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

(1)

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

where

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

Left part of equation (1) can be rewritten in terms of Sandface flowrates:

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

(3)

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

(4)	$\begin{array}{l}\displaystyle G^{\downarrow}_w = B_w \cdot C^{\downarrow}_W\end{array}$

(5)	$\begin{array}{l}\displaystyle G^{\downarrow}_g = B_g \cdot C^{\downarrow}_G\end{array}$

where

Bo, Bg, Bw
Yg,p
Yw,p

Derivation

(6)

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

(7)

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

(8)

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

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:

(9)

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

which is equivalent to (2).

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See Also

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Production Targets Optimisation @model

See Also