.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/hydrology/plot_05_dupuit_unconfined_flow.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_hydrology_plot_05_dupuit_unconfined_flow.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_hydrology_plot_05_dupuit_unconfined_flow.py:


Dupuit's unconfined flow (1863)
===============================

Jules Dupuit assumed that groundwater beneath a gently sloping water table
moves horizontally with a velocity uniform on each vertical, driven by the
slope of the water table. Darcy's law then gives a discharge proportional to
the thickness times the slope, which is constant between two rivers, so the
square of the water-table height varies linearly and the profile is a
parabola. The example draws the profile and checks that every section carries
the same discharge.

.. GENERATED FROM PYTHON SOURCE LINES 13-34



.. image-sg:: /api/gallery/hydrology/images/sphx_glr_plot_05_dupuit_unconfined_flow_001.png
   :alt: Dupuit: a parabolic water table (dotted: straight lines)
   :srcset: /api/gallery/hydrology/images/sphx_glr_plot_05_dupuit_unconfined_flow_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.hydrology import darcy_flux, dupuit_discharge, dupuit_head

    length, conductivity = 500.0, 2e-4
    x = np.linspace(0, length, 501)
    fig, ax = plt.subplots()
    for downstream in (2.0, 8.0, 14.0):
        head = dupuit_head(x, 20, downstream, length)
        section = darcy_flux(conductivity, np.gradient(head**2, x)) / 2  # h dh/dx = d(h^2)/dx / 2
        discharge = dupuit_discharge(conductivity, 20, downstream, length)
        assert np.allclose(section[1:-1], discharge, rtol=1e-9)
        ax.plot(x, head, label=f"q' = {discharge * 86400:.2f} m^2/day")
        ax.plot(x, 20 + (downstream - 20) * x / length, ":", color="0.6")
    ax.set(xlabel="distance from the upstream river [m]", ylabel="water-table height above the base [m]")
    ax.set_title("Dupuit: a parabolic water table (dotted: straight lines)")
    ax.legend()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.101 seconds)


.. _sphx_glr_download_api_gallery_hydrology_plot_05_dupuit_unconfined_flow.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../lite/lab/index.html?path=api/gallery/hydrology/plot_05_dupuit_unconfined_flow.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_dupuit_unconfined_flow.ipynb <plot_05_dupuit_unconfined_flow.ipynb>`

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

      :download:`Download Python source code: plot_05_dupuit_unconfined_flow.py <plot_05_dupuit_unconfined_flow.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_05_dupuit_unconfined_flow.zip <plot_05_dupuit_unconfined_flow.zip>`


.. only:: html

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

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