.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "content/examples/20-published/plot_richards_celia1990.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_content_examples_20-published_plot_richards_celia1990.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_content_examples_20-published_plot_richards_celia1990.py:


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

.. GENERATED FROM PYTHON SOURCE LINES 42-112



.. image-sg:: /content/examples/20-published/images/sphx_glr_plot_richards_celia1990_001.png
   :alt: Mixed Method, Head-Based Method
   :srcset: /content/examples/20-published/images/sphx_glr_plot_richards_celia1990_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    /home/vsts/work/1/s/simpeg/flow/richards/simulation.py:321: UserWarning:

    cell_gradient_BC is deprecated and is not longer used. See cell_gradient







|

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    import discretize
    from simpeg import maps
    from simpeg.flow import richards


    def run(plotIt=True):
        M = discretize.TensorMesh([np.ones(40)])
        M.set_cell_gradient_BC("dirichlet")
        params = richards.empirical.HaverkampParams().celia1990
        k_fun, theta_fun = richards.empirical.haverkamp(M, **params)
        k_fun.KsMap = maps.IdentityMap(nP=M.nC)

        bc = np.array([-61.5, -20.7])
        h = np.zeros(M.nC) + bc[0]

        def getFields(timeStep, method):
            timeSteps = np.ones(int(360 / timeStep)) * timeStep
            prob = richards.SimulationNDCellCentered(
                M,
                hydraulic_conductivity=k_fun,
                water_retention=theta_fun,
                boundary_conditions=bc,
                initial_conditions=h,
                do_newton=False,
                method=method,
            )
            prob.time_steps = timeSteps
            return prob.fields(params["Ks"] * np.ones(M.nC))

        Hs_M010 = getFields(10, "mixed")
        Hs_M030 = getFields(30, "mixed")
        Hs_M120 = getFields(120, "mixed")
        Hs_H010 = getFields(10, "head")
        Hs_H030 = getFields(30, "head")
        Hs_H120 = getFields(120, "head")

        if not plotIt:
            return
        plt.figure(figsize=(13, 5))
        plt.subplot(121)
        plt.plot(40 - M.gridCC, Hs_M010[-1], "b-")
        plt.plot(40 - M.gridCC, Hs_M030[-1], "r-")
        plt.plot(40 - M.gridCC, Hs_M120[-1], "k-")
        plt.ylim([-70, -10])
        plt.title("Mixed Method")
        plt.xlabel("Depth, cm")
        plt.ylabel("Pressure Head, cm")
        plt.legend(
            (r"$\Delta t$ = 10 sec", r"$\Delta t$ = 30 sec", r"$\Delta t$ = 120 sec")
        )
        plt.subplot(122)
        plt.plot(40 - M.gridCC, Hs_H010[-1], "b-")
        plt.plot(40 - M.gridCC, Hs_H030[-1], "r-")
        plt.plot(40 - M.gridCC, Hs_H120[-1], "k-")
        plt.ylim([-70, -10])
        plt.title("Head-Based Method")
        plt.xlabel("Depth, cm")
        plt.ylabel("Pressure Head, cm")
        plt.legend(
            (r"$\Delta t$ = 10 sec", r"$\Delta t$ = 30 sec", r"$\Delta t$ = 120 sec")
        )


    if __name__ == "__main__":
        run()
        plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 4.995 seconds)

**Estimated memory usage:**  9 MB


.. _sphx_glr_download_content_examples_20-published_plot_richards_celia1990.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_richards_celia1990.ipynb <plot_richards_celia1990.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_richards_celia1990.py <plot_richards_celia1990.py>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_