Modelling facility for field-average formation pressure
at any time moment as response to production flowrates history, which in case of
MBO fluid takes form
: LaTeX Math Block |
---|
anchor | MatBal |
---|
alignment | left |
---|
| \phi_n(p) = \frac{B_o - R_s \, B_g}{1- R_s \, R_v} \cdot F_O
+\frac{ B_g - R_v \, B_o}{1- R_s \, R_v} \cdot F_G
+B_w \, F_W |
| LaTeX Math Block |
---|
| \phi_n = \exp \left[ c_\phi \, (p-p_i) \right] \approx 1 + c_\phi \, (p-p_i) + 0.5 \, c^2_\phi \, (p-p_i)^2 |
|
LaTeX Math Block |
---|
| F_O = V_\phi^{-1} \, \delta \, Q_O + F_{Oi} |
| LaTeX Math Block |
---|
| F_{Oi} = \frac{s_{oi}}{B_{oi}} + \frac{R_{vi}\, s_{gi}}{B_{gi}} |
| LaTeX Math Block |
---|
| \delta \, Q_O = - Q^{\uparrow}_O |
|
LaTeX Math Block |
---|
| F_G = V_\phi^{-1} \, \delta \, Q_G + F_{Gi} |
| LaTeX Math Block |
---|
| F_{Gi} = \frac{R_{si}\, s_{oi}}{B_{oi}} + \frac{ s_{gi}}{B_{gi}} |
| LaTeX Math Block |
---|
| \delta \, Q_G = Q^{\downarrow}_G - Q^{\uparrow}_G + Q^{\downarrow}_{GCAP} |
|
LaTeX Math Block |
---|
| F_W = V_\phi^{-1} \, \delta \, Q_W + F_{Wi} |
| LaTeX Math Block |
---|
| F_{Wi} = \frac{ s_{wi}}{B_{wi}} |
| LaTeX Math Block |
---|
| \delta \, Q_W = Q^{\downarrow}_W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ} |
|
where
| | LaTeX Math Inline |
---|
body | --uriencoded--Q%5e%7B\uparrow%7D_O(t) |
---|
|
| |
LaTeX Math Inline |
---|
body | V_\phi = V \cdot \phi_i |
---|
|
| initial open pore volume of the main pay (excluding the aquifer and gas cap) | LaTeX Math Inline |
---|
body | --uriencoded--Q%5e%7B\uparrow%7D_G(t) |
---|
|
| |
| | LaTeX Math Inline |
---|
body | --uriencoded--Q%5e%7B\uparrow%7D_W(t) |
---|
|
| |
| pore compressibility | LaTeX Math Inline |
---|
body | --uriencoded--Q%5e%7B\downarrow%7D_W(t) |
---|
|
| |
LaTeX Math Inline |
---|
body | --uriencoded--s_%7Bwi%7D |
---|
|
| initial water saturation | LaTeX Math Inline |
---|
body | --uriencoded--Q%5e%7B\downarrow%7D_G(t) |
---|
|
| |
LaTeX Math Inline |
---|
body | --uriencoded--s_%7Bgi%7D |
---|
|
| | LaTeX Math Inline |
---|
body | --uriencoded--Q%5e%7B\downarrow%7D_%7BWAQ%7D(t) |
---|
|
| Cumulative water influx from Aquifer Expansion by the time moment |
LaTeX Math Inline |
---|
body | --uriencoded--s_%7Boi%7D |
---|
|
| initial oil saturation: LaTeX Math Inline |
---|
body | --uriencoded--s_%7Boi%7D = 1 - s_%7Bwi%7D - s_%7Bgi%7D |
---|
|
| LaTeX Math Inline |
---|
body | --uriencoded--Q%5e%7B\downarrow%7D_%7BGCAP%7Dt) |
---|
|
| Cumulative gas influx from Gas Cap expansion by the time moment |
|
|
|
|
| | | |
| | | |
| | | |
The MatBal equation
LaTeX Math Block Reference |
---|
|
can be re-written in the following popular form:
LaTeX Math Block |
---|
anchor | MatBal_formula |
---|
alignment | left |
---|
| p = p_i + \frac{\delta Q}{c_\phi \, V_\phi} + \delta p_i |
| LaTeX Math Block |
---|
anchor | MatBal_formula |
---|
alignment | left |
---|
| \delta p_i = \frac{ B_{og} \, F_{Oi} + B_{go} \, F_{Gi} + B_w \, F_W -1}{c_\phi} |
|
LaTeX Math Block |
---|
| B_{og} = \frac{B_o - R_s \, B_g}{1- R_s \, R_v} |
| LaTeX Math Block |
---|
| B_{go} = \frac{ B_g - R_v \, B_o}{1- R_s \, R_v} |
|
where
The MatBal equation
LaTeX Math Block Reference |
---|
|
can be complemented by constant
PI model of Bottom-Hole Pressure ( for
producers and
LaTeX Math Inline |
---|
body | p^{\downarrow}_{wf}(t) |
---|
|
for
injectors): LaTeX Math Block |
---|
anchor | BHP_PROD |
---|
alignment | left |
---|
| p^{\uparrow}_{wf, k}(t) = p(t) - {J^{\uparrow}_k}^{-1} \cdot \frac{dQ^{\uparrow}_k}{dt} |
| LaTeX Math Block |
---|
anchor | BHP_INJ |
---|
alignment | left |
---|
| p^{\downarrow}_{wf, \, j}(t) = p(t) - {J^{\downarrow}_j}^{-1} \cdot \frac{dQ^{\downarrow}_j}{dt} |
|
where | where |
LaTeX Math Inline |
---|
body | p^{\uparrow}_{wf, \, k}(t) |
---|
|
| | LaTeX Math Inline |
---|
body | p^{\downarrow}_{wf, \, j}(t) |
---|
|
| |
| cumulative offtakes from -th producer by the time moment | | cumulative intakes to -th injector by the time moment |
| | | |
In practice there is no way to measure the external influx
LaTeX Math Inline |
---|
body | Q^{\downarrow}_{GC}(t) |
---|
|
and
LaTeX Math Inline |
---|
body | Q^{\downarrow}_{AQ}(t) |
---|
|
so that one need to model them and calibrate model parameters to fit available data on
production flowrates history and
formation pressure data records.
There is a list of various analytical Aquifer Drive and Gas Cap Drive models which are normally related to pressure dynamics
:
which closes equation
LaTeX Math Block Reference |
---|
|
for the pressure
.
Approximations
In some specific cases equation
LaTeX Math Block Reference |
---|
|
can be explicitly integrated with the accuracy sufficient for practical applications: | Low pressure dry gas |
---|
LaTeX Math Inline |
---|
body | \{ \phi_e = {\rm const}, \ c_t = {\rm const} \} |
---|
|
| LaTeX Math Inline |
---|
body | c_t = c_r + \frac{1}{p} \sim \frac{1}{p} |
---|
|
|
LaTeX Math Block |
---|
| p(t) = p_i + \frac{\Delta Q(t)}{V_\phi \cdot c_t} |
| LaTeX Math Block |
---|
| p(t) = p_i \exp \left[ \frac{\Delta Q(t)}{V_\phi} \right] |
|
where
The above approximations sometime allow using simple graphical methods for rough estimation of drainage volume
and associated
Hydrocarbon Reserves.
See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Material Balance Analysis (MatBal)
[ Material Balance Pressure Plot ][ FMB Pressure @model]
[ Derivation of Material Balance Pressure @model ]
[ Modified Black Oil fluid @model (MBO) ]