Peak Oil: Testing Hubbert’s Curve via Theoretical Modeling

Abstract

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

Appendices

Appendix A: The Rate Difference

The rate difference, Δr, as defined by (Brandt, 2007) is

$$ \Updelta r = r_{\rm inc}-r_{\rm dec}, $$

where the rate of increase r _inc and rate of decrease r _dec are determined by fitting Equation (A.1) to the production data (Brandt, 2007).

$$ P^{\prime}(t) = \left\{ \begin{array}{lll} e^{r_{{\rm inc}}(t-T_{\rm start})} & \hbox{if} & t \leq T_{\rm peak} \\ P^{\prime}(T_{\rm peak})e^{-r_{\rm dec}(t-T_{\rm peak})} & \hbox{if} & t > T_{\rm peak} \end{array} \right. $$

(A.1)

where P′(t) is production in (barrels/year) and T _peak is the year the production peaks (year) and r _inc and r _dec are the rate constants (year⁻¹) and T _start is the year production was 1 barrel a year Brandt (2007). In calculating the rate difference of the theoretical model equation (A.1) was altered to

$$ P^{\prime}(t) = \left\{ \begin{array}{lll} P(40)e^{r_{{\rm inc}}(t-40)} & \hbox{if} & t \leq T_{\rm peak} \\ P^{\prime}(T_{\rm peak})e^{-r_{\rm dec}(t-T_{\rm peak})} & \hbox{if} & t > T_{\rm peak} \end{array} \right. $$

(A.2)

where P(40) is the production of oil estimated by the theoretical model in the year 1900. Using Equation (A.2) the rate difference for the pessimistic case was 0.0184 (years⁻¹), optimistic case was 0.0179 (years⁻¹), and the ideal case was 0.0217 (years⁻¹).

Appendix B: Coefficient of Determination

The coefficient of determination, R ², was used to measure the accuracy of the discovery model to the data.

$$ R^2=\sum_{n}\frac{[y_f(n)-\bar{y}_{a}]^2-[y_f(n)-y_a(n)]^2} {[y_f(n)-\bar{y}_{a}]^2} $$

For the pessimistic case URR = 2 trillion barrels and for the optimistic case URR = 3 trillion barrels. For the ideal case, the URR was a variable. The best fit was found by varying b _t, t _t (and URR for ideal case) to obtain the highest R ² value. The constants are shown in Table B.1. The actual data for the pessimistic and optimistic cases was truncated to the year 1966, as the optimists claim that oil reserves found in the past will grow.

Table B.1 The URR, b _t, and t _t for the 3 Cases

Appendix C: Determining the Constants

The method to determine valid estimates for the constants $k_{\rm w},\, w_{l_{\rm T}},$ and P ₀ in the reservoir production model is given in this section. The best data found is unfortunately state based rather than reservoir based data from EIA (2007). The production model was used with several cycles to model the production as a function of time, and the number of wells as a function of cumulative production for various states as shown in Figs. C.1–C.4.

Figure C.1

(a) The number of wells as a function of cumulative production for Alaska and (b) Production as a function of time for Alaska

Figure C.2

(a) The number of wells as a function of cumulative production for Alabama and (b) Production as a function of time for Alabama

Figure C.3

(a) The number of wells as a function of cumulative production for Nevada and (b) Production as a function of time for Nevada

Figure C.4

(a) The number of wells as a function of cumulative production for South Dakota and (b) Production as a function of time for South Dakota

Note our model assumes that no wells are shut down, and instead exponentially decay and although are still on-line, are in reality producing no significant quantity of oil. This is the reason for the poor fit of the well model for Nevada and South Dakota. Now observe that there is only one sensible option for the $w_{l_{\rm T}}$ constants, since these values need to match the actual total wells. The k _w values and P ₀ values determine the rate of increase in the wells model and are determined by trial and error so that the wells model and production model fit the data as accurately as possible. Whilst the values used produce reasonably accurate results, we need to check that the initial production values P ₀ correspond to the actual initial production. Unfortunately the initial production for all the wells is not known; however, the number of wells as a function of size and time is known (EIA 2007) and the model’s predictions were compared with the actual data for the four states, as shown in Figs. C.5–C.8 (Note that the size of the wells from EIA, 2007 is explained in Table C.1).

Table C.1 The Size of Wells from EIA (2007)

Figure C.5

The number of wells as a function of size and time for Alaska. (a) Actual and (b) Model

Figure C.6

The number of wells as a function of size and time for Alabama. (a) Actual and (b) Model

Figure C.7

The number of wells as a function of size and time for Nevada. (a) Actual and (b) Model

Figure C.8

The number of wells as a function of size and time for South Dakota. (a) Actual and (b) Model

The Figs. C.5–C.8 indicate a reasonable fit and hence the initial well productions P ₀ can be assumed to be reasonable estimates. The constants k _w, P ₀, and $w_{l_{\rm T}}$ used in the state models are shown in Table C.2.

Table C.2 The Constants for Various States

Now if we ignore the outlier of 32 for South Dakota, the average for k _w is 10.7 and this value is assumed to be constant in the world model; including the outlier the average becomes 12. By plotting w _T vs. URR _l /P ₀ we obtained the linear relation $w_{l_{\rm T}} = 0.072 URR_{l}/P_0$ which is shown in Fig. C.9. The linear relationship is expected, as increasing the size of the reservoir would increase the total number of wells needed. Equally if we have two reservoirs of the same size we could either have a small number of wells with a large initial production P ₀ or a large number of wells with a small initial production P ₀. Hence the linear relationship between $w_{l_{\rm T}}$ and URR _l /P ₀ was expected.

Figure C.9

$w_{l_{\rm T}}$ versus URR _l /P ₀

The value for P ₀ appears to have a great deal of variability. However, by analyzing the other U.S. state wells sizes from EIA (2007), we observe that Alaska along with Federal Pacific and Federal Gulf, have abnormally large wells compared to the rest of the U.S., we hence considered the Alaskan well production data as an outlier and ignored the data. Taking the initial production from Nevada, South Dakota, and Alabama, we obtain an average of 18.3 ML/year, which places it in category 16 in the EIA sizes. Hence we assumed that the values of the constants were P ₀ = 18.3 ML/year, k _w = 10.7, and $w_{l_{\rm T}}=0.072 URR_{l}/P_0.$

Appendix D: Amended Model

The amended model $C_{{\rm p}_{\rm mod}}(t)$ is determined from the method explained in Mohr and Evans (2007), and formally is:

$$ C_{{\rm p}_{\rm mod}}(t)=\left\{ \begin{array}{lll} C_{\rm p}(t) & \hbox{if} & t\leq 118 \\ f_1(t) & \hbox{if} & 118 < t\leq 123 \\ f_2(t) & \hbox{if} & 123 < t \leq 129 \\ f_3(t) & \hbox{if} & 129 < t\leq 145 \\ f_4(t) & \hbox{if} & 145 < t\leq t_2 \\ C_{\rm p}(t+(t_1-t_2)) & \hbox{if} & t_2 < t \end{array} \right. . $$

Now f ₁(t), f ₂(t), and f ₃(t) are small polynomials fitted to the production data using least squares method, and formally are:

$$ \begin{aligned}f_1(t) & = -0.82t+121.7 \\ f_2(t)& =0.43t-32.3\\ f_3(t)& = 0.007t^2-1.6t+109.5 \end{aligned} $$

the f ₄(t) is a 3rd degree polynomial. The polynomial was determined by the literature method explained generally in Mohr and Evans (2007). Specifically f ₄(t) is the 3rd degree polynomial such that the following equations are solved:

$$ \begin{aligned} f_4(145) &=p(145)\\ f'_4(145) & = p'(145)\\ f''_4(145) & = p''(145)\\ f_4(t_2) &=C_{\rm p}(t_1)\\ f'_4(t_2) &=C'_{\rm p}(t_1)\\ \int\limits_{t_0}^{t_1}{C_{\rm p}(t)dt} &=\int\limits_{145}^{t_2}{p(t)dt}. \end{aligned} $$

where t ₀ ≈ 138 and p(t) is a polynomial to replicate the long-term historic trend and is p(t) = −0.004t ² + 1.5t−104.6. With the list of equations solved, we obtain t ₁ = 153.5, t ₂ = 162, and f ₄(t) = −0.0012t ³ + 0.51t ² − 73.4t + 3515.6 for the ideal case. For the pessimistic case it was t ₁ = 147.2, t ₂ = 153.6, and f ₄(t) = −0.0041t ³ + 1.77t ² − 256.2t + 12353.8. t ₁ = 177.8, t ₂ = 196.4, and f ₄(t) = −0.00009t ³ + 0.036t ² − 4.3t + 178.8 for the optimistic case.