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titleDetailing


LaTeX Math Inline
body\sigma = \frac{k \, h}{\mu}

transmissibility

LaTeX Math Inline
body\mu

dynamic fluid viscosity

LaTeX Math Inline
body\chi = \frac{k}{\mu} \, \frac{1}{\phi \, c_t}

pressure diffusivity

LaTeX Math Inline
bodyc_t = c_r + c

total compressibility

LaTeX Math Inline
bodyk

absolute permeability

LaTeX Math Inline
body{c_r}

pore compressibility

LaTeX Math Inline
body{\phi}

porosity

LaTeX Math Inline
bodyc

fluid compressibility




Physical Model

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Radial fluid flowHomogenous reservoirInfinite boundary

Zero wellbore radius

Slightly compressible fluid flowConstant rateConstant Skin

LaTeX Math Inline
bodyp(t, r)

LaTeX Math Inline
bodyM(r, p)=M =\rm const

LaTeX Math Inline
body\phi(r, p)=\phi =\rm const

LaTeX Math Inline
bodyh(r)=h =\rm const

LaTeX Math Inline
bodyc_r(r)=c_r =\rm const

LaTeX Math Inline
bodyr \rightarrow \infty

LaTeX Math Inline
bodyr_w = 0

LaTeX Math Inline
bodyc_t(r,p) = \rm const

LaTeX Math Inline
bodyq_t = \rm const

LaTeX Math Inline
bodyS = \rm const


Mathematical Model

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titleDefinition



LaTeX Math Block
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\frac{\partial p}{\partial t}  = \chi \, \left( \frac{\partial^2 p}{\partial r^2} + \frac{1}{r} \frac{\partial p}{\partial r} \right)



LaTeX Math Block
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p(t = 0, {\bf r}) = p_i



LaTeX Math Block
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p(t, r \rightarrow \infty ) = p_i



LaTeX Math Block
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\frac{\partial p(t, r )}{\partial r} \bigg|_{r \rightarrow 0} = \frac{q_t}{\sigma \, d}



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