# Composite IPR Curves

This methods is used with oil wells with water production

07-23-2022

This method combine the Vogel method for oil flow and the constant productivity index for water flow. The IPR curves is determined geometrically from a equations group that consider water and oil flow.

The next figure is used to determine the equation that allow calculate the flowing bottom hole pressure with a total rate.

The composite IPR can be divide in 3 regions.

1. Between 0 and the flow rate at the bubble point, $$0 < q_t <q_b$$ . In this region the rate and flowing bottom hole pressure have a linear relationship.

$P_{wt}=P_r-\frac{q_t}{J}$

The region between the flow rate at the bubble point to the maximum oil flow rate, $$q_b < q_t < q_{o max}$$ . At a total flow rate, the flowing bottom hole pressure is defined by:

$P_{wt}=F_o(P_{wtoil})+F_w(P_{wfwater})$

Using Vogel equation for oil and constant productivity index for water, the flowing bottom hole pressure at the total flow rate is:

$P_{wt}=F_w(P_r-\frac{q_t}{J})+F_o(0.125)P_b\left[-1+\sqrt{81-80\left[\frac{q_t-q_b}{q_{omax}-q_b}\right]}\right]$

The interval between the maximum oil flow rate and the maximum total flow rate, $$q_{o max} < q_t < q_{t max}$$ . In this interval, the composite IPR curve would have a constant slope, since the curve is mostly affected by water production. So, $$tan \beta$$ must be determined for calculating the flowing bottom-hole pressure at a total rate as follows:

1. Take a total flow rate is very close to the maximum oil flow rate:

$q_t=0.999 q_omax$

1. Since the difference between $$q_t$$ and $$q_{omax}$$ is very small, we can assume that $$\alpha_2 = \alpha_1$$ and $$\beta_2 = \beta_1$$ and the tangent of these angles can be calculated geometrically in the shaded triangle.

$tan \beta_1=CD/CG$

$tan \alpha_1=CG/CD$

CD is the difference between the flowing bottom hole pressure at point D, $$P_{wfD}$$ and the flowing bottom hole pressure ar point C, $$P_{wfC}$$

$CD=P_{wfD}-P_{wfc}$

Point D lies on the composite IPR curve, so:

$P_{wfD}=F_o(P_{wfoil})+F_w(P_{wfwater})$

Or, from Figure 1:

$P_{wfD}=F_o*P_{wfB}+F_w*P_{wfE}$

$P_{wfB}=0.125(P_b)*\left[-1+\sqrt{81-80\left[\frac{0.999q_{omax}-q_b}{q_{omax}-q_b}\right]}\right]$

$P_{wfE}=P_r-\frac{0.999q_{omax}}{J}$

Therefore: $P_{wfD}=F_w\left(P_r-\frac{0.999q_{omax}}{J}\right)+F_o(0.125)P_b*\left[-1+\sqrt{81-80\left[\frac{0.999q_{omax}-q_b}{q_{omax}-q_b}\right]}\right]$

From Figure 1, $$P_{wfC}=P_{wfG}$$ where G also lies on the composite IPR curve for $$qt=q_{omax}$$

$P_{wfG}=F_o(P_{wfoil}+F_w(P_{wfwater}))$

At $$q_t=q_{omax}$$, $$P_{wfoil}=0$$; therefore:

$P_{wfG}=F_w(P_{wfwater})=F_w\left(Pr-\frac{q_{omax}}{J}\right)$ $P_{wfC}=P_{wfG}=F_w\left(P_r-\frac{q_{omax}}{J}\right)$

Substituting equation, yields:

$CD=P_{wfD}-P_{wfC}=F_w\left(\frac{0.001q_{omax}}{J}\right)+F_o(0.125)P_b\left[-1+\sqrt{81-80\left[\frac{0.999q_{omax}-q_b}{q_{omax}-q_b}\right]}\right]$

CG is the difference between $$q_t$$ and $$q_{omax}$$, therefore:

$CG=q_{omax}-0.999q_{omax}=0.001q_{omax}$

The maximum total flow rate (for the composite IPR curve) can be calculated by usgin the following equation:

$q_{tmax}=q_{omax}+F_w(P_r-\frac{q_{omax}}{J})(tan \alpha)$

### Calculation os the total flow rate ar certain flowing bottom-hole pressures for the composite IPR curve

The composite IPR curve can be divided into three intervals, and in every interval, the total flow rate at certain flowing bottom-hole pressures can be calculated as follows:

1. For pressures between reservoir pressure and the bubble-point pressure, $$P_b < P_{wf} < P_r$$, the toal flow rate can be calculated by using the following equation:

$q_t=J(P_r-P_{wf})$

1. For pressures between the bubble-point pressure and te flowing bottom-hole pressure where the oil flow rate is equal to the maximum rate, $$P_{efG} < P_{wf} < P_b$$, the total flow rate is:

for $$B \neq 0$$ $q_t=\frac{-C+\sqrt{C^2-4B^2}D}{2B^2}$ for $$B=0$$ $q_t=D/C$

where:

$A=\frac{P_{wf}+0.125F_oP_r}{0.125F_0P_b}$ $B=\frac{F_w}{0.125F_oP_bJ}$ $C=2(A)(B)+\frac{80}{q_{omax}-q_b}$ $D=A^2-(80)\frac{q_b}{q_{omax}-q_b}-81$

1. For pressures between P{wfg} and 0, that is $$0 < P_{wf} < P_{efG}$$, the total flow rate is:

$q_t=\frac{P_{wfG}+q_{omax}(tan \beta)-P_{wf}}{tan\beta}$

### Example Problem

#### Reservoir pressure greater than the bubble-point pressure

Give data:

Reservoir pressure, $$P_r=2500 \ psi$$ Bubble-point pressure, $$P_b=2100 \ psi$$ Test data: Flowing bottom hole pressure = $$P_b=2300 \ psi$$ Flow rate (total), $$q_t= 500 \ b/d$$

Calculate: Determine the composite IPR curves for $$F_w = 0.5$$

Code

First, using all the equation above, calculate total flow rates and flowing bottom hole pressure at this rate

library(ggplot2)

Py <- 2550
Pb <- 2100
Pwf <- 2300
Qo <- 500
Fw <- 0.5

#Since Pwf_test > Pb
IP <- Qo/(Py-Pwf)
qb <- IP*(Py-Pb)
qomax <- qb+(2*Pb)/1.8

CD <- (1-Fw)*0.125*Pb*(sqrt(81-80*(0.999*qomax-qb)/(qomax-qb))-1)+
Fw*(Py-(0.999*qomax)/IP)-Fw*(Py-qomax/IP)

CG <- 0.001*qomax

tana <- CG/CD
tanb <- 1/tana

qtmax <- qomax+Fw*(Py-qomax/IP)*tana
PwfG <- Fw*(Py-qomax/IP)

[1] "j =  2"
[1] "qb =  900"
[1] "qomax =  3233.33333333333"
[1] "tana =  0.409688827872138"
[1] "tanb =  2.44087690941891"
[1] "qtmax =  3424.521453007"
[1] "PwfG =  466.666666666667"

Now, we can define some function to do the intervals calculations, with this we can use a $$P_{wf}$$ vector to estimate flow rate.

A_IPR_c <- function(Pwf,Fw,Pb,Py){
A <- (Pwf+0.125*(1-Fw)*Pb-Fw*Py)/(0.125*(1-Fw)*Pb)
return(A)
}

B_IPR_c <- function(Fw,Pb,J){
B <- Fw/(0.125*(1-Fw)*Pb*J)
return(B)
}

C_IPR_c <- function(Pwf,Fw,Pb,Py,J,qomax,qb){
A <- (Pwf+0.125*(1-Fw)*Pb-Fw*Py)/(0.125*(1-Fw)*Pb)
B <- Fw/(0.125*(1-Fw)*Pb*J)
C <- 2*A*B+(80)/(qomax-qb)
return(C)
}

D_IPR_c <- function(Pwf,Fw,Pb,Pr,J,qomax,qb){
A <- (Pwf+0.125*(1-Fw)*Pb-Fw*Py)/(0.125*(1-Fw)*Pb)
D <- A^2-80*(qb/(qomax-qb))-80
return(D)
}


After that, we can define a $$P_{wf}$$ vecto and apply the equation according the intervals

pwf <- c(0,200 ,350 ,600, 1000 ,1400, 1500, 1700, 2100, 2300, 2400, 2550)

A <- A_IPR_c(pwf,Fw,Pb,Py)
B <- B_IPR_c(Fw,Pb,IP)
C <- C_IPR_c(pwf,Fw,Pb,Py,IP,qomax,qb)
D <- D_IPR_c(pwf,Fw,Pb,Py,IP,qomax,qb)

qt <- ifelse(pwf >= Pb, IP*(Py-pwf),
ifelse(pwf >= PwfG & pwf < Pb,
ifelse(rep(B, length(pwf)) != 0 ,
(-C+sqrt(C^2-4*B^2*D))/(2*B^2), -D/C),
ifelse(pwf < PwfG, (PwfG+qomax*tanb-pwf)/tanb  , 0)))

IPR <- data.frame(pwf = pwf, qt = qt)

ggplot(IPR, aes(x = qt, y = pwf)) +
geom_line()

knitr::kable(IPR, caption = "Composite IPR, FW = 0.5")

Table 1: Composite IPR, FW = 0.5
pwf qt
0 3424.521
200 3342.584
350 3281.130
600 3135.673
1000 2738.169
1400 2159.923
1500 1995.359
1700 1646.911
2100 900.000
2300 500.000
2400 300.000
2550 0.000

Reference: Brown, K. (1984) The technology of artificial lift methods Vol. 4

### Citation

Vazquez (2022, July 23). Chato Solutions: Composite IPR Curves. Retrieved from https://www.chatosolutions.com/posts/2022-07-23-picomposite/
@misc{vazquez2022composite,
}