11.3 Description of the CANSAZ-3D Module

The three-dimensional model, Combined Analytical-Numerical SAturated Zone 3-D (CANSAZ-3D), simulates the three-dimensional steady state groundwater flow and contaminant transport, and forms the saturated zone module of the EPACMTP. CANSAZ-3D describes advective-dispersive transport in an aquifer with a steady state flow field and a patch source of contaminants at the water table. This model consists of two modules: (a) a module for the ground water flow in the saturated zone, and (b) a module for the contaminant migration through the saturated zone. The flow module estimates the hydraulic head and the flow velocities in an aquifer of constant thickness, whereas the transport module simulates the transport of dissolved contaminants, and estimates the contaminant concentrations at a receptor well. The flow module considers three-dimensional steady state flow, with an optional two-dimensional simulation. On the other hand, the transport module considers advection, hydrodynamic dispersion, equilibrium sorption, zero-order production, and first-order decay.

The important assumptions in the groundwater flow module are as follows:

• the aquifer is homogeneous
• the groundwater flow is steady-state, isothermal, and governed by Darcy's Law,
• the fluid is slightly compressible and homogeneous,
• the contribution of recharge from the unsaturated zone is small relative to the regional flow in the aquifer, and the saturated aquifer thickness is large relative to the head difference that establishes the regional gradient, and
• the principal directions of the hydraulic conductivity tensors are aligned with the Cartesian coordinate system.

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

• the flow field is at a steady state,
• the aquifer is homogeneous and initially contaminant free,
• adsorption of contaminants onto solid phase is described by the Freundlich equilibrium isotherm,
• chemical/bio-chemical degradation of the contaminant can be described by a first-order process,
• the mass flux of contaminants through the source is constant,
• the chemical is dilute and is present in the solution or adsorbed onto the soil matrix,
• hydrodynamic dispersion can be described as a Fickian process, and
• the saturated zone is a chemically buffered system with constant and uniform geochemical characteristics.

A brief description of the groundwater flow module and of the transport module follows. Additional details can be found in the EPACMTP background document [64].

11.3.1 Formulation of the Groundwater Flow Module

The governing equation for steady state flow in three dimensions is

$$K_x \frac{\partial^{2}\!H}{\partial^{2}x} + K_y \frac{\partia... ...\partial^{2}y} + K_z \frac{\partial^{2}\!H}{\partial^{2}z} = 0$$ (11.8)

where $H$ is the hydraulic head (L), and $K_x$, $K_y$, and $K_{z}$ are hydraulic conductivities (L/T). The boundary conditions are given by

$$H(0,y,z) = H_1$$

$$H(x_L,y,z) = H_2$$

$$\frac{\partial H}{\partial y}(x,0,z) = 0$$

$$\frac{\partial H}{\partial y}(x,\pm\frac{y_{_L}}{2},z) = 0$$ (11.9)

$$\frac{\partial H}{\partial z}(x,y,0) = 0, \mbox{\ \ \ \ \ and}$$

$$-K_Z\frac{\partial H}{\partial z}(x,y,B) = \left\{\begin{array}{l... ..._{_D}}{2} \le y \le \frac{y_{_D}}{2}\\ I_r \mbox{elsewhere} \end{array}\right.$$

where $x_L$, $y_L$ and $B$ are length, width and thickness of the aquifer system, $x_d$, $x_u$ and $y_{_D}$ are the upstream and downstream coordinates and the width of the source, $I$ is the infiltration rate through the rectangular surface patch source, and $I_r$ is the recharge rate at the water table outside the patch area. The $V_i$'s are the Darcy velocities obtained from the FECTUZ model, and are given by

$$\displaystyle V_{l} = -K_{l}\frac{d c}{d l},\ \ \ \mbox{where $l$ corresponds to $x,y$\space or $z$ }$$ (11.10)

These equations can be solved by either finite difference technique or finite element methods, by numerically discretizing the aquifer region of interest into three-dimensional elements. In addition, CANSAZ-3D can also simulate a 2-D groundwater flow in the $x\!\!-\!\!y$ plane. The present implementation of the EPACMTP model uses the finite difference technique for a 2-D solution. The details on the implementation of the finite element and finite difference techniques in CANSAZ-3D are presented in background document [64].

11.3.2 Formulation of the Saturated Zone Contaminant Migration Module

The three dimensional transport of contaminants in an aquifer can be described by

$${\frac{\partial }{\partial x_i}\left(D_{ij}\frac{\partial c_p}{\partial x_j}\right) - V_i\frac{\partial c_p}{\partial x_i}}$$

$$= \phi Q_p \lambda_p c_p + \phi R_p \frac{\partial c_p}{\partial t}... ...lambda_m c_m, \ \begin{array}{rcl} p & = & 1,n_c\\ i,j & = & 1,2,3 \end{array}$$ (11.11)

where $c_p$ (M/L$^3$) is the concentration of the $p$th component species in the $n_c$ member decay chain, $\lambda_p$ (1/T) and $R_p$are the first order decay and retardation coefficients, $Q_p$ and $Q_m$ are correction factors to account for sorbed phase decay of species $p$ and parent $m$, respectively, and $\phi$ is the aquifer effective porosity. The dispersion coefficients $D_{ij}$ are given by

$$D_{ii} = \alpha_{_L} \frac{V_i^2}{\vert V\vert} + \alpha_{_T} \frac{V_j^2}{\vert V\vert} + \alpha_{_V} \frac{V_k^2}{\vert V\vert} + \phi D^*$$

$$D_{ij} = (\alpha_{_L} - \alpha_{_T}) V_i V_j / \vert V\vert$$ (11.12)

where $\alpha_{_L},\alpha_{_T}$ and $\alpha_{_V}$ are the longitudinal, horizontal transverse and vertical dispersivity (L) respectively, and $D^*$ is the effective molecular diffusion coefficient (L$^2$/T). The retardation factor $R$ and the decay coefficient $Q$ are given as in Equation D.6 in Appendix D.2.2. The factors $\xi_{pm}, Q_m$ and $\lambda_m$ are the same as described in Appendix D.2.2. Here $i,j$ and $k$ denote the $x,y$ and $z$ directions respectively, and the Einstein summation convention is used to simplify the notation.

The initial and boundary conditions for the problem are as follows:

$$\begin{array}{lcc} \mbox{\makebox[0cm][l]{upstream boundary:}} & ... ... x < x_u,\ x > x_d,\ y < \frac{y_{_D}}{2},\ y > \frac{y_{_D}}{2}\\ \end{array}$$ (11.13)

11.3.2.1 Solution Method for the Transport Module

For steady state cases, the concentration in a receptor well is estimated by applying the de Hoog algorithm [47]. For time dependent simulations, the governing equation is solved by using the Laplace Transform Galerkin (GLT) technique, developed by Sudicky [196]. CANSAZ-3D has options for either a finite difference or a finite element solution. A detailed description of the solution method can be found in the background document [64].