.. _doc.MUSTANG.diffu:

Diffusion within sediment and at interface
++++++++++++++++++++++++++++++++++++++++++++

* **Diffusion into the sediment** is computed resolving following equations 1 (after :ref:`doc.MUSTANG.consol` processes or not)

   | :math:`\frac{\partial (dzs_k*\varphi_k*Cs_k)}{\partial t}=+F_k-F_{k-1}`     (Equation 1A)
   |
   | :math:`F_k=\phi i_k^{t+dt}*Ds_k*\frac{Cs_{k+1}^{t+dt}-Cs_{k}^{t+dt}}{dzsi_k}-Wsi_k*(c_{ex}Cs_{k+1}^{t+dt}+f_{ex}Cs_{k}^{t+dt})`     (Equation 1B)   
   
* with :
   * :math:`Cs_k^{t+dt}` : dissolved substance concentration into layer k and at time t+dt (Mass/m^3[pw] [pw:pore water])
   * :math:`F_k` : Flux at interface between layer k and layer k+1 (Mass/T/m^2[sed])
   * :math:`dzs_k` : thickness of layer k (m[sed])
   * :math:`dzsi_k` : intermediate thickness between Cs_k and Cs_{k+1} (m[sed])
   * :math:`\varphi_k^{t}` : porosity into the layer k at time t (m^3[pw]/m^3[sed])
   * :math:`\phi i_k^{t}` : intermediate porosity at the interface between the layer k and the layer k+1 at time t (m^3[pw]/m^3[sed])
   * :math:`Ds_k` : Effective dispersion coefficient (m^2[sed]/T), corrected by tortuosity :math:`\theta` (see Eq 1C)
   * :math:`Wsi_k` : transfert rate of pore water at the interface between the layer k and the layer k+1  [m^3[pw]/m^2[sed]/T)
   * :math:`f_{ex},c_{ex}` : factors upstream decentering for advection (:math:`f_{ex}=1` : completely upstream evaluation)
  
   * :math:`Ds_k=\frac{D_0}{\theta^2}` and :math:`\theta^2=1-\ln(\varphi^2)=1-2*ln(\varphi)` (Boudreau,1997)
   
.. image:: FIG/SEDIM/diffusion_sed1.jpg

* **At bottom** , F(k-1)=0. 
* **At sediment surface** (water-sediment interface) (k=ksmax) : 
   * :math:`F_k=\beta (C_{wat}^{t+dt}-Cs_k^{t+dt}) - Wsi_k*Cs_k^{t+dt}`
   * Three expressions are proposed here (dependent of **choice_flx_diffsed** given in *paraMUSTANG.txt*) 
		* **choice_flx_diffsed =1** : Fick law with diffusive sublayer supposed to be = half the thickness of the bottom layer 
		        * ==>> :math:`\beta=2.*\frac{Ds_k}{dzsi_{ksmax}+dz(1)}`  
		* **choice_flx_diffsed given =2** : Fick law with diffusive sublayer supposed to be = distance epdifi (given in *parasedim.txt*)
				* ==>> :math:`\beta=\frac{Ds_k}{0.5*dzsi_{ksmax}+epdifi}`  
		* **choice_flx_diffsed given =3** : formulation proposed by Boudreau (1997)
				* ==>> :math:`\beta=\frac{Ds_k}{\delta_e}`  
				* :math:`\delta_e`  : thickness of an effective diffusive sublayer that has a completely linear gradient and has the same flux as the real diffusive sublayer of thickness :math:`\delta_e` (see fig)
				* :math:`\beta=0.0889 U_* Sc^{-0.704}`  : formulation of Shaw and Hanratty (1977)  in Boudreau (1997) (U_* is the shear velocity and Sc is the Schmidt number)