Resource Appraisal and the Effect of Price and Technology Noel D. Uri Office of Energy Use Analysis/EIA, Department of Energy, Washington, D.C. 20461
This paper endeavors to estimate the cumulative level of discovery and production of crude oil in the United States. Two functional specifications are suggested and estimated. The results indicate that between 193 and 203 billion barrels will ultimately be recoverable, of which 117 billion barrels have been Droduced through 1978. The energy problem confronting the United States is quite succinctly put in the National Energy Plan ( I ) : The demand for energy is increasing, while the supplies of crude oil and natural gas are diminishing. This observation serves to highlight the need for accurate resource estimates of the amount of crude oil reserves that are producible. Such estimates will indicate the extent to which energy supply will be dependent on imports with the consequent implications for national economic security and the American lifestyle (some analysts have suggested we will run out of domestic crude oil by the mid- 1980’s). This paper is concerned with this issue of assessing the nature and extent of domestic crude oil resources that remain available for development using a combination of two standard techniques. One technique for crude oil resource estimation involves an engineering approach and uses projections of drilling, discovery, and production processes t o infer the quantity of recoverable oil remaining in the ground. A second technique is an econometric approach and uses a structural model to suggest what future supply can be achieved by producers as they respond to price changes and technological improvements. These two approaches are combined to analyze the issue of interest.
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 technology. 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 t o increase in the future, despite the fact that older wells are declining and future efforts will face 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 be only 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 a t all.
One specialized form of engineering projection that has been employed is to use a production-history profile 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 final 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 behavior 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. Estimating Maximum Recoverable Reserves From the record of annual production, dQp/dt, the cumulative production, Qp, can be obtained. From the value of cumulative production and proved reserves, Qr, for any given year, the cumulative proved discoveries, Qd, are defined by: Qd
=
Qp
-I-Q r
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 dQpldt begins a t zero and then, after passing one or more maxima, ultimately returns to zero. In parallel, the cumulative production curve begins a t zero and increases monotonically with time until it finally levels off asymptotically to the ultimate quantity, Q*, indicating total recoverable resources. The relations between the curves of cumulative production, proved reserves, and cumulative proved discoveries for a single-cycle production history 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 single-cycle 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 a t . Because of the similarity between the two curves, the discovery curves give an approximate preview of the behavior of the production curve by the lead time interval 4 t . That is, one may determine
This article not subject to US. Copyright. Published 1980 American Chemical Society
Volume 14, Number 3, March 1980
283
Figure 1. Variation with time of proved reserves, cumulative production, and cumulative proved discoveries
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, dQrldt, 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 a t 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: Qr
= Qd
dQr/dt = dQd/dt
- dQp/dt
(2)
When proved reserves reach their maximum, the rate of increase of proved reserves is zero. T h a t is:
The curves in Figure 2 give little information about the magnitude 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 a t any given time. In particular, after the peak in the rate of discoveries has been reached, the peak in proved reserves may be expected to occur at about At/& and that in the rate of production a t about At later. In order to obtain analytical derivatives for the three curves, it is necessary to fit them to explicit functional relations. One such relation, the logistic curve, has been used effectively by others (Schanz (2)). The form of this equation is:
+ ae-b(t-fO))
(4)
where Qt denotes the cumulative quantity of crude oil discovered or produced up to period t ; t - t o is the time after a reference t o ; and Q*, a , and b are constants to be estimated. As t increases, Q approaches the value Q* asymptotically. Hence, the curve of Q as a function of time begins at zero, rises initially exponentially, then slows down in its growth rate, passes its inflection point, and eventually levels off asymptotically to a n ultimate maximum value Q*. The derivative of Equation 4 with respect to time, Le., the rate of production from 1 year to the next, is just:
where c = b/Q*. (This is easily demonstrated by taking the derivative of Qt in Equation 4 and doing some simple algebraic manipulations.) 284
One of the disadvantages of the logistic specification is the fact that it is symmetric with respect to time. This assumption has been effectively criticized by Deffeyes, Mayer, and Watson (3) and others. The contention is correctly made that there is just no reason to believe that resource discovery and production should be the same on either side of the temporal midpoint. Consequently, a functional form which is asymmetric is introduced with the objective of comparing the robustness of the resulting estimates of Q*. Thus, the Gompertz curve which has a positive skew will be used. Its cumulative distribution is given by:
- Qp
the time derivatives of which are:
Qt = Q*/(1
Figure 2. Variation of rates of production, of proved discovery, and of rate of increase of Droved reserves
Environmental Science & Technology
where the terms are as previously defined. The rate of discovery and production between successive periods will be given as:
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 (Lakhani ( 4 ) ) .Note that the logistic curve reaches its maximum when actual discovery or production reaches 50% of the maximum level. Either the logistic specification or the Gompertz specification can be used t o estimate the maximum discovery or production of crude oil 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. T o 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. I t is reasonable to suppose that it is a function of price (in both the current and previous periods), 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., Uri ( 5 ) ) .Technological change is likewise a potentially important factor. One has t o look only briefly a t the history of crude oil production to note the significant impact technological change has had on secondary and tertiary recovery techniques (Blair (6)). This leads to the suggestion that Equations 5 and 7 should be reformulated as:
im
3
140 -
120 -
loo 80-
60-
40-
....
0
.*
20..*.e
.*
.
.*
.. ...
.*
. .'
.. * **
..***
.
.*
.e'
**
e**
e**
**.**.*e******** ~
1900
1910
I
I
I
I
1920
1033
1940
1950
I
I
1960
1970
1980
Year
Figure 3.
Historical cumulative discoveries
Billions of Barrels
160
140
120
100
80
60
40
l! 0 Year
Figure 4.
Historical cumulative production
and
is that the parameter estimates (beyond the Q*) do not change in the future. Estimation Technique
where Pt-, denotes the well-head price of crude oil in the period t - i; Tt denotes a measure of technological change from period t - 1 to period t; X I , Xz,Y1,and Yz 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 to use the estimates together with asymptotic forecasts of price and technological change to yield an estimate of the maximum discovery or production of crude oil in the United States. The only assumption imposed on this process
T o 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 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 Volume 14, Number 3, March 1980
285
8
7-
6-
5-
4 1
*
*
0.0
3-
* * 2-
e
* .. *
**
* .
0.0
1-
***
** ***e****
e*
* * ** e
0 . 0
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3.0-
*
0
.
:*e* **
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**
2.0-
em
8 1.5-
J*
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ar
.5-
0 1860
,&
1880
1900
1920
I 1940
I
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1960
1980
I 2000
2
!O
Year
Figure 6. Historical production rate
show this to be a realistic supposition. As a result, the estimation is done separately for each equation and each model.
Data and Structure The data on discoveries and production used in the estimation were obtained from the American Petroleum Institute, American Gas Association, and Canadian Petroleum Institute ( 7 ) . 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. T o give a better appreciation of how the data relate to the suggested theoretical functional specifications, historical data on cumulative discoveries, cu286
Environmental Science & Technology
mulative production, the rate of discovery, and the rate of production are given in Figures 3 through 6, respectively. Constant dollar (1975) average well-head crude oil price data were obtained from the Department of Energy (8). 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 twoperiod lag (Le., the current period price and two lagged values and 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, since it is felt that the impact of technology on discovery and production will eventually exhaust
Table 1. Coefficient Estimates for the Logistic Model a 1. discovery 2. production a
b
XI1
XI2
XI3
X 2
Pb
0.2346 (0.0112) 0.6799 (0.2696)
2.1532 (0.8987) 2.2368 (1.2256)
3.0033 (0.3 298) 2.6720 (0.95 74)
1.8046 (1.0327) 2.0939 (0.8241)
0.1531 (0.0037) 0.3323 (0.0151)
0.8090 (0.1152) 0.8295 (0.1095)
D.W.
R2
2.34
0.8817
2.13
0.9546
Standard errors of estimates in parentheses. p denotes the serial correlation coefficient. Durbin-Watson statistic.
Table II. Coefficient Estimates for the Gompertz Model a 9
1. discovery 2. production a
0.2566 (0.0 133) 0.1780 (0.1016)
Y11
0.4823 (0.2041) 0.5793 (0.1913)
Y21
0.9362 (0.3546) 0.7642 (0.3 6 75)
Y31
Y2
Pb
0.1397 (0.06 19) 0.2201 (0.1024)
0.3971 (0.0337) 0.521 1 (0.179 1)
0.8352 (0.1079) 0.0356 (0.0 180)
D.W.
R2
1.92
0.9773
2.07
0.9217
Standard errors of estiamtes in parentheses. p denotes the serial correlation coefficient. Durbin-Watson statistic.
itself, the measure defined as l/e(t-to) was used, where t and to are as previously defined and the complete term is just Tt . Empirical Results The logistic model characterized in Equation 5a and the Gompertz model characterized in Equation 7a were estimated in the fashion previously indicated. The results are presented in Table I for the logistic model and in Table I1 for the Gompertz model, with both fitting the data well as indicated by the high R2.Standard errors of the estimates are given in parentheses. T o be consistent with the earlier work of Hubbert (9, IO), 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 Durbin-Watson statistic testing for the existence of serial correlation after the adjustment indicates its absence.) The signs of the coefficients are what one would a priori expect. For example, an increase in price would make exploration of potentially marginal reserves economically feasible, as well as increase the degree of intensiveness of production through the use of more expensive techniques. Additionally, the results indicate that, technologically, there is a maximum feasible level of reserves and production given the price of crude. Conclusions The foregoing 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 o f t is allowed to approach infinity, then the value of Q*, Le., cumulative reserves and cumulative production in the limit, will approach the value X l l P t X21Pt-1 X31Pt-2 for the logistic model and the value YI1 log Pt + Ys1 log Pt-l Y31 log Pt-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, $29 per barrel (Kilgore and Rodekohr ( I I ) ) , the Q* will approach a value for the logistic model of 189.97 billion barrels for discovery and 193.47 billion barrels for production. Similarly, for the Gompertz model the values for discovery and production are 201.91 and 203.08 billion barrels, respectively. This is somewhat larger than the 170 billion barrels assumed by Hubbert (9) and the 154-178 billion barrels estimate of Mayer e t al. ( 3 ) .The difference, of course, is accounted for by the methodology employed. (Note that through the end of 1978,117 billion barrels of crude oil have been produced.) The results support 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 of crude oil of about $8 per barrel to $29 per barrel will ultimately be on the order of 20 to 30 billion barrels. Whether it is advantageous to encourage a price increase of this order of magnitude, or whether a price of $29 per barrel will ever be reached as backstop technologies take over, is outside the scope of 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. As a final observation, there is nothing in the results to suggest the superiority of the Gompertz model specification over the logistic approach. Consequently, there is no reason to more readily accept the 193 billion barrel estimate over the 203 billion barrel estimate. The results differ because the model specifications differ. Model specifications different than the logistic and Gompertz would in all likelihood produce different results (in a statistical sense). The essence of the results obtained here, then, is that regardless of the model specified, all such estimates reflecting price and technological effects will be strictly greater than the estimates derived from a pure engineering approach. The variation in the estimates here simply outlines the range of inherent uncertainty. Acknowledgment The author thanks Joanne Walter for excellent secretarial assistance. Literature Cited (1) Executive Office of the President “ T h e National Energy Plan”; US.Government Printing Office: Washington, D.C., 1977. (2) Schanz, J. J. Resources 1978,58, 1-20. (3) Deffeyes, K. S.; Mayer, L. S.; Watson, G. S.“Modelling the Rates of Domestic Crude Oil Discovery and Production”; Department of Statistics, Princeton University: Princeton, N.J., February 1979. (4) Lakhani, H. Technol. Forecasting Social Change 1975, 7, 3355. (5) Uri, N. D., manuscript in preparation. (6) Blair, J. M . “ T h e Control of Oil”; Random House: New York, 1976. ( 7 ) American Petroleum Institute, American Gas Association, Canadian Petroleum Association “Reserves of Crude Oil, Natural Gas Liquids, and Natural Gas in the United States and Canada as of December 31,1977”; American Petroleum Institute: Washington, D.C., 1978. ( 8 ) Department of Energy “Historical Review of Domestic Oil and Gas Exploration Activity”; Department of Energy: Washington, D.C., 1978. (9) Hubbert, M. K. NAS-NRC Publ. 1962, No. 1000-D. (10) Hubbert, M. K. “U.S. Energy Resources, A Review as of 1972”; US.Government Printing Office: Washington, D.C., 1974. (11) Kilgore, C.; Rodekohr, M. “An Evaluation of World Oil Prices”; Department of Energy: Washington, D.C., May 1978.
Received for reoieu M a l 18, 1979. Accepted Nocember 26, 1979 Volume 14, Number 3, March 1980
287
