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InputsOutputs

LaTeX Math Inline
bodyq_t

total sandface rate

LaTeX Math Inline
bodyp(t,r)

reservoir pressure

LaTeX Math Inline
body{p_i}

initial formation pressure

LaTeX Math Inline
body{p_{wf}(t)}

well bottomhole pressure



LaTeX Math Inline
body\sigma

transmissibility,

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

LaTeX Math Inline
body\chi

pressure diffusivity,

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



LaTeX Math Inline
bodyS

skin-factor

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Expand
titleDetailing


LaTeX Math Inline
bodyk

absolute permeability

LaTeX Math Inline
bodyc_t

total compressibility,

LaTeX Math Inline
bodyc_t = c_r + c

LaTeX Math Inline
bodyh

effective thickness

LaTeX Math Inline
body{c_r}

pore compressibility

LaTeX Math Inline
body\mu

dynamic fluid viscosity

LaTeX Math Inline
bodyc

fluid compressibility

LaTeX Math Inline
body{\phi}

porosity




Physical Model

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Radial fluid flowHomogenous reservoir
Infinite
Finite reservoir flow boundary

Zero wellbore radius

Slightly compressible fluid flowConstant rateConstant skin

LaTeX Math Inline
bodyp(t, {\bf r}) \rightarrow p(

t,

r)

LaTeX Math Inline
body{\bf r} \in ℝ^2 = \{ x, y\}

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

r_e

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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Expand
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
\left[ r\frac{\partial p(t, r )}{\partial r} \right]_{r \rightarrow r_w} = \frac{q_t}{2 \pi \sigma}



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