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=1+y_t}^{365+y_t} G_t = \sum_{t=1+y_t}^{365+y_t} 
\left( G_t^{+} - G_t^{-} \right)

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}

where

years	assessment period	days	running time in the form of the number of days past the start of production	years	number of whole years past the start of production by the current moment
–	discount rate
volume/day	oil production rate for -th producer	cash/volume	cost of produced oil treatment and transportation from -th wellhead to CTM	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 CTM	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 CTM	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 and usual subject to engineering restrictions:

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)

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)

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

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

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

\sum_{p=1}^{N^{\uparrow}_P} q^{\uparrow}_{L, p}(t) \leq q^{\uparrow}_{LMAX}

q^{\downarrow}_{WMIN}(t) \leq 
\sum_{i=1}^{N^{\downarrow}_W} q^{\downarrow}_{W, i}(t) 
\leq q^{\downarrow}_{WMAX}

\sum_{j=1}^{N^{\downarrow}_G} q^{\downarrow}_{G, j}(t)
\leq q^{\downarrow}_{GMAX}

q^{\downarrow}_{WMIN}(t) = \sum_{p=1}^{N^{\uparrow}_P} q^{\uparrow}_{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_{WS} \cdot q_{WS}(t)
- C_{GS} \cdot q_{GS}(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} )

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

q^{\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