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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. | | 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] |
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| [[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:
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| :<math>\frac{\partial \theta}{\partial t}= \frac{\partial}{\partial z}
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| \left[ K(\theta) \left (\frac{\partial \psi}{\partial z} + 1 \right) \right]\
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| </math>
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| where
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| :<math>K</math> is the [[hydraulic conductivity]],
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| :<math>\psi</math> is the [[pressure head]],
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| :<math>z</math> is the [[elevation]] above a vertical [[datum (geodesy)|datum]],
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| :<math>\theta</math> is the [[water content]], and
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| :<math>t</math> is [[time]].
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| 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).
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| ==Derivation==
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| Here we show how to derive the Richards equation for the vertical direction in a very simplistic form.
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| 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,
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| put in mathematical language:
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| :<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>
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| Put in the 1D form for the direction <math>\hat{k}</math>:
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| :<math>\frac{\partial \theta}{\partial t}= -\frac{\partial}{\partial z} q </math>
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| Horizontal flow in the horizontal direction is formulated by the empiric law of Darcy:
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| :<math>q= - K \frac{\partial h}{\partial z} </math>
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| Substituting ''q'' in the equation above, we get:
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| :<math>\frac{\partial \theta}{\partial t}= \frac{\partial}{\partial z} \left[ K \frac{\partial h}{\partial z}\right] </math> | |
| Substituting for ''h'' = ''ψ'' + ''z'':
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| :<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>
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| 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.
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| ==Formulations==
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| 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.
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| 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''.
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| ===''Head-based''===
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| :<math> C(h)\frac{\partial h}{\partial t}= \nabla \cdot K(h) \nabla h </math>
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| Where ''C(h)'' [1/L] is a function describing the rate of change of saturation with respect the hydraulic head:
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| :<math> C(h) \equiv \frac{\partial \theta }{\partial h} </math>
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| This function is called 'specific moisture capacity' in the literature, and could be determined for
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| 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>
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| ===''Saturation-based''===
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| :<math> \frac{\partial \theta }{\partial t}= \nabla \cdot D(\theta) \nabla \theta </math>
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| Where ''D''(''θ'') [L<sup>2</sup>/T] is 'the soil water diffusivity':
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| :<math> D(\theta) \equiv \frac{ K(\theta) }{C(\theta)} \equiv \frac{\partial h}{ \partial \theta} </math>
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| {{reflist}}
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| [[Category:Soil physics]]
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| [[Category:Hydrology]]
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| [[Category:Partial differential equations]]
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