The general form of objective function for production targets optimisation is given by:

(1)

$\begin{array}{l}\displaystyle G(t) = \sum_{p=1}^{N^{\uparrow}} \left[ R_O \cdot q^{\uparrow}_{O, p} + R_G \cdot q^{\uparrow}_{G, p} \right] - \sum_{p=1}^{N^{\uparrow}} C^{\uparrow}_L \cdot q^{\uparrow}_{L, p} - \sum_{p=1}^{N^{\uparrow}} C_O \cdot q^{\uparrow}_{O, p} - \sum_{p=1}^{N^{\uparrow}} C_G \cdot q^{\uparrow}_{G, p} - \sum_{p=1}^{N^{\uparrow}} C_W \cdot q^{\uparrow}_{W, p} - \sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_W \cdot q^{\downarrow}_{W, i} - \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_G \cdot q^{\downarrow}_{G, i} \rightarrow \rm max\end{array}$

where

$\begin{array}{l}q^{\uparrow}_{O, p}\end{array}$	oil production rate for $\begin{array}{l}p\end{array}$ -th producer, volume/day	$\begin{array}{l}C_O\end{array}$	cost of oil treatment at surface, cash/volume	$\begin{array}{l}R_O\end{array}$	oil price, cash/volume
$\begin{array}{l}q^{\uparrow}_{G, p}\end{array}$	gas production rate for $\begin{array}{l}p\end{array}$ -th producer, volume/day	$\begin{array}{l}C_G\end{array}$	cost of gas treatment at surface, cash/volume	$\begin{array}{l}R_G\end{array}$	gas price, cash/volume
$\begin{array}{l}q^{\uparrow}_{W, p}\end{array}$	water production rate for $\begin{array}{l}p\end{array}$ -th producer, volume/day	$\begin{array}{l}C_W\end{array}$	cost of water treatment at surface, cash/volume	$\begin{array}{l}N^{\uparrow}\end{array}$	number of producers
$\begin{array}{l}q^{\uparrow}_{L, p}\end{array}$	liquid production rate for $\begin{array}{l}p\end{array}$ -th producer, volume/day	$\begin{array}{l}C^{\uparrow}_L\end{array}$	cost of fluid lift, cash/volume	$\begin{array}{l}N^{\downarrow}_W\end{array}$	number of water injectors
$\begin{array}{l}q^{\downarrow}_{W, i}\end{array}$	water injection rate for $\begin{array}{l}i\end{array}$ -th water injector, volume/day	$\begin{array}{l}C^{\downarrow}_W\end{array}$	cost of water injection, cash/volume	$\begin{array}{l}N^{\downarrow}_G\end{array}$	number of gas injectors
$\begin{array}{l}q^{\downarrow}_{G, i}\end{array}$	gas injection rate for $\begin{array}{l}i\end{array}$ -th gas injector, volume/day	$\begin{array}{l}C^{\downarrow}_G\end{array}$	cost of gas injection, cash/volume	$\begin{array}{l}t\end{array}$	time

This can be rewritten in terms of sandface flowrates:

(2)	$\begin{array}{l}\displaystyle G = \sum_{p=1}^{N_{\rm prod}} G^{\uparrow}_{ut,p} \cdot q^{\uparrow}_{t, p} - \sum_{i=1}^{N_{W, \rm inj}} G^{\downarrow}_w \cdot q^{\downarrow}_{w, i} - \sum_{j=1}^{N_{G, \rm inj}} G^{\downarrow}_g \cdot q^{\downarrow}_{g, i} \rightarrow \rm max\end{array}$

where

$\begin{array}{l}\displaystyle G^{\uparrow}_{t,p}\end{array}$

$\begin{array}{l}\displaystyle G^{\downarrow}_w = B_w \cdot C_{W, \rm inj}\end{array}$

$\begin{array}{l}\displaystyle G^{\downarrow}_g = B_g \cdot C_{G, \rm inj}\end{array}$

Derivation

(3)

$\begin{array}{l}\displaystyle G = \sum_{p=1}^{N_{\rm prod}} \left[ (R_O -C_O) \cdot q^{\uparrow}_{O, p} + (R_G-C_G) \cdot q^{\uparrow}_{G, p} - C_L \cdot q^{\uparrow}_{L, p} - C_W \cdot q^{\uparrow}_{W, p} \right] - \sum_{i=1}^{N_{W, \rm inj}} C_{W, \rm inj} \cdot q^{\downarrow}_{W, i} - \sum_{j=1}^{N_{G, \rm inj}} C_{G, \rm inj} \cdot q^{\downarrow}_{G, i} \rightarrow \rm max\end{array}$

(4)

$\begin{array}{l}\displaystyle G = \sum_{p=1}^{N_{\rm prod}} \left[ \left[ (R_O -C_O) + (R_G-C_G) \cdot GOR \right] \cdot q^{\uparrow}_{O, p} - (C_L + C_W \cdot Y_w) \cdot q^{\uparrow}_{L, p} \right] - \sum_{i=1}^{N_{W, \rm inj}} C_{W, \rm inj} \cdot q^{\downarrow}_{W, i} - \sum_{j=1}^{N_{G, \rm inj}} C_{G, \rm inj} \cdot q^{\downarrow}_{G, i} \rightarrow \rm max\end{array}$

(5)

$\begin{array}{l}\displaystyle G = \sum_{p=1}^{N_{\rm prod}} \left[ \left[ (R_O -C_O) + (R_G-C_G) \cdot GOR \right] \cdot (1-Y_w) - (C_L + C_W \cdot Y_w) \right] \cdot q^{\uparrow}_{L, p} - \sum_{i=1}^{N_{W, \rm inj}} C_{W, \rm inj} \cdot q^{\downarrow}_{W, i} - \sum_{j=1}^{N_{G, \rm inj}} C_{G, \rm inj} \cdot q^{\downarrow}_{G, i} \rightarrow \rm max\end{array}$

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