Applicability of Hubbert model to global mining industry Interpretations and insights Lucas Riondet123 Daniel Suchet4 Olivier Vidal5Jos e Halloy3

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Applicability of Hubbert model to global mining industry:

Interpretations and insights

Lucas Riondet1,2,3*, Daniel Suchet4, Olivier Vidal5Jos´e Halloy3,**

1Univ. Grenoble Alpes, CNRS, Grenoble INP, G-SCOP, 38000 Grenoble, France

2I2M Bordeaux, UMR 5295, Institut de Chamb´ery, 73370 Le Bourget du Lac, France

3Universit´e Paris Cit´e, CNRS, LIED UMR 8236, F-75006 Paris, France.

4Institut du Photovolta¨ıque d’Ile de France IPVF UMR 9006, CNRS, Ecole

Polytechnique, 91120 Palaiseau, France.

5Univ. Grenoble Alpes, CNRS, Institut des sciences de la Terre, Grenoble, France

* lucas.riondet@grenoble-inp.fr ** jose.halloy@u-paris.fr

This work has been submitted to PLOS Sustainability and Transformation.

Abstract

The Hubert’s model has been introduced in 1956 as a phenomenological description of

the time evolution of US oil fields production. It has since then acquired a vast

notoriety as a conceptual approach to resource depletion. It is often invoked nowadays

in the context of the energy transition to question the limitations induced by the

finitude of mineral stocks. Yet, its validity is often controversial despite its popularity.

This paper offers a pedagogical introduction to the model, assesses its ability to describe

the current evolution of 20 mining elements, and discusses the nature and robustness of

conclusions drawn from Hubbert’s model considered either as a for cast or as a foresight

tool. We propose a novel way to represent graphically these conclusions as a ”Hubbert’s

map” which offers direct visualization of their main features.

October 6, 2022 1/36

arXiv:2210.02298v1 [physics.soc-ph] 5 Oct 2022

Contents

1 Introduction 3

1.1 Objectives................................... 3

1.2 Defining the Hubbert model . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 How to identify variables and parameters in the mining context? 6

2.1 Evaluation of P(t) .............................. 7

2.2 Evaluation of Q(t)and Q0......................... 7

2.3 Evaluation of τ................................ 7

2.4 Evaluation of K................................ 7

3 The Hubbert map representation 8

4 Materials and methods 9

4.1 USGSdata .................................. 9

4.2 Computation and curve fitting . . . . . . . . . . . . . . . . . . . . . . . 9

5 Results 11

5.1 A detailed application to Copper . . . . . . . . . . . . . . . . . . . . . . 11

5.2 Application to elements with copper-like extraction evolution . . . . . . 12

5.3 Application to multi-trends elements . . . . . . . . . . . . . . . . . . . . 13

5.3.1 Definition of chemical elements with Hubbertian trend . . . . . . 14

5.3.2 Chemical elements with multiple annual production trends . . . 16

5.3.3

Elements whose extraction trend cannot be described by a Hubbert

model ................................. 18

6 Discussion 19

6.1 A predictive reading of the model . . . . . . . . . . . . . . . . . . . . . . 19

6.2 A foresight reading of the model . . . . . . . . . . . . . . . . . . . . . . 21

6.3 Phenomena and constraints affecting the desirability of scenarios . . . . 23

6.3.1 Materials demand for the energy and industrial transitions . . . 23

6.3.2 The planetary boundaries framework and the phosphorus case . 24

6.3.3 Social pressure and (mid and) long-term planning . . . . . . . . 24

7 Conclusion 25

8 Declaration of competing interest 25

9 Credit authorship contribution statement 25

A Supporting information 29

B Appendix: Raw data 30

C Appendix: Parameter K definition in multiple trends case 31

D Appendix: Hubbertian trend elements 32

E Appendix: Exponential trend elements 33

F Appendix: Multiple trends elements 35

G Appendix: Silver and Rare Earths 36

October 6, 2022 2/36

1 Introduction

Access and use of resources represent a key issue which has been exacerbated along the

rising and the growing complexity of modern societies. A popular way of looking at this

issue may take roots in the writings of the economist Thomas Malthus (1766-1834) [1]

and of the mathematician philosopher Nicolas de Condorcet (1743-1794) [2], both

conceptualizing that resources needs could not be covered in the future. In their works,

population is considered as the main factor causing the food resources decline and

agricultural outputs were linearly increasing. Establishing that population tended to

rise faster than yields, Malthus predicted massive food shortage for the following

century, while Condorcet submitted the concept of ”progress” which should make

production means improving at the same time as population growth. Their conclusions

differ broadly, due to their specific conception of societies and of technological impacts.

In other words, their models are not only scientific objects but also support a societal

vision. During the 20th century technical progress has brought about rising of

production and then has turned the limitation of producing rate problematic to concern

about limitation of resources themselves. Moreover, technical progress also has made

fossil energy production a centerpiece of society, supplanting then conditioning food

supply. With the increasing share of fossil fuel, draining from a finite stock of resources,

the issue raised by Malthus and Condorcet naturally turned towards the energy sector.

This concern is first epitomized in 1865 by the Coal question of W. Jevons, which

foresees many of our contemporary challenges [3]. But the most celebrated approach is

undoubtedly the 1956 work of Marion King Hubbert [4]. While crude oil extractions in

the US were booming, with an output doubling every 9 years, Hubbert addressed the

question of the continuation of the observed trends. Considering a model building on

Verhulst equation also called

logistic curve

[5] [6], Hubbert envisioned a peak of the US

conventional oil production 10 years in the future, followed by a rapid decrease. The

simplicity of the model, the audaciousness of the conclusions and the remarkable

accuracy of this trajectory up to the mid-2000’ ensured the reputation of this work [7].

Since this seminal study, a large amount of work has been dedicated to investigate

the ramification of a depletion dynamics with various types of models. Based on system

dynamics, the The limit to Growth [8] defended a holistic approach on a global scale

and promoted to widen the scope of exhaustion models, including to mineral resources.

The model was based on curves similar to the Verhulst one, the Gompertz function, and

has popularized depletion curves. In the continuity, the cumulative property of depletion

curves implying that the sum of several production peaks gives a general peak shape is

still being used to refine peak resources predictions on regional and global scale [9] [10].

Moreover, since 2000s and in the context of the energy transition mineral resources

and/or demand have been increasingly studied with forecasting models using logistic

curves or at least sigmoid curves [11] [12] [13]. Thus, even if a maximum of interest for

oil peak happened during 2010s [14], depletion curves are still prevalent and used to

support decisions, notably in energy transition management.

As a consequence, and with recycling processes being minor so far in terms of mass

flow, the mining sector evolution provides a good description of the current

management of non-energy resources and its supposed depletion.

1.1 Objectives

Due to its history and the availability of data on global scale, the topic of mining

production is a good study case for interrogating the use of future oriented model and

investigate the possible interpretation of the results.

In that respect, and given the popularity of the concept of peak production, the

Hubbert model has been chosen to illustrate two possible interpretative approaches: the

October 6, 2022 3/36

forecasting and foresight perspectives.

In a forecasting interpretation, the future already exists and is inevitable. As such it

can be predicted with regard to uncertainties associated to the model and simplifying

assumptions.

In a foresight interpretation, the model depicts potential futures (i.e scenarios) and

assesses their plausibility as well as their desirability. It is a decision support thinking.

The group of scenarios we consider here is to continue mining in a business-as-usual

mode. The only constraint considered that is limiting is the geological availability of the

chemical elements considered, all things being equal. We therefore consider that the

demand for mining products will continue to grow, that energy, technological or

geopolitical factors are not limiting in these scenarios. Of course, these are simple and

basic scenarios that could be enriched later in other studies. They highlight the

geological constraint in a world where the demand for mineral resources does not

weaken, which corresponds to the extrapolations of the current outlook.

Therefore, this work proposes a pedagogical case study of the application of Hubbert

model on a global scale and on the mining industry. First, we define the basic

assumptions and mathematical properties of the model. Then, we explore its

applicability to 20 mining elements including copper, with a methodology similar to the

one used historically. This application uses mining resources estimation and is

associated to a predictive perspective. In addition, Hubbert models with varying input

parameters are applied to highlight an equivalence zone where models fit historical data

with the same accuracy. This work is closer to a foresight implementation of the model.

A new representation has been built to facilitate a prospective reading of the model.

Finally, the outcomes and the difference between the two approaches are exposed in the

discussion part.

1.2 Defining the Hubbert model

The Hubbert model offers a phenomenological description for the time evolution of the

cumulative production

(

) (or equivalently, the annual extraction

(

) =

dQ/dt

) from

a finite stock. One strength of the models resides in its utmost simplicity. The model is

based on the logistic growth equation proposed by P.F. Verhulst [5].

The three mathematical hypothesis of the model are:

1. There is an amount K, defined as the maximum quantity of resources extracted

throughout the course of the study such as:

K=Z∞

−∞

P(t)·dt =Q(t= +∞)−Q(t=−∞) (1)

where Q(t=−∞) = 0. Note that the stock considered here represents the

quantity that will be effectively extracted at the end of time, and which is

assumed to be fixed. This quantity is also called the Ultimately Recoverable

Resources (URR).

The annual production

(

) is proportional to the quantity already extracted

(

)

and to the quantity still to be produced, which are the remaining resources

R(t) = K−Q(t).

3. The proportionality coefficient 1/(Kτ) is constant, with τa characteristic time

parameter related to the dynamic of the production.

With these three assumptions, we define that the cumulative production Q(t) satisfies

the Verhulst’s differential equation, as follows:

October 6, 2022 4/36

dQ(t)

| {z }

Annual production

K·1

τ·Q(t)

|{z}

Cumulative production

·(K−Q(t))

| {z }

Remaining resources

(2)

The solution to Verhulst’s equation is the so-called logistic function and the resulting

expressions of the production variables

(

) and

(

) constitute what we refer to

the Hubbert model, presented on Eq (3).

∀t∈[ti,∞[,











Q(t) = K

1 + K

−1!·exp −(t−t0)

τ!

P(t) = dQ(t)

R(t) = K−Q(t)

(3)

The model is then fully characterized with the two parameters τand K, and one

initial condition at time t0for the cumulative production such as Q(t0) = Q0.

The typical time evolution of these functions are drawn in Fig 1 with the initial

condition Q(0) = 0.

Fig 1. Typical time evolution of the Hubbert model solution. The figure

shows normalized values. The cumulative production, Q(t), shows a sigmoidal shape

(blue curve, right scale). The scale is normalized by

. Its asymptotic value is

. The

remaining resources, R(t), is a symmetrical sigmoid with an asymptotic value of 0

(dashed blue curve). The annual production (black line, left scale), P(t), presents a

symmetrical bell shape with a maximum that will be referred too as a ”peak” (black

dot). The scale is normalized by the peak value.

Several stages can be identified in this dynamics. In the first stage of extraction (at

times such that K >> Q(t)), production increases exponentially with a growth rate

given by τ. When cumulative extraction becomes non negligible compared to K,

production slows down, and reaches a maximum (the so-called ”peak”). Note that the

peak is reached when the cumulative production Q(tpeak) is equal to half of K- or

equivalently, when a quantity K/2 remains to be extracted. The peak does not imply

the depletion of the resource under study, but indicates the beginning of the reduction

of the annual production. Finally, symmetrically to the first phase, production

decreases exponentially at the same rate

and tends towards zero when the cumulative

production approaches the maximum extractable quantity.

October 6, 2022 5/36

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ApplicabilityofHubbertmodeltoglobalminingindustry:InterpretationsandinsightsLucasRiondet1,2,3*,DanielSuchet4,OlivierVidal5JoseHalloy3,**1Univ.GrenobleAlpes,CNRS,GrenobleINP,G-SCOP,38000Grenoble,France2I2MBordeaux,UMR5295,InstitutdeChambery,73370LeBourgetduLac,France3UniversiteParisCite,CNRS,LIED...

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Applicability of Hubbert model to global mining industry Interpretations and insights Lucas Riondet123 Daniel Suchet4 Olivier Vidal5Jos e Halloy3

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