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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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A_e \, h_e \int_{p_i}^p \phi_e(p) \, c_t(p) \, dp = \Delta Q (t) = Q^{\downarrow}_t(t) - Q^{\uparrow}_t(t) + Q^{\downarrow}_{GC}(t) + Q^{\downarrow}_{AQ}(t) |
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c_t(s,p) = c_r + c_w s_w + c_o s_o + c_g s_g + s_o [ R_{sp} + (c_r + c_o) R_{sn} ] + s_g [ R_{vp} + R_{vn}(c_r + c_g) ] |
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s_w(t) = s_{wi} + (1-s_{wi}) \cdot \rm RFO(t)/E_S |
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s_g(t) = s_{wi} + (1-s_{wi}) \cdot \rm RFG(t)/E_S |
...
\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}_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}_W - Q^{\uparrow}_W + Q^{\downarrow}_{WAQ} |
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where
| | | | \Delta Q | --uriencoded--Q%5e%7B\uparrow%7D_O(t) |
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| A_e | | | | Q^{\uparrow}_t| --uriencoded--Q%5e%7B\uparrow%7D_G(t) |
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| h_e | | | | Q^{\downarrow}_t| --uriencoded--Q%5e%7B\uparrow%7D_W(t) |
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| _e(p) | 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 | | Q^{\downarrow}_{GC}| --uriencoded--Q%5e%7B\downarrow%7D_G(t) |
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| Cumulative gas injection by the time moment cumulative gas influx from Gas Cap Expansion | c_ | (p) |
| LaTeX Math Inline |
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| body | --uriencoded--s_%7Bgi%7D |
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| | | Q^{\downarrow}_{AQ}| --uriencoded--Q%5e%7B\downarrow%7D_%7BWAQ%7D(t) |
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| cumulative Cumulative water influx from Aquifer Expansion by the time moment | R_{sn}, \; R_{vn} | |
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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_%7BGCAP%7Dt) |
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| Cumulative gas influx from Gas Cap expansion by the time moment |
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| | | Solution GOR as functions of and temperature reservoir pressure T | |
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| body | --uriencoded--s_%7Bwi%7D | B_w(p) |
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| Initial water saturationOil and Gas Recovery Factor | | |
The MatBal equation
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is often can be complemented by constant
PI model of Bottom-Hole Pressure ( for
producers and
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| body | p^{\downarrow}_{wf}(t) |
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for
injectors):...
which closes equation
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for the pressure
.
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Approximations
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In some specific cases equation
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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 \cdot c_t 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)
[ Material Balance Pressure Plot ][ FMB Pressure @model]
[ Derivation of Material Balance Pressure @model ]
[ Modified Black Oil fluid @model (MBO) ]
