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XCRM – Liquid-Control Cross-well Capacitance Resistance Model
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p_e \ (t) = p_{nr} \ (0) + \gamma_n^{-1} \cdot \sum_m \left( Q^{\uparrow}_{nm} + Q^{\downarrow}_{nm} \ \right) |
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p_{{\rm wf}, n} \ \ \ (t) = p_e \ (t) + J_n^{-1} \cdot q_n(t) |
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Q^{\uparrow}_{nm} \ =
\ - \ f^{\uparrow}_{O,nm} \ \cdot B_{ob} \cdot \, Q^{\uparrow}_O
\ - \ f^{\uparrow}_{G,nm} \ \cdot B_{go} \cdot Q^{\uparrow}_G
\ - \ f^{\uparrow}_{W,nm} \ \cdot B_w \cdot Q^{\uparrow}_W
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Q^{\downarrow}_{nm} \ =
f^{\downarrow}_{G,nm} \ \cdot B_{go} \cdot Q^{\downarrow}_G
\ + \ f^{\downarrow}_{W,nm} \ \cdot B_w \cdot Q^{\downarrow}_W
\ + \ B_{go} \cdot Q^{\downarrow}_{GCAP} \
\ + \ B_w \cdot Q^{\downarrow}_{WAQ}
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Q_m(t) = \int_0^t q_m(t) \, dt |
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B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v} |
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B_{go} = \frac{ B_g - R_v \, B_o}{1- R_s \, R_v} |
The objective function is:
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E[ \ \tau_n, \gamma_n, f_{nm} \ ] = \sum_k \sum_n \left[ {\rm w}_e \cdot \left( p_{e,n} \ \ (t_k) - \tilde p_{e,n} \ \ (t_k) \right)^2
+ {\rm w}_{\rm wf} \ \ \cdot \left( p_{{\rm wf},n} \ \ (t_k) - \tilde p_{{\rm wf},n} \ \ (t_k) \right)^2 \right] \rightarrow \min |
The constraints are:
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J_n \geq 0 , \quad \gamma_n \geq 0, \quad f_{nm} \geq 0 , \quad \sum_i^{N^{\uparrow}} f_{nm} \leq 1 |
In regular case , the initial formation pressure at datum is the same for all wells: | LaTeX Math Inline |
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body | --uriencoded-- p_%7Bnr%7D(0) = p_i = %7B\rm const%7D, \ \forall nSome extensions to conventional CRM model can be found in XCRM – Liquid-Control Cross-well Capacitance Resistance Model @model.See Also
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Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Capacitance Resistance Model (CRM)
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