1. Factors Affecting Permeability 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 primary sources that were not cited by NExT. Note that some slides have a notes section.

2. 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, p atm upstream pressure, just a few psi higher than p atm 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 p mean , we obtain absolute (equivalent liquid) permeability

3. 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

4. Non-Darcy Flow - Klinkenberg Effect As p mean , gas permeability approaches absolute permeability

5. Non-Darcy Flow - Klinkenberg Effect Klinkenberg correction for k air depends on mean flowing pressure –correction ratio shown is for p mean = 1 atm

6. Non-Darcy Flow - Klinkenberg Effect Klinkenberg correction for k air is more important for low absolute permeability

7. 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

8. Non-Darcy Flow - Turbulence Recalling Darcy’s equation for gas flow, (z  g )=Constant For laboratory flow experiments we can assume T=Tsc and z=1 For Darcy flow, plotting (q g,sc p sc )/A vs. (p 1 2 -p 2 2 )/(2L) results in straight line. line passes through origin [when q g,sc =0, then (p 1 2 -p 2 2 )=0] slope = k/  g behavior departs from straight line under turbulent flow conditions (high flow velocity)

9. Non-Darcy Flow - Turbulence Slope = k  0 0 (p 1 2 - p 2 2 ) 2 L q p sc A Darcy flow Non-Darcy flow

10. 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)

11. 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/k gas ) vs. q g,sc k gas determined from Darcy’s Law (incorrectly assuming Darcy flow) and is a function of q g,sc –intercept = (1/k abs ); absolute permeability (1/kgas), (1/md) Intercept = [1/k abs ] Slope = [(  g,sc )/(  g A)] q g,sc

12. Non-Darcy Flow - Forchheimer Equation Non-Darcy Coefficient, , is an empirically determined function of absolute permeability –For Travis Peak (Texas) NOB=Net Overburden