PNG 550
Reactive Transport in the Subsurface

9.5 At the long time scale: chemical weathering

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If you have paid attention to a roadside cut, you will notice that the color, texture or structure along the direction of flow varies. This is because rocks at different depth have been subject to different extent of chemical weathering, resulting in gradients in mineral composition, reaction tendency, and porosity. Soils are generated over thousands to millions of years of chemical and physical alteration of their parent bedrock, which typically have much lower porosity and permeability than soils. Why and how are soils at different depth weathered differently? How do different parameters control chemical weathering?

Chemical and Physical Property Evolution

Mineral volume fraction

Using mass conservation principle, the code calculates the change in mass and volume in each mineral phase, which updates individual mineral volume fractions using the following equation:

\begin{equation}\phi_{j m, t+\Delta t}=\phi_{j m, t}+V_{j m} \times r_{j m, t+\Delta t} \times \Delta t\end{equation}

Where $\phi_{j m, t}$ is the volume fraction of each mineral phase jm, Vjm is the molar volume of mineral jm, $r_{j m, t+\Delta t}$ is the the reaction rate of the mineral jm at time $t+\Delta t$. This can be done for each grid block in the domain.

Porosity

The porosity at any time $t$ can be updated as follows:

\begin{equation}\phi_{t}=1-\sum_{j m-1}^{m t o t} \phi_{j m, t}\end{equation}

Where $\phi_t$ is the porosity at time $t$ and mtot is the number of solid phases.

Reactive surface area

The reactive surface area for mineral jm at time $t$, $A_{j m, t}$ is calculated based on the change in porosity and mineral volume fraction compared to their values at initial time 0 (Steefel et al., 2015):

\begin{equation}A_{j m, t}=A_{j m, 0}\left(\frac{\phi_{t}}{\phi_{0}}\right)\left(\frac{\phi_{j m, t}}{\phi_{j m, 0}}\right)^{\frac{2}{3}}\end{equation}

Permeability

The local permeability in individual grid blocks is calculated using local porosity values from equation and the Kozeny-Carman equation (Steefel et al,. 2015):

\begin{equation}\frac{K_{t+\Delta t}}{K_{t}}=\left(\frac{\phi_{t+\Delta t}}{\phi_{t}}\right)^{3} \times\left(\frac{1-\phi_{t}}{1-\phi_{t+\Delta t}}\right)^{2}\end{equation}

Where $K_{t+\Delta t}$ is the permeability at time $t+D t$ updated from time $t$. With updated permeability, flow velocities can be updated using Darcy's law. In this example, however, we use constant flow velocity for simplicity.

Quantification of chemical weathering: mass transfer coefficient $\tau$

Within a soil profile, $\tau$ is used to assess the effect of the chemical and physical alteration. Note that this is different from the time scale $\tau$ that we used previously on dimensionless numbers. We have to stick to this notation because the chemical weathering community uses $\tau$ for mass transfer coefficient, which is calculated by:

\begin{equation}\tau_{i, j}=\frac{c_{j, w} c_{i, p}}{c_{j, p} c_{i, w}}-1\end{equation}

where $\tau_{i, j}$ is the mass transfer coefficient of element j relative to reference element i. cj,w and cj,p are the concentration of element j in weathered soil and in parent rock, respectively. ci,w and ci,p are the concentration of element i in weathered soil and in parent rock, respectively. The element i is considered as an immobile reference element to exclude the effects of the physical forces such as expansion/compaction. Titanium and zirconium are usually considered as immobile elements in calculating mass transfer coefficient. A negative $\tau_{i, j}$ value indicates mass loss while a positive $\tau_{i, j}$ value indicates mass enrichment for element j relative to element i. When $\tau_{i,j}=-1$, element j has been completely depleted as a result of the chemical weathering process. In calculating $\tau$ of K and Si relative to Zr, it is typically assumed that the soil column always has a constant concentration of immobile reference element Zr, which means $\frac{c_{Z r, p}}{c_{Z r, w}}=1$ all the time.