11.2 Description of the FECTUZ Module

The model for the Finite Element and semi-analytical Contaminant Transport in the Unsaturated Zone (FECTUZ) is a one-dimensional flow and transport code, that simulates vertical water flow and solute transport through the unsaturated zone above an unconfined aquifer. This model describes the migration of contaminants downward from a disposal unit, typically a landfill or surface impoundment, through the unsaturated zone underlying the disposal unit, to an unconfined aquifer. This model can simulate cases where the flow and transport are one-dimensional, in the downward vertical direction; the flow and transport are driven by the seepage from a landfill or a surface impoundment, which is assumed to occur at a constant rate. Flow is assumed to be always at steady state, while either transient or steady-state solute transport simulations can be performed. FECTUZ consists of two modules: a flow module and a solute transport module, and there are certain assumptions incorporated into both the unsaturated flow module and the unsaturated transport module.

The important assumptions in the flow module are as follows:

• the flow of the fluid phase is one-dimensional, isothermal, steady and governed by Darcy's law, and is not affected by the presence of dissolved chemicals, the fluid is slightly compressible and homogeneous,
• the soil profile consists of one or more, individually uniform, incompressible soil layers, and
• the effects of hysteresis in soil constitutive relations are negligible.

The important assumptions incorporated in the transport module are as follows:

• advection and dispersion are one-dimensional,
• transport in the porous medium is governed by Fick's law,
• sorption reactions can be described by Freundlich equilibrium isotherm,
• the effects of bio-chemical decay can be described by first-order degradation and zero-order production reactions, and
• fluid properties are independent of concentrations of contaminants.

A brief description of the mathematical formulation of the flow module and of the transport module is presented here. Additional details on the model formulation and the solution method can be found in the EPACMTP background document [64].

11.2.1 Steady State Flow Module Formulation

The governing equation for the steady state flow module of FECTUZ is given by Darcy's Law:

$$-K_sk_{rw}\left[\frac{d \psi}{d z} - 1\right] = I \ \ \ \ \ ;\ \ \ \ \ \psi_l = 0$$ (11.1)

where $\psi$(L) is the pressure head, $z$(L) is the depth coordinate which is taken positive downward, $K_s$(L/T) is the saturated hydraulic conductivity, $k_{rw}$ is the relative permeability, $I$(L/T) is the infiltration rate, and $l$(L) is the thickness of the unsaturated zone. The symbols inside the parentheses denote the dimensions of the physical quantities.

The permeability-water content relation is given assumed to follow the Mualem-van Genuchten model [205], and is given by

$$k_{rw} = S_e^{1/2}\left[1-{\left(1-S_e^{1/\gamma}\right)^\gamma}\right]^2$$ (11.2)

where

$$S_e = \left\{ \begin{array}{lcl} \left[1 + (-\alpha\psi)^\bet... ...gamma}& , & \psi < 0\\ 1 & , & \psi \ge 0 \end{array}\right.$$ (11.3)

where the parameters $\alpha, \beta,$ and $\gamma$ are soil specific shape parameters. Further, parameters $\beta$ and $\gamma$ are related through the equation $\gamma = 1 - 1/\beta$. Descriptive statistical values for $\alpha$and $\beta$ have been determined by Carsel and Parrish [30] for 12 soil classifications.

11.2.1.1 Solution Method for Flow Module

The solution of Equation D.1 involves the following steps:

• substitution of Equations D.2 and D.3 into Equation D.1,
• replacement of the derivative in Equation D.1 with a backward finite difference approximation, and
• solution of the resultant equation using a combined Newton-Raphson and bi-section method.

In this method, the unsaturated zone is discretized into a number of one-dimensional segments, with that a high resolution close to the water table and also close to layer interfaces for layered soils. The surface impoundment boundary condition is solved by using the Darcy's Law, and iteratively solving for the infiltration rate through the surface impoundment liner.

  
11.2.2 Solute Transport Module Formulation

The one-dimensional transport of solute species is modeled in FECTUZ using the following advection-dispersion equation:

$$\frac{\partial }{\partial z}\left[D\frac{\partial c_i}{\parti... ...\lambda_i C_i - \sum^{M}_{m=1} \theta \xi_{im}Q_m\lambda_m c_m$$ (11.4)

where $c_{i}$ is the concentration of the $i$th species (M/L$^3$), $D$ is the apparent dispersion coefficient (L$^2$/T), $V$ is the Darcy velocity (L/T) obtained from the solution of the flow equation, $R$ is the retardation factor, $\lambda_i$ is the first order decay constant (1/T), $Q$ is a coefficient to incorporate decay in the sorbed phase, and the summation on the right-hand side of Equation D.4 represents the production due to the decay of parent species, where $M$ is the total number of parents. The coefficient $\xi_{im}$ is a constant related to the decay reaction stoichiometry, representing the fraction of $m$th species that decays to the $i$th daughter species.

The dispersion coefficient $D$ is defined as

$$D = \alpha_LV + \theta D^*$$ (11.5)

where $\alpha_L$ is the longitudinal dispersivity (L) and $D^*$is the effective molecular diffusion coefficient (L$^2$/T). $R$signifies the effect of equilibrium sorption, whereas Q signifies the effect of decay. They are given by

 

$$R = 1 + \frac{\rho_{_b}}{\theta}k_1\eta c^{\eta -1}$$

$$Q = 1 + \frac{\rho_{_b}}{\theta}k_1 c^{\eta -1}$$ (11.6)

where $k_1$ and $\eta$ are nonlinear Freundlich parameters.

The initial and boundary conditions of the one-dimensional transport problem can be given by one of the following equations:

$$c_i(z,0) = c_i^{in}$$

$$-D\frac{\partial c_i}{\partial z}(0,t) = v(c_i^0(t) - c_i)$$

$$c_I(0,t) = c_i^0(t)$$ (11.7)

$$\frac{\partial c_i}{\partial t} (l,t) = 0$$

where $c_i^{in}$ is the initial concentration of the $i$th species, $c_i^0(t)$ is the leachate concentration emanating from the disposal facility, $z$ is the downward vertical coordinate, and $l$ is the depth of the water table.

11.2.2.1 Solution Method for Solute Transport Module

In the FECTUZ solute transport model, three solution methods are employed based on the complexity of the problem: (a) for single species, steady-state simulations involving a linear adsorption isotherm, an analytical solution is used, (b) for transient or steady-state decay chain simulations with linear sorption, a semi-analytical method is used, and (c) for cases involving nonlinear sorption, a numerical finite element solution is used.