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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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Constant rate production

LaTeX Math Inline
bodyq_t = \rm const

Radial fluid flow
Radial fluid flowHomogenous reservoirInfinite boundary

Zero wellbore radius

Slightly compressible fluid flowConstant rate production

LaTeX Math Inline
bodyp(t,

{\bf

r

}

)

Slightly compressible fluid flow

LaTeX Math Inline
body

c_t

M(r, p)=

c_r +c

M =\rm const

Homogeneous reservoir

LaTeX Math Inline
body

M

\phi(

{\bf

r

}

, p)=

M

\phi =\rm const

LaTeX Math Inline
body

\phi

h(

{\bf

r

}, p

)=

\phi

h =\rm const

LaTeX Math Inline
body

h({\bf r})=h

c_r(r)=c_r =\rm const

Infinite boundary

LaTeX Math Inline
bodyr \rightarrow \infty

LaTeX Math Inline
bodyr_w = 0

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

LaTeX Math Inline
bodyq_t = \rm const


Mathematical Model

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titleDefinition



LaTeX Math Block
anchor52112
alignmentleft
\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
anchor88AEG
alignmentleft
p(t = 0, {\bf r}) = p_i



LaTeX Math Block
anchor3MUX9
alignmentleft
p(t, r \rightarrow \infty ) = p_i



LaTeX Math Block
anchorEM415
alignmentleft
\frac{\partial p(t, r )}{\partial r} \bigg|_{r \rightarrow 0} = \frac{q_t}{\sigma \, d}



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