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 |
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| G^{\uparrow}_{t,k} = \frac{\left[ (R_O - C^{\uparrow}_{O,k}) + (R_G - C^{\uparrow}_{G,k}) \cdot Y_{gG,k} \right] \cdot (1- Y_{wW,k})
- C^{\uparrow}_{L,k} - C^{\uparrow}_{W,k} \cdot Y_{wW,k} }
{B_{w,k} Y_{wW,k} + \left[ (B_{o,k} - R_{s,k} B_{g,k}) + (B_{g,k} - R_{v,k} B_{o,k}) \, Y_{gG,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 | 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}(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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| Expand |
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| Panel |
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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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| G(t) = \sum_{p=1}^{N^{\uparrow}_P} \left[
\left[ (R_O - C^{\uparrow}_{O,p}) + (R_G - C^{\uparrow}_{G,p}) \cdot Y_{gG,p} \right] \cdot q^{\uparrow}_{O, p}
- C^{\uparrow}_{L,p} - C^{\uparrow}_{W,p} \cdot Y_{wW,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_{gG,p} \right] \cdot (1- Y_{wW,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 | 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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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_{wW,p} }
{B_w Y_{wW,p} + \left[ (B_o - R_s B_g) + (B_g - R_v B_o) \, Y_{gG,p} \right] \cdot (1-Y_{wW,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} ) |
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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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