The Oil Shock Model with Dispersive Discovery- Simplified

Originally posted at peak oil climate and sustainability

Some changes have been made to this post see after the Excel File link.  Below figure 10.

The Oil Shock Model was first developed by Webhubbletelescope and is explained in detail in The Oil Conundrum. (Note that this free book takes a while to download as it is over 700 pages long.) The Oil Shock Model with Dispersive Discovery is covered in the first half of the book. I have made a few simplifications to the original model in an attempt to make it easier to understand.

Figure 1

In a previous post I explained convolution and its use in modelling oil output in the Bakken/Three Forks and Eagle Ford LTO (light tight oil) fields. Briefly, an average hyperbolic well profile (monthly oil output) is combined with the number of new wells completed each month by means of convolution to find a model of LTO output.

In the Oil Shock model the maximum entropy probability distribution is analogous to the average well profile and the annual oil discoveries are analogous to the number of new well completions in my LTO models.

The maximum entropy probability distribution is used when we know there is a probability distribution but we have very little information about what it looks like. In this case, the only assumption that is made is that a probability distribution exists and that it has a positive mean and a standard deviation. The maximum entropy probability distribution sets the mean equal to the standard deviation and has the form

m=k*exp(-k*t)

where

m=probability that a discovery will become a producing reserve t years after discovery,

k=a constant set at 0.05 in my models, where 1/k is the mean number of years from discovery to producing reserve (20 years), and

t=years from discovery to producing reserve.

This is also called the negative exponential distribution see

http://en.wikipedia.org/wiki/Exponential_distribution

The maximum entropy principle was first proposed by E.T. Jaynes see

http://en.wikipedia.org/wiki/Edwin_Thompson_Jaynes

If 1000 kb was discovered when t=0 and if m=0.049(or 4.9%) when t=1, then 1000*0.049=49 kb of new producing reserves are added to cumulative producing reserves in year 1. (Where year 1 means one year from the date of discovery.)

Chart below shows the Maximum Entropy Probability (MaxEnt), m vs year from first discovery.

Figure 2

Figure 3

http://www.theoildrum.com/node/10009

The figure above is from Jean Laherrere’s final post at the Oil Drum (figure 7) at the link above. The figure below is figure 9 from the same Oil Drum post.

Figure 4

These two charts were used to estimate backdated discoveries of proved plus possible (2P) C+C excluding extra heavy(XH) oil reserves from 1901 to 2010, I read the data from the charts as best I could. (The green curves in both figure 3 and figure 4.) In my opinion Jean Laherrere provides the best oil discovery estimate that is publicly available.

In an attempt to match Jean Laherrere’s model, I used Webhubbletelescope’s dispersive discovery model and fit this model to the discovery data.

The dispersive discovery model describes cumulative discovery, D in the following equation

D=U/(1+C/t^6) where

U=URR=2,200,000 million barrels (for the model presented here),

C=constant determined by best fit to discovery data, in this case C=800 trillion, and

t= year where t=0 is 1870, t=1 is 1871, etc.

The rationale behind this equation is developed in The Oil Conundrum in Chapter 9, pp 167-177, particularly pp. 170-171 equations 9-25 and 9-28. The combination of those two equations is the basis for the cumulative discovery(D) relationship above.

The chart below shows the model fit to Jean Laherrere’s discovery data. The vertical axis is millions of cumulative barrels discovered.

Figure 5

The following chart shows yearly discoveries(real), the centered 25 year average for discoveries and the dispersive discovery model. Vertical axis is millions of barrels per year.

Figure 6

The discovery model can be convolved with the maximum entropy probability distribution to find the new producing reserves (n) that are added to the cumulative producing reserves (P) each year using a simple spreadsheet to add it all up. In the chart below the vertical axis is millions of barrels per year of new producing reserves (reserves which begin producing in a given year.)

Figure 7

The next step is to find the cumulative producing reserves, P. Each year oil is extracted from P and new reserves (n) are added.

P2=P1+n-e where

P2 are the cumulative producing reserves at the end of year 2

P1 are the cumulative producing reserves at the end of year 1

n= new producing reserves added to P1 in year 2

e= oil extracted (or produced) from P1 in year 2

r=e/P1= extraction rate

Actual production data for C+C-XH is used to determine the extraction rate necessary for the model to match the output data. The chart below shows the cumulative producing reserves and extraction rate for C+C less extra heavy oil which matches the output data from 1960 to 2014.

The model from 2015 to 2050 is based on the underlying model for new producing reserves added each year and the assumed extraction rates. These rise a little from 2015 to 2021 from 5.5% to 5.8% and then remain flat. Over the 2009 to 2015 period extraction rates rose from 4.7% to 5.5%.

Figure 8

Below is Jean Laherrere’s estimate of 2P technical reserves for C+C-XH, from the Oil Drum Post referenced above.

Figure 9

In 2010 the model producing reserves are about 62% of the 2P technical reserves estimated by Jean Laherrere(850 Gb in 2010).

For more mature regions, such as the US and Norway, producing reserves are about 78% to 80% of 2P reserves when we assume 2P reserves are about 33% higher than 1P reserves for the US (Norway reports 2P reserves, but the US EIA reports 1P reserves).

This model uses a separate model for extra heavy oil because the time it takes to develop extra heavy oil resources is different from C+C-XH. There are 500 Gb of extra heavy oil URR and 2200 Gb of C+C-XH URR for a World total C+C URR of 2700 Gb. This C+C URR matches Jean Laherrere’s February 2013 estimate, see

http://aspofrance.viabloga.com/files/JL_2013_oilgasprodforecasts.pdf

pages 21 and 22 suggest a possible C+C less extra heavy URR of 2600 Gb, but it is decided that the creaming curve gives more reliable results and a URR of 2200 Gb is used. On page 35, figure 53 the world C+C forecast is shown. Figure 1 below is the Oil Shock Model result with a URR of 2700 Gb and the assumptions outlined above ( for future extraction rates, time from discovery to producing reserves, and the discovery model matching discovery data).

Figure 1

Any changes in these underlying assumptions would change the model output, in particular future extraction rates and discoveries are unknown and could be either higher or lower than assumed in the scenario presented. Note that the extra heavy oil model is more conservative than Jean Laherrere’s model which peaks at about 15 Mb/d in 2070, my extra heavy oil model peaks at about 10 Mb/d in 2050 (compared with 11.5 Mb/d in 2050 for Jean Laherrere’s extra heavy oil model). The overall C+C model I present is somewhat more pessimistic than Jean Laherrere’s model, for example C+C output is about 6 Mb/d lower in 2050.

The following chart shows the discovery model, new producing reserves, and C+C-XH output in Gb per year.

Figure 10

In the original Oil shock model there were several stages between discovery and production called fallow, build, and mature. Each stage involved convolution similar to the convolution I used here with the maximum entropy probability distribution and the discovery data, but in the original model there were three such maxent probability distributions and three convolutions rather than just one.

It seemed too complicated to explain all that so I collapsed the fallow, build, and maturation stages of the original model into a single convolution where we go directly from the discovery stage to the new producing reserve stage (which is essentially the same as the mature reserves of the original model.)

In a future post I will show how potential reserve growth could lead to a higher URR of C+C less extra heavy oil than suggested by Jean Laherrere. This implies that the model presented here may be a little on the pessimistic side.

Excel Spreadsheet with model at this link.

I made an error calculating the ratio of US producing reserves to 2P reserves.

If we assume 2P reserves are 25% higher than 1P reserves then the 1996 to 2011 average for the US is about 61% rather than the 78 to 80% I claimed in the original post.  As the US is one of the most mature oil provinces in the World, it is unlikely that the World average producing reserve to 2P reserve ratio would be higher than the US, but the model presented suggests exactly that and needs to be reworked, this is due to my initial error calculating the producing reserve to 2P reserve ratio in the US.

If we change the value of k for the maximum entropy probability distribution to 0.04 (from 0.05 in the original model) and adjust the extraction rates so that output from 1960 to 2014 matches EIA data we get the following.

 

peak oil barrel