Fig. 1. Multi-layer well schematic

(1)	$\begin{array}{l}\displaystyle q = \sum_{k=1}^N q_k\end{array}$

(2)	$\begin{array}{l}\displaystyle p_{wf} = p_e - q/J\end{array}$

(3)	$\begin{array}{l}\displaystyle J = \sum_{k=1}^N J_k\end{array}$

(4)	$\begin{array}{l}\displaystyle p_e = J^{-1} \cdot \sum_{k=1}^N J_k \, (p_k- \delta p_k)\end{array}$

where

$\begin{array}{l}q\end{array}$	total subsurface flowrate of the well
$\begin{array}{l}J\end{array}$	total well productivity Index
$\begin{array}{l}p_e\end{array}$	apparent formation pressure of dual-layer formation
$\begin{array}{l}p_{wf} = p_{wf, k_{ref}}\end{array}$	bottom-hole pressure at the top of the reference layer $\begin{array}{l}k_{reff}\end{array}$
$\begin{array}{l}p_{wf,k} = p_{wf} +\delta p_k\end{array}$	bottom-hole pressure at the top of the $\begin{array}{l}k\end{array}$ -th layer
$\begin{array}{l}\delta p_k\end{array}$	wellbore pressure loss between $\begin{array}{l}k\end{array}$ -th layer and reference layer $\begin{array}{l}k_{reff}\end{array}$
$\begin{array}{l}q_k\end{array}$	total subsurface flowrate of the $\begin{array}{l}k\end{array}$ -th layer
$\begin{array}{l}p_k\end{array}$	formation pressure of the $\begin{array}{l}k\end{array}$ -th layer
$\begin{array}{l}J_k\end{array}$	productivity Index of the $\begin{array}{l}k\end{array}$ -th layer

The above equations are valid for both producers $\begin{array}{l}q>0\end{array}$ and injectors $\begin{array}{l}q< 0\end{array}$ .

In many practical cases:

(5)	$\begin{array}{l}\displaystyle \delta p_k = \rho \, g \, h_k\end{array}$

where

$\begin{array}{l}\rho\end{array}$	wellbore fuid density
$\begin{array}{l}g\end{array}$	gravity constant
$\begin{array}{l}h_k = TVDSS_k - TVDSS_{k_{ref}}\end{array}$	true vertical height between $\begin{array}{l}k\end{array}$ -th layer and reference layer $\begin{array}{l}k_{reff}\end{array}$

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