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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Richards equation ： ウィキペディア英語版 Richards equation The Richards' equation represents the movement of water in unsaturated soils, and was formulated by Lorenzo A. Richards in 1931. It is a non-linear partial differential equation, which is often difficult to approximate since it does not have a closed-form analytical solution. Darcy's law was developed for saturated flow in porous media; to this Richards applied a continuity requirement suggested by 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: :$\backslash frac=\; \backslash frac$ \left(K(\theta) \left (\frac + 1 \right) \right )\ where :$K$ is the hydraulic conductivity, :$\backslash psi$ is the pressure head, :$z$ is the elevation above a vertical datum, :$\backslash theta$ is the water content, and :$t$ 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: :$\backslash frac=\; \backslash vec\; \backslash cdot\; \backslash left(\backslash sum\_^n\}\}\; -\; \backslash sum\_^m\}\}\; \backslash right)$ Put in the 1D form for the direction $\backslash hat$: :$\backslash frac=\; -\backslash frac\; q$ Horizontal flow in the horizontal direction is formulated by the empiric law of Darcy: :$q=\; -\; K\; \backslash frac$ Substituting ''q'' in the equation above, we get: :$\backslash frac=\; \backslash frac\; \backslash left(K\; \backslash frac\backslash right)$ Substituting for ''h'' = ''ψ'' + ''z'': :$\backslash frac=\; \backslash frac\; \backslash left(K\; \backslash left\; (\; \backslash frac\; +\; \backslash frac\; \backslash right\; )\; \backslash right)\; =\; \backslash frac\; \backslash left(K\; \backslash left\; (\; \backslash frac\; +\; 1\; \backslash right\; )\; \backslash right)$ We then get the equation above, which is also called the mixed form of Richards' equation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Richards equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.