Transport Models

The transport component of Amanzi models a set of physical processes that lead to movement of dissolved and solid contaminants in the subsurface, treating the chemical and microbiological reactions that can affect the transport rate through a retardation effect as a separate set of processes. The governing PDE for the component \(C_i\) reads

\[\frac{\partial (\phi s_l C_i)}{\partial t} + \nabla \cdot \boldsymbol{J}_i^{\text{adv}} = Q_i - \nabla \cdot \boldsymbol{J}_i^{\text{disp}} - \nabla \cdot \boldsymbol{J}_i^{\text{diff}}\]

where \(\phi\) is the porosity, \(s_l\) is the liquid saturation, \(\boldsymbol{J}_i^\text{adv}\) is the advective flux, \(\boldsymbol{J}_i^\text{disp}\) is the dispersive flux, \(\boldsymbol{J}_i^\text{diff}\) is the diffusive flux (often grouped with the dispersive flux), and \(Q_i\) is the summation of the various source terms.

Advective Flux

Advection involves the translation in space of dissolved or suspended material at the rate of movement of the bulk fluid phase. The advective flux of a dissolved species \(C_i\) in porous media can be described mathematically as

\[\boldsymbol{J}_i^\text{adv} = \boldsymbol{q}_l C_{i},\]

where \(\boldsymbol{q}_l\) is the Darcy velocity.

Dispersive Flux

Dispersion of a dissolved species involves its spreading along tortuous pathways in a porous medium caused by mixing effects. Dispersion takes place in the direction of the flow (longitudinal) and normal to the flow (transverse). A conventional Eulerian Fickian representation of dispersion is assumed, which may be taken as the asymptotic limiting form of the dispersion tensor. The dispersive flux has the form

\[\boldsymbol{J}_i^\text{disp} = - \phi s_l \boldsymbol{D} \nabla C_i,\]

where \(\boldsymbol{D}\) denotes the dispersion tensor. The dispersion tensor takes different forms depending on whether the media is isotropic or anisotropic. For an isotropic medium with no preferred axis of symmetry the dispersion tensor has the well-known form:

\[\boldsymbol{D} = \alpha_T \|\boldsymbol{v}\| \boldsymbol{I} + \left(\alpha_L-\alpha_T \right) \frac{\boldsymbol{v} \boldsymbol{v}}{\|\boldsymbol{v}\|},\]

characterized by the two parameters \(\alpha_L\) [m] and \(\alpha_T\) [m] referred to as the longitudinal and transverse dispersivity, respectively. The vector \(\boldsymbol{v}\) [m/s] denotes the average pore velocity, and \(\boldsymbol{I}\) is the identity matrix.

Diffusive Flux

Molecular diffusion is often indistinguishable from mechanical dispersion as a process operating in porous media, and thus the two are often lumped together to form a hydrodynamic dispersion term. Molecular diffusion is an entropy-producing process in which the random motion of molecules causes spreading or homogenization of a concentration field. Atomistic representations of molecular diffusion capture this random motion, but continuum models of the kind considered here typically represent only the average behavior of the molecules.

Molecular diffusion is usually described in terms of Fick’s First Law, which states that the diffusive flux is proportional to the concentration gradient. Since water-rock interaction commonly takes place in porous materials, it is important to account for the effect of tortuosity, which is defined as the ratio of the path length the solute would follow in water alone, \(L\), relative to the tortuous path length it would follow in porous media, \(L_e\):

\[\tau_{L} = (L/L_e)^2.\]

The diffusive flux, then, is given by

\[\boldsymbol{J}_{i}^\text{diff} = - \phi s_l D_i \tau_{L} \nabla C_i.\]

where \(D_i\) is referred to as the diffusion coefficient and is specific to the chemical component considered as indicated by the subscript \(i\).

Boundary Conditions

A first-type or Dirichlet condition involves specification of a fixed value of the concentration, \(C_i\) at the boundary location:

\[C_i(\boldsymbol{x}, t) = C_{i,0}(\boldsymbol{x}, t) \qquad \boldsymbol{x} \in \Gamma^{in},\quad t > 0,\]

where \(C_0\) is a given function. Pure advective transport requires to set up the Dirichlet boundary conditions only on the inflow boundary of the computational domain.

A second-type or Neumann boundary condition involves specification of the flux

\[J_i(\boldsymbol{x}, t) = J_{i,0}(\boldsymbol{x}, t), \qquad \boldsymbol{x} \in \partial \Omega,\quad t > 0,\]

where \(J_{i,0}\) is a given flux function. The Dirichlet boundary condition on the outflow part of the computational domain may result in parabolic and/or exponential boundary layers; therefore it should be used with caution.

Initial Conditions

An initial condition specifies concentration at time \(T=0\) inside the computational domain:

\[C_i(\boldsymbol{x}, 0) = C_{i,0}(\boldsymbol{x}) \qquad \boldsymbol{x} \in \Omega.\]

Source Terms

The source term \(Q_i\) is a given function specifying usually location of wells inside the computational domain:

\[Q_i(\boldsymbol{x}, t) = Q_{i,0}(\boldsymbol{x},t), \qquad \boldsymbol{x} \in \Omega,\quad t > 0.\]