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This type of flow is called radial fluid flow and a type library model provides a reference for radial fluid flow diagnostics.

Inputs & Outputs

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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
bodyd

reservoir channel width



LaTeX Math Inline
body\sigma

transmissibility

LaTeX Math Inline
body\chi

pressure diffusivity


Expand
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

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

Slightly compressible fluid flow

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

Homogeneous reservoir

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

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

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

Infinite boundary

LaTeX Math Inline
bodyr \rightarrow \infty


Mathematical Model

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Expand
titleDefinition



LaTeX Math Block
anchor52112
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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
anchor3MUX9
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p(t, r \rightarrow \infty ) = p_i



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



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