modeling us petroleum production using standard and discounted exponential growth approaches

SAN FRANCISCO STATE UNIVERSITY

Modeling US Petroleum Production Using Standard and Discounted Exponential

Growth Approaches

Mark Ciotola

5/20/2010

Business 899

Supervised by Prof. Ramesh Bollapragada

SFSU BUS 899 May 2010

M. Ciotola

ii

ABSTRACT

The nature of petroleum deposits, their location and utilization is discussed. Economic factors and

general trends are also identified. Standard quantitative forecasting techniques are utilized to

forecast U.S. petroleum production. The concept of the Hewett-Hubbert curve, commonly known

as “peak oil”, is introduced. Existing methods of generating Hewett-Hubbert distributions are

demonstrated. Then, a discounted exponential growth method motivated by a proposed e th Law

of Thermodynamics is shown to generate Hewett-Hubbert distributions. The discounted

exponential growth method is then utilized to model U.S. petroleum production. Errors produced

by several standard quantitative forecasting techniques and the discounted exponential growth

method are compared.


M. Ciotola

iii

Contents

EXECUTIVE SUMMARY...............................................................................................................1

SECTION 1: BACKGROUND REGARDING GLOBAL OIL GEOLOGY AND ECONOMICS.2

1.1 Why is there so much interest in oil?.......................................................................................2

1.2 What is oil and how is it formed?............................................................................................ 2

1.3 Where is oil located in the world?........................................................................................... 4

1.4 How Much Oil is There?..........................................................................................................5

1.5 How Much Oil Does the Global Economy Require? ..............................................................6

1.6 Why has Oil Become So Expensive? What Are the Current Trends on Oil............................7

1.7 Context for Production Forecasting....................................................................................... 11

SECTION 2: CONTEMPORARY PRODUCTION FORECASTING....................................... 12

2.1 Introduction............................................................................................................................11

2.2 Naive and Faith-Based Methods............................................................................................ 12

2.3 “Drill, Baby, Drill!”—Future Holes In the Ground Approach.............................................. 13

2.4 Trend Approaches, Generally................................................................................................ 13

2.5 Single Exponential Smoothing...............................................................................................14

2.6 Exponential Smoothing: Winters Method with Trend and Seasonality.................................14

2.7 Exponential Smoothing: Holt Method with Trend for Annual Data..................................... 15

2.9 Time Series Decomposition................................................................................................... 15

2.9 Regression..............................................................................................................................16

SECTION 3: DISCOUNTED EXPONENTIAL GROWTH HYPOTHESIS.................................17

Introduction..................................................................................................................................17

3.1 A Brief Introduction to Thermodynamics Fundamental Principles.......................................17

3.1.1 First Law of Thermodynamics.........................................................................................17

3.1.2 The Second Law of Thermodynamics............................................................................. 18

3.2 A Simple Thermal Conductor Bridging A Thermal Potential............................................... 20

3.3 Simple heat engine................................................................................................................. 24

3.4 Reproducing heat engines...................................................................................................... 30

3.5 Bring it all together: Generating a Hewett-Hubbert Curve Using Thermodynamics........... 32

3.6 Reproducing heat engines requiring maintenance................................................................. 33

3.7 Applying Discounted Exponential Growth to an actual example..........................................35

4 APPLICATION OF DISCOUNTED EXPONENTIAL APPROACH TO PETROLEUM

PRODUCTION............................................................................................................................38

4.1 Regional Field Hewett-Hubbert Approach............................................................................ 38


M. Ciotola

iv

4.2 Hubbert basic Formula...........................................................................................................39

4.3 Discounted Exponential Growth Approach .......................................................................... 40

4.4 Discounted Exponential Growth Modeled With Iterative Program...................................... 40

4.5 Implementing Discounted Exponential Growth Modeled Analytically................................ 42

4.5.1 Efficiency Term Modeled As Simple Linear Function................................................... 43

4.5.2 Efficiency Term Modeled In Terms of Heat Flow.......................................................... 44

4.5.3 Efficiency Term Modeled In Terms of Entropy Increase................................................45

4.5.4 Comparison With Logistic Function............................................................................... 46

5 CONCLUSIONS ......................................................................................................................... 46

REFERENCES................................................................................................................................53

APPENDICES.................................................................................................................................56

Minitab Logs

Single Exponential Smoothing 1980-99

US Petroleum Production 1970-99 Decomposition

US Petroleum Production 1980-99 Decomposition

US Petroleum Production 1970-99 Regression/Trend—Linear w/Annual Data

US Petroleum Production 1970-99 Regression/Trend—Quadratic w/Annual Data

US Petroleum Production 1979-99 Regression/Trend—Linear w/Monthly Data w/ Forecast of

40 periods

US Petroleum Production 1979-99 Regression/Trend—Linear w/Monthly Data w/ Forecast of

122 periods

Minitab Charts

Single Exponential Smoothing 1980-99; Without Forecast to 2008

Single Exponential Smoothing 1980-99; With Forecast to 2008

US Petroleum Production 1980-99 Time Series Decomposition; Components

US Petroleum Production 1980-99 Time Series Decomposition; With Forecast to 2008

US Petroleum Production 1980-99 Time Series Decomposition; Seasonality

US Petroleum Production 1970-99 Regression/Trend—Linear w/Annual Data

US Petroleum Production 1970-99 Regression/Trend—Quadratic w/Annual Data

Excel Spreadsheets (may be in separate file)

US Petroleum Production 1920-2008 Naive w/Monthly

US Petroleum Production 2000-2008 Winters w/Monthly

US Petroleum Production 1970-2008 Decomposition w/Monthly

EXECUTIVE SUMMARY

The nature of petroleum deposits, their location and utilization is discussed. Economic factors and

general trends are also identified. Standard quantitative forecasting techniques are utilized to

forecast U.S. petroleum production. The concept of the Hewett-Hubbert curve, commonly known

as “peak oil”, is introduced. Such curves have been used since 1929 to model the production of

limited mineral resources such as metals or oil.1 Existing methods of generating Hewett-Hubbert

distributions are demonstrated. Then, a discounted exponential growth method motivated by a

proposed e th Law of Thermodynamics is shown to generate Hewett-Hubbert distributions. Then

it is shown how Hewett-Hubbert curves can be generated from fundamental thermodynamic

principles, using reproducing heat engines that require maintenance. The discounted exponential

growth method is then utilized to model U.S. petroleum production. Errors produced by several

standard quantitative forecasting techniques versus the discounted exponential growth method are

compared. The Winter’s Method applied to monthly data demonstrates seasonality in petroleum

production. Regression utilizing a quadratic trend is shown to reduce error when applied to

annual data. The discounted exponential growth method produces reasonable agreement over

upside modeling, but less so for downside modeling, although this may be due to error induced by

the manual choice of parameters and potential profile.

1 The author also uses Hubbert curves to model a wide range of social phenomena such as historical dynasties and economic bubbles.


M. Ciotola

2

SECTION 1: BACKGROUND REGARDING GLOBAL OIL GEOLOGY AND ECONOMICS

1.1 Why is there so much interest in oil?

Oil, also known as petroleum, is an essential resource in today's industrialized

society. Oil provides gasoline, airplane fuel and heating oil and is also the basis of much

of the chemical industry. Oil is an essential part of the modern food production used to

power farm machinery and to transport food from the farm to the consumer.

Oil is a topic of increasing importance. Prices for oil and gasoline have

skyrocketed. International competition for oil reserves has intensified. Why is there so

much interest in oil these days? Internationally, oil is the most used energy source.

Demand for oil increases along with economic development as people around the world

strive for improved standards of living. There are other fuels and materials, but oil is still

the most efficient for many energy needs.

1.2 What is oil and how is it formed?

Oil is a mixture of hydrocarbons which are molecules made up of carbon and

hydrogen atoms. The formation of oil is complicated and involves a series of steps and

conditions that typically require a 100 million years to complete. First, organic matter

from dead plants, animals and plankton fall to the bottom of shallow sea basins. If

organic material is deposited at a high rate in large quantities, it will become preserved.

Eventually, the deposited matter sinks deeper and deeper due to geologic forces as well

as the weight of the accumulating matter above it. As the matter sinks thousands of feet


M. Ciotola

3

under the ground, it becomes hotter and the organic material is literally "cooked" until it

become oil. Heat breaks down the organic matter into long hydrocarbon molecules.

Eventually oil is formed (Chapman, 1983; Riva, 1983; Deffeyes, 2001).

Just like food, oil can become over-cooked. If the matter sinks deeper than 15,000

feet, the increasing heat breaks apart the long chains of hydrocarbons that then become

natural gas. Oil can only be recovered in certain circumstances. The oil must form in

areas of porous, sedimentary rock. Nearly all oil found in commercially valuable oil

deposits has migrated from its original source rock into other rocks called traps that are

"capped" by a layer of non-porous rock such as a salt dome (Riva, 1983; Deffeyse, 2001).

Oil wells drill into oil traps to extract oil. Many discoveries are too small to be

commercially exploited. Even if a discovery is commercially viable, it requires about 10

years to develop into a productive oil field. An oil field can include a few wells or many

wells. There are only a few dozen really large oil fields in the world. (Deffeyse, 2001).

Not all oil is the same but differs due to its composition. All oil contains some

impurities such as sulfur. Oil that contains little or no sulfur is called "sweet." It is

slightly more valuable than oil that contains more sulfur. In addition, oil can contain

different types of hydrocarbons. Oil that contains shorter hydrocarbons is called "light"

and is slightly more valuable than heavier hydrocarbons. Some oil contains significant

amounts of wax. Wax itself is valuable, but lighter oil is better for making gasoline and

tends to be more valuable, especially in the United States. Oil varies widely, but different

regions tend to have different qualities. Oil from Texas and nearby states tends to be light


M. Ciotola

4

and sweet. Fortunately, this is the best kind to refine into gasoline, given the huge

demand for gasoline in the United States (Chapman, 1983; Deffeyse, 2001).

1.3 Where is oil located in the world?

Oil can only be formed under specific circumstances, so that oil deposits are only

present in limited areas. Typically, past or present oceans and a source of organic matter

were required. Organic matter only forms in the presence of nutrients. Erosion from land

masses is an important source of nutrients, so living organisms are largely present on land

masses or in the ocean areas adjacent to land masses. Today, most oil deposits are found

either on land that was previously an ocean or in ocean areas that are or were adjacent to

land masses.

The Middle-East contains the largest source of oil deposits. Much of the Middle-

East used to be a shallow-sea, but its position between several large land masses such as

Africa and Asia has caused it at times to be squeezed. This has produced complicated

geologic conditions including folding and fracturing, and had allowed the oil deposits to

migrate into the present oil fields. Most Middle East oil deposits are found in Saudi

Arabia, Kuwait, Iraq and Iran (Riva, 1983).

In North America, most oil is found either near the ocean coastlines or in a central

area (Texas, Oklahoma, and Kansas) roughly adjacent to the Gulf of Mexico. The broad

continental shelf along North America's southeastern coast and the Gulf of Mexico

contain large oil deposits. Southern California coastal and adjacent areas also contain

significant oil deposits. Northern Alaska is another major location of oil deposits. Most

oil deposits in Alaska already being exploited. Despite great fanfare, the Alaskan


M. Ciotola

5

National Wildlife Refuge contains relatively little additional oil (Chapman, 1983;

Deffeyse, 2001).

The European North Sea is also a prodigious source of oil deposits, primarily

portions that lie between the United Kingdom and Norway. Former Soviet Union states

and Indonesia are other important sources. An emerging source is the East Asian coastal

areas, which have not yet been significantly exploited due to political issues. Although

new Asian sources will be important, it is unlikely that there will be any new, extremely

large oil fields found there comparable with existing large sources such as the Middle

East oil fields (Chapman, 1983; Riva, 1983; Deffeyse, 2001).

1.4 How Much Oil is There?

Every few months, newspaper headlines make dramatic claims, such as "the U.S.

will run out of oil in 10 years" or "New Oil Field Discovered Will Supply Oil Needs for

100 years." These headlines are often misleading. One has to look at the facts behind

those figures to make sense of the global oil supply.

The supply of oil is typically expressed in terms of reserves. One measure is proven

reserves which concerns existing economically viable oil fields. The amount of proven

reserves can change due to either new discoveries or significant changes in the price of

oil. In 2004, there were over 1 trillion barrels of proven reserves (see below).

Table 1—Estimated Proved Reserves of Oil, 2009

In Billions of Barrels (EIA, 2009)

Region or Country Proved Reserves

Africa 117.064

Canada 178.092


M. Ciotola

6

Central and South America 122.687

China 16.000

India 5.625

Middle East 745.998

Russia 60.000

United States 21.317

Other 75.424

Total 1342.207

There are several widely-varying estimates for reserves. Other measures of oil

supply include anticipated reserve growth and undiscovered oil. All of these measures

involve uncertainty and there is considerable debate over how to calculate oil supply.

Most estimates place the total oil supply between 2 to 3 trillion barrels. (Ahlbrandt, 2002;

Deffeyes, 2001; EIA, 2004).

1.5 How Much Oil Does the Global Economy Require?

To understand the significance of the oil supply figures, it is important to know

about present consumption of oil. Consumption figures contain less uncertainty than do

supply figures. The United States is presently the largest consumer of petroleum,

followed by Europe, the Former Soviet Union, China and Japan (see below).

Table 2—Consumption of Oil, 2008

In Billions of Barrels (EIA, 2009)

Country or Region Annual Demand

China 2.867

Europe 5.897

India 1.073

Japan 1.747

Russia 1.059

United States 7.121

Other 11.450

Total 31.214


M. Ciotola

7

Total world consumption is about 30 billion barrels per year. At just the present

rate of consumption, proven reserves, if taken alone, would be exhausted in about 40

years. However, if the total oil supply (including anticipated reserve growth and

undiscovered oil) is 2.5 trillion barrels, then the present rate of consumption could be

sustained for about 80 years.

1.6 Why has Oil Become So Expensive? What Are the Current Trends on Oil

Consumption and New Discoveries?

Although present oil consumption figures are well known, forecasting future oil

consumption is dependent upon future economic development, technology, politics and

oil supply and therefore are highly uncertain. What is certain is that consumption of oil

has been dramatically increasing over the past few years.

Skyrocketing demand for oil is driving a bidding war and record prices. Oil prices

have exceeded $57 per barrel. But analysts do not think the oil rise is over. "We could

see it at $70 some time this year," declared Rob Laughlin, an analyst at Man Financial, a

hedge fund group, to the BBC's Today program (BBC News, March 9, 2005), and oil hit

much higher prices in 2008. What are the factors driving this demand? What is the

nature of price system limitations that are resulting in dramatically increasing prices?

Growing industrialization of China and India, as well as other nations, is

increasing demand for oil. China has rapidly increased its quantity of automobiles. In

addition, oil is used as a feedstock for many common materials used in manufactured

products such as plastics and Styrofoam. As the United States, Canada and other

countries import greater amounts of manufactured products from China, the Chinese


M. Ciotola

8

demand for oil feedstocks likewise increases. China accounted for 15% of the growth in

oil demand in 2005 year and 40% over the four years prior to that (BBC News, March 9,

2005). China imports 47% of its petroleum as of 2007 (BBC News, 2007). India is

rapidly growing as well and is yet another factor in the increased demand for oil. In

addition, residents of the U.S. have accumulated large, gas-guzzling vehicles over the

past decade.

Will the acceleration in demand continue at its present rate? An important

companion indicator is demand for steel. Although Chinese demand for steel

skyrocketed in 2004, demand has dramatically decreased this year to the point where

China has surplus steel-making capacity. Part of this decrease may be due to a slowing

construction industry. China had made serious efforts to slow this sector. However, if

part of the demand is a cooling of the automobile sector, then acceleration in demand for

oil will diminish (Also see EIA, March 8, 2005). Nevertheless, demand is still expected

to increase. Other countries are still showing strong economic growth rates as well.

Another factor driving demand is an expectation of oil shortages within the next

few years. Countries that expect shortages are attempting to protect themselves by

hoarding oil stocks. Laughlin observed considerable speculation in the oil market. He

added that countries such as China are building up stocks of oil even though the price is

high at the moment. "They are just guzzling at the moment," he said (BBC News, March

18, 2005). The present hoarding has artificially increased present demand levels. The

U.S. already has a strategic oil reserve but has not been dramatically increasing its stocks.

Increased competition for oil products has resulted in oil reaching nominally

record prices. However, it is a common understanding by economists that greater prices


M. Ciotola

9

will drive the development of additional supplies that will mitigate prices. The problem

with this understanding is that supply of a natural resource cannot be arbitrarily

increased. As early as 1798, economist T. R. Malthus pointed out this principle in his

studies population and agriculture. In contemporary terms, Exxon chairman and chief

executive Lee Raymond told analysts: "We are in the mode where the fundamentals of

supply and demand really don't drive the price. High oil prices fail to lift oil stock (BBC

News, 18 March, 2005)." For example, China's output only rose 2% last year.

Improved technology such as Global Positioning Systems (GPS) and advanced seismic

techniques have been utilized to discover additional reserves. To give economists some

credit, the amount of recoverable oil in the ground is partially a factor of price. Despite

record use of oil, reserved have actually slightly increased over the past year. Higher

prices justify higher recovery costs.

Unfortunately newly-discovered reserves tend to be located in environmentally or

politically sensitive areas. "Much of the new production in the 1990s has come from the

developing countries of Latin America, West Africa, the non-OPEC Middle East, and

China." In fact, production from such sources is expected to increase "from 46.7 million

barrels per day in 2001 to 64.6 million barrels per day in 2025" (EIA, 2004). To put that

in perspective, "U.S. oil demand averaged 20.5 million barrels per day in 2004 (EIA,

2005)."

Also, new technologies are better at discovering offshore oil deposits. The

potential for environmentally disastrous leaks is significant. The Artic is another

controversial source of oil. "Arctic basins with about 25% of undiscovered oil resources"

(Ahlbrandt, 2002). Despite the political fanfare over drilling the Artic Wildlife Reserve in


M. Ciotola

10

Alaska, that region contained relatively little oil. The bulk of Alaska's oil resources are

already being exploited (Deffeyes, 2001).

In addition, such reserves are more expensive to exploit since they are offshore or

in remote areas (EIA, 2004). Deposits in Africa and South America are often in remote

jungles. Offshore deposits require expensive drilling rigs and deep drilling. Artic areas

typically involve drilling through hundreds of meters of permafrost.

Yet despite increased reserves, oil prices have still dramatically increased. Causes

of these increases can be traced to increased competition for oil products, the expectation

in future increases in demand as well as the inability of the oil production infrastructure

to keep pace with growing demand.

There is still plenty of oil in the ground for at least the next few decades, so why

have prices increased so dramatically? The reason is only partly related to supply. Oil in

the ground is a much different matter than immediate supply. Immediately prior to the

U.S. 2004 Presidential election, Saudi Arabia was producing every barrel it could

squeeze out of its wells, yet prices remained high. The problem is two-fold. For the

immediate timeframe, the oil production, transportation and refining infrastructure is

relatively fixed. When demand increases quickly, the infrastructure operates at full

capacity, and so excessive demand creates a bidding war.

Infrastructure can be expanded over a period of several years. However, in a

business environment that focuses on quarterly profits, infrastructure expansion is

undesirable. Recall that 10 years is typically required to develop an oil discovery into a

productive oil field. Pipelines and refineries are expensive to build. So infrastructure

expansion creates short-term losses rather than short-term profits. CEO compensation


M. Ciotola

11

rarely emphasizes the company's performance 10 years in the future. So relatively little

business incentive exists to expand infrastructure to meet increased demand. However,

governments can help to develop infrastructure through political pressure and incentives.

As infrastructure grows to meet demand, prices increases will stabilize. Nevertheless, the

long-term price of oil can be expected to increase as demand increases but supply

remains limited.

1.7 Context for Production Forecasting

Oil requires about 100 million years to form, about 10 years to develop and

exploit, and less than ten seconds to burn. Most oil is found in past or present ocean

environments such as the Middle East, the Gulf of Mexico, the North Sea or adjacent

areas such as Texas. There is an estimated 2 to 3 trillion barrels of commercially

exploitable oil already discovered or anticipated to be discovered. Present oil

consumption is 30 billion barrels per year, but has been dramatically increasing over the

past few years. Presently high oil prices are due to increased demand exceeding

production and delivery infrastructure as well as investor speculation that demand for oil

will continue to skyrocket. As infrastructure catches up with increased demand, price

increases should slow, but not disappear. There are sufficient oil reserves to meet demand

for at least the next few decades, although present levels of consumption cannot be

maintained indefinitely.

SECTION 2: CONTEMPORARY PRODUCTION FORECASTING

2.1 Introduction


M. Ciotola

12

Several contemporary methods exist for oil production forecasting, ranging from

sophisticated to simple, from scientific to faith-based. There may be a commonly-

believed analogy regarding oil forecasting to the Heisenberg Uncertainty Principle in

Physics: the act of forecasting oil production will affect the actual level of oil production

(author observation). Therefore, forecasting petroleum production is fraught with

political and even religious implications. Putting politics largely aside, this section will

attempt to catalog major forecasting techniques, and discuss the strengths and weaknesses

of each from physical and quantitative viewpoints.

Actual petroleum production data is obtained by the Energy Information

Administration (EIA), a division of the United States Department of Energy. EIA data is

a widely-recognized benchmark for U.S. production data (the author found no significant

controversies regarding EIA data when concerning U.S. production).

2.2 Naive and Faith-Based Methods

The Naive Method results in a forecast that is identical to data from the most

recent observed period. Specifically, “naive forecasts assume that recent periods are the

best predictors of the future” (Hanke and Wichern, 2005). So the Naive Method would

forecast that oil production will continue at the same level as in the most recent period,

such as the last year. Some faith-based methods propose a physical mechanism to support

the naive method, that supernatural forces replace oil that is removed from the ground,

thereby ensuring a perpetual supply (e.g., see Akers, 2007). Naive and faith-based

methods are not unreasonable approaches for rough forecasts within a period of a few


M. Ciotola

13

years, since new oil production infrastructure can take several years to build, and new oil

fields can easily take a decade or more to develop (see Section 1).

[See Appendices for Excel printout]

2.3 “Drill, Baby, Drill!”—Future Holes In the Ground Approach

The “Future Holes in the Ground” approach states that future oil production is a

function of the quantity of oil fields developed and wells placed in the ground. Since it

typically takes a decade to develop a new oil field, this approach is no better than the

naive method for shorter periods. For periods ranging from a decade to a few decades,

this approach may be effective (author opinion), since there still exist large, relatively

unexploited oil fields, such as in Iran (Deffayse, 2001), but since there are only a limited

quantity of such fields, this approach will likely not provide accurate forecasts be

effective beyond a few decades. This approach assumes that demand will rise to meet

production; the popularity of the Hummer and SUVs as a response to the relatively cheap

petroleum bubble of the 1990s supports this assumption (author observation). It also

assumes that the oil industry can drill as many holes as it desires, as long as the U.S.

public supports military intervention (available in unlimited quantities) in other countries

where needed, such as Iraq and Iran. This assumption is in part valid (the U.S. did invade

Iraq), but not entirely valid (any such plans that have existed to invade Iran have so far

not occurred). This approach can be summed up by a popular U.S. political mantra:

“Drill, Baby, Drill!”

2.4 Trend Approaches, Generally


M. Ciotola

14

Trend approaches identify trends in historic data and then extrapolates those

trends into forecasts (text). Such trends, when using averaging and accounting for cycles,

are often effective for forecasting where linear, quadratic and even exponential trends

exist (Hanke and Wichern, 2005). However, they may fall short when there are more

complex mechanisms affecting the data over time. Complex, multi-component

exponential curves can change quickly (e.g., see Meadows et al, 1972).

2.5 Simple Exponential Smoothing

Simple exponential smoothing method attempts to forecast data by averaging data

for several past periods, so that single period variations do not skew forecasts. However,

smoothing is in a manner that is biased towards most recent data (Hanke and Wichern,

2005). For example, simple exponential smoothing might help minimize the impact of a

one-time hurricane event.

[See Appendices for Excel and Minitab printouts]

2.6 Exponential Smoothing: Winters Method with Trend and Seasonality

The Winters method involves exponential smoothing, but it also captures trend

and seasonality components (Hanke and Wichern, 2005). The Winters method improves

forecast when seasonal data is involved. Since seasonality is involved in U.S. petroleum

production, the Winters method is utilized here for analyzing monthly petroleum

production data.


M. Ciotola

15

[See Appendices for Excel and Minitab printouts]

2.7 Exponential Smoothing: Holt Method with Trend for Annual Data

The Holt method is similar to the Winters method, but does not take seasonality

into account. The Holt method involves exponential smoothing, but it also captures a

trend component. The Holt method can account for linear trends or even simple

exponential or quadratic trends. The Holt method is most appropriate to analyze to annual

data where any seasonality have been averaged out when used to analyze petroleum

production.

[See Excel and Minitab printouts]

2.8 Time Series Decomposition

Decomposition involves separating the components of time series data by trend

and seasonality. Minitab was utilized to perform decomposition on U.S. petroleum

production data from January 1970 to December 1999. (See Minitab printout and graphs

and Excel data table in Appendices).

Seasonality was indicated, but it was not a smooth change from month to month.

Production was especially low during January; this could be due to difficulties

transporting oil from Alaska’s Prudhoe Bay during especially cold weather (author

speculation), since crude oil can solidify when cold.


M. Ciotola

16

A forecast was then made from January 2000 to December 2009. (See Minitab

printout and graphs and Excel data table in Appendices).

SUMMARY OF MINITAB PRINTOUT

Multiplicative Model

Fitted Trend Equation: Yt = 282374 - 420.937*t Seasonal Indices

Month Index Month Index 1 1.02314 2 0.92934 3 1.02463 4 0.99138 5 1.01979 6 0.98262

7 1.01041 8 1.00651 9 0.98042 10 1.02164 11 0.98736 12 1.02275

Accuracy Measures

MAPE 3 MAD 7402 MSD 89458039

2.9 Regression

Decomposition involves separating the components of time series data by trend and

seasonality. Minitab was utilized to perform decomposition on U.S. petroleum production data

from January 1970 to December 1999. (See Minitab printout, graphs and Excel data table in

Appendices). Both linear and quadratic trends were modeled. The quadratic trend provided a

better fit (lower MSE, etc.).


M. Ciotola

17

MINITAB PRINTOUT

LINEAR:

Fitted Trend Equation: Yt = 3552631 - 40120.1*t Accuracy Measures

MAPE 4.27524E+00 MAD 1.23858E+05 MSD 2.59980E+10

MINITAB PRINTOUT

QUADRATIC:

Fitted Trend Equation: Yt = 3322570 + 3016*t - 1391*t**2

Accuracy Measures

MAPE 3.46434E+00 MAD 1.04881E+05 MSD 1.73331E+10

SECTION 3: DISCOUNTED EXPONENTIAL GROWTH HYPOTHESIS

Introduction

This paper assumes an e th Law of Thermodynamics which states that systems

will tend to configure themselves in a manner that maximizes the rate of entropy

production.2 Therefore systems will attempt to grow exponentially where possible.

2 For arguments and examples in support of an e th Law of Thermodynamics, see M. Ciotola, Physical History and Economics, Second Edition, 2009. For earlier references, see M. Ciotola, Physical History and Economics, 2003 and M. Ciotola, LASE, 2001. The Law of Maximum Entropy production proposed by R. Swenson is related.


M. Ciotola

18

3.1 A Brief Introduction to Thermodynamics Fundamental Principles

Thermodynamics is a branch of physics that concerns the flow of heat energy and

the ability to convert energy into work.

3.1.1 First Law of Thermodynamics

The First Law of Thermodynamics requires that energy can neither be created nor

destroyed. In other words, energy is conserved. This simply means that if heat flows from

one object to another, the quantity of heat leaving the first object must equal the quantity

of heat entering the second object.3 In other words, any heat lost by one body must be

gained by another body. This can be expressed by the following equation, where E is

energy of a system as a function of time t:

E (t) = 0

A simple example that demonstrates the First Law is to mix a quantity of cool

water with an equal quantity of hot water. If the water is kept in insulated containers

before and after the mixing, then the temperature of the final mixture will be the mean of

the temperatures of the original constituents (there may be a slight variation due to

evaporation or escaped heat). In other words, the total amount of heat energy remained

the same despite the mixing and temperature changes. The First Law of Thermodynamics

does not require that heat will flow at all, merely that if energy does flow, then it must

still be conserved.

3 The phrase “conservation of energy” has a much different meaning than the common phrase “conserving energy”. The latter refers to consuming less of useful forms of energy such as coal or petroleum.


M. Ciotola

19

3.1.2 The Second Law of Thermodynamics

The Second Law of Thermodynamics states that if heat does flow, then overall, a

quantity called entropy will tend to increase.4

Entropy before heat flow Entropy after heat flow,

or

S(t0) S(t1),

where S is entropy and t is time.

Entropy has a precise meaning in physics that can be found in advanced textbooks

dedicated to thermodynamics. For now, we just need to recognize that when a closed

system has reached equilibrium, its entropy is at a maximum. The further away a closed

system is from equilibrium, the lower its entropy will tend to be. An example of a system

at equilibrium is in the above water-mixing example after the water has been mixed and

all becomes the same temperature.

Stated another way, the entropy of an isolated system away from equilibrium shall

tend to increase.5,6 A corollary is that a system will approach a state of maximum entropy

4 Specifically, the entropy of an isolated system shall tend to increase. A more precise definition is that “any large system in equilibrium will be found in the macrostate with the greatest multiplicity (aside from fluctuations that are normally too small to measure).” D. Schroeder, An Introduction to Thermal Physics. San Francisco: Addison-Wesley, 2002. 5 A more precise definition is that “any large system in equilibrium will be found in the macrostate with the greatest multiplicity (aside from fluctuations that are normally too small to measure).” D. Schroeder, An Introduction to Thermal Physics. San Francisco: Addison-Wesley, 2002. 6The Second Law states that the entire universe is moving towards greater entropy. The entire universe can be viewed as an isolated system.


M. Ciotola

20

if given enough time.7 A system in a state of maximum entropy is in essence a system in

equilibrium. However, the Second Law does not describe the rate at which entropy shall

be produced, nor how long it would take a system to produce maximum entropy.

There are commonly several situations, in addition to heat flow, where entropy

increases. When Many chemical reactions result in increases of entropy, such as when

gasoline is combusted to propel an automobile. Entropy is increased when substances

become more mixed even where no chemical reaction occurs, such as when helium and

neon gasses become mixed together. Chemical reactions, such as burning coal and oil or

metabolizing sugars also results in entropy production.

3.2 A Simple Thermal Conductor Bridging A Thermal Potential

Most introductory physics textbooks do have an example concerning

thermodynamics that involves time.8 This example concerns a simple thermal conductor

through which energy flows from a warmer reservoir to the cooler reservoir. In most

textbooks, the term reservoir refers to a body whose temperature remains constant

regardless of how much heat energy flows in or out of it.9 However, in this paper, the

temperature of a reservoir will change depending on whether energy is added to or

removed from it.

7 Recall our simple thermal conductor example. As heat energy moves from the warmer to the cooler region, entropy is produced. 8 One can infer the passage of time by multiplying the calculated heat flow by time. However, this is example is not really time dependent. The heat flow remains constant regardless of how much time passes in this idealized example. It is nevertheless a good approximation for many real situations. 9 Heat flow is also proportional to the difference in two temperatures that the thermal conductor bridges. This difference has nothing to do with the conductors themselves. Heat flows through a thermal conductor in proportion to the area of the conductor as well as its thermal conductivity. More heat will flow through a broad conductor than a narrow one. Also, more heat will flow through a material with a high thermal conductivity such as aluminum than through a material with low thermal conductivity such as wood. Heat flow is inversely proportional to the conductor's length. More heat will flow through a shorter conductor than a long one. This is known as Fourier’s heat conduction law


M. Ciotola

21

Recalling the First Law of Thermodynamics, any heat lost by one body must be

gained by another body. The quantity of heat energy lost by the warmer reservoir is

identical to the quantity of heat energy gained by the cooler reservoir:

Heat lost by warmer reservoir = Heat gained by cooler reservoir

Recalling the Second Law of Thermodynamics, if heat flows from the warmer

reservoir to the cooler reservoir, then the total entropy of this system increases. The

entropy of a cooler reservoir increases as its temperature increases. Conversely, if a

warmer reservoir’s temperature decreases as it loses heat energy, then its entropy will

decrease.

In this example, the heat lost by the warmer reservoir is gained by the cooler

reservoir. The magnitude of the entropy increase of the cooler body is greater than the

magnitude of entropy loss by the warmer body, so the total entropy of the combined

warmer reservoir - cooler reservoir system increases.

Picture a well-insulated container of hot water and another of cold water.10 The

container of hot water constitutes a warmer body, which we will call the hot reservoir.

The container of cold water represent a cooler body, which we will call the cold

reservoir. A copper bar thermally bridges the two containers so that heat can flow from

the hot container to the cold container.

Entropy = Entropy (hot reservoir) + entropy (cold reservoir)

10 If ice water is used, then energy due to the phase change of melting ice must also be accounted for.


M. Ciotola

22

Potential entropy is represented by the difference between the quantity of entropy

when equilibrium is reached minus the quantity of entropy before any heat flow takes

place. This will be a positive number. An expression for potential entropy in this case is

the following equation, where m is the mass of the water in each reservoir, c is the

specific heat of water, T is temperature, and R is potential entropy, f refers to final, i

refers to initial, H refers to hot and C refers to cold:

R = mH cwater ln ( TH f / TH i ) + mC cwater ln ( TC f / TC i )

It should be noted that

TH f = TC f

which is the equivalent of stating that the a system has reached equilibrium. In

this context, the terms Hot and Cold refer to specific containers rather than the state of

those containers.

Entropy at the beginning is:

S = mH cwater TH i + mC cwater TC i

Entropy at end is:

S = mH cwater TH f + mC cwater TC f


M. Ciotola

23

Or more simply,

S = ( mH + mC ) cwater T f

Heat flow through a thermal conductor is proportional to the area of the conductor

as well as its thermal conductivity. More heat will flow through a broad conductor than a

narrow one. Also, more heat will flow through a material with a high thermal

conductivity such as aluminum than through one with low thermal conductivity such as

wood. Heat flow is inversely proportional to the conductor's length. Thus, more heat will

flow through a short conductor than a long one.

Heat flow is also proportional to the difference in two temperatures that the

thermal conductor bridges. This difference in temperatures has nothing to do with the

conductors themselves. A greater temperature difference will provide a greater heat flow

across a given conductor, regardless of the characteristics of that conductor.

The equation for heat flow through a conductor is known as Fourier’s Conduction

Law11, where Q is heat flow, k is a thermal constant, A is conductor area, T is

temperature, x is conductor length and t is time:

Q / t = k A T / x

11 Electrical engineers will find this equation similar to an arrangement of Ohms Law, where electric current is proportional to voltage divided by resistance (I = V/R).


M. Ciotola

24

So we know how much heat will flow as a function of the temperature difference

between the two reservoirs. Since we know that the original temperature difference is, we

know the initial condition that dictates the initial heat flow. Since we know the initial heat

flow, we can deduce temperature changes in the reservoirs that then dictate future heat

flow. We can therefore determine how much of the potential entropy is consumed as a

function of time by monitoring each reservoir to determine how much entropy has been

created up to any given point of time. Potential entropy is shown as a function of time in

Figure 2. In the following figures, data was simulated using 1 kg reservoirs of water at

about 273 K and about 373 K, neglecting phase changes.

FIGURE 2 Simple Conductor: Potential Entropy Remaining versus Time

(Simulated data)


M. Ciotola

25

3.3 Simple heat engine

A common example used to illustrate the Second Law is the Heat Engine. To

power a heat engine to function, heat must flow across a temperature difference,12 from a

warmer region to a cooler one. A heat engine uses temperature differences to perform

work. When heat flows to power a heat engine, part of the available energy is put into

work and the remainder results in waste heat. An example is the temperature difference

between a hot flame and a cool tank of water being used in a steam engine.

No engine turns all of the heat flow into work. That would imply 100%

efficiency, which is impossible in theory13

as well as practice, regardless of how well the

engine is constructed. The Second Law of Thermodynamics tells that even the best

engines will produce entropy along with work. The best efficiency that an ideal engine

can achieve is known as its Carnot Efficiency. The Carnot Efficiency is simply the

difference between the warmer and cooler temperature divided by the warmer

temperature. In reality, most engines are a great deal less efficient than even the Carnot

efficiency. Several modern means exist to utilize higher order energy. The equation for

Carnot Efficiency is:

Carnot Efficiency = 1 – (cooler temperature/warmer temperature)

Absolute temperatures must be used to calculate the Carnot Efficiency. Absolute

temperatures are measured from absolute zero, which is the lowest possible temperature

12 Incidentally, a heat engine is a system that has pure physical aspects as well as social aspects. 13 Unless the cold reservoir is at a temperature of absolutely zero, which is nearly impossible, statistically speaking.


M. Ciotola

26

in theory, and has never been quite obtained in practice. Such temperatures are measured

in a kind of degree called Kelvin. 0° Celsius equals about 273 Kelvin.

An example is the temperature difference between a hot flame and a cool tank of

water being used in a steam engine. Then, part of the available energy is used to perform

work and the remainder is exhausted as waste heat. For instance, a steam engine could

contain a piston that converts some of the heat flow into a cyclic in-out motion that

represents work done upon a load, such as a flywheel wheel. Steam released into cooler

air represents waste heat. When waste heat is created, an intangible quantity called

entropy is produced. The more the heat engine works, the more entropy it will produce.14

If heat flows from a warmer object to a cooler object (where no engine is

involved), no work results, but entropy is still produced (or you could say that the entropy

of the system under consideration increases). For example, thermal conduction results in

lots of entropy production but little work. A thermal conductor can be thus thought of as

a lazy heat engine. If a heat engine operates at less than the Carnot ideal, entropy

increases. This is the case for all real heat engines.

A simple heat engine "heat engine" is shown in Figure 3. Most technical details

have been omitted. Here, a tank of water that is presumably heated into high-pressure

steam by a flame or other energy source. The tank contains a piston that converts some of

that heat energy into a cyclic in-out motion that represents work done upon a load, here

represented by the wheel. The little puffs of steam represents the waste heat.

14 In theory, a heat engine is not required to produce entropy if the temperature of the cold region is absolute zero (which is about – 273° C). In practice, such a low temperature is physically impossible.


M. Ciotola

27

FIGURE 3 Simple heat engine

Heat engines need to work between warmer and cooler heat reservoirs. Warmer

heat reservoirs can be flames, hot air or steam, for example. Cooler heat reservoirs can be

ice, cold air or cool water, for example. Air at room temperature can serve as either kind

of reservoir depending on how hot or cold the other reservoir is.

Here we see a heat engine working between a warmer reservoir and a cooler

reservoir, left and right, respectively (Fig. 4). Warmth is represented by greater height

and redder shading (if your version is in color). The higher and redder the left heat

reservoir, the hotter is it. Conversely, coolness is represented by lower height and bluer

shading. The lower and bluer the cooler heat reservoir, the colder it is. Our heat engine

begins operating between a quite hot and a quite cold reservoir as shown here (Figure 4).

FIGURE 4 Heat engine operating between thermal reservoirs

As the heat engine continues to operate, the warmer heat reservoir becomes less hot and

the cool reservoir becomes less cold (Figure 5).


M. Ciotola

28

FIGURE 5 Thermal reservoirs partially depleted

FIGURE 6 Thermal reservoirs completely depleted

Eventually, both the warmer and cooler heat reservoirs reach the same

temperature (Figure 6).15 When this happens, no more work is possible. The heat engine

is no longer operable.

In the real work, all sorts of mechanisms can function as heat engines, such as

bacteria, cattle, humans, automobiles, generators and even organizations and whole

societies. They are driven by a potential which is typically ultimately thermodynamic in

origin and produce work.

Here, the calculation of potential entropy is the same.

R = mH cwater ln ( TH f / TH i ) + mC cwater ln ( TC f / TC i )

15 At this point, the reservoirs are said to be in thermal equilibrium.


M. Ciotola

29

Potential is represented by the amount of entropy when equilibrium is reached minus

entropy at beginning of time.

Entropy at beginning is:

S = mH cwater TH i + mC cwater TC i

Entropy at end is:

S = mH cwater TH f + mC cwater TC f

However, here is where the examples diverge. Instead of a simple thermal

conductor bridging the thermal difference following Fourier’s Conduction Law, we are

now considering a heat engine. The amount of heat transferred by the heat engine is a

fixed about that we set arbitrarily. Once set, it does not change. For example, we could

set the heat engine to transfer 10 MJ per day.

Although the heat engine produces work, that work cannot exceed the amount of

energy transferred. In fact, it will be less.

Q = heat removed from hot reservoir – heat put into work

The percentage of energy removed that is turned into work is the engine’s

efficiency. The efficiency for the engine is a function of potential. The higher the

potential, the higher the efficiency.


M. Ciotola

30

Figure 7 shows the work produced at any given time by a simple heat engine.

(This heat engine works at 50% of the Carnot efficiency; real heat engines are often even

much less efficient). Notice that as time passes, work decreases. Although a constant

amount of heat continues to be removed from the hot reservoir, nevertheless, the

temperature difference between reservoirs is continually decreasing. This means that

efficiency is continually decreasing, which results in the heat engine performing less

work as time passes. When the system reaches equilibrium and there is no longer any

temperature difference, then efficiency will become zero, and no further work will be

possible. This is shown in figure 7 when the curve intersects the time axis.

FIGURE 7 Simple Heat Engine: Production versus Time

(Simulated data)

3.4 Reproducing heat engines


M. Ciotola

31

The work produced by heat engines can be used to build additional heat engines.

These additional heat engines can produce proportionately more work to produce even

more heat engines. This idea is pictured here (Figure 8).

FIGURE 8 Heat engines begetting additional heat engines

The growth of heat engines is then exponential, at least until limiting factors come

into play. This is a key point. Because heat engines can beget heat engines, an

exponential increase in the rate of entropy creation can take place.

Here, entropy creation is proportional to the quantity of heat engines. The e th

Law (also known as fast entropy) favors exponential growth in entropy production, so

fast entropy favors the "spontaneous" appearance and endurance of heat engines. Under

the Second Law alone, the spontaneous appearance of a heat engine is possible, but

improbable. Fast entropy then utilizes those improbable appearances to create self-

sustaining, exponentially growing systems.

As with the simple heat engine example, the amount of heat transferred by each

heat engine is a fixed about that we set arbitrarily.


M. Ciotola

32

Let’s assume 300 J of the work is required to build new heat engines. Let’s further

assume new engines are produced instantaneously at the end of each day and put into

commission at the beginning of each next day. Then heat transferred for the second day

will be due to two heat engines. On the third day due to even more heat engines. So as

long as the potential remains high, the quantity of heat engines, this the quantity of heat

transfer grows exponentially. (Note that the data was simulated in seconds rather than

days; in a real mining region Hubbert curve, building new “engines” may require days or

even years. However, the underlying concepts still apply).

3.5 Bring it all together: Generating a Hewett-Hubbert Curve Using

Thermodynamics

Despite the growing population of heat engines, since the amount of potential is

limited, then the efficiency will begin to drop. Consequently, the amount of work each

heat engine produces drop, even though it still transfers the same amount of energy. So

for a while, the amount of heat engines and total heat transferred may continue to

increase, but the rate of increase will slow down. Eventually efficiency drops to a point

where production drops to zero. In this case, the drop-off is quite sudden and may happen

instantaneously. This results of the situation are shown in Figure 9. Partial heat engines

were allowed to be built in this simulation.


M. Ciotola

33

FIGURE 9 Reproducing Heat Engine: Production versus Time

(Simulated data)

3.6 Reproducing heat engines requiring maintenance

In real situations, heat engines themselves will require work to be maintained. For

example, if the “heat engines” in an old-fashioned mining region are humans, they must

be fed energy-containing food. So part of the work produced must be used to maintain the

heat engine. (In the case of a gold mine, some of the gold is used to purchase food). This

work is a drag. If there is not enough work to maintain the heat engines, the amount of

functioning heat engines and possibly also their efficiency will drop even further (beyond

that due to the second law).

This situation is shown in Figure 10. Note that unlike the immediately above

example (shown in Figure 9), there is not dearly as steep of a drop-off of production.

However, neither is the curve symmetric. It takes longer to build up production, and less

time for it to fall at the end of the Hubbert curve.


M. Ciotola

34

The data simulated in Figures 10 and 11 was generated in a similar manner to the

example in section 3.4, excerpt that the maintenance cost was subtracted from the amount

of work available to build heat engines. If there was not sufficient work to maintain the

engines, then the quantity of engines was decreased until it matched the amount that

could be maintained at a given point of time.

FIGURE 10 Reproducing Heat Engine Requiring Maintenance: Potential Entropy Remaining versus Time

(Simulated data)


M. Ciotola

35

FIGURE 11 Reproducing Heat Engine Requiring Maintenance: Potential Entropy Remaining versus Time

(Simulated data)

3.7 Applying Discounted Exponential Growth to an actual example

This paper assumes an e th Law of Thermodynamics which states that systems

will tend to configure themselves in a manner that maximizes the rate of entropy

production.16 Therefore systems will attempt to grow exponentially where possible.

Hewett-Hubbert (HH) curves actually represent statistical distributions regardless

of how they are generated. HH curves generally represent situations where there are a

considerable number of actors, such as in the case of regional oil production or mining.17

16 For arguments and examples in support of an e th Law of Thermodynamics, see M. Ciotola, Physical History and Economics, Second Edition, 2009. For earlier references, see M. Ciotola, Physical History and

Economics, 2003 and M. Ciotola, LASE, 2001. The Law of Maximum Entropy production proposed by R.

Swenson is related. 17 Such curves apparently originated with D. F. Hewett in 1929 for European metal mining. A crude form of such curves were used by the Technical Alliance and Technocracy, Inc. in the early 1930s, of which M. King Hubbert was a co-founder.


M. Ciotola

36

This allows for the formation of a Hewett-Hubbert distribution, which in turn, makes

thermodynamic modeling possible. Thermodynamic modeling is essentially a form of

statistical modeling. If there is a sufficient amount of independent actors, then an HH

curve can be modeled as a deterministic limiting case, but their statistical nature should

always be remembered.

With the e th Law in mind, we start to model an HH curve by using an pure

exponential growth function, representing output versus time. Let us call production P,

and time t, so that

P = et

In reality, there are additional multipliers (called parameters or constants), but we need

not consider those for this paper. Often, they need to be determined empirically.

However, such a function will quickly grow and approach infinity. However, in

the case of Hewett-Hubbert curves, there are limiting factors such as the amount of a

fixed resource in the ground, as well as increasing average effort needed to acquire each

additional unit of production. The rising rate of additional effort required (on average) to

produce each additional unit represents decreasing efficiency. So we need to discount

pure exponential growth by this decreasing efficiency. A simple way to do so is to

multiply the exponential growth function by efficiency. If we call efficiency d, and d is a

function of time, then

P = et D(t)


M. Ciotola

37

Efficiency is itself a function of a thermodynamic potential driven by the e th

Law. Such a potential might be manifested by a vigorously sought after resource that

only exists in a limited quantity, such as in the case of a region of accessible18 gold

reserves.

A quick way19 to model efficiency is to divide cumulative production so far by the

original total amount of resource (i.e. total possible production) and subtract the result

from one. For example, if the total original amount of the resource is G and the amount

produced thus far is F, then the efficiency could be expressed as

D(t) = 1 – F(t) / G

where F is a function of time. The resulting equation is a Hewett-Hubbert curve:

P = et ( 1 – F(t) / G )

Figure 1 shows a Hubbert curve for transferred gold and silver production in

Spanish New World territories from the period of about 1516 to about 1650. A power

profile is shown in HH curve format for the Habsburg dynasty that governed Spain

during this period.

18 In this context, the term “accessible” refers to the amount of the resource than can be extracted using

technology and other means that available either at present or during the period of exploitation. 19

There are better ways to determine efficiency, but this method allows for a quick first approximation. A

more fundamental approach is shown later in this paper.


M. Ciotola

38

FIGURE 1 Imports of New World Gold and Silver to 1665

(data from Gibson)

We can also generate a Hewett-Hubbert Curve directly from fundamental

thermodynamic principles, as will be shown later.

4 APPLICATION OF DISCOUNTED EXPONENTIAL APPROACH TO PETROLEUM PRODUCTION

This section will compare the strengths and weaknesses of each approach with

each other. Then both methods will be combines to determine if residuals can be further

reduced.

4.1 Regional Field Hewett-Hubbert Approach

In the early 1900s, David Foster Hewett modeled metal production in European

regions as a downward-facing curve with a peak where maximum metal production

occurred, and beginning and end-points where significant production began and ended.


M. Ciotola

39

(Hewett, 1921). M. King Hubbert refined this model and applied it to petroleum

production (Hubbert, 1947, etc.). Although Hubbert was employed by Shell Oil during

much of his work, Hubbert’s approach has been viciously criticized. Most of the

objections have been political or faith-based. However, other objections have been

physical, in that Hubbert originally only counted production for the lower 48 U.S. states

and did not include Alaska, foreign source or deep-water sources. (Deffayse, etc.).

Although such arguments claim to invalidate Hubbert’s methodology, all they really do

physically is to shift the production peak to a later point of time.

4.2 Hubbert basic Formula

An initial approach to create a Hubbert formula is simply to create an inverted

parabola in the form of

y = -x2,

(Hubbert, 1980) and then fit the parabola as best as possible to the data obtained thus far.

A more sophisticated approach is to fit the data to a normal distribution such as:

y = e-t^2.

This approach works reasonably well as a first approximation. Yet uncertainly as to

where to locate the peak of the distribution can involve considerable debate and result in


M. Ciotola

40

widely divergent forecasts. (See Deffeyse, 2001). Hubbert eventually developed a

somewhat different formula, but one which essentially produced a bell curve.

4.3 Discounted Exponential Growth Approach

The discounted exponential growth approach models a system as exhibiting pure

exponential growth, but then discounts that growth by a diminishing efficiency that is a

partial function of remaining potential. This approach can be modeled iteratively or

analytically.

4.4 Discounted Exponential Growth Modeled With Iterative Program

The easiest way to utilize the discounted exponential growth approach is to run

the model iteratively though a quantitative processing platform such as Microsoft Excel

or Python. This approach is demonstrated as follows. Work is calculated for each period,

and then is used to “construct” additional heat engines. (In the case of petroleum, a heat

engine could be analogous to an oil well). This process is ideally repeated until infinity,

but in practice is run until production drops below the initial level.

A Python program was written to generate the model. However, due to the limited

number of data points, the modeled was recreated using Microsoft Excel.

The e th Law approach is used to generate a Hewett-Hubbert distribution to model

data from M. K. Hubbert’s 1949 paper in Science magazine (see Figure 12). Here, only

the initial stages of a curve have formed, chiefly the pure exponential growth stage. Note

that this data only goes up to about 1947. The purpose of the following oil production

charts is to illustrate the general e th Law methodology and how it might be utilized to


M. Ciotola

41

analyze oil production. There are people with more experience in petroleum modeling

with the author, and they might be able to produce a more precise application of the e th

Law.

FIGURE 12 Model data compared with data up to 1947 used by Hubbbert

The e th Law approach is then used to generate a Hewett-Hubbert distribution to model

U.S. oil production (see Figure 13). Here, more of a curve is shown, including the peak

and part of the decline.


M. Ciotola

42

FIGURE 13 Model data compared with actual data for US Oil Production versus Year

4.5 Implementing Discounted Exponential Growth Modeled Analytically

Expressing the discounted exponential growth approach analytically is more

challenging. A simple model involving a pure exponential growth function multiplied by

a basic efficiency function is conceptually reasonable, but provides insufficient

flexibility.

The first term on the right represents pure exponential growth (e.g. see below).

The Second term on the right represents efficiency. This function successfully produces a

bell-shaped curve as shown below. For application to an actual case, appropriate

parameters in the form of constants will need to be included, such as to represent initial

conditions.


M. Ciotola

43

The pure exponential growth function can be modeled either as a function of time

or of cumulative production. We will model it here as a function of cumulative

production. This term is fairly straightforward.

y = k • ex,

where k represents the initial starting condition.

4.5.1 Efficiency Term Modeled As Simple Linear Function

The simplest approach is to create efficiency as a simple linear function with a

negative slope.

y= - mx + b,

where m is the slope. Such a linear approach captures improvements in economies of

scale and better technology, thereby appropriately mitigating the initial steep drop-off of

intrinsic efficiency. A linear approach will be utilized for our model.

So we will could the following type of formula to model our approach:

y = ex mx + b( )


M. Ciotola

44

Admittedly, this curve provides too steep an intersection with the x axis, but for

petroleum, that point will be of relatively little interest. For petroleum, b represents initial

efficiency, while m represents the steepness of the efficiency term. (Perhaps the

efficiency term should be replaced with the logistic equation (e.g. beginning at 1 and

ending at zero).

4.5.2 Efficiency Term Modeled In Terms of Heat Flow

The efficiency term is more challenging. We know that it starts high (but at less

than one) and ends low (but not necessarily at zero). A direct approach is:

y = 1 – (Tc/Th)

Where the difference Th - Tc represents a potential difference, or the “punch” of the

potential. Yet what are Tc and Th in the context of petroleum production? They could be

said to represent cost and benefit, respectively. Th could represent the selling price of oil,

while Tc could represent the cost of production. A high Th and low Tc would represent a

high net benefit (or, in the economics sense, “economic rent”). Th = Tc represents no net

benefit. Generally speaking, efficiency is proportionate to net benefit.

Recognizing that the reservoir temperatures may themselves be exponential

functions of the potential, then the full equation can be expressed as follows:

y =peak

parameter( )ekx 1 TcTh( ) Th+(T Th )e ( 0.001 T ( x 6)

Tc+(T Tc )e ( 0.001 T ( x 6)( )( )


M. Ciotola

45

or

y =peak

parameter( )ekx 1 T TpT +Tp( ) (T +Tp )+( Tp )e ( 0.001 T ( x 6)

(T Tp )+(Tp )e ( 0.001 T ( x 6)( )( )

Th is analogous to benefit. Tc is analogous to cost. Here, T = (Th + Tc)/2. T can possibly

be viewed as market price.

FIGURE 14 Production Versus Time Involving Efficiency Discounted Exponential Growth

4.5.3 Efficiency Term Modeled In Terms of Entropy Increase

If heat flow is translated into entropy increase, then it is possible to simply divide

cumulative entropy production so far by the original total amount of potential entropy

and subtract the result from 1. For example, if the total original amount of the potential


M. Ciotola

46

entropy is R0 and the amount produced thus far is St, then the efficiency could be

expressed as

E(t) = 1 – S(t) / R0

where F is a function of time. The full equation becomes:

P = et ( 1 – S(t) / R0 )

4.5.4 Comparison With Logistic Function

The above equation is similar to the logistic function:

dP/dt = P(t) ( 1 – P(t) / N )

where dP/dt represents population change with respect to time, P represents population,

and N represents a constant known as the carrying capacity. Hubbert uses the logistics

equation in part of his model (Hubbert, 1980), However, The logistics equation replaces

the exponential term with different term and does itself not produce a bell-shaped curve,

but rather approaches the constant N as t approaches infinity. This similarity suggests that

the logistics equation itself can be derived using thermodynamics.

5 COMPARISONS AND CONCLUSIONS


M. Ciotola

47

This paper has shown that it is possible to generate curves that resemble Hubbert

from well-known thermodynamic principles, even if the e th Law is not fully accepted.

This approach was then used to model transferred gold and silver (analogous to

production) from Spain’s former New World territories. Application of the model gave a

fair visual fit. The thermodynamic approach provides a fundamental approach that can

then be generalized to apply to a wide range of large-scale human phenomena.

Standard forecasting approaches produced varying results (see table, below; at

most, only six significant digits are shown since that is the level of precision of the data).

The naive method rapidly produced higher error with increasing time. Decomposition

showed seasonality, including a substantial drop in production during February.

Forecasting using linear trend analysis greatly reduced the error, especially within a

decade of forecasts. Forecasting using quadratic trend analysis provided a noticeable

improvement over use of the linear trend.

Naive (monthly; 1970-1999)

Exponential Smoothing (monthly; 1980-1999)

Decom- position (monthly)

Linear Trend (annual)

Quadratic Trend (annual; 1970-1999)

Description

MAPE 4.14479% 2% 3% 4.27524% 3.46434%

MAD, MAE

8,988.93 5590 7,402.00 1.23858E+05 1.04881E+05

MSD, MSE

123,243,000 5,6421,900 89,458,000 2.59980E+10 1.73331E+10

Regression utilizing a quadratic trend is most similar standard method to the

discounted exponential growth method. This discounted exponential growth method

provides reasonable agreement for modeling the initial Hubbert (1949) upside data, given

the wild swings of the great depression and World War II, but was arguably less effective


M. Ciotola

48

in modeling downside data (see table below), although this may be due to error induced

by the manual choice of parameters. Also, this forecast could possibly be improved by

utilizing a linear potential.

Quadratic Trend (annual; 1970-1999)

Discounted Exponential Growth; Iterative; Upside (annual, 1920-47)

Discounted Exponential Growth; Iterative; Downside (annual, 1970-99)

Description

MAPE 3.46434% 11.26% 7.886707%

MAD, MAE

104,881 313405

218,980

MSD, MSE

1.73331E+10 1.43036E+11

4.68387E+11

Areas for future study and improvement could involve studying the sensitivity of

error to the selection of the time intervals modeled and forecasted under each of the

methods.


M. Ciotola

49

REFERENCES: Ahlbrandt, Thomas S. (2002), Future oil energy resources of the world. U. S. Geological Survey, Denver, CO, United States. International Geology Review. 44; 12, Pages 1092-1104. 2002. Winston & Son. Silver Spring, MD, United States. Akers, Keith (2007), The Jesus Family Tomb -- and Peak Oil. http://keithakers.com/Jesus-Family-Tomb-Peak-Oil.htm, last viewed on May 5, 2010. BBC News, "China's global hunt for oil", March 9, 2005. http://news.bbc.co.uk/2/hi/business/4191683.stm, last viewed on April 6, 2010. BBC News, "China's gets foreign oil incentives", March 1, 2007. http://news.bbc.co.uk/2/hi/business/6407337.stm, last viewed on April 6, 2010. BP (formerly British Petroleum) (2004), "Proven Oil Reserves", Geographical, 0016-741X, July 1, 2004, Vol.76, Issue 7 (which relies upon BP Statistical Review of World Energy, June 2003). Burt, J. A., "The thermal-wave lens", Can. J. Phys. 64: 1053, 198681995. Carroll, B. and D. Ostlie (2007), An Introduction to Modern Astrophysics, 2

nd Ed. Pearson Addison-Wesley.

Chapman, R.E., Oil Geology. Amsterdam: Elsevier, 1983. Ciotola, M. (1997), "San Juan Mining Region Case Study: Application of Maxwell-Boltzmann Distribution Function", Journal of Physical History and Economics, Vol 1. Ciotola, M. (2001), "Factors Affecting Calculation of L", Kingsley, S., R. Bhathal, ed.s, Conference Proceedings, International Society for Optical Engineering (SPIE), Vol. 4273. Ciotola, M. (2002), Hurtling Towards Heat Death (talk delivered at San Francisco State University). September,. Ciotola, M. (2003), Physical History and Economics. Pavilion Press, San Francisco. Ciotola, M. (2006), "Thermodynamic Perspective on Profits", North American Technocrat, Vol. 5, Issue 18. Deffeyes, Kenneth S. (2001), Hubbert's Peak. Princeton: Princeton University Press. Energy Information Administration, U.S. Oil Production figures, 2009, 2010. Energy Information Administration, World Petroleum Consumption, Annual Estimates, 1980-2008, 2009. Energy Information Administration, International Energy Outlook 2004. Energy Information Administration, February 2010 International Petroleum Monthly, March 10, 2010.


M. Ciotola

50

Energy Information Administration, Short-Term Energy Outlook, March 8, 2005. Energy Information Administration, Petroleum Marketing Annual 2003. Energy Information Administration, Oil and Gas Supply Module Documentation, 2004. Energy Information Administration, World Proved Crude Oil Reserves, 2009. Garraty J. A. and P. Gay (1972), Columbia History of the World. Dorset Press (Harper and Rowe). Georgescu-Roegen, N. (1971), The Entropy Law and the Economic Process. Harvard University Press. Gibson, C. (1966), Spain in America. Harper and Row. Hanke, John E., Wichern, Dean W. (2005), Business Forecasting, 8

th Ed. Prentice Hall.

Heilbroner, R. L. (1980), The Worldly Philosophers 5

th Ed. (Touchstone (Simon and Schuster).

Hewett, D. F. (1929), Cycles in Metal Production, Technical Publication 183. The American Institute of of Mining and Metallurgical Engineers. Hubbert, M. K. (1936), "Man-Hours and Distribution", Technocracy. Series A, No. 8. Hubbert, M. K. (1947), "Energy from Fossil Fuels", Science. Vol. 109. Hubbert, M. King, 1980. J. Lunine, (2005), Astrobiology, A Multidisciplinary Approach. Pearson Addison Wesley. Mazour, A. G., and J. M. Peoples (1975), Men and Nations, A World History, 3

rd Ed., Harcourt, Brace, Jovanovich.

Meadows, D, and D. Meadows, J. Randers and W. Behrens III (1972), Limits to Growth, Club of Rome. Prigogine, I. (1967), Introduction to Thermodynamics of Irreversible Processes, Wiley, New York, pp. 67. ff. Riva, Joseph P. (1983), World Oil and Reserves, Boulder, Colorado: Westview Press, Inc. Schroeder, D. V. (2000), Introduction to Thermal Physics. Addison Wesley Longman. Stowe, K. (1984), Introduction to Statistical Mechanics and Thermodynamics. John Wiley & Sons. United States Geological Survey, 2010, etc. Note on references:


M. Ciotola

51

The author independently conceived the statements made in this essay, with the exceptions, of course, of the 1st and 2nd Laws of Thermodynamics and its application to heat engines, and Hubbert’s peak production hypothesis. However, so many people have written on thermodynamics, that the author does not claim the statements are new. Nevertheless, to the author’s knowledge, the author is the first to synthesize all of these statements into an integrated whole. The author has subsequently been introduced to relevant the prior work of R. Swenson and I. Prigogine who have themselves discovered some of the major pieces.


M. Ciotola

52

APPENDICES TIME SERIES DECOMPOSITION FOR C2 (data source: U.S. Energy Information Administration) Multiplicative Model Data C2 Length 240 NMissing 0 Fitted Trend Equation: Yt = 282374 - 420.937*t Seasonal Indices Period Index 1 1.02314 2 0.92934 3 1.02463 4 0.99138 5 1.01979 6 0.98262

7 1.01041 8 1.00651 9 0.98042 10 1.02164 11 0.98736 12 1.02275

Accuracy Measures

MAPE 3 MAD 7402 MSD 89458039

Forecasts Period Forecast Period Forecast 241 185114 242 167753 243 184521 244 178117 245 182792 246 175715 247 180261 248 179141

249 174084 250 180974 251 174485 252 180309 253 179946 254 163059 255 179346 256 173109 257 177641


M. Ciotola

53

258 170751 259 175157 260 174057 261 169132 262 175813 263 169498 264 175143 265 174778 266 158365 267 174170 268 168102 269 172490 270 165788 271 170053 272 168973 273 164179 274 170652 275 164511 276 169977 277 169610 278 153670 279 168995 280 163094


M. Ciotola

54

US PETROLEUM PRODUCTION 1979-1999 TIME SERIES DECOMPOSITION Minitab log (data source: U.S. Energy Information Administration) Time Series Decomposition for C2 Multiplicative Model Data C2 Length 360 NMissing 0 Fitted Trend Equation: Yt = 294539 - 278.633*t Seasonal Indices Period Index Period Index

1 1.01828 2 0.92982 3 1.02459 4 0.99078 5 1.02156 6 0.98313

7 1.01172 8 1.01052 9 0.98093 10 1.02217 11 0.98828 12 1.01822

Accuracy Measures

MAPE 5 MAD 11057 MSD 197402283

Forecasts Period Forecast Period Forecast 361 197498 362 180083 363 198152 364 191336 365 196996 366 189310

367 194534 368 194022 369 188067 370 195690 371 188927 372 194366


M. Ciotola

55

373 194094 374 176974 375 194726 376 188023 377 193580 378 186023 379 191151 380 190643 381 184787 382 192272 383 185622 384 190962 385 190689 386 173865 387 191300 388 184710 389 190165 390 182736 391 187768 392 187264 393 181507 394 188854 395 182318 396 187557 397 187284 398 170756 399 187875 400 181398 401 186749 402 179449 403 184385 404 183885 405 178228 406 185437 407 179013 408 184153 409 183879 410 167647 411 184449 412 178085 413 183333 414 176162 415 181003 416 180507 417 174948 418 182019 419 175709 420 180748

421 180475 422 164538 423 181023 424 174772 425 179918 426 172874 427 177620 428 177128 429 171668 430 178601 431 172404 432 177344 433 177070 434 161429 435 177597 436 171459 437 176502 438 169587 439 174237 440 173749 441 168388 442 175183 443 169100 444 173939 445 173665 446 158320 447 174171 448 168147 449 173086 450 166300 451 170854 452 170370 453 165108 454 171766 455 165796 456 170535 457 170261 458 155211 459 170745 460 164834 461 169671 462 163013 463 167471 464 166992 465 161828 466 168348 467 162491 468 167130


M. Ciotola

56

469 166856 470 152102 471 167320 472 161521 473 166255 474 159726 475 164089 476 163613 477 158549 478 164930 479 159187 480 163725 481 163451 482 148993


M. Ciotola

57

US PETROLEUM PRODUCTION 1980-1999 TIME SERIES DECOMPOSITION Minitab log (data source: U.S. Energy Information Administration) Time Series Decomposition for C2 Multiplicative Model Data C2 Length 240 NMissing 0 Fitted Trend Equation: Yt = 282374 - 420.937*t Seasonal Indices

Period Index Period Index

1 1.02314

2 0.92934 3 1.02463 4 0.99138 5 1.01979 6 0.98262

7 1.01041 8 1.00651 9 0.98042 10 1.02164 11 0.98736 12 1.02275

Accuracy Measures

MAPE 3 MAD 7402 MSD 89458039

Forecasts Period Forecast Period Forecast 241 185114 242 167753 243 184521 244 178117 245 182792 246 175715 247 180261

248 179141 249 174084 250 180974 251 174485 252 180309 253 179946 254 163059


M. Ciotola

58

255 179346 256 173109 257 177641 258 170751 259 175157 260 174057 261 169132 262 175813 263 169498 264 175143 265 174778 266 158365 267 174170

268 168102 269 172490 270 165788 271 170053 272 168973 273 164179 274 170652 275 164511 276 169977 277 169610 278 153670 279 168995 280 163094


M. Ciotola

59

US PETROLEUM PRODUCTION 1979-1999 REGRESSION: LINEAR MODEL Minitab log (data source: U.S. Energy Information Administration) Trend Analysis for C2 Data C2 Length 30 NMissing 0 Fitted Trend Equation: Yt = 3552631 - 40120.1*t Accuracy Measures

MAPE 4.27524E+00 MAD 1.23858E+05 MSD 2.59980E+10


M. Ciotola

60

US PETROLEUM PRODUCTION 1970-1999 REGRESSION: QUADRATIC MODEL Minitab log (data source: U.S. Energy Information Administration) Trend Analysis for C2 Data C2 Length 30 NMissing 0 Fitted Trend Equation: Yt = 3322570 + 3016*t - 1391*t**2 Accuracy Measures

MAPE 3.46434E+00 MAD 1.04881E+05 MSD 1.73331E+10


M. Ciotola

61

US PETROLEUM PRODUCTION BY MONTH 1970-1999 REGRESSION: LINEAR MODEL Minitab log (data source: U.S. Energy Information Administration) Results for: Minitab_US_Petr_1970_1999_mo.MTW Trend Analysis for C2 Data C2 Length 360 NMissing 0 Fitted Trend Equation: Yt = 294447 - 278.207*t Accuracy Measures

MAPE 5 MAD 12128 MSD 234705915

Forecasts Period Forecast Period Forecast 361 194014 362 193736 363 193458 364 193180 365 192902 366 192623 367 192345 368 192067 369 191789 370 191511 371 191232 372 190954 373 190676 374 190398 375 190120 376 189841 377 189563

378 189285 379 189007 380 188729 381 188450 382 188172 383 187894 384 187616 385 187338 386 187059 387 186781 388 186503 389 186225 390 185946 391 185668 392 185390 393 185112 394 184834


M. Ciotola

62

395 184555 396 184277 397 183999 398 183721 399 183443 400 183164


M. Ciotola

63

US PETROLEUM PRODUCTION BY MONTH 1970-1999 REGRESSION: LINEAR MODEL Minitab log (data source: U.S. Energy Information Administration) Trend Analysis Plot for C2 Trend Analysis for C2 Data C2 Length 360 NMissing 0 Fitted Trend Equation: Yt = 294447 - 278.207*t Accuracy Measures

MAPE 5 MAD 12128 MSD 234705915

Forecasts Period Forecast 361 194014 362 193736 363 193458 364 193180 365 192902 366 192623 367 192345 368 192067 369 191789 370 191511 371 191232 372 190954 373 190676 374 190398 375 190120

376 189841 377 189563 378 189285 379 189007 380 188729 381 188450 382 188172 383 187894 384 187616 385 187338 386 187059 387 186781 388 186503 389 186225 390 185946


M. Ciotola

64

391 185668 392 185390 393 185112 394 184834 395 184555 396 184277 397 183999 398 183721 399 183443 400 183164 401 182886 402 182608 403 182330 404 182052 405 181773 406 181495 407 181217 408 180939 409 180661 410 180382 411 180104 412 179826 413 179548 414 179270 415 178991 416 178713 417 178435 418 178157 419 177878 420 177600 421 177322 422 177044 423 176766 424 176487 425 176209 426 175931 427 175653 428 175375 429 175096 430 174818 431 174540 432 174262 433 173984 434 173705 435 173427 436 173149

437 172871 438 172593 439 172314 440 172036 441 171758 442 171480 443 171202 444 170923 445 170645 446 170367 447 170089 448 169810 449 169532 450 169254 451 168976 452 168698 453 168419 454 168141 455 167863 456 167585 457 167307 458 167028 459 166750 460 166472 461 166194 462 165916 463 165637 464 165359 465 165081 466 164803 467 164525 468 164246 469 163968 470 163690 471 163412 472 163134 473 162855 474 162577 475 162299 476 162021 477 161742 478 161464 479 161186 480 160908 481 160630 482 160351


M. Ciotola

65

GRAPHS 2.5 Simple Exponential Smoothing (monthly data, 1980-99: without and with forecast)

240216192168144120967248241

300000

275000

250000

225000

200000

175000

150000

Index

C2

Alpha 0.246885

Smoothing Constant

MAPE 2

MAD 5590

MSD 56420905

Accuracy Measures

Actual

Fits

Variable

Smoothing Plot for C2Single Exponential Method

2502252001751501251007550251

300000

275000

250000

225000

200000

175000

150000

Index

C2

Alpha 0.246885

Smoothing Constant

MAPE 2

MAD 5590

MSD 56420905

Accuracy Measures

Actual

Fits

Forecasts

95.0% PI

Variable

Smoothing Plot for C2Single Exponential Method

2.8 Time Series Decomposition (monthly data, 1980-99: components; with forecast)

24019214496481

280000

240000

200000

160000

Index

24019214496481

280000

240000

200000

160000

Index

24019214496481

1.1

1.0

0.9

Index

24019214496481

20000

0

-20000

Index

Component Analysis for C2Multiplicative Model

Original Data

Seasonally Adjusted Data

Detrended Data

Seas. Adj. and Detr. Data

2802522241961681401128456281

300000

275000

250000

225000

200000

175000

150000

Index

C2

MAPE 3

MAD 7402

MSD 89458039

Accuracy Measures

Actual

Fits

Trend

Forecasts

Variable

Time Series Decomposition Plot for C2Multiplicative Model

2.8 Time Series Decomposition, cont. (monthly data, 1980-99: seasonal analysis)

121110987654321

1.02

0.99

0.96

0.93

121110987654321

7.5

5.0

2.5

0.0

121110987654321

1.1

1.0

0.9

121110987654321

20000

0

-20000

Seasonal Analysis for C2Multiplicative Model

Seasonal Indices

Percent Variation by Season

Detrended Data by Season

Residuals by Season

[See next page for further charts]

2.9 Regression (annual data, 1970-99: linear trend; quadratic trend)

30272421181512963

3600000

3400000

3200000

3000000

2800000

2600000

2400000

2200000

2000000

Index

C2

MAPE 4.27524E+00

MAD 1.23858E+05

MSD 2.59980E+10

Accuracy Measures

Actual

Fits

Variable

Trend Analysis Plot for C2Linear Trend Model

Yt = 3552631 - 40120.1*t

30272421181512963

3600000

3400000

3200000

3000000

2800000

2600000

2400000

2200000

2000000

Index

C2

MAPE 3.46434E+00

MAD 1.04881E+05

MSD 1.73331E+10

Accuracy Measures

Actual

Fits

Variable

Trend Analysis Plot for C2Quadratic Trend Model

Yt = 3322570 + 3016*t - 1391*t**2

modeling us petroleum production using standard and discounted exponential growth approaches

Documents