# JD

## Brief

JD - dimensionless productivity index[1], inverse of dimensionless pressure (based on average pressure) which contains the type of flow regime, boundary condition, drainage shape and stimulation [2].

## Math & Physics

${J_D} = \frac{1}{\bar{P}_D}$

From the Darcy's law for the unfractured well the JD is:

Well in circular drainage area Well in a drainage area with the shape factor ${C_A}$[2]
Steady state ${J_D} = \frac{1}{ln{\frac{r_e}{r_w}-\frac{1}{2}+S}}$ ${J_D} = \frac{1}{\frac{1}{2}ln{\frac{4.5A}{C_A{r_w}^2}+S}}$
Pseudo steady state ${J_D} = \frac{1}{ln{\frac{r_e}{r_w}-\frac{3}{4}+S}}$ ${J_D} = \frac{1}{\frac{1}{2}ln{\frac{2.25A}{C_A{r_w}^2}+S}}$

Some typical ${C_A}$ values: circle 31.6, square 30.88 [3].

### Oil

${J_D} = \frac{141.2 B \mu}{kh} \frac{q}{\bar{P} - P_{wf}} = \frac{141.2 B \mu}{kh} J$
${q} = \frac{kh}{141.2 B \mu} (\bar{P} - P_{wf}) J_D$

### Gas

$J_D=\frac{1422 \times 10^3\ T_R}{kh} \frac{q_g}{P_{\bar{P}}-P_{P_{wf}}}$

## Maximum $J_D$

The undamaged unstimulated vertical well potential in a pseudo steady radial flow in a circular drainage area:

${J_D}_{max} \approx \frac{1}{ln{\frac{500}{0.1}-\frac{3}{4}+0}} \approx 0.1287$

The maximum possible stimulated well potential for pseudo steady linear flow is:

${J_D}_{max}= \frac{6}{\pi} \approx 1.91$ , see 6/π stimulated well potential

The maximum possible stimulated well potential for steady state linear flow is:

${J_D}_{max}= \frac{4}{\pi} \approx 1.27$ , see 4/π stimulated well potential

## Nomenclature

$B$ = formation volume factor, bbl/stb
$C_A$ = Dietz shape factor, dimensionless
$J$ = productivity index, stb/psia
$J_D$ = dimensionless productivity index, dimensionless
$kh$ = permeability times thickness, md*ft
$\bar{P}$ = average reservoir pressure, psia
$\bar{P}_D$ = dimensionless pressure (based on average pressure), dimensionless
$P_{\bar{P}}$ = average reservoir pseudopressure, psia2/cP
$P_{wf}$ = well flowing pressure, psia
$P_{P_{wf}}$ = average well flowing pseudopressure, psia2/cP
$q$ = flowing rate, stb/d
$q_g$ = gas rate, MMscfd
$r_w$ = wellbore radius, ft
$r_e$ = drainage radius, ft
$S$ = skin factor, dimensionless
$T$ = temperature, °R

### Greek symbols

$\mu$ = viscosity, cp