Modelling facility for field-average average formation pressure
at at any time moment as response to production flowrates history, which in case of MBO fluid takes form:| LaTeX Math Block |
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| anchor | MatBal |
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\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 |
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| \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 |
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| F_O = V_\phi^{-1} \, \delta \, Q_O + F_{Oi} |
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| F_{Oi} = \frac{s_{oi}}{B_{oi}} + \frac{R_{vi}\, s_{gi}}{B_{gi}} |
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| \delta \, Q_O = - Q^{\uparrow}_O |
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| F_G = V_\phi^{-1} \, \delta \, Q_G + F_{Gi} |
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| F_{Gi} = \frac{R_{si}\, s_{oi}}{B_{oi}} + \frac{ s_{gi}}{B_{gi}} |
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| \delta \, Q_G = Q^{\downarrow}_G - Q^{\uparrow}_ |
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G + Q^{\downarrow}_{GCAP} |
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| F_W = V_\phi^{-1} \, \delta \, Q_W + F_{Wi} |
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| F_{Wi} = \frac{ s_{wi}}{B_{wi}} |
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| \delta \, Q_W = Q^{\downarrow}_ |
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W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ} |
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where
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| body | --uriencoded--Q%5e%7B\uparrow%7D_O(t) |
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| body | V_\phi = V \cdot \phi_i |
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| initial open pore volume of the main pay (excluding the aquifer and gas cap) | | LaTeX Math Inline |
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| body | --uriencoded--Q%5e%7B\uparrow%7D_G(t) |
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| body | --uriencoded--Q%5e%7B\uparrow%7D_W(t) |
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| pore compressibility | | LaTeX Math Inline |
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| body | --uriencoded--Q%5e%7B\downarrow%7D_W(t) |
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| body | --uriencoded--s_%7Bwi%7D |
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| initial water saturation | | LaTeX Math Inline |
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| body | --uriencoded--Q%5e%7B\downarrow%7D_G(t) |
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| LaTeX Math Inline |
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| body | --uriencoded--s_%7Bgi%7D |
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| body | --uriencoded--Q%5e%7B\downarrow%7D_%7BWAQ%7D(t) |
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| Cumulative water influx from Aquifer Expansion by the time moment |
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| body | --uriencoded--s_%7Boi%7D |
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| initial oil saturation: | LaTeX Math Inline |
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| body | --uriencoded--s_%7Boi%7D = 1 - s_%7Bwi%7D - s_%7Bgi%7D |
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| body | --uriencoded--Q%5e%7B\downarrow%7D_%7BGCAP%7Dt) |
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| Cumulative gas influx from Gas Cap expansion by the time moment |
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The MatBal equation
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can be complemented by constant PI model of Bottom-Hole Pressure ( for producers and ...
for injectors):| LaTeX Math Block |
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| anchor | BHP_PROD |
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P^p^{\uparrow}_{wf, k}(t) = p(t) - {J^{\uparrow}_k}^{-1} \cdot \frac{dQ^{\uparrow}_k}{dt} |
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| anchor | BHP_INJ |
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| alignment | left |
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P^p^{\downarrow}_{wf, \, j}(t) = p(t) - {J^{\downarrow}_j}^{-1} \cdot \frac{dQ^{\downarrow}_j}{dt} |
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| where | where |
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p_i = p(0) | initial formation pressure| p^{\uparrow}_{wf, \, k}(t) |
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\Delta Q (t) | cumulative reservoir fluid balance P^uparrowfield-average producerstP^{wf}field-average BHP in injectorsQ^downarrowcumulative intakes by the time moment t(t)tphi_e(p)effective porosity as function of formation pressure p(t) |
In practice there is no way to measure the external influx
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| body | Q^{\downarrow}_{GC}(t) |
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...
and
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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
:| LaTeX Math Inline |
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bodyAQcumulative volumetric inflow from aquifer expansion | | | | |
= Q^{\downarrow}_{GC}(p(t)) |
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| Q^{\downarrow}_{AQ}(t) = Q^{\downarrow}_{AQ}(p(t)) |
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which closes equation
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for the pressure .Approximations
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In some specific cases equation For low compressibility rocks and fluids
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| body | \{ \phi_e = {\rm const}, \ c_t = {\rm const} \} |
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the MatBal equation | LaTeX Math Block Reference |
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can be explicitly integrated with the accuracy sufficient for practical applications: | Low pressure dry gas |
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| body | --uriencoded--c_t = c_\phi + c_%7B\rm fluid%7D = %7B\rm const%7D |
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| body | --uriencoded--c_g = \sim \frac%7B1%7D%7Bp%7D |
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| p(t) = p_i + \frac{\Delta Q(t)}{V_ |
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where
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bodyV_e = A_e \, h_e \, \phi_e | drainage volume | For ideal dry gas:
| | LaTeX Math Block |
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| p(t) = p_i \exp \left[ \frac{\Delta Q(t)}{V |
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...
where
is Cumulative Voidage Replacement Balance (CVRB):| LaTeX Math Block |
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| \Delta Q = - \frac{B_o - R_s \, B_g}{1- R_s \, R_v} \cdot \, Q^{\uparrow}_O + \frac{ B_g - R_v \, B_o}{1- R_s \, R_v} \cdot \, \left( Q^{\downarrow}_G - Q^{\uparrow}_G + Q^{\downarrow}_{GCAP} \right) + B_w \, \left( Q^{\downarrow}_W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ} \right) |
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The above approximations sometime allow using simple graphical methods for rough estimation of drainage volume
and associated Hydrocarbon Reserves.See Also
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Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Material Balance Analysis (0D or MatBal)MatBal)
[ Material Balance Pressure Plot ][ FMB Pressure @model]
[ Derivation of Material Balance Pressure @model ]
[ Modified Black Oil fluid @model (MBO) ]
