Operational calculus - Revision history

en>Bender235: /* References */clean up; http->https or pr-URI, per VPP using AWB

2014-09-15T06:04:56Z

References: clean up; http->https or pr-URI, per VPP using AWB

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	~~{{About\|relative permeability in multiphase flow in porous media\|relative magnetic permeability\|permeability (electromagnetism)}}~~		Hi there, I am Andrew Berryhill. Mississippi is where her home is but her spouse wants them to move. The favorite pastime for him and his kids is fashion and he'll be beginning something else along with it. He is an info officer.<br><br>Here is my web-site; online psychic chat ([http://www.herandkingscounty.com/content/information-and-facts-you-must-know-about-hobbies http://www.herandkingscounty.com/])

	~~In [[multiphase flow]] in [[porous media]]~~, ~~the '''relative permeability''' of a phase is a dimensionless measure of the effective permeability of that phase~~. It is ~~the ratio of the effective permeability of that phase to the absolute permeability. It can be viewed as an adaptation of [[Darcy's law]] to multiphase flow.~~

	~~For two-phase flow in porous media given steady-state conditions, we can write~~

	~~:<math>q_i = -\frac{\kappa_i}{\mu_i} \nabla P_i \qquad \text{for} \quad i=1,2</math>~~

	where ~~<math>q_i</math>~~ is ~~the flux, <math>\nabla P_i</math> is the pressure drop, <math>\mu_i</math> is the viscosity~~. The ~~subscript <math>i</math> indicates that the parameters are for phase <math>i</math>.~~

	~~<math>\kappa_i</math> is here the '''phase permeability''' (i.e., the '''effective permeability''' of phase <math>i</math>), as observed through the equation above.~~

	~~'''Relative permeability''', <math>\kappa_{\mathit{ri}}</math>,~~ for ~~phase <math>i</math> is then defined from <math>\kappa_i = \kappa_{\mathit{ri}}\kappa</math> as~~

	~~:<math>\kappa_{\mathit{ri}} = \kappa_i / \kappa</math>~~

	where <math>\kappa</math> is the [[Permeability (fluid)\|permeability]] of the porous medium in single-phase flow, i.e., the '''[[Permeability (fluid)\|absolute permeability]]'''. Relative permeability must be between zero and ~~one.~~

	~~In applications, relative permeability~~ is ~~often represented as a function of [[Water content\|water saturation]], however due to [[Hysteresis\|capillary hysteresis]], one often resorts to one function or curve measured under [[drainage]]~~ and ~~one measured under [[imbibition]].~~

	Under this approach, the flow of each phase is inhibited by the presence of the other phases. Thus the sum of relative permeabilities over all phases is less than 1. However, apparent relative permeabilities larger than 1 have been obtained since the Darcean approach disregards the viscous coupling effects derived from momentum transfer between the phases (see assumptions below). This coupling could enhance the flow instead of inhibit it. This has been observed in heavy oil petroleum reservoirs when the gas phase flows as bubbles or patches (disconnected).<ref>M.C. Bravo, M. Araujo / International Journal of Multiphase Flow 34 (2008) 447–460</ref>

	~~==Assumptions==~~
	~~The above form for Darcy~~'s law is sometimes also called Darcy's extended law, formulated for horizontal, one-dimensional, [[immiscible]] multiphase flow in homogeneous and [[isotropic]] porous media. The interactions between the fluids are neglected, so this model assumes that the solid porous media and the other fluids form a new porous matrix through which a phase can flow, implying that the fluid-fluid interfaces remain static in steady-state flow, which is not true, but this approximation has proven useful anyway.

	~~Each of the phase saturation must~~ be ~~larger than the irreducible saturation, and each phase is assumed continuous within the porous medium.~~

	~~==Approximations==~~

	~~Based on experimental data, simplified models of relative permeability as a function of [[water saturation]] can be constructed~~.

	~~===Corey-type===~~
	~~An often used approximation of relative permeability~~ is ~~the Corey correlation which is [[power law]] in the water saturation <math>S_w</math>.<ref>~~
	~~{{cite journal\|author=R.H. Brooks and A.T. Corey\|title=Hydraulic properties of porous media\|journal=Hydrological Papers\|volume=3\|publisher=Colorado State University\|year=1964}}</ref><ref>~~
	~~{{cite journal\|author=A.T. Corey\|title=The Interrelation Between Gas and Oil Relative Permeabilities\|journal=Prod~~. ~~Monthly\|date=Nov 1954\|volume=19\|issue=1\|pages=38–41}}~~
	~~</ref> If <math>S_\mathit{wi}</math> (also denoted <math>S_\mathit{wir}</math>, or <math>S_\mathit{wr}</math>, or~~ <~~math~~>~~S_\mathit{wc}~~<~~/math~~>) is ~~the irreducible (minimal) water saturation, and <math>S_\mathit{orw}</math> is the residual (minimal) oil saturation after water flooding, we can define a normalized~~ (~~or scaled) water saturation value~~
	~~[[Image:SwNormalisation.svg\|right\|300px\|thumb\|Normalization of water saturation values]]~~
	[~~[Image~~:~~CoreyExampleForRelativePermeability.png\|300px\|thumb\|Example of Corey approximation in normalized <math>S_w</math> coordinates, here <math>N_\mathit{o}<~~/~~math> = <math>N_\mathit{w} = 2<~~/~~math>~~.]]
	~~:<math>S_\mathit{wn}(S_w) = \frac{S_w - S_\mathit{wi}}{1-S_\mathit{wi} - S_{orw}}<~~/~~math>~~

	~~The Corey correlations of the relative permeability of oil and water are then~~
	~~:<math>K_\mathit{row}(S_{w}) = (1-S_\mathit{wn}(S_w))^{N_\mathit{o}}<~~/~~math> and~~
	~~:<math>K_\mathit{rw}(S_{w})=K{_\mathit{rw}^o}S{_\mathit{wn}(S_w)}^{N_\mathit{w}}</math>~~
	~~when the permeability basis is oil with irreducible water present.~~

	~~We note the desired properties~~
	~~:<math>\begin{align}~~
	~~K_\mathit{row}(S_\mathit{wi}) & = 1 & K_\mathit{row}(1~~-~~S_\mathit{orw}) & = 0 \\~~
	~~K_\mathit{rw}(S_\mathit{wi}) &= 0 & K_\mathit{rw}(1-S_\mathit{orw}) &= K_\mathit{rw}^o~~
	~~\end{align}</math>~~

	The empirical parameters <math>N_\mathit{o}</math> and <math>N_\mathit{w}</math> can be obtained from measured data either by optimizing to analytical interpretation of measured data, or by optimizing using a core flow numerical simulator to match the experiment(often called history matching). <math>N_\mathit{o}</math> = <math>N_\mathit{w} = 2</math> is sometimes appropriate.The physical property <math>K_\mathit{rw}^o</math> is called the end point of the water relative permeability, and it is obtained either before or together with the optimizing of <math>N_\mathit{o}</math> and ~~<math>N_\mathit{w}</math>.~~

	~~In case of gas~~-~~water system or gas~~-~~oil system there are Corey correlations similar to the oil~~-~~water relative permeabilities correlations shown above.~~

	~~===LET~~-~~type===~~
	~~The Corey approximation only has one degree of freedom for the oil relative permeability and two degrees of freedom for the water permeability (in <math>S_{wn}</math>).~~
	~~The LET~~-correlation <ref>{{cite journal\|author=Lomeland F., Ebeltoft E. and Hammervold Thomas W.\|title=A New Versatile Relative Permeability Correlation\|journal=Reviewed Proceedings of the 2005 International Symposium of the SCA, Abu Dhabi, United Arab Emirates, October 31 - ~~November 2, 2005, SCA 2005-32}}</ref> adds more degrees of freedom in order to accommodate the shape of measured relative permeability curves in SCAL experiments.~~
	~~[[Image~~:~~LETexampleForRelativePermeability.png\|300px\|thumb\|Example of LET-correlation with L,E,T all equal to 2, and <math>K_\mathit{rw}^o = 0.6<~~/~~math>. Normalized <math>S_w<~~/~~math> coordinates~~.]]

	~~The LET-type approximation is described by 3 parameters L, E, T~~. ~~The correlation for water and oil relative permeability with water injection is thus~~
	~~:<math>K_\mathit{rw}=\frac{{K_\mathit{rw}^o}{S_\mathit{wn}}^{L_\mathit{w}}}{{S_\mathit{wn}}^{L_\mathit{w}}+{E_\mathit{w}}{(1-S_\mathit{wn})}^{T_\mathit{w}}}<~~/~~math>~~
	~~and~~
	~~:<math>K_\mathit{row}=\frac{(1-S_\mathit{wn}~~)~~^{L_o}}{{(1-S_\mathit{wn})^{L_o}}+{E_\mathit{o}}{S_\mathit{wn}}^{T_\mathit{o}}}</math>~~
	~~written using the same <math>S_w</math> normalization as for Corey.~~

	Only <math>S_\mathit{wi}</math> , <math>S_\mathit{orw}</math> and <math>K_\mathit{rw}^o</math> have direct physical meaning, while the parameters ''L'', ''E'' and ''T'' are empirical. The parameter ''L'' describes the lower part of the curve, and by similarity and experience the ''L''-values are comparable to the appropriate Corey parameter. The parameter ''T'' describes the upper part (or the top part) of the curve in a similar way that the ''L''-parameter describes the lower part of the curve. The parameter ''E'' describes the position of the slope (or the elevation) of the curve. A value of one is a neutral value, and the position of the slope is governed by the ''L''- and ''T''-parameters. Increasing the value of the ''E''-parameter pushes the slope towards the high end of the curve. Decreasing the value of the ''E''-parameter pushes the slope towards the lower end of the curve. Experience using the LET correlation indicates that the parameter, ''L'' ≥ 1, ''E'' > 0 and ''T'' ≥ 0.5.

	~~In case of gas-water system or gas-oil system there are LET correlations similar to the oil-water relative permeabilities correlations shown above.~~

	~~==See also==~~

	* [[Permeability (earth sciences)]]
	* [[Capillary pressure]]
	* [[Imbibition]]
	* [[Drainage]]
	* [[Buckley–Leverett equation]]

	~~==References==~~
	~~{{reflist}}~~

	~~{{DEFAULTSORT:Relative Permeability}}~~
	~~[[Category:Fluid dynamics]]~~
	~~[[Category:Porous media]]~~

en>Vatsal19: /* Principle */

2013-12-07T06:12:41Z

Principle

← Older revision		Revision as of 08:12, 7 December 2013
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	~~Friends contact him Royal Seyler~~. ~~Climbing~~ is ~~what adore doing~~. The ~~job I~~'ve been ~~occupying~~ for ~~years~~ is ~~a bookkeeper but I~~'~~ve already applied~~ for ~~an additional~~ one. ~~Arizona~~ is ~~her beginning location~~ and ~~she will never move~~.<br><br>~~Here~~ is ~~my web site -~~ [~~http~~://~~Chilevisual~~.~~net~~/~~index~~.~~php?mod~~=~~users~~&~~action~~=~~view~~&id=~~7096 http~~://~~Chilevisual~~.~~net~~/]		{{About\|relative permeability in multiphase flow in porous media\|relative magnetic permeability\|permeability (electromagnetism)}}

			In [[multiphase flow]] in [[porous media]], the '''relative permeability''' of a phase is a dimensionless measure of the effective permeability of that phase. It is the ratio of the effective permeability of that phase to the absolute permeability. It can be viewed as an adaptation of [[Darcy's law]] to multiphase flow.

			For two-phase flow in porous media given steady-state conditions, we can write

			:<math>q_i = -\frac{\kappa_i}{\mu_i} \nabla P_i \qquad \text{for} \quad i=1,2</math>

			where <math>q_i</math> is the flux, <math>\nabla P_i</math> is the pressure drop, <math>\mu_i</math> is the viscosity. The subscript <math>i</math> indicates that the parameters are for phase <math>i</math>.

			<math>\kappa_i</math> is here the '''phase permeability''' (i.e., the '''effective permeability''' of phase <math>i</math>), as observed through the equation above.

			'''Relative permeability''', <math>\kappa_{\mathit{ri}}</math>, for phase <math>i</math> is then defined from <math>\kappa_i = \kappa_{\mathit{ri}}\kappa</math> as

			:<math>\kappa_{\mathit{ri}} = \kappa_i / \kappa</math>

			where <math>\kappa</math> is the [[Permeability (fluid)\|permeability]] of the porous medium in single-phase flow, i.e., the '''[[Permeability (fluid)\|absolute permeability]]'''. Relative permeability must be between zero and one.

			In applications, relative permeability is often represented as a function of [[Water content\|water saturation]], however due to [[Hysteresis\|capillary hysteresis]], one often resorts to one function or curve measured under [[drainage]] and one measured under [[imbibition]].

			Under this approach, the flow of each phase is inhibited by the presence of the other phases. Thus the sum of relative permeabilities over all phases is less than 1. However, apparent relative permeabilities larger than 1 have been obtained since the Darcean approach disregards the viscous coupling effects derived from momentum transfer between the phases (see assumptions below). This coupling could enhance the flow instead of inhibit it. This has been observed in heavy oil petroleum reservoirs when the gas phase flows as bubbles or patches (disconnected).<ref>M.C. Bravo, M. Araujo / International Journal of Multiphase Flow 34 (2008) 447–460</ref>

			==Assumptions==
			The above form for Darcy's law is sometimes also called Darcy's extended law, formulated for horizontal, one-dimensional, [[immiscible]] multiphase flow in homogeneous and [[isotropic]] porous media. The interactions between the fluids are neglected, so this model assumes that the solid porous media and the other fluids form a new porous matrix through which a phase can flow, implying that the fluid-fluid interfaces remain static in steady-state flow, which is not true, but this approximation has proven useful anyway.

			Each of the phase saturation must be larger than the irreducible saturation, and each phase is assumed continuous within the porous medium.

			==Approximations==

			Based on experimental data, simplified models of relative permeability as a function of [[water saturation]] can be constructed.

			===Corey-type===
			An often used approximation of relative permeability is the Corey correlation which is [[power law]] in the water saturation <math>S_w</math>.<ref>
			{{cite journal\|author=R.H. Brooks and A.T. Corey\|title=Hydraulic properties of porous media\|journal=Hydrological Papers\|volume=3\|publisher=Colorado State University\|year=1964}}</ref><ref>
			{{cite journal\|author=A.T. Corey\|title=The Interrelation Between Gas and Oil Relative Permeabilities\|journal=Prod. Monthly\|date=Nov 1954\|volume=19\|issue=1\|pages=38–41}}
			</ref> If <math>S_\mathit{wi}</math> (also denoted <math>S_\mathit{wir}</math>, or <math>S_\mathit{wr}</math>, or <math>S_\mathit{wc}</math>) is the irreducible (minimal) water saturation, and <math>S_\mathit{orw}</math> is the residual (minimal) oil saturation after water flooding, we can define a normalized (or scaled) water saturation value
			[[Image:SwNormalisation.svg\|right\|300px\|thumb\|Normalization of water saturation values]]
			[[Image:CoreyExampleForRelativePermeability.png\|300px\|thumb\|Example of Corey approximation in normalized <math>S_w</math> coordinates, here <math>N_\mathit{o}</math> = <math>N_\mathit{w} = 2</math>.]]
			:<math>S_\mathit{wn}(S_w) = \frac{S_w - S_\mathit{wi}}{1-S_\mathit{wi} - S_{orw}}</math>

			The Corey correlations of the relative permeability of oil and water are then
			:<math>K_\mathit{row}(S_{w}) = (1-S_\mathit{wn}(S_w))^{N_\mathit{o}}</math> and
			:<math>K_\mathit{rw}(S_{w})=K{_\mathit{rw}^o}S{_\mathit{wn}(S_w)}^{N_\mathit{w}}</math>
			when the permeability basis is oil with irreducible water present.

			We note the desired properties
			:<math>\begin{align}
			K_\mathit{row}(S_\mathit{wi}) & = 1 & K_\mathit{row}(1-S_\mathit{orw}) & = 0 \\
			K_\mathit{rw}(S_\mathit{wi}) &= 0 & K_\mathit{rw}(1-S_\mathit{orw}) &= K_\mathit{rw}^o
			\end{align}</math>

			The empirical parameters <math>N_\mathit{o}</math> and <math>N_\mathit{w}</math> can be obtained from measured data either by optimizing to analytical interpretation of measured data, or by optimizing using a core flow numerical simulator to match the experiment(often called history matching). <math>N_\mathit{o}</math> = <math>N_\mathit{w} = 2</math> is sometimes appropriate.The physical property <math>K_\mathit{rw}^o</math> is called the end point of the water relative permeability, and it is obtained either before or together with the optimizing of <math>N_\mathit{o}</math> and <math>N_\mathit{w}</math>.

			In case of gas-water system or gas-oil system there are Corey correlations similar to the oil-water relative permeabilities correlations shown above.

			===LET-type===
			The Corey approximation only has one degree of freedom for the oil relative permeability and two degrees of freedom for the water permeability (in <math>S_{wn}</math>).
			The LET-correlation <ref>{{cite journal\|author=Lomeland F., Ebeltoft E. and Hammervold Thomas W.\|title=A New Versatile Relative Permeability Correlation\|journal=Reviewed Proceedings of the 2005 International Symposium of the SCA, Abu Dhabi, United Arab Emirates, October 31 - November 2, 2005, SCA 2005-32}}</ref> adds more degrees of freedom in order to accommodate the shape of measured relative permeability curves in SCAL experiments.
			[[Image:LETexampleForRelativePermeability.png\|300px\|thumb\|Example of LET-correlation with L,E,T all equal to 2, and <math>K_\mathit{rw}^o = 0.6</math>. Normalized <math>S_w</math> coordinates.]]

			The LET-type approximation is described by 3 parameters L, E, T. The correlation for water and oil relative permeability with water injection is thus
			:<math>K_\mathit{rw}=\frac{{K_\mathit{rw}^o}{S_\mathit{wn}}^{L_\mathit{w}}}{{S_\mathit{wn}}^{L_\mathit{w}}+{E_\mathit{w}}{(1-S_\mathit{wn})}^{T_\mathit{w}}}</math>
			and
			:<math>K_\mathit{row}=\frac{(1-S_\mathit{wn})^{L_o}}{{(1-S_\mathit{wn})^{L_o}}+{E_\mathit{o}}{S_\mathit{wn}}^{T_\mathit{o}}}</math>
			written using the same <math>S_w</math> normalization as for Corey.

			Only <math>S_\mathit{wi}</math> , <math>S_\mathit{orw}</math> and <math>K_\mathit{rw}^o</math> have direct physical meaning, while the parameters ''L'', ''E'' and ''T'' are empirical. The parameter ''L'' describes the lower part of the curve, and by similarity and experience the ''L''-values are comparable to the appropriate Corey parameter. The parameter ''T'' describes the upper part (or the top part) of the curve in a similar way that the ''L''-parameter describes the lower part of the curve. The parameter ''E'' describes the position of the slope (or the elevation) of the curve. A value of one is a neutral value, and the position of the slope is governed by the ''L''- and ''T''-parameters. Increasing the value of the ''E''-parameter pushes the slope towards the high end of the curve. Decreasing the value of the ''E''-parameter pushes the slope towards the lower end of the curve. Experience using the LET correlation indicates that the parameter, ''L'' ≥ 1, ''E'' > 0 and ''T'' ≥ 0.5.

			In case of gas-water system or gas-oil system there are LET correlations similar to the oil-water relative permeabilities correlations shown above.

			==See also==

			* [[Permeability (earth sciences)]]
			* [[Capillary pressure]]
			* [[Imbibition]]
			* [[Drainage]]
			* [[Buckley–Leverett equation]]

			==References==
			{{reflist}}

			{{DEFAULTSORT:Relative Permeability}}
			[[Category:Fluid dynamics]]
			[[Category:Porous media]]

en>Suslindisambiguator: /* References */

2012-08-17T14:06:25Z

References

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Friends contact him Royal Seyler. Climbing is what adore doing. The job I've been occupying for years is a bookkeeper but I've already applied for an additional one. Arizona is her beginning location and she will never move.<br><br>Here is my web site - [http://Chilevisual.net/index.php?mod=users&action=view&id=7096 http://Chilevisual.net/]