# Peaking Dynamics of the Production Cycle of a Nonrenewable Resource

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

- The resource is defined as an initial fixed stock. In the case of crude oil, it is the ensemble of the extractable resources (“URR” (ultimate recoverable resources));
- The extraction and processing of the resource transform it into a stock called “capital” that aggregates all the economic entities created by the process. The flow from the resource stock to the capital stock is called “production”;
- The resource stock is depleted proportionally to the size of the capital stock;
- The capital stock grows in proportion to the amount of the remaining resources, and to the amount of available capital;
- The capital stock is depleted at a rate proportional to the size of the stock (“depreciation”).

_{1}R − k

_{2}RC

_{2}RC − k

_{3}C

_{1}= 0, and the first equations (Equations (1)) of the system become as follows:

_{2}RC

_{2}RC) divided by the energy expended (“invested”) by the capital stock (that is, the flow out of the C stock (−k

_{3}C)). The result is as follows:

_{2}R/k

_{3}

_{c}= 1, with the “c” subscript referring to the capital peak.

_{p}. It can be approximately determined considering that the peak occurs when about half of the URR has been extracted. Hence, R

_{p}= ½R(0) and EROI

_{p}= ½ηk

_{2}R

_{p}/k

_{3}. Alternatively, the EROI

_{p}can be determined by setting to zero the second derivative of Equation (3) and combining the result with Equation (4). As discussed in [16], the result is that EROI

_{p}= k

_{2}C

_{p}/k

_{3}+ 1, where, again, the “p” subscript indicates the values of the variables at the production peak. Because all the factors in this expression are larger than zero, it follows that the EROI at the production peak is larger than one. This result tells us that the production peaks earlier than the capital accumulation. This is a behaviour that agrees with the historical data.

_{3}C can be neglected. This assumption is equivalent to assuming that the growth of the stocks is exponential during the early stages of the cycle. In this case, the equations of the model are as follows:

_{2}RC

dC/dt = ηk

_{2}RC

_{0}− R), with the “0” subscript indicating the value of the variables at the start of the cycle. Therefore:

_{2}η R (R

_{0}− R)

_{0}/(1 + exp(R

_{0}k

_{2}η(t − t

_{0})))

_{0}, the exponential is equal to 1 and we have R = ½R

_{0}. This means that half of the resource has been extracted. The equation tp = t

_{0}corresponds to the peak time from the start of the exploitation cycle. This can be demonstrated by taking the second derivative of the equation for R and setting it to zero, noting that the peak corresponds to the flex of the resource curve. This interpretation implies that the production curve is symmetric, which is a reasonable approximation in many historical cases. This equality has been extensively used in early studies on oil production to estimate the data of the peak [17]. Finally, note that for R = R

_{0}, we have t = −∞.

_{s}(“t-start”)) that we take as the start of the cycle. We can take t

_{0}= 0; therefore, the “time to peak” (TtP) is −t

_{s}. The remaining resources at t = t

_{s}are defined as Rs.

_{0}/(1 + exp(R

_{0}k

_{2}ηts))

_{s}if we know the value of the Rs/R

_{0}that we call “F”, which can be calculated from the historical data. It follows that:

_{0}k

_{2}ηts) = 1/F − 1

_{0}k

_{2}ηts = ln(1/F − 1)

_{s}= ln(1/F − 1)/(R

_{0}k

_{2}η)

_{s}= ln(1/F − 1)/(k

_{3}EROI

_{0})

_{p}= k

_{3}/k

_{2}+ C

_{p}

_{c}= k

_{3}/k

_{2}

_{pc}is equal to C

_{p}. Because the two peaks are close to each other, we can discretize the first equation of the SCLV model and write ΔR

_{pc}/Δt = k

_{2}R

_{p}C

_{p}. Substituting, we have the following:

_{2}R

_{p}

_{2}. Taking into account that EROI = ηk

_{2}R/k

_{3}, and that the peak occurs at approximately R

_{p}= ½ R(0), it follows that:

_{p}· k

_{3}/k

_{2}R

_{p})

_{0}· k

_{3})

## 3. Results

#### 3.1. The Mousetrap Experiment

- k
_{2}= 0.0604 (s^{−1}*nballs^{−1}); - η = 2;
- k
_{3}= 3.31 (s^{−1}); - R
_{0}= 50.7 (n traps); - C
_{0}= 0.6 (n balls); - TtP = 1.2 s (time to production peak);
- TtC = 1.5 s (time to capital peak);
- 2 × 0.0604/3.31 = 0.036.

_{c}). We determined earlier on that this ratio must be equal to 1 if the two stocks are measured in the same units. Now, using the values obtained in the fitting, and noting that the number of untriggered traps at the peak capital is equal to ca. 26, we have EROI

_{c}= 2 × 0.06 × 26/3.31 = 0.95, which is approximately correct.

_{0}(the EROI at the start of the cycle). Using the same formula as before, we find that the EROI

_{0}= 1.82. Correctly, it is larger than one; otherwise, the chain reaction could not have started. We may also estimate the EROI

_{p}(the EROI at the peak of the production curve). From the formula EROI

_{p}= k

_{2}C

_{p}/k

_{3}+ 1, using C

_{p}= 11, we have EROI

_{p}= 1.02, which is larger than the EROI

_{c}, as it should be.

_{0}k

_{2}η). F can be estimated by noting that the experiment starts with 50 traps, each loaded with 2 balls. At the start, one extra ball is dropped on the trap array—which corresponds to one-half of an additional trap. So, R

_{s}/R

_{0}is 50/50.5 = 0.99. The result is TtP = 0.77 s, which is correct in terms of the order of magnitude, but smaller than the actual value of TtP = 1.15 s.

_{2}R

_{p}. With R

_{p}= 27 and k

_{2}= 0.0604, we have Δt = 0.6 s. Again, this value can be considered correct as an order of magnitude, but it is double the actual value, which is about 0.3 s.

#### 3.2. Model Comparison

_{2}.

_{2}is evident. This approximation remains valid if we express the time to peak as a function of the value of the EROI at the start of the cycle (EROI

_{0}), as shown in Figure 6.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Abbreviation | Explanation |

C | Capital stock |

CAS | Complex adaptive system |

EROI | Energy return for energy invested |

EROI_{0} | EROI at the beginning of the production cycle |

EROI_{p} | EROI at the production peak |

EROI_{c} | EROI at the peak of the capital stock |

R | Resource stock |

SCLV | Single-cycle Lotka–Volterra |

TtP | Time to peak |

URR | Ultimate recovery resources |

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**Figure 1.**An interpretation of the two-stock model of the extraction of a mineral resource according to the conventions of system dynamics. The rectangles indicate stocks, while the double-edged connectors indicate flows. Single arrows indicate the effect of the parameters drawn using Vensim™ software. k

_{1}, k

_{2}, and k

_{3}represent, respectively, the regeneration rate, production rate, and depreciation rate, and they are deeply explained in [6,14]. Figure drawn using Vensim ©.

**Figure 2.**A qualitative illustration of the typical bell-shaped “Hubbert curve” simulated using the SCLV model. The vertical scale is the production of a nonrenewable resource in arbitrary units. The horizontal scale is also in arbitrary time units. This curve may represent the smoothed case of the production of crude oil in a large producing region [1]. It has also been observed for the production of other resources, such as for fisheries [5].

**Figure 3.**Qualitative behaviour of the parameters of the SCLV model. The X-scale is in arbitrary time units, while the Y-scale is in arbitrary energy units. The meanings of the terms are explained in the text.

**Figure 4.**Results of the “mousetrap experiment” according to the EROI model. he data shown are the result of averaging three experimental runs. “Untriggered Traps” are the resources, “Flying Balls” are the capital, and the production is the number of flying balls generated per unit time (Figure 3).

**Figure 6.**The values of the timing of the peak (TtP) as a function of the system EROI at the start, according to the model calculation and logistic approximation. The parameters are those for the “mousetrap” model described before. The results are consistent with those obtained in the previous section. The EROI

_{0}in the mousetrap experiment was 1.8, the experimentally observed TtP was 1.2 s, while the calculated peak using the logistic approximation was 0.8 s. These results are approximately reproduced by the calculations.

**Figure 7.**The effect of the variable initial EROI on the distance between peak production and peak capital. The parameters of the model are the same as those for the “mousetrap” model described in the text. The “Full model” data were calculated by solving the differential equations of the model by iterative methods. The “approximate” model describes a model in which the production peak time was calculated using the approximation described in the text, and the capital peak from solving the equations by iterative calculations. The actual distance of the two peaks is ca. 0.2 s for EROI(0) = 1.8.

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**MDPI and ACS Style**

Perissi, I.; Lavacchi, A.; Bardi, U. Peaking Dynamics of the Production Cycle of a Nonrenewable Resource. *Sustainability* **2023**, *15*, 6920.
https://doi.org/10.3390/su15086920

**AMA Style**

Perissi I, Lavacchi A, Bardi U. Peaking Dynamics of the Production Cycle of a Nonrenewable Resource. *Sustainability*. 2023; 15(8):6920.
https://doi.org/10.3390/su15086920

**Chicago/Turabian Style**

Perissi, Ilaria, Alessandro Lavacchi, and Ugo Bardi. 2023. "Peaking Dynamics of the Production Cycle of a Nonrenewable Resource" *Sustainability* 15, no. 8: 6920.
https://doi.org/10.3390/su15086920