(1)	$\begin{array}{l}\displaystyle p_e \ (t) = p_{nr} \ (0) + \gamma_n^{-1} \cdot \sum_m \left( Q^{\uparrow}_{nm} + Q^{\downarrow}_{nm} \ \right)\end{array}$

(2)	$\begin{array}{l}\displaystyle p_{{\rm wf}, n} \ \ \ (t) = p_e \ (t) + J_n^{-1} \cdot q_n(t)\end{array}$

(3)	$\begin{array}{l}\displaystyle Q^{\uparrow}_{nm} \ = \ - \ f^{\uparrow}_{O,nm} \ \cdot B_{ob} \cdot \, Q^{\uparrow}_O \ - \ f^{\uparrow}_{G,nm} \ \cdot B_{go} \cdot Q^{\uparrow}_G \ - \ f^{\uparrow}_{W,nm} \ \cdot B_w \cdot Q^{\uparrow}_W\end{array}$

(4)	$\begin{array}{l}\displaystyle Q^{\downarrow}_{nm} \ = f^{\downarrow}_{G,nm} \ \cdot B_{go} \cdot Q^{\downarrow}_G \ + \ f^{\downarrow}_{W,nm} \ \cdot B_w \cdot Q^{\downarrow}_W \ + \ B_{go} \cdot Q^{\downarrow}_{GCAP} \ \ + \ B_w \cdot Q^{\downarrow}_{WAQ}\end{array}$

(5)	$\begin{array}{l}\displaystyle Q_m(t) = \int_0^t q_m(t) \, dt\end{array}$

(6)	$\begin{array}{l}\displaystyle B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v}\end{array}$

(7)	$\begin{array}{l}\displaystyle B_{go} = \frac{ B_g - R_v \, B_o}{1- R_s \, R_v}\end{array}$

The value of $\begin{array}{l}\gamma_n\end{array}$ can be linked to drainable volume $\begin{array}{l}V_{\phi, n}\end{array}$ as:

(8)	$\begin{array}{l}\displaystyle \gamma_n = c_{t,n} \cdot V_{\phi, n} = (c_r + s_{w,n} \cdot c_w + s_{o,n} \cdot c_o + s_{g,n} \cdot c_g) \cdot \phi_n \cdot V_n\end{array}$

(9)	$\begin{array}{l}\displaystyle s_{w,n} + s_{o,n} + s_{g,n} = 1\end{array}$

In case of Water Injector : $\begin{array}{l}s_{o,n} = s_{or}, \quad s_{g,n} = 0, \quad s_{w,n} = 1 - s_{or}\end{array}$ .

In case of Gas Injector: $\begin{array}{l}s_{o,n} = 0, \quad s_{g,n} =1- s_{wcg}, \quad s_{w,n} = s_{wcg}\end{array}$ .

The objective function is:

(10)

$\begin{array}{l}\displaystyle E[ \ \tau_n, \gamma_n, f_{nm} \ ] = \sum_k \sum_n \left[ {\rm w}_e \cdot \left( p_{e,n} \ \ (t_k) - \tilde p_{e,n} \ \ (t_k) \right)^2 + {\rm w}_{\rm wf} \ \ \cdot \left( p_{{\rm wf},n} \ \ (t_k) - \tilde p_{{\rm wf},n} \ \ (t_k) \right)^2 \right] \rightarrow \min\end{array}$

The constraints are:

(11)	$\begin{array}{l}\displaystyle J_n \geq 0 , \quad \gamma_n \geq 0, \quad f_{nm} \ \geq 0 , \quad \sum_m f^{\uparrow}_{O, nm} \ \leq 1 , \quad \sum_m f^{\uparrow}_{G, nm} \ \leq 1, \quad \sum_m f^{\downarrow}_{W, nm} \ \leq 1, \quad \sum_m f^{\downarrow}_{G, nm} \ \leq 1\end{array}$

In regular case , the initial formation pressure at datum is the same for all wells: $\begin{array}{l}p_{nr}(0) = p_i = {\rm const}, \ \forall n\end{array}$

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See Also

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XCRM – Liquid-Control Cross-well Capacitance Resistance Model @model

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