The plot of water production rate $\begin{array}{l}q_W\end{array}$ (along y-axis) against the oil production rate $\begin{array}{l}q_O\end{array}$ (along x-axis).

It can be used for express Watercut Diagnostics of thief water production.

The mathematical model of the thief water production from aquifer is based on the following equation:

(1)	$\begin{array}{l}\displaystyle q_W = a \, \cdot q_O + b\end{array}$

(2)	$\begin{array}{l}\displaystyle a = J^{-1}_{1O} \cdot ( J_{1W} + J_{2W})\end{array}$

(3)	$\begin{array}{l}\displaystyle b = J_{2W} \cdot (p^_2 - p^_1)\end{array}$

where

		$\begin{array}{l}q_W\end{array}$	water production rate	$\begin{array}{l}q_O\end{array}$	oil production rate
$\begin{array}{l}p^*_1\end{array}$	formation pressure in petroleum reservoir	$\begin{array}{l}J_{1W}\end{array}$	water productivity index of petroleum reservoir	$\begin{array}{l}J_{1O}\end{array}$	oil productivity index of petroleum reservoir
$\begin{array}{l}p^*_2\end{array}$	formation pressure in aquifer	$\begin{array}{l}J_{2W}\end{array}$	water productivity index of aquifer

For the case of aquifer pressure is higher than that of petroleum reservoir: $\begin{array}{l}b > 0 \Leftrightarrow p^*_2 > p^*_1\end{array}$

For the case of aquifer pressure is lower than that of petroleum reservoir: $\begin{array}{l}b < 0 \Leftrightarrow p^*_2 < p^*_1\end{array}$

In practical applications, the equation (1) is often considered through the weighted average values:

(4)	$\begin{array}{l}\displaystyle \langle q_W \rangle = a \, \cdot \langle q_O \rangle + \, b\end{array}$

where

$\begin{array}{l}\langle q_W \rangle, \ \langle q_O \rangle\end{array}$

are weighted average of $\begin{array}{l}q_W\end{array}$ and $\begin{array}{l}q_O\end{array}$

There are different ways to calculate weighted average of the dynamic variable, for example:

$\begin{array}{l}\displaystyle \langle A \rangle_t \ = \frac{1}{t} \int_o^t A(t) \, dt\end{array}$

$\begin{array}{l}\displaystyle \langle A \rangle_q \ = \frac{1}{Q(t)} \int_o^t A(t) \, q(t) \, dt\end{array}$

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See Also