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

2025-04-27 0 0 9.6MB 36 页 10玖币
侵权投诉
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
Q
(
t
) (or equivalently, the annual extraction
P
(
t
) =
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).
2.
The annual production
P
(
t
) is proportional to the quantity already extracted
Q
(
t
)
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)
dt
| {z }
Annual production
=1
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
Q
(
t
),
P
(
t
) and
R
(
t
) constitute what we refer to
the Hubbert model, presented on Eq (3).
∀t∈[ti,∞[,
Q(t) = K
1 + K
Q0
−1!·exp −(t−t0)
τ!
P(t) = dQ(t)
dt
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
K
. Its asymptotic value is
K
. 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
τ
and tends towards zero when the cumulative
production approaches the maximum extractable quantity.
October 6, 2022 5/36
摘要：

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

展开>> 收起<<
Applicability of Hubbert model to global mining industry Interpretations and insights Lucas Riondet123 Daniel Suchet4 Olivier Vidal5Jos e Halloy3.pdf

共36页,预览5页

还剩页未读， 继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

相关推荐

更多
立即下载
分类：图书资源 价格：10玖币 属性：36 页 大小：9.6MB 格式：PDF 时间：2025-04-27

开通VIP享超值会员特权

  • 多端同步记录
  • 高速下载文档
  • 免费文档工具
  • 分享文档赚钱
  • 每日登录抽奖
  • 优质衍生服务

作者详情

相关内容

更多

热门标签

高考理综 人际关系 力的合成 配电装置 动力学连接体 全宋诗作者索引 公务员考试
/ 36
评分 收藏
分享
立即下载
客服
关注