Constitutive laws

The mass transport equation is based on the constitutive laws for the capacity $c$ and the hydraulic conductivity $k$.

The capacity $c$ is defined as

$$c=-\rho\frac{\partial \theta}{\partial P_{c}}$$ (274)

where $\theta$ is the volumetric water content ( $\theta=\frac{V_{w}}{V_{T}}$ with $V_{w}$ the volume of water, and $V_{T}$ the total volume) which is calculated by a modified version of van Genuchten’s retention model. Note that the presence of a crack in an element does not influence the capacity in the present model.

The volumetric water content is

$$\theta = S_{e} (\theta_{s} - \theta_{r})+\theta_{r}$$ (275)

where $\theta_{r}$ and $\theta_{s}$ are the residual and saturated water contents corresponding to effective saturation values of $S_{e} = 0$ and $S_{e} = 1$, respectively.

The effective degree of saturation $S_{e}$ is defined as

$$S_{e} = \left\{ \begin{array}{ll} \frac{\theta_{m}-\theta_{r}... ...{c(aev)} \\ 1 & \mbox{if } P_{c}<P_{c(aev)} \end{array}\right.$$ (276)

where $\theta_{m}$ is an additional model parameter and $P_{c(aev)}$ is the air-entry value of capillary pressure which separates saturated ( $P_{c}<P_{c(aev)}$) from unsaturated states ( $P_{c} \ge P_{c(aev)}$). It is intuitive that the smaller the pore size of the mate- rial, the larger the value of $P_{c(aev)}$ will be.

The relative conductivity $\kappa_{r}$ is a function of the effective degree of saturation and is defined as

$$\kappa_{r} = \sqrt{S_{e}}\left(\frac{1-\left[1-\left(\frac{S_{e}}... ...\theta_{r}}{\theta_{s}-\theta_{r}}}\right)^{\frac{1}{m}}\right]^{m}}\right)^{2}$$ (277)

If $\theta_{m} = \theta_{s}$, we have $\frac{\theta_{m}-\theta_{r}}{\theta_{s}-\theta_{r}}=1$ : the equation reduces to the expression of the relative conductivity of the original van Genuchten model.

The model parameters are summarized

Table 62: Material for unsaturated flow in lattice models - summary.

Description	Material for fluid transport in lattice models
Record Format	latticetransmat num(in) # d(rn) # k(rn) # vis(rn) # contype(in) # thetas(rn) # thetar(rn) # paev(rn) # m(rn) # a(rn) # thetam(rn) # [ ctor(rn) #]
Parameters	- num material model number
	- d fluid mass density
	- k permeability ($m^{2}$)
	- vis dynamic viscosity ($Pa.s$)
	- contype unsaturated flow allowed when contype=1
	- thetas saturated water content
	- thetar residual water content
	- paev air-entry value of capillary pressure
	- m van Genuchten parameter
	- a van Genuchten parameter
	- thetam additional model parameter for the modified version of van Genuchten’s retention model
	- ctor coefficient of tortuosity ( $ctor=\frac{1}{\tau}\le1$)
Supported modes	2dMassLatticeTransport