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

(1)	$\begin{array}{l}\displaystyle G = \sum_t \frac{G_t}{1+r^t} \rightarrow \rm max\end{array}$

(2)

$\begin{array}{l}\displaystyle G_t = \sum_{k=1}^{N^{\uparrow}_P} \left[ R_O(t) \cdot q^{\uparrow}_{O, k}(t) + R_G(t) \cdot q^{\uparrow}_{G, k}(t) \right] - \sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{L,k} \cdot q^{\uparrow}_{L, k}(t) - \sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{O,k} \cdot q^{\uparrow}_{O, k} (t) - \sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{G,k} \cdot q^{\uparrow}_{G, k} (t) - \sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{W,k} \cdot q^{\uparrow}_{W, k}(t) - \sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_{W,j} \cdot q^{\downarrow}_{W, i}(t) - \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot q^{\downarrow}_{G, j}(t)\end{array}$

where

$\begin{array}{l}q^{\uparrow}_{O, k}(t)\end{array}$	volume/day	oil production rate for $\begin{array}{l}k\end{array}$ -th producer	$\begin{array}{l}C^{\uparrow}_{O,k}\end{array}$	cash/volume	cost of produced oil treatment and transportation from $\begin{array}{l}k\end{array}$ -th wellhead to CMS	$\begin{array}{l}R_O(t)\end{array}$	cash/volume	oil selling price
$\begin{array}{l}q^{\uparrow}_{G, k}(t)\end{array}$	volume/day	gas production rate for $\begin{array}{l}k\end{array}$ -th producer	$\begin{array}{l}C^{\uparrow}_{G,k}\end{array}$	cash/volume	cost of produced gas treatment and transportation from $\begin{array}{l}k\end{array}$ -th wellhead to CMS	$\begin{array}{l}R_G(t)\end{array}$	cash/volume	gas selling price
$\begin{array}{l}q^{\uparrow}_{W, k}(t)\end{array}$	volume/day	water production rate for $\begin{array}{l}k\end{array}$ -th producer	$\begin{array}{l}C^{\uparrow}_{W,k}\end{array}$	cash/volume	cost of produced water treatment and transportation from $\begin{array}{l}k\end{array}$ -th wellhead to CMS	$\begin{array}{l}N^{\uparrow}_P\end{array}$	counts	number of producers at $\begin{array}{l}t\end{array}$
$\begin{array}{l}q^{\uparrow}_{L, k}(t)\end{array}$	volume/day	liquid production rate for $\begin{array}{l}k\end{array}$ -th producer	$\begin{array}{l}C^{\uparrow}_{L, k}\end{array}$	cash/volume	cost of fluid lift from reservoir to the $\begin{array}{l}k\end{array}$ -th wellhead, cash/volume	$\begin{array}{l}N^{\downarrow}_W\end{array}$	counts	number of water injectors at $\begin{array}{l}t\end{array}$
$\begin{array}{l}q^{\downarrow}_{W, i}(t)\end{array}$	volume/day	water injection rate for $\begin{array}{l}i\end{array}$ -th water injector	$\begin{array}{l}C^{\downarrow}_{W,i}\end{array}$	cash/volume	cost of water injection, including purchase, treatment, transportation and pumping into $\begin{array}{l}i\end{array}$ -th well	$\begin{array}{l}N^{\downarrow}_G\end{array}$	counts	number of gas injectors at $\begin{array}{l}t\end{array}$
$\begin{array}{l}q^{\downarrow}_{G, i}(t)\end{array}$	volume/day	gas injection rate for $\begin{array}{l}i\end{array}$ -th gas injector	$\begin{array}{l}C^{\downarrow}_{G,j}\end{array}$	cash/volume	cost of gas injection, including purchase, treatment, transportation and pumping into $\begin{array}{l}i\end{array}$ -th well	$\begin{array}{l}t\end{array}$	days	time

The objective function (1) can be rewritten in terms of Surface flowrates $\begin{array}{l}\{ q^{\uparrow}_L, q^{\downarrow}_W, q^{\downarrow}_G \}\end{array}$ :

(3)	$\begin{array}{l}\displaystyle G_t = \sum_{p=1}^{N^{\uparrow}_P} C^{\uparrow}_{OGW}(t) \cdot q^{\uparrow}_{L, p}(t) - \sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_{W,j} \cdot q^{\downarrow}_{W, i}(t) - \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot q^{\downarrow}_{G, j}(t)\end{array}$

(4)	$\begin{array}{l}\displaystyle C^{\uparrow}_{OGW}(t) = \left[ (R_O(t) - C^{\uparrow}_{O,p}) + (R_G(t) - C^{\uparrow}_{G,p}) \cdot Y_{g,p}(t) \right] \cdot (1- Y_{w,p}(t)) - C^{\uparrow}_{L,p} - C^{\uparrow}_{W,p} \cdot Y_{w,p}(t)\end{array}$

where

$\begin{array}{l}Y_{w,k}(t) = q_{W,k} / q_{L,k}\end{array}$	Watercut in $\begin{array}{l}k\end{array}$ -th well
$\begin{array}{l}Y_{g,k}(t) = q_{G,k} / q_{O,k}\end{array}$	Gas-Oil Ratio in $\begin{array}{l}k\end{array}$ -th well

Usually, each well has a fixed range of its rate variations:

(5)	$\begin{array}{l}\displaystyle q^{\uparrow}_{LMIN, p} \leq q^{\uparrow}_{L, p}(t) \leq q^{\uparrow}_{LMAX, p}\end{array}$

(6)	$\begin{array}{l}\displaystyle q^{\downarrow}_{WMIN, i} \leq q^{\downarrow}_{W, i}(t) \leq q^{\downarrow}_{WMAX, i}\end{array}$

(7)	$\begin{array}{l}\displaystyle q^{\downarrow}_{GMIN, j} \leq q^{\downarrow}_{G, j}(t) \leq q^{\downarrow}_{GMAX, j}\end{array}$

The objective function (3) can be further rewritten in terms of Sandface flowrates $\begin{array}{l}\{ q^{\uparrow}_t, q^{\downarrow}_w, q^{\downarrow}_g \}\end{array}$ :

(8)

$\begin{array}{l}\displaystyle G = \sum_{k=1}^{N^{\uparrow}_P} G^{\uparrow}_{t,k} \cdot q^{\uparrow}_{t, k} - \sum_{i=1}^{N^{\downarrow}_W} G^{\downarrow}_{w,i} \cdot q^{\downarrow}_{w, i} - \sum_{j=1}^{N^{\downarrow}_G} G^{\downarrow}_{g,j} \cdot q^{\downarrow}_{g, j} \rightarrow \rm max\end{array}$

(9)

$\begin{array}{l}\displaystyle G^{\uparrow}_{t,k} = \frac{\left[ (R_O - C^{\uparrow}_{O,k}) + (R_G - C^{\uparrow}_{G,k}) \cdot Y_{g,k} \right] \cdot (1- Y_{w,k}) - C^{\uparrow}_{L,k} - C^{\uparrow}_{W,k} \cdot Y_{w,k} } {B_{w,k} Y_{w,k} + \left[ (B_{o,k} - R_{s,k} B_{g,k}) + (B_{g,k} - R_{v,k} B_{o,k}) \, Y_{g,k} \right] \cdot (1-Y_{w,k})}\end{array}$

(10)	$\begin{array}{l}\displaystyle G^{\downarrow}_{w,i} = B_{w,i} C^{\downarrow}_{W,i}\end{array}$

(11)	$\begin{array}{l}\displaystyle G^{\downarrow}_{g,i} = B_{g,i} \cdot C^{\downarrow}_{G,i}\end{array}$

where

$\begin{array}{l}B_{w,k} = B_w(p_{wf,k}(t))\end{array}$	Water FVF for $\begin{array}{l}k\end{array}$ -th well	$\begin{array}{l}p_{wf,k}(t)\end{array}$	BHPin $\begin{array}{l}k\end{array}$ -th well
$\begin{array}{l}B_{o,k} = B_o(p_{wf,k}(t))\end{array}$	Oil FVF for $\begin{array}{l}k\end{array}$ -th well	$\begin{array}{l}R_{s,k} = R_s(p_{wf,k}(t))\end{array}$	Solution GOR in $\begin{array}{l}k\end{array}$ -th well
$\begin{array}{l}B_{g,k} = B_g(p_{wf,k}(t))\end{array}$	Gas FVF for $\begin{array}{l}k\end{array}$ -th well	$\begin{array}{l}R_{v,k} = R_v(p_{wf,k}(t))\end{array}$	Vaporized Oil Ratio in $\begin{array}{l}k\end{array}$ -th well

Derivation

(12)

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

(13)

$\begin{array}{l}\displaystyle G(t) = \sum_{p=1}^{N^{\uparrow}_P} \left[ \left[ (R_O - C^{\uparrow}_{O,p}) + (R_G - C^{\uparrow}_{G,p}) \cdot Y_{g,p} \right] \cdot q^{\uparrow}_{O, p} - C^{\uparrow}_{L,p} - C^{\uparrow}_{W,p} \cdot Y_{w,p} \cdot q^{\uparrow}_{L, p} \right] - \sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_{W,j} \cdot q^{\downarrow}_{W, i} - \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot q^{\downarrow}_{G, j}\end{array}$

(14)

$\begin{array}{l}\displaystyle G(t) = \sum_{p=1}^{N^{\uparrow}_P} \left[ \left[ (R_O - C^{\uparrow}_{O,p}) + (R_G - C^{\uparrow}_{G,p}) \cdot Y_{g,p} \right] \cdot (1- Y_{w,p}) - C^{\uparrow}_{L,p} - C^{\uparrow}_{W,p} \cdot Y_{w,p} \right] \cdot q^{\uparrow}_{L, p} - \sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_{W,j} \cdot q^{\downarrow}_{W, i} - \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot q^{\downarrow}_{G, j}\end{array}$

Translating $\begin{array}{l}q^{\downarrow}_{W, i}\end{array}$ and $\begin{array}{l}q^{\downarrow}_{G, j}\end{array}$ to Sandface flowrates $\begin{array}{l}q^{\downarrow}_{w, i}\end{array}$ and $\begin{array}{l}q^{\downarrow}_{g, j}\end{array}$ with formation volume factor and substituting liquid production rate $\begin{array}{l}q^{\uparrow}_{L, p}\end{array}$ from

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one arrives to:

(15)

$\begin{array}{l}\displaystyle G(t) = \sum_{p=1}^{N^{\uparrow}_P} \frac{\left[ (R_O - C^{\uparrow}_{O,p}) + (R_G - C^{\uparrow}_{G,p}) \cdot Y_{g,p} \right] \cdot (1- Y_{w,p}) - C^{\uparrow}_{L,p} - C^{\uparrow}_{W,p} \cdot Y_{w,p} } {B_w Y_{w,p} + \left[ (B_o - R_s B_g) + (B_g - R_v B_o) \, Y_{g,p} \right] \cdot (1-Y_{w,p})} \cdot q^{\uparrow}_{t, p} - \sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_{W,j} \cdot B_w \cdot q^{\downarrow}_{w, i} - \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot B_g \cdot q^{\downarrow}_{g, j}\end{array}$

which is equivalent to (8).

Depending on Lift mechanism the rates in equation (8) may be set directly or calculated from THP and formation pressure $\begin{array}{l}p_e\end{array}$ (which is a usual case in injection wells):

(16)	$\begin{array}{l}\displaystyle q^{\uparrow}_{t, k} = J_{t,k} \cdot ( p_{e,k} - p_{wf,k} )\end{array}$

(17)	$\begin{array}{l}\displaystyle G^{\downarrow}_{w,i} = J_{w,i} \cdot ( p_{wf,i} - p_{e,i} )\end{array}$

(18)	$\begin{array}{l}\displaystyle G^{\downarrow}_{g,i} = J_{g,i} \cdot ( p_{wf,i} - p_{e,i} )\end{array}$

Producing wells may spontaneously vary between Constant rate production: q_L = const and Constant pressure production: pwf = const (see Constant rate production: q_L = const for alternation details).

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