# How Insoluble Inclusions and Intersecting Layers Affect the Leaching Process within Potash Seams
## 2 Materials and Methods
### Extended Interchange Approach
However, Mineral 3 has a lower dissolution rate, and can therefore only be equilibrated with the solution in the pink subregion. Following this reasoning, one would end up with an additional subregion for each mineral, which is of course not practical. Instead, the permeated part within each cell is split up into minerals that are in equilibrium with the solution (equilibrated minerals \(Min_{eq}\)) and such that are not (non-equilibrated minerals \(Min_{eq}\); Figure 3b). Thereby, the volume of the permeated part \(V_{per}\) is determined by the most soluble mineral (Mineral 2, green area). Outside of this area, the salt rock is still dry and the composition \(Min_{dy}\) corresponds to the originally unaffected potash seam. In contrast, minerals within the permeated area \(V_{per}\) are surrounded by solution, whereby it is not yet in equilibrium with all minerals. Minerals that remain undissolved during the interchange due to their lower dissolution rates (Mineral 1 and parts of Mineral 3 within the green area, Figure 3a) are assigned to the non-equilibrated minerals \(Min_{eq}\). In contrast, precipitations from Solution B belong to the minerals equilibrated with the solution \(Min_{eq}\). In the case of partly permeated cells, three different types of dissolution can occur (Figure 3b): the first one is interchange with the (fully permeated) adjacent cells according to Equation (2). For this, all three mineral compositions \(Min_{dy}\), \(Min_{eq}\) and \(Min_{eq}\), are added together to determine \(Min_{A}\). \(\Phi_{A}\) is the average porosity of the cell calculated from the volume of \(Min_{A}\) related to the cell volume. If the partly permeated cell shows a higher average porosity than the fully permeated one, the interchange of minerals is possible in the reverse direction as well. The dissolved amount of each mineral is split among \(Min_{dy}\), \(Min_{eq}\) and \(Min_{eq}\), according to their contribution to the total amount of this mineral. The solution transferred in return is mixed with the pre-existing solution in the permeated part. The chemical reactions resulting from the mixing and the following equilibration with \(Min_{eq}\) are determined after the interchange (Figure 1). In addition to the interchange, internal dissolution processes within cells containing dry (\(Min_{dy}\)) or non-equilibrated (\(Min_{eq}\)) minerals need to be taken into account (Figure 3b). The interface area between a dry part and the adjacent permeated part(s) is assumed to be constant over time. At the beginning, when the cell is completely dry, the adjacent permeated area is the neighbor cell and dry minerals can only be dissolved via interchange. Accordingly, the interface area is the interface between both cells, which is the cell width multiplied by its height. As \(V_{per}\) increases, \(Min_{dy}\) makes up a smaller part of the total mineral amount within the partly permeated cell and its interchange amount decreases. At  
Figure 3: Sketch of the interchange at the dissolution front (**a**) resulting in a partly permeated cell (**b**). The permeated part \(V_{per}\) contains minerals \(Min_{eq}\) that are in equilibrium with Solution A and minerals \(Min_{eq}\) that are not. In addition to interchange with adjacent Cells (1), the internal dissolution of non-equilibrated minerals (2) and dry Minerals (3) is considered following Equations 2,4,5, (modified from [PERSON] et al. [6]). the same time, the interface area between the dry and permeated parts within the partly permeated cell increases, enhancing the internal dissolution of dry minerals. To account for that, the permeation state of the cell \(S_{per}\) (-) is calculated referring to the volume of the permeated part \(V_{par}\) in relation to the cell volume \(V_{cell}\):  
\[S_{per}=V_{per}/V_{cell} \tag{3}\]  
The higher the \(S_{per}\), the more the dry rock is dissolved into the solution of its own cell (by internal dissolution) instead of the adjacent cell (by interchange).