p_{e,n} \ (t) = p_{i,n} \ (0) + \gamma_n^{-1} \cdot  \sum_m \left(  Q^{\uparrow}_{nm} +  Q^{\downarrow}_{nm} \ \right)

p_{{\rm wf}, n} \ \ \ (t) = p_e \ (t) + 0.5 \cdot J_{On}^{-1} \cdot \left[  q_{On}(t)   + f_{nn} \cdot \frac{\mu_W}{\mu_O} \cdot \frac{k_{ro}(s_w)}{k_{rw}(s_w)} \cdot q_{Wn}(t)   \right]

sw = \frac{1}{ \frac{B_o}{B_w} \cdot \frac{1-Y_w}{Y_w}  + 1}, \quad Y_w = \frac{q_W}{q_W + q_O}

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

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}

Q_m(t) =  \int_0^t q_m(t) \, dt

B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v}

B_{go} = \frac{ B_g - R_v \, B_o}{1- R_s \, R_v}

where are Dynamic fluid properties.

The value of cumulative Gas Cap influx is modelled as in Gas Cap Drive @model.

The value of cumulative Aquifer influx is modelled as in Aquifer Drive Models (most popular being Carter-Tracy model for infinite-volume aquifer and Fetkovich for finite-volume aquifer).

In case of Water Injector : .

In case of Gas Injector: .

The objective function is:

E[ \ \tau_n, \gamma_n, f_{nm} \ ] = \sum_n {\rm w}_k \sum_k  \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

where are the weight coefficients for formation pressure and bottom-hole pressure correspondingly

and are the the weight coefficients for time (usually the weight of the later times is higher than that for early times).

The constraints are:

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

Normally, the initial formation pressure at datum is the same for all wells: .

The value of can be linked to the Dynamic drainage volume of a well as:

\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

s_{w,n} + s_{o,n} + s_{g,n} = 1

See Also