The general form of objective function for production targets optimisation is given by:
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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
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where
| volume/day | oil production rate for -th producer | | cash/volume | cost of produced oil treatment and transportation from -th wellhead to CMS | | cash/volume | oil selling price |
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| volume/day | gas production rate for -th producer | | cash/volume | cost of produced gas treatment and transportation from -th wellhead to CMS | | cash/volume | gas selling price |
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| volume/day | water production rate for -th producer | | cash/volume | cost of produced water treatment and transportation from -th wellhead to CMS | | counts | |
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| volume/day | liquid production rate for -th producer | | cash/volume | cost of fluid lift from reservoir to the -th wellhead, cash/volume | | counts | number of water injectors at |
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body | q^{\downarrow}_{W, i} |
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| volume/day | water injection rate for -th water injector | | cash/volume | cost of water injection, including purchase, treatment, transportation and pumping into -th well | | counts | number of gas injectors at |
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body | q^{\downarrow}_{G, i} |
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| volume/day | gas injection rate for -th gas injector | | cash/volume | cost of gas injection, including purchase, treatment, transportation and pumping into -th well | | months | time |
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The objective function
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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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:
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G(t) = \sum_{p=1}^{N^{\uparrow}_P} C^{\uparrow}_{OGW} \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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C^{\uparrow}_{OGW}(t) = \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}
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where
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body | Y_{w,k} = q_{W,k} / q_{L,k} |
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body | Y_{g,k} = q_{G,k} / q_{O,k} |
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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^{\uparrow}_{L, p} \leq q^{\uparrow}_{LMAX, p}, \
q^{\downarrow}_{WMIN, i} \leq q^{\downarrow}_{W, i} \leq q^{\downarrow}_{WMAX, i}, \
q^{\downarrow}_{GMIN, j} \leq q^{\downarrow}_{G, j} \leq q^{\downarrow}_{GMAX, j} |
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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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 |
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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} |
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G^{\downarrow}_{g,i} = B_{g,i} \cdot C^{\downarrow}_{G,i} |
where
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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}(t)) |
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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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borderColor | wheat |
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bgColor | mintcream |
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borderWidth | 7 |
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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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| 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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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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| 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 |
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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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| . |
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Depending on Lift mechanism the rates in equation
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may be set directly or calculated from
THP and
formation pressure (which is a usual case in
injection wells):
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q^{\uparrow}_{t, k} = J_{t,k} \cdot ( p_{e,k} - p_{wf,k} ) |
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G^{\downarrow}_{w,i} = J_{w,i} \cdot ( p_{wf,i} - p_{e,i} ) |
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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 ]