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

G = \sum_{y=1}^{N_y} \frac{AG_y}{(1+r)^y} \rightarrow \rm max

AG_y = \sum_t G_t = \sum_t G_t^{+} - G_t^{-}

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]

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)

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)

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}

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)

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}

q^{\uparrow}_{LMIN, p} \leq q^{\uparrow}_{L, p}(t) \leq  q^{\uparrow}_{LMAX, p}

q^{\downarrow}_{WMIN, i} \leq q^{\downarrow}_{W, i}(t) \leq  q^{\downarrow}_{WMAX, i}

q^{\downarrow}_{GMIN, j} \leq q^{\downarrow}_{G, j}(t) \leq q^{\downarrow}_{GMAX, j}

where

years	assessment period	days	day within a given year	–	discount rate
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
volume/day	water supply/disposal rate	cash/volume	cost of water supply	cash/volume	cost of water disposal
volume/day	gas supply/disposal rate	cash/volume	cost of gas supply	cash/volume	cost of gas disposal
volume/day	water injection rate for -th water injector	cash/volume	cost of water injection, including treatment, transportation and pumping into -th well	counts	number of water 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	counts	number of gas injectors at

The objective function can be rewritten in terms of Surface flowrates :

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^{\uparrow}_{WS,k} \cdot q^{\uparrow}_{WS}(t)
-C^{\uparrow}_{GS} \cdot q^{\uparrow}_{GS}(t)

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)

where

	Watercut in -th well
	Gas-Oil Ratio in -th well

The objective function can be further 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} - C^{\uparrow}_{WS,k} \cdot q^{\uparrow}_{WS}(t) \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
	Oil FVF for -th well		Solution GOR 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 .

Depending on Lift mechanism the rates in equation may be set directly or calculated from THP and formation pressure (which is a usual case in injection wells):

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

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

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

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).

See Also