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

(1)	$\begin{array}{l}\displaystyle G = \sum_{y=1}^{N_y} \frac{AG_y}{(1+r)^y} \rightarrow \rm max\end{array}$

(2)	$\begin{array}{l}\displaystyle AG_y = \sum_{t=1+y_t}^{365+y_t} G_t = \sum_{t=1+y_t}^{365+y_t} \left( G_t^{+} - G_t^{-} \right)\end{array}$

(3)	$\begin{array}{l}\displaystyle G_t^{+} = \sum_{k=1}^{N^{\uparrow}_P} \left[ R_O(t) \cdot q^{\uparrow}_{O, k}(t) + R_G(t) \cdot q^{\uparrow}_{G, k}(t) \right]\end{array}$

(4)

$\begin{array}{l}\displaystyle G_t^{-} = \sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{L,k} \cdot q^{\uparrow}_{L, k}(t) +\sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{O,k} \cdot q^{\uparrow}_{O, k} (t) +\sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{G,k} \cdot q^{\uparrow}_{G, k} (t) +\sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{W,k} \cdot q^{\uparrow}_{W, k}(t) +\sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_{W,j} \cdot q^{\downarrow}_{W, i}(t) +\sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot q^{\downarrow}_{G, j}(t) + C_{WS} \cdot q_{WS}(t) + C_{GS} \cdot q_{GS}(t)\end{array}$

(5)	$\begin{array}{l}\displaystyle q_{WS}(t) = \sum_{i=1}^{N^{\downarrow}_W} q^{\downarrow}_{W, i}(t) - \sum_{k=1}^{N^{\uparrow}_P} q^{\uparrow}_{W, k}(t)\end{array}$

(6)	$\begin{array}{l}C_{WS}(t)= \begin{cases} C^{\uparrow}_{WS}(t), & \mbox{if } q_{WS}(t)>0 \\ C^{\downarrow}_{WS}(t), & \mbox{if } q_{WS}(t)< 0 \end{cases}\end{array}$

(7)	$\begin{array}{l}\displaystyle q_{GS}(t) = \sum_{j=1}^{N^{\downarrow}_G} q^{\downarrow}_{G, j}(t) - \sum_{k=1}^{N^{\uparrow}_P} q^{\uparrow}_{G, k}(t)\end{array}$

(8)	$\begin{array}{l}C_{GS}(t)= \begin{cases} C^{\uparrow}_{GS}(t), & \mbox{if } q_{GS}(t) > 0 \\ C^{\downarrow}_{GS}(t), & \mbox{if } q_{GS}(t) > 0 \end{cases}\end{array}$

where

$\begin{array}{l}N_y\end{array}$	years	assessment period	$\begin{array}{l}t\end{array}$	days	running time in the form of the number of days past the start of production $\begin{array}{l}t=0\end{array}$	$\begin{array}{l}y_t\end{array}$	years	number of whole years past the start of production by the current moment $\begin{array}{l}t\end{array}$
$\begin{array}{l}r\end{array}$	–	discount rate
$\begin{array}{l}q^{\uparrow}_{O, k}(t)\end{array}$	volume/day	oil production rate for $\begin{array}{l}k\end{array}$ -th producer	$\begin{array}{l}C^{\uparrow}_{O,k}(t)\end{array}$	cash/volume	cost of produced oil treatment and transportation from $\begin{array}{l}k\end{array}$ -th wellhead to CTM	$\begin{array}{l}R_O(t)\end{array}$	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	$\begin{array}{l}C^{\uparrow}_{G,k}(t)\end{array}$	cash/volume	cost of produced gas treatment and transportation from $\begin{array}{l}k\end{array}$ -th wellhead to CTM	$\begin{array}{l}R_G(t)\end{array}$	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	$\begin{array}{l}C^{\uparrow}_{W,k}(t)\end{array}$	cash/volume	cost of produced water treatment and transportation from $\begin{array}{l}k\end{array}$ -th wellhead to CTM	$\begin{array}{l}N^{\uparrow}_P(t)\end{array}$	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	$\begin{array}{l}C^{\uparrow}_{L, k}(t)\end{array}$	cash/volume	cost of fluid lift from reservoir to the $\begin{array}{l}k\end{array}$ -th wellhead, cash/volume
$\begin{array}{l}q_{WS}(t)\end{array}$	volume/day	water supply/disposal rate	$\begin{array}{l}C^{\uparrow}_{WS}(t)\end{array}$	cash/volume	cost of water supply	$\begin{array}{l}C^{\downarrow}_{WS}(t)\end{array}$	cash/volume	cost of water disposal
$\begin{array}{l}q_{GS}(t)\end{array}$	volume/day	gas supply/disposal rate	$\begin{array}{l}C^{\uparrow}_{GS}(t)\end{array}$	cash/volume	cost of gas supply	$\begin{array}{l}C^{\downarrow}_{GS}(t)\end{array}$	cash/volume	cost of gas disposal
$\begin{array}{l}q^{\downarrow}_{W, i}(t)\end{array}$	volume/day	water injection rate for $\begin{array}{l}i\end{array}$ -th water injector	$\begin{array}{l}C^{\downarrow}_{W,i}(t)\end{array}$	cash/volume	cost of water injection, including treatment, transportation and pumping into $\begin{array}{l}i\end{array}$ -th well	$\begin{array}{l}N^{\downarrow}_W(t)\end{array}$	counts	number of water 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	$\begin{array}{l}C^{\downarrow}_{G,j}(t)\end{array}$	cash/volume	cost of gas injection, including purchase, treatment, transportation and pumping into $\begin{array}{l}i\end{array}$ -th well	$\begin{array}{l}N^{\downarrow}_G(t)\end{array}$	counts	number of gas injectors at $\begin{array}{l}t\end{array}$

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

(9)

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

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

(11)	$\begin{array}{l}\displaystyle 0 \leq q^{\uparrow}_{L, p}(t) \leq q^{\uparrow}_{LMAX, p}(t)\end{array}$

(12)	$\begin{array}{l}\displaystyle 0 \leq q^{\downarrow}_{W, i}(t) \leq q^{\downarrow}_{WMAX, i}(t)\end{array}$

(13)	$\begin{array}{l}\displaystyle 0 \leq q^{\downarrow}_{G, j}(t) \leq q^{\downarrow}_{GMAX, j}(t)\end{array}$

(14)	$\begin{array}{l}\displaystyle \sum_{p=1}^{N^{\uparrow}_P} q^{\uparrow}_{L, p}(t) \leq q^{\uparrow}_{LMAX}\end{array}$

(15)	$\begin{array}{l}\displaystyle q^{\downarrow}_{WMIN}(t) \leq \sum_{i=1}^{N^{\downarrow}_W} q^{\downarrow}_{W, i}(t) \leq q^{\downarrow}_{WMAX}\end{array}$

(16)	$\begin{array}{l}\displaystyle \sum_{j=1}^{N^{\downarrow}_G} q^{\downarrow}_{G, j}(t) \leq q^{\downarrow}_{GMAX}\end{array}$

(17)	$\begin{array}{l}\displaystyle q^{\downarrow}_{WMIN}(t) = \sum_{p=1}^{N^{\uparrow}_P} q^{\uparrow}_{W, p}(t)\end{array}$

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

Sandface Formalism

The objective function (9) can be further rewritten in terms of Sandface flowrates $\begin{array}{l}\{ q^{\uparrow}_t, q^{\downarrow}_w, q^{\downarrow}_g \}\end{array}$ :

(18)

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

(19)

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

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

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

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

(22)

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

(23)

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

(24)

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

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 (Liquid production rate = qL:1) one arrives to:

(25)

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

which is equivalent to (18).

Depending on Lift mechanism the rates in equation (18) 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):

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

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

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

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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