| LaTeX Math Block |
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| p_{e,n} \ (t) = p_{i,n} \ (0) + \gamma_n^{-1} \cdot \sum_m \left( Q^{\uparrow}_{nm} + Q^{\downarrow}_{nm} \ \right) |
| | LaTeX Math Block |
|---|
| p_{{\rm wf}, n} \ \ \ (t) = p_e \ (t) + 0.5 \, \left[ q_O(t)/J_{On} + f_{nn} \cdot q_W(t)/ J_{Wn} \right] |
|
| LaTeX Math Block |
|---|
| 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
|
|
| LaTeX Math Block |
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| 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}
|
|
|
| LaTeX Math Block |
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| Q_m(t) = \int_0^t q_m(t) \, dt |
| | LaTeX Math Block |
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| B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v} |
| | LaTeX Math Block |
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| B_{go} = \frac{ B_g - R_v \, B_o}{1- R_s \, R_v} |
|
where
| LaTeX Math Inline |
|---|
| body | B_o, \, B_g, \, B_w, \, R_s, \, R_v |
|---|
|
are
Dynamic fluid properties.
The value of cumulative Gas Cap influx
| LaTeX Math Inline |
|---|
| body | --uriencoded--Q%5e%7B\downarrow%7D_%7BGCAP%7D |
|---|
|
is modelled as in
Gas Cap Drive @model.
The value of cumulative Aquifer influx
| LaTeX Math Inline |
|---|
| body | --uriencoded--Q%5e%7B\downarrow%7D_%7BGCAP%7D |
|---|
|
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 :
| LaTeX Math Inline |
|---|
| body | --uriencoded-- s_%7Bo,n%7D = s_%7Bor%7D \ , \quad s_%7Bg,n%7D = 0 \ , \quad s_%7Bw,n%7D = 1 - s_%7Bor%7D |
|---|
|
.
In case of Gas Injector:
| LaTeX Math Inline |
|---|
| body | --uriencoded-- s_%7Bo,n%7D = 0 \ , \quad s_%7Bg,n%7D = 1 - s_%7Bwcg%7D \ , \quad s_%7Bw,n%7D = s_%7Bwcg%7D |
|---|
|
.
The objective function is:
| LaTeX Math Block |
|---|
|
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
| LaTeX Math Inline |
|---|
| body | --uriencoded--%7B\rm w%7D_e + %7B\rm w%7D_%7B\rm wf%7D = 1 |
|---|
|
are the weight coefficients for
formation pressure and
bottom-hole pressure correspondingly
and
| LaTeX Math Inline |
|---|
| body | --uriencoded--%7B\rm w%7D_k = %7B\rm w%7D(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:
| LaTeX Math Block |
|---|
|
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:
| LaTeX Math Inline |
|---|
| body | --uriencoded-- p_%7Bi,n%7D(0) = p_i = %7B\rm const%7D, \ \forall n |
|---|
|
.
The value of
can be linked to the
Dynamic drainage volume of a well | LaTeX Math Inline |
|---|
| body | --uriencoded--V_%7B\phi, n%7D |
|---|
|
as:
| LaTeX Math Block |
|---|
| \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 |
| | LaTeX Math Block |
|---|
| 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 ]
