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proxy model of Productivity Index for stabilised reservoir flow.

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alignmentleft
J  = \frac{q}{p_{\rm frm} - p_{wf}} = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} - \epsilon + S} = \frac{2 \pi \cdot \frac{k \, h}{\mu} }{ \ln \frac{r_e}{r_w} - \epsilon + S}

where

LaTeX Math Inline
bodyq

depending on application may mean a total sandface flowrate (

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bodyq_t
) or a product of surface flowrate and FVF (
LaTeX Math Inline
bodyq = q_{\rm srf} B
)

...

LaTeX Math Inline
bodyp_{wf}

LaTeX Math Inline
bodyp_{\rm frm}

depending on application may mean a drain-boundary formation pressure (

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bodyp_e
) or drain-area formation pressure (
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...

bodyp_r
)

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body\sigma

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bodyr_w

wellbore radius

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bodyr_e

distance to a drainarea boundary

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bodyS

total skin

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body\epsilon

a model parameter depending on Productivity Index definition and boundary type (

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body\epsilon =\{ 0, \, 0.5, \, 0.75 \}
, see Table 1 below)


In case of homogeneous reservoir with only one vertical well producing the Dupuit PI @model is the exact analytical solution of Reservoir Flow Model (RFM).


Table 1. Variations to Dupuit PI @model depending  on the reservoir flow regime and the definition/application of Productivity Index.


Drain-area Productivity Index

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bodyJ_r = \frac{q}{p_r - p_{wf}}

...

Drain-boundary Productivity Index 

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bodyJ_e = \frac{q}{p_e - p_{wf}}


Steady State flow regime (SS)
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alignmentleft
q
J_r  =
J
 \frac{2 \pi \sigma}{ \
, (p_r - p_{wf})

corresponding to linear IPR with constant productivity index :

3AIXS
ln \frac{r_e}{r_w} - 0.5 + S}
LaTeX Math Block
alignmentleft
J_e  = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w}  + S}

 
Pseudo-Steady State flow regime (PSS)
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alignmentleft
J_r  = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} - 0.75 + S}
LaTeX Math Block
LaTeX Math Block
anchor
alignmentleft
J_e  = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} - 
+
0.5 
\epsilon
+ S}

...



For the fractured vertical well the geometrical skin-factor 

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bodyS_G
is related to Fracture half-length 
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body

...

X_f
as:

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...

anchorXf

...

alignment

...

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body\epsilon = 0.5

...

left
S_G = -\ln \left(\frac{X_f}{2\, r_w} \right)



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alignmentleft
J  = \frac{q}{p_{\rm frm} - p_{wf}} = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} - \epsilon + S} =  \frac{2 \pi M \cdot h}{ \ln \frac{r_e}{r_w} - \epsilon + S}  = \frac{2 \pi k_{abs} \cdot h}{ \ln \frac{r_e}{r_w} - \epsilon + S}  \cdot M_r  = T \cdot M_r(s_w, s_g)


See also

...

Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing / Pressure Testing

Reference

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Dupuit, J., Etudes theoriques et pratiques sur le mouvement des eaux dans les canaux decouverts et a travers les terrains permeables, 2eme edition; Dunot, Paris, 1863.

...

LaTeX Math Inline
body\epsilon = 0.75

...