## 4 Stability Analysis of Difference Equations
### Results and Discussion  
By comparing the simulated value with the physical one, mutual verification of digital and physical models can be performed. In the comparison, the upstream pressures under four gradients are taken. They are 0 MPa, 10 MPa, 20 MPa, and 30 MPa, respectively. The time to get the last data point is 30,000 s, 18,000 s, 25,200 s, and 25,200 s, so the numerical simulation is the same as the measured time in the laboratory. A comparison of the recovery  
Figure 5: Dense matrix pore network model. (**a**) Matrix block generated by the algorithm, and (**b**) pore network inside the matrix block.  
Figure 6: Example of pressure field simulation in a depletion process. (**a**) Pressure distribution (3 days), and (**b**) pressure distribution (6 days).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Level Factor**} & \multirow{2}{*}{**Pressure (MPa)**} & \multirow{2}{*}{**Viscosity (mPa-s)**} & \multirow{2}{*}{**Permeability (mD)**} & \multirow{2}{*}{**Length (m)**} & \multirow{2}{*}{**Time (h)**} \\ \hline
1 & 10 & 0.6 & 0.075 & 3.25 & 2.5 \\
2 & 20 & 1.2 & 0.15 & 6.5 & 5 \\
3 & 30 & 1.8 & 0.225 & 9.75 & 7.5 \\
4 & 40 & 2.4 & 0.3 & 13 & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Influencing factors and level selection table.  
ratio of the theoretical and measured curves under upstream pressure was made and shown in Figure 7.  
From the comparison of Figure 7, it can be observed that the established numerical model is very close to the actual core physical properties. The final exhaustion yields are basically coincidental, which indicates the effectiveness of digital core technology.  
(1) The theoretical and measured recovery factors have a large error in the early stage of exhaustion mining. However, as the mining time increases, the theoretical model and the measured recovery data begin to approach. At the end of exhaustion, the theoretical and measured results are basically coincident. The reason for this is that the digital core ignores interfacial tension between gas and oil, thus, resulting in a different flow state at the end of a digital one compared to a physical one.  
(2) The relative error between the numerical model and physical experiment is not large, but the four sets of contrast curves show the high pressure of the previous theoretical model. The reason is that, in the early stage of exhaustion mining, the pressure drop of the theoretical model has not yet propagated upstream. Therefore, the upstream pressure remains unchanged during the next period of mining start.  
Additionally, during the numerical simulation, an interesting phenomenon in which some cracks have both open and closed states is observed. This phenomenon gives a microscopic interpretation clearly of the starting pressure gradient of a tight reservoir. As shown in Figure 8a, a plan view of the model at a pressure differential of 10 MPa (b) corresponding to a plan view of the model at a pressure of 40 MPa. It can be observed that the micro-cracks within the marked range are in a closed state when the pressure difference of failure is 10 MPa. However, it is turned on under the depletion pressure difference of 40 MPa. It explains, at a microscopic level, why there is a starting pressure gradient in a tight sandstone reservoir in the case of a low pressure difference.  
Furthermore, 16 calculations of the orthogonal test were performed according to the selected five factors and four levels, and the simulation results are shown in Table 4.  
The calculation results on the orthogonal test were analyzed by considering the absolute deviation of each parameter, and the results are listed in Table 5.  
It can be seen from the results in Table 5 that the change of pressure has the largest influence on tight oil recovery and the change of viscosity has the smallest one. Furthermore, the parameters in order of sensibility are pressure, length, permeability, time, and viscosity.  
Figure 7: Comparison of numerical simulation and physical experiment under different upstream pressures.