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

G(t) = \sum_{k=1}^{N^{\uparrow}_P} \left[ R_O \cdot q^{\uparrow}_{O, k} + R_G \cdot  q^{\uparrow}_{G, k} \right] 
- \sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{L,k} \cdot q^{\uparrow}_{L, k}
- \sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{O,k} \cdot q^{\uparrow}_{O, k} 
- \sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{G,k} \cdot q^{\uparrow}_{G, k} 
- \sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{W,k} \cdot q^{\uparrow}_{W, k}
- \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

where

volume/day	oil production rate for -th producer	cash/volume	cost of produced oil treatment and transportation from -th wellhead to CMS	cash/volume	oil selling price
volume/day	gas production rate for -th producer	cash/volume	cost of produced gas treatment and transportation from -th wellhead to CMS	cash/volume	gas selling price
volume/day	water production rate for -th producer	cash/volume	cost of produced water treatment and transportation from -th wellhead to CMS	counts	number of producers at
volume/day	liquid production rate for -th producer	cash/volume	cost of fluid lift from reservoir to the -th wellhead, cash/volume	counts	number of water injectors at
volume/day	water injection rate for -th water injector	cash/volume	cost of water injection, including purchase, treatment, transportation and pumping into -th well	counts	number of gas injectors at
volume/day	gas injection rate for -th gas injector	cash/volume	cost of gas injection, including purchase, treatment, transportation and pumping into -th well	months	time

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

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

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})}

G^{\downarrow}_{w,i} = B_{w,i} C^{\downarrow}_{W,i}

G^{\downarrow}_{g,i} = B_{g,i} \cdot C^{\downarrow}_{G,i}

where

Water FVF for -th well	BHPin -th well	Watercut in -th well
Oil FVF for -th well	Solution GOR in -th well	Gas-Oil Ratio in -th well
Gas FVF for -th well	Vaporized Oil Ratio in -th well

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}

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}

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}

Translating and to Sandface flowrates and with formation volume factor and substituting liquid production rate from one arrives to:

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}

which is equivalent to .

The rates in equation may be set directly or calculated from the preset of THP (which is a usual practice in injection wells):

Assuming constant productivity index at each time step equation can be rewritten in terms of formation pressure and botomhole pressure

q^{\uparrow}_{t, k} = J_{t,k} \cdot ( p_{e,k} - p_{wf,k} )

G^{\downarrow}_{w,i} = J_{w,i} \cdot ( p_{w,i} - p_{e,i} )

G^{\downarrow}_{g,i} = J_{g,i} \cdot ( p_{g,i} - p_{e,i} )

See Also