occur, the quantity that will be produced, and how it will be distributed over time. This is done by curve-fitting techniques using an appropriate functional specification. This specification is the subject of the analysis. Before turning to it, however, a brief review of the theoretical justification for the engineering projection methodology is in order. E s t i m a t i n g m a x i m u m r e c o v e r a b l e reserves From the record of annual production, dQp/dt, the cululatire production, Qp, can be obtained. From the value of cumulative production and proved reserves, Qr, for any given year, the cumulative proved discoveries, Qa, are defined by:
Oa = Op + Qr
(1)
That is, the oil whose discovery may be said to have been proved during any given year is the sum of the oil already produced plus the remaining proved reserves. During the complete cycle of production, the curve of the rate of production dQp/dt, begins at zero and then, after passing one or more maxima, ultimately returns to zero. In parallel, the cumulative production curve begins at zero and increases monotonically with time until it finally levels off asymptotically to the ultimate quantity, Q*, indicating total recoverable resources.
120' Appl. Math. Modelling, 1982, Vol. 6, April
/ t ~ /
d
~
/ Cumul producti otoiv0p en,
o
0
Time Variation with time of proved reserves,cumulative production, and cumulative proved discoveries Figure I
The relations between the curves of cumulative production, proved reserves, and cumulative proved discoveries for a single-cycle production lfistory are shown in Figure 1. For a small area, the production curve for the complete cycle may involve more than one major cycle. In a large area (e.g., the aggregate United States), the production-rate curve is the composite of the production from all its components, both old and new. In such an instance, production irregularities at the micro-level tend to cancel out so that for such an area the production history shows every promise of giving a comparatively smooth singlecycle curve. There is a close resemblance between the cumulative discovery curve and the cumulative production curve, except that in the mid-range the discovery curve precedes that of production by a nearly constant time interval At. Because of the similarity between the two curves, the discovery curves give an approximate preview of the behaviour of the production curve by the lead time interval At. That is, one may determine approximately how much oil will be produced At years later by examining what the discovery curve is currently doing. This is demonstrated in Figure 2. A third curve, that of the rate of increase of proved reserves, dQr/dt, is also of interest. There is a positive period representing the interval during which proved reserves are increasing and a negative period during which they are decreasing. The point at which the rate of increase is equal to zero coincides with the intersection between the rate-of-discovery and the rate-of-production curves. This result may be obtained from equation (1) by noting that:
Or = aa - ap the derivatives of which are:
dQr/dt = daa/dt -- dQp/dt
(2)
When proved reserves reach their maximum, the rate of increase of proved reserves is zero. That is:
dQr/dt = 0
and
dQa/dt = dQp/dt
(3)
The curves in Figure 2 give little information about the maghitude of the complete cycle until the maximum value of the rate of increase of the proved reserves is reached.
After that, the three curves taken together give an increasingly accurate estimate of the degree of advancement reached over the cycle at any given time. In particular, after the peak in the rate of discoveries has been reached, the
Appl. Math. Modelling, 1982, Vol. 6, April 123 the Gompertz curve which has a positive skew, will be used. Its cumulative distribution is given by: Qt = Q *ab(t - to)
(6)
The rate of discovery and production between successive periods will be given as: dat - - = gQt(log Q* - l o g at) dt
dOr/dt" ~ / / Figure2
Variation of rates of production, of proved discovery, and of rate of increase of proved reserves
peak in proved reserves may be expected to occur at about At/2, and that in the rate of production at about At later. In order to obtain analytical derivatives for these three curves, it is necessary to fit them to explicit functional rleations. Unfortunately, however, such relations are not theurgically supplied. There is simply no theoretical justification for selecting one specification over another. 2 Nevertheless, there is considerable support for the notion that when the production history of a nation is far enough advanced that it begins to fit something like a logistic curve (as is currently the case in the U.S.), one should abandon the other approaches in favour of the more rigorous exploration-history method (Cook, a p. 145). This is the tactic adopted here. The logistic curve, has been used effectively by others (see Schanz4 for a survey). The form of this equation is a t = a*/(1 + a e - b ( t - t ° ) )
(4)
where Qr denotes the cumulative quantity of crude oil discovered or produced up to period t; t - - t o is the time after a reference to; and Q*, a and b are constants to be estimated. (Note that if to is not chosen arbitrarily, a fourth parameter must be introduced.) As t increases, Q approaches the value Q* asymptotically. Hence, the curve of Q as a function of time begins at zero, rises initially approximately exponentially, then slows down in its growth rate, passes its inflection point, and eventually levels off asymptotically to an ultimate maximum value Q*. The derivative of equation (4) with respect to time, i.e., the rate of production from one year to the next, is just: dOt= cat(Q* - at) dt
(5)
where c = b/Q*. (This is easily demonstrated by taking the derivative of Qt in equation (4) and doing some simple algebraic manipulations.) One of the disadvantages of the logistic specification is that it is symmetric with respect to time. This assumption has been effectively criticized by Mayer et al. s and others. The contention is correctly made that there is no reason to believe that resource discovery and production should be the same on either side of the temporal midpoint. This suggests that other functional forms, in the hope of gaining a truly objective estimate, should be tried. While it is impossible to consider all of these, one that is assymetric is introduced with the goal of comparing the robustness (in the statistical sense) of the resulting estimates of Q*. Thus,
(7)
where g = -- log b (log denoting logarithmic transformation to the base e). With the Gompertz curve, growth in discovery or production rises rapidly to its maximum rate which occurs when actual discovery or production equals 37% of the maximum cumulative level. Thereafter, growth rate at any point above the maximum is greater than that equally distant point below the maximum. 6 (Note that the logistic curve reaches its maximum slope when actual discovery or production reaches 50% of the maximum level.) Either the logistic specification or the Gompertz specification can be used to estimate the maximum discovery or production of crude off in the United States. For comparative purposes, both will be used. Additionally, either the cumulative function or the first derivative of the cumulative function can be estimated. To simplify the estimation the latter form is selected. Before turning to the actual estimation, one additional factor should be considered. There is no reason to suppose that the asymptotic value Q* is static. It is reasonable to suppose that it is a function of price (both current and lagged), technological change, etc. In the case of price, it is a well established fact that price is causally related to the level of production (see e.g., UriC). Technological change is likewise a potentially important factor. One has to look only briefly at the history of crude oil production to note the significant impact technological change has had on secondary and tertiary recovery techniques, a This leads to the suggestion that equation (5) and equation (7) should be reformulated as: --=
1-Qt
~
Otliet_i+ot 2
(5a)
i=o and dt
gQt
~1i logPt_i +/32 log Tt--log Qt (7a) i
where P t - i denotes the well-head price of crude oil in period t -- i; Tt denotes a measure of technological change from period t -- 1 to period t; all ,/31i, (i = 0, 1 , . . . , k), a 2 and/32 are parameters to be estimated; and the other terms are as previously defined. What now becomes the objective is to estimate relations (5a) and (7a) and use the estimates together with asymptotic forecasts of price and technological change to yield estimates of the maximum discovery and production of crude oil in the United States. The only assumption imposed on this process is that the parameter estimates (beyond the Q*) do not change in the future. Estimation technique To estimate equation (7a), ordinary least squares with an adjustment for serial correlation could be employed. This is not feasible for equation (5a), however, because of its
Appl. Math. Modelling, 1982, Vol. 6, April 121
Domestic crude oil resource appraisal: N. D. Uri
nonlinear nature. Consequently, a maximum likelihood approach adjusting for serial correlation is used to estimate both equations to provide an element of consistency. Second, not all of the information contained in the theoretical considerations is used. Specifically, if both the discovery and production curves are estimated, then some efficiency will be gained if the asymptotic values are constrained to be equal subject, of course, to an appropriate hypothesis test. Preliminary results did not show this to be a realistic supposition. As a result, the estimation is done separately for each equation and each model. D a t a and s t r u c t u r e The data on discoveries and production used in the estimation were obtained from the American Gas Association. 9 This is the conventional source. Data on cumulative production are available back to 1920 but reliable data on cumulative proved discoveries only began in 1945. Consequently, for consistency reasons, the years 1945 through 1978 serve as the sample period. Constant dollar (1975) average well-head crude oil price data were obtained from the Department of Energy. z° In introducing price as a measure to induce increased discovery and production since there is not a full immediate response to price variation, various lengths of lag were tried. A two period lag (the current period price and two lagged values (i.e., k = 2 in equations (5a) and (7a))) was judged to be most appropriate on a significance and goodness-of-fit basis. Specification of a good measure of technological change proved elusive. There are no measures of efficiency improvement, drilling, enhanced recovery, etc., generally available. Therefore, it is felt that the impact of technology on discovery and production will eventually exhaust itself. After some preliminary analysis, a measure defined as 1/e ( t - to) was used where t and to are as previously defined and the complete term is just Tt. E m p i r i c a l results The logistic model characterized in equation (5a) and the Gompertz model characterized in equation (7a), were
Table I
estimated in the fashion previously indicated. The results are presented in Table I for the logistic model and in Table 2 for the Gompertz model. Standard errors of the estimates are given in parentheses. To be consistent with the earlier work of Hubbert, z~' ~2 to was taken to be 1900. All of the coefficient estimates for both model specifications are significantly different from zero at the 95% level. Additionally, serial correlation was adjusted for, being initially a problem as indicated by the statistical significance of p. The signs of the coefficient are what one would expect. For example, an increase in price would make exploration of potentially marginal reserves economically feasible, as well as increasing the degree of intensitiveness of production through the use of more expensive techniques. Further, the full impact of a price increase is such that the increase in discovery and production activity is greatest at a lag of one period (i.e., the coefficient is largest). (Note that no a priori restrictions were placed on the lag structures.) Additionally, the results indicate that, technologically, there is a maximum feasible level of reserves and production regardless of the price of crude. The quantity of crude oil available is indeed scarce and will become more so implying that the law of diminishing returns applies to it as well as to technology. Conclusions The present analysis was undertaken in an effort to obtain estimates of cumulative reserves and cumulative production of crude oil in the aggregate United States as a function of price and technological change. What do the results indicate in this regard? If the value of t is allowed to approach infinity (that is, consider the ultimate quantity of recoverable reserves), then the value of Q*, i.e., cumulative reserves and cumulative production in the limit, will approach the value oqoPt + t~,Pt_ l + a l 2 P t - 2 for the the logistic model and the value/31o logPt +/311 log Pt_ l + /3~2 log P r - 2 raised to the power e for the Gompertz model. If it is assumed that the price of crude oil will reach a maximum value of, say $45.43 per barrel, 13 then Q* will approach a value for the logistic model of 175.32 billion
Coefficient estimates for the logistic model (standard errors of estimates are given in parentheses)
2 Production
R2
0.6059
2.17
0.8765
0.6391 (0.2487)
2.30
0.9411
p (serial correlation coefficient)
DurbinWatson statistic
R2
~1o
CEIl
t'Yl2
(~2
1.3620 (0.6875)
1.4107 (0.3404)
1.0646" (0.4552)
0.1722 (0.0897)
(0.2576)
(0.4033)
1.3210
0.2890
(0.0754)
1.4040 (0.7172)
1.0719
(0.1497)
(0.6211 )
(0.1301 )
b 1 Discovery
DurbinWatson statistic
p (serial correlation coefficient)
0.2041 0.1003)
Table 2 Coefficient estimates for the Gompertz model (standard errors of estimates in parentheses)
1 Discovery
0.1882 (0.0899)
0.4717 (0.2171 )
0.5891 (0.2601)
0.3011 (0.1163)
0.1656 (0.0543)
0.6100 (0.2940)
1.92
0.9773
2 Production
0.0796 (0.0223)
0.4011 (0.1893)
0.6129
0.3445 (0.1243)
0.4207 (0.2014)
0.3179 (0.1121)
2.16
0.9~3
(0.3002)
12i
A p p l . Math'. Modelling, 1982, Vol. 6, April
Domestic crude oil resource appraisal: N. D. Uri
barrels for discovery and 172.49 billion barrels for production. Similarly for the Gompertz model the values for discovery and production are 180.73 billion barrels and 188.40 billion barrels, respectively. Within the arena of statistical inference, these estimates are remarkably close. These estimates, however, are somewhat larger than the 170 billion barrels assumed by Hubbert, the 154-178 billion barrels estimated of Mayer et al. s and the 159 billion barrel estimate obtained by Uri. 2 The difference of course, is accounted for by the methodology employed. In each of these previous situations no explicit account is taken of price or technology induced expansions of the resource base. The per capita consumption of crude oil is directly related to the standard of living in the U.S) 4 Energy consumption per capita varies in approximately a linear fashion with per capita income.IS Today we are intensely aware of these dependencies and to the extent that higher prices and technological change will induce an expanded resource base of crude oil, fluctuations in the standard of living due to supply interruptions can be minimized by substituting domestic production for supplies from uncertain foreign sources. The results supported the hypothesis that a higher real price of oil will result in increased aggregate production. The additional production resulting from an increase of the 1978 price o f crude oil of about $8 per barrel to $45 per barrel will ultimately be on the order of 20 to 30 billion barrels. Whether it is advantageous to encourage a price increase o f this order of magnitude, or whether a price of $45 per barrel will ever be reached as a backstop technologies take over, is outside the scope o f the present investigation. What is relevant is the observation that, based on historical performance, cumulative reserves and cumulative production can be expanded with an increase in the price of crude oil. It must be realized in making these projections that past experience is being projected into the future. Insofar as the past does not adequately represent the future as producer and governmental behaviour modulates, then the estimates are likely to be in error. This is particularly problematical in the current situation given the large degree of uncertainty surrounding energy related issues. This should not be an absolute deterrent, however, to making policy inferences. Indeed the implicit price elasticity coming from the logistic model is 0.063 and that coming from the Gompertz model is 0.090. That is, if the domestic price o f crude oil is allowed to expand to the level forecast,
then for each 1% increase in the domestic price, one will observe between a 0.06 and 0.09% increase in the amount of crude oil that can be feasibly produced. Thus, decontrol of the domestic price of crude oil will result in a smaller future dependence on foreign crude oil. Up to the end of 1978, the United States had produced 117 billion barrels from its resource base. Consequently, there are in the neighbourhood of 5 5 - 6 0 billion barrels still in place. If we continue at our current pace of production, this should last for 18 to 20 years. Hopefully, this will prove an adequate length of time to develop and refine alternative technologies.
References 1 Committee on Mineral Resources and the Environment, Mineral Resources and the Environment, National Academy of Science, Washington, D.C., 1975 2 Uri, N. D. 'Two models for estimating undiscovered oil reserves', Appl. Energy, (in press) 3 Cook, E. 'Forecasting depletion', in 'Background readings on energy policy', (Committee on Ways and Means, ed.) U.S. Government Printing Office, Washington, D.C., 1975 4 Schanz, J. J. 'Oil and gas resouxces: welcome to uncertainty', Resources, 1978, 58, 1 5 Mayer, L. S. et al. 'Modeling the rates of domestic crude oil discovery and production:, Department of Statistics, Princeton University, February 1979 6 I_;akhanJ,H. Technol. Forecasting and Social Change, 1975, 7, 33 7 Uri, N. D. 'Price, quantity and causality in the production of crude petroleum in the United States', Energy Sources, (in press)
8 9
Blair,J. M. 'The control of oil', Random House, New York, 1976 American Petroleum Institute, American Gas Association, Canadian Petroleum Association, Reserves o f Crude Oil, Natural Gas Liquids, and Natural Gas in the United States and Canada as of December 31, 1977, American Petroleum
II
Institute, Washington, 1978 Department of Energy, 'Historical review of domestic oil and gas exploration activity', Department of Energy, Washington, D.C., 1978 Hubbert, M. K. Energy Resources, A Report to the Committee
12
on Natural Resources: National Academy o f Science-National Research Council, Pub. 1000-D, Washington, D.C., 1962 Hubbert, M. K. 'U.S. Energy resources, A review as of 1972',
I0
U.S. Government Printing Office, Washington, D.C., 1974 Data Resources, Inc., 'Energy review: Summer 1979:, Data Resources, Inc., Lexington, 1979 14 Uri, N. D. 'Energy, GNP and causality: a statistical look at the issue', Energy Communication , (in press) 15 Daxmstadter, J. et al. 'How industrial societies use energy', Johns Hopkins University Press, Baltimore, MD, 1977 13
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