A proxy model of watercut Y_W in producing well with reservoir saturation $\begin{array}{l}s=\{ s_w, \, s_o, \, s_g \}\end{array}$ and reservoir pressure $\begin{array}{l}p_e\end{array}$ :

(1)	$\begin{array}{l}\displaystyle {\rm Y_{Wm}} = \frac{1 - \epsilon_g}{1 + \frac{M_{ro}}{M_{rw}} \cdot \frac{B_w}{B_o} }\end{array}$

(2)	$\begin{array}{l}\displaystyle \epsilon_g = \frac{A}{q_t} \cdot M_{ro} \cdot \left[ \frac{\partial P_c}{\partial r} + (\rho_w-\rho_o) \cdot g \cdot \sin \alpha \right]\end{array}$

where

$\begin{array}{l}B_w(p_e)\end{array}$	Water formation volume factor	$\begin{array}{l}B_o(p_e)\end{array}$	Oil formation volume factor	$\begin{array}{l}s\end{array}$	Reservoir saturation $\begin{array}{l}\{ s_w, \, s_o, \, s_g \}\end{array}$
$\begin{array}{l}M_{rw}(s)\end{array}$	Relative water mobility	$\begin{array}{l}M_{ro}(s)\end{array}$	Relative oil mobility	$\begin{array}{l}p_e\end{array}$	Current formation pressure
$\begin{array}{l}\rho_w\end{array}$	Water density	$\begin{array}{l}\rho_o\end{array}$	Oil density	$\begin{array}{l}g\end{array}$	Standard gravity constant
$\begin{array}{l}q_t\end{array}$	Total sandface flowrate	$\begin{array}{l}A\end{array}$	Cross-sectional flow area	$\begin{array}{l}\alpha\end{array}$	Deviation of flow from horizontal plane
$\begin{array}{l}P_c(s)\end{array}$	capillary pressure

If capillary effects are not high $\begin{array}{l}P_c \rightarrow 0\end{array}$ or saturation does not vary along the streamline substantially $\begin{array}{l}\displaystyle \frac{\partial s_w}{\partial r} \rightarrow 0\end{array}$ , then $\begin{array}{l}\displaystyle \frac{\partial P_c}{\partial r} = \dot P_c \cdot \frac{\partial s_w}{\partial r} \approx 0\end{array}$ .

If flow is close to horizontal $\begin{array}{l}\sin \alpha \rightarrow 0\end{array}$ then gravity effects are vanishing too: $\begin{array}{l}(\rho_w-\rho_o) \cdot g \cdot \sin \alpha \approx 0\end{array}$ .

In these cases (1) simplifies to:

(3)	$\begin{array}{l}\displaystyle {\rm Y_{Wm}} = \frac{1}{1 + \frac{M_{ro}}{M_{rw}} \cdot \frac{B_w}{B_o} } = \frac{1}{1 + \frac{k_{ro}}{k_{rw}} \cdot \frac{\mu_w }{\mu_o } \cdot \frac{B_w}{B_o}}\end{array}$

The models (1) and (3) can also be used in production analysis assuming homogeneous reservoir water saturation $\begin{array}{l}s_w\end{array}$ :

(4)	$\begin{array}{l}\displaystyle s_w(t) = s_{wi} + (1-s_{wi}) \cdot \rm E_{Dow}(t) = s_{wi} + (1-s_{wi}) \cdot \rm RFO(t)/E_S\end{array}$

where

$\begin{array}{l}\displaystyle {\rm RFO} = \frac{Q^{\uparrow}_O}{V_{\rm STOIIP}}\end{array}$	current oil recover factor
$\begin{array}{l}Q^{\uparrow}_O\end{array}$	cumulative oil production
$\begin{array}{l}V_{\rm STOIIP}\end{array}$	STOIIP
$\begin{array}{l}E_S\end{array}$	sweep efficiency
$\begin{array}{l}s_{wi}\end{array}$	initial water saturation
$\begin{array}{l}s_{orw}\end{array}$	residual oil saturation to water sweep

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