A proxy model of Productivity Index for stabilised reservoir flow in homogeneous reservoir:

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

where

transmissibility

a model parameter depending on Productivity Index definitions and boundary type (, see table below)



Drain-area Productivity Index 

Drain-boundary Productivity Index 

Steady State flow regime (SS)


J_r  = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + 0.5 + S}



J_e  = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w}  + S}


 Pseudo-steady State flow regime (PSS)


J_r  = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + 0.75 + S}



J_e  = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + 0.5 + S}




The relation between the total sandface flowrate , bottomhole pressure  and field-average formation pressure  during the stabilized reservoir flow:

corresponding to linear IPR with constant productivity index :

J_r = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + \epsilon+ S}

where

transmissibility

for Steady State (SS) well flow regime

for Pseudo Steady State (PSS) well flow regime


See also

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

Reference

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.