Hubbert's Curve in OilWorld

Well, after having a big Iraq week around here, time for something a little bit different.  This is the first post in what I hope will be an occasional series exploring a little further the ideas behind the Hubbert Theory.  I have made some attempt to make this post accessible to non-specialists, but I'm not too confident I'm going to succeed in reaching altogether non-technical folks, so don't feel bad if you're in that category and this doesn't make much sense.

There has been quite a bit of controversy about the Hubbert Curve - the specific mathematical form that Hubbert used for his bell shaped curve of oil production over time.  This is technically the derivative of the logistic curve.  There are some people who seem to believe it's useless, while others think it's an inviolate law of nature that cannot be contravened.  To me, it's a rather rough approximation that's useful in some limited cases, and so badly inaccurate as to be worthless in others.

At the end of the day, it's important to understand that oil production is to a significant degree a function of human behavior and choices - people decide where to drill, when to start producing the resulting oil, whether to go as fast as possible or save some for the grandchildren, when to shut the wells in to raise prices, when to piss off the international community and get put under a sanctions regime, and when to fight wars and blow up their neighbor's oil production infrastructure.  These complex human processes will never be fully described by some simple closed form mathematical equation.  Therefore, demands for rigorous first principles mathematical derivations are kind of besides the point - Seldon's laws will only ever appear in science fiction.

However, what I want to do in this post is to set up a toy model world, the OilWorld of the title, in which Hubbert's Curve is a rigorously accurate description of oil production.  (For the knowledgeable, this is just a thin disguise for the SI model of epidemiology).  However, I've set it up in a way that I hope in future posts, as time allows, to consider various variants.  The hope is to aid the reader's intuition for when the Hubbert model has a prayer of being useful, and when it really doesn't.

Ok, so here are the rules of OilWorld.  The geography of OilWorld consists of a square array of square tiles.  Some tiles have oil under them, and most don't.  The inhabitants of OilWorld have somehow discovered oil and built their first oil rig.  A small region of OilWorld is shown up above (the green tiles have oil under them, the white ones don't).

The rules of drilling in OilWorld are as follows.  Each year, it's possible for a rig to drill a single square.  Unfortunately, the inhabitants of OilWorld have yet to invent seismic exploration technology, so they can't tell where the oil is and they have no choice but to pick a random square.  If they pick a green square, they spend the year exploiting all the oil under that square (which coincidentally is just the right amount to be produced in one year).  With the profits from that oil, they are then able to build a second rig, which is ready to begin drilling at the start of the next year.  In that case, both rigs pick a random square at the start of the next year.

If a rig picks a white square, then there is no profit and no new rig.  Instead, the rig just moves on and tries again the following year.

Now this may seem a very strange and artificial model.  But it does capture some features of the real world - there's only so much oil under the ground, it's not easy to find, you only get to use more infrastructure if your previous efforts with the infrastructure you have were profitable, and the more infrastructure you have, the faster you can find the oil as long there's still some there to find.  And it's also the simplest possible model I can think of that will give a Hubbert curve for oil production.  We can add various tweaks to move it in the direction of realism at later times and see which tweaks really mess up the Hubbert-like behavior, and which ones don't.  (I don't have this program of posts fully worked out yet, so I may very well end up getting surprised too).

Ok, so let's see why this gives a Hubbert curve (and here is where I'm afraid we have to leave the non-mathematical reader completely - but if you are really determined not to miss out, you could try skipping ahead a few paragraphs and see if you can pick up the thread again after the nasty equations stop whizzing past your ears and the simulation graphs start).

Suppose that a fraction a(t) of squares have oil under them at time t.  Suppose also at time t, we have N(t) rigs. Call the time step dt (one year in the discrete version outlined above, but in a continuous version of the theory could be arbitrary). Now if either dt or a are small compared to the overall problem, we can neglect collisions and the expected number of squares of oil found in a given time step is just N(t)a(t)dt. But this is also the number of new rigs that will be created. So we have



dN = N(t)a(t)dtor

dN/dt = Na
Now, by the rules up above, every time we find a square of oil, we create a new rig. Therefore, if the size of the grid is m on each side, we have that


N(t) = m²(a(0) - a(t))or

a(t) = a(0) - N(t)/m²

Substituting for a above, and relabelling a(0) as a₀, we thus get


dN/dt = a₀ * N(1 - N/(a₀ m²))This is the logistic equation for N, meaning that N, the number of rigs, will have a sigmoid (S-shaped) form.  The cumulative amount of oil produced likewise, since each rig is created from the profit of that oil.  Mathematically the solution is


N(t) = a₀ m² e^a₀t /(a₀ m² + e^a₀t - 1) Now, to check this, it's very simple (assuming you know how to write computer programs!) to implement a simulation of OilWorld.  I include my version in an appendix below.  I ran the simulation on a grid of 400x400 cells, with 10% of them having oil.  The following plot shows the fraction of the oil produced in each simulated year on the left scale, and the rig count - both simulated and the logistic curve form above - on the right hand scale.

You can see the excellent agreement between the logistic curve and the simulated rigs, and the nice Hubbert-y peak for the oil production.  Finally, the post wouldn't be complete without showing you the Hubbert linearization:

A nice straight line, just the way it should be.

Appendix - simulation code in C

#define gridSize 2000
#define oilProb  0.1

int oilPresent[gridSize][gridSize];

int main(int argc, char* argv[])
{
  int i, j, rig, oilCellsFound;
  int       year                = 0;
  double    randMax             = pow(2, 31) - 1;
  int       rigCount            = 1;
  int       rigCountNextStep    = 1;
  int       oilTotal            = 0;
  int       oilTotalFound       = 0;
  
  // Initialize the simulation arrays.
  for(i = 0; i < gridSize; i++)
    for(j = 0; j < gridSize; j++)
     {
      if(random()/randMax < oilProb)
       {
        oilPresent[i][j] = 1;
        oilTotal++;
       }
      else
        oilPresent[i][j] = 0;
     }
  
  // Run the simulation
  printf("Year\tOil Perc\tRigs at End\n");
  while(oilTotalFound < oilTotal)  //outer loop over years till we got everything
   {
    oilCellsFound = 0;
    for(rig = 0; rig < rigCount; rig++)
     {
      int guessI = random()%gridSize;
      int guessJ = random()%gridSize;
      if(oilPresent[guessI][guessJ])
       {
        oilCellsFound++;
        oilTotalFound++;
        rigCountNextStep++;
        oilPresent[guessI][guessJ] = 0;
       }
     }
    printf("%d\t%.5f%%\t%d\n", year, ((double)oilCellsFound)/oilTotal*100.0, rigCountNextStep);
    year++;
    rigCount = rigCountNextStep;
   }
  
  return 0; 
}