A specific IPR model:

(1)	$\begin{array}{l}\displaystyle \frac{q_O}{q_{O, \rm max}} = 1 - a \, \frac{p_{wf}}{p_r} - (1-a) \Bigg(\frac{p_{wf}}{p_r} \Bigg)^2 \quad , \quad p_b > p_r > p_{wf}, \quad \{ 0< a< 1 \}\end{array}$

where

$\begin{array}{l}q_O\end{array}$	surface production rate of rate
$\begin{array}{l}q_{O, \rm max}\end{array}$	Absolute Open Flow (AOF) for oil
$\begin{array}{l}p_{wf}\end{array}$	bottom-hole pressure (BHP)
$\begin{array}{l}p_r\end{array}$	drainarea formation pressure
$\begin{array}{l}p_b\end{array}$	bubble-point pressure
$\begin{array}{l}a\end{array}$	model parameter $\begin{array}{l}\{ 0< a< 1 \}\end{array}$

This is a generalisation of Vogel model and reduces to Vogel model for $\begin{array}{l}a = 0.2\end{array}$ .

It also goes as a partial case of the Wiggins IPR @ model with $\begin{array}{l}\{ a_1 = a, \, a_2 = 1-a, \, a_k = 0, \forall k> 2 \}\end{array}$

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See also