Type Curves Oil and Gas: Projecting the Production Decline Rate

In this lesson, you’ll learn how to use 3rd party data, as well as company-provided figures, to approximate the decline rate of an “average well” in the Pennsylvania region – and you’ll build in support for different EURs and IP rates.

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Type Curves Oil and Gas: Projecting the Production Decline Rate

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Table of Contents:

  • 1:29: Lesson Overview and Why the Decline Rate Matters
  • 5:08: Finding Historical Data on Decline Rates
  • 11:20: Making Estimates for Year 1 Production
  • 12:42: Using Goal Seek to Fit the Decline Rate Data to UPL’s Wells
  • 17:37: Building in Support for EUR and IP Rate Sensitivity Toggles
  • 20:36: Using a MIN Function to Handle Different EUR Cases
  • 26:24: Calculating Totals and Oil/Gas/NGL Split
  • 29:07: Recap & Summary

Type Curves Oil and Gas: Projecting the Decline Rate for PUD Reserves


Okay. Hello, and welcome to our fourth lesson in this module on the proved undeveloped, probable, and possible reserves in the Pennsylvania region, which, of course, we’re using as an example. The Wyoming region is going to be largely the same. Some of the numbers will be different. But it’s exactly the same logic and formulas for the most part.

So we’re going through this in depth and then we’re really just going to copy and paste this over later on, and then apply it to the Wyoming region and tweak a few of the numbers. So what are we doing in this step of the process? Well, in the previous lesson, remember, we had just gone through and projected the wells drilled by proved undeveloped, probable, and possible reserves.

So, we broke down the number of wells drilled that we projected each year, and categorized them according to the different reserve types and the potential future locations in each of those types. So what we’re going to do now is figure out what the decline rate looks like. Now, remember, this is very different than what we did in the proved developed region over here, or in the proved developed reserves, really.


Because in this case, we didn’t have enough granular detail to say, “Okay, this well is five years old. This well is two years old. This one is brand new. Let’s aggregate all those decline rates overtime and figure out what it may look like.” For proved developed producing and non-producing reserves, companies really don’t disclose enough detail to do that. So the best you can really do is, sort of assume an overall decline rate as we did in this region.

But it’s a different story when you’re drilling for new wells in these reserves that do not yet have any wells in the ground as is the case here. So, what we are going to do is assume a single decline rate for the entire Pennsylvania region for proved undeveloped, probable, and possible reserves.

Now, we could attempt to assume different rates in different years for different reserve types in different regions, which is a whole lot of numbers to look at. But the truth is, it’s pretty much impossible to do that. It’s just too granular. It’s too difficult and there’s just no data that detailed to support these numbers. Even as it is right now, as you’ll see, it’s sort of a huge stretch to get a lot of the numbers that we’re going to go through here in the first place.


So we’re going to leave it at this and assume a single decline rate, a single type of well for this region of Central Pennsylvania, and just go with it going forward. Now, to do this we’re going to need to look at third-party data sources. We’re going to need to see if other companies in their investor presentations or if Ultra Petroleum in its investor presentations discloses data on this.

Then, what we’re going to do is try to fit that data as best we can to the specific estimated ultimate recovery in an average well in this region, and also the number of production years we’re assuming. Again, there isn’t a magical way to figure out all this data.

A lot of it is trial and error. A lot of it’s looking at other sources and sort of triangulating the data to figure out what might be a reasonable assumption. Then, we also have to build in support for sensitivities. Now, on the Assumptions tab, remember we had set up these sensitivity toggles here for things like working interest, royalty rate, estimated ultimate recovery, and IP rate.


So, we have to make sure that when we change around variables like these, these are actually reflected in our decline rate data. The really important thing here is that if we change around the estimated ultimate recovery, or the IP rate, or both. We have to make sure that we don’t suddenly start getting very odd numbers.

What this means is that for the estimated ultimate recovery, if this goes up or down we’re going to have to scale up our production levels in each year up or down, and we’re going to have to be really careful about the way we do it. Because if we don’t do it in the correct way, nothing else here is going to work correctly. So we have to be very, very careful of that.

Now, with the IP rate, a similar issue; the problem here is that if the IP rate goes up to a very high level in the first year, for example, which it would because of the initial production rate. So if it goes up to a very high level, let’s say, we could end up with a scenario where we produce everything in that average well, and then we run out. But if we don’t set up our model correctly, we could still model it so that we’re still producing something even after we have gone through the entire estimated ultimate recovery of the well.


So, we’re going to have to be very careful to put in the proper checks for these. It’s very common to make mistakes here and to leave these out, but we want our model to be as flexible as possible, so we’re going to include both of these.

So, how are we going to do it? Well, it’s a seven-step process. I should rename this to step number eight. We shouldn’t have two sevens there. So it’s an eight-step process and we’re going to start by looking up data first. Then, assume an initial production rate for year one.

Then, take the rest of this data; use Goal Seek with whatever we found to try to fit the estimated ultimate recovery of the well we have. Then we’re going to copy and paste the values all as hard-coded values in the Decline Rate column, and then we’ll build in support for sensitivities after that.

Then, we will make sure that even if the initial production rate changes, our model still works. So we’re going to have to use a MIN function in the Pre-Adjustment Total column. Then we’ll calculate the total column based on the pre-adjustment numbers, and then our adjustment factor. So if the estimated ultimate recovery goes up or down, we want to make sure that’s supported here as well.


Then we’ll check the numbers a little bit for different adjustment factors, different IP rates, and so on and so forth. So let’s get started with this. Step number one, where do we find this data? How can we possibly get this? Well, the bad news is that the company doesn’t give a whole lot of useful information. I’ve already looked through their entire Investor Relations section and a lot of other sources.

So I’ll save you some time and tell you that the one useful source is this May 2013 presentation, at least in this current time period. If you go to page 35 of that presentation, conservative type curve estimates. So what they’re doing here is showing you what the production curves in the Pennsylvania region…

These counties are both roughly in Central Pennsylvania – what these look like from one to 360 days, and then 360 to 720 days; so, roughly over two years. What is the key take away from this? Well, the key takeaway is there’s a massive decline just in the first year. Look at this. It’s going from seven to around two by the end of the year.


So think about what a massive drop that represents. Even in some of the less extreme ones, it’s still a pretty big drop just in that first year, and then after that it really flattens out quite a bit from year two onward. Now, there’s still going to be a substantial percentage drop. But in the grand scheme of things, it’s going to be significantly smaller than this year one drop.

So if we wanted to, we could attempt to go in and get all the data from this, but in this case, it’s a graph. It doesn’t actually give us numbers. So it’s not terribly helpful. So another approach is to look at Equity Research, and if you look at this Canaccord report, let’s do a search for “decline rate”. Let’s keep going.

Okay, so look at this, “Steep initial production decline rates in Pinedale and Marcellus.” So they’re saying that Marcellus, where we are now, Pennsylvania, has a decline rate of 50%, and then 30% in the second year. Whereas, Pinedale in Wyoming has 60% and then 40%; so this is a good data point to have. This is what Equity Research is at.


I’ll save you some time. They don’t give a whole lot more information than that, so we’re pretty much done. This is just another way to check the data and check some of our numbers against what other people are saying.

So we have that, and then a third source is to look at what other companies in this region are doing. So this other company, EQT Corporation, it’s a bigger, more diversified company. But they also have some good data in their investor presentation, which I’ve linked to below this video, on the decline rates and what their production curves look like.

So, let’s go up, and I actually just went to the end of the presentation. I’m going to go up and show you where some of these are. Okay, so here we go. So take a look at this. In the Marcellus shale, they’re giving us their daily production rates, the decline rates and this is more helpful because look at this. The first 100 months are represented. So that is around eight years’ worth of data; whereas, the Ultra Petroleum presentation only really gave us two years of data.


So we have their decline curves over here. If we wanted to make a very, very simple estimate, we could attempt to eyeball this and say, “Okay, well what should it be on average over year one? What should it be on average over year two, and come up with some type of estimate like that. But as usual, there’s actually a better way to do it.

Because if you go to the Investor Relations section of their site, they have data on the type curves in the Marcellus region for Southwestern, Northern West Virginia, Central Pennsylvania, and so on. So of these regions, Central Pennsylvania is the best fit for our company. That’s where most of their operations are.

If you go and look at a map; I’m not going to go into that level of detail, but if you want to go look at a map of those counties, you can do that. What’s striking about this is look at this. Month 1: 213,088, and then Month 12: 58,000; this is a massive drop just in the span of a single year, and this is why it’s so important to take that initial production rate, and not just multiply it by 365, but to actually discount that by a pretty good factor, because the actual annual production here is going to be far lower than that 24-hour initial production rate times the 365 or 366 days in the year.


So, what I’ve done over here just to save us some time is I’ve summed up the 1,000 cubic feet equivalent produced in each year. So I’ve summed these all up just by using a SUMIF function in Excel, and I have all the data here, and then I have our decline rates over here as well.

So, we have all this, and you can go through and do these calculations yourself. I just don’t want to show you something quite this boring, because we already have all the data, and it’s really just a matter of summing up everything here.

So, 55% decline in the first year, which roughly matches what Equity Research said. They said around a 50% rate, and then we have close to a 30% rate. Equity Research said a 40% rate. So it seems like overall we are in the ballpark here, even though our numbers do not exactly match up with what other sources had.


Now, the other important thing to note is the 30-day average initial production rate. So if you go to page 12 of their investor presentation… So, if you look at this, they have 7,970, an average 30-day I-period for one type of well, and then about 6,500 for the other one. So if we roughly take the midpoint of that range of 6,500 and around 8,000 or 7,900, whatever you want to use, that’s how I got to this 7,300 number there.

So, multiply this by 365 and it says, “Okay, if that actually continues past 30 days for all 365 days of the year, this is how much we’ll actually produce,” and of course that didn’t happen in real life. Here’s how much they actually produced.

So if we take this number and multiply it by 365, we get to this 2.6 billion cubic feet, or cubic feet equivalent number; whereas, what actually happened in real life was only around 1.2 billion cubic feet in the first year. That’s exactly because of what I just showed you, that month over month there is a sharp production decline here, as is the case with almost all types of shale, oil and gas.


So, our approach here is going to be assuming that the year one production is a percentage of that IP rate times 365. It comes out to around 45% here. It might be lower than this. It may be higher than this. But it would be odd, for example, if this was, say, 90%. But it would also be odd if this were something like 10%. So usually it’s somewhere in the middle of this range.

So, with this data in hand, here’s what we can do now. Now that we have some sense of the data, we can go back to this. We finished off step one here, and what we want to do now is start by figuring out what this initial rate in year one is going to be. Then, after that, we’ll look at the rest of the data and try to fit it to the estimated ultimate recovery we have in our model.

So, let’s go down here, and so the first thing we need to do is calculate what the applied annual rate is at this initial production rate. So, let’s just multiply… Actually, let’s go up and get our initial production rate from all the way up here, and I got to the wrong cell. Let’s go up and take our initial production rate from here. Anchor that and multiply it by 365.


Then what percentage should we assume for this? We could make a number of different assumptions. We could make it something close to 45%. I’m going to be a little bit more aggressive here and say 60%. Again, this is based on a lot of tweaking of the numbers after the fact and sort of massaging the data and making sure that it roughly matches the graph in the investor presentation.

So, we’re assuming a difference from what EQT had in its presentations, but still something that’s roughly in that range of, say, 40% to 60% for this year one production. So, we have that. Now what we can do is check off step number two, and then for step number three we can start figuring this out and use Goal Seek to fit this data to what we’re looking for.

We are going to assume that this takes place over a 40 or 41-year time frame. So at the end, when we add up our pre-adjustment total right here, we want to make sure this adds up to the estimated ultimate recovery for an average well in this region.


So for this decline rate, where can we get the data? Well, let’s go back to the Marcellus data for EQT and take all these decline rates. We have to be careful here. They have a lot more years than we do. So, I’m going to copy these and paste as values. So we have that. You can see how it really flattens out over time here.

So here’s what I’m going to do. For the pre-adjustment total, let’s take the 365-day rate at the IP rate and multiply it by this difference of 60%, right there. Then, take the previous number and multiply it by one plus the decline rate. Copy this down, and then add this up so we have this. Now, what we want to do here is pull in a few other numbers. Let’s get the total number from all the way up at the top, from where we have the estimated ultimate recovery.


We have that, and then the remainder here. So this is just going to be equal to the total minus the subtotal right here. So, we have this in place, and on the surface this might seem reasonable. The problem with this data, though, is it doesn’t exactly match up. We have an estimated ultimate recovery of around 7.6 billion or 7.7 billion cubic feet equivalent. But here we’re only producing around 7.2 billion, and then the rest comes in the remainder period.

So, that’s not necessarily wrong. But it is clear that we should be assuming a slightly different rate so that all of this gets produced within these 40 years, which is what our goal is here. We’re trying to target it so that each of these wells has a production lifespan of about 40 years based on the fact that in another region of the same shale these wells were producing for around 50 years.

So, it seems reasonable that something in a 40 to 50-year range would work for this and that there probably should not be so much left over in the Remainder column, or the Remainder row right here. So here’s what we’re going to do.


We’re going to use the GOAL SEEK function. I’m going to enter up here “factor”, and for now I’m just going to say “one”. What I’m going to do here is create a separate column for these decline rates and multiply each one by the factor up here, and let’s copy this down. Okay, I forgot to anchor the factor part. So we have that, and now I’m going to change the pre-adjustment total so that it’s pulling from this column instead.

So we have that. Copy this down. Then for the subtotal, so we want to figure out what this factor is so that we can apply it to these numbers and figure out what the correct hard-coded values over here should be so that our subtotal matches the estimated ultimate recovery right here. So, the first thing I’m going to do is copy and paste this estimated ultimate recovery as a value, and this is so that I can copy and paste it for use in the GOAL SEEK function.


Let’s go down here. Go in the subtotal; ‘Alt + A’ for data, then ‘W’ for what-if analysis, then ‘G’ for Goal Seek. So we’re going to set cell AU170 to value. I’m going to paste in the hard-coded estimated ultimate recovery right there by changing cell factor. So we have that. Let’s press Okay, and so there we have it.

So there we have our factor. So we have a slight discount to the decline rates that EQT had in its data. So what we’re going to do now is take these rates, and now that we clearly produced the estimated ultimate recovery over these 40 or 41 years, this matches up and we have the rates that we’re looking for.

So, essentially we’re just taking the rates from another company and adjusting them downward or upward as necessary so that we produce the total available in this average well at the end of the 40 or 41-year lifespan. So, let’s copy and paste these as values now and I’ll copy… Let’s get rid of the factor here because we don’t need it anymore, and then let’s change these back to be pulling from the AT column right here.


We have that. Let’s copy and paste these as values over here under Decline Rate. We have that, and so now we have our new set of values right here. Now, that is pretty much all we’re going to do for this step of fitting the decline rate data to the specific estimated ultimate recovery we have. We still have a few things to finish here, namely the fact that we have to allow for this adjustment factor.

We have to allow for different estimated ultimate recoveries and different initial production rates as well. So, for now I’m going to check off step number three because we’ve done that, and step number four we’ve already done as well. We’ve copy and pasted those in. What we have to do now is, as I say over here, we have to build in support for this adjustment factor that we’re linking to.

As I say here, the estimated ultimate recovery is really a big open question in this case study because we’ve gotten contradictory data from different sources. If you remember from our data gathering lessons, we got to an estimated ultimate recovery here that is in line with what the company has in its presentations, but which implies total reserves that are much too large.


So, something odd is clearly going on here and there are a lot of questions over the correct rate to use. As a result, we are going to make sure above all else that, for example, if we change the estimated ultimate recovery to ten billion cubic feet, well we still want to produce the correct amount each year. We still want to use the same decline rates.

So, what that means basically is that these numbers have to change. We’re going to have to scale these numbers up, and if this is 10%, let’s say; well, we have to make sure that these numbers also all go up by 10%. So for our reference I have the baseline estimated ultimate recovery, and then this adjustment factor tells us how much bigger or smaller it will be overall.

So, let’s say, for example, that we go back to our assumptions and we go to our sensitivity toggle and I change this to a 10% increase. So what happens here, well this is 10%. The estimated ultimate recovery is 8.4 billion cubic feet equivalent, and look at this. We now have something huge in the remainder over here, which isn’t necessarily a mistake.


But if the estimated ultimate recovery is 10% bigger, then, all else being equal, these production values each year should also be about 10% bigger. Fortunately, there’s an easy way to fix this, which is that you just take the pre-adjustment total and then you multiply it by one plus the adjustment factor up here.

We’re not multiplying it directly in this column over here because with these columns we are multiplying by one plus the decline rate each year. So, if we also factored in the adjustment factor here, it would mess up the math. There could be a way to do this effectively, but it’s just too complicated to set up and too confusing. So I’m making it in this separate column instead.

So, we have this. Let’s make sure we anchor the adjustment factor right there and copy this down. So we have this and somehow the formula in here has gotten messed up, so I’m fixing the decimal places.


Now, total I can also copy over here and its okay to leave this one as is for now because these numbers are going to be off anyway. It’s really these that we care about. So, if I take total minus subtotal we get zero for the remainder, which is really what we wanted to do here in the first place.

So, to scale this up and down, let’s just try another number here. Let’s go back to our Assumptions tab. Let’s say I change this to negative 10%. Well, in this case it scales down as appropriate, although this data is still off. So, this is good news. It looks like this is at least working correctly. I’m going to go and set this back to 10% for now, just so we have that.

So this is a good step in the right direction. We might think we’re almost done and we do have the adjustment factor support built in. But we have one other thing to do which is to make sure that we are also not producing more than the total amount of estimated ultimate recovery here. Remember that in the Proved Developed Producing Build what we did is we took the MIN function and we always check the beginning reserves and the implied daily production to figure this out.


Well, we need to do something similar here because if, for example… Let’s say, I’ll illustrate the dangers of what we have here so far. Let’s go up and let’s say that the initial production rate goes up to a ridiculously high number. Let’s say it goes up to 20. So, 20 million cubic feet equivalent per day, and let’s go down and see what happens.

So you can see the problem here. We get negatives in the Remainder column and the reason we get negatives is because this column is not adjusting properly. What really happens here is that the estimated ultimate recovery gets used up very early on in year three or year four here, or something like that.

So really, we should just stop production after that. But we’re not checking for that case, and as a result of course we get negatives in our Remainder column here, which we do not want by any means. You want that to either be zero or positive. So we’re going to have to add in a check for that. Now, the way to do this, or at least one way to do it, we’re going to leave this variable alone right now.


Then what we’re going to do is go down here. First we are going to change around a few things. First, this estimated ultimate recovery, instead of linking to the number here which has already been influenced by sensitivities, what we’re going to do instead is link to the number on our Data tab.

So, if we link to the number on the Data tab, the advantage here is that this is sort of the raw number before it’s been adjusted for sensitivities. What it also means is that no matter what the IP rate is, it’ll still work with this base data. So, I’m going to take this number and then multiply it by Units.

So really what’s going on here is if we go in and modify the IP rate, well this is what it looks like with the baseline estimated ultimate recovery and the IP rate modification built in. Then, this column is what it looks like when you have this one, which already reflects the changed IP rate, and then this one will reflect the changed estimated ultimate recovery. So, we’re sort of looking at these changes in two steps for the sensitivity.


I realize it may be a little bit confusing to follow. But think about it like this. We are making sure this column supports a different IP rate, and then this one will support a different estimated ultimate recovery by scaling up everything proportionately. With this one, we’re not scaling up anything because we’re working with the baseline estimated ultimate recovery right here. I should label this “Base Estimated Ultimate Recovery”.

So, we’re not scaling up anything. But we will be applying the MIN functions here to make sure we never go over this 7,694 number. To figure this out, what we want to do is take the MIN between the estimated ultimate recovery minus however much we’ve produced so far, and our annual production number.

So in year zero, period zero, really year one, we’re just going to take our estimated ultimate recovery, and take the minimum between that and then our annual production right here. So what we can do is just move around this number. We have that.


Then, for the rest of these you have to do something slightly different, which is take the MIN between the implied number, and then the estimated ultimate recovery minus the summation of everything so far. So, this one, as usual, we have to anchor with ‘F4’, and then this one I’m going to anchor the row part of the 129 right there. Then, let’s copy this down and hope this works.

Okay, so we have a few issues here. One thing is we need to change this total to link to the base estimated ultimate recovery instead. But let’s just go through and test this and see if it works with a changed IP rate. So, let’s go up and let’s say that we changed the IP rate to 20.

What happens now? Well, take a look at this. So, with an IP rate of 20, we produce exactly however much we have and we were done by year two, really year three here, period two, year three, because our IP rate is so much higher to begin with.


Then, look at over here. We scale this up proportionally, but we’re still done by period two. So this appears to work correctly and if you want you could change around a lot of other things here, and it’ll still work correctly. So, for example, you could go in and you could change around the decline rate.

So you could make a decline rate here of negative 10%, and look at this. Now, it correctly only goes to year 11, and if you make the decline rates even lighter, well it keeps going over a shorter and shorter time period and you produce more and more of this up front. This keeps working even when the adjustment factor is different.

So, you can play around with this yourself if you want to see more about this. If you change something like the year one production as a percent of the total 365 days times the IP rate, well look at this. Again, the same thing happens and it adjusts appropriately and everything here works correctly. So that’s really the purpose of the schedule.

This one is so that we can change around the IP rate, the annual production levels, the decline levels, and still have it work with our baseline estimated ultimate recovery. Then, this column supports the fact that we may have different adjustment factors if the estimated ultimate recovery itself changes.


So we have this in place now, and what I’m going to do is just go up and check off step number six here. So, we have our MIN function added in and all this works correctly; by the way, when we get to the probable and possible reserves, all this will be a lot easier because we can just copy and paste and link directly to the values up here.

So what we’re going to do now is we’ve actually already calculated the total column. So we don’t need to do anything else there. What we do need to do is divide it into oil versus gas versus NGLs. So, we’re going to take our total and then we’re going to multiply it by the percent oil that we’re assuming, which of course is right there, and then the percent gas we’re assuming. So let’s take the total and let’s get the percent oil right there and divide by our conversion factor, and then NGLs, let’s get the number over there.


Then multiply it by the percent NGLs and divide it by our conversion factor. So we have that. So we have all those in place, and now let’s copy this down. Once again, I’m going to fix the formatting. So we have that in place. Let’s also flush out the subtotals here at the bottom and the remainder, and all that logic.

We have that and for the subtotal and remainder we can just copy these formulas over. Let’s fix the decimals. So, really we now have all of that in place, and it pretty much works as you would expect.


So, not that complicated in terms of the math formulas, but it’s a little complicated to wrap your head around building and support for different sensitivities, different IP rates, different decline rates, if we want to change that, different production rates, different estimated ultimate recoveries. Again, the whole point of this is just to make our model as flexible as possible.

So, step seven is done and then step eight, we could go in and do this. But to be honest, we’ve been checking this all along. So I’m not going to worry about it too much. If you really want to, though, you can go back and you can change around all these numbers. You can change this back to zero, verify that. Of course, this still works as intended. You can change around the year one production as a percent of the IP Rate times 360.

You can see here, this makes it sort of messed up because now we have a whole lot that gets produced in the remainder period. But that’s okay because then we can just change the decline rates. So, let’s say I change the decline rate and make this negative 5%, negative 5%, and look at this. Now, all these numbers change as well, and everything here is still calculated correctly. So, you can go in and change around, play around with these numbers on your own if you want. I am not going to worry about it too much.


I’m just going to leave this set to zero for now, and that’s pretty much it, all we’re going to do with this particular production decline curve. So, just to recap this lesson, this is important because decline rates differ in different regions and we could make a very granular estimate, separate it by reserve type by region. But we’re just doing it by region in this case because that’s the best data that we have.

To do this, we have to look at third-party data sources and fit the data to what we have. So what we did here specifically is look at the Marcellus shale data for another company, EQT Corporation. We took the decline rates from there, and we also looked at their year one production as a percent of the IP rate times 365 days.

Then, we made our initial guesstimates for in this pre-adjustment total column. Then we used the GOAL SEEK function to figure out the correct decline rates right here and then we had to make other adjustments. So, if the estimated ultimate recovery changes we have to adjust up or down, this column over here, the actual total that we’re going to be using in calculations.


Whereas, if the IP rate changes or the decline rates change, we want to wrap this whole column, all these, in MIN functions, so that we never end up producing more than our estimated ultimate recovery. To do that, we basically take the year and then we subtract the total so far, and then we compare that to the annual number. We use whichever one is less there to make sure we never produce more than whatever is in this actual well.

So we have that and then we calculated the total at the bottom, and then we checked the numbers a little bit. We also sort of did that as we went through this exercise. So that’s it for this lesson. Coming up next we’re going to now aggregate all this and look at the production across all wells drilled in this reserve type, the PUD, or proved undeveloped reserve type in Pennsylvania.

So we’re going to flush out this section over here for the net wells drilled and the production each year. Then we’re going to apply very, very similar logic to the probable and possible reserves and you’ll see how we can really copy and paste, and in many cases just link to some of these numbers above for that part of this case study as well.

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