.. _doc.MUSTANG.consol:

Consolidation Process
+++++++++++++++++++++
 
* The mixed-sediment consolidation model, detailed in Grasso et al. (2015), is based on Toorman’s (1996) unifying theory for sedimentation and consolidation of several classes of sediment. 
* Following Merckelbach’s derivation of Gibson equation, and using as state variable the mass concentration of each sediment class Ci, the mass conservation equation during consolidation can be written as:

   :math:`\frac{\partial C_i}{\partial t}=+\frac{\partial}{\partial z} [\frac{k}{\rho_w}C_i \triangle (load)]`     (Equation 2)

with

   :math:`\triangle (load)=C \frac{\rho_s - \rho_w}{\rho_s} + \frac{1}{g} \frac{\partial \sigma'}{\partial z}`     
       
where :
     | C is the sediment total mass concentration, assuming the same grain density :math:`\rho_s` for all sediment classes *i*, 
     | *k* is the permeability (m/s), 
     | :math:`\rho_w` is the water density, 
     | *g* the gravity 
     | :math:`\sigma'` the effective stress. 

* In order to account for segregation due to polydispersity during sedimentation, the sand settling velocity was chosen as the maximum between the sedimentation rate in Eq.2 and the hindered settling velocity :math:`Ws_{si}` hindered of the sand class *si* considered. 
* The mud fraction, however, is only driven by the sedimentation rate in Eq. 2, so that finally the following equation 3 is solved:

   :math:`\frac{\partial C_i}{\partial t}=+\frac{\partial}{\partial z} [C_i MAX(\frac{k}{\rho_w}C_i \triangle (load),Ws_{si,hindered}]`   (Equation 3)

* We used a segregation formulation based on the relative mud concentration (:math:`C_{relmud}`):

   :math:`C_{relmud}=\frac{C_{mud}}{1-\varphi_{sand}}=\varphi_{relmud} \rho_s`   (Equation 4)

with :
     | :math:`C_{mud}` the mass concentration of mud (clay and silt) 
     | :math:`\varphi_{sand}` the volumetric concentration of sand (grain diameter > 63 µm), to express the hindered settling of sand class *si* as:

 :math:`Ws_{si,hindered}=Ws_{si} [1-\frac{C_{relmud}}{C_{relmud_{crit}}}]^p`   (Equation 5)

where :
    | :math:`Ws_{si}` is the non-hindered settling velocity estimated by Souslby's (1997) formulation and the power *p* is defined as 4.65 according to Richardson and Zaki's (1954) observations. 
    | :math:`C_{relmud_{crit}}` is an empirical parameter calibrated in order that the sand settling becomes hindered by fine (muddy) particles when their relative concentration get close to a threshold value.

* The resolution of Eq.5 requires the specification of two constitutive relationships for the permeability and the effective stress, respectively (e.g. Alexis et al. 1992; Toorman 1999).
* The permeability constitutive relationship is computed in coupling two formulations.
* The first is related to the void ratio e (e.g. Bartholomeeusen et al. 2002; Le Hir et al. 2011), which reads:

   :math:`k_e=k_1 e^{k_z}`   (Equation 6)

* and the second is related to the relative volume fraction of fine particles relmud (see Eq.5), based on the fractal theory presented by Merckelbach and Kranenburg (2004), expressed as:

   :math:`k_{\varphi}=K_k \varphi \stackrel{-n}{relmud}`   (Equation 7)

with 
    | :math:`n = \frac{2}{3-n_f}` 
    | and :math:`n_f` is the fractal number that characterizes the distribution of solids in the sediment. 

* Similarly, this fractal theory enabled to compute the effective stress as:

   :math:`\sigma'=K_d \varphi \stackrel{-n}{relmud}`   (Equation 8)

where :math:`k_1, k_2, K_k, K_d, n` are empirical parameters.