# How Insoluble Inclusions and Intersecting Layers Affect the Leaching Process within Potash Seams
## 3 Results
### Peclet and Damkohler Numbers  
In order to determine whether the systems are transport- or reaction-dominated, and if the transport is dominated by advection or diffusion, the dimensionless parameters of Peclet (Pe) and Damkohler (Da) can be used, allowing for a classification into four different cases which show different temporal and spatial evolutions of the leaching zone [6]. The Peclet number is determined from the flow velocity \(v\), the diffusion coefficient \(D_{f}\) and the characteristic length \(l\), following Equation (7). Pe has to be calculated individually for each cell, because \(v\) varies in space (and time). \(l\) is approximated with the current width of the leaching zone, which varies over time in the vertical direction. \[Pe=v\cdot l/D_{f} \tag{7}\]  
To determine if the system was dominated by advection (Pe > 2) or diffusion (Pe < 2), the median of all permeated cells was determined. All homogeneous and heterogeneous scenarios showed a median Peclet number which was clearly above two over the entire modeling period, i.e., all systems were advection-dominated. This means that diffusion was negligible as a transport process within the leaching zone, and the Damkohler number, which is defined as the ratio between reaction rate and transport velocity, could be calculated from the flow velocity \(v\) according to Equation (8). To calculate the reaction rate of each mineral, Equation (1) was used, considering the saturations of the inflowing solution. The studies of [PERSON] et al. [33] indicate that at the same flow velocity, low saturations cause transport-dominated systems (Da > 1), whereas high saturations result in reaction-dominated systems (Da < 1). The solution at the left boundary was NaCl-saturated; therefore, the reaction rate for halite was zero. In the cases of sylvite and carnallite, _rate_ corresponds to \(k_{max}\) because the inflowing solution is highly undersaturated with respect to both minerals. The transport velocity corresponds to the flow velocity calculated by TRANSE. \[Da=reaction\ rate/transport\ velocity=k_{max}/v\qquad\qquad(if\ Pe>2) \tag{8}\]  
A Damkohler number for the system can be derived from \(k_{max}\) and the median of all flow velocities within the permeated area. The latter is divided into upper and lower halves in cases of intersecting halite layers. Figure 8 shows that the median always ranges between \(10^{-6}\) m/s and \(10^{-8}\) m/s. This means that for \(k_{max}\) = 5 \(\times\)\(10^{-4}\) cm/s, Da is always above one, whereas for \(k_{max}\) = 5 \(\times\)\(10^{-6}\) cm/s, Da falls below one if the flow velocity is above 5 \(\times\)\(10^{-8}\) m/s. In the case of homogeneous potash seams, this holds true during the first 2 years of simulation (Figure 8a,b). After that, the decrease in flow velocity results in a Damkohler number slightly above one, eventually equaling 1.4. If the potash seam is intersected by a horizontal halite layer, it takes 3 years to reach Da = 1 because the average flow velocity is slightly higher at the beginning (Figure 8a). Later, its decrease is stronger; thus, Da = 1.4 is eventually reached as well. Similar evolutions can be observed within the upper half of the potash seam if the intersecting halite layer is inclined. However, after 7-8 years, the flow velocity decrease becomes faster and Da becomes higher. On the other hand, flow velocities within the lower half are constantly lower for a +10\({}^{\circ}\) inclination and constantly higher for a \(-10^{\circ}\) inclination. In the latter case, Da rises above one only after 13 years. In the case of halite inclusions, it partly takes more than 2 years to reach Da = 1, e.g., for the 17:0.5 distribution, and the evolution of the average flow velocity is less regular, resulting in several crossings of Da = 1 over time, e.g., for the 34:1 distribution (Figure 8b). However, the overall evolution of flow velocities is relatively similar to the homogeneous case. In contrast, \(k_{max}\) = 5 \(\times\) 10\({}^{-4}\) cm/s leads to larger deviations in flow velocity (Figure 8c,d).