Storage equation for fully saturated analysis

The hydraulic pressure head $h$ follows from the storage equation:

$$C \; \dot{h} = ( k^p_1 \frac{\partial^2 h}{\partial {x_1}^2} + ... ...{\partial {x_3}^2} ) + \frac{\partial v_i}{\partial x_i} - \alpha \dot{T} + f$$

Primary unknown is the hydraulic pressure head groundflow_pressure. Further notation: $C$ group_groundflow_capacity; $k^p_i$ group_groundflow_permeability in $i$-direction (intrinsic permeability); $x_i$ space coordinate; $v_i$ material velocity (if present); $\alpha$ group_groundflow_expansion is the expansion coefficient of the groundwater for temperature changes. The equation is given for space coordinates following material velocities $v_i$ (if present).

Groundflow velocities

The groundflow velocities, after initializing groundflow_velocity, follow from:

$${v_i}^{{\rm g}} = k^p_i \frac{\partial h}{\partial x_i}$$

Total groundwater pressure

The total groundwater pressure, or pore-pressure, is for example needed to calculate the total stresses in soils. The total groundwater pressure follows from:

$$p_{\rm total} = h - \rho g z$$

where $g$ is the gravitational acceleration, and $\rho$ is the groundflow_density (Please notice that $g$ and $z$ typically are negative numbers).

Tochnog considers pressure a pore pressure of $p=0$, or positive, as indication that there is in fact no water pressure, so the porous soil skeleton is filled with air. In this case. the total soil stress is only composed by the effective stress of the soil skeleton.

The total stress in soils follows from: total soil stress = effective soil stress + total groundwater pressure. This will only be done for isoparametric finite elements which have groundflow data specified.

Static groundwater pressure

The static pressure due to gravity is:

$$p_{\rm static} = \rho g \Delta z$$

where the $\Delta z$ is the distance to the groundwater level, the phreatic level. The phreatic level needs to be specified with the groundflow_phreatic_level record. If that groundflow_phreatic_level record is not specified, the static pressure part is not used, so that the static pressure becomes zero.

Dynamic groundwater pressure

The dynamic groundwater pressure follows from

$$p_{\rm dynamic} = p_{\rm total} - p_{\rm static}$$

Boundary conditions

If the groundwater velocity is 0 normal to an edge (say at the interface with a rock layer it is zero), then you should prescribe nothing on that edge (Tochnog will then take care of that boundary condition for you).

At the phreatic level where the groundflow meets free air the hydraulic pressure head should become $\rho g z$. You can either set this yourself by using bounda_dof combined with bounda_time or else demand that Tochnog automatically does it for you by activating the option groundflow_phreatic_bounda.

At edges where you have some other hydraulic head you need to specify that head yourself with bounda_dof and bounda_time records.

If gravity is not of importance, e.g. in biomechanics where the storage equation is used to model fluid transport in soft tissues, the hydraulic pressure head $h$ is equal to the total pressure, and thus is zero at edges where the water meets the free air. In this case, set $h$ to zero by using bounda_dof combined with bounda_time.

Postprocessing

For all printing, plotting etc. you normally get the hydraulic pressure head $h$ since it is the primary unknown solved in the storage equation. The total pressure, static pressure and dynamic pressure are obtained using the post_calcul option.

Naming conventions

Following conventional naming, we remind the user that the capacity depends on the porosity $n$ and water compressibility $\beta$:

$$C = n \; \beta$$

and for the (intrinsic) permeability:

$$k^p_i = \frac{k_i}{\rho \; \vert g \vert }$$

where $k_i$ is the hydraulic conductivity in $i$-direction.