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Excerpt Include
Aquifer Drive
Aquifer Drive
nopaneltrue


Inputs & Outputs

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InputsOutputs

LaTeX Math Inline
bodyp(t)

field-average formation pressure at time moment

LaTeX Math Inline
bodyt

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

cumulative subsurface water influx from aquifer

LaTeX Math Inline
bodyp_i

initial formation pressure

LaTeX Math Inline
bodyq^{\downarrow}_{AQ}(t) = \frac{dQ^{\downarrow}_{AQ}}{dt}

subsurface water flowrate from aquifer

LaTeX Math Inline
bodyJ_{AQ}

aquifer Productivity Index

LaTeX Math Inline
body\tau

aquifer relaxation time
Expand
titleDetailing
Detailing Inputs

LaTeX Math Inline
body\displaystyle J_{AQ} = \frac{\theta}{2\pi} \cdot \frac{2 \pi \sigma}{\ln \frac{A_{AQ}}{A_e}+0.75}


aquifer Productivity Index

LaTeX Math Inline
body\theta

central angle of net pay area ↔ aquifer contact

LaTeX Math Inline
body\sigma

aquifer transmissibility

LaTeX Math Inline
bodyA_e

net pay area

LaTeX Math Inline
bodyA_{AQ}

aquifer area

LaTeX Math Inline
body\displaystyle \tau = \frac{V_{AQ} \, c_t}{J_{AQ}}

aquifer relaxation time

LaTeX Math Inline
bodyc_t=c_r +c_w

aquifer total compressibility

LaTeX Math Inline
bodyc_r

aquifer pore compressibility 

LaTeX Math Inline
bodyc_w

aquifer water compressibility

LaTeX Math Inline
bodyV_{AQ} = A_e \cdot h \cdot \phi

aquifer volume 

LaTeX Math Inline
bodyh

aquifer effective thickness

LaTeX Math Inline
body\phi

aquifer porosity


Physical Model

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Radial Composite Reservoir

Image Modified

Const Productivity IndexAquifer
LaTeX Math Block
alignmentleft
J_{AQ} = \frac{q_{AQ}}{p_{AQ}(t)-p(t)} = \rm const
Pseudo Steady State Flow
LaTeX Math Block
alignmentleft
p_{AQ}(t) = p_i - \frac{Q_{AQ}(t)}{V_{AQ} \cdot c_t}





Fig. 1. Fetkovich aquifer drive schematic



Mathematical Model

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LaTeX Math Block
anchorFetkovich_PSS
alignmentleft
\frac{d Q^{\downarrow}_{AQ}}{dt} + \frac{1}{\tau} Q^{\downarrow}_{AQ} = J \cdot ( p_i - p(t))
LaTeX Math Block
anchor1
alignmentleft
q^{\downarrow}_{AQ}(t)=\frac{d Q^{\downarrow}_{AQ}}{dt}
Expand
titleDerivation
Panel
bgColorCORNSILK

Assumption #1 = Const Productivity Index Aquifer:

LaTeX Math Block
alignmentleft
q_{AQ} = \frac{d Q_{AQ}}{dt} = J_{AQ} \cdot ( p_{AQ}(t) - p(t))


Assumption #2 = Pseudo Steady State Flow:

LaTeX Math Block
alignmentleft
p_{AQ}(t) = p_i - \frac{Q_{AQ}}{V_{AQ} c_t}


Eliminating

LaTeX Math Inline
bodyp_{AQ}(t)
one arrives to
LaTeX Math Block Reference
anchorFetkovich_PSS
.

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