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Excerpt

A proxy model of Productivity Index for stabilised reservoir flow.

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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 (
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bodyq = q_{\rm srf} B
)

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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
)

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body\sigma

formation transmissibility

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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

LaTeX Math Inline
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 

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


Steady State flow regime (SS)
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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
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J_r  = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + 0.75 + S}
LaTeX Math Block
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J_e  = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + 0.5 + S}



For the fractured vertical well the geometrical skin-factor 

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bodyS_G
is related to Fracture half-length 
LaTeX Math Inline
bodyX_f
as:

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



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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)

...