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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 |
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anchor | MatBal |
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alignment | left |
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| \phi_n(p) = \frac{B_o - R_s \, B_g |
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)G() G (G- ) = \exp \left[ c_\phi \, ( |
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1- R_sp-p_i) \right] \approx 1 + c_\phi \, |
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R_v=+ 0.5 \, c^2_\phi \, (p-p_i)^2 |
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Ge^\phi^{-1} \, \delta \, Q_O + |
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\left[ | LaTeX Math Block |
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| F_{Oi} = \frac{s_{oi}}{B_{oi}} + \frac{R_{vi}\, s_{gi}}{B_{gi}} |
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\right] | LaTeX Math Block |
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| \delta \, Q_O = - Q^{\uparrow}_O |
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Ge^\phi^{-1} \, \delta \, Q_G + |
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\left[ | LaTeX Math Block |
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| F_{Gi} = \frac{R_{si}\, s_{oi}}{B_{oi}} + \frac{ s_{gi}}{B_{gi}} |
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\right] | LaTeX Math Block |
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| \delta \, Q_G = Q^{\downarrow}_G - Q^{\uparrow}_G + Q^{\downarrow}_{GCAP} |
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Ge^\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}_W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ} |
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LaTeX Math Block |
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\phi_n = \exp \left[ c_\phi \, (p-p_i) \right] = 1 + c_\phi \, (p-p_i) + 0.5 \, c^2_\phi \, (p-p_i)^2 where
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body | --uriencoded--Q%5e%7B\uparrow%7D_O(t) |
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LaTeX Math Inline |
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body | V_e\phi = V \cdot \phi_i |
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| initial open pore volume of the main pay initial oil+gas pay drainage volume (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\uparrow%7Ddownarrow%7D_W(t) |
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| Cumulative water productioninjection by the time moment |
LaTeX Math Inline |
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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_WG(t) |
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| Cumulative water gas injection by the time moment |
LaTeX Math Inline |
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body | --uriencoded--s_%7Bgi%7D |
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body | --uriencoded--Q%5e%7B\downarrow%7D_G%7BWAQ%7D(t) |
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| Cumulative gas injectionwater influx from Aquifer Expansion by the time moment |
LaTeX Math Inline |
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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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| LaTeX Math Inline |
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body | --uriencoded--Q%5e%7B\downarrow%7D_%7BWAQ%7D(t%7BGCAP%7Dt) |
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| Cumulative water gas influx from Aquifer Expansion Gas Cap expansion by the time moment |
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body | --uriencoded--Q%5e%7B\downarrow%7D_%7BGCAP%7Dt) | R_s(p) |
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| Solution GOR as functions of Cumulative gas influx from Gas Cap expansion by the time moment reservoir pressure t |
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The MatBal equation
LaTeX Math Block Reference |
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is often can be complemented by constant
PI model of Bottom-Hole Pressure ( for
producers and
LaTeX Math Inline |
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body | p^{\downarrow}_{wf}(t) |
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for
injectors):...
which closes equation
LaTeX Math Block Reference |
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for the pressure
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Approximations
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In some specific cases equation
LaTeX Math Block Reference |
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can be explicitly integrated with the accuracy sufficient for practical applications:\{ --uriencoded--c_t = c_\phi + c_ |
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e = { const}, \ c_t = {\rm const} \}fluid%7D = %7B\rm const%7D |
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t c_r + frac{1}{p} \frac{1}{p} |
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| p(t) = p_i + \frac{\Delta Q(t)}{V_ |
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e | LaTeX Math Block |
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| p(t) = p_i \exp \left[ \frac{\Delta Q(t)}{V_ |
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e where
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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 This allows using simple graphical methods for estimating 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 (MatBal)
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[ Derivation of Material Balance Pressure @model ]
[ Modified Black Oil fluid @model (MBO) ]