# Parametric Analysis for 3D Modeling of Consolidation-Induced Solute Transport Using OpenFOAM
## 2 Theoretical Model  
### Three-Dimensional Governing Equation  
This study integrates [PERSON]'s consolidation theory [17] and builds on the previous 1D model [16; 18] to derive the three-dimensional storage equation. \[S_{r}n\beta\frac{\partial p}{\partial t}+S_{r}\
abla\cdot\mathbf{v_{s}}=\frac {1}{\rho_{w}g}\
abla\cdot(\mathbf{K}\cdot\
abla p), \tag{1}\]  
where \(p\), \(S_{r}\), \(n\), and \(\rho_{w}\) denote excess pore pressure, saturation, initial porosity, and density of pore water, respectively. The displacement of solid particles, \(\mathbf{u_{s}}\), determines their velocity, \(\mathbf{v_{s}}\). \[\mathbf{v_{s}}=\frac{\partial\mathbf{u_{s}}}{\partial t}=\bigg{(}\frac{ \partial u_{x}}{\partial t},\frac{\partial u_{y}}{\partial t},\frac{\partial u _{z}}{\partial t}\bigg{)}. \tag{2}\]  
The hydraulic conductivity tensor, \(\mathbf{K}\), characterizes the capacity of water to flow through the soil and is expressed as  
\[\mathbf{K}= \begin{pmatrix}K_{x}&0&0\\ 0&K_{y}&0\\ 0&0&K_{z}\end{pmatrix} . \tag{3}\]In an unsaturated soil, air can be freely distributed within the pores or dissolved in water. This presence of air alters the compressibility of the pore fluid, denoted as \(\beta\), which is expressed as follows [20]:  
\[\beta=\frac{S_{r}}{k_{\text{z}ro}}+\frac{1-S_{r}+r_{h}S_{r}}{(p_{a}+p_{0})}, \tag{4}\]  
where the bulk modulus of pore water \(k_{\text{z}ro}\) is fixed at 2000 MPa, the volumetric air fraction dissolved in water \(r_{h}\) is set at 0.02 [16], and both the apparent pressure \(p_{a}\) and atmospheric pressure \(p_{0}\) are set to 100 kPa. The displacement equation is based on the force balance equation, incorporating the soil's self-weight in the \(z\)-direction, with stress-strain relationships described by [PERSON]'s law [21]. \[\
abla\cdot[\mu\
abla\mathbf{u_{s}}+\mu(\
abla\mathbf{u_{s}})^{T}+\lambda \text{Itr}(\
abla\mathbf{u_{s}})]- \begin{bmatrix}0\\ 0\\ (1-n)(p_{s}-S_{r}p_{w})g\
abla\cdot\mathbf{u_{s}}\end{bmatrix} =\
abla p, \tag{5}\]  
where the identity tensor is symbolized as \(\mathbf{I}\). The Lame parameters, \(\mu\) and \(\lambda\), are functions of the elastic modulus \(E\) and Poisson ratio \(\
u\), with the following relationships:  
\[\mu=\frac{E}{2(1+\
u)},\quad\text{and}\quad\lambda=\frac{\
u E}{(1+\
u)(1-2\
u )}. \tag{6}\]  
After solving the consolidation equation, pore water pressure and soil displacement are integrthe solute transport model through [PERSON]'s law [22], \(\mathbf{v_{f}}=-\frac{1}{\rho_{u}gS_{r}n}\mathbf{K}\cdot\
abla p+\mathbf{v_{s}}\). This model accounts for both fluid and particle velocities as well as the changing porosity over time. Conservation of the mass of the solute is considered in both the liquid and solid phases. As a result, a coupled solute transport equation that incorporates a linear adsorption process is established. \[ \begin{split}[S_{r}n+(1-n)\rho_{s}K_{d}]\frac{\partial c_{f}}{ \partial t}=& S_{r}n\mathbf{D}\cdot\
abla^{2}c_{f}+S_{r}n\
abla c_{f} \cdot\
abla\cdot\mathbf{D}\\ &-S_{r}n\mathbf{v_{f}}\cdot\
abla c_{f}-[(1-n)\rho_{s}K_{d}] \mathbf{v_{s}}\cdot\
abla c_{f}\\ &+(S_{r}n\beta\frac{\partial p}{\partial t}+S_{r}n\beta\mathbf{v_ {f}}\cdot\
abla p)c_{f}.\end{split}  \tag{7}\]  
In (7), \(c_{f}\) represents the solute concentration, \(K_{d}\) represents the distribution coefficient, and \(\mathbf{D}\) is the hydrodynamic dispersion coefficient tensor, which combines mechanical dispersion and molecular diffusion.