Lab-1.5-final.docx

LAB 1.5 MODELLING OIL PRODUCTION ABSTRACT In the experiment, the U.S. and world crude oil production was modelled using a logistic growth model from year 1920-24 to 2005-08. In the U.S. crude oil production, two models with carrying capacities 200- and 300-billion barrels were compared to the actual data and theoretical model resulting the model with 200 carrying capacity to be more accurate. On the other hand, for the world oil production, two models with carrying capacities, 2.1 and 3 trillion barrels were also determined along with the time when the rate of production of oil reaches their maximum, which were both 5 years from 1920-24, 1925-29. It was also concluded that an increase in price affects these models by the increase in growth rate and decreasing the carrying capacity.

I. INTRODUCTION According to Deffeyes (2001), global oil production reached a peak sometime during last decade. After the peak, the world’s production of crude oil will fall, never to rise again. The world will not run out of energy, but developing alternative energy sources on a large scale will take at least 10 years. The slowdown in oil production may already be beginning; the current price fluctuations for crude oil and natural gas may be the preamble to a major crisis. There are two things that are clear about crude oil. One is that we use a lot of it. The world consumption of crude oil is approximately 80 million barrels per day, and world consumption grew by 3.4% in 2004. The other is that the earth’s oil reserves are finite. The processes that created the crude oil that we use today are fairly well understood. There may be significant deposits of crude oil yet to be discovered, but it is a limited resource. Governments, economists, and scientists argue endlessly about almost every other aspect of oil production. Exactly how much oil is left in the earth and what fraction of that oil can or will ever be removed is difficult to estimate and has significant financial ramifications. Substantial disagreement on oil policy is not surprising. Predictions of the decline in production are notoriously difficult, and it is easy to find examples of such predictions that ended up being absurdly wrong. On the other hand, sometimes predictions of decline in production are accurate. In Hubbert’s Peak, Kenneth Deffeyes recounts the work of geologist M. King Hubbert. Hubbert fits a logistic model to the production data for crude oil in the United States. Using production data up to the mid 1950s along with approximations of the total amount of recoverable crude oil, Hubbert predicted that production would peak in the U.S. in the 1970s. Research said, he was right. The method he used will thus be integrated in this experiment (Blanchard, Devaney, & Hall, 2011). Logically, the decline in supply of crude oil will certainly result in an increase in price of oil products. This price increase will provide more funds for crude oil production, perhaps slowing the rate of decline. Hence, this factor is mostly considered in modelling oil productions. In this lab we will model the U.S. and world crude oil production using a logistic model, where the carrying capacity represents the total possible recoverable crude oil.

II. OBJECTIVES The general objective of the study is to model the U.S. and world crude oil production using a logistic model. Specifically, the study aims to: A. Model the crude oil production of the U.S. by assuming that the total amount of recoverable crude oil in the U.S. are 200 and 300 billion barrels; B. Model the world crude oil production based on estimates of total recoverable crude oil (past and future) of 2.1 trillion barrels and of 3 trillion barrels; i. Using the models, predict when do the rate of production of oil reaches its maximum; ii. Describe how price increase might affect the predictions of the model for world oil production; and iii. Modify the model to reflect these assumptions. III. METHODOLOGY The development of the logistic model of U.S. and world crude oil productions will be based on the following data shown in the table below. Table 1: Oil production per five year periods in billions of barrels Year

U.S. Oil

World Oil

Year

U.S. Oil

World Oil

1920-24

2.9

4.3

1965-69

15.8

65.4

1925-29

4.2

6.2

1970-74

17.0

93.9

1930-34

4.3

7.0

1975-79

15.3

107

1935-39

5.8

9.6

1980-84

15.8

101

1940-44

7.5

11.3

1985-89

15.2

104

1945-49

9.2

15.2

1990-94

12.9

110

1950-54

11.2

22.4

1995-99

11.5

118

1955-59

12.7

31.9

2000-04

10.4

126

1960-64

13.4

44.6

2005-08

7.4

107

The differential equation for logistic growth is dP P   k 1   P dt N 

where dP dt

: the change between U.S. Oil production per five year periods

N

: the recoverable crude oil

P

: Population

k

: growth-rate coefficient for population P

Using separation of variables, it can be shown that the solution of the logistic equation is as follows: dP P   k 1   P dt N  dP N P  k P  dt  N  dP k  dt PN  P  N

Integrating left hand side by partial fractions,

1 A B   PN  P  P N  P  1   N  P A  BP 1  AN  AP  BP At constant P: 0  A  B A B

At P = 0: 1  AN  A0  B0 1  AN 1 N 1 B N A

Thus,

1 1 1 1      PN  P  N  P N  P  dP 1  dP dP      PN  P  N  P N  P   1 N

 dP

dP



1

  P  N  P    N  kdt

 ln P   ln N  P   k t  C  N P ln    k t  C  P 

Transforming to anti-logarithmic form, the equation gives, NP  e  kt e C P NP  Ce  kt P

At t = 0; P = 0; N  P0  Ce 0  C P0

Substituting it to the equation N P  Ce  kt P N  1  Ce  kt P

We obtain P t  

N 1  Ce  kt

where C is an arbitrary constant that represents the parameter which depends on the initial condition P0.

Also, using the Logistic Differential Equation, dP P   kP1   dt N 

We transform it into a linear equation, by dividing P,

1 P 1 P Hence,

dP P   k 1   dt  N dP kP k dt N

y

1 dP P dt

mslope  

k N

xP

 y  int ercept  b  k Using this equation we will be able to get the value of m (slope) and k (y-intercept) by plotting the values of

1 dP along the y-axis and P along the x-axis. p dt

Firstly, using the method above, the parameter values for a logistic differential equation that fit the crude oil production data for the U.S. (see Table 1) are determined. Secondly, the model of the crude oil production of U.S. assuming that the total amount of recoverable crude oil in the U.S. is 200 billion barrels is calculated. This assumption includes what has already been recovered and serves the role of the carrying capacity in the logistic model. This will also be repeated by replacing 200 billion barrels with 300 billion barrels. Thirdly, the model of the world crude oil production based on estimates of total recoverable crude oil (past and future) of 2.1 trillion barrels and of 3 trillion barrels are also made. (Both of these estimates are commonly used. They are based on differing assumptions concerning what it means for crude oil to be “recoverable.”). Aside from these, the time when the models’ rate of production of oil reaches its maximum is also predicted. And lastly, a description of how price increase might affect the predictions of the model for world oil production are explained.

IV. RESULTS AND DISCUSSIONS A. U.S. Crude Oil Production Model The first order of business is finding the parameter values for a logistic differential equation that fits the crude oil production data for the US. These data come from column two of table 1. Using Excel,

1 dP is calculated and plotted against P. This was done in order to determine k, p dt

actual N, and the arbitrary constant C.

Table 2: U.S. Oil Production Raw Data Time

Population

P

1/P

dP/dt

1/P*dP/dt

0

2.9

2.9

0.34482759

0.84

0.289655

5

4.2

7.1

0.14084507

0.86

0.121127

10

4.3

11.4

0.0877193

1.16

0.101754

15

5.8

17.2

0.05813953

1.5

0.087209

20

7.5

24.7

0.04048583

1.84

0.074494

25

9.2

33.9

0.02949853

2.24

0.066077

30

11.2

45.1

0.02217295

2.54

0.056319

35

12.7

57.8

0.01730104

2.68

0.046367

40

13.4

71.2

0.01404494

3.16

0.044382

45

15.8

87

0.01149425

3.4

0.03908

50

17

104

0.00961538

3.06

0.029423

55

15.3

119.3

0.00838223

3.16

0.026488

60

15.8

135.1

0.00740192

3.04

0.022502

65

15.2

150.3

0.00665336

2.58

0.017166

70

12.9

163.2

0.00612745

2.3

0.014093

75

11.5

174.7

0.0057241

2.08

0.011906

80

10.4

185.1

0.00540249

1.48

0.007996

85

7.4

192.5

0.00519481

---

---

Graphing it, we get

Figure 1: U.S Oil Production

1 dP versus P p dt

Given the trendline equation y  0.0008 x  0.1241 m

k N

 0.1241 N N  155 .125

 0.0008 

Getting C, C

N  Po Po

155 .125  2.9 2 .9 C  52 .49138

C

The theoretical logistic model then derived from the actual data is Pt  

N 1  Ce  kt 155 .125 Pt   1  52.49138 e 0.1241t

Plotting it,

Figure 2: Graphical representation of the U.S. Oil Production using the Theoretical Model

Figure 3: Comparison between Actual Data and Theoretical Model In figure 3, the theoretical model fit with the actual data from time 0 to 17, approximately. From that point onwards, they started deviating from each other. That being said, it can be concluded that predicting both the growth rate and the total amount of recoverable crude oil from the data is difficult. Hence, a given N with values 200 and 300 billion barrels are assumed. The models derived from these, will then be fitted against the actual production values to determine which is more accurate.

i. Using 200 billion barrels as the total amount of recoverable crude oil in the U.S. At year 1920-24, we let P(0) = 2.9 and t = 5 years to calculate C and k which are the parameters values to get the logistic equation. For U.S. Oil production, the initial condition is assumed: at year 1920-24, we let P(0) = 2.9 and t = 5 years. From this data, the value of growth-rate constant, k, and the arbitrary constant, C, is determined; 1  N  Po ln  k  Po

   1  200  2.9  k  ln   5  2.9  k  0.8438000954 k  0.8438 t

For C, C

N  P0 P0

200  2.9 2.9 C  67.96551724 C  67.9655

C

Hence, the model is P

Sketching t versus P, we get

200 1  67.9655 e 0.8438t

Figure 4: Graphical representation of the U.S. Oil Production with N= 200 billion barrels It was stated by Blanchard et al. (2011) that P(t) increases if 0 < P < N. The same is observed in the figure above, the curve increased upward given the conditions 0 < P < N from Table 1. This model approached and leveled off at the carrying capacity N = 200 billion barrels. ii. Using 300 billion barrels as the total amount of recoverable crude oil in the U.S. The differential logistic equation with 300 billion barrels carrying capacity is dP P    k 1  P dt  300 

Determining the parameter value of k and C given that the total amount of recoverable crude oil N, 1  N  Po ln  k  Po

   1  300  2.9  k  ln   5  2.9  k  0.9258716091 k  0.9259 t

Getting C,

C

N  P0 P0

300  2.9 2.9 C  102 .4482759 C  102 .448

C

The model is then P

300 1  102 .448e 0.9259t

Graphing it,

Figure 5: Graphical representation of the U.S. Oil Production with N= 300 billion barrels

iii. Comparison between the Models

Figure 6: U.S. Oil Production Models with different N In Figure 8, it can be observed that by varying the value of N, the point where the curve approaches equilibrium also changes. At N = 200, the curve slowly increased until it leveled off at 200. The same can be said for N = 300. Hence, increasing N also increases the maximum amount of barrels of crude oil that can be recovered in U.S. However, it was a bit far compared to the actual and theoretical model. The curves immediately reached N at time 24 years from year 1920-24. Hence, certain factors may cause them to be inaccurate. That being said, the model with N = 200 is more accurate since its nearer to the actual data and the theoretical model derived from it. B. World Crude Oil Production Model i. Using 2.1 trillion barrels as the total amount of recoverable crude oil in the world The next stage of the report is to repeat the analysis for the World Oil supply. The differential model used is the same as the logistic model of U.S. oil, though certain parameters will be changed. First, k and C will be calculated, then the logistic model equation will be determined. Aside from that, the time when the rate of production of oil reaches its maximum are also predicted. This will all start from the differential logistic equation dP P    k 1  P dt  2100 

In this analysis, 2.1 trillion barrels will be used as N. For this production, the initial conditions are: at year 1920-24, P(0) = 4.3 and t = 5 years, calculating for k

1  N  Po ln  k  Po

   1  2100  4.3  t  ln   5  4.3  k  1.237805577 k  1.238 t

Determining C C

N  P0 P0

2100  4.3 4.3 C  487 .372093 C  487.372 C

Substituting, we get the equation P

2100 1  487 .372 e 1.238t

Plotting it,

Figure 7: Graphical representation of the World Oil Production with N = 2.1 trillion barrels

To predict when do the model’s rate of production of oil reaches its maximum, we should first note that for a logistic function, the maximum dP/dt occurs when P = N/2 and time to reach maximum rate of change is equal to t

1 N  Po ln k Po

Hence, for the world crude oil production, the time to peak oil production is 1  2100  4.3  ln   1.238  4.3  t  5 years

t

Overall, the model predicted that 5 years from 1920-24, which was 1925-29, the world crude oil production will reach its peak. On this year, N 2 2100 P 2 P  1050 billion barrels

P

Thus, annual production is expected to be 1.05 trillion barrels. ii. Using 3 trillion barrels as the total amount of recoverable crude oil in the world Using the total amount of recoverable crude oil N equal to 3 trillion barrels dP P    k 1  P dt  3000 

With the same set of initial conditions as that of N = 2.1 trillion barrels, the value of k is calculated as 1  N  Po ln  k  Po

   1  3000  4.3  t  ln   5  4.3  k  1.309263637 k  1.309 t

For C

C

N  P0 P0

3000  4.3 4.3 C  696 .6744186 C  696 .674

C

The logistic equation generated is then P

3000 1  696.67e 1.309t

Graphing the model,

Figure 8: Graphical representation of the World Oil Production with N = 3 trillion barrels Calculating the time for the models’ rate of production of oil to reach its maximum t

1 N  Po ln k Po

1  3000  4.3  ln   1.309  4.3  t  5 years

t

The model predicted that 5 years from 1920-24, which was also 1925-29, the world crude oil production will reach its peak.

N 2 3000 P 2 P  1500 billion barrels

P

Hence, the annual production is expected to be 1.5 trillion barrels. iii. Effect of Price Increase Economically, the decline in production of crude oil will certainly result in an increase in price of oil products. This price increase will provide more funds for crude oil production, perhaps slowing the rate of decline. This increase in price affects the model by the increase in the growth rate and decreasing the limiting value, or the total recoverable crude oil.

V. CONCLUSION In the experiment, the U.S. and world crude oil production was modelled using a logistic growth model from year 1920-24 to 2005-08. In the U.S. crude oil production, two carrying capacities were given, which were 200- and 300-billion barrels. Their growth rates k were determined to be 0.8438 and 0.9259, respectively. The models were compared to the actual data and the theoretical model, also a logistic model generated from the actual data having a carrying capacity of 155.125 billion barrels, to determine their accuracy. Unfortunately, they were both far from these two curves. That being said, the model with carrying capacity of 200 is more accurate, since it was the nearest of the two. On the other hand, for the world oil production, two carrying capacities were also given. These are 2.1 and 3 trillion barrels. The model of these two were determined along with the time when the rate of production of oil reaches their maximum. These two models predicted an equal value which was 5 years from 1920-24. 1925-29. And lastly, in determining the effect of price increase, it was concluded that an increase in price affects the model by the increase in the growth rate and decreasing the limiting value, or the total recoverable crude oil.

VI. REFERENCES [1] Blanchard, P., Devaney, R. L., & Hall, G. R. (2011). Differential Equations . Boston: CENGAGE Learning. [2] Deffeyes, K. S. (2001). Hubbert’s Peak The Impending World Oil Shortage. New Jersey: Princeton University Press.