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.. _sphx_glr_getting_started_01_Critical_Porosity_and_Gassmann_Fluidsubtitution.py:


Critical porosity model and Gassmann Fluid substitution
=======================================================

This pages gives a simple examples of using the classes ``EM`` and ``Fluid`` of ``rockphypy`` to compute Critical porosity model and perform Gassmann Fluid substitution. 

.. GENERATED FROM PYTHON SOURCE LINES 8-15

.. code-block:: python3


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








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.. 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 23-38

Critical porosity model
~~~~~~~~~~~~~~~~~~~~~~~

Remember Nur’s hypothesis: There is a critical (structure-dependent)
porosity at which the framework stiffness goes to zero. The simple yet 
powerful critical porosity model is defined as: 

.. math::
        K_{\text {dry }}=K_{0}\left(1-\frac{\phi}{\phi_{\mathrm{c}}}\right)

.. math::
        \mu_{\text {dry }}=\mu_{0}\left(1-\frac{\phi}{\phi_{\mathrm{c}}}\right)

where :math:`K_0` and :math:`\mu_0` are the mineral bulk and shear moduli.


.. GENERATED FROM PYTHON SOURCE LINES 41-65

Gassmann Fluid substitution
~~~~~~~~~~~~~~~~~~~~~~~~~~~
Compute dry frame moduli using rock physics models such as critical porosity 
model is straightforward as shown above. To compute saturated elastic moduli, Gassmann's explicit equations for fluid substition can be applied. Gassmann's relations not only allows us to perform fluid 
substition (when one fluid is replaced
with another), but to predict saturated-rock moduli from dry-rock moduli, and # vice versa. 

Here we show how to compute saturated moduli via Gassmann thoery. 

The equation we are gonna use is 

.. math::
        K_{\text {sat }}=K_{\text {dry }}+\frac{\left(1-K_{\text {dry }} / K_{0}\right)^{2}}{\phi / K_{\mathrm{fl}}+(1-\phi) / K_{0}-K_{\text {dry }} / K_{0}^{2}}


This is one of the equivelent expressions of the well known GS-Biot theory for fluid substitution:

.. math::
        \frac{K_{\text {sat }}}{K_{0}-K_{\text {sat }}}=\frac{K_{\text {dry }}}{K_{0}-K_{\text {dry }}}+\frac{K_{\text {fl }}}{\phi\left(K_{0}-K_{\text {fl }}\right)}, \quad \mu_{\text {sat }}=\mu_{\text {dry }}


Now Let's firstly estimate the effective dry-rock and water saturated moduli # using critical porosity model. Assume that the rock is Arenite with 40% 
critical porosity.


.. GENERATED FROM PYTHON SOURCE LINES 68-76

.. code-block:: python3


    # specify model parameters
    phic=0.4
    phi=np.linspace(0.001,phic,100,endpoint=True) # solid volume fraction = 1-phi
    K0, G0= 37,44
    Kw = 2.2
    Kg = 0.5








.. GENERATED FROM PYTHON SOURCE LINES 77-83

.. code-block:: python3


    # Compute dry-rock moduli
    K_dry, G_dry= EM.cripor(K0, G0, phi, phic)
    # saturate rock with water 
    Ksat, Gsat = Fluid.Gassmann(K_dry,G_dry,K0,Kw,phi)








.. GENERATED FROM PYTHON SOURCE LINES 84-97

.. code-block:: python3


    # plot
    # sphinx_gallery_thumbnail_number = 1
    plt.figure(figsize=(6,6))
    plt.xlabel('Porosity')
    plt.ylabel('Bulk modulus [GPa]')
    plt.title('V, R, VRH, HS bounds')
    plt.plot(phi, K_dry,label='dry rock K')
    plt.plot(phi, Ksat,label='saturated K')

    plt.legend(loc='best')
    plt.grid(ls='--')




.. image-sg:: /getting_started/images/sphx_glr_01_Critical_Porosity_and_Gassmann_Fluidsubtitution_001.png
   :alt: V, R, VRH, HS bounds
   :srcset: /getting_started/images/sphx_glr_01_Critical_Porosity_and_Gassmann_Fluidsubtitution_001.png
   :class: sphx-glr-single-img





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We can see from the figure that effective bulk modulus increases when the 
rock is saturated.



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   **Total running time of the script:** (0 minutes 0.592 seconds)


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