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Oil Field Models, Decline Rates and Convolution

Oil Field Models, Decline Rates and Convolution thumbnail

The eventual peak and decline of light tight oil (LTO) output in the Bakken/ Three Forks play of North Dakota and Montana and the Eagle Ford play of Texas are topics of much conversation at the Peak Oil Barrel and elsewhere.

The decline rates of individual wells are very steep, especially early in the life of the well (as much as 75% in the first year for the average Eagle Ford well), though the decline rates become lower over time and eventually stabilize at around 6 to 7% per year in the Bakken.

What is not obvious is that for the entire field (or play), the decline rates are not as steep as the decline rate for individual wells. I will present a couple of simple model to illustrate this concept.

Much of the presentation is a review of ideas that I have learned from Rune Likvern and Paul Pukite (aka Webhubbletelescope), though any errors in the analysis are mine.

A key idea underlying the analysis is that of convolution. I will attempt an explanation of the concept which many people find difficult.

At Wikipedia there is a fairly mathematical presentation of the concepts which often confuses people.  There are a couple of nice visuals to convey the concept as well see this page.

In the visual below a function f (in blue) is convolved with a function g (in red) to produce a third function (in black) which we could call h where h=f*g and the asterisk represents convolution, just as a + symbol is used to represent addition.

Convolution of box signal with itself2.gif
Convolution of box signal with itself2” by Convolution_of_box_signal_with_itself.gif: Brian Amberg
derivative work: Tinos (talk) – Convolution_of_box_signal_with_itself.gif. Licensed under CC BY-SA 3.0via Wikimedia Commons.

I think the best way to present convolution is with pictures. Chart A below shows a relationship between oil output (in barrels per month) and months from the first oil output for the average well in an unspecified LTO play.

This relationship is a simple hyperbola of the form q=a/(1+kt), where a and k are constants of 13,000 and 0.25 respectively, t is time in months, and q is oil output.

Chart A is often referred to as a well profile. The values for the constants were chosen to make the well profile fairly similar to an Eagle Ford average well profile. EUR30 is the estimated ultimate recovery from this average well over a 30 year well life.


Chart B shows the relationship between the number of new wells that begin producing each month and the months from the start of production for the entire field.


The convolution of the relationship shown in chart A and the relationship shown in chart B results in a third relationship shown in Chart C below, oil output vs. months from start of field output. Output has been converted to kb/d from barrels per month.


It is indeed strange that two very different shapes (a hyperbola and a trapezoid) would combine to form the shape shown in chart C.   A spreadsheet can be downloaded here, with the scenario above laid out.
What was surprising to me when I first tried this analysis was that a combination of the average well profile with the number of wells added each month reproduced the oil output data fairly closely.

To clarify this further, I have created a simple model. As before, we have a hyperbolic well profile in chart 1 (slightly different than chart A above) and the number of new wells added each month in chart 2, but in chart 2 this is over a short 6 month period. After that time no more new wells are added.



In the chart below I show the output for each group of wells that begins production in successive months. The output from all wells starting production in month 1 are labelled “month 1 wells”, there are 6 of these groups up to “month 6 wells”. The number of wells added each month is shown as a dashed line read off the right axis. Remember that 30 wells are added each month from month 1 to month 6 so output for “month x wells” will be 30 times month 1 of the well profile in month x and 30 times month 2 of the well profile in month x+1, etc.


The convolution of Chart 1 and Chart 2 results in Simple oil model 1 shown below.


This model is very simple in order to present how the principle works in a clear manner. When the annual decline rate for the “field” is compared to the average well’s annual decline rate, they are very similar for this simple 6 month model. More realistic models are presented later for comparison.

Note that month zero in the chart below is the month of maximum annual decline rate, for the average well the maximum annual decline rate happens in month 13 and for the field it occurs in month 18, the curves have been shifted to the left by 13 and 18 months so that the maximum decline rates match up at month zero for easy comparison.


The spreadsheet for simple model 1 can be downloaded here.

A second simple model with the number of wells added each month rising from 5 new wells per month to 30 new wells per month over 6 months and then falling back to no wells added by month 12 is shown below.




Note that the “month 7 wells” output curve is the same as the “month 5 wells“ output curve, but shifted 2 months to the right. Likewise month 8 is month 4 shifted 4 months to the right and this same symmetry is true for months 9 and 3(6 month shift right), months 10 and 2, and months 11 and 1 where the shift right in the curve is equal to the difference in the month when the well started production (8 months and 10 months for the last two cases respectively).

When all of these 11 curves are added up for each month (the convolution of the “well output of the average new well” chart and the “number of new wells added per month” chart) we get the Simple Oil Model 2 chart below.


Simple model 2 can be downloaded here.

I now present a different model with a higher EUR well profile (than in chart A) and a lower rate of addition of new wells (than in chart B). This model’s well profile is similar to the average North Dakota Bakken well profile.



The convolution of the two charts above results in the field output shown below.


How does the annual field decline rate compare to the average new well annual decline rate in this case? In the chart below we see that a slower decrease in the rate that new wells are added causes the annual field decline rate to be only 22% at most, about 3 times lower than the maximum annual well decline rate.


The spreadsheet for the model above can be downloaded here.

As this result is rather counterintuitive, I will try another modification to the model. The well profile remains unchanged, but there is a steeper reduction in the rate that new wells are added to field production.

Such a scenario could occur if there was a steep drop in oil prices as in the early 1980s. It will also occur if there is a decrease in new well productivity which will reduce profits and the incentive to add more wells.
The well profile chart is unchanged, the other two charts are as follows:



Even in this case the maximum annual field decline rates are less than half the maximum well decline rate. This is because we have almost 15,000 wells added over an 11 year period and their decline behavior in the aggregate is much different than that of an individual well. See chart below.


Note that the field decline rate is very high, close to a 30% maximum rate in this scenario. If the rate that new wells are added drops to zero over a 1 to 2 year period and no further wells are added, we would expect the field decline to behave like the gray curve in the chart above.  Spreadsheet for the 5.6 Gb scenario can be downloaded here.

Earlier I mentioned that when I first tried this method I was surprised that such a simple model could accurately match output from the Bakken or Eagle Ford fields.

Using data from the North Dakota Industrial Commission(NDIC) on oil output, the number of new wells added per month, and individual well data(from Rune Likvern initially and lately from Enno Peters) I attempted to match scenarios initially presented by Rune Likvern at the Oil Drum.

Below I present the well profile and number of new wells added each month.



When the two charts above are combined (convolved) we get the output curve below.


Note that the sharp drop off in the number of producing wells added each month is not very realistic and is an artifact of the way I set up these simple models for illustration (they end at 130 months so the number of producing wells had to be ramped down very quickly).

Such a scenario would be more likely if there was a sharp rise in well costs, or a sharp drop in oil prices or new well productivity (EUR). The field decline rate is somewhat similar to the previous scenario, rising quickly to a 28% annual decline rate which falls to 10% after 5 years and to 7% in 8 years.


This simple Bakken model can be downloaded here.

A fairly realistic scenario for the North Dakota Bakken (it is a little on the low end of likely scenarios) is presented now for comparison to the model above. This scenario has an ERR (economically recoverable resource) of 5.3 Gb where the more likely range is 7 to 9 Gb, based on USGS estimates. The average well profile and number of new wells added each month are below.



When we convolve the two charts above the following model output results. The match to the data is surprisingly good.


The annual field decline rate and well decline rate are shown below. In this case the maximum annual field decline is about 16% in 2021 and falls to 8% by 2026 and to 5% in 2031, the maximum annual well decline rate is 61%, the well decline rate is shown for a well starting production in Dec 2013.


The spreadsheet with this more realistic model is quite large (18 MB) so those with limited bandwidth may want to skip it.  The realistic Bakken model can be downloaded here.

For the Eagle Ford play I was able to collect data on single well leases from the Railroad Commission of Texas, data on the number of producing wells in the play and output data. I developed an average well profile (shown below) and combined it with the number of new wells added each month to produce an output chart.

Note that the output chart is for crude only and does not include condensate.



The two charts above are combined (or convolved) to give the output chart below.


Note that there is about 20% of Eagle Ford output that is condensate, when this condensate is added to the URR above for crude only we get a URR of 5.1 Gb of C+C.

As in the case of the North Dakota Bakken/Three Forks the match between the model and data is surprisingly good considering the simplicity of the model and the complexity of the real world.


Oil field output can be simulated with the convolution of the average well profile of newly added wells and the number of new wells added each month. I presented several simple models to demonstrate this concept.  An obvious weakness for any attempt at forecasting is that the future average well profile may change over time and the number of new wells added in any future month is unknown.

The decline rate of a field of wells will tend to be considerably lower than the decline rate of the individual well. The field decline rate depends on several factors: the decline rate of individual wells, the total number of wells in the field, the period of time over which these older wells were added (whether the period was long or short), and finally the rate at which the number of new wells added decreases as the field begins to decline.

Several models were presented showing how the field decline rate might vary under differing circumstances.

The concepts presented were applied to scenarios which simulated both the North Dakota Bakken and Eagle Ford shale plays with fairly good precision.

In a future post I plan to show how the convolution of two mathematical functions is used to develop the Oil Shock Model.

peak oil barrel

15 Comments on "Oil Field Models, Decline Rates and Convolution"

  1. Guthrum on Wed, 25th Jun 2014 12:33 pm 

    What does all this actually mean in real-world speak??

  2. penury on Wed, 25th Jun 2014 1:24 pm 

    I am not smart enough to answer your question. The only thing I got out of the article is that the field will decline but it is impossible to say how fast based upon current information.

  3. rockman on Wed, 25th Jun 2014 1:31 pm 

    G – This is a great and admiral effort to characterize the play. But he also makes the point that the ultimate models are not prediction tools. The basic problem is that none of the plays are “fields” in the accepted definition. Wells and fields (once fully developed) have a decline rate. But TRENDS, like the Bakken and EFS, don’t have decline rates per se. Perhaps more appropriately characterized by a “development rate”. And the development rate, as the author points out, is a function of a number of factors. But the decline rate of the individual existing wells isn’t very relevant. One could often characterize individual shale wells as their on “field”.

    Consider how little drilling there was in the Bakken and EFS earlier despite the fact everyone in the oil patch knew the oil was there and the technology already existed. That was shown in the very productive fractured Austin Chalk play in Texas that was heavily drilled horizontally during the 90’s.

    So what changed? Easy answer: oil prices increased 300% and US companies had few other opportunities. So why was the surge in US oil production not predicted? Another easy answer: nearly no one was predicting a 300% increase in oil prices.

    So take a guess of future oil prices. Yours is a good as anyone’s IMHO. So consider you assume continued high prices. Unfortunately that’s not the final answer. Regardless of how high prices might get it doesn’t create drilling opportunities if we’ve drilled most of the viable location. Every oil play ever developed on the planet eventually collapsed to some degree. The Austin Chalk play I mentioned has some activity today but nothing like during the boom. And how big was the AC boom in the 90’s? It eventually covered an area 5X as much as the current EFS trend. But it had its geographic limit just like the Bakken and EFS will eventually.

    So now you know how to accurately predict future production of any of those trends: you just have to accurately predict future oil prices as well as the geologic limit of the undrilled areas of the trend.

    Easy peazy. LOL. Models are easy to make. Accurate predictions aren’t as easy. As I have pointed out before: building models is like masturbation: there’s nothing wrong with doing either as long as you don’t start believing they’re the real thing.

  4. mtadd on Wed, 25th Jun 2014 1:49 pm 

    At steady-state, oil field production is proportional to number of wells developed per month, assuming the average individual well profile is constant.

  5. nemteck on Wed, 25th Jun 2014 3:27 pm 

    A very good article. Its main purpose is to show how the oil production is a function of decline rate and the addition of new wells.

    Another influence on the overall field production is that the new added wells are, in average, not that productive as the inner core of the field.

  6. rockman on Wed, 25th Jun 2014 4:44 pm 

    mtadd – “…assuming the average individual well profile is constant.” Which, in fact, has never happened in any trend in the entire history of conventional and unconventional oil/NG development in the USA. The oil patch has a habit of developing the better wells first. The locations with lower potential sit fallow waiting for better prices and/or improved tech. Which is exactly why there was a boom in shale production when oil prices increased. All the locations being developed today weren’t drilled 10 years even though they would have had the same production profile then as today. Just as there are areas in both the Bakken and EFS today which have insufficient production profiles that can be justified at current oil prices.

    A good example would be the Bakken test drilled a year ago that had a very disappointing production profile: turns out it wasn’t just a poor producer…the Bakken formation wasn’t even present in the area. So much for constant production profiles down the road. LOL.

  7. Nony on Wed, 25th Jun 2014 10:42 pm 

    How ’bout that ND Bakken pipeline that just got approved? I remember a lot of harrumphing about the canceled projects. But one just went forward. Gotta be fair, gotta give credit. That is steel in the ground from the “most conservative people in the oil patch”. 😉

    P.s. Piccolo and Rune’s predictions look pretty lame and doomerish now. Did they go to the million barrel day celebration and buy everyone a round? 😉

  8. J-Gav on Thu, 26th Jun 2014 3:53 am 

    Pretty cool stuff. I learn something every day.

    Then again, as Rockman says, “Models are easy to make. Acccurate predictions aren’t as easy.”

  9. rockman on Thu, 26th Jun 2014 8:06 am 

    J-Gav: Despite seeming to pick on models they can be useful. If constructed properly we can determine which of the assumptions are the most sensitive. If one models with Assumption A, use the extreme possibilities and the results don’t create a significantly different answer then we can stop debating Assumption A. OTOH if variations in Assumption B produces wildly different outcomes then that factor needs to be studied intensely.

    That’s why I dislike seeing simple models presented with that sensitivity analysis. Time and again I’ve seen model assumptions varied until the desired outcome was generated. A very common abuse of modeling in the oil patch in my experience:”OK…model says the well isn’t economic to drill. Then change A, B and C and see if it works then. Alright…it works now!!! Let’s drill it!” Seen it done more times then I can guess.

  10. Nony on Thu, 26th Jun 2014 12:15 pm 

    Rockman, I would never abuse an NPV model! (In a way that you could catch it). 😉

  11. shiftshaper on Thu, 26th Jun 2014 12:18 pm 

    Rockman, Is it possible that at $135 a barrel or more there might be a huge elephant find somewhere out there that we don’t know about? Or is all the oil that we know about pretty much already discovered? Could there be some in the Artic we don’t know about? I know for the most part estimates tend to be high…Also could there be large finds that are being kept secret so as to wait for prices to climb to a certain amount and then…Aha! look what we found!…..

  12. Nony on Thu, 26th Jun 2014 12:29 pm 

    DC, good stuff, man. I appreciate how you change people’s views in non-confrontational way. I think Ron is finally understanding the RRC versus EIA corrections (and that YES they do matter after more than the first month).

    For me the portfolio aspect of combining these wells is intuitive. Decline rate (both absolute and percentage) drops a lot after the first year. So the longer you’ve been developing the field/and the more your OVERALL production includes old wells, the lower your overall decline rate is. (The less sensitive it is too new well additions.) I think a lot of peakers have just a reflexive shale-is-evil bias and don’t think through the portfolio aspects.

    Anyways, like I say…you are good at dripping on people and getting them to move/learn without having to make it a direct confrontation of ideas. Bet that serves you well in corporate world.

  13. rockman on Thu, 26th Jun 2014 2:49 pm 

    Nony – I said I don’t like it when someone uses a very biased model. But I didn’t say I didn’t know how to do it…and do it very well. I’ve turned chicken sh*t into chicken salad more times than I can remember. I’ve smoked some of the top third party auditors in the biz and have been handsomely rewarded for it.

    Which is why it’s impossible for anyone to effectively blow smoke up my ass these days. I like to think of myself as the equivalent of an anti-sniper sniper. LOL.

  14. Nony on Fri, 27th Jun 2014 8:49 am 

    Rock, any way you cut it, there’s a million bpd coming out of the Bakken and even more out of the EF. That is not a small development and pretty stunning ramp ups (rate of growth).

    The peakers poo-pooed those fields/trends/continuous oil accumulations/whatevers all the way up. They went with what they wanted to believe rather than objective analysis. Picollo was wrong, Rune was wrong. and they were wrong in a particular direction. Like someone who messes up giving change…in a particular direction.

  15. Nony on Fri, 27th Jun 2014 4:13 pm 

    Rock, I actually think you’re missing Dennis’s main point by discussing the issue of model complexity, tweakability, etc. This is not a modeling post per se. It’s main topic is not the predictive model. It’s to show the concept of how average decline of a summation of wells differs from that of an individual well. An intuitive concept that many commenters don’t grok.

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