18.41 derivation

Brief

18.41 is the well know constant which is used for converting from the Darcy's law units to the metric units in the well's inflow equations.

For example Darcy's law for the single-phase flow is as follows:

$q_o = \frac{1}{18.41} \times \frac{k_oh}{B_o\mu_o} \times \Delta P \times J_D$

The derivation of the 18.41 constant is given below.

Math and Physics

Darcy's law:

$q = -\frac{kA}{\mu} \frac{dP}{dL}$

In Darcy's units:

$\frac{[cm^3]}{[sec]} = - \frac{[D][cm^2]}{[cP]} \frac{[atm]}{[cm]}$

Converting to the metric units:

$\frac{[cm^3] \frac{[m^3]}{[1000000 cm^3]}}{[sec] \frac{[day]}{[86400 sec]}} = - \frac{[D] \frac{[1000 mD]}{[D]}[cm^2] \frac{[m^2]}{[10000 cm^2]}}{[cP]} \frac{[atm]}{[cm] \frac{[m]}{[100 cm]}}$

So:

$\frac{[m^3]}{[day]} \frac{86400}{1000000} = - \frac{[mD][m^2]}{[cP]} \frac{[atm]}{[m]} \frac{1000 \times 100}{10000}$

And:

$\frac{[m^3]}{[day]} = - C_{LF} \frac{[mD][m^2]}{[cP]} \frac{[atm]}{[m]}$

where

$C_{LF} = \frac{1000 \times 100}{10000} \frac{1000000}{86400} = \frac{10000000}{86400} = 115.7407407$

For the radial flow:

$C_{RF} = \frac{C_{LF}}{2\pi} = \frac{115.7407407}{2\pi} = 18.42$

Note that the published and commonly used constant in 18.41.

Nomenclature

$A$ = Darcy's law cross-sectional area, cm²

$B_o$ = oil formation volume factor, m^3/m^3

$C_{LF}$ = linear flow units conversion constant

$C_{RF}$ = radial flow units conversion constant

$h$ = effective feet of oil pay, m

$J_D$ = dimensionless productivity index, dimensionless

$k$ = Darcy's law permeability, d

$k_o$ = effective permeability to oil, md

$L$ = Darcy's law length, cm

$P$ = Darcy's law pressure, atm

$\Delta P$ = drawdown, atm

$q$ = Darcy's law flow rate, cm³/sec

$q_o$ = oil flow rate, m^3/d

Greek symbols

$\mu$ = Darcy's law oil viscosity, cp

$\mu_o$ = oil viscosity, cp

See Also

Darcy's law
141.2 derivation
18.41 derivation