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Modelling facility for field-average formation pressure 

LaTeX Math Inline
bodyp(t)
 at any time moment 
LaTeX Math Inline
bodyt
 as response to production flowrates history:


LaTeX Math Block
anchor

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MatBal
alignmentleft

...

 

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A_e \, h_e \

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anchorSo
alignmentleft

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int_{p_i}^p \phi_e(p) \, c_t(p) \, dp  = \Delta Q (t) =  Q^{\downarrow}_t(t) - 

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Q^{\uparrow}_

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t(t)

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 +

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anchorSg
alignmentleft

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 Q^{\downarrow}_

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{GC}(t) + 

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Q^{\downarrow}_{

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WAQ}(t

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)

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s_w + s_o + s_g = 1

where

LaTeX Math Inline
bodyp_i = p(0)

LaTeX Math Inline
body\Delta Q (t)

LaTeX Math Inline
bodyA_e

LaTeX Math Inline
bodyQ^{\uparrow}_t(t)

full-field cumulative offtakes by the time moment

LaTeX Math Inline
bodyt

LaTeX Math Inline
bodyh_e

LaTeX Math Inline
bodyQ^{\downarrow}_t(t)

full-field cumulative intakes by the time moment

LaTeX Math Inline
bodyt

LaTeX Math Inline
body\phi_e(p)

effective porosity as function of formation pressure 

LaTeX Math Inline
bodyp(t)
 

LaTeX Math Inline
bodyQ^{\downarrow}_{GC}(t)

cumulative volumetric inflow from Gas Cap Expansion

LaTeX Math Inline
bodyc_t(p)


total compressibility as function of formation pressure 

LaTeX Math Inline
bodyp(t)

LaTeX Math Inline
bodyQ^{\downarrow}_{WAQ}(t)

cumulative volumetric inflow from Aquifer Expansion

The direct consequence of the above equations:

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anchorMatBal
alignmentleft

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The MatBal equation 

LaTeX Math Block Reference
anchorMatBal
  is often complemented by constant PI  model of Bottom-Hole Pressure (
LaTeX Math Inline
bodyp^{\uparrow}_{wf}(t)
 for producers and 
LaTeX Math Inline
bodyp^{\downarrow}_{wf}(t)
 for injectors):

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