Material for unsaturated flow in lattice models - LatticeTransMat

Lattice transport elements have to be used with this material model. A positive sign is assumed for liquid tension, unlike the convention of soil mechanics which assumes compression positive.

These transport elements are idealised as one-dimensional conductive pipes. The gradient of hydraulic head, which governs flow rate along each transport element, is determined from the capillary pressures $P_{c}$ at the two nodes.

The mass balance equation describes the change in moisture inside a porous element as a consequence of liquid flow and solid-liquid retention. It leads to the following partial differential equation

$$c\frac{\partial P_{c}}{\partial t} + k (\nabla(\frac{P_{c}}{g}-\rho z)) =0$$ (267)

where $P_{c}$ is the capillary pressure, $c$ is the mass capacity function( $s^{2}m^{-2}$), $k$ is the Darcy hydraulic conductivity ($ms^{-1}$), $\rho$ is the fluid mass density, $g$ is the acceleration of gravity, $z$ is the capillary height and t is the time.

The hydraulic conductivity $k$ consists of

$$k=k_{0} + k_{c}$$ (268)

where $k_{0}$ is the hydraulic conductivity of the intact material and $k_{c}$ is the additional conductivity due to cracking.

Darcy hydraulic conductivity $k_{0}$ is defined as

$$k_{0}=\frac{\rho g}{\mu}\kappa$$ (269)

where $\mu$ is the dynamic viscosity ($Pa.s$), $\kappa$ is the permeability also called intrinsic conductivity($m^{2}$), and $\kappa_{r}$ is the relative conductivity. $\kappa_{r}$ is a function of the effective degree of saturation.

The cracking part is

$$k_{c}=\xi\frac{\rho g}{\mu}\frac{w^{3}_{c}}{12h} \kappa_{r}$$ (270)

where $\xi$ is a tortuosity factor taking into account the roughness of the crack surface, $w_{c}$ is the equivalent crack opening of the dual mechanical lattice and $h$ is the length of the dual mechanical element.