CHAPTER 06.04: NONLINEAR REGRESSION: Harmonic Decline Curve Model: Transformed Data: Example: Part 2 of 2
So as you can see that I need to first calculate my zi values. So the summation zi values are simply the inverse of the qi values, so those are the summation of 1 divided by qi values. So in that case, it'll be 1 divided by 260, plus 1 divided by 189, plus 1 divided by 120, plus 1 divided by 87, plus 1 divided by 75, because those are the values which are given to me, these are the individual 1 values, I have to take the inverse of each q value, and add them up to get the summation of zi values, and this number here turns out to be 0.042298. So similarly I'm going to calculate my other summations which I need. I need the summation of all the time values, so that gives me 2, plus 6, plus 10, plus 14, plus 18, that's all the t values which I have, that gives me 50. So similarly I can, let's suppose I'm going to calculate my ti zi summation which I also need, and that'll be the values of the t values, which is 2, then the inverse of the q values, which is 1 divided by 260 for the first one, the next one will be 6, times 1 divided by 189, the next one will be 10, times 1 divided by 120, so all these values which you are seeing here, this is the t value and this is the corresponding inverse of the q values which are given for the data, so that's all I'm doing here, so this will be 14 . . . it'll be 14, times 1 divided by 87, plus 18, times 1 divided by 75, and that value turns out to be 0.52369. So we need one more summation, so let me just go through the whole thing, as opposed to just telling you to do as homework. We need summation of ti squared, that's simply taking each individual t value, and squaring it, and those values turn out to be 660. So these are the summations which I need in my linear regression formula which I wrote for a0 and a1. So let me go ahead and find out what those a0 and a1 values are. So I'm going to find my a1, a1 i n, n is the number of data points which I have, which is 5 in this case, then the summation of zi ti, which is 0.52369, minus summation of zi, summation, ti, summation of zi is 0.042298, and the summation of ti is 50, then this whole thing is getting divided by n times summation, ti squared, which is 660, and n is 5, so I can substitute that, n is 5, and then I have 660, minus summation, ti, squared, which is 50 squared. So these are all the summations which I needed, and those are the ones which I showed you in the previous board here, and this a1 value which you are getting here turns out to be 6.2944 times 10 to the power -4. And now you need to calculate a0, which is the z-bar, minus a1 t-bar. The z-bar is nothing but the average values of z. The z value is 0.042298 divided by 5, because I have five data points, a1 I just found, 6.2944 times 10 to the power -4, and t-bar is the average value of the t values, which is 50 is the total value, divided by 5, and this a0 value turns out to be equal to 0.0021652. So these are the constants which are coming from the z versus t data formula, but that's not what we are looking for, we are looking for the values of a and b of the harmonic decline curve, and we know that b is nothing but 1 divided by a0. b is nothing but 1 divided by a0, so it will b 1 divided by 0.0021652, and that value turns out to be 461.85, so that's what I get for the value of b. Now, a is nothing but . . . a will be, a is equal to b times a1, it is b times a1, that's what a is, and I know b, which I just found out, which is 461.85, and a1 I just found out to be 6.2944 times 10 to the power -4, and that value turns out to be 0.29071. So what I have been able to do is I have been able to use the transformed data to use my linear regression formulas to calculate my a0 and a1, and then backtrack my values of a and b of the harmonic decline curve. So what that tells me is that the harmonic decline curve which I have is now q is equal to b, 1 plus a t, and b I just found out will be 461.85, divided by 1 plus a, which is 0.29071, times t. So the two questions which were asked of me was to find out what will be the production at 60 months, and the second question was, hey, what will be the . . . at what time will the production be 20 barrels per day, so that you can figure out when to abandon the well. So if you look at part A, we want to find out what is the value of the . . . how much in barrels per day are being produced at 60 months? So I'm just going to plug in 60 into the formula which I just derived for the harmonic decline curve, and this value here is turning out to be approximately 25 barrels per day. So that's what you're getting, so at 60 months, you will have about 25 barrels per day. The second question was that at what point is your production 20 barrels per day? So you know that it is going to be more than 60 months, because at 60 months you are still producing 25 barrels per day, so it has to go a little bit further to go to 20 barrels per day. So how we're going to do that is by saying 20 is equal to 461.85, divide by 1 plus 0.29071 t. So from here I find out what t is by saying 1 plus 0.29071 t is equal to 461.85 divided by 20, and t will be equal to 461.85, divided by 20, minus 1, divided by 0.29071, that's simple algebra which you are trying to do there, and this time here turns out to be approximately 76 months. So what you are figuring out here is that you are getting the time that when the number of barrels which will be produced will be 20 will be 76 months, so that's when you would abandon the well if that was the . . . that was the limit you would you would like it to have, 20 barrels per month. And that's the end of this segment.