Slice sampling - Revision history

en>Magioladitis: Replace unicode entity nbsp for character [NBSP] (or space) per WP:NBSP + other fixes, replaced: → (2) using AWB (10331)

2014-07-29T05:13:30Z

Replace unicode entity nbsp for character [NBSP] (or space) per WP:NBSP + other fixes, replaced: → (2) using AWB (10331)

← Older revision		Revision as of 07:13, 29 July 2014
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	~~Andrew Simcox~~ is ~~the name his mothers~~ and ~~fathers gave him and he totally loves this name~~. I ~~am presently~~ a ~~journey agent~~. ~~To climb~~ is ~~some thing I really appreciate~~ doing~~. Mississippi is where her home is~~ but ~~her husband wants them to transfer~~.<br><br>~~My webpage~~ [~~http~~://~~checkmates~~.co.~~za/index~~.~~php?do=~~/~~profile-56347~~/~~info~~/ ~~tarot~~ readings]		The author's name is Andera and she believes it seems quite good. Credit authorising is how she tends to make a living. For a while I've been in Alaska but I will have to transfer in a yr or two. What me and my family adore is doing ballet but I've been taking on new issues lately.<br><br>Look at my web page :: [https://Www-ocl.gist.ac.kr/work/xe/?document_srl=605236 love psychic readings]

en>Mindmatrix: revert - rm promotional link; this is not a web directory

2014-02-22T18:07:00Z

revert - rm promotional link; this is not a web directory

← Older revision		Revision as of 20:07, 22 February 2014
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	~~The '''Richards equation''' represents~~ the ~~movement of water in [[Vadose zone\|unsaturated]] soils,~~ and ~~was formulated by [[Lorenzo A. Richards]] in 1931.<ref>{{cite journal\|author=Richards, L~~.~~A. \|year=1931 \|title=Capillary conduction of liquids through porous mediums \|journal=Physics \|volume=1 \|issue=5 \|pages=318–333 \|doi=10.1063/1.1745010 \|bibcode = 1931Physi...1..318R }}</ref> It is~~ a ~~[[Nonlinearity\|non-linear]] [[partial differential equation]], which is often difficult to approximate since it does not have a [[Closed-form expression\|closed-form]] analytical solution~~.		Andrew Simcox is the name his mothers and fathers gave him and he totally loves this name. I am presently a journey agent. To climb is some thing I really appreciate doing. Mississippi is where her home is but her husband wants them to transfer.<br><br>My webpage [http://checkmates.co.za/index.php?do=/profile-56347/info/ tarot readings]

	[[Darcy's law]] was developed for saturated flow in porous media; to this Richards applied a continuity requirement suggested by [[Edgar Buckingham\|Buckingham]], and obtained a ''general partial differential equation describing water movement in unsaturated non-swelling soils''. The transient state form of this flow equation, known commonly as Richards equation:

	~~:<math>\frac{\partial \theta}{\partial t}= \frac{\partial}{\partial z}~~
	~~\left[ K(\theta) \left (\frac{\partial \psi}{\partial z} + 1 \right) \right]\~~
	~~</math>~~
	~~where~~
	~~:<math>K</math> is the [[hydraulic conductivity]],~~
	~~:<math>\psi</math> is the [[pressure head]],~~
	~~:<math>z</math> is the [[elevation]] above a vertical [[datum (geodesy)\|datum]],~~
	~~:<math>\theta</math>~~ is ~~the [[water content]], and~~
	~~:<math>t</math> is [[time]]~~.

	~~Richards equation~~ is ~~equivalent to the [[groundwater flow equation]], which~~ is ~~in terms of hydraulic head (''h''), by substituting ''h'' = ''ψ'' + ''z'', and changing the storage mechanism~~ to ~~dewatering~~. ~~The reason for writing it in the form above is for convenience with boundary conditions (often expressed in terms of pressure head, for example atmospheric conditions are ''ψ'' = 0).~~

	~~==Derivation==~~
	~~Here we show how to derive the Richards equation for the vertical direction in a very simplistic form.~~
	~~Conservation of mass says the rate of change of saturation in a closed volume is equal to the rate of change of the total sum of fluxes into and out of that volume,~~
	~~put in mathematical language:~~
	~~:<math>\frac{\partial \theta}{\partial t}= \vec{\nabla} \cdot \left(\sum_{i=1}^n{\vec{q}_{i,\,\text{in}}} - \sum_{j=1}^m{\vec{q}_{j,\,\text{out}}} \right)</math>~~

	~~Put in the 1D form for the direction <math>\hat{k}</math>:~~
	~~:<math>\frac{\partial \theta}{\partial t}= -\frac{\partial}{\partial z} q </math>~~
	~~Horizontal flow in the horizontal direction is formulated by the empiric law of Darcy:~~
	~~:<math>q= - K \frac{\partial h}{\partial z}~~ <~~/math~~>
	~~Substituting ''q'' in the equation above, we get:~~
	:<~~math~~>~~\frac{\partial \theta}{\partial t}= \frac{\partial}{\partial z} \left~~[ ~~K \frac{\partial h}{\partial z}\right] </math>~~
	~~Substituting for ''h'' = ''ψ'' + ''z''~~:
	:<math>\frac{\partial \theta}{\partial t}= \frac{\partial}{\partial z} \left[ K \left ( \frac{\partial \psi}{\partial z} + \frac{\partial z}{\partial z} \right ) \right] = \frac{\partial}{\partial z} \left[ K \left ( \frac{\partial \psi}{\partial z} + 1 \right ) \right]</~~math>~~

	We then get the equation above, which is also called the mixed form <ref>{{cite journal\|author=Celia et al. \|year=1990 \|title=A general Mass-Conservative Numerical Solution for the Unsaturated Flow Equation \|journal=Water Resources Research \|volume=26 \|issue=7 \|pages=1483–1496\|bibcode = 1990WRR....26.1483C \|doi = 10.1029/WR026i007p01483 }}</ref> of Richard equation.

	~~==Formulations==~~
	~~The Richards Equation appears in many articles in the environmental literature due to the fact that it describes the flow in the interface between fully saturated aquifers and surface water and~~/~~or the atmosphere~~. ~~It also appears in pure mathematical journals due to the fact that it has non-trivial solutions. Usually, it is presented in one of three forms.~~
	~~The ''mixed form'' containing the pressure and the saturation is discussed above. It can also appear in two other formulations: ''head-based'' and ''saturation-based''~~.

	~~===''Head-based''===~~
	~~:<math> C(h)\frac{\partial h}{\partial t}= \nabla \cdot K(h) \nabla h </math>~~
	~~Where ''C(h)'' [1~~/~~L] is a function describing the rate of change of saturation with respect the hydraulic head:~~
	~~:<math> C(h) \equiv \frac{\partial \theta }{\partial h} </math>~~
	~~This function is called 'specific moisture capacity' in the literature, and could be determined for~~
	~~different soil types using curve fitting and laboratory experiments measuring the rate of infiltration of water into soil column, as described for example in Van Genuchten, 1980~~.~~<ref>{{cite journal\|author=Van Genuchten, M. Th. \|year=1980 \|title~~=A-Closed Form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils\|journal=Soil Science Society of America Journal \|volume=44 \|pages=892–898 \|doi=10.2136/sssaj1980.03615995004400050002x}}</~~ref>~~

	~~===''Saturation~~-~~based''===~~
	~~:<math> \frac{\partial \theta }{\partial t}= \nabla \cdot D(\theta) \nabla \theta </math>~~
	~~Where ''D''(''θ'') [L<sup>2<~~/~~sup>~~/~~T] is 'the soil water diffusivity':~~
	~~:<math> D(\theta) \equiv \frac{ K(\theta) }{C(\theta)} \equiv \frac{\partial h}{ \partial \theta} </math>~~

	~~{{reflist}}~~

	~~[[Category:Soil physics]]~~
	~~[[Category:Hydrology]]~~
	~~[[Category:Partial differential equations]~~]

en>Schnellp: Clarified introduction.

2013-09-17T16:49:56Z

Clarified introduction.

← Older revision		Revision as of 18:49, 17 September 2013
Line 1:		Line 1:
	~~Hi there~~. ~~Allow me begin by introducing the writer~~, ~~her~~ title is ~~Sophia~~. ~~Her family life~~ in ~~Ohio~~. ~~To perform lacross~~ is the ~~thing I love most~~ of ~~all~~. ~~Office supervising~~ is ~~where her main income arrives from~~.<br><br>~~Feel free to surf to my weblog~~: ~~psychic readers~~ ([~~http:~~/~~/www~~.~~prograd~~.~~uff~~.br/~~novo~~/~~facts~~-~~about~~-~~growing~~-~~greater~~-~~organic~~-~~garden Full Record~~])		The '''Richards equation''' represents the movement of water in [[Vadose zone\|unsaturated]] soils, and was formulated by [[Lorenzo A. Richards]] in 1931.<ref>{{cite journal\|author=Richards, L.A. \|year=1931 \|title=Capillary conduction of liquids through porous mediums \|journal=Physics \|volume=1 \|issue=5 \|pages=318–333 \|doi=10.1063/1.1745010 \|bibcode = 1931Physi...1..318R }}</ref> It is a [[Nonlinearity\|non-linear]] [[partial differential equation]], which is often difficult to approximate since it does not have a [[Closed-form expression\|closed-form]] analytical solution.

			[[Darcy's law]] was developed for saturated flow in porous media; to this Richards applied a continuity requirement suggested by [[Edgar Buckingham\|Buckingham]], and obtained a ''general partial differential equation describing water movement in unsaturated non-swelling soils''. The transient state form of this flow equation, known commonly as Richards equation:

			:<math>\frac{\partial \theta}{\partial t}= \frac{\partial}{\partial z}
			\left[ K(\theta) \left (\frac{\partial \psi}{\partial z} + 1 \right) \right]\
			</math>
			where
			:<math>K</math> is the [[hydraulic conductivity]],
			:<math>\psi</math> is the [[pressure head]],
			:<math>z</math> is the [[elevation]] above a vertical [[datum (geodesy)\|datum]],
			:<math>\theta</math> is the [[water content]], and
			:<math>t</math> is [[time]].

			Richards equation is equivalent to the [[groundwater flow equation]], which is in terms of hydraulic head (''h''), by substituting ''h'' = ''ψ'' + ''z'', and changing the storage mechanism to dewatering. The reason for writing it in the form above is for convenience with boundary conditions (often expressed in terms of pressure head, for example atmospheric conditions are ''ψ'' = 0).

			==Derivation==
			Here we show how to derive the Richards equation for the vertical direction in a very simplistic form.
			Conservation of mass says the rate of change of saturation in a closed volume is equal to the rate of change of the total sum of fluxes into and out of that volume,
			put in mathematical language:
			:<math>\frac{\partial \theta}{\partial t}= \vec{\nabla} \cdot \left(\sum_{i=1}^n{\vec{q}_{i,\,\text{in}}} - \sum_{j=1}^m{\vec{q}_{j,\,\text{out}}} \right)</math>

			Put in the 1D form for the direction <math>\hat{k}</math>:
			:<math>\frac{\partial \theta}{\partial t}= -\frac{\partial}{\partial z} q </math>
			Horizontal flow in the horizontal direction is formulated by the empiric law of Darcy:
			:<math>q= - K \frac{\partial h}{\partial z} </math>
			Substituting ''q'' in the equation above, we get:
			:<math>\frac{\partial \theta}{\partial t}= \frac{\partial}{\partial z} \left[ K \frac{\partial h}{\partial z}\right] </math>
			Substituting for ''h'' = ''ψ'' + ''z'':
			:<math>\frac{\partial \theta}{\partial t}= \frac{\partial}{\partial z} \left[ K \left ( \frac{\partial \psi}{\partial z} + \frac{\partial z}{\partial z} \right ) \right] = \frac{\partial}{\partial z} \left[ K \left ( \frac{\partial \psi}{\partial z} + 1 \right ) \right]</math>

			We then get the equation above, which is also called the mixed form <ref>{{cite journal\|author=Celia et al. \|year=1990 \|title=A general Mass-Conservative Numerical Solution for the Unsaturated Flow Equation \|journal=Water Resources Research \|volume=26 \|issue=7 \|pages=1483–1496\|bibcode = 1990WRR....26.1483C \|doi = 10.1029/WR026i007p01483 }}</ref> of Richard equation.

			==Formulations==
			The Richards Equation appears in many articles in the environmental literature due to the fact that it describes the flow in the interface between fully saturated aquifers and surface water and/or the atmosphere. It also appears in pure mathematical journals due to the fact that it has non-trivial solutions. Usually, it is presented in one of three forms.
			The ''mixed form'' containing the pressure and the saturation is discussed above. It can also appear in two other formulations: ''head-based'' and ''saturation-based''.

			===''Head-based''===
			:<math> C(h)\frac{\partial h}{\partial t}= \nabla \cdot K(h) \nabla h </math>
			Where ''C(h)'' [1/L] is a function describing the rate of change of saturation with respect the hydraulic head:
			:<math> C(h) \equiv \frac{\partial \theta }{\partial h} </math>
			This function is called 'specific moisture capacity' in the literature, and could be determined for
			different soil types using curve fitting and laboratory experiments measuring the rate of infiltration of water into soil column, as described for example in Van Genuchten, 1980.<ref>{{cite journal\|author=Van Genuchten, M. Th. \|year=1980 \|title=A-Closed Form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils\|journal=Soil Science Society of America Journal \|volume=44 \|pages=892–898 \|doi=10.2136/sssaj1980.03615995004400050002x}}</ref>

			===''Saturation-based''===
			:<math> \frac{\partial \theta }{\partial t}= \nabla \cdot D(\theta) \nabla \theta </math>
			Where ''D''(''θ'') [L<sup>2</sup>/T] is 'the soil water diffusivity':
			:<math> D(\theta) \equiv \frac{ K(\theta) }{C(\theta)} \equiv \frac{\partial h}{ \partial \theta} </math>

			{{reflist}}

			[[Category:Soil physics]]
			[[Category:Hydrology]]
			[[Category:Partial differential equations]]

76.103.248.19: /* Example */

2012-06-16T02:26:29Z

Example

New page

Hi there. Allow me begin by introducing the writer, her title is Sophia. Her family life in Ohio. To perform lacross is the thing I love most of all. Office supervising is where her main income arrives from.<br><br>Feel free to surf to my weblog: psychic readers ([http://www.prograd.uff.br/novo/facts-about-growing-greater-organic-garden Full Record])