.. _doc.sedim.formul: Formulations of sediment module SEDI_MARS3D -------------------------------------------- .. image:: FIG/SEDIM/sedim_fig3.jpg * :ref:`doc_erosion` :math:`=f_1(\tau_s,\tau_e, composition\ density\ of\ sediment\ surface)` * :ref:`doc_depot` :math:`=W_s . f_2(\tau_s,\tau_d). C_{bottom\ extrapolated}` * :ref:`doc_settling` :math:`W_s=f_3(C,turbulence,salinity)` * :ref:`doc_erosion` :math:`\tau_e=f_4(consolidation)` * :ref:`doc_floculation` * :ref:`doc_consolidation` * :ref:`doc_diffused` * :ref:`doc_bioturb` (not operational) | | * There are 2 options (2 strategies) : **key_sedim_mudsand** or **key_sedim_mixsed**. but, this last option **key_sedim_mixsed is highly recommended** and the only one presented in this documentation. The new module **MUSTANG** (which will replace SEDI-MARS) only use the option **key_sedim_mixsed** (:ref:`doc.module.MUSTANG`). * The different computation steps for sediment module are : * 1. computing bottom shear stress :math:`=\tau_s` f(U,waves, roughness)(:ref:`doc_erosion`) * 2. computing settling rate in water column (:ref:`doc_settling`) and (:ref:`doc.sedim.ws`) * 3. computing settling trends (:ref:`doc_settling`) * 4. extrapolation of sand concentration near bottom (:ref:`doc_sedim_extrap`) * 5. sediment consolidation and diffusion into sediment layers (:ref:`doc_consolidation` and :ref:`doc_diffused`) * 6. computing total mass eroded and suspending (:ref:`doc_erosion`) * 7. after particules transport in water column, computing of effective deposit, updating sediment layers (:ref:`doc_depot`) | see :ref:`doc.sedim.organig`) .. _doc_erosion: Erosion Process +++++++++++++++ * **Erosion flux E** : :math:`E=E_0*\alpha*[\frac{\tau-\tau_{ce}}{\tau_{ce}}]^n` * :math:`E_0` and n depend on the mud fraction in sediment * :math:`fr_{mud} < frv1` : sandy sediment (frv1=0.3 typically = *frmudcr1* calculated from *coef_frmudcr1* in *parasedim.txt*) * :math:`fr_{mud} > frv2` : muddy sediment (frv2=0.7 typically = *frmudcr2* given in *parasedim.txt*) * the **critical erosion shear stress** of erosion :math:`\tau_{ce}` depends on the content of sand / mud and on sediment consolidation. It may be modulated by heterometry (hide / show) * :math:`\alpha` is a correction factor * :math:`\alpha=1.+corfluer1*\max(0,corfluer2-Csed_{total-surf})` (*corfluer1* et *corfluer2* given in *parasedim.txt*) * For muddy sediment : * :math:`E_0=E0_{mud}` (given in *parasedim.txt*) * :math:`\tau_{ce}=x1_{toce}* C_{mud}^{neros_{mud}}` (parameters given in *parasedim.txt*) * For sandy sediment : * :math:`E_0=E0_{sand}` * :math:`E0_{sand}` given in *parasedim.txt* or estimated by formulations depending on the option chosen * *E0_sand_option* = 0 ==> E0_sand= E0_sand_para read in this namelist *parasedim.txt* * *E0_sand_option* = 1 ==> E0_sand evaluated with Van Rijn (1984) formulation * *E0_sand_option* = 2 ==> E0_sand evaluated with erodimetry formulation :math:`E0_{sand}=\min(0.27,1000*d50-0.01)*(\tau-\tau_{ce})^{n_{eros-sand}}` * :math:`\tau_{ce}=` estimated from sand characteristics * For sand/mud mixing : Erosion flux depends on proportions of the mixture .. note:: Several options are available but all are questionable and should be tested and used carefully * The option choice is given by *ero_option* in *parasedim.txt* * *ero_option* = 0 : pure mud behavior (for all particles and whatever the mixture) * *ero_option* = 1 : linear interpolation between sand and mud behavior, depend on proportions of the mixture * *ero_option* = 2 : formulation derived from that of J. Vareilles (2013) * *ero_option* = 3 : formulations proposed by B. Mengual (2015) with exponential coefficients depend on proportions of the mixture * Erosion flux is applied in proportion to the mass of the considered fraction of sediment * A lateral erosion process of a wet or dry cell was introduced into the model. * Parameters *coef_erolat*, *l_erolat_wet_cell* and *coef_tenfon_lat* are used for this option., given in *parasedim.txt* +--------------------------------------------------------------------------------------+--------------------------------------------+ | | **dry cell** : | .. image:: FIG/SEDIM/lateral_erosion.jpg | | | :math:`Ero_{lat}=(\alpha_{lat} * U_{neighb}^2-\tau_{ce})*H_{neighb}` | | | | | | | | **wet cell** (if *l_erolat_wet_cell = .TRUE.*) | | | | :math:`Ero_{lat}=(\beta_{lat} * \alpha_{lat} * U_{neighb}^2-\tau_{ce})*\delta_H` | | | | | | | | with :math:`\alpha_{lat}` =coef_tenfon_lat | | | | with :math:`\beta_{lat}` =coef_erolat | | | | with :math:`U_{neighb}` =water current in the neigboring cell | | | | with :math:`H_{neighb}` =depth in the neighboring cell | | | | with :math:`\delta_H` =depth difference between eroded cell and the neighboring | | +--------------------------------------------------------------------------------------+--------------------------------------------+ .. _doc_depot: Deposit Process +++++++++++++++ * first deposit of gravels, then sands, then muds * deposit is caracterised by the mass fraction of each variable ( :math:`f_{rdep}(iv)` ) .. image:: FIG/SEDIM/sedim_deposit.jpg * A sliding process of the fluid mud was introduced into the model; mud slides if the slope is steep and deposits towards lower neighboring cells according to the slope .. image:: FIG/SEDIM/glissement_vase.jpg .. _doc_settling: Setlling Process +++++++++++++++++ * Settling process depend on settling velocity, which varies according to the particulate material. * a modelling strategy is proposed with a choice of some formulations for defining the settling velocity W_s for **MUD variables**. (:ref:`doc.sedim.ws`). * SAND variable have settling velocities W_s wich depend on particle diameter. * None constitutive SORBED variables have the same settling velocity as the associated constitutives particulate variable. * For the none constitutive variables which are not sorbed on constitutive particulate variable (type NoCP), settling velocity is evaluated in the same manner as MUD variables. * Settling processes are treated implicitly in water transport equations. | First, deposition flux trends are evaluated for each variable before advection computations : | :math:`Fl_{w2s}^a= W_s . f_2(\tau_s,\tau_d)` (m/s) | Then after advection resolution, effective deposit is computed with new concentrations in water | :math:`Fl_{w2s}^b= Fl_{w2s}^a*C_{bottom}` (Masse/m2/s) | and sediment layers are updated * For variables which have high settling velocity, vertical transport is computed using several sub time step in order to avoid instabilities. .. _doc_sedim_extrap: Extrapolation of sand variable near bottom (Rouse profile) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ .. _doc_sandbottom: Treatment of sand as 2D variable in water column (*key_sandbottomcell*) +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ .. _doc_diffused: Diffusion within sediment and at interface ++++++++++++++++++++++++++++++++++++++++++++ * **Diffusion into the sediment** is computed resolving following equations 1 (after :ref:`doc_consolidation` 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 *parasedim.txt*) * *choice_flx_diffsed given* =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) .. _doc_bioturb: Bioturbation within sediment +++++++++++++++++++++++++++++ * Bioturbation is the result of all activities of the meio and macro-fauna living in the water-sediment interface or in the upper layers of the sediment. * Bioturbation processes include : * "surface biodiffusion" | resulting of benthic organisms living in the first centimeters of the sediment. | Their movement causes the mechanical homogenization of the substrate and randomly in three dimensions; * "bioirrigation" generated by agencies that construct galleries or burrows in the sediment. These biodiffuseurs gallery provide irrigation sediment creating water currents to respiratory and food purposes. * "bioadvection" induced by organisms that ingest sediment particles in depth (anoxic zone) and discharge their fecal pellets on the sediment surface. This transport facing up induces a direct link between two non-adjacent geochemical and different strata. The bioadvection may also be downwardly directed * **But in a first step, bioturbation is considered here only as an apparent biodiffusion mixing coefficient Db**. The coefficient Db at each point and in each sediment layer is evaluated based on an average profile with a maximum bioturbation intensity (Db1 in m^2.s^-1) near the interface to a specified depth (zdbm, m), then a decrease in depth according to a slope coefficient (xdb2 without unit) to a maximum depth of bioturbation (zdb0 in m), beyond which the bioturbation is considered null (Figure) .. image:: FIG/SEDIM/fig_bioturb_Db.jpg .. note:: These processes are available in SEDIMARS but **not operational**. They mus be tested and developped by researchers. * For dissolved substances, bioturbation processes are resolved simultaneously with the diffusion process in the sediment (diffusion routine) * For particulate substances, * either bioturbation processes are not involved * or (to be checked) resolved simultaneously with the consolidation process, but explicitly in time with fractionnary time step; and if consolidation is not taking account, bioturbation for particulate variables are resolved alone in the consolidation routine. * Parameters for bioturbation are given in parasedim.txt (:ref:`doc_parasedim`) | **l_bioturb** : boolean set to .true. si taking into account bioturbation diffusion in sediment | **xbioturbmax** : max diffusion coefficient by bioturbation Db (in surface) | **xbioturbk** : coef (slope) for bioturbation coefficient between max Db at sediment surface and 0 at bottom | **dbiotu0** : max depth beneath the sediment surface below which there is no bioturbation | **dbiotum** : sediment thickness wherein the diffusion-bioturbation coefficient Db is constant (max) | **frmud_db_min** : mud fraction limit (min) below which there is no Bioturbation | **frmud_db_max** : mud fraction limit (max) above which the bioturbation coefficient Db is maximum (muddy sediment) .. _doc_consolidation: Consolidation of sediment +++++++++++++++++++++++++ * The mixed-sediment consolidation model, detailed in Grasso et al. (submitted), 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. .. _doc_morphocoupl: Morphodynamic coupling ++++++++++++++++++++++ .. _doc_floculation: Floculation +++++++++++