Changes

==Theoretical background==

==Theoretical background==

The fundamental relationship given by Henry <ref name=pt05r44>Darcy, H., 1856, Les Fontaines Publiques de la Ville de Dijon: Paris, Victor Dalmont, p. 590–594.</ref> is the basis for permeability determination. Darcy's law originates from the interpretation of the results of the flow of water through an experimental apparatus, shown in Figure 1. In this experiment, water was allowed to flow downward through the sand pack contained in an iron cylinder. Manometers located at the input and output ends measured fluid pressures, which were then related to flow rates to obtain the following fundamental Darcy's law:

 

The fundamental relationship given by Henry <ref name=pt05r44>Darcy, H., 1856, Les Fontaines Publiques de la Ville de Dijon: Paris, Victor Dalmont, p. 590–594.</ref> is the basis for permeability determination. Darcy's law originates from the interpretation of the results of the flow of water through an experimental apparatus, shown in [[:file:permeability_fig1.png|Figure 1]]. In this experiment, water was allowed to flow downward through the sand pack contained in an iron cylinder. Manometers located at the input and output ends measured fluid pressures, which were then related to flow rates to obtain the following fundamental Darcy's law:

:<math>q = KA \frac{\Delta h}{L}</math>

:<math>q = KA \frac{\Delta h}{L}</math>

* ''L'' = length (cm)

* ''L'' = length (cm)

The units in which permeability is typically expressed are the ''darcy'' (d) and ''millidarcy'' (md). A permeability of 1 d allows the flow of 1 cm³ per second of fluid with 1 cP (centipoise) viscosity through a cross-sectional area of 1 cm² when a pressure gradient of 1 atm/cm is applied. This definition unfortunately contains nonconsistent units, as pressure is expressed in atmospheres rather than in fundamental units. Lowman et al.<ref name=pt05r104>Lowman, S. W., 1972, Definition of selected groundwater terms—revisions and conceptual refinements: U., S. Geological Survey Water Supply Paper 1988, 21 p.</ref>, however, have redefined the darcy unit in the mks system in which square meters represents the standard dimension of permeability. The millidarcy, which is one-thousandth of a darcy, is commonly used in core analysis and oilfield operations.

The units in which permeability is typically expressed are the ''darcy'' (d) and ''millidarcy'' (md). A permeability of 1 d allows the flow of 1 cm³ per second of fluid with 1 cP (centipoise) viscosity through a cross-sectional area of 1 cm² when a pressure gradient of 1 atm/cm is applied. This definition unfortunately contains nonconsistent units, as pressure is expressed in atmospheres rather than in fundamental units. Lowman et al.<ref name=pt05r104>Lowman, S. W., 1972, Definition of selected groundwater terms—revisions and conceptual refinements: U. S. Geological Survey Water Supply Paper 1988, 21 p.</ref>, however, have redefined the darcy unit in the mks system in which square meters represents the standard dimension of permeability. The millidarcy, which is one-thousandth of a darcy, is commonly used in core analysis and oilfield operations.

 

==Factors controlling permeability==

==Factors controlling permeability==

===Pore geometry===

===Pore geometry===

Permeability is a function of the geometry of the pore structure of the porous media. Permeability is controlled in sandstone by grain size, grain orientation, packing arrangement, cementation, clay content, bedding, and grain size distribution and sorting. In carbonates, permeability is a function of the degree of mineral alteration (such as dolomitization), [[porosity]] development, and fractures. Figure 2 shows the relationship among permeability, sorting, and grain size.

Permeability is a function of the geometry of the pore structure of the porous media. Permeability is controlled in sandstone by grain size, grain orientation, packing arrangement, cementation, clay content, bedding, and grain size distribution and sorting. In carbonates, permeability is a function of the degree of mineral alteration (such as dolomitization), [[porosity]] development, and fractures. [[:file:permeability_fig2.png|Figure 2]] shows the relationship among permeability, sorting, and grain size.

 

[[file:permeability_fig1.png|thumb|{{figure number|1}}Modified schematic diagram of Darcy's experimental apparatus. (Modified from <ref name=pt05r56>Folk, R. L., 1959, Practical petrographic classification of limestones: AAPG Bulletin, v. 43, p. 1–38.</ref>.)]]

 

[[file:permeability_fig2.png|thumb|{{figure number|2}}Relationship among permeability, sorting, and grain size. (From <ref name=pt05r124>Pettijohn, F. J., 1975, Sedimentary rocks, 3rd ed.: New York, Harper and Row, p. 628.</ref>; after Krumbein and Monk, 1942.)]]

===Bedding===

===Bedding===

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 the chapters on “Porosity” and “Core-Log Transformations and Porosity-Permeability Relationships” in Part 5.)

===Confining pressure===

===Confining pressure===

Permeability decreases with increasing confining pressure. Unconsolidated or poorly lithified rock undergoes much greater permeability reduction under confining pressure than well-consolidated rock. As shown in Figure 3, a greater percentage of permeability reduction is typically observed in lower permeability rock than in higher permeability rock. To determine permeability-stress relationships, which are representative of ''in situ'' reservoir conditions, permeability measurements should be made on selected samples at a series of confining pressures. Jones<ref name=pt05r86>Jones, S. C., 1988, Two-point determinations of permeability and PV versus net confining stress: Society of Petroleum Engineer Formation Evaluation, v. 3, p. 235–241.</ref> has recently presented a method that allows a two-point determination of a permeability-stress model that reduces the required number of permeability measurements under confining stress for permeability-stress prediction.

Permeability decreases with increasing confining pressure. Unconsolidated or poorly lithified rock undergoes much greater permeability reduction under confining pressure than well-consolidated rock. As shown in [[:file:permeability_fig3.png|Figure 3]], a greater percentage of permeability reduction is typically observed in lower permeability rock than in higher permeability rock. To determine permeability-stress relationships, which are representative of ''in situ'' reservoir conditions, permeability measurements should be made on selected samples at a series of confining pressures. Jones<ref name=pt05r86>Jones, S. C., 1988, Two-point determinations of permeability and PV versus net confining stress: Society of Petroleum Engineer Formation Evaluation, v. 3, p. 235–241.</ref> has recently presented a method that allows a two-point determination of a permeability-stress model that reduces the required number of permeability measurements under confining stress for permeability-stress prediction.

 

==Permeability determination==

==Permeability determination==

Liquid and gas permeability can be determined on core samples in the laboratory. However, gas permeability is determined most frequently because sample preparation is simplified and the analytical procedure is fairly rapid. Two methods currently exist for gas permeability determination: steady-state and unsteady-state.

Liquid and gas permeability can be determined on core samples in the laboratory. However, gas permeability is determined most frequently because sample preparation is simplified and the analytical procedure is fairly rapid. Two methods currently exist for gas permeability determination: steady-state and unsteady-state.

===Gas permeability by steady-state method===

===Gas permeability by steady-state method===

A simplified schematic diagram of the steady-state apparatus is shown in Figure 4a. The apparatus includes a pressurized gas cylinder, a Hassler core holder, and a flowmeter. The apparatus is designed to ensure that no restrictions exist in flow lines that could cause a pressure drop between the core face and the pressure gauges. To determine air permeability, a clean, dried core sample is first placed in the core holder and pressure is applied to the rubber sleeve to seal it to the core. Air is then injected at a constant pressure until gas production rate and pressure stabilize. The pressure differential between the two ends of the core and flow rate are recorded for permeability calculation using the integrated form of Darcy's law for a compressible fluid. Thus,

A simplified schematic diagram of the steady-state apparatus is shown in [[:file:permeability_fig4.png|Figure 4a]]. The apparatus includes a pressurized gas cylinder, a Hassler core holder, and a flowmeter. The apparatus is designed to ensure that no restrictions exist in flow lines that could cause a pressure drop between the core face and the pressure gauges. To determine air permeability, a clean, dried core sample is first placed in the core holder and pressure is applied to the rubber sleeve to seal it to the core. Air is then injected at a constant pressure until gas production rate and pressure stabilize. The pressure differential between the two ends of the core and flow rate are recorded for permeability calculation using the integrated form of Darcy's law for a compressible fluid. Thus,

:<math>k_{a} = \frac{2000p_{a}\mu q_{a}L}{(p_{1}^{2} - p_{2}^{2})A}</math>

:<math>k_{a} = \frac{2000p_{a}\mu q_{a}L}{(p_{1}^{2} - p_{2}^{2})A}</math>

:<math>V_{a} = \frac{qa}{A}, \mbox{ cm/sec}</math>

:<math>V_{a} = \frac{qa}{A}, \mbox{ cm/sec}</math>

[[file:permeability_fig4.png|thumb|{{figure number|4}}Schematic diagram of (a) steady-state and (b) unsteady-state apparatus.]]

Measurements are usually made at several gas flow rates to ensure that flow conditions satisfy Darcy's law. In practice, gas permeability is calculated from the slope of the plot of ''V''_a versus (''p''₁² - ''p''₂²)/''L'', which results in a straight line passing through the origin as long as the conditions for Darcy's flow are maintained. The steady-state method has been the industry standard for many years because it is a convenient technique and the equipment is easy to operate.

Measurements are usually made at several gas flow rates to ensure that flow conditions satisfy Darcy's law. In practice, gas permeability is calculated from the slope of the plot of ''V''_a versus (''p''₁² - ''p''₂²)/''L'', which results in a straight line passing through the origin as long as the conditions for Darcy's flow are maintained. The steady-state method has been the industry standard for many years because it is a convenient technique and the equipment is easy to operate.

[[file:permeability_fig5.png|left|thumb|{{figure number|5}}Typical pressure drawdown plot. (Modified from <ref name=pt05r85 />.)]]

===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 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 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"