A proxy model of Productivity Index for stabilised reservoir flow.

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

where

$\begin{array}{l}q\end{array}$	depending on application may mean a total sandface flowrate ( $\begin{array}{l}q_t\end{array}$ ) or a product of surface flowrate and FVF ( $\begin{array}{l}q = q_{\rm srf} B\end{array}$ )
$\begin{array}{l}p_{wf}\end{array}$	bottomhole pressure
$\begin{array}{l}p_{\rm frm}\end{array}$	depending on application may mean a drain-boundary formation pressure ( $\begin{array}{l}p_e\end{array}$ ) or drain-area formation pressure ( $\begin{array}{l}p_r\end{array}$ )
$\begin{array}{l}\sigma\end{array}$	formation transmissibility
$\begin{array}{l}r_w\end{array}$	wellbore radius
$\begin{array}{l}r_e\end{array}$	distance to a drainarea boundary
$\begin{array}{l}S\end{array}$	total skin
$\begin{array}{l}\epsilon\end{array}$	a model parameter depending on Productivity Index definition and boundary type ( $\begin{array}{l}\epsilon =\{ 0, \, 0.5, \, 0.75 \}\end{array}$ , see Table 1 below)

In case of homogeneous reservoir with only one vertical well producing the Dupuit PI @model is the exact analytical solution of Reservoir Flow Model (RFM).

Table 1. Variations to Dupuit PI @model depending on the reservoir flow regime and the definition/application of Productivity Index.

Drain-area Productivity Index, $\begin{array}{l}J_r = \frac{q}{p_r - p_{wf}}\end{array}$

Drain-boundary Productivity Index $\begin{array}{l}J_e = \frac{q}{p_e - p_{wf}}\end{array}$

Steady State flow regime (SS)

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

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

Pseudo-Steady State flow regime (PSS)

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

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

For the fractured vertical well the geometrical skin-factor $\begin{array}{l}S_G\end{array}$ is related to Fracture half-length $\begin{array}{l}X_f\end{array}$ as:

(1)	$\begin{array}{l}\displaystyle S_G = -\ln \left(\frac{X_f}{2\, r_w} \right)\end{array}$

$\begin{array}{l}\displaystyle J = \frac{q}{p_{\rm frm} - p_{wf}} = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} - \epsilon + S} = \frac{2 \pi M \cdot h}{ \ln \frac{r_e}{r_w} - \epsilon + S} = \frac{2 \pi k_{abs} \cdot h}{ \ln \frac{r_e}{r_w} - \epsilon + S} \cdot M_r = T \cdot M_r(s_w, s_g)\end{array}$

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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