Productivity Index @model

The most accurate prediction of Productivity Index is based on numerical reservoir flow simulations.

Apart from this there are numerous proxy models covering popular categories of reservoir flow regimes, dynamic reservoir properties and well-reservoir contacts.

One of the most popular proxy model is Dupuit PI @model which covers a very wide range of stabilised reservoir flow.

Dupuit PI @model

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

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

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Productivity Index @model