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The general form of objective function for production targets optimisation is given by:
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| G = \sum_{y=1}^{N_y} \frac{AG_y}{(1+r)^y} \rightarrow \rm max
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| AG_y = \sum_t G_t = \sum_t G_t^{+} - G_t^{-}
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| anchor | GtPlus |
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| alignment | left |
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| 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]
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| anchor | GtMinus |
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| alignment | left |
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| 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)
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| 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)
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| C_{WS}(t)= \begin{cases}
C^{\uparrow}_{WS}(t), & \mbox{if } q_{WS}(t)>0
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C^{\downarrow}_{WS}(t), & \mbox{if } q_{WS}(t)<0
\end{cases}
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| 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)
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| C_{GS}(t)= \begin{cases}
C^{\uparrow}_{GS}(t), & \mbox{if } q_{GS}(t) > 0
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C^{\downarrow}_{GS}(t), & \mbox{if } q_{GS}(t) > 0
\end{cases}
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where
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| anchor | RateLimit |
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| alignment | left |
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| q^{\uparrow}_{LMIN, p} \leq q^{\uparrow}_{L, p}(t) \leq q^{\uparrow}_{LMAX, p} |
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| anchor | RateLimit |
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| alignment | left |
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| q^{\downarrow}_{WMIN, i} \leq q^{\downarrow}_{W, i}(t) \leq q^{\downarrow}_{WMAX, i} |
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| anchor | RateLimit |
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| alignment | left |
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| q^{\downarrow}_{GMIN, j} \leq q^{\downarrow}_{G, j}(t) \leq q^{\downarrow}_{GMAX, j} |
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where
| years | assessment period | | days | day within a given year | | – | discount rate |
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| years | assessment period | | days | day within a given year | | – | discount rate |
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| body | --uriencoded--q%5e%7B\uparrow%7D_%7BO, k%7D(t) |
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| volume/day | oil production rate for -th producer | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\uparrow%7D_%7BO,k%7D(t) |
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| cash/volume | cost of produced oil treatment and transportation from -th wellhead to CMS | | cash/volume | oil selling price |
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| body | --uriencoded--q%5e%7B\uparrow%7D_%7BG%7BO, k%7D(t) |
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| volume/day | gas oil production rate for -th producer | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\uparrow%7D_%7BG%7BO,k%7D(t) |
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| cash/volume | cost of produced gas treatment oil treatment and transportation from -th wellhead to CMS | | cash/volume | gas selling priceoil selling price |
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| body | --uriencoded--q%5e%7B\uparrow%7D_%7BW%7BG, k%7D(t) |
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| volume/day | water gas production rate for -th producer | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\uparrow%7D_%7BW%7BG,k%7D(t) |
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| cash/volume | cost of produced water treatment gas treatment and transportation from -th wellhead to CMS | | LaTeX Math Inline |
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| body | --uriencoded--N%5e%7B\uparrow%7D_PR_G(t) |
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| counts | cash/volume | gas selling pricenumber of producers at |
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| t | | LaTeX Math Inline |
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| body | --uriencoded--q%5e%7B\uparrow%7D_%7BL%7BW, k%7D(t) |
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| volume/day | liquid water production rate for -th producer | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\uparrow%7D_%7BL%7BW,k%7D(t) |
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| cash/volume | cost of fluid lift from reservoir to the produced water treatment and transportation from -th wellhead, cash/volume to CMS | | q | %7BWS%7D | volume/day | counts | number of producers at water supply/disposal rate |
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| C%5e%7B | %7BWS%7D | cash/volume | cost of water supplyvolume/day | liquid production rate for -th producer | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\ |
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| downarrow%7D | %7BWS%7D | cash/volume | water disposalfluid lift from reservoir to the -th wellhead, cash/volume | |
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| body | --uriencoded--q_%7BWS%7D--uriencoded--q_%7BGS%7D(t) |
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| volume/day | gas water supply/disposal rate | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\uparrow%7D_%7BGS%7D%7BWS%7D(t) |
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| cash/volume | cost of gas water supply | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\downarrow%7D_%7BGS%7D%7BWS%7D(t) |
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| cash/volume | cost of gas disposalwater disposal |
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| q%5e%7B\downarrow%7D_%7BW, i%7D | volume/day | water injection rate for gas supply/disposal rate | | i | | --uriencoded--C%5e%7B\uparrow%7D_%7BGS%7D(t) |
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| cash/volume | cost of gas supply | -th water injector | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\downarrow%7D_ |
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| %7BW,i%7D | cash/volume | cost of gas disposal |
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| body | --uriencoded--q%5e%7B\downarrow%7D_%7BW, i%7D(t) |
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| volume/day | water injection rate for -th water injector | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\downarrow%7D_%7BW,i%7D(t) |
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| cash/volume | cost of water injection, including treatment, transportation and pumping into -th well | | LaTeX Math Inline |
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| body | --uriencoded--N%5e%7B\downarrow%7D_W(t) |
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| counts | number of water injectors at |
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| body | --uriencoded--q%5e%7B\downarrow%7D_%7BG, i%7D(t) |
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| volume/day | gas injection rate for -th gas injector | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\downarrow%7D_%7BG,j%7D(t) |
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| cash/volume | cost of gas injection, including purchase, treatment, transportation and pumping into -th well | | LaTeX Math Inline |
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| body | --uriencoded--N%5e%7B\downarrow%7D_G(t) |
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| counts
| number of gas injectors at |
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...
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| 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) |
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| 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)
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where
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| body | --uriencoded--Y_%7Bw,k%7D(t) = q_%7BW,k%7D / q_%7BL,k%7D |
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| body | --uriencoded--Y_%7Bg,k%7D(t) = q_%7BG,k%7D / q_%7BO,k%7D |
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Usually, each well has a fixed range of its rate variations:
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| anchor | RateLimit |
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| alignment | left |
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q^{\uparrow}_{LMIN, p} \leq q^ \cdot (1- Y_{w,p}(t))
- C^{\uparrow}_{L, |
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(t)\leq q^LMAX | LaTeX Math Block |
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| anchor | RateLimit |
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| alignment | left |
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q^{\downarrow}_{WMIN, i} \leq q^{\downarrow}_{W, i}(t) \leq q^{\downarrow}_{WMAX, i} |
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| anchor | RateLimit |
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| alignment | left |
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q^{\downarrow}_{GMIN, j} \leq q^{\downarrow}_{G, j}(t) \leq q^{\downarrow}_{GMAX, j}
where
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| body | --uriencoded--Y_%7Bw,k%7D(t) = q_%7BW,k%7D / q_%7BL,k%7D |
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| body | --uriencoded--Y_%7Bg,k%7D(t) = q_%7BG,k%7D / q_%7BO,k%7D |
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The objective function
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can be further rewritten in terms of
Sandface flowrates | LaTeX Math Inline |
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| body | \{ q^{\uparrow}_t, q^{\downarrow}_w, q^{\downarrow}_g \} |
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:
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