The objective function LaTeX Math Block Reference |
---|
| can be further rewritten in terms of Sandface flowrates LaTeX Math Inline |
---|
body | \{ q^{\uparrow}_t, q^{\downarrow}_w, q^{\downarrow}_g \} |
---|
| :\right] = \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{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 |
---|
| 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} \ | rm inj}} C_{W, \rm inj} \cdot q^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})} |
LaTeX Math Block |
---|
| G^{\downarrow}_{w,i} = B_{w,i} C^{\downarrow}_{W,i} |
LaTeX Math Block |
---|
| G^{\downarrow}_{g,i} | i}
-= B_{g,i} \cdot C^{\downarrow}_{G,i} |
where LaTeX Math Inline |
---|
body | B_{w,k} = B_w(p_{wf,k}(t)) |
---|
|
| | | |
---|
LaTeX Math Inline |
---|
body | B_{o,k} = B_o(p_{wf,k}(t)) |
---|
|
| | LaTeX Math Inline |
---|
body | R_{s,k} = R_s(p_{wf,k}(t)) |
---|
|
| |
---|
LaTeX Math Inline |
---|
body | B_{g,k} = B_g(p_{wf,k}(t)) |
---|
|
| | LaTeX Math Inline |
---|
body | R_{v,k} = R_v(p_{wf,k}(t)) |
---|
|
| |
---|
Expand |
---|
|
Panel |
---|
borderColor | wheat |
---|
bgColor | mintcream |
---|
borderWidth | 7 |
---|
|
LaTeX Math Block |
---|
| G(t) = \sum_{ |
|
| jNN^{\uparrow}_P} \left[
(R_O - C^{\uparrow}_{O,p}) \cdot q^{\uparrow}_{O, p} + (R_G - C^{\uparrow}_{G,p}) \cdot |
|
| \rm inj}} C_{G, \rm inj} 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}_{ |
|
| G \rightarrow \rm max
- \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot q^{\downarrow}_{G, j}
|
|
|
| 343G1N_rm produparrow}_P} \left[
\left[ (R_O - |
|
| C_O C^{\uparrow}_{O,p}) + (R_G |
|
| -C_G - C^{\uparrow}_{G,p}) \cdot |
|
| GOR 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}
|
LaTeX Math Block |
---|
| 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_L -C_WC^{\uparrow}_{L,p} - C^{\uparrow}_{W,p} \cdot Y_ |
|
| w){w,p}
\right] \cdot q^{\uparrow}_{L, p}
- \sum_{i=1}^{N^{\downarrow}_W} |
|
| \right]C^{\downarrow}_{W,j} \cdot q^{\downarrow}_{W, i}
- \sum_{j=1}^{N^{\downarrow}_G} C^{\downarrow}_{G,j} \cdot q^{\downarrow}_{G, j}
|
Translating LaTeX Math Inline |
---|
body | q^{\downarrow}_{W, i} |
---|
| and LaTeX Math Inline |
---|
body | q^{\downarrow}_{G, j} |
---|
| to Sandface flowrates LaTeX Math Inline |
---|
body | q^{\downarrow}_{w, i} |
---|
| and LaTeX Math Inline |
---|
body | q^{\downarrow}_{g, j} |
---|
| with formation volume factor and substituting liquid production rate from LaTeX Math Block Reference |
---|
anchor | qL |
---|
page | Liquid production rate = qL |
---|
| one arrives to: LaTeX Math Block |
---|
| G(t) = \sum_{p=1}^ |
|
| {N{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} \ |
|
| rm inj}} Ccdot Y_{W,p} }
{B_w Y_{W,p} |
|
| \rm inj} + \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{N^{\downarrow}_G} C^{\downarrow}_{G, |
|
| \rm inj}} C_{G, \rm inj} \cdot j} \cdot B_g \cdot q^{\downarrow}_{g, j}
|
which is equivalent to LaTeX Math Block Reference |
---|
| . |
|
Depending on Lift mechanism the rates in equation LaTeX Math Block Reference |
---|
| may be set directly or calculated from THP and formation pressure (which is a usual case in injection wells): LaTeX Math Block |
---|
| q^{\uparrow}_{t, k} = J_{t,k} \cdot ( p_{e,k} - p_{wf,k} ) |
LaTeX Math Block |
---|
| q^{\downarrow}_{w,i} = J_{w,i} \cdot ( p_{wf,i} - p_{e,i} ) |
LaTeX Math Block |
---|
| q^{\downarrow}_{ | Grightarrow \rm max
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).
|