# How Insoluble Inclusions and Intersecting Layers Affect the Leaching Process within Potash Seams
## 2 Materials and Methods
### Extended Interchange Approach
Equation (4) is used to determine the amount of mineral \(n\) dissolved from \(Min_{dry}\) into the solution within the permeated part:  
\[Min_{sol,dry,n}=rate_{n}\cdot Min_{dry,n}/\left(di\cdot\left(1-S_{per}\right) \right)\cdot\Phi_{per}\cdot S_{per}\cdot dt \tag{4}\]  
Generally, Equation (4) is identical to Equation (2), but multiplied by the permeation state \(S_{per}\). _rate_ is calculated from the saturation state within the partly permeated cell using Equation (1). In order to determine the affected amount of mineral \(n\) if a width of 1 cm of the dry part is dissolved, the present amount, \(Min_{dry,n}\) (mol), is divided by the diameter of the dry part perpendicular to the dissolution front. Therefore, the cell diameter \(di\) (cm) is now multiplied by the volume ratio of the dry part (\(I\)-\(S_{per}\)). The porosity within the dry part is nearly zero, simplifying the term (\(\Phi_{B}\)-\(\Phi_{A}\))/(\(I\)-\(\Phi_{A}\)) in Equation (2) to \(\Phi_{B}\), which equals the average porosity of the permeated part \(\Phi_{per}\). The dissolved amount of each mineral \(Min_{sol,dry}\) is added to \(Min_{eq}\). The same applies to precipitations resulting from the equilibration after the interchange (Figure 1). Similar to the interchange of dry minerals, the highest dissolution rate according to Equation (1) determines the volume that is newly added to the permeated part \(V_{per}\) (Figure 3). For minerals with lower dissolution rates, the difference between the amount within this volume and the dissolved amount is added to \(Min_{eq}\). Finally, the internal dissolution of non-equilibrated minerals (\(Min_{neq}\)) has to be taken into account (Figure 3b). These minerals belong to the permeated part of a cell and are already surrounded by solution. However, when the area is permeated for the first time, their dissolution rate is too low to dissolve all minerals. Basically, there are two possible reasons for this: either the maximum dissolution rate \(k_{max}\) is (much) lower than that of the other minerals, or the solution is already (nearly) saturated with these. In the latter case, one could add them to \(Min_{eq}\) without changing the solution or rock composition of the permeated part. However, the solution composition changes over time, and originally saturated minerals may become undersaturated. In this case, an immediate equilibration could induce errors because the lower dissolution rates would be neglected. To account for the variation of dissolution rates, \(Min_{neq}\) is added stepwise to the solution. The dissolved amount \(Min_{sol,neq}\) (mol) of each mineral is determined by Equation (5):  
\[ \begin{array}{ll}Min_{sol,neq,n}&=rate_{n}\cdot Min_{neq,n}/V_{neq}\cdot contact \_area\cdot dt\\ &=rate_{n}\cdot Min_{neq,n}/V_{neq}\cdot factor\cdot di\cdot 1m\cdot\Phi_{eq} \cdot S_{per}\cdot dt\end{array}  \tag{5}\]  
The rate law is similar to those used for interchange (Equation (2)) and the internal dissolution of dry minerals (Equation (4)). Again, _rate_ (cm/s) is calculated from the saturation state within the partly permeated cell according to Equation (1). \(Min_{neq}\) (mol) divided by the volume of the non-equilibrated minerals \(V_{neq}\) (cm\({}^{3}\)) and multiplied by the contact area between non-equilibrated minerals and solution (cm\({}^{2}\)) gives the amount of mineral \(n\) if a width of 1 cm of the non-equilibrated minerals is dissolved. \(V_{neq}\) is calculated from the mineral amounts and densities in \(Min_{neq}\), neglecting small porosities. However, the contact area between non-equilibrated minerals and solution is hard to determine, because the pore structure and spatial arrangement of the different minerals in the original rock are unknown. Therefore, a variable called _factor_ (-) is introduced. If it is set to 1, a rectangular contact area similar to that between \(Min_{dry}\) and the solution (Equation (4)) is assumed. This can be seen as a minimum value.