One of the most general form of IPR model below bubble-point pressure:

(1)	$\begin{array}{l}\displaystyle \Psi^n(p_b) - \Psi^n(p_{wf}) = (q/q_\mathrm{max} - q_b/q_\mathrm{max}) \left[ \frac{1}{(1-q_b/q_\mathrm{max})} \cdot \Psi^n(p_b) + (1-q/q_\mathrm{max}) \cdot H(q)\right]\end{array}$

(2)

$\begin{array}{l}\displaystyle H(q) = \frac{\sum_{k=0}^m a_k \, (q/q_\mathrm{max})^k}{\sum_{k=0}^m b_k \, (q/q_\mathrm{max})^k} = \frac{a_0 + a_1 \cdot (q/q_\mathrm{max}) + a_2 \cdot (q/q_\mathrm{max})^2 + ... + a_m \cdot (q/q_\mathrm{max})^m} {b_0 + b_1 \cdot (q/q_\mathrm{max}) + b_2 \cdot (q/q_\mathrm{max})^2 + ... + b_m \cdot (q/q_\mathrm{max})^m}\end{array}$

(3)	$\begin{array}{l}\displaystyle p_b > p_r > p_{wf}\end{array}$

where

$\begin{array}{l}p_{wf}\end{array}$	bottom-hole pressure (BHP)
$\begin{array}{l}p_b\end{array}$	bubble-point pressure
$\begin{array}{l}q\end{array}$	sandface flowrate
$\begin{array}{l}q_\mathrm{max}\end{array}$	Absolute Open Flow (AOF)
$\begin{array}{l}\Psi(p)\end{array}$	pseudo-pressure function specific to fluid type
$\begin{array}{l}n\end{array}$	pseudo-pressure curvature
$\begin{array}{l}a\end{array}$	laminar flow coefficient
$\begin{array}{l}b\end{array}$	turbulent flow coefficient

It needs well tests at least three different rates to assess $\begin{array}{l}\{ a \, , \, b, \, p_r \}\end{array}$ but obviously more tests will make assessment more accurate.

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PADE IPR @ model

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References