.. _examples_FLOW_Richards_1D_Celia1990:

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..    SimPEG/Examples/__init__.py    ..
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FLOW: Richards: 1D: Celia1990
=============================

There are two different forms of Richards equation that differ
on how they deal with the non-linearity in the time-stepping term.

The most fundamental form, referred to as the
'mixed'-form of Richards Equation Celia1990_

.. math::

    \frac{\partial \theta(\psi)}{\partial t} - \nabla \cdot k(\psi) \nabla \psi - \frac{\partial k(\psi)}{\partial z} = 0
    \quad \psi \in \Omega

where \\(\\theta\\) is water content, and \\(\\psi\\) is pressure head.
This formulation of Richards equation is called the
'mixed'-form because the equation is parameterized in \\(\\psi\\)
but the time-stepping is in terms of \\(\\theta\\).

As noted in Celia1990_ the 'head'-based form of Richards
equation can be written in the continuous form as:

.. math::

    \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

However, it can be shown that this does not conserve mass in the discrete formulation.

Here we reproduce the results from Celia1990_ demonstrating the head-based formulation and the mixed-formulation.

.. _Celia1990: http://www.webpages.uidaho.edu/ch/papers/Celia.pdf


.. plot::

    from SimPEG import Examples
    Examples.FLOW_Richards_1D_Celia1990.run()

.. literalinclude:: ../../../SimPEG/Examples/FLOW_Richards_1D_Celia1990.py
    :language: python
    :linenos: