# Initial Conditions

DORiE computes transient solutions, and hence needs a solution from which it can start. There are multiple ways of specifying an initial condition. All have in common that the data provided must be interpolated on the respective solution grid function space. Depending on the actual input data, this means that information can be lost and specific features can be distorted. Users are responsible to ensure that the solution grid function space and the input data for initial conditions match in this sense.

Initial conditions can generally be stated in several physical quantities, as long as the respective quantity has a unique transformation to the solver solution quantity.

Initial condition input is controlled entirely via the Configuration File.

Note

The initial condition is projected onto the actual solution function space. Depending on grid shape and resolution, function space (order), and interpolator (if applicable), the resulting solution may vary greatly from the actual input data.

## Input Types

This is an overview of all input types for initial conditions. They are controlled by the initial.type key and shared between all models unless otherwise specified.

### Analytic

• type = analytic

An analytic function $$f(\vec{p})$$ which depends on the physical position $$\vec{p}$$. The function must be defined via the key initial.equation. For parsing the input expression, we use muparser which supports a set of common mathematical functions. Additionally, the following variables can be used:

Available variables:
• x: X-coordinate $$p_1 \, [\mathrm{m}]$$.

• y: Y-coordinate $$p_2 \, [\mathrm{m}]$$.

• z: Z-coordinate $$p_3 \, [\mathrm{m}]$$ (only in 3D).

• h: Height above origin. Synonymous to y in 2D and z in 3D.

• pi: Mathematical constant $$\pi$$.

• dim: Number of physical dimensions.

Tip

Assuming the target quantity is the matric head (see Transformation Types), typical initial conditions for a Richards model are

• Hydrostatic equilibrium: A vertical gradient of $$-1$$ and a fixed value <v> at height $$h = 0 \, \mathrm{m}$$:

initial.equation = -h + <v>

• Gravity flow: Constant value.

Tip

The expression for a gaussian pulse of solute concentration centered at $$\vec{p} = [0.5, 0.5]^T \, \mathrm{m}$$ and variance of $$\sigma^2 = \left( 0.1 \, \mathrm{m} \right)^2$$ is:

initial.equation = <m> * exp(-((x-0.5)^2+(y-0.5)^2)/(2.*0.1^2)) / (2*pi*0.1^2)


where <m> is the total solute mass of the pulse $$m_s \, [\text{kg}]$$.

### Dataset

• type = data

Load the initial condition from a data file initial.file by opening the dataset initial.dataset inside this file. The data is interpreted as function $$f(\mathbf{p})$$ of the physical position $$\mathbf{p}$$ using one of the available interpolators, which can be chosen using the setting initial.interpolation. The input data is automatically streched to match the grid extensions.

Note

For FEorder > 0, linear interpolation is recommended.

Supported file extensions:

• .h5: H5 data file. initial.dataset may be a file-internal path to the target dataset.

### Stationary (Richards only)

• type = stationary

Compute the stationary part of the Richards equation,

$- \nabla \cdot \left[ K \left[ \nabla h_m - \mathbf{\hat{g}} \right] \right] = 0 \ ,$

and use the solution as initial condition for the transient simulation. The boundary conditions at the simulation start time are used to solve the problem. No further input is needed, but the boundary condition file might need to be adjusted.

Note

As this computes the actual solution, no transformation is applied. The config file key initial.quantity is ignored.

Tip

If you want the simulation to start with different boundary conditions than those used for computing the stationary initial condition, specify boundary conditions that end with the simulation start time and start at the same time or earlier, like so (assuming the simulation starts at $$t=0\,\text{s}$$):

upper:
index: 1
conditions:
# Applied for computing the initial condition
for_ic:
type: Neumann
value: -5e-6
time: [0.0, 0.0]
# Applied during the entire transient simulation
for_sim:
type: Dirichlet
value: -6
time: 0.0


## Transformation Types

This is an overview of the transformation types of all models. They are controlled via the initial.quantity key.

### Richards

Initial condition tranformations for the Richards solver.

No Transformation
• quantity = matricHead

The input data is directly interpreted as matric head $$f = h_m \, [\text{m}]$$.

Water Content to Matric Head
• quantity = waterContent

The input data is interpreted as water content, $$f = \theta_w \, [\text{-}]$$, and transformed into matric head via the parameterization of the medium.

Values greater than the porosity $$\phi$$ and less than the residual water content $$\theta_r$$ are automatically clamped to fit the allowed range. Additionally, any input value $$f(x_0)$$ at some position $$x_0$$ on the grid will result in a saturation greater zero, $$\Theta (x_0) > 0$$, to avoid divergence of the matric head towards negative infinity.

### Transport

Initial condition tranformations for the Transport solver.

No Transformation
• quantity = soluteConcentration

The input data is directly interpreted as solute concentration, $$f = c_w [\mathrm{kg}/\mathrm{m}^d]$$, where $$d$$ indicates the spatial dimensions of the grid.