## 4 Results and Analysis
### Seepage Simulation and Calculation of Absolute Permeability Coefficient Tensor
#### 4.3.2 Absolute Permeability Tensor Calculation  
When calculating the absolute permeability coefficient tensor, the boundary conditions were set in the same way as those in the seepage simulation, except that periodic boundary conditions were imposed on the bottom inlet boundary III\({}^{\prime}\) and outlet boundary I\({}^{\prime}\). In order to make the model converge when calculating, the maximum number of iterations was set to 100,000 and the convergence accuracy was set to 0.001 \(\upmu\)m\({}^{2}\). The absolute permeability tensor calculation module in AVIZO software was run, as shown in Figure 15. After 65,540 iterations to finally reach the iteration accuracy, the vertical absolute permeability coefficient tensor of the loess in the simulated area was finally calculated, as shown in Table 3. The absolute permeability coefficient of the Malan loess sample was 0.2848 \(\upmu\)m\({}^{2}\) in the X-direction, 0.03156 \(\upmu\)m\({}^{2}\) in the Y-direction, and 0.07378 \(\upmu\)m\({}^{2}\) in the Z-direction. X-direction (vertical) absolute permeability coefficient was 9.02 times higher than the Y-direction (horizontal) absolute permeability coefficient and 3.86 times higher than the Z-direction (horizontal) absolute permeability coefficient. The absolute permeability coefficients along the Z-direction were on the same order of magnitude as those along the Y-direction.  
After obtaining the value of absolute permeability k, the permeability coefficient \(K\) of the loess sample can be calculated based on Equation (7), where \(k\) is the absolute permeability (unit: \(\upmu\)m\({}^{2}\)), \(\rho_{w}\) is the density of water (unit: g/cm\({}^{3}\)), and \(\mu\) is the dynamic viscosity of the liquid (unit Pa-s).  
\[K=k\frac{\rho_{w}g}{\mu} \tag{7}\]  
The X-direction (vertical) permeability coefficient of Malan loess sample was 2.791 \(\times\) 10\({}^{-4}\) cm/s, Y-direction (horizontal) permeability coefficient was 3.093 \(\times\) 10\({}^{-5}\) cm/s, and Z-direction (horizontal) permeability coefficient was 7.23 \(\times\) 10\({}^{-5}\) cm/s. The absolute permeability of Malan loess was calculated from 0.13 to 7.81 \(\upmu\)m\({}^{2}\) by the pore network  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{k_{x}}\) (\(\upmu\)m\({}^{2}\)) & \(\mathbf{k_{y}}\) (\(\upmu\)m\({}^{2}\)) & \(\mathbf{k_{z}}\) (\(\upmu\)m\({}^{2}\)) & **Eigen Value** \\ \hline
0.284803 & \(-\)0.030083 & \(-\)0.124343 & 0.346743 \\ \(-\)0.029247 & 0.031562 & 0.025184 & 0.034523 \\ \(-\)0.123694 & 0.28359 & 0.07378 & 0.008879 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Absolute permeability tensor.  
Figure 15: Iterative accuracy of the permeability coefficient tensor.  
model method by [PERSON] et al. The permeability coefficients of Malan loess ranged from 1.28 \(\times\) 10\({}^{-4}\) cm/s to 7.65 \(\times\) 10\({}^{-3}\) cm/s. In this study, the absolute permeability coefficients and permeability coefficients of Malan loess samples were calculated within this interval using this method of volume averaging, which indicates that it is feasible to calculate the absolute permeability of Malan loess using the volume averaging method.