Math 337 Project 1
Modelling Oil Production
In this project we will model US and world crude oil production using a logistic model, where the carrying capacity represents the
total possible recoverable crude oil.
Some of our parameter values will come from the data presented in Figure 1 below. The values in the table are in millions of barrels.
Year | US Oil | World Oil | Texas Oil
1920-24 | 2.9 | 4.3
1925-29 | 4.2 | 6.2
1930-34 | 4.3 | 7.0
1935-39 | 5.8 | 9.6
1940-44 | 7.5 | 11.3
1945-49 | 9.2 | 15.2
1950-54 | 11.2 | 22.4
1955-59 | 12.7 | 31.9
1960-64 | 13.4 | 44.6
1965-69 | 15.8 | 65.4
1970-74 | 17.0 | 93.9
1975-79 | 15.3 | 107
1980-84 | 15.8 | 101 | .009874
1985-89 | 15.2 | 104 | .010606
1990-94 | 12.9 | 110 | .008821
1995-99 | 11.5 | 118 | .007102
2000-04 | 10.4 | 122.2 | .005656
2005-09 | 9.4 | 130.7 | .005426
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. That data comes from column two of table 1. While we are here, we also will need parameter values for a logistic differential equation that fits the crude oil production data for the world. That data will come from column three of table 1.
For our differential equation, we will use the equation for logistic growth from the textbook. This equation is: where is the recoverable crude oil.
Before rushing to a solution, lets do some paper and pencil to determine k. We assume that dP/dt is the change between the first and second line of Table 1 for US Oil. will be taken as the population in line two of the table for US Oil. The equation then reads . Solving this for , we get: . Using this equation yields for , .
Let's see what happens when we try to solve the differential equation.
The graphs are the same, the only difference being the upper limit, now 300 as opposed to 200.
Now we generate the same set of graphs except using 300 billion barrla as the amount of recoverable crude.
We once again run the same equation, this time using 400 barrels of recoverable crude
The next stage of the report is to repeat the analysis for the World Oil supply. The model is exactly the same, we need only change the values for
the amount of recoverable crude oil. The first analysis will use 2,1 trillion barrels. We will repeat the analysis for 3 trillion barrels.
First order of business though is to determine for the world oil supply. Using the numbers from
the first two rows of the table for world oil, using them exactly as we did for analylizing the US supply.
The equation we use is . Solving for yields . This yields
Now we are ready to run the equations.
Repeating the analysis for 3 billion barrels of world recoverable crude oil.
And repeating the graph for 3.1 billion barrels of reoverable crude
The first of each set was run using the Sage generated solution. The second set of graphs was run using the solution
we found by hand.
Okay. In the above graph, we tried to predict when production would reach a maximum. The graph shows production should reach maximum
in very near 50 years.
The above error code is baffling! I cannot even find where the code is, let alone correct it. Sage has great potential
but problems likethis should never happen!
Now we will generate a graph using the solution found by hand
Now we need to try to determine when Texas oil production will reach maximum.
From the looks of the above graph, Texas oil prodection will reach maximum in about 80 years,
Considering the change in costs, it is the opinion of this author that production will likely be directly proportional to
cost. If we use to represent cost, then should represent future production numbers.