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proxy model of Productivity Index for stabilised reservoir flow.

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}



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_{\rm frm}

depending on application may mean a drain-boundary formation pressure ( p_e) or drain-area formation pressure ( p_r)



wellbore radius


distance to a drainarea boundary


total skin


a model parameter depending on Productivity Index definition and boundary type ( \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 IndexJ_r = \frac{q}{p_r - p_{wf}}

Drain-boundary Productivity Index  J_e = \frac{q}{p_e - p_{wf}}

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}

For the fractured vertical well the geometrical skin-factor  S_G is related to Fracture half-length  X_f as:

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

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


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.

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