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(1) |
p_e \ (t) = p_{nr} \ (0) + \gamma_n^{-1} \cdot \sum_m \left( Q^{\uparrow}_{nm} + Q^{\downarrow}_{nm} \ \right) |
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(2) |
p_{{\rm wf}, n} \ \ \ (t) = p_e \ (t) + J_n^{-1} \cdot q_n(t) |
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(3) |
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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(4) |
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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(5) |
Q_m(t) = \int_0^t q_m(t) \, dt |
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(6) |
B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v} |
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(7) |
B_{go} = \frac{ B_g - R_v \, B_o}{1- R_s \, R_v} |
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where
B_o, \, B_g, \, B_w, \, R_s, \, R_v are Dynamic fluid properties.
The value of cumulative Gas Cap influx
Q^{\downarrow}_{GCAP} is modelled as in Gas Cap Drive @model.
The value of cumulative aquifer influx
Q^{\downarrow}_{GCAP} is modelled as in Aquifer Drive Models (most popular being Carter-Tracy model).
The value of
\gamma_n can be linked to the Dynamic drainage volume of a well
V_{\phi, n} as:
(8) |
\gamma_n = c_{t,n} \cdot V_{\phi, n} = (c_r + s_{w,n} \cdot c_w + s_{o,n} \cdot c_o + s_{g,n} \cdot c_g) \cdot \phi_n \cdot V_n |
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(9) |
s_{w,n} + s_{o,n} + s_{g,n} = 1 |
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In case of Water Injector :
s_{o,n} = s_{or}, \quad s_{g,n} = 0, \quad s_{w,n} = 1 - s_{or}.
In case of Gas Injector:
s_{o,n} = 0, \quad s_{g,n} =1- s_{wcg}, \quad s_{w,n} = s_{wcg}.
The objective function is:
(10) |
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:
(11) |
J_n \geq 0 , \quad \gamma_n \geq 0, \quad f_{nm} \ \geq 0 , \quad \sum_m f^{\uparrow}_{O, nm} \ \leq 1 , \quad \sum_m f^{\uparrow}_{G, nm} \ \leq 1, \quad \sum_m f^{\downarrow}_{W, nm} \ \leq 1, \quad \sum_m f^{\downarrow}_{G, nm} \ \leq 1 |
In regular case , the initial formation pressure at datum is the same for all wells:
p_{nr}(0) = p_i = {\rm const}, \ \forall n
See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Capacitance Resistance Model (CRM) / Capacitance-Resistivity Model (CRM) @model
Production – Injection Pairing @ model
[ Slightly compressible Material Balance Pressure @model ]