# How Insoluble Inclusions and Intersecting Layers Affect the Leaching Process within Potash Seams
## 2 Materials and Methods
### Extended Interchange Approach  
In [PERSON] et al. [6], constant dissolution rates were used to calculate the interchange of salt minerals. However, it is known that close to equilibrium, the dissolution rates decrease considerably [16; 17; 18]. Based on the results of [PERSON] et al. [19], a linear saturation dependency is assumed:  
\[rate=k_{max}\cdot(1-\Omega)\qquad\quad with\ \Omega=\frac{Q}{K}=10^{SI} \tag{1}\]  
where \(k_{max}\) (cm/s) is the dissolution rate in an infinitely dilute solution. Q is the activity product and K the equilibrium constant. The saturation index SI (-) results from log(Q/K). Leaching zones can contain several primary and secondary minerals that differ in maximum dissolution rate \(k_{max}\) as well as in saturation state [20]; therefore, an average dissolution rate for the potash seam (Figure 2a) is not applicable here. Instead, the dissolution rate according to Equation (1) and the amount of minerals dissolved into an adjacent cell \(Min_{sol}\) (mol) need to be calculated individually for each mineral (Figure 2b). Thereby, \(rate\) (cm/s) always refers to the saturation state of the solution into which the mineral is dissolved. Between Cell A and Cell B, the amount of interchanged mineral \(n\) can be calculated with Equation (2):  
\[Min_{sol,n}=rate_{n}\cdot Min_{A,n}/di\cdot(\Phi_{B}-\Phi_{A})/(1-\Phi_{A}) \cdot dt \tag{2}\]  
The amount of mineral \(n\) present in Cell A, \(Min_{A,n}\) (mol), divided by the cell diameter perpendicular to the dissolution front \(di\) (cm), gives the affected amount of mineral \(n\) if the width of 1 cm is dissolved from Cell A. To take into account the contact area between the rock in Cell A and the solution in Cell B, this amount is multiplied with the porosity difference between both cells (\(\Phi_{B}-\Phi_{A}\)) and divided by the volume fraction of the rock in Cell A (\(I-\Phi_{A}\)). If \(\Phi_{A}\geq\Phi_{B}\), the interchange is zero, i.e., minerals can only be dissolved into adjacent cells with higher porosity. \(dt\) (s) is the length of the simulation time step. If the solution in Cell B is already saturated with respect to \(n\), \(rate_{n}\) becomes zero and the mineral is not dissolved. The interchange takes place at each interface between two cells within the model. The volumes of all minerals dissolved at one interface are cumulated to determine the amount of solution that is transferred in return to keep the volume of both cells constant (Figure 2). At the dissolution front, the interchange approach enables originally dry cells to receive solution for the first time from an adjacent cell (Figure 3a). The solution needs to be undersaturated with at least one of the minerals from the dry rock; otherwise, no interchange takes place. Newly permeated cells are divided into a permeated and a dry sub-cell to avoid immediate drying out due to high solid-fluid-ratios (Figure 3b). In cases of constant dissolution rates, all minerals in the permeated part were assumed to be in equilibrium with the solution of the cell, whereas minerals in the dry part were not considered in the subsequent calculation of chemical reactions [6]. However, as shown in  
Figure 2: Sketch of the interchange approach with (**a**) constant and (**b**) variable dissolution rates. A defined amount of each mineral according to Equation (2) is dissolved from a cell with low porosity (Cell A) into an adjacent cell with higher porosity (Cell B). These amounts are cumulated, and the solution is transferred into the cell with lower porosity to fill the newly formed pore space (modified from [PERSON] et al. [6]). Figure 3a, the varying dissolution rates challenge the identification of a clear borderline between dry and permeated or equilibrated and non-equilibrated parts, respectively. It can be seen that Mineral 2 has the highest dissolution rate, and therefore limits the region of increased porosity where mineral solution occurs (green). Within this region, it is reasonable to equilibrate the solution with regard to Mineral 2.