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Lindsey Heagy
53 lines
1.7 KiB
ReStructuredText
53 lines
1.7 KiB
ReStructuredText
.. _examples_FLOW_Richards_1D_Celia1990:
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.. --------------------------------- ..
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.. ..
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.. THIS FILE IS AUTO GENEREATED ..
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.. ..
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.. SimPEG/Examples/__init__.py ..
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.. ..
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.. --------------------------------- ..
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FLOW: Richards: 1D: Celia1990
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=============================
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There are two different forms of Richards equation that differ
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on how they deal with the non-linearity in the time-stepping term.
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The most fundamental form, referred to as the
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'mixed'-form of Richards Equation Celia1990_
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.. math::
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\frac{\partial \theta(\psi)}{\partial t} - \nabla \cdot k(\psi) \nabla \psi - \frac{\partial k(\psi)}{\partial z} = 0
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\quad \psi \in \Omega
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where \\(\\theta\\) is water content, and \\(\\psi\\) is pressure head.
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This formulation of Richards equation is called the
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'mixed'-form because the equation is parameterized in \\(\\psi\\)
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but the time-stepping is in terms of \\(\\theta\\).
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As noted in Celia1990_ the 'head'-based form of Richards
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equation can be written in the continuous form as:
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.. math::
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\frac{\partial \theta}{\partial \psi}\frac{\partial \psi}{\partial t} - \nabla \cdot k(\psi) \nabla \psi - \frac{\partial k(\psi)}{\partial z} = 0 \quad \psi \in \Omega
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However, it can be shown that this does not conserve mass in the discrete formulation.
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Here we reproduce the results from Celia1990_ demonstrating the head-based formulation and the mixed-formulation.
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.. _Celia1990: http://www.webpages.uidaho.edu/ch/papers/Celia.pdf
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.. plot::
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from SimPEG import Examples
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Examples.FLOW_Richards_1D_Celia1990.run()
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.. literalinclude:: ../../../SimPEG/Examples/FLOW_Richards_1D_Celia1990.py
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:language: python
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:linenos:
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