[Doc] Document the dispersion transport parametererizations

parent 0ded6c21
......@@ -168,16 +168,30 @@ porosity, :math:`\theta_s = \phi`.
Transport Parameterizations
^^^^^^^^^^^^^^^^^^^^^^^^^^^
The hydrodynamic dispersion tensor :math:`\mathsf{D}_\mathsf{hd} \,
[\mathrm{m}^2\mathrm{s}^{-1}]` is
Regarless the parameterization, the transport simulation always computes the
microscopic péclet number, for which it requires the characteristic pore length,
the molecular diffusion, and the fluid velocity. The latter is directly
provided by the richards simulation while the other two have to be specified
for each volume:
Permanent parameters:
* ``mol_diff``: Molecular diffusion
:math:`\mathsf{D}^\mathsf{m} \, [\mathrm{m}^2\mathrm{s}^{-1}]`
* ``char_length``: Characteristic pore length :math:`\ell \, [\mathrm{m}]`
The hydrodynamic dispersion tensor :math:`\mathsf{D}^\mathsf{hd} \,
[\mathrm{m}^2\mathrm{s}^{-1}]` is the main parameter to provide in the
transport simulation. Below you will find several parameterization for this.
.. object:: Constant parameterization
Implements the following parameterization function:
In this case, the hydrodynamic dispersion tensor is inserted directly
component-by-compoment. Altough the hydrodynamic tensor must be symmetric
because of the physics, DORiE does not check the symmetry of tensors.
.. math::
\mathsf{D}_\mathsf{hd} = \text{const}.
\mathsf{D}^\mathsf{hd}_{ij} = \text{const}.
* ``type: Dhd_const``
......@@ -193,6 +207,8 @@ The hydrodynamic dispersion tensor :math:`\mathsf{D}_\mathsf{hd} \,
transport:
type: Dhd_const
parameters:
mol_diff:
char_length:
hydrodynamic_disp_xx:
hydrodynamic_disp_xy:
hydrodynamic_disp_xz:
......@@ -203,13 +219,18 @@ The hydrodynamic dispersion tensor :math:`\mathsf{D}_\mathsf{hd} \,
hydrodynamic_disp_zy:
hydrodynamic_disp_zz:
(Alternative) YAML template:
A more readable alternative is possible. In this case, the
``hydrodynamic_disp`` key is a list of values containing all values
(:math:`\text{dim}\times\text{dim}`) of the tensor. The list is row-wise
oriented.
.. code-block:: yaml
transport:
type: Dhd_const
parameters:
mol_diff:
char_length:
hydrodynamic_disp: [ ... ]
.. object:: Power law parameterization
......@@ -233,26 +254,36 @@ The hydrodynamic dispersion tensor :math:`\mathsf{D}_\mathsf{hd} \,
transport:
type: Dhd_pl
parameters:
mol_diff:
char_length:
alpha:
gamma:
.. object:: Superposition parameterization
Implements the following parameterization function:
The hydrodynamic dispersion tensor is said to be the superposition of
several diffusion/dispersion processes. In DORiE this possible by summing
several valid parameterizations types. Currently we provide
parameterizations for the *effective diffusion*
:math:`\mathsf{D}^\mathsf{eff}` and for the *effective hydromechanic tensor*
:math:`\mathsf{D}^\mathsf{hm}` identified by the key prefixes ``Deff`` and
``Dhm`` respectively.
.. math::
\mathsf{D}_\mathsf{hd} = \mathsf{D}_\mathsf{hm}+I\mathsf{D}_\mathsf{eff}.
\mathsf{D}_\mathsf{hd} = \mathsf{D}^\mathsf{hm}+I\mathsf{D}^\mathsf{eff}.
* ``type: <Dhm_type> + <Deff_type>``
Each of the types are furthere parameterized and can be found below.
.. object:: Constant effective diffusion parameterization
Implements the following parameterization function:
In this case, the effective diffusion is inserted directly.
.. math::
\mathsf{D}_\mathsf{eff} = \text{const}.
\mathsf{D}^\mathsf{eff} = \text{const}.
* ``Deff_type: Deff_const``
......@@ -267,7 +298,10 @@ The hydrodynamic dispersion tensor :math:`\mathsf{D}_\mathsf{hd} \,
transport:
type: <Dhm_type> + Deff_const
parameters:
mol_diff:
char_length:
eff_diff:
# <Dhm_type> parameters ...
.. object:: Milligton-Quirk I effective diffusion parameterization
......@@ -275,7 +309,7 @@ The hydrodynamic dispersion tensor :math:`\mathsf{D}_\mathsf{hd} \,
.. math::
\mathsf{D}_\mathsf{eff} = \mathsf{D}_\mathsf{m}\frac{\theta_s^{7/3}}{\phi^{2/3}}.
\mathsf{D}^\mathsf{eff} = \mathsf{D}^\mathsf{m}\frac{\theta_s^{7/3}}{\phi^{2/3}}.
where the volumetric water content :math:`\theta_s \, [-]`
is automatically obtained from richards simulation.
......@@ -295,7 +329,9 @@ The hydrodynamic dispersion tensor :math:`\mathsf{D}_\mathsf{hd} \,
type: <Dhm_type> + Deff_MQ1
parameters:
mol_diff:
char_length:
phi:
# <Dhm_type> parameters ...
.. object:: Milligton-Quirk I effective diffusion parameterization
......@@ -323,11 +359,14 @@ The hydrodynamic dispersion tensor :math:`\mathsf{D}_\mathsf{hd} \,
type: <Dhm_type> + Deff_MQ1
parameters:
mol_diff:
char_length:
phi:
# <Dhm_type> parameters ...
.. object:: Constant effective hydromechanic dispersion tensor parameterization
Implements the following parameterization function:
In this case, the effective hydromechanic dispersion tensor is inserted
directly.
.. math::
......@@ -347,6 +386,8 @@ The hydrodynamic dispersion tensor :math:`\mathsf{D}_\mathsf{hd} \,
transport:
type: Dhm_const + <Deff_type>
parameters:
mol_diff:
char_length:
eff_hydromechanic_disp_xx:
eff_hydromechanic_disp_xy:
eff_hydromechanic_disp_xz:
......@@ -356,15 +397,22 @@ The hydrodynamic dispersion tensor :math:`\mathsf{D}_\mathsf{hd} \,
eff_hydromechanic_disp_zx:
eff_hydromechanic_disp_zy:
eff_hydromechanic_disp_zz:
# <Deff_type> parameters ...
(Alternative) YAML template:
A more readable alternative is possible. In this case, the
``eff_hydromechanic_disp`` key is a list of values containing all values
(:math:`\text{dim}\times\text{dim}`) of the tensor. The list is row-wise
oriented.
.. code-block:: yaml
transport:
type: Dhm_const + <Deff_type>
parameters:
mol_diff:
char_length:
eff_hydromechanic_disp: [ ... ]
# <Deff_type> parameters ...
.. object:: Effective hydromechanic dispersion tensor for isotropic media parameterization
......@@ -378,8 +426,8 @@ The hydrodynamic dispersion tensor :math:`\mathsf{D}_\mathsf{hd} \,
* ``Dhm_type: Dhm_iso``
Parameters:
* ``lambda_l``: Longitudinal dispersivity :math:`\lambda_l \, [...]`
* ``lambda_t``: Transverse dispersivity :math:`\lambda_t \, [...]`
* ``lambda_l``: Longitudinal dispersivity :math:`\lambda_l \, [\mathrm{m}^2\mathrm{s}^{-1}]`
* ``lambda_t``: Transverse dispersivity :math:`\lambda_t \, [\mathrm{m}^2\mathrm{s}^{-1}]`
YAML template:
......@@ -388,8 +436,11 @@ The hydrodynamic dispersion tensor :math:`\mathsf{D}_\mathsf{hd} \,
transport:
type: Dhm_iso + <Deff_type>
parameters:
mol_diff:
char_length:
lambda_l:
lambda_t:
# <Deff_type> parameters ...
Scalings
......
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