## Example Simulation

### Model Parameters

In this section, we simplify the shale gas reservoir with complex micro-scale fractures into a combination of as a dual porosity continuum media and a discrete fracture media. Based on the discrete fracture model, the artificial fracture can be simplified as a surface element by using a reduction dimensional method. The data for the formation and well properties used in the simulations are shown in Table 3.

**Table 3.** The parameter set for the shale gas reservoir model.

The spatial arrangement of multi-section fractured horizontal wells is shown in Figure 6. The half-length of the artificial fractures is 200 m.

**Figure 6.** A numerical model of a multi-section fractured horizontal well.

### Results Analysis

#### Pressure Distribution in Artificial Fractures

Figure 7 shows the pressure distribution in the artificial fractures at the beginning of production. The pressure distribution in the artificial fractures is related to parameters such as fracture aperture and permeability. As can be seen from the figure, due to the high conductivity of the artificial fractures, the pressure in the fracture rapidly decreases. A drawdown pressure is created between the artificial fractures and the matrix–micro-fracture system, so that gas flows from the matrix–micro-fracture system into artificial fractures and gas is produced by production well.

**Figure 7.** Pressure distribution in artificial fractures. (a) T = 0 d; (b) T = 1 d; (c) T = 2 d; (d) T = 3 d.

#### Gas Desorption Process

Based on the physical model parameters, the production of shale gas multi-section fractured horizontal wells is simulated. It has been shown in Figure 8 that during the first three years of production, the decline of pressure in the reservoir is mainly concentrated in the area that is near the wellbore and the hydraulic fracture faces, while the pressure drop in the outer area is very small. It shows that the produced gas mainly comes from free gas and desorption gas in the stimulated volume.

**Figure 8.** Reservoir pressure distribution at different production times. (a) T = 1a; (b) T = 3a; (c) T = 5a; (d) T = 10a.

Figure 9 shows the average reservoir pressure, gas production rate, and cumulative gas production at different Langmuir volumes. We found that the desorption process has the effect of supplementing the reservoir pressure, but the effect is not significant. Since the gas production rate is affected not only by the physical properties of the reservoir, but also by the pressure distribution, the gas desorption process has limited supplementary effects on pressure, and the impact on the gas production rate is not significant. At the same time, as the Langmuir volume increases, the cumulative gas production gradually increases.

Figure 9 shows the average reservoir pressure, gas production rate, and cumulative gas production at different Langmuir volumes. We found that the desorption process has the effect of supplementing the reservoir pressure, but the effect is not significant. Since the gas production rate is affected not only by the physical properties of the reservoir, but also by the pressure distribution, the gas desorption process has limited supplementary effects on pressure, and the impact on the gas production rate is not significant. At the same time, as the Langmuir volume increases, the cumulative gas production gradually increases.

**Figure 9.** Influence of gas desorption on horizontal well productivity. (a) Average reservoir pressure at different Langmuir volumes; (b) Gas production rate and cumulative gas production at different Langmuir volumes.

#### The Klinkenberg effect and Diffusive Gas Transport

Figure 10a–c show the Knudsen number distribution of shale gas reservoirs in fractured horizontal wells at the same time of production, under different shale matrix permeabilities. As can be seen from the figure, the closer to the artificial fractures, the larger the Knudsen number. This is due to the negative correlation between Knudsen number and pressure, so the lower the pressure, the larger the Knudsen number. At the same time, when the shale permeability decreases, the pressure drop of the artificial fractures becomes larger and the pressure drop funnel becomes steeper. Therefore, the closer the pressure gets to artificial fractures, the greater the increase of the Knudsen number.

**Figure 10.** Field distribution of Knudsen number and matrix pressure in shale gas reservoir. (a) km = 10^{−5} mD, Kn; (b) km = 10^{−4} mD, Kn; (c) km = 10^{−3} mD, Kn; (d) km = 10^{−5} mD, pm; (e) km = 10^{−4} mD, pm; (f) km = 10^{−3} mD, pm.

Figure 10d–f show the matrix pressure distribution of shale gas reservoirs in fractured horizontal wells at the same time of production, under different shale matrix permeabilities. It can be seen from the figure that the pressure of the artificial fractures falls fastest, and the closer to artificial fractures, the lower reservoir pressure. Comparing the reservoir pressures under different shale permeability conditions, the lower the shale permeability, the faster the pressure of the artificial fractures drops and the fewer reservoirs are used, resulting in steeper pressure drop funnels. This is because for shale reservoirs with low permeability, it is difficult for gas to flow in such dense porous media, so the gas stored in the shale cannot be added to the artificial fractures in time when the gas in the fracturing fractures. When the gas in the artificial fractures is recovered, the pressure in the fracture rapidly decreases. Compared to shales containing nano-micro pores, gases stored in the fractures and the region near fractures are more likely to be produced to make the pressure drop faster.

Figure 11 shows the curve of the gas production rate and cumulative gas production for multi-section fractured horizontal wells with different shale permeability. As can be seen from the figure, the gas production rate and cumulative gas production increase with the increase of shale permeability, and the growth rate also increases. However, compared with the production rate and cumulative production (without considering diffusion and slippage effects), the increment of gas production rate and cumulative production (considering the diffusion and slippage effects), decreases with increasing shale permeability. This shows that when shale permeability becomes smaller (pore size decreases), Knudsen diffusion and slippage effects have a greater impact on the daily gas production and the cumulative production of fracturing horizontal wells.

**Figure 11.** The effect of Knudsen diffusion and the Klinkenberg effect on productivity. (a) Production rate at different matrix permeability; (b) Cumulative gas at different matrix permeability.

#### Artificial Fracture Morphology

Based on the above numerical model, we change the number of fractured sections (Figure 12a–c) and the half-length of artificial fractures (Figure 12d–f) to simulate the production of shale gas. It can be seen from Figure 12g that as the half-length of artificial fractures increases, the gas production rate and cumulative gas production also increase. However, the increasing rate in the gas production rate and cumulative gas production has gradually decreased.

The main reason for this is that as the half-length of the artificial fractures increases, the multi-fracture interference becomes severer. Therefore, as the half-length of artificial fractures increases, the increasing rate in the gas production rate and cumulative gas production decreases.

**Figure 12.** The effect of horizontal well parameters on productivity. (a) section number = 3; (b) section number = 5; (c) section number = 7; (d) fracture half-length = 100 m; (e) fracture half-length = 150 m; (f) fracture half-length = 200 m; (g) Three fractured sections, T = 5a; (h) Four fractured sections, T = 5a.

From Figure 12h, it can be seen that the number of sections has an important influence on the gas production rate and cumulative gas production. With the increase in the section number, the gas production rate and cumulative gas production also increase. It is worth noting that as the section number increases, the rate of decline in gas production rate also increases. Similar to the previous situation, the main reason is that with the increase of the section number, the multi-fracture interference becomes severer. As a result, the larger section number, the faster the gas production rate declines.

Through analysis, it is found that excessive half-length of fractures and section numbers will generate strong multi-fracture interference, which will have a negative impact on the productivity of horizontal wells.

Therefore, for a horizontal well fracturing design, the half-length of fractures and section number should not be pursued blindly, but the parameters of horizontal wells should be optimized to reduce the multi-fracture interference.

## Conclusions

In this paper, based on the matrix–micro-fracture continuous dual model and discrete fracture model, a mathematical model of the shale gas reservoir considering a multi-scale flow mechanisms is established.

The numerical calculation format using a cell-centered variable arrangement of shale gas three-dimensional flow based on the finite volume element method is deduced. In this case, the variables and their associated quantities are stored in the centroids of the control elements. Thus, the elements are the same as the discretization elements; in general, the method is second-order accurate, because all quantities are calculated at the element and face centroids.

Talyor series expansion can be used to reconstruct the variations within the cell. Another advantage of the cell-centered formulation is that it allows the use of general polygonal elements without the need for pre-defined shape functions. This permits a straightforward implementation of a full multigrid strategy.

The artificial fracture is expressed by the two-dimensional surface, and the wellbore is expressed by a one-dimensional solid based on the dimension reduction method. The finite volume element method is used to solve the multi-section fractured horizontal well productivity and pressure distribution.

Through the analysis of the simulation results, it is found that the model can reflect the initial production of shale gas and its characteristics of rapid decline. The analysis shows that the gas desorption of shale gas has a great impact on reserves, which in turn have a supplementary effect on the reservoir pressure. On one hand, with the prolongation of production time, the proportion of desorption is increased. On the other hand, shale gas production is mainly affected by the scope of stimulated volume.

According to the development process of shale gas reservoirs, a numerical model of a stimulated reservoir volume fractured horizontal well is established. The analysis shows that the pressure will rapidly decrease in artificial fractures.

The desorption process has a great influence on the geological reserves, but has a limited impact on the productivity of horizontal wells. With the decrease of the pore size of the matrix, the Klinkenberg effect and gas diffusion have a greater impact on shale gas productivity. When the matrix permeability is greater than 0.01 mD, those flow mechanisms has no significant effect on the productivity. Compared with the fracture half-length, the section number has a greater impact on the productivity of shale gas. However, the excessive half-length of the fracture and the section number all induce multi-fracture interference. Therefore, the horizontal well parameters need to be optimized.

The parameters of the artificial fracture network can be conveniently adjusted and the factors affecting the productivity can be analyzed. The research content of this paper has certain theoretical and practical significance for the volume fractured design of shale gas reservoirs and the reasonable evaluation of production capacity.

### Author Contributions

Conceptualization, X.C.; Data curation, C.T.; Formal analysis, C.T.; Investigation, C.T.; Resources, X.C.; Software, P.Y.; Supervision, Z.D.; Validation, J.W.; Writing–original draft, C.T.; Writing–review & editing, J.W.

#### Funding

This work was supported by the visiting scholar program by the China Scholarship Council (No. 201608515035) and Texas Tech University, National Major Projects China (No. 2016ZX05048-002), National Major Projects China (No. 2016ZX05010-002-002), The Fund of SKL of Petroleum Resources and Prospecting, Beijing (No. PRP/open 1501), and the National Natural Science Foundation of China (Grant No. 51474179).

#### Conflicts of Interest

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

### Nomenclature

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