| (1) |
p_{e,n} \ (t) = p_{i,n} \ (0) + \gamma_n^{-1} \cdot \sum_m \left( Q^{\uparrow}_{nm} + Q^{\downarrow}_{nm} \ \right) |
|
| (2) |
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] |
|
| (3) |
s_{wn} = \left[ \frac{B_o}{B_w} \cdot \frac{1-Y_{Wn}}{Y_{Wn}} + 1 \right]^{-1}, \quad Y_{Wn} = \frac{q_{Wn}}{q_{Wn} + q_{On} } |
|
| (4) |
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
|
|
| (5) |
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}
|
|
| (6) |
Q_m(t) = \int_0^t q_m(t) \, dt |
|
| (7) |
B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v} |
|
| (8) |
B_{go} = \frac{ B_g - R_v \, B_o}{1- R_s \, R_v} |
|
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 for infinite-volume aquifer and Fetkovich for finite-volume aquifer).
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:
| (9) |
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
{\rm w}_e + {\rm w}_{\rm wf} = 1 are the weight coefficients for formation pressure and bottom-hole pressure correspondingly
and
{\rm w}_k = {\rm w}(t_k) are the the weight coefficients for time (usually the weight of the later times is higher than that for early times).
The constraints are:
| (10) |
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:
p_{i,n}(0) = p_i = {\rm const}, \ \forall n.
The value of
\gamma_n can be linked to the Dynamic drainage volume of a well
V_{\phi, n} as:
| (11) |
\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 |
|
| (12) |
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 ]
