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

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

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

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

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

for each fluid phase individually: $\begin{array}{l}p_{e9,o}, \, p_{e9,g}, \, p_{e9,w}\end{array}$

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

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

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

Oil Sandface Productivity Index $\begin{array}{l}J_o = \frac{q_o}{p_{e,o} - p_{wf}}\end{array}$

Gas Sandface Productivity Index $\begin{array}{l}J_g = \frac{q_g}{p_{e,g} - p_{wf}}\end{array}$

Water Sandface Productivity Index $\begin{array}{l}J_ w = \frac{q_q}{p_{e,w} - p_{wf}}\end{array}$

(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 surface flowrate $\begin{array}{l}q_L\end{array}$

Oil Surface Productivity Index $\begin{array}{l}J_O = \frac{q_O}{p_r - p_{wf}}\end{array}$

Gas Surface Productivity Index $\begin{array}{l}J_G = \frac{q_G}{p_r - p_{wf}}\end{array}$

Water Surface Productivity Index $\begin{array}{l}J_W = \frac{q_W}{p_r - p_{wf}}\end{array}$

Liquid Surface Productivity Index $\begin{array}{l}J_L = \frac{q_L}{p_r - p_{wf}}\end{array}$

(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 $\begin{array}{l}A_e\end{array}$

$\begin{array}{l}q_t = q_w+q_o+q_g\end{array}$

Total Sandface Productivity Index: $\begin{array}{l}J_t = \frac{q_t}{p_e - p_{wf}}\end{array}$

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