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_{wn})}{k_{rw}(s_{wn})} \cdot q_{Wn}(t) \right] |
| s_{wn} = \left[ 1 + \frac{B_o}{B_w} \cdot \frac{q_{On}}{f_{nn} \cdot q_{Wn}} \right]^{-1} |
|
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
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Capacitance Resistance Model (CRM)
[ Capacitance-Resistivity Model (CRM) @model ][ Slightly compressible Material Balance Pressure @model ]
[ Dynamic fluid properties ]