Factors AffectingPermeability Determination

Non-Darcy Flow

Some figures in this section are from “Fundamentals of Core Analysis,” Core Laboratories, 1989.Some slides in this section are from NExT PERF Short Course Notes, 1999.Some slides appear to have been obtained from unknown primarysources that were not cited by NExT. Note that some slides have a notes section.

Air Permeability Measurement

• Measurement of permeability in the laboratory is most commonly done with air– Convenient and inexpensive– Problem: low values of mean flowing pressure

• downstream pressure, patm

• upstream pressure, just a few psi higher than patm

• At low mean flowing pressure, gas slippage occurs– Diameter of flow path through porous media approaches the “mean free

path” of gas molecules• mean free path is a function of molecule size• mean free path is a function of gas density

– Increasing mean flowing pressure results in less slippage• as pmean, we obtain absolute (equivalent liquid) permeability

Non-Darcy Flow - Gas Slippage

• Liquid flow and gas flow at high mean flowing pressure is laminar– Darcy’s Law is valid– flow velocity at walls is zero

• At low mean flowing pressure gas slippage occurs– Non-Darcy flow is observed– flow at walls is not zero

• Klinkenberg developed a method to correct gas permeability measured at low mean flowing pressure to equivalent liquid permeability

Non-Darcy Flow - Klinkenberg Effect

• As pmean, gas permeability approaches absolute permeability

Non-Darcy Flow - Klinkenberg Effect

• Klinkenberg correction for kair depends on mean flowing pressure– correction ratio shown is for pmean = 1 atm

Non-Darcy Flow - Klinkenberg Effect

• Klinkenberg correction for kair is more important for low absolute permeability

Non-Darcy Flow - High Flow Rates• In the field, gas wells exhibit non-Darcy flow at high flow rates• At high flow velocity, inertial effects and turbulence become

important, and cause non-Darcy flow– inertial effect

Non-Darcy Flow - Turbulence

• Recalling Darcy’s equation for gas flow, (zg )=Constant

• For laboratory flow experiments we can assume T=Tsc and z=1

• For Darcy flow, plotting (qg,sc psc)/A vs. (p12-p2

2)/(2L) results in straight line.

• line passes through origin [when qg,sc =0, then (p12-p2

2)=0]• slope = k/ g

• behavior departs from straight line under turbulent flow conditions (high flow velocity)

22

21

gsc

scscg, pp

μz21

pTT

LAkq

2L

pppA

μkq

22

21

scgscg,

Non-Darcy Flow - Turbulence

Slope = k

00 (p1

2- p22)

2 L

q psc

A

Darcy flow

Non-Darcyflow

Non-Darcy Flow - Forchheimer Equation

• Forchheimer proposed a flow equation to account for the non-linear effect of turbulence by adding a second order term

– Note that unit corrections factors would be required for non-coherent unit systems.

– As flow rate decreases, we approach Darcy’s Law (2nd order term approaches zero)

2g

ggg

Aq

ρβAq

kμ

dsdp

Non-Darcy Flow - Forchheimer Plot

• Based on Forchheimer’s Equation a plotting method was developed to determine absolute permeability even with Non-Darcy effects– (1/kgas) vs. qg,sc

• kgas determined from Darcy’s Law (incorrectly assuming Darcy flow) and is a function of qg,sc

– intercept = (1/kabs); absolute permeability

(1/k

gas)

, (1/

md)

Intercept = [1/kabs]

Slope = [(g,sc)/(gA)]

qg,sc

Non-Darcy Flow - Forchheimer Equation

• Non-Darcy Coefficient, , is an empirically determined function of absolute permeability– For Travis Peak (Texas)

NOB=Net Overburden