Definition specifics on formation pressure (Pe) and productivity index (PI) 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 define formation pressure and productivity index.

This difference is summarized in the table below:

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

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

Productivity Index $\begin{array}{l}J\end{array}$

DM

9-cell formation pressure

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

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

Oil sandface flowrate $\begin{array}{l}q_o\end{array}$

Gas sandface flowrate $\begin{array}{l}q_g\end{array}$

Water sandface flowrate $\begin{array}{l}q_w\end{array}$

sandface 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 formation pressure estimate within the drainage area Ae

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

Oil surface flowrate $\begin{array}{l}q_O\end{array}$

Gas surface flowrate $\begin{array}{l}q_G\end{array}$

Water surface flowrate $\begin{array}{l}q_W\end{array}$

Liquid flowrate $\begin{array}{l}q_L\end{array}$

surface 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: $\begin{array}{l}J_{OW} = \frac{q_{OW}}{p_r - p_{wf}}\end{array}$

WT

Boundary-average formation pressure estimate along the boundary of drainage area Ae

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

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:

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

Total sandface 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 (Pe) and productivity index (PI) between Dynamic Modelling, Well Flow Performance and Well Tests