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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
body{t}

time

LaTeX Math Inline
bodyk

absolute permeability

LaTeX Math Inline
body{r}

radial direction

LaTeX Math Inline
body{\phi}

porosity
LaTeX Math Inline
bodyc_t = c_r + c
,
LaTeX Math Inline
body{c_r}
,
LaTeX Math Inline
bodyc


total, pore and fluid compressibility



Physical Model

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Radial fluid flowHomogenous reservoirInfinite boundary

Zero wellbore radius

Slightly compressible fluid flowConstant rate production

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(p) = c_r +c = \rm const

LaTeX Math Inline
bodyq_t = \rm const



Mathematical Model

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



LaTeX Math Block
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p(t=0,r) = p_i



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



LaTeX Math Block
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\left[ r \frac{\partial p}{\partial r} \right]_{r=0} = - \frac{q_t}{2 \pi \sigma}


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