Using the Lightboard for an Exam Review Session with Complicated Mathematical Problems

ALEX: So here we're gonna be working on question two for the midterm review. In this case, we're going to be finding stock tank oil in place for a reservoir. We're gonna be basically given four different data points in the reservoir regions a B C and D. The values are listed here, so you have a saturation of oil, which is SO. You have your net pay zone which is HN. Your gross pay zone, which is HG. Your oil formation volume factor which is BO. And you have your porosity which is phi.

So in this case we're gonna be doing two different methods. We're gonna be doing volumetric method, which you'll see first, and then you're gonna see iso-contour method. And this will give you an example of why one way may be better than the other way and how to approach a problem like this. And this is a very common problem in reservoir engineering because you're not able to drill a well in every single location you want to. It cost money and so you have to work with data you already have in the reservoir. And so you have to make sure you understand how to use those values properly. Um so in this case, like I said we have these values given and so what I'm gonna do first is calculate the average values. So basically for a saturation of oil for example, I'll add all the SOs up and just divide by four because there's four regions. So and it's the same thing for each property so I'm just going to give you those averages right here. So the saturation of oil average is going to be 70%. Our net pay zone average is going to be 72.5 feet. Our gross pay zone average is going to be 112.5 feet. Our formation volume factor of oil average is gonna be 1.2. And our porosity average is going to be 16.2%.
 
Using these values, we can then find what our stock tank oil is in place using a few equations. So the first equation I want to talk about is equation 4.03 that you'll find in your notes. This is the gross volume of your pay zone. Which is going to be and 43,560, what I'm about to write, is just a unit conversion to go from acres to feet. So one acre equals 43,560 feet squared. So we have 43,560. And then we're multiplying by our area of our reservoir which is forty acres. So basically this is like for region A, B, C and D it's encompassed within that. And so we multiply that by here because we basically want to get that into feet squared. So we can calculate the volume. And then we're gonna multiply by our gross pay zone.

You might ask like what's the difference between gross pay and net net pay zone. You'll notice that you'll learn like in classes like 440, PNG 440 for example, you may have like a certain water saturation and a part of your pay zone, which really you won't be able to get much production from, or it's not very good for to recover from, so you don't really include that in your net pay zone, but it's including your gross pay zone. And so we multiplied by our gross pay zone average. Which is the 112.5. So when we do that, we are going to get 43,560 times 40 times HG average, which is 1 12.5. Now we will be using this, I'm just gonna keep it as it is so I can show you how things cancel. We're gonna then plug this in to an equation that solves for our stock tank oil in place. Which this is going to be equation 4.04A in your notes. So that equation is this. Which this is what we just found. And then we're gonna have our net pay zone average. Divided by our gross pay zone average. Multiplied by our porosity average. Multiplied by our oil saturation average. And then we're going to divide this all by 5.615. Like I said, because this right here, when you solve for this, this will be units of feet cubed, and so when we find stock tank oil in place, we want it in terms of barrels, or in this case would be stock tank barrels, because you see I'll divide by a BO. So this will be our BO average. So when you multiply or divide by your Bo average you're basically converting your barrels. It's like stock tank barrels.So you basically want what it would be at your production facility. How much oil you'd have. So this is just gonna be this equation right here divided by this. It's a little, it looks a little off, but it's just your gross volume of your rock. Times your average of net pay zone. Divided by gross pay zone. Times porosity average. Times well oil saturation average. Divided by 5.615. Times formation volume factor average. And when we do this calculation, like I said, these values right are here or just these values right here, just plug it in for them. We find that N, another reason why I kept it like this, as you'll see this HG right here, is the same thing as this  HG, so it cancels. So basically this right here, and this right here, cancel. So that's why I kind of wanted to show it like that. And so, when I write this out, we're going to have 43,560. Times our area in acres, which is 40. Times, we're gonna have our net pay zone average which is going to be 72.5. Then we're going to be multiplying this all by porosity average and oil saturation. So our porosity average is 0.162. Our oil saturation average is 0.7. And we're dividing this by 5.615. Like I said previously, you're using the 5.615 to convert from cubic feet to barrels and then the BOs convert from barrels to stock tank barrels. So you know 5.615 times your BO average which is 1.2. And when we do this calculation, we find that N is going to be equal to 2,132,584.59 STB. This can also be written as 2,132.584 MSTB or 2.132584 MMSTB. So our MM is millions, M is a thousand. And so this is basically what our original oil would be in place if we used volumetric method. And now since we did this, we can talk about iso contour method which is our other method of approach for evaluating different points in the reservoir.

So for this we're going to have an equation 4.05. It's gonna be a little bit different, in this case, we're going to ignore gross volume and gross pay zone. Just like I said it cancels anyway and because how this approach works it's a little weird in terms of having different HGs. Because we're not averaging A, B, C, and D anymore. We're actually gonna find a value for an iso-contour of region A, region B, region C, region D individually.

And so the equation for this we'll be the following. So this could be G or O. This is just in terms of the phase you're working with so in this case we're just working with oil. So we're gonna have porosity times the net pay zone times SO divided by BO. As you can see it's very similar to what N is. It just doesn't include the VGRV or the HG like I said. So in this we can calculate this for region A, B, C, and D. So basically I'll just show you what, how you calculate for region A and it'll be the same way for the other three regions.  

So for ICO of A, it's just going to equal the porosity A, 0.10. The net pay zone for a is 75 feet. The oil saturation is 0.6. And the formation volume factor is 1.2.

When you do this calculation you're going to get that ICOfor region A is 3.075. And now for B it's gonna be the same way, just using Bs values now. It's going to be 10.046. For C, it's going to be 5.250. And for region D, it's going to be 8.727.

And so these are just some, we look at them as constants. So it's a parameter for that region and so once we have these values, we can actually average them together. Kind of like what volumetric did, but now we're doing each region individually instead of like connecting them. So when we do that. It's it over here. We're gonna add all four of these values up. Divided by four. And we  get 6.7745. And similar to how we solved for the N, which is original stock tank oil in place. We had all this information times VGRV. And so like I said, the HN over HG porosity SO BO, it's all part of the ICO except like HG because it cancels.

And so basically what we can do with this, right here, is just plug it in for the N equation. So N is going to equal 43,560 times 40. Because our area is still 40 acres. Times our ICO average, which is the 6.7745. And then we're going to divide by 5.615. Because everything else that's not included here is already calculated for when you compare it to N. So this is going to equal 2,102,206.38 STB. Or you can write it as 2,102.206 MSTB. Or 2.102 MMSTB.

And so as you can see, these values are very close to each other, but there is a main reason why iso-contour method is preferred and you're going to see that with the next example. Because one of each property in each of these regions is gonna be very low to where it's not really shown when you're taking the averages, when you doing the volumetric method. So you're gonna be getting a lot more oil than what you would if you do the iso-contour method. And so I'm going to write those values up now.

Okay, so in this case, we have three regions now and I'm going to show you why that iso-contour method is a lot better than the volumetric method. So in the last case, the numbers were somewhat close. I think it was about 20,000 STB or so different, or in this case it'll be a lot bigger difference.

So we're given region A, B, and C. If you notice like region A, for example, it's porosity is approximately zero percent and region B will have the oil saturation approximately equal to zero percent and region C will have the net pay zone equal to approximately zero feet. And so we can like, just think about in terms of like very small. Like say your porosity is like one times 10 to the negative twelve or something like that, like there may be a little bit of porosity but it's gonna be so little that there'd be such little oil in that area. And you can look at that for all three regions because each have a porosity of zero. But when you're taking the average of all three porosity, as you'll see, it doesn't really mimic or show that you have a zero porosity in one of your regions. And so for when we do our averages, just like before I'll be the same way. Just add all three up and divide by three.

So our oil saturation will be 0.50. Our net pay zone average will be 51.67 feet. Our gross average is 113.33 feet. Our BO average is going to be 1.23 BBL for STB. And our porosity average is going to be .117. So 11.75 percent. And like using equations showed earlier, let's solve the volumetric of oil first. So first like we can use the VGRV which is going to be somewhat similar to this last case. So we're gonna have the unit conversion again of 43,560 times our area which is 40 acres. So our area staying the same for this case, still 40 acres. And then our gross pay zone is going to be a 113. Our average gross pay zone 113.33 feet. And so then when we solve for N. We really, just like before, we're gonna use the average properties again. So we have the VGRV. And then we're going to multiply this by our net pay zone average over our gross pay zone average. So like the 113.33s will cancel. So let us like cancel them right now. And then we're going to multiply by the average porosity and then average oil saturation times 0.50  And then just like before we're gonna divide this by the unit conversion of feet cubed barrels. Which is 5.615, when you divide by 5.615 anyway. And then we're going to multiply by our BO average which is going to be 1.23. And when we do this calculation, you find that your original oil, original stock tank oil in place is going to be 762,583.35 STB. Which before is around two million or so. So let's say like maybe a third or so less or two-thirds less than what it was originally. Now we will see here, in the iso-contour method, when you take each ridge region individually, you'll get a value that's negligible. Basically zero.

So let's do ICO of A just like before it's gonna be like the same process for A, region A, B, and C. So I'll show the calculations for A. So for A it's going to be 70. Because HN is 70. Times our porosity which is approximately zero so like very small. Multiplied by our oil saturation which is going to be 0.75. And then we're going to divide that by our formation volume factor of oil which is 1.2. And so like I said, zero, it's approximately zero for your porosity so this is going to end up being zero. And it'll be the same thing for region B and C. Because as I said, like with B, your oil saturation is approximately zero. So you can say this is also zero. And the same thing with C, your net pay zone is approximately zero so, we can say it's about zero. And then so when you calculate N using the average of your iso-contour, you find that the average of this is just zero plus zero plus zero all divided by three, so it's just zero. So when you're calculating your oil in place using iso-contour method, you're going to have N times your average ICO which is just zero, times like the VGRV term which is the 43,560 times 40. And like I said before the HGs cancels, so we can just ignore that in this case. And then that's just going to be divided by 5.615. So as you can see, this is zero.

And this makes sense too. This is what you should expect for your oil in your reservoir. Just because if region A, neither of these regions have the capability of having any oil present because at region A your porosity is approximately 0% so there will be no room for any fluid volume at all. In region B the oil saturation is approximately zero so it may be all water. So that porosity is 20%. So the water saturation may be a 100% or like 99.99% and so on. And region C, our net pay zone is zero feet, so there is no actual region where there's going to be any oil in terms of producible oil anyway.

And so, as you can see, the zeros are very different than 750 mm stock tank barrels. And so that's why the iso-contour method is a lot more accurate. Because by doing the average volumetric method, you're going to find that there will be oil in this reservoir, but as you can see in each region, there wouldn't be any oil so that's why the ISO contour method is better.