Darcy equation for oil field units:

Linear flow

Radial low

Klinkenberg effect

The gas permeability measured at low mean pressures (which is typically used in the lab) by using the

equation derived in class is usually higher than the absolute permeability of the porous media because

of an electro-kinetic phenomena known as Klinkenberg (1941) effect.

The reason for excess gas permeability is the slippage of gas molecules during flow in micro scale pore

space of the rock. When the size of the pore space and the mean free path of gas molecules are close to

each other, the chance of gas slippage during flow in porous media increases. Therefore, two conditions

give rise to gas slippage:

1- When gas pressure is low

At low gas pressures the mean free path of gas molecules increases, and become close to the

size of pore space. This results in gas slippage and consequently higher gas permeability.

However, this usually occurs during laboratory measurement of permeability by a gas, and does

not happen at the high mean pressures of a petroleum reservoir.

2- When the pore size of the rock is very small

Gas slippage may occur in shale gas reservoir even at high gas pressures. This is due to the

existence of extremely small pores (1-100 nanometers) in shale sediments.

Klinkenberg suggested the following equation for relating gas permeability to absolute permeability of a

porous media:

where

Figure 1 shows that the gas permeability obtained by measured pressure data and the equation derived

in class, is linearly related to reciprocal of mean gas pressure. The slop and intercept of this line can be

used to obtain the absolute permeability of the rock (kL).

Figure 1. Klinkenberg permeability correction (Peters, in press)

Figure 2. Permeability of a core sample to hydrogen, nitrogen and carbon dioxide. Absolute permeability

of the core to isooctane = 2.55 md (from Klinkenberg, 1941).

Non-Darcy flow

One of the main assumptions of Darcy’s law is laminar flow, where the turbulent-inertial forces are

negligible compared with viscous forces. The Reynolds number for flows in porous media is defined by

Re = (ρ v Dp ) / µ

Where

v = Darcy velocity, cm/s

Dp= mean grain diameter of the granular porous medium, cm

ρ = fluid density, gm/cm3

µ = fluid viscosity, poise

Figure 3 shows the fanning friction factor (which can be considered as a measure of pressure drop in

porous media) versus Reynolds number in a log-log plot. We observe that when Reynolds number

exceeds the value of about 1 the measured data deviate from the straight line, and the deviation

increases by increasing the Reynolds number. Therefore, Darcy low is only valid when Re<1.

Figure 3. Fanning friction factor for flow in porous media (Rose, 1945)

The following plot classifies the flow in porous media based on the value of Reynolds number.

Figure 4. Classification of flow in porous media (Peters, in press).