# Dispersive Mass Flow in 3D 

Dispersive as well as diffusive solute transport are driven by a **concentration gradient.** 


## Concentration Gradient

In natural systems, the concentration gradient is usually not constant in space and the simple expression $\Delta C/L$ can therefore **not be used.** Rather, the space derivative of concentration $dC/dx$ **has to be used** to represent **gradual concentration changes** in an appropriate manner.

The dispersive mass flow in 1D is calculated as 

$$
\begin{align}
& J_{disp,h} = -n_e \cdot A \cdot D_{hyd} \cdot \frac{dC}{dx}
& j_{disp,h} = -n_e \cdot A \cdot D_{hyd} \cdot \frac{dC}{dx}
\end{align}
$$



For 2D or 3D solute transport the concentration gradient has to be a **vector to account for magnitude and direction.** This is archived via partial derivates of concentration with respect to Cartesian coordinates. 

$$ gradC = \nabla C = 
\left(
    \begin{array}{c}
        \frac{\partial C}{\partial x}\\
        \frac{\partial C}{\partial y}\\
        \frac{\partial C}{\partial z}
    \end{array}
\right)$$ 

"gradC" is a vector pointing in the **direction of the steepest increase of C.** (When hydrogeologists speak of the concentration gradient, they usually refer to the direction of steepest decrease of solute concentration.)



## Dispersive Mass Flow in Isotropic Aquifers

If 2D or 3D solute transport is considered, dispersive mass flow and dispersive mass flux are vectors with two or three components.

$$
\begin{align}
& J_{disp,h} = 
    \left(
        \begin{array}{c}
            J_{disp,hx}\\
            J_{disp,hy}\\
            J_{disp,hz}
        \end{array}
    \right)
& j_{disp,h} = 
    \left(
        \begin{array}{c}
            j_{disp,hx}\\
            j_{disp,hy}\\
            j_{disp,hz}
        \end{array}
    \right)
\end{align}
$$

As you can see the relationships with the concentration gradient correspond to the previous section: 

$$ 
\begin{align}
& J_{disp,h} = -n_e \cdot A \cdot D_{hyd} \cdot gradC
& j_{disp,h} = -n_e \cdot  D_{hyd} \cdot gradC 
\end{align}
$$

The components of the dispersive mass flow depend on the conditions in the aquifer. 

In case of an **isotropic** aquifer we can describe $D_{hyd}$ as $D_{hyd} = \alpha \cdot |v| + n_e \cdot D$. 
In **homogeneous** systems dispersivity **$\alpha$ is a constant**, 
whereas in **heterogeneous** ones the dispersivity **depends on the directions** ($\alpha = \alpha(x,y,z) $).

## Dispersive Mass Flow in Anisotropic Aquifers 

In all practical cases of 2D and 3D solute transport, **dispersion depens on direction.** Therefore for anisotropic conditions!
Hence, $D_{hyd}$ is not a scalar quantity but a **matrix** (tensor) **which relates the concentration gradient** (vector) **to the dispersive mass flow** (vector). If the principal axes of the dispersion tensor $D_{hyd}$ coincide with the axes of a Cartesian coordinate system and groundwater flow is uniform along the x-axis, the dispersive mass flux is given by 

$$ 
\left(
    \begin{array}{c}
        j_x\\
        j_y\\
        j_z
    \end{array}
\right)
= -n_e
\left(
    \begin{array}{ccc}
        \alpha_L \cdot v_x + n_e \cdot D & 0 & 0\\
        0 & \alpha_{Th} \cdot v_x + n_e \cdot C & 0\\
        0 & 0 & \alpha_{Tv} \cdot v_x + n_e \cdot D
     \end{array}
\right)\cdot
\left(
    \begin{array}{c}
        \frac{\partial C}{\partial x} \\
        \frac{\partial C}{\partial y} \\
        \frac{\partial C}{\partial z} 
    \end{array}
\right)
$$




with $\alpha_L$ = longitudinal dispersivity, $\alpha_{Th}$ = horizontal transverse dispersivity, $\alpha_{Tv}$ = vertical transverse dispersivity.

In most cases we have $\alpha_L > \alpha_{Th} > \alpha_{Tv}$ and the values differ by about one order of magnitude. However, this is only a rule of thumb.


```python

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