A proxy model of Productivity Index for stabilised reservoir flow. where
q depending on application may mean a total sandface flowrate (
q_t) or a product of surface flowrate and FVF (
q = q_{\rm srf} B)
p_{wf}
p_{\rm frm} depending on application may mean a drain-boundary formation pressure (
p_e) or drain-area formation pressure (
p_r)
\sigma
r_w
r_e
S
\epsilon a model parameter depending on Productivity Index definition and boundary type (
\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,
J_r = \frac{q}{p_r - p_{wf}} Drain-boundary Productivity Index
J_e = \frac{q}{p_e - p_{wf}}
J = \frac{q}{p_{\rm frm} - p_{wf}} = \frac{2 \pi \sigma}{ \ln \frac{r_e}{r_w} + \epsilon + S}
wellbore radius distance to a drainarea boundary total skin
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}
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.