Correlation between Water-Oil-Ratio (WOR) and Water Oil Balance Ratio (WOBR):

$\begin{array}{l}\displaystyle {\rm WOR} = \Phi \left( \frac{ \Omega_W^{\downarrow} \cdot Q_W^{\downarrow} - \Omega_W^{\uparrow} \cdot Q_W^{\uparrow} } { EUOR - Q_O^{\uparrow} } \right)\end{array}$

where $\begin{array}{l}\Phi()\end{array}$ is some function.

For example, one can select $\begin{array}{l}\Phi(x) = x^n\end{array}$ and search for the exponent $\begin{array}{l}n\end{array}$ , good water percentages $\begin{array}{l}\Omega_W^{\downarrow}\end{array}$ , $\begin{array}{l}\Omega_W^{\uparrow}\end{array}$ and even adjust $\begin{array}{l}EUOR\end{array}$ in order to achieve the best model $\begin{array}{l}{\rm WOR}^*(t)\end{array}$ fit to the WOR history:

$\begin{array}{l}\displaystyle \sum_t \left\| {\rm WOR}(t) - {\rm WOR}^*(t) \right\| \rightarrow 0\end{array}$

$\begin{array}{l}\displaystyle {\rm WOR}^*(t) = \left[ \frac{ \Omega_W^{\downarrow} \cdot Q_W^{\downarrow}(t) - \Omega_W^{\uparrow} \cdot Q_W^{\uparrow}(t) } { EUOR - Q_O^{\uparrow}(t) } \right]^n\end{array}$

where $\begin{array}{l}\left\| x \right\|\end{array}$ is the norm of $\begin{array}{l}x\end{array}$ .

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WOR vs WOBR @model

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