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Tuesday, September 20, 2005

Oil Shock Model - Continuous

I bit the bullet and updated the oil shock model to instead do a grind-it-out numerical integration. As it stood, the original code got a little too intricate for my tastes; yet it served it's purpose in providing me some valuable insight. Overall, the results duplicate that of the previous algorithm, plus or minus some numerical error.

The code at the bottom of the post solves the following differential equation:
R(t+dt) = R(t) + (T(t) - R(t) * E(t)) * dt

P(t) = E(t) * R(t)
R(t) = Current reserves
T(t) = Triangular discovery curve
E(t) = Extraction rate (yearly or daily)
P(t) = Yearly (or daily) Production
The equation basically states that the reserves accumulate by discovery but deplete by extraction proportional to the amount available for extraction. This latter proportionality allowed me to do an exponential convolution, but the book-keeping of a variable extraction rate caused by temporal oil shocks made the algorithm a bit unwieldy. With a numerical integration, I can apply the proverbial mathematical sledge hammer to get at the results.

One can also now see how this differs from the logistic formulation favored by other analysts:
Q(t+dt)= Q(t) + k*Q(t)*(1-Q(t))*dt
(where Q is the cumulative production as a proportion of the ultimately recovered resource, and k is a constant that sets the width of the peak) Yet I still don't understand what this means in terms of a physical model.

I kept the same temporal oil shocks in the model but I made their duration match the span of the political crisis in real terms.

It helps to know the dates at which the oil shocks occurred. The oil embargo started in late 1973 and lasted until the middle of 1974. The Iranian hostage crisis started in late 1979 and the early 80's recession officially ended by 1984. The gulf war started in 1990 and it's associated recession ended in 1992. The change in extraction rate was interpolated over each of these intervals to make the suppressions in the curve a bit smoother.

with Text_IO;
procedure Shock_Model_Continuous is

type Shock is record
Year : Float; -- Year of shock
Rate : Float; -- Rate of extraction
end record;
type Shocks is array (Natural range <>) of Shock;

Triangle_Width : constant := 87.0; -- Triangle window
Volume : constant := 2400.0; -- In BBls

S : constant Shocks :=
((1944.0, 0.060), -- Start of data
(1974.0, 0.060), -- Start of oil embargo
(1974.5, 0.051), -- End of oil embargo
(1980.0, 0.051), -- Start of Iran hostage crisis
(1984.0, 0.034), -- End of recession
(1990.0, 0.034), -- Start of Gulf War
(1992.0, 0.030), -- End of recesssion
(2100.0, 0.030));

-- Make the shock a continuous function
-- by interpolating between points in the list
function Interpolate_Shocks (Year : Float) return Float is
K : Integer := S'Last - 1;
for J in S'First + 1 .. S'Last loop
if S (J).Year > Year then
K := J - 1;
end if;
end loop;
return S (K).Rate +
(S (K + 1).Rate - S (K).Rate) * (Year - S (K).Year) /
(S (K + 1).Year - S (K).Year);
end Interpolate_Shocks;

-- Creates a symmetric window for dicovered reserves
function Triangle_Window
(Year : Float;
Start : Float;
Length : Float)
return Float
Y : constant Float := Year - Start;
Half : constant Float := Length / 2.0;
if Y < 0.0 or Y > Length then
return 0.0;
end if;
if Y < Half then
return Y / Half;
return (Length - Y) / Half;
end if;
end Triangle_Window;

C : Float := 0.0; -- Reserve
R : Float; -- Extraction Rate
P : Float; -- Production Rate
DT : constant Float := 0.01; -- 1/100th of year
Start : constant Float := S (S'First).Year;
Finish : constant Float := S (S'Last).Year;
Time : Float := Start;
K : constant Float := Volume / Triangle_Width * 2.0;
V : Float := 0.0; -- Volume of oil extracted
R := Interpolate_Shocks (Time);
C := C +
K * Triangle_Window (Time, Start, Triangle_Width) * DT -
C * R * DT;
P := R * C / 365.0; -- BBls/day
V := V + R * C * DT;
if Time > Float (Integer (Time)) - DT / 2.0 and
Time < Float (Integer (Time)) + DT / 2.0
then -- Print output only once per year
(Integer'Image (Integer (Time)) & "," & P'Img & "," & V'Img);
end if;
Time := Time + DT;
exit when Time > Finish;
end loop;
end Shock_Model_Continuous;

The output plotted against the BP data.


Professor Blogger @whut said...

I didn't know I was in the prediction game :)

Yes, that point is spiking up. It looks like it is almost completely due to increase in OPEC and Russia production.
OPEC went through a bit of a lull between 2000 & 2003 before jumping up while Russia has been steadily increasing since 2000.

2000 - 2001 - 2002 - 2003 - 2004
31354 30628 28855 30686 32927 OPEC
35583 35541 36056 35870 35916 Non-OPEC
08013 08659 09533 10499 11417 Russia
74950 74828 74443 77054 80260 Total

21517 21297 21426 21161 20732 OECD

2:47 PM  
Professor Anonymous Anonymous said...

As I pointed out in my comment to the previous version of "Oil Schock Model", the upward trend in 2003 and 2004 is suspicious. It might well be that we will have the peak this or next year. The probability for 2007 is quite high.

6:47 AM  
Professor Blogger @whut said...

Suspicious is right. The only way I can model this is with a "reverse shock"; which means that the extraction rate crept up a little bit the last couple of years. How could that happen?
Remember that the extraction rate in the shock model is set proportionately to how much discovered reserves remain.

Could it be that the Saudi's are upping their extraction rate? Or did they just bring new reserves on line? The latter would counter the trends of recent years. The former would say they are desperately pulling out as much as they can.

12:06 PM  
Professor Blogger JD said...

I didn't know I was in the prediction game :)

Waffler. I'm looking forward to mocking you when your pseudo-science "prediction" turns out to be wrong.

6:58 PM  
Professor Blogger @whut said...

Thanks John Denver, I'm on a Rocky Mountain High. And thank god you're a country bumpkin.

9:04 PM  
Professor Anonymous Anonymous said...

Don't even pay attention to JD. He is a complete troll. He is not worthy of our time.

What you are doing here is very interesting, wht.

2:20 AM  
Professor Blogger @whut said...

A troll indeed. And a schizophrenic one at that.

7:53 PM  

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