Changes

===[[Porosity]]===

===[[Porosity]]===

Several attempts have been made in the past to derive a general relationship between porosity and permeability. Prominent among these relationships is the work of Kozeny<ref name=pt05r98>Kozeny, J. S., 1927, Uber Kapillare Leitung des Wassers im Boden (Aufstieg, Versickerung und Anwendung auf die Bewasserung): S.-Ber. Wiener Akad. Abt. II a, v. 136, p. 271–306.</ref>, which considered the porous media as a bundle of capillary tubes of equal length. Modifications to account for tortuosity of flow paths in the porous media have been proposed, including the Carman-Kozeny model (1938). Unfortunately, only qualitative results have been obtained using these permeability-porosity relationships because of the complexity of the geometry of the porous media.

Several attempts have been made in the past to derive a general relationship between porosity and permeability. Prominent among these relationships is the work of Kozeny,<ref name=pt05r98>Kozeny, J. S., 1927, Uber Kapillare Leitung des Wassers im Boden (Aufstieg, Versickerung und Anwendung auf die Bewasserung): S.-Ber. Wiener Akad. Abt. II a, v. 136, p. 271–306.</ref> which considered the porous media as a bundle of capillary tubes of equal length. Modifications to account for tortuosity of flow paths in the porous media have been proposed, including the Carman-Kozeny model (1938). Unfortunately, only qualitative results have been obtained using these permeability-porosity relationships because of the complexity of the geometry of the porous media.

Berg<ref name=pt05r25>Berg, R. R., 1970, Method for determining permeability from reservoir rock properties: Transactions Gulf Coast Association of Geological Societies, v. 20, p. 303–317.</ref> suggested that a better understanding of the properties of the rock that control size, shape, and continuity of the rock is the key to relating fluid flow properties to reservoir rock properties. Qualitatively, it is reasonable to assume that permeability should increase with increase in porosity in unfractured reservoirs without significant diagenetic alterations. In fact, it has been shown that there is a relationship between porosity and permeability within units with the same hydraulic properties<ref name=pt05r8>Amaefule, J. O., Keelan, D. K., Kersey, D. G., Marschall, D. M., 1988, Reservoir description—a practical synergistic engineering and geological approach based on analysis of core data: 63rd SPE Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Houston, TX, October 2–5, SPE 18167.</ref>. (For more on porosity, see the chapters on “Porosity” and “Core-Log Transformations and Porosity-Permeability Relationships” in Part 5.)

Berg<ref name=pt05r25>Berg, R. R., 1970, Method for determining permeability from reservoir rock properties: Transactions Gulf Coast Association of Geological Societies, v. 20, p. 303–317.</ref> suggested that a better understanding of the properties of the rock that control size, shape, and continuity of the rock is the key to relating fluid flow properties to reservoir rock properties. Qualitatively, it is reasonable to assume that permeability should increase with increase in porosity in unfractured reservoirs without significant diagenetic alterations. In fact, it has been shown that there is a relationship between porosity and permeability within units with the same hydraulic properties.<ref name=pt05r8>Amaefule, J. O., Keelan, D. K., Kersey, D. G., Marschall, D. M., 1988, Reservoir description—a practical synergistic engineering and geological approach based on analysis of core data: 63rd SPE Annual Technical Conference and Exhibition of the Society of Petroleum Engineers, Houston, TX, October 2–5, SPE 18167.</ref> (For more on porosity, see [[Porosity]] and [[Core-log transformations and porosity-permeability relationships]].)

[[file:permeability_fig3.png|left|thumb|{{figure number|3}}Effect of net confining stress on permeability. (After <ref name=pt05r8 />.)]]

[[file:permeability_fig3.png|left|thumb|{{figure number|3}}Effect of net confining stress on permeability. (After <ref name=pt05r8 />.)]]

When gas is used to determine permeability at low mean pressure, the resistance to flow from drag is very low, resulting in “gas slip conditions.” Consequently, permeability calculated from Darcy's law will be too high and must be corrected using the Klinkenberg<ref name=pt05r97>Klinkenberg, L. J., 1941, The permeability of porous media to liquid and gases, in Drilling and Production Practices: Washington, D., C., American Petroleum Institute, p. 200–211.</ref> model. When gas permeability is corrected for slippage effects at the fluid/pore wall interface, it is called equivalent, nonreactive, liquid permeability or Klinkenberg permeability.

When gas is used to determine permeability at low mean pressure, the resistance to flow from drag is very low, resulting in “gas slip conditions.” Consequently, permeability calculated from Darcy's law will be too high and must be corrected using the Klinkenberg<ref name=pt05r97>Klinkenberg, L. J., 1941, The permeability of porous media to liquid and gases, in Drilling and Production Practices: Washington, D., C., American Petroleum Institute, p. 200–211.</ref> model. When gas permeability is corrected for slippage effects at the fluid/pore wall interface, it is called equivalent, nonreactive, liquid permeability or Klinkenberg permeability.

At high flow rates, gas flowing through porous media accelerates at pore throats and decelerates in pore bodies, giving rise to what is called inertial effects. Non-Darcy flow has been described by Forchheimer<ref name=pt05r58>Forchheimer, P. H., 1901, Wasserbewegung durch Boden: Zeitschrift Verein Deutscher Ingenieure, v. 45, n. 50, p. 1781–1788.</ref>, who presented modifications.

At high flow rates, gas flowing through porous media accelerates at pore throats and decelerates in pore bodies, giving rise to what is called inertial effects. Non-Darcy flow has been described by Forchheimer,<ref name=pt05r58>Forchheimer, P. H., 1901, Wasserbewegung durch Boden: Zeitschrift Verein Deutscher Ingenieure, v. 45, n. 50, p. 1781–1788.</ref> who presented modifications.

===Laboratory methods for permeability determination===

===Laboratory methods for permeability determination===

===Gas permeability by unsteady-state method===

===Gas permeability by unsteady-state method===

Aronofsky<ref name=pt05r21>Aronofsky, J. S., 1954, Effect of gas slip on unsteady flow of gas through porous media: Journal of Applied Physics, v. 25, n. 1, p. 48–53., 10., 1063/1., 1721519</ref> has discussed the theory of transient permeability measurements, and the development of transient state permeameters has been discussed by Wallick and Aronofsky<ref name=pt05r160>Wallick, G. C., Aronofsk, J. S., 1954, Effects of gas slip on unsteady flow of gas through porous media—experimental verification.: Transactions of the American Institute of Mining and Engineering, v. 201, p. 322–324.</ref>, Champlin (1962), Morris<ref name=pt05r115>Morris, W. L., 1953, Assignor, Philips Petroleum Co. Portable Permeameter: U., S. Patent No. 2,633,015, March 23.</ref>, and Jones<ref name=pt05r85>Jones, S. C., 1972, Rapid accurate unsteady-state klinkenberg permeameter: Society of Petroleum Engineers Journal, v. 12, p. 383–397., 10., 2118/3535-PA</ref>.

Aronofsky<ref name=pt05r21>Aronofsky, J. S., 1954, Effect of gas slip on unsteady flow of gas through porous media: Journal of Applied Physics, v. 25, n. 1, p. 48–53., 10., 1063/1., 1721519</ref> has discussed the theory of transient permeability measurements, and the development of transient state permeameters has been discussed by Wallick and Aronofsky,<ref name=pt05r160>Wallick, G. C., Aronofsk, J. S., 1954, Effects of gas slip on unsteady flow of gas through porous media—experimental verification.: Transactions of the American Institute of Mining and Engineering, v. 201, p. 322–324.</ref>, [[Champlin (1962)]], Morris,<ref name=pt05r115>Morris, W. L., 1953, Assignor, Philips Petroleum Co. Portable Permeameter: U., S. Patent No. 2,633,015, March 23.</ref> and Jones.<ref name=pt05r85>Jones, S. C., 1972, Rapid accurate unsteady-state klinkenberg permeameter: Society of Petroleum Engineers Journal, v. 12, p. 383–397., 10., 2118/3535-PA</ref>

A schematic diagram of the unsteady-state Klinkenberg permeameter (Jones, 1990) is shown in [[:file:permeability_fig4.png|Figure 4b]]. The permeameter works on the principle of transient analysis of pressure pulse decay in which Klinkenberg permeability is determined as a function of gas (ideally helium) pressure decay. This equipment consists of a reference cell of known volume that charges the core sample with gas. A downstream valve vents the gas pressure, and pressure change as a function of time is recorded. A typical pressure drawdown plot (Jones, 1990) is shown in [[:file:permeability_fig5.png|Figure 5]]. Advantages of the unsteady-state method include the ability to determine simultaneously (from Figure 5) the Klinkenberg permeability (''k''_∞ helium slippage factor (β_He), and the inertial coefficient (β). A comparison of the steady-state method to the unsteady-state method is presented in Table 1.

A schematic diagram of the unsteady-state Klinkenberg permeameter [[(Jones, 1990)]] is shown in [[:file:permeability_fig4.png|Figure 4b]]. The permeameter works on the principle of transient analysis of pressure pulse decay in which Klinkenberg permeability is determined as a function of gas (ideally helium) pressure decay. This equipment consists of a reference cell of known volume that charges the core sample with gas. A downstream valve vents the gas pressure, and pressure change as a function of time is recorded. A typical pressure drawdown plot [[(Jones, 1990)]] is shown in [[:file:permeability_fig5.png|Figure 5]]. Advantages of the unsteady-state method include the ability to determine simultaneously (from Figure 5) the Klinkenberg permeability (''k''_∞ helium slippage factor (β_He), and the inertial coefficient (β). A comparison of the steady-state method to the unsteady-state method is presented in Table 1.

{| class = "wikitable"

{| class = "wikitable"

===Liquid permeability by unsteady-state method===

===Liquid permeability by unsteady-state method===

A technique based on pulse decay analysis<ref name=pt05r6>Amaefule, J. O., Masuo, S. T., 1986, Use of [[capillary pressure]] data for rapid evaluation of formation damage or [[stimulation]]: Society of Petroleum Engineers Paper No. 12475.</ref> has been developed recently to determine effective permeability to liquid for low quality reservoir rocks. The authors reviewed computational techniques and experimental protocols for liquid permeability determination. A technique that allows the simultaneous determination of liquid permeability and compressibility was also developed. A detailed discussion of this technique is beyond the scope of this chapter, therefore, interested readers are referred to the paper by <ref name=pt05r6 />).

A technique based on pulse decay analysis<ref name=pt05r6>Amaefule, J. O., Masuo, S. T., 1986, Use of [[capillary pressure]] data for rapid evaluation of formation damage or [[stimulation]]: Society of Petroleum Engineers Paper No. 12475.</ref> has been developed recently to determine effective permeability to liquid for low quality reservoir rocks. The authors reviewed computational techniques and experimental protocols for liquid permeability determination. A technique that allows the simultaneous determination of liquid permeability and compressibility was also developed. A detailed discussion of this technique is beyond the scope of this article, therefore, interested readers are referred to<ref name=pt05r6 />.

===Permeability averaging and uncertainty determination===

===Permeability averaging and uncertainty determination===

It is necessary to average permeability determined for each pay zone to obtain permeability distribution. The most commonly used method to average horizontal permeability is the arithmetic average. Comparison of core permeabilities shows that arithmetic average permeabilities values generally agree with well test permeabilities.

It is necessary to average permeability determined for each pay zone to obtain permeability distribution. The most commonly used method to average horizontal permeability is the arithmetic average. Comparison of core permeabilities shows that arithmetic average permeabilities values generally agree with well test permeabilities.

Systematic and/or random errors may affect the accuracy of permeability determined from any method, whether laboratory core or well test analysis. Uncertainty in the models used for permeability determination and input variables can result only in random errors if the same analytical technique, equipment calibration, and quality control scenario are considered. Amaefule and Keelan<ref name=pt05r5>Amaefule, J. O., Keelan, D. K., 1989, Stochastic approach to computation of uncertainty in petrophysical properties: SC Conference Paper No. 8907.</ref> have shown that random errors can be addressed through stochastic modeling in which uncertainty can be assigned to the independent variables by multiple measurements and statistical calculations. Typically, accuracy of measured permeabilities decline at low and high values and are usually within ±5%<ref name=pt05r89>Keelan, D. K., 1971, A critical review of core analysis techniques: 22nd Annual Technical Meeting of the Petroleum Society of the Canadian Institute of Mining, Calgary, Banff, Alberta, June 2–5, Paper No. 7612, p. 1–13.</ref>.

Systematic and/or random errors may affect the accuracy of permeability determined from any method, whether laboratory core or well test analysis. Uncertainty in the models used for permeability determination and input variables can result only in random errors if the same analytical technique, equipment calibration, and quality control scenario are considered. Amaefule and Keelan<ref name=pt05r5>Amaefule, J. O., Keelan, D. K., 1989, Stochastic approach to computation of uncertainty in petrophysical properties: SC Conference Paper No. 8907.</ref> have shown that random errors can be addressed through stochastic modeling in which uncertainty can be assigned to the independent variables by multiple measurements and statistical calculations. Typically, accuracy of measured permeabilities decline at low and high values and are usually within ±5%.<ref name=pt05r89>Keelan, D. K., 1971, A critical review of core analysis techniques: 22nd Annual Technical Meeting of the Petroleum Society of the Canadian Institute of Mining, Calgary, Banff, Alberta, June 2–5, Paper No. 7612, p. 1–13.</ref>

==See also==

==See also==