.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "getting_started/02_Bounds.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_02_Bounds.py>`
        to download the full example code

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

.. _sphx_glr_getting_started_02_Bounds.py:


Elastic bounds
===============

This example shows the comparison between different elastic bounds to Nur's cirtical porosity model: Voigt and Reuss Bounds, Voigt–Reuss–Hill Average,Hashin-Shtrikmann bounds 

.. GENERATED FROM PYTHON SOURCE LINES 8-16

.. 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 17-22

.. code-block:: python3

    import rockphypy # import the module for rock physics
    from rockphypy import EM # import the "effective medium" EM module 
    # import the 'Fluid' module 
    from rockphypy import Fluid








.. GENERATED FROM PYTHON SOURCE LINES 23-73

Voigt and Reuss Bounds
~~~~~~~~~~~~~~~~~~~~~~
At any given volume fraction of constituents, the effective modulus will fall between the bounds. In the lecture we learn that the **Voigt upper bound** :math:`M_v` of N phases are defined as:

.. math::
      M_V=\sum_{i=1}^{N}f_iM_i 


The **Reuss lower bound** is 

.. math::
       \frac{1}{M_R} =\sum_{i=1}^{N}\frac{f_i}{M_i}


where :math:`f_i` is the volume fraction of the ith phase and :math:`M_i` is the elastic bulk or shear modulus of the ith phase.


Voigt–Reuss–Hill Average Moduli Estimate
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 This average is simply the arithmetic average of the Voigt upper
bound and the Reuss lower bound.

.. math::
      M_{VRH}=\frac{M_V+M_R}{2}


:bell: 
  For fluid-solid composite, calculation of Reuss, Voigt and VRH bounds are straightforward. Here we give a generalized function to compute effective moduli of N-phases composite: 


Hashin-Shtrikmann bounds
~~~~~~~~~~~~~~~~~~~~~~~~
The Voigt and Reuss bounds are simply arithmetic and harmonic averages. The **Hashin-Shtrikman bounds** are stricter than the Reuss-Voigt bounds. The *two-phase* HS bounds can be written as: 

.. math::
      K^{\mathrm{HS} \pm}=K_{1}+\frac{f_{2}}{\left(K_{2}-K_{1}\right)^{-1}+f_{1}\left(K_{1}+\frac{4}{3} \mu_{1}\right)^{-1}}


.. math::
      \mu^{\mathrm{HS} \pm}=\mu_{1}+\frac{f_{2}}{\left(\mu_{2}-\mu_{1}\right)^{-1}+2 f_{1}\left(K_{1}+2 \mu_{1}\right) /\left[5 \mu_{1}\left(K_{1}+\frac{4}{3} \mu_{1}\right)\right]}


where the superscript +/− indicates upper or lower bound respectively. :math:`K_1` and :math:`K_2` are the bulk moduli of individual phases; :math:`\mu_1` and :math:`\mu_2` are the shear moduli of individual phases; and :math:`f_1` and :math:`f_2` are the volume fractions of individual phases. The upper and lower bounds are computed by interchanging which material is termed 1 and which is termed 2. The expressions yield the upper bound when the stiffest material is termed 1 and the lower bound when the softest material is termed 1. The expressions shown in the lectures represents the HS bound for *fluid-solid composite*. 

Examples
~~~~~~~~
Let's compute effective bulk and shear moduli of a water saturated rock using different bound models. 

Here we also make a comparision with Nur's critical porosity model as introduced in the previous example. 


.. GENERATED FROM PYTHON SOURCE LINES 76-95

.. code-block:: python3



    # specify model parameters
    phi=np.linspace(0,1,100,endpoint=True) # solid volume fraction = 1-phi
    K0, G0= 37,44 # moduli of grain material
    Kw, Gw= 2.2,0 # moduli of water 
    # VRH bounds
    volumes= np.vstack((1-phi,phi)).T
    M= np.array([K0,Kw])
    K_v,K_r,K_h=EM.VRH(volumes,M)
    # Hashin-Strikmann bound 
    K_UHS,G_UHS= EM.HS(1-phi, K0, Kw,G0,Gw, bound='upper')
    # Critical porosity model
    phic=0.4 # Critical porosity
    phi_=np.linspace(0.001,phic,100,endpoint=True) # solid volume fraction = 1-phi
    K_dry, G_dry= EM.cripor(K0, G0, phi_, phic)# Compute dry-rock moduli
    Ksat, Gsat = Fluid.Gassmann(K_dry,G_dry,K0,Kw,phi_)# saturate rock with water









.. GENERATED FROM PYTHON SOURCE LINES 96-112

.. code-block:: python3



    # plot
    plt.figure(figsize=(6,6))
    plt.xlabel('Porosity')
    plt.ylabel('Bulk modulus [GPa]')
    plt.title('V, R, VRH, HS bounds')
    plt.plot(phi, K_v,label='K Voigt')
    plt.plot(phi, K_r,label='K Reuss = K HS-')
    plt.plot(phi, K_h,label='K VRH')
    plt.plot(phi, K_UHS,label='K HS+')
    plt.plot(phi_, Ksat,label='K CriPor')
    plt.legend(loc='best')
    plt.grid(ls='--')





.. image-sg:: /getting_started/images/sphx_glr_02_Bounds_001.png
   :alt: V, R, VRH, HS bounds
   :srcset: /getting_started/images/sphx_glr_02_Bounds_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 113-115

**Reference**: Mavko, G., Mukerji, T. and Dvorkin, J., 2020. The rock physics handbook. Cambridge university press.



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

   **Total running time of the script:** (0 minutes 0.189 seconds)


.. _sphx_glr_download_getting_started_02_Bounds.py:

.. only:: html

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




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

      :download:`Download Python source code: 02_Bounds.py <02_Bounds.py>`

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

      :download:`Download Jupyter notebook: 02_Bounds.ipynb <02_Bounds.ipynb>`


.. only:: html

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

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