Definition specifics on formation pressure and productivity index in between Dynamic Modelling, Well Flow Performance and Well Tests

Despite of terminological similarity there is a big difference in the way Dynamic Modelling, Well Flow Performance and Well Testing deal with formation pressure and flowrates which results in a difference in productivity index definition and corresponding analysis.

This difference is summarized in the table below:

Formation pressure, $\begin{array}{l}p\end{array}$

Flow rate, $\begin{array}{l}q\end{array}$

Producivity Index, $\begin{array}{l}J\end{array}$

DM

field-average pressure within the 9-cell area $\begin{array}{l}A_{e9}\end{array}$

(1)	$\begin{array}{l}\displaystyle p_{e9, \ i,j} = \frac{1}{9} \sum_{k=i-1}^{i+1} \sum_{l=j-1}^{j+1} p_{k,l} = \frac{1}{9} ( p_{i,j} + p_{i, \, j+1} + p_{i, \, j-1} + p_{i-1, \, j} + p_{i-1, \, j} + p_{i-1 \, j-1} + p_{i+1, \, j+1} + p_{i-1 \, j+1} + p_{i+1, \, j-1} )\end{array}$

phase flowrate at sandface: $\begin{array}{l}\{ q_w, \, q_o, \, q_g \}\end{array}$

(each fluid phase separately)

phase productivity index:

$\begin{array}{l}J_ w = \frac{q_q}{p_r - p_{wf}}\end{array}$ , $\begin{array}{l}J_o = \frac{q_o}{p_r - p_{wf}}\end{array}$ , $\begin{array}{l}J_g = \frac{q_g}{p_r - p_{wf}}\end{array}$

WFP

field-average pressure within the drainage area $\begin{array}{l}A_e\end{array}$

(2)	$\begin{array}{l}\displaystyle p_r = \frac{1}{A_e} \iint_{A_e} p(x,y,z) dS\end{array}$

surface component flowrate $\begin{array}{l}\{ q_W, q_O, q_G \}\end{array}$

(each fluid component separately)

and sometimes liquid flowrate $\begin{array}{l}q_{\rm liq} = q_W + q_O\end{array}$

fluid component productivity index:

$\begin{array}{l}J_W = \frac{q_W}{p_r - p_{wf}}\end{array}$ , $\begin{array}{l}J_O = \frac{q_O}{p_r - p_{wf}}\end{array}$ , $\begin{array}{l}J_G = \frac{q_G}{p_r - p_{wf}}\end{array}$

and sometimes liquid productivity index: $\begin{array}{l}J_{\rm liq} = \frac{q_{\rm liq}}{p_r - p_{wf}}\end{array}$

WT

average pressure value at the boudary of drainage area $\begin{array}{l}A_e\end{array}$

(3)	$\begin{array}{l}\displaystyle p_e = \frac{1}{L_e} \int_0^{L_e} p(x,y,z) dl\end{array}$

where $\begin{array}{l}L_e\end{array}$ is the boundary of drainage area $\begin{array}{l}A_e\end{array}$

total flowrate at sandface:

$\begin{array}{l}q_t = B_w \, q_W + B_o \, q_O + B_g \, ( q_G - R_s q_O)\end{array}$ – for Black Oil

$\begin{array}{l}q_t = B_w \, q_W + \frac{B_o - R_s B_g}{1 - R_v R_s} \, q_O + \frac{B_g - R_v B_o}{1 - R_v R_s} \, q_G\end{array}$ – for Volatile Oil

or $\begin{array}{l}\{ W, \, O, \, G \}\end{array}$ pseudo-components of Compositional Model

total multiphase productivity index: $\begin{array}{l}J_t = \frac{q_t}{p_e - p_{wf}}\end{array}$

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Definition specifics on formation pressure and productivity index in between Dynamic Modelling, Well Flow Performance and Well Tests