.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "getting_started/09_Berrymann_SC.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_getting_started_09_Berrymann_SC.py>`
        to download the full example code

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

.. _sphx_glr_getting_started_09_Berrymann_SC.py:


Berryman Self Consistent 
========================
Berryman Self consistent approach for calculating the elastic moduli of N phase composite

.. GENERATED FROM PYTHON SOURCE LINES 8-15

.. code-block:: python3


    import numpy as np 
    import matplotlib.pyplot as plt
    plt.rcParams['font.size']=14
    plt.rcParams['font.family']='arial'









.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: python3


    import rockphypy # import the module 
    from rockphypy import EM 
    from rockphypy import Fluid








.. GENERATED FROM PYTHON SOURCE LINES 22-31

Berryman's self-consistent approximations for N-phase composites are described by the equations: 

.. math::
       \sum_{\mathrm{i}=1}^N x_{\mathrm{i}}\left(K_{\mathrm{i}}-K_{\mathrm{SC}}^{e f f}\right) P^{e f f \mathrm{i}}=0 

.. math::
        \sum_{\mathrm{i}=1}^N x_{\mathrm{i}}\left(\mu_{\mathrm{i}}-\mu_{\mathrm{SC}}^{e f f}\right) Q^{e f f \mathrm{i}}=0



.. GENERATED FROM PYTHON SOURCE LINES 33-38

The question is as follows: Calculate the self-consistent effective bulk and shear moduli, :math:`K_{eff}^{SC}` and :math:`G_{eff}^{SC}`, for a water-saturated rock consisting of spherical quartz grains (aspect ratio :math:`\alpha` = 1) and total porosity 0.3. The pore space consists of spherical pores :math:`\alpha` = 1 and thin penny-shaped cracks (:math:`\alpha` = 1e^−2). The thin cracks have a porosity of 0.01, whereas the remaining porosity (0.29) is made up of the spherical pores.


There are three phases in the composite, i.e. quartz grain, water filled spherical pore, and water filled thin cracks. We can use the method ``EM.Berryman_sc`` to easily solve this exercise to find the effective moduli of the composite.


.. GENERATED FROM PYTHON SOURCE LINES 41-55

.. code-block:: python3


    # three phase K=[K1,K2,K3]
    K=[37,2.25,2.25]
    # G=[G1,G2,G3]
    G=[44,0,0]
    # X=[frac1, frac2, frac3]
    X=[0.7,0.29,0.01]
    # Alpha= [alpha1, alpha2, alpha3]
    Alpha=[1,1,0.01]

    # effective moduli
    K_sc,G_sc= EM.Berryman_sc(K,G,X,Alpha)









.. GENERATED FROM PYTHON SOURCE LINES 56-60

.. code-block:: python3



    print('K_eff and G_eff of the composite are {:.2f} GPa and {:.2f} GPa, respectively'.format(K_sc,G_sc))





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

 .. code-block:: none

    K_eff and G_eff of the composite are 16.80 GPa and 11.56 GPa, respectively




.. GENERATED FROM PYTHON SOURCE LINES 61-63

The effective moduli of the two phase composite as a function of the volume fraction of the soft fluid filled crack can be calculated as follow:  


.. GENERATED FROM PYTHON SOURCE LINES 63-77

.. code-block:: python3


    # two phase K=[K1,K2]
    K=[37,2.25]
    # G=[G1,G2]
    G=[44,0,]
    Alpha=[1,0.1]

    frac = np.linspace(0,1,50)
    K_eff = np.zeros(frac.size)
    G_eff = np.zeros(frac.size)
    for i, val in enumerate(frac):
        X=[1-val, val]
        K_eff[i],G_eff[i]= EM.Berryman_sc(K,G,X,Alpha)








.. GENERATED FROM PYTHON SOURCE LINES 78-87

.. code-block:: python3


    # sphinx figure 
    # sphinx_gallery_thumbnail_number = 1
    plt.figure(figsize=(5,5))
    plt.plot(frac,K_eff,'-k',lw=3,label='K_eff')
    plt.plot(frac,G_eff,'-b',lw=2,label='G_eff')
    plt.xlabel('Volume fraction of soft phase')
    plt.ylabel('Effective modulus')
    plt.legend()



.. image-sg:: /getting_started/images/sphx_glr_09_Berrymann_SC_001.png
   :alt: 09 Berrymann SC
   :srcset: /getting_started/images/sphx_glr_09_Berrymann_SC_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x7f70f23f97f0>



.. GENERATED FROM PYTHON SOURCE LINES 88-91

As can be shown in the figure, the effective shear modulus of the two phase composite becomes 0 when the volume fraction of the soft phase is approximately 45% for an aspect ratio of 0.1. this prediction is very similar to the critical porosity model which predicts a suspension of grain in the fluid when the porosity exceeds about 0.4. 

However, feel free to change the aspect ratio for the soft phase from 0.1 to 0.01, then the modelling results of the effective shear modulus becomes zero when the volume fraction of the soft phase is approximately 12%. 


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

   **Total running time of the script:** (0 minutes 0.384 seconds)


.. _sphx_glr_download_getting_started_09_Berrymann_SC.py:

.. only:: html

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




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

      :download:`Download Python source code: 09_Berrymann_SC.py <09_Berrymann_SC.py>`

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

      :download:`Download Jupyter notebook: 09_Berrymann_SC.ipynb <09_Berrymann_SC.ipynb>`


.. only:: html

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

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