The general form of objective function for production targets optimisation is given by:
(1) | G = \sum_{y=1}^{N_y} \frac{AG_y}{(1+r)^y} \rightarrow \rm max |
(2) | AG_y = \sum_t G_t |
(3) | G_t = G_t^{+} - G_t^{-} |
(4) | 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] |
(5) | 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) |
(6) | 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) |
(7) | 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} |
(8) | 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) |
(9) | 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
N_y | years | assessment period | t | days | day within a given year | r | – | |
---|---|---|---|---|---|---|---|---|
q^{\uparrow}_{O, k}(t) | volume/day | oil production rate for k-th producer | C^{\uparrow}_{O,k}(t) | cash/volume | R_O(t) | cash/volume | oil selling price | |
q^{\uparrow}_{G, k}(t) | volume/day | gas production rate for k-th producer | C^{\uparrow}_{G,k}(t) | cash/volume | R_G(t) | cash/volume | gas selling price | |
q^{\uparrow}_{W, k}(t) | volume/day | water production rate for k-th producer | C^{\uparrow}_{W,k}(t) | cash/volume | N^{\uparrow}_P(t) | counts | number of producers at t | |
q^{\uparrow}_{L, k}(t) | volume/day | liquid production rate for k-th producer | C^{\uparrow}_{L, k}(t) | cash/volume | cost of fluid lift from reservoir to the k-th wellhead, cash/volume | |||
q_{WS}(t) | volume/day | water supply/disposal rate | C^{\uparrow}_{WS}(t) | cash/volume | cost of water supply | C^{\downarrow}_{WS}(t) | cash/volume | cost of water disposal |
q^{\uparrow}_{GS}(t) | volume/day | gas supply rate | C^{\uparrow}_{GS}(t) | cash/volume | cost of gas supply | C^{\downarrow}_{GS}(t) | cash/volume | cost of gas disposal |
q^{\downarrow}_{W, i}(t) | volume/day | water injection rate for i-th water injector | C^{\downarrow}_{W,i}(t) | cash/volume | cost of water injection, including treatment, transportation and pumping into i-th well | N^{\downarrow}_W(t) | counts | number of water injectors at t |
q^{\downarrow}_{G, i}(t) | volume/day | gas injection rate for i-th gas injector | C^{\downarrow}_{G,j}(t) | cash/volume | cost of gas injection, including purchase, treatment, transportation and pumping into i-th well | N^{\downarrow}_G(t) | number of gas injectors at t |
The objective function (1) can be rewritten in terms of Surface flowrates \{ q^{\uparrow}_L, q^{\downarrow}_W, q^{\downarrow}_G \}:
(10) | 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) |
(11) | 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) |
(12) | 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) |
(13) | 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} |
(14) | 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) |
(15) | 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
Y_{w,k}(t) = q_{W,k} / q_{L,k} | Watercut in k-th well |
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Y_{g,k}(t) = q_{G,k} / q_{O,k} | Gas-Oil Ratio in k-th well |
Usually, each well has a fixed range of its rate variations:
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The objective function (10) can be further rewritten in terms of Sandface flowrates \{ q^{\uparrow}_t, q^{\downarrow}_w, q^{\downarrow}_g \}:
(19) | 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 |
(20) | 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})} |
(21) | G^{\downarrow}_{w,i} = B_{w,i} C^{\downarrow}_{W,i} |
(22) | G^{\downarrow}_{g,i} = B_{g,i} \cdot C^{\downarrow}_{G,i} |
where
B_{w,k} = B_w(p_{wf,k}(t)) | Water FVF for k-th well | p_{wf,k}(t) | BHPin k-th well |
---|---|---|---|
B_{o,k} = B_o(p_{wf,k}(t)) | Oil FVF for k-th well | R_{s,k} = R_s(p_{wf,k}(t)) | Solution GOR in k-th well |
B_{g,k} = B_g(p_{wf,k}(t)) | Gas FVF for k-th well | R_{v,k} = R_v(p_{wf,k}(t)) | Vaporized Oil Ratio in k-th well |
Depending on Lift mechanism the rates in equation (19) may be set directly or calculated from THP and formation pressure p_e (which is a usual case in injection wells):
(27) | q^{\uparrow}_{t, k} = J_{t,k} \cdot ( p_{e,k} - p_{wf,k} ) |
(28) | G^{\downarrow}_{w,i} = J_{w,i} \cdot ( p_{wf,i} - p_{e,i} ) |
(29) | 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).
See Also
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 ]