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 |
G_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^{\uparrow}_{WS} \cdot q^{\uparrow}_{WS}(t) + C^{\uparrow}_{GS} \cdot q^{\uparrow}_{GS}(t) |
q^{\uparrow}_{WS}(t) = \sum_{i=1}^{N^{\downarrow}_W} q^{\downarrow}_{W, i}(t) - \sum_{k=1}^{N^{\uparrow}_P} q^{\uparrow}_{W, k}(t) > 0 |
q^{\uparrow}_{GS}(t) = \sum_{j=1}^{N^{\downarrow}_G} q^{\downarrow}_{G, j}(t) - \sum_{k=1}^{N^{\uparrow}_P} q^{\uparrow}_{G, k}(t) >0 |
where
years | assessment period | days | day within a given year | – | ||||
---|---|---|---|---|---|---|---|---|
volume/day | oil production rate for -th producer | cash/volume | cash/volume | oil selling price | ||||
volume/day | gas production rate for -th producer | cash/volume | cash/volume | gas selling price | ||||
volume/day | water production rate for -th producer | cash/volume | 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 rate | cash/volume | cost of water supply | |||||
volume/day | gas supply rate | cash/volume | cost of gas supply | |||||
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 | 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) |
q^{\uparrow}_{WS}(t) = \sum_{i=1}^{N^{\downarrow}_W} C^{\downarrow}_{W,j} \cdot q^{\downarrow}_{W, i}(t) - \sum_{k=1}^{N^{\uparrow}_P} C^{\uparrow}_{W,k} \cdot q^{\uparrow}_{W, k}(t) |
q^{\uparrow}_{WS}(t) = \sum_{i=1}^{N^{\downarrow}_W} q^{\downarrow}_{W, i}(t) - \sum_{k=1}^{N^{\uparrow}_P} q^{\uparrow}_{W, k}(t) > 0 |
q^{\uparrow}_{GS}(t) = \sum_{j=1}^{N^{\downarrow}_G} q^{\downarrow}_{G, j}(t) - \sum_{k=1}^{N^{\uparrow}_P} q^{\uparrow}_{G, k}(t) > 0 |
where
Watercut in -th well | |
Gas-Oil Ratio in -th well |
Usually, each well has a fixed range of its rate variations:
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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 |
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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: qL = const and Constant pressure production: pwf = const (see Constant rate production: qL = const for alternation details).
Petroleum Industry / Upstream / Production / Field Development Plan
Subsurface Production / Well & Reservoir Management / [ Production Targets ]
Subsurface E&P Disciplines / Production Technology
[ Constant rate production: qL = const ] [ Constant pressure production: pwf = const ]