Oil Field Models, Decline Rates and Convolution

This post is by Dennis Coyne

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 3.0 via 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.

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188 Responses to Oil Field Models, Decline Rates and Convolution

  1. Enno says:

    Mike, you mentioned “Dennis, I think produced water will be the single biggest threat to predicted EURs from tight oil resources.”

    Until thus far I ignored the water data in the NDIC reports, but it is available. I don’t know the accuracy, but just glancing at the data there don’t seem to be major errors in it.
    Below I post 3 graphs:
    1) Cumulative oil production per starting year. The same graph I posted before, showing the average oil production for wells starting production in a certain year.
    2) Cumulative water production per starting year. This graph is exactly the same as 1), except that it shows the average cumulative water production, instead of oil production.
    3) Total water production in North Dakota. This shows that quite some water was already produced with the old wells in North Dakota, until recently. Water production initially slowly increased with the new oil production, but as also can be seen from graph 2), newer wells are producing more water.

    I hope Mike or someone else can chime in with thoughts on this average well/overall water production.

    • Enno says:

      The last graph was slightly incorrect, here the update.

      • Dennis Coyne says:

        Hi Enno,

        Very interesting thanks.

        I think the water becomes more important toward the end of the wells life. So the ratio of oil to water and how it changes from year 3 to year 7 of the well’s life (as we only have about 7 years of data for Bakken wells).

        At this point we just don’t know, but Mike’s intuition is likely correct when these horizontal fracked wells get down to 7 b/d (or possibly even 10 b/d if costs rise faster than oil prices) the wells will be capped. The question then becomes how quickly do we get down to 10 b/d, I am doubtful that the terminal decline rate (exponential tail of well profile) is any larger than 10%. If that is correct the 10 b/d level is reached at about 20 years and the new well EUR decreases by about 7%. Those last 10 years from 20 to 30 years are very low output so it does not really change thing very much.

        • Watcher says:

          NoDak needs to think about this in the context of Mike’s comment.

          When flow gets low, the big company wants to sell it. They will sell the well to some newly formed LLC that will collect money for a year or so as it flows 7 bpd and then 6 and then 5 and then 4 — at which point, being an LLC with no other assets, they walk away.

          And there is the well, unplugged and leaking residual radioactive stage markers onto the surface.

          NoDak should refuse to allow purchase of wells by LLCs without plug and abandon escrow.

          • Mike says:

            Watcher, in Texas all operators must now post a bond with the Texas Railroad Commission to cover the cost of plugging abandoned wells. Several years ago the cost of drilling permits was raised significantly and those additional funds were earmarked for plugging orphaned wells in Texas. Also, it is next to impossible to assign, or convey, plugging liability from one operator to another without bonds being in place at the time of conveyance.

            I suspect that N.D. has, like most of the world, taken the TRRC’s lead in this regard, as well as most regulatory matters. I hope so, anyway.


        • Mike says:

          Hi, Enno, that’s some good work; thank you. Cleary the oil to water ratio for North Dakota wells began to increase at the time the Bakken was ramping up and that makes perfect sense to me given the enormous number of wells drilled and that few of those wells make water initially, other than small amounts of induced frac water. Dennis is correct, typically produced water begins to increase as reservoir depletion occurs.

          I guess for all the ranting I’ve done about the costs of water production and the effects of water on late-life well economics, it occurs to me that these tight oil wells do not produce from conventional reservoirs that often are associated with structural traps and that often contain underlying connate water. In other words, each tight oil well produces only that oil in place around the wellbore, within the frac radius, that it created for itself. Each shale well more or less makes a space for itself in dense shale, drains that space, and that’s that. Its hard to think of the Bakken as one vast, homogenous reservoir, like for instance, Ghawar where an ocean of oil use to sit on an ocean of water. The big question to me is how much interstitial water there is bound in the shale matrix that will begin to increase over time and reduce OWR. I don’t know, others surely do.

          In Texas we do not report water production each month from individual wells. I would not expect a public company needing desperately to sell stock and raise investment capital to want to volunteer something as ominous as increasing produced water rates. So water is like a 4 letter word in the shale patch and it seldom gets uttered. As I drive up and down S. Texas highways, however, the vacuum truck traffic hauling water to disposal systems is enormous and growing. There are waiting lines going into disposal facilities, 24/7. Not all of that is flowback frac water, I guarantee.

          If N.D, on the other hand, does require monthly water production on a well by well, or unit by unit basis, it would be fascinating to see if, and to what extent, the OWR is declining on individual tight oil wells over time.

          • Enno says:

            Thanks Mike & others for your comments, very interesting.

            • Dennis Coyne says:

              Hi Mike,

              I really appreciate your comments I have learned a lot from what you have said. Please continue to point out any mistakes I make.

              I am no oil man and can learn much from people who actually produce oil.

              • Mike says:

                Thanks, Dennis. We’ll learn together, all of us. So we can take the message of peak production rates to the street.

          • Jeffrey J. Brown says:

            There is also a question as to often and to what extent that the massive frac jobs break into water bearing zones, and as pressure depletion continues around the fractures, water encroachment from water bearing zones increases.

            • Mike says:

              Westexas, goodonya. Along the EF trend, updip from the shelf margin, there are lots of growth faults associated with sediment slumping towards the basin. These faults often have lots of displacement and grow with depth. In the overlying Austin Chalk carbonate, these growth faults can be connected to the underlying Edwards ocean, so to speak, and lots of very, very salty water was produced from the Chalk when wells were frac’d too close to these big faults. Now and then we’ll hear of a shale well in the EF that is frac’d and makes no hydrocarbons and mucho 150,000 ppm chlorides water…from the Edwards. It is a very good point you bring up.

    • SuddenDebt says:

      That is some excellent charts and information.

      To me there are two observations worth noticing;
      1) Looking at wells by vintage it appears as water production has grown faster than oil production.
      2) It could be interesting to see how the water oil ratio (or its inverse) developed with time. Water to oil ratio appears to be growing with time.

      I do not know about the chemical composition from the produced water, but somehow the produced water is disposed of.
      Water normally flows more willingly than oil.

    • Enno says:

      The chart below shows the ratio of the water produced vs oil produced, using the same data. The strange peaks/drops at the end of the tails are indeed in the data. I can’t really explain it, except that the number of wells at the last 12 points in each tail is dropping, because wells from those years did not reach that age yet, and therefore the last point only contains the January wells from that year, which can cause some erratic behavior (low denominator).

      • SuddenDebt says:

        Thanks a lot for your work and sharing!

        From your chart(s) there appear to be one trend (barring any noise in the data); the water to oil ratio has increased with time both for the “older” legacy wells, but also for the newer wells.
        Water, oil and natural gas flows co-mingled in the wells and looking at your charts it appears as with time, total liquid flow for “newer” wells has increased and so has the water to oil ratio.

      • Dennis Coyne says:

        Thanks Enno.

        Great stuff. Seems like it leveled off in 2013 (similar level to 2012), 2014 probably too early to tell.

      • Mike says:

        Yikes, Enno. I don’t like what I see in your data.

        I wonder; I said in Texas we do not report water production per well every month and that is true. We do, however, estimate, loosely, water production per well each annual quarter. I wonder if N.D. does not do the same thing?

  2. Old farmer mac says:

    The inestimable Mr Lynch is still explaining why guys who majored in the sciences and have spent their life working in the sciences are ignorant of the realities of oil over at Forbes whereas the economists know all about it.

    My maternal grand father once had a beautiful 62 Chevy two door hardtop that really was driven only to church and the doctors office. He bought it new after retiring and it had only a few thousand miles on it in the mid eighties when he decided to park it under an old shade tree instead of the carport as usual.We all tried to explain that the tree was rotten and just about dead and thus in danger of falling on the car but the old fellow got more stubborn by the day being close on to a hundred and would not listen.If we had not said anything he would probably have put it back in the carport.

    He reminds me of the cornucopian element of businessmen who do not believe something that comes out of a hole in the ground will ever run short.

  3. Dean F. says:


    “The Railroad Commission of Texas’ estimated final production for April 2014 is 72,632,766 barrels of crude oil”, which means 2421092 bbl/day.

    My corrected model gave (reported in the comments of the previous post):
    2455818 bbl/day using the correction with absolute numbers
    2461750 bbl/day using the correction with percentages

    Not bad… I am quite close ^_^ !

    • Dean F. says:

      P.S. these numbers are for oil only, no condensate (RRC does not give final estimates for condensate)

    • Dennis Coyne says:

      Hi Dean,

      I think if you look back at older reports, you will find that the RRC usually underpredicts even with the adjustment factor.

      • Dean F. says:

        Thanks. I will have a look at them

      • Dean F. says:

        I considered for curiosity the data before September 2013 because, for those data, the EIA data and my corrected data show a difference which is smaller than 1%. I found that for 2013 data, the adjusted RRC data are usually underestimated with respect to EIA and my corrected data of an amount close to 5%. For 2012 data, the underestimation ranged between 5% and 10%. However, the level of underestimation tends to decrease over time (I suppose because the oil growth in Texas is decreasing and oil production will probably reach a maximum at the end of 2014 / beginning of 2015).

  4. robert wilson says:

    Fun at Forbes——excerpts

    …Michael Lynch, 2 days ago
    We don’t plant petroleum but we explore for it.

    Rob Rob 2 days ago

    And, as you showed in the graph in Peak Oil 1 – discoveries from exploration peaked in the 1960′s and have declined each decade since. Does that not demonstrate that oil is not “effectively infinite” as some economists have postulated?

    Michael Lynch, 1 day ago
    That graph is one of the problems: ‘discoveries’ actually means ‘currently estimated discoveries’. Thus, the amount of oil discovered shown is underestimated. Reserves have not declined because of upward revisions to field size.


    Robert Wilson 2 hours ago
    Nature planted the petroleum as noted in Hartmann’s book “The Last Hours of Ancient Sunlight”.——

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