# Differences in Performance of Models for Heterogeneous Cores during Pulse Decay Tests
## 4 Comparison of Dual-Porosity and Dual-Permeability Models
### Effect of Matrix-Fracture Permeability Ratio  
The interporosity flow coefficient is defined as follows [30]:  
\[\lambda=\alpha L^{2}\frac{k_{\text{mi}}}{k_{\text{f}}}=\alpha L^{2}\frac{k_{ \text{m}}}{\beta k_{\text{f}}}, \tag{1}\]  
where \(k_{\text{mi}}\) is the intrinsic permeability (m\({}^{2}\)) of the matrix. In the dual-permeability model, \(k_{\text{m}}\) is the average permeability of the cross section. \(\beta\) is the ratio of the cross-sectional area occupied by the matrix. For the layered matrix, its shape factor can be written as [30]  
\[\alpha=\frac{12}{h_{\text{m}}^{2}}, \tag{2}\]  
Figure 7: Effect of the storativity ratio on pulse decay tests for dual-permeability cores. (**a**) Pressure history. (**b**) Pressure derivative history.  
where \(h_{\rm m}\) is the thickness of the layered matrix (m). Therefore, the following holds.  
\[\lambda=\frac{12L^{2}}{\beta h_{\rm m}^{2}}\frac{k_{\rm m}}{k_{\rm f}}. \tag{3}\]  
Generally, the diameter of the sample is 0.5 to 1 times its length, and \(h_{\rm m}\) is less than half of the diameter. The cross-sectional area of the low permeability layer is generally larger than the cross-sectional area of the high permeability layer, at least not too much lower. Therefore, \(\beta\) is generally not less than 0.5. If the sample has only one penetrating fracture and the diameter is equal to the length, then \(h_{\rm m}=0.5L\). Assuming \(k_{\rm m}/k_{\rm f}=10^{-2}\), \(\lambda\) can reach 0.98. It both \(k_{\rm m}\) and \(k_{\rm f}\) are average cross-sectional permeabilities. If the sample has multiple layers of alternating high and low permeability layers, \(h_{\rm m}\) is smaller, and \(L/h_{\rm m}\) will take a larger value. Therefore, the interporosity flow coefficient \(\lambda\) may have a higher value, not the same as the range of interporosity flow coefficients at the reservoir scale.  
The numerical modeling results of pulse decay tests for dual-permeability and dual-porosity models with different matrix-fracture permeability ratios are shown in Figure 8. When the interporosity flow coefficient is small (\(\lambda=0.01\)), the pressure and pressure derivatives of the two models coincide in the early (\(t_{\rm D}<1\)) and equilibrium (\(t_{\rm D}>10^{3}\)) stages, and they only have a slight difference in the interporosity flow (\(1<t_{\rm D}<10^{3}\)) stage. When the matrix-fracture permeability ratio is less than \(10^{-5}\), the results of the two models coincide substantially. If the interporosity flow coefficient is large (\(\lambda=1\)), the interporosity flow occurs in the early stage (\(t_{\rm D}<1\)), and the matrix-fracture permeability has the same effect on the pressure and pressure derivative as the interporosity flow coefficient is small. Only when the value is less than \(10^{-3}\) do the pressure and pressure derivative curves coincide for the two models. Even when the results of the two models do not completely coincide, their difference is very small. Therefore, when the matrix-fracture permeability ratio is less than \(10^{-2}\), the difference between the dual-permeability and dual-porosity models can be ignored.  
Figure 8: _Cont._  
Figure 8: Effect of matrix-fracture permeability ratio on pulse decay tests for dual-permeability cores. (**a**) Pressure history for \(\lambda=0.01\). (**b**) Pressure derivative history \(\lambda=0.01\). (**c**) Pressure history for \(\lambda=1\). (**d**) Pressure derivative history for \(\lambda=1\).