# Differences in Performance of Models for Heterogeneous Cores during Pulse Decay Tests
## Appendix B Dual-Permeability Model and Its Numerical Solution
### Numerical Method  
The sample is divided into \(N\) segments with spatial interval \(\Delta x_{\text{D}}\). Assuming the time interval is \(\Delta t_{\text{D}}\), and the governing Equations (100) and (101) can be discretized into  
\[\frac{1}{\Delta x_{\text{D}}}\Bigg{(}\frac{p_{\text{fD},j+1}^{n+1}-p_{\text{ fD},j}^{n+1}}{\Delta x_{\text{D}}}-\frac{p_{\text{fD},j}^{n+1}-p_{\text{fD},j-1}^{n+ 1}}{\Delta x_{\text{D}}}\Bigg{)}+\lambda_{\text{D}}\Big{(}p_{\text{mD},j}^{n +1}-p_{\text{fD},j}^{n+1}\Big{)}=\omega\frac{p_{\text{fD},j}^{n+1}-p_{\text{ fD},j}^{n}}{\Delta t_{\text{D}}}, \tag{104}\]  
\[\frac{1}{\Delta x_{\text{D}}}\frac{k_{\text{m}}}{k_{\text{f}}}\Bigg{(}\frac{p _{\text{mD},j+1}^{n+1}-p_{\text{mD},j}^{n+1}}{\Delta x_{\text{D}}}-\frac{p_{ \text{mD},j}^{n+1}-p_{\text{mD},j-1}^{n+1}}{\Delta x_{\text{D}}}\Bigg{)}- \lambda_{\text{D}}\Big{(}p_{\text{mD},j}^{n+1}-p_{\text{fD},j}^{n+1}\Big{)}=( 1-\omega)\frac{p_{\text{mD},j}^{n+1}-p_{\text{mD},j}^{n}}{\Delta t_{\text{D}}} \tag{105}\]  
Boundary conditions can be discretized into  
\[\frac{p_{\text{fD},1}^{n+1}-p_{\text{fD},0}^{n+1}}{\Delta x_{\text{D}}}A_{ \text{u}}+\frac{p_{\text{mD},1}^{n+1}-p_{\text{mD},0}^{n+1}}{\Delta x_{\text{D }}}A_{\text{u}}\frac{k_{\text{m}}}{k_{\text{f}}}=\frac{p_{\text{uD}}^{n+1}-p _{\text{uD}}^{n}}{\Delta t_{\text{D}}}, \tag{106}\]  
\[-\frac{p_{\text{fD},N}^{n+1}-p_{\text{fD},N-1}^{n+1}}{\Delta x_{\text{D}}}A_{ \text{d}}-\frac{p_{\text{mD},N}^{n+1}-p_{\text{mD},N-1}^{n+1}}{\Delta x_{\text{ D}}}A_{\text{d}}\frac{k_{\text{m}}}{k_{\text{f}}}=\frac{p_{\text{dD}}^{n+1}-p_{ \text{dD}}^{n}}{\Delta t_{\text{D}}}. \tag{107}\]  
Equations (104) and (105) can be simplified to  
\[-\tau p_{\text{fD},j+1}^{n+1}+(2\tau+\omega+\lambda_{\text{D}}\Delta t_{\text {D}})p_{\text{fD},j}^{n+1}-\tau p_{\text{fD},j-1}^{n+1}-\lambda_{\text{D}} \Delta t_{\text{D}}p_{\text{mD},j}^{n+1}=\omega p_{\text{fD},j}^{n}, \tag{108}\]  
\[-\tau\frac{k_{\text{m}}}{k_{\text{f}}}p_{\text{mD},j+1}^{n+1}+\left(2\tau\frac {k_{\text{m}}}{k_{\text{f}}}+\lambda_{\text{D}}\Delta t_{\text{D}}+1-\omega \right)p_{\text{mD},j}^{n+1}-\tau\frac{k_{\text{m}}}{k_{\text{f}}}p_{\text{mD},j -1}^{n+1}-\lambda_{\text{D}}\Delta t_{\text{D}}p_{\text{fD},j}^{n+1}=(1- \omega)p_{\text{mD},j}^{n} \tag{109}\]  
Boundary conditions Equations (107) and (108) can be simplified to  
\[-\theta A_{\text{u}}p_{\text{fD},1}^{n+1}-\theta A_{\text{u}}\frac{k_{\text{m}} }{k_{\text{f}}}p_{\text{mD},1}^{n+1}+\left(\theta A_{\text{u}}+\theta A_{\text{u }}\frac{k_{\text{m}}}{k_{\text{f}}}+1\right)p_{\text{uD}}^{n+1}=p_{\text{uD}}^{n}, \tag{110}\]  
\[\left(\theta A_{\text{d}}+\theta A_{\text{d}}\frac{k_{\text{m}}}{k_{\text{f}}}+1 \right)p_{\text{dD}}^{n+1}-\theta A_{\text{d}}p_{\text{fD},N-1}^{n+1}-\theta A_ {\text{d}}\frac{k_{\text{m}}}{k_{\text{f}}}p_{\text{mD},N-1}^{n+1}=p_{\text{dD}}^ {n}. \tag{111}\]  
In the above equation, \(p_{\text{mD},0}=p_{\text{fD},0}=p_{\text{u}}\), \(p_{\text{mD},N}=p_{\text{fD},N}=p_{\text{d}}\). the above equations can form a linear equation of rank \(2(N+1)\) and be solved by a numerical method.