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\displaystyle


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G^{\uparrow}_{t,p} 



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body
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G^{\downarrow}_w = B_w \cdot C^{\downarrow}_W
 



LaTeX Math Inlinebody\displaystyle
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G^{\downarrow}_g = B_g \cdot C^{\downarrow}_G




Expand
titleDerivation


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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} \cdot q^{\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} \cdot q^{\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
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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} \cdot q^{\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}

Substituting liquid production rate 

LaTeX Math Inline
bodyq^{\uparrow}_{L, p}
 from 
LaTeX Math Block Reference
anchorqL
pageLiquid production rate
 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} \cdot q^{\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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