 So to find the oil viscosity at pressures above the bubble point, we're going to be using equations 3.56 and 3.57. So the viscosity of oil is going to equal to our viscosity of our oil at bubble point, which we have right here, times our pressure of interest, which for our case is going to be 4,000 psi divided by our bubble point pressure, which is going to be 2,634 to the C power. And the constant C comes from the equation 2.6 times the pressure of interest, which is going to be 4,000. So it's going to be P to the 1.187 e to the negative 11.513 plus 8.98 times 10 to the negative fifth times your pressure of interest again, which is going to be 4,000. So C is going to equal to, so C will be equal to 0.3424. And plugging the C into the equation above and using the viscosity of oil at bubble point 0.3778, we're going to get 0.3778. Remember, the pressure will be 4,000 psi. That's the pressure of interest here divided by our bubble point pressure. So we get our viscosity of our oil at 4,000 psi. It's going to be equal to 0.4358 centerpiece. And once we find that, we are pretty much done with all the oil properties. Next, we can start working on the gas properties. And like I said earlier, there is no free gas above bubble point. So there is no such thing as viscosity of your gas. There's no gas formation volume factor density of gas. Those really don't exist when your pressure is above bubble point. So we kind of ignore them for the PVT table. So the only pressures of interest we're going to have for the gas properties is going to be at 2,000 psi. And so the first thing we'll do is we'll find our compressibility factor, which also known as Z factor. So this is used a couple steps. So the first thing you're going to need to do is find what your critical pressure is and what your critical temperature is of the fluid. And you do this by using your specific gravity of your gas. It's going to be in equations 3.60 and 3.61. And once you find what PPC is, so your critical pressure of your gas and then your critical pressure of your gas, you're going to be finding your pseudo-reduced temperature and your pseudo-reduced pressure, which is just your pressure of interest or your temperature of interest divided by your critical pressure and your critical temperature. So I'll write that on the board. So to find your pseudo-reduced temperature, it's going to be the temperature of your, well, it's going to be a 580 ranking. So for these, we're going to be using ranking. So it's going to be 580 and it's going to be divided by your TPC. And then so for T, sorry, PPR, so here, so for PPR, it's going to equal P over PPC. And an important note is kind of out of the scope of this class is, so for our case for standing cats anyway, you'll see the temperature, your reduced temperature is always going to be greater than 1. You can have a pseudo-reduced temperature less than 1, but in this isotherm, we'll actually be passing through like the two-phase region, which then it gets crazy with Z-factor, then you have to worry about the Z-factor of liquids, but I don't believe we'll be worrying about that. So our case is if your TPR is less than 1, you might want to check your units to make sure you're using ranking or something might be wrong. And remember to get to ranking, you have your Fahrenheit, you add 460 to have your Fahrenheit is to get to ranking. And so we can find what our TPC is and our PPC is using equation 3.60 and 3.61 where our input is a specific gravity of gas. So for PPC, we're going to have the equation 756.8 minus 131 times specific gravity of gas. For a lot of these things we've talked about, you're mainly going to see these in 410 stuff we've done so far. And now stuff like we're doing with Z-factor, you'll start seeing also 475, PNG 410, PNG 475, maybe a little bit of PNG 405. And these concepts will build on each other as you get to later classes like 430 and 480. But this is actually a really good introduction to those classes. You already have some background information going in. So this is this equation that we're going to have minus 3.6 have specific gravity of the gas squared. And so for our TPC, it's going to be 169.2 plus 349.5 times specific gravity of gas minus 74 times specific gravity of gas squared. So our specific gravity of gas is going to be 0.7. And so you're only going to have one PPC so critical temperature and critical pressure of the fluid for the scope of this class. And so for our PPC, we will be getting PPC will equal 663.336 PSIA. Critical temperature will be your TPC will equal 377.59 ranking. And so using these equations up here for your pseudo-reduced pressure and pseudo-reduced temperature, we know that the temperature is going to be 580. So we can just plug that in right now. TPR is going to equal 580 over our TPC, which is going to be 377.6. Which is going to give us a value about 1.5366. And then for our PPR, it's going to be, like I said, our pressure of interest is going to be below the bubble point because these don't really exist above the bubble point because you're not going to have free gas. So that's why we're going to do it for 2000 PSI. So our PPR is going to be 2000 PSI over 663.336, which is going to equal, and these are units of PSI going to equal about 3. And these are the reduced properties of their unit list because you're dividing by the same, or have the same units on the numerator and denominator. And so once you have your reduced temperature and reduced pressure, you look at your standing catch chart to find where they intersect to find your Z value. And so on your standing catch chart, it's only valid for, like I said, reduced temperatures above one. So how Z works, or I guess this is Z, this is going to be your pseudo-reduced pressure. So each line you see on your standing catch chart is for a different isotherm, meaning you have the same temperature. So for this case, we have one temperature, so it just means we'll have one line. And so it might look something like this. And so as you notice with standing catch, there's two different axes. There's a top and the bottom. The reason there's two different ones is to model the graph in different ways. So like your top of your graph will be for, like, lower pseudo-reduced pressure. So say from, like, here to here. And then at the bottom of your graph, you have greater pseudo-reduced pressures, which is going to, like, model from, like, here to here. And so it's just a different way to have the graph so, like, models different parts of your compressibility factor. So for our case, we have a pretty low pseudo-reduced pressure, so it's going to be falling within the first area of the range. And so when we look at standing catch chart, we find that our z factor for 2000 psi will be 0.825. So pretty close to 0.83 for our z factor. And so this z factor will be used to find things such as your density of your gas. So I can just write that up here, your equation. So your density of your gas is going to be pressure times your molecular weight of your gas divided by z, r, t, where t is in ranking, z is your compressibility factor you just found, r is a constant, which is 10.73, p is your pressure of interest, which in this case is going to be 2000 psi, and your molecular weight, I'll show you how to calculate right now. So you have a specific gravity of your gas, which is relative to your density of air, which is 28.97 pounds per pound mole, I believe is the, yeah, pound per pound mole. So it's going to be your molecular weight of your gas divided by your molecular weight of air, which is going to be equal 28.97 pounds per pound mole. This is a little different for liquids. For liquids, your specific gravity is relative to the density of water. And so it would be specific gravity like oil, for example, is equal to your density or oil divided by density of water, which water density is going to be one gram per cc. And so that's why oil floats on top of water, meaning it's less dense, and that's why your specific gravity, the gravities of your oil you see are less than one. And so for example, we found r is to be 0.81556. So to get the molecular weight of our gas, we know what our gamma g is, it's 0.7. We know what our molecular weight of our air is, it's 28.97. We just multiply the two to get our molecular weight of our gas. So the molecular weight of the gas will be 20.279 pound per pound mole. And then we can plug that into this equation right here to find what our density is at 2000 psi of the gas. So we know our pressure is 2000. Molecular weight is 20.279. We know that z is 0.825. We know that r is 10.73, and r is just a conversion factor. And then temperature is going to be in ranking, which is going to be 580. Oh, that's right, that's right. We've got density of the gas to be 7.8994 pound per foot cubed. And another thing you may see in the notes a lot of times is 5.615. This is just a conversion to go between cubic feet and barrels. And so basically if you have a value that's let's say you have one barrel, but you want to turn it into cubic feet, you just multiply this by 5.615. Because the units of this are the cubed and a barrel. And similarly if you have like say two cubic feet and you want to convert the barrels, you just divide by the 5.615. All that is is a unit conversion factor. From here we can find our gas formation volume factor, which is BG. And so this is somewhat similar. You'll see this used in later classes like 475, 410, and especially in like 430. And so BG, I'll just write it up here while I have some space. BG is going to be equal to, like I said you can have this like in feet cubed, or you can have it like in terms of like barrels for like SCF. However you would like to notation, it just depends if you want to have the 5.615 in there or not. So BG is going to equal Z times your temperature of your reservoir fluid times your pressure at standard conditions which is 14.7 psi. Your temperature at standard conditions which is 60 degrees Fahrenheit or 520 degrees ranking. And then times your pressure of your reservoir. So in our case for 2000 psi we're going to have 0.825 times like temperature of reservoir which is going to be 580 because this has to be in ranking. 580 times 14.7 divided by 520 times like I said it's going to be a 2000 psi. So our BG is going to be 0.00676. And because we didn't have the 5.615 in this case it's going to be feet cubed per SCF. If you want it in terms of barrel per SCF you just have a 5.615 term right there to make into barrels per SCF. And so if you think about it it makes sense that your BG should be a pretty low value because gas is very compressible and so like if you have a large volume of gas you can compress it to a small volume. But if you just let it out to like room temperature conditions like it's going to expand a lot so your relative volume of your gas in the reservoir compared to gas at standard conditions is going to be very small. So for equations such as this and these they can be derived from the real gas. The difference between real gas and ideal gas is real gas has a compressibility factor Z which really all it is is a correction factor to get it towards like what it would be like as ideal. So that's what like Z factor is it's just a correction factor. And so with this equation for example your molecular weight ZRTP and rho G if you take the N over the other side you have V over N and then N can be looked at in terms of mass over molecular weight. And so you have this N over here so this molecular weight can actually be multiplied over to the left hand side so you have P times V times molecular weight equals Z times mass RT and since you have mass here and you have V over here you can divide your V over and so we know that mass over volume is going to be density and so rearranging the equation you can get density equals your pressure times molecular weight divided by ZRT. And then from this point we can actually find what our viscosity of our gas would be. So the molecular weight of gas is going to be used to find the viscosity of gas and so we had our molecular weight of gas to be 20.279 so I'm just going to write that over here. 20.279.