# Parameterization

Specifying the domain properties requires input of the soil architecture (in relation to the grid) and of the soil parameters. The latter incorporate the subscale physics, the representation of the soil hydraulic and solute properties below the REV scale. DORiE requires several input files for retrieving this information, depending on the type of grid used for the computation.

A YAML parameter file is always required. It specifies a set of parameterizations, along with a medium index. This index is used to reference the medium specified in the architecture files, or with the “global” indices given in the config file.

Additionally, you can specify settings for the small-scale heterogeneities of the soil. They require an input via H5 datasets that contain appropriate scaling factors. You can conveniently create such datasets using the random field generator of the command line interface.

## YAML Parameter File

This file is used to specify the parameterization for each medium inside the simulated domain. It follows a simple hierarchical syntax. The file path and name must be specified via the <model>.parameters.file key of the config file.

The top-level mapping must contain the key volumes. This key contains a sequence of arbitrary parameterization names. These, in turn, contain the medium index, and the model type (richards or transport). The medium index must be an integer. Each model key contains the parameterization type, and the actual parameters.

Note

Parameterization data for model transport is only required if the model is actually enabled in the config file settings, via simulation.mode = richards+transport.

Heterogeneities throughout the entire domain are set via the top level key scaling. It must contain a supported scaling type, which may default to none. In case an actual scaling is to be applied, the section must contain the key data, which specifies the datasets required for this type of scaling.

Warning

Transport parameters like porosity or characteristic length currently are not affected by small scale heterogoeneities.

Every scaling_section has the same keys: The H5 filepath is given by file, and the internal dataset path by dataset. You can then choose an interpolation method for the dataset. You may optionally insert values for the physical extensions of the dataset and the offset of its (front) lower left corner from the origin. If at least one of them is omitted, the inserted field is automatically scaled to match the maximum grid extensions. This also works for irregular grid shapes.

volumes:
<name-0>:
index: <int>
richards:
type: <string>
parameters:
<param-name-0>: <float>
# more parameters ...

# if transport mode is enabled ...
transport:
type: <string>
parameters:
<param-name-0>: <float or sequence>
# more parameters ...

# more volumes ...

scaling:
type: <string>
data:
<scaling_section>:
file: <string>
dataset: <string>
interpolation: <string>
# optional:
extensions: <sequence>
offset: <sequence>

# more scaling sections ...


You find a documentation of the parameterizations implemented in DORiE along with the parameters they require below. The parameterization type must match a static parameterization name or a valid combination of them. Likewise, the parameters must match the names of parameters these objects use.

## Scaling Field Datasets

One or more datasets in possibly multiple HDF5 files are required for creating a medium with small-scale heterogeneities. The datasets must consist of floating-point values and must have the same dimensionality as the grid. Other than that, there is no restriction on their shape because they are inserted into Interpolators. The interpolators supply scaling factor values ased on positions on the grid and thus create a grid-independent representation. Scaling field input works the same for any supported grid type.

During setup, the program reads the interpolator values at the barycenter of every grid cell on the coarsest grid configuration. These values are stored and referenced with the grid cell. If the cell is refined, the same scaling factors apply in all its child cells.

## Supported Parameter File Types

This section lists the available types for parameterizations, scalings, and interpolators in a understandable format.

### Richards Parameterizations

As the soil porosity $$\phi \, [-]$$ and the residual water content $$\theta_r \, [-]$$ vary between soil types, it is convenient to define the soil water saturation $$\Theta \, [-]$$ by

$\Theta (\theta_w) = \frac{\theta_w - \theta_r}{\phi - \theta_r}, \quad \Theta \in \left[ 0, 1 \right],$

where $$\theta_w \, [-]$$ is the volumetric soil water content. One typically assumes that the entire pore space can be filled up with water, hence we set the saturated water content $$\theta_s \, [-]$$ equal to the porosity, $$\theta_s = \phi$$. This relation can be manipulated in certain local scalings.

#### Mualem–van Genuchten

Implements the following parameterization functions:

$\begin{split}\Theta (h_m) &= \left[ 1 + \left[ \alpha h_m \right]^n \right]^{-1+1/n} \\ K (\Theta) &= K_0 \Theta^\tau \left[ 1 - \left[ 1 - \Theta^{n / \left[ n-1 \right]} \right]^{1-1/n} \right]^2\end{split}$
• type: MvG

Parameters:
• theta_r: Residual water content $$\theta_r$$

• theta_s: Saturated water content / Porosity $$\theta_s$$

• k0: Saturated conductivity $$K_0 \, [\text{ms}^{-1}]$$

• alpha: Air-entry value $$\alpha \, [\text{m}^{-1}]$$

• n: Retention curve shape factor $$n$$

• tau: Skew factor $$\tau$$

YAML template:

richards:
type: MvG
parameters:
theta_r:
theta_s:
k0:
alpha:
n:
tau:


### Transport Parameterizations

Regardless of the parameterization, the transport model 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 model while the other two have to be specified for each volume:

Permanent parameters:
• mol_diff: Molecular diffusion $$D_m \, [\text{m}^2\text{s}^{-1}]$$

• char_length: Characteristic pore length $$\ell \, [\text{m}]$$

Note

We support two options for specifying tensors through the YAML syntax. You may either specify every entry of the tensor with a dedicated key, like

<param>_xx: <value_xx>
<param>_xy: <value_xy>
# ...


or give an entire sequence that will be mapped to the respective entries,

<param>: [<value_xx>, <value_xy> # , ...
]


The sequence is interpreted as a flattened tensor with row-major layout.

Warning

You must omit any component containing the spatial direction z in a 2D setup.

The hydrodynamic dispersion tensor $$\mathsf{D}_\text{hd} \, [\text{m}^2\text{s}^{-1}]$$ is the main parameter to provide in the transport model. Below you will find several parameterization for this.

#### Constant

In this case, the hydrodynamic dispersion tensor is inserted directly component-by-compoment.

Note

From a physical point of view, the hydrodynamic tensor must be symmetric, but the user input is not verified by DORiE in this regard.

$\mathsf{D}_\text{hd} = \text{const}.$
• type: Dhd_const

Parameters:
• hydrodynamic_disp_<ij>: (i, j)-th component of the hydrodynamic dispersion tensor, $$\left( \mathsf{D}_\text{hd} \right)_{ij} \, [\text{m}^2\text{s}^{-1}]$$, or

• hydrodynamic_disp: Flattened hydrodynamic dispersion tensor $$\mathsf{D}_\text{hd} \, [\text{m}^2\text{s}^{-1}]$$.

YAML template:

transport:
type: Dhd_const
parameters:
mol_diff:
char_length:

# sequence notation, or...
hydrodynamic_disp: [ ]

# component notation
hydrodynamic_disp_xx:
hydrodynamic_disp_xy:
hydrodynamic_disp_xz:
hydrodynamic_disp_yx:
hydrodynamic_disp_yy:
hydrodynamic_disp_yz:
hydrodynamic_disp_zx:
hydrodynamic_disp_zy:
hydrodynamic_disp_zz:


#### Power Law

Implements the following parameterization function:

$D_\text{hd} = \gamma D_m \operatorname{pe}^\alpha.$
• type: Dhd_pl

Parameters:
• gamma: Scale the power law $$\gamma \, [-]$$

• alpha: Exponent of the power law $$\alpha \, [-]$$

• mol_diff: Molecular diffusion $$D_m \, [\text{m}^2\text{s}^{-1}]$$

The Peclét number $$\operatorname{pe}$$ is specified in the config file.

YAML template:

transport:
type: Dhd_pl
parameters:
mol_diff:
char_length:
alpha:
gamma:


#### Superposition

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 $$D_\text{eff}$$ and for the effective hydromechanic tensor $$\mathsf{D}_\text{hm}$$ identified by the key prefixes Deff and Dhm respectively.

$\mathsf{D}_\text{hd} = \mathsf{D}_\text{hm} + D_\text{eff}.$
• type: <Dhm_type> + <Deff_type>

Each of the types are further parameterized and can be found below.

##### Effective Diffusion
###### Constant Effective Diffusion

In this case, the effective diffusion is inserted directly,

$D_\text{eff} = \text{const}.$
• Deff_type: Deff_const

Parameters:

• eff_diff: Effective diffusion $$D_\text{eff} \, [\text{m}^2\text{s}^{-1}]$$

YAML template:

transport:
type: <Dhm_type> + Deff_const
parameters:
mol_diff:
char_length:
eff_diff:
# <Dhm_type> parameters ...

###### Milligton-Quirk I Effective Diffusion

Implements the following parameterization function:

$D_\text{eff} = D_m \frac{\theta_w^{7/3}}{\phi^{2/3}}.$

where the volumetric water content $$\theta_w \, [-]$$ is automatically obtained from the Richards model.

• Deff_type: Deff_MQ1

Parameters:
• mol_diff: Molecular diffusion $$D_m \, [\text{m}^2\text{s}^{-1}]$$

• phi: Soil porosity $$\phi \, [-]$$

YAML template:

transport:
type: <Dhm_type> + Deff_MQ1
parameters:
mol_diff:
char_length:
phi:
# <Dhm_type> parameters ...

###### Milligton-Quirk II Effective Diffusion

Implements the following parameterization function:

$D_\text{eff} = D_m \frac{\theta_w}{\phi^{2/3}}.$

where the volumetric water content $$\theta_w \, [-]$$ is automatically obtained from the Richards model.

• Deff_type: Deff_MQ2

Parameters:
• mol_diff: Molecular diffusion $$D_m \, [\text{m}^2\text{s}^{-1}]$$

• phi: Soil porosity $$\phi \, [-]$$

YAML template:

transport:
type: <Dhm_type> + Deff_MQ1
parameters:
mol_diff:
char_length:
phi:
# <Dhm_type> parameters ...

##### Effective Hydromechanic Dispersion
###### Constant Effective Hydromechanic Dispersion Tensor

In this case, the effective hydromechanic dispersion tensor is inserted directly.

$\mathsf{D}_\text{hm} = \text{const}.$
• Dhm_type: Dhm_const

Parameters:
• eff_hydromechanic_disp_<ij>: (i, j)-th component of the hydromechanic dispersion tensor, $$\left(\mathsf{D}_\text{hm}\right)_{ij} \, [\text{m}^2\text{s}^{-1}]$$, or

• eff_hydromechanic_disp: Flattened hydromechanic dispersion tensor $$\mathsf{D}_\text{hm} \, [\text{m}^2\text{s}^{-1}]$$.

YAML template:

transport:
type: Dhm_const + <Deff_type>
parameters:
mol_diff:
char_length:

# sequence notation, or...
eff_hydromechanic_disp: [ ]

# component notation
eff_hydromechanic_disp_xx:
eff_hydromechanic_disp_xy:
eff_hydromechanic_disp_xz:
eff_hydromechanic_disp_yx:
eff_hydromechanic_disp_yy:
eff_hydromechanic_disp_yz:
eff_hydromechanic_disp_zx:
eff_hydromechanic_disp_zy:
eff_hydromechanic_disp_zz:

# <Deff_type> parameters ...

###### Effective Hydromechanic Dispersion Tensor for Isotropic Media

Implements the following parameterization function:

$\left( \mathsf{D}_\text{hm} \right)_{ij} = (\lambda_l-\lambda_t)\frac{v_i v_j}{\lvert \mathbf{v} \rvert} + \delta_{ij}\lambda_t \lvert \mathbf{v} \rvert,$

where $$\mathbf{v} \, [\text{ms}^{-1}]$$ is the local fluid velocity and $$\delta_{ij}$$ is the Kronecker delta.

• Dhm_type: Dhm_iso

Parameters:
• lambda_l: Longitudinal dispersivity $$\lambda_l \, [\text{m}^2\text{s}^{-1}]$$

• lambda_t: Transverse dispersivity $$\lambda_t \, [\text{m}^2\text{s}^{-1}]$$

YAML template:

transport:
type: Dhm_iso + <Deff_type>
parameters:
mol_diff:
char_length:
lambda_l:
lambda_t:
# <Deff_type> parameters ...


### Scalings

Every scaling_section has the following layout:

<scaling_section>:
file: <string>
dataset: <string>
interpolation: <string>
# optional:
extensions: <sequence>
offset: <sequence>


The setting interpolation accepts any of the implemented Interpolators. The values of extensions and offset are sequences containing the coordinates in the respective spatial dimensions.

Note

extensions and offset of the scaling field will be set to match the grid extensions automatically, if at least one of these keys is omitted. Only omitting the respective values of both keys will lead to a parser error.

#### Miller Scaling

Assumes geometric similarity between scaled regions. Applies the following scaling:

$\begin{split}\Theta &= \Theta (h_m \cdot \xi_M)\\ K &= K (\Theta) \cdot \xi_M^2\end{split}$
• type: Miller

Scaling sections:
• scale_miller: Scaling factor $$\xi_M$$ to be applied onto matric head and conductivity simultaneously.

YAML template:

type: Miller
data:
scale_miller:
# ...


#### Miller Porosity Scaling

Applies porosity scaling in addition to regular Miller scaling, violating its geometric similarity assumption.

$\begin{split}\Theta &= \Theta (h_m \cdot \xi_M) \\ K &= K (\Theta) \cdot \xi_M^2 \\ \phi &= \theta_s - \xi_\theta\end{split}$
• type: MilPor

Scaling sections:
• scale_miller: Scaling factor $$\xi_M$$ applied onto matric head and conductivity simultaneously.

• scale_porosity: Scaling summand $$\xi_\theta$$ subtracted from the saturated conductivity.

YAML template:

type: MilPor
data:
scale_miller:
# ...
scale_porosity:
# ...