A proxy model of Productivity Index for stabilised reservoir flow in homogeneous reservoir:

$\begin{array}{l}\displaystyle J = \frac{q}{p_{frm} - p_{wf}} = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + \epsilon + S}\end{array}$

where

$\begin{array}{l}\sigma\end{array}$	transmissibility
$\begin{array}{l}\epsilon\end{array}$	a model parameter depending on Productivity Index definitions and boundary type

Drain-area Productivity Index

Drain-boundary Productivity Index

$\begin{array}{l}\displaystyle J_r = \frac{q}{p_r - p_{wf}}\end{array}$

$\begin{array}{l}\displaystyle J_e = \frac{q}{p_e - p_{wf}}\end{array}$

SS

$\begin{array}{l}\displaystyle J_r = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + 0.5 + S}\end{array}$

$\begin{array}{l}\displaystyle J_e = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + S}\end{array}$

PSS

$\begin{array}{l}\displaystyle J_r = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + 0.75 + S}\end{array}$

$\begin{array}{l}\displaystyle J_e = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + 0.5 + S}\end{array}$

The relation between the total sandface flowrate $\begin{array}{l}q\end{array}$ , bottomhole pressure $\begin{array}{l}p_{wf}\end{array}$ and field-average formation pressure $\begin{array}{l}p_r\end{array}$ during the stabilized reservoir flow:

corresponding to linear IPR with constant productivity index :

(1)	$\begin{array}{l}\displaystyle J_r = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + \epsilon+ S}\end{array}$

where

$\begin{array}{l}\sigma\end{array}$	transmissibility
$\begin{array}{l}\epsilon = 0.5\end{array}$	for Steady State (SS) well flow regime
$\begin{array}{l}\epsilon = 0.75\end{array}$	for Pseudo Steady State (PSS) well flow regime

Reference

Dupuit, J., Etudes theoriques et pratiques sur le mouvement des eaux dans les canaux decouverts et a travers les terrains permeables, 2eme edition; Dunot, Paris, 1863.

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Dupuit PI @model

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