The general form of objective function for production targets optimisation is given by:
(1) | 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
q^{\uparrow}_{O, k} | volume/day | oil production rate for k-th producer | C^{\uparrow}_{O,k} | cash/volume | R_O | cash/volume | oil selling price | |
---|---|---|---|---|---|---|---|---|
q^{\uparrow}_{G, k} | volume/day | gas production rate for k-th producer | C^{\uparrow}_{G,k} | cash/volume | R_G | cash/volume | gas selling price | |
q^{\uparrow}_{W, k} | volume/day | water production rate for k-th producer | C^{\uparrow}_{W,k} | cash/volume | N^{\uparrow}_P | counts | number of producers at t | |
q^{\uparrow}_{L, k} | volume/day | liquid production rate for k-th producer | C^{\uparrow}_{L, k} | cash/volume | cost of fluid lift from reservoir to the k-th wellhead, cash/volume | N^{\downarrow}_W | counts | number of water injectors at t |
q^{\downarrow}_{W, i} | volume/day | water injection rate for i-th water injector | C^{\downarrow}_{W,i} | cash/volume | cost of water injection, including purchase, treatment, transportation and pumping into i-th well | N^{\downarrow}_G | counts | number of gas injectors at t |
q^{\downarrow}_{G, i} | volume/day | gas injection rate for i-th gas injector | C^{\downarrow}_{G,j} | cash/volume | cost of gas injection, including purchase, treatment, transportation and pumping into i-th well | t | months | time |
Left part of equation (1) can be rewritten in terms of Sandface flowrates:
(2) | 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 |
(3) | 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})} |
(4) | G^{\downarrow}_{w,i} = B_{w,i} C^{\downarrow}_{W,i} |
(5) | 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 | Y_{w,k} = q_{W,k} / q_{L,k} | Watercut in 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 | Y_{g,k} = q_{G,k} / q_{O,k} | Gas-Oil Ratio 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 |
The rates in equation (2) may be set directly or calculated from THP and formation pressure p_e (which is a usual case in injection wells):
(10) | q^{\uparrow}_{t, k} = J_{t,k} \cdot ( p_{e,k} - p_{wf,k} ) |
(11) | G^{\downarrow}_{w,i} = J_{w,i} \cdot ( p_{wf,i} - p_{e,i} ) |
(12) | 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 ]