# 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:

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

with

\(\triangle (load)=C \frac{\rho_s - \rho_w}{\rho_s} + \frac{1}{g} \frac{\partial \sigma'}{\partial z}\)

- where :
- C is the sediment total mass concentration, assuming the same grain density \(\rho_s\) for all sediment classes
*i*,*k*is the permeability (m/s),\(\rho_w\) is the water density,*g*the gravity\(\sigma'\) 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 \(Ws_{si}\) 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:

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

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

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

- with :
- \(C_{mud}\) the mass concentration of mud (clay and silt)\(\varphi_{sand}\) the volumetric concentration of sand (grain diameter > 63 µm), to express the hindered settling of sand class
*si*as:\(Ws_{si,hindered}=Ws_{si} [1-\frac{C_{relmud}}{C_{relmud_{crit}}}]^p\) (Equation 5)

- where :
- \(Ws_{si}\) 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.\(C_{relmud_{crit}}\) 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:

\(k_e=k_1 e^{k_z}\) (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:

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

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

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

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

where \(k_1, k_2, K_k, K_d, n\) are empirical parameters.