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The general form of objective function for production targets optimisation is given by:
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y=1}^{N_y} \frac{AG_y}{(1+r)^y} \rightarrow \rm |
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| 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)
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| LaTeX Math Block |
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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, |
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k}(t) + R_G(t) \cdot q^{\uparrow}_{G, |
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| LaTeX Math Block |
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| anchor | GtMinus |
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| alignment | left |
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| G_t^{-} =
\sum_{ |
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,k} \cdot q^{\uparrow}_{L, |
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,k} \cdot q^{\uparrow}_{O, |
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,k} \cdot q^{\uparrow}_{G, |
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{N^{\downarrow}_W} C^{\downarrow}_{W,j} \cdot q^{\downarrow}_{W, |
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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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| LaTeX Math Block |
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| q_{WS}(t) = \sum_{i=1}^{N^{\downarrow}_W} q^{\downarrow}_{W, i} |
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{N^{\uparrow}_P} q^{\uparrow}_{ |
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| 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}
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| LaTeX Math Block |
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| q_{GS}(t) = \sum_{j=1}^{N^{\downarrow}_G} q^{\downarrow}_{G, |
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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
\\
C^{\downarrow}_{GS}(t), & \mbox{if } q_{GS}(t) > 0
\end{cases}
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where
where
| | years | assessment period | | days | running time in the form of the number of days past the start of production | | | number of whole years past the start of production by the current moment |
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| – | discount rate |
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| LaTeX Math Inline |
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| body | --uriencoded--q%5e%7B\uparrow%7D_%7BO, k%7D(t) |
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| volume/day |
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pC_O| --uriencoded--C%5e%7B\uparrow%7D_%7BO,k%7D(t) |
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| cash/volume | cost of produced oil treatment and transportation from |
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q^{\uparrow}_{G, p} | | cash/volume | oil selling price |
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| LaTeX Math Inline |
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| body | --uriencoded--q%5e%7B\uparrow%7D_%7BG, k%7D(t) |
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| volume/day | |
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pC_G | cost of gas treatment| --uriencoded--C%5e%7B\uparrow%7D_%7BG,k%7D(t) |
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| cash/volume | cost of produced gas treatment and transportation from -th wellhead to CTM | | cash/volume | gas selling price |
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q^{\uparrow}_{W, p} | | --uriencoded--q%5e%7B\uparrow%7D_%7BW, k%7D(t) |
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| volume/day | water production rate for -th producer | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\uparrow%7D_%7BW,k%7D(t) |
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| cash/volume | cost of produced water treatment and transportation from -th wellhead to CTM | | LaTeX Math Inline |
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| body | --uriencoded--N%5e%7B\uparrow%7D_P(t) |
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| counts | |
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| body | --uriencoded--q%5e%7B\uparrow%7D_%7BL, k%7D(t) |
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| volume/day | |
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water p | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\uparrow%7D_%7BL, k%7D(t) |
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| cash/volume | cost of fluid lift from reservoir to the -th wellhead, cash/volume | |
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| LaTeX Math Inline |
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| body | --uriencoded--q_%7BWS%7D(t) |
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| volume/day | water supply/disposal rate | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\uparrow%7D_%7BWS%7D(t) |
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| cash/volume | cost of water supply | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\downarrow%7D_%7BWS%7D(t) |
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| cash/volume | cost of water disposal |
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| body | --uriencoded--q_%7BGS%7D(t) |
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| volume/day | gas supply/disposal rate | | LaTeX Math Inline |
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| body | --uriencoded--C%5e%7B\uparrow%7D_%7BGS%7D(t) |
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| cash/volume | cost of gas supply | |
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C_W | cost of water treatment | | --uriencoded--C%5e%7B\downarrow%7D_%7BGS%7D(t) |
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| cash/volume | cost of gas disposal |
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| LaTeX Math Inline |
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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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| LaTeX Math Inline |
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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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The objective function
| LaTeX Math Block Reference |
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can be rewritten in terms of Surface flowrates | LaTeX Math Inline |
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| body | \{ q^{\uparrow}_L, q^{\downarrow}_W, q^{\downarrow}_G \} |
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and usual subject to engineering restrictions:
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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_{WS} \cdot q_{WS}(t)
- C_{GS} \cdot q_{GS}(t) |
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| LaTeX Math Block |
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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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| LaTeX Math Block |
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| anchor | RateLimit |
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| alignment | left |
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| 0 \leq |
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body | q^{\uparrow}_{L, p}(t) \leq q^{\uparrow} |
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| anchor | RateLimit |
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| alignment | left |
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| 0 \leq q^{\downarrow}_{W, i}(t) \leq q^{\downarrow}_{WMAX, i}(t) |
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| anchor | RateLimit |
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| alignment | left |
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| 0 \leq q^{\downarrow}_{G, j}(t) \leq q^{\downarrow}_{GMAX, j}(t) |
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| anchor | RateLimit |
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| alignment | left |
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| \sum_{p=1}^{N^{\uparrow}_P} q^{\uparrow}_{L, p}(t) \leq q^{\uparrow}_{LMAX}
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| LaTeX Math Block |
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| anchor | RateLimit |
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| alignment | left |
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| q^{\downarrow}_{WMIN}(t) \leq
\sum_{i=1}^{N^{\downarrow}_W} q^{\downarrow}_{W, i}(t)
\leq q^{\downarrow}_{WMAX}
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| LaTeX Math Block |
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| anchor | RateLimit |
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| alignment | left |
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| \sum_{j=1}^{N^{\downarrow}_G} q^{\downarrow}_{G, j}(t)
\leq q^{\downarrow}_{GMAX} |
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| LaTeX Math Block |
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| anchor | RateLimit |
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| alignment | left |
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| q^{\downarrow}_{WMIN}(t) = \sum_{p=1}^{N^{\uparrow}_P} q^{\uparrow}_{W, p}(t)
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where
liquid production rate for | LaTeX Math Inline |
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| body | --uriencoded--Y_%7BW,k%7D(t) = q_%7BW,k%7D / q_%7BL,k%7D |
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p producer |
| LaTeX Math Inline |
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| body | --uriencoded--Y_%7BG,k%7D(t) = q_%7BG,k%7D / q_%7BO,k%7D |
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C_L | cost of fluid lift
| Expand |
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The objective function | LaTeX Math Block Reference |
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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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:| LaTeX Math Block |
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| 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 |
| LaTeX Math Block |
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| 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})} |
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| G^{\downarrow}_{w,i} = B_{w,i} C^{\downarrow}_{W,i} |
| LaTeX Math Block |
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| G^{\downarrow}_{g,i} = B_{g,i} \cdot C^{\downarrow}_{G,i} |
where | LaTeX Math Inline |
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| body | B_{w,k} = B_w(p_{wf,k}(t)) |
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| body | B_{o,k} = B_o(p_{wf,k} |
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| body | R_{s,k} = R_s(p_{wf,k}(t)) |
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| body | B_{g,k} = B_g(p_{wf,k}(t)) |
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| body | R_{v,k} = R_v(p_{wf,k}(t)) |
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| Expand |
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| borderColor | wheat |
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| bgColor | mintcream |
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| borderWidth | 7 |
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| LaTeX Math Block |
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| 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}
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| LaTeX Math Block |
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| 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}
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| 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}
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Translating | LaTeX Math Inline |
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| body | q^{\downarrow}_{W, i} |
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and |
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| LaTeX Math Inline |
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| body | q^{\downarrow}_{G, j} |
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to Sandface flowrates | LaTeX Math Inline |
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| body | q^{\downarrow}_{w, i} |
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gas injection rate for
-th gas injector...
and | LaTeX Math Inline |
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| body | q^{\downarrow}_{g, j} |
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with formation volume factor and substituting liquid production rate from | LaTeX Math Block Reference |
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| anchor | qL |
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| page | Liquid production rate = qL |
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one arrives to:| LaTeX Math Block |
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| 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}
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which is equivalent to | LaTeX Math Block Reference |
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Depending on Lift mechanism the rates in equation | LaTeX Math Block Reference |
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may be set directly or calculated from THP and formation pressure (which is a usual case in injection wells):| LaTeX Math Block |
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| q^{\uparrow}_{t, k} = J_{t,k} \cdot ( p_{e,k} - p_{wf,k} ) |
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| q^{\downarrow}_{w,i} = J_{w,i} \cdot ( p_{wf,i} - p_{e,i} ) |
| LaTeX Math Block |
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| q^{\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). |
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See Also
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Petroleum Industry / Upstream / Production / Field Development Plan
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Subsurface E&P Disciplines / Production Technology
[ Constant rate production: qL = const ] [ Constant pressure production: pwf = const ]
