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Excerpt

A proxy model of Productivity Index for stabilised reservoir flow.

LaTeX Math Block
alignmentleft
J  = \frac{q}{p_{\rm frm} - p_{wf}} = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + \epsilon + S}

where

LaTeX Math Inline
bodyq

depending on application may mean a total sandface flowrate (

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

LaTeX Math Inline
bodyp_{wf}

bottomhole pressure  

LaTeX Math Inline
bodyp_{\rm frm}

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

LaTeX Math Inline
bodyp_e
) or drain-area formation pressure (
LaTeX Math Inline
bodyp_r
)

LaTeX Math Inline
body\sigma

formation transmissibility

LaTeX Math Inline
bodyr_w

wellbore radius

LaTeX Math Inline
bodyr_e

distance to a drainarea boundary

LaTeX Math Inline
bodyS

total skin

LaTeX Math Inline
body\epsilon

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

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


The Dupuit PI @model is actually exact for homogeneous reservoir.


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


Drain-area Productivity Index, 

LaTeX Math Inline
bodyJ_r = \frac{q}{p_r - p_{wf}}

Drain-boundary Productivity Index Index 

LaTeX Math Inline
bodyJ_e = \frac{q}{p_e - p_{wf}}


Steady State flow regime (SS)


LaTeX Math Block
alignmentleft
J_r  = \frac{2 \pi \sigma}{ \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)


LaTeX Math Block
alignmentleft
J_r  = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + 0.75 + S}



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



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