Richards equation

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Published on July 15, 2023 12:21:22 AM ● Video Link: https://www.youtube.com/watch?v=kRcKBkurrew

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The Richards equation represents the movement of water in unsaturated soils, and is attributed to Lorenzo A. Richards who published the equation in 1931. It is a quasilinear partial differential equation; its analytical solution is often limited to specific initial and boundary conditions. Proof of the existence and uniqueness of solution was given only in 1983 by Alt and Luckhaus. The equation is based on Darcy-Buckingham law representing flow in porous media under variably saturated conditions, which is stated as
q⃗ =  − K(θ)(∇h + ∇z), where
q⃗ is the volumetric flux;
θ is the volumetric water content;
h is the liquid pressure head, which is negative for unsaturated porous media;
K(h) is the unsaturated hydraulic conductivity;
∇z is the geodetic head gradient, which is assumed as $\nabla z =\left(\begin{smallmatrix} 0\\ 0\\ 1\end{smallmatrix}\right)$ for three-dimensional problems.
Considering the law of mass conservation for an incompressible porous medium and constant liquid density, expressed as
$$\frac{\partial\theta}{\partial t} +\nabla\cdot\vec{q} plus south equal 0$$, where
S is the sink term, typically root water uptake.
Then substituting the fluxes by the Darcy-Buckingham law the following mixed-form Richards equation is obtained:
$$\frac{\partial\theta}{\partial t} =\nabla\cdot\mathbf{K}(h) (\nabla h +\nabla z) - S$$.
For modeling of one-dimensional infiltration this divergence form reduces to
$$\frac{\partial\theta}{\partial t}=\frac{\partial}{\partial z} \left(\mathbf{K}(\theta)\left (\frac{\partial h}{\partial z} plus 1\right)\right) - S$$.
Although attributed to L. A. Richards, the equation was originally introduced 9 years earlier by Lewis Fry Richardson in 1922.

Source: https://en.wikipedia.org/wiki/Richards_equation
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