Material models for lattice transport elements

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

(132)\[c\frac{\partial P_{c}}{\partial t} + k\mbox{div}(\nabla(\frac{P_{c}}{g}-\rho z)) =0\]

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

(133)\[k=k_{0} + k_{c}\]

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

(134)\[k_{0}=\frac{\rho g}{\mu}\kappa\]

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

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

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.

One-dimensional transport element

The discrete form of the differential equation for mass transport for a one-dimensional transport element is

(136)\[\alpha_{e}P_{c} + C_{e}\frac{\partial P_{c}}{\partial t} = f_{e}\]

where \(P_{c}\) is a vector containing the nodal values of the capillary pressure, \(\alpha_{e}\) is the conductivity matrix, \(C_{e}\) is the capacity matrix and \(f_{e}\) is the nodal flow rate vector (\(kg.m^{3}\)).

The capacity matrix is

(137)\[\begin{split}C_{e} = \frac{Al}{12}c \left( \begin{array}{ c c } 2 & 1 \\ 1 & 2 \end{array} \right)\end{split}\]

where \(c\) is the capacity of the material, \(l\) is the length of the transport element and \(A\) is the cross-sectional area of the transport element.

The conductivity matrix is defined as

(138)\[\begin{split}\alpha_{e} = \frac{A}{l}\frac{k}{g} \left( \begin{array}{ c c } 1 & -1 \\ -1 & 1 \end{array} \right)\end{split}\]

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

(139)\[c=-\rho\frac{\partial \theta}{\partial P_{c}}\]

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

(140)\[\theta = S_{e} (\theta_{s} - \theta_{r})+\theta_{r}\]

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

(141)\[\begin{split}S_{e} = \left\{ \begin{array}{ll} \frac{\theta_{m}-\theta_{r}}{\theta_{s}-\theta_{r}}\left(1+\left(\frac{ P_{c}}{a}\right)^{\frac{1}{1-m}}\right)^{-m} & \mbox{if } P_{c} \ge P_{c(aev)} \\ 1 & \mbox{if } P_{c}<P_{c(aev)} \end{array} \right.\end{split}\]

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 material, 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

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

If \(\theta_{m} = \theta_{s}\), we have :math:` 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 84 Model Parameters

Description	Material for fluid transport in lattice models
Record Format
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 (\(ctor=\frac{1}{\tau}\le1\))

Table 85 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
Tests/Examples	lm/lattice2dcrackinput.in, lm/lattice3d_mt1.in, lm/lattice3d_mt2.in, lm/latticetransmat.in