## Discussion and Analysis

### Flow Regimes

In order to obtain the main flow regimes of the improved five-region flow model, the type curves of the pressure-transient response were plotted by employing pseudo-steady inter-porosity flow in the SRV region.

The related parameters are listed in Table 1. Figure 2 shows the pressure-transient response of MFHWs in shale gas reservoirs. There are five flow stages on the type curves: (1) bilinear flow in each hydraulic fracture and in the SRV region (region 1), where the pressure derivative curve’s slope is 1/4 (α = 1, and df = 2); (2) first linear flow in the SRV region, where the pressure derivative curve shows a straight line with a slope of 1/2 (α = 1, and df = 2); (3) inter-porosity and fractal-anomalous diffusion in the SRV region; (4) second linear flow from the USRV to SRV region, where the pseudo-pressure derivative curve presents a straight line with a slope of 1/2 (α = 1, and df = 2); and (5) pseudo-steady flow (boundary control flow), where the pseudo-pressure and pseudo-pressure derivative curves are all represented by straight lines with a unit slope.

**Figure 2.*** Transient pressure type curves of multiple fractured horizontal wells (MFHWs) in a shale gas reservoir.*

### Sensitivity Analysis

In the corresponding sensitivity analysis, firstly, one relevant parameter was changed while keeping the other parameters at their original values. Then, all the relevant parameters were changed at the same time. Model parameters were given values in the simulation by referring to relevant literature [6,12,16,17,23,25,26], and they are listed in Table 2.

**Table 2.** Model parameters.

Figure 3 shows that the fracture conductivity mainly affects the early flow stages. The greater the fracture conductivity is, the smaller the gas flow resistance is, and the smaller pressure consumption is with the same production. It is not difficult to see that the fracture conductivity mainly influences the pressure and pressure derivative curves in the bilinear flow and first linear flow stages. With an increase in the fracture conductivity, the duration of the bilinear flow stage decreases and the duration of the first linear flow stage increases. As seen in Figure 3, when FCD = 25, only the first linear flow regime can be observed.

**Figure 3.** Effect of fracture conductivity on type curves.

Figure 4 demonstrates the type curves of the pressure and pressure derivative for MFHWs in a shale gas reservoir with various anomalous diffusion exponent (α) and tortuosity index (θ) values. As can be seen, one intersection point exists between the anomalous diffusion and classical diffusion pressure derivative curves. At the early bilinear and linear flow stages, the pressure and pressure derivative for α < 1 or θ > 0 (anomalous diffusion) are smaller than those for α = 1 or θ = 0 (classical diffusion). When the value of α increases (θ decreases), the pressure and its derivative will also increase. The reason for this is that anomalous diffusion delays the performance of pressure derivative behaviors. However, after the inter-porosity flow stage, with different α values, the difference will be more obvious, and the trend is the opposite. In other words, a decrease in α (θ increasing) causes the pressure and its derivative to increase over time. This accounts for the characteristic of sub-diffusion (slower flow) when α < 1 or θ > 0 (anomalous diffusion).

**Figure 4.** Effect of the anomalous diffusion exponent on type curves.

Figure 5 shows that the mass fractal dimension of induced fractures (Hausdorff index) has a significant effect on the pressure behavior at almost all the stages, except for the wellbore storage stage. Overall, the smaller the mass fractal dimension is, the larger the gas flow resistance is and the greater the pressure consumption is with the same production. As can be seen, the locations of the type curves are higher with a smaller df. The reason for this is that a smaller df value represents more resistance in the complex induced fractures.

**Figure 5.** Effect of mass fractal dimension on type curves.

Figure 6, Figure 7 and Figure 8 demonstrate the influences of the adsorption factor, apparent permeability coefficient, and inter-porosity flow coefficient on the type curves of MFHWs. As shown in Figure 6, the adsorption factor mainly influences the position of the type curves at the inter-porosity flow stage.

A larger adsorption factor represents a stronger adsorption and production capacity and therefore makes the “concave” appear wider and deeper on the type curves. Figure 7 shows the effect of the apparent permeability coefficient on the transient pressure response. The apparent permeability has a similar effect to that of the inter-porosity coefficient in Figure 8. The total seepage and diffusion ability of the shale matrix is represented by the apparent permeability coefficient. The smaller the apparent permeability coefficient or inter-porosity coefficient is, the later the “depression” appears on the type curves.

**Figure 6.** Effect of the adsorption factor on type curves.

**Figure 7.** Effect of the apparent permeability coefficient on type curves.

**Figure 8.** Effect of the inter-porosity flow coefficient on type curves.

Figure 9 shows the impact of the stress sensitivity factor on the pressure-transient response of MFHWs. It can be seen that stress sensitivity affects the whole flow stage, and it has a greater impact in the late time period. The reason for this is that the pressure drop becomes greater in the late time period. The greater the stress sensitivity is, the higher the positions of the pressure and pressure derivative curves are. This depicts the weaker seepage capacity.

**Figure 9.** Effect of the stress sensitivity factor on type curves.

As shown in Figure 10, when all the factors are changed at the same time from a smaller parameter group (1) to a larger parameter group (2), the positions of type curves for parameter group (3) are obviously lower than the positions of type curves for parameter group (4). This indicates that when all the factors become larger, the final pressure drop becomes smaller. The reason for this is that most factors with greater values, such as FCD, α, df σm, βt, and λ, can have positive effects by making the pressure consumption smaller, and only γ∗D has the opposite influence on pressure and pressure derivatives.

**Figure 10.** Effect of characteristic factors on type curves (FCD, α, df, σm, βt, λ, and γ∗D).

## Case Study

This section shows an application of the presented model in a fractured horizontal well (A1) of an actual shale gas field in the Sichuan basin, which has 12 fractures evenly distributed along its horizontal wellbore. The depth of well A1 is 880 m and the thickness of the shale layer is 76 m. The production was 2400 cubic meters per day for 16 h, and then it was shut down for 73 h during the pressure build-up test. For more details, refer to the related literature [16]. After transferring the build-up testing data to dimensionless forms, the actual log-log curves were plotted.

As shown in Figure 11, the improved five-region flow model proposed in this work was applied to match the build-up testing data and was able to perfectly match the real testing data by adjusting the relevant parameters. The results of the interpretation are listed in Table 3.

The results reveal that hydraulic fracturing greatly increases the permeability of the fractured zone and produces complex induced fractures with fractal features.

**Figure 11.** Type curve matching for well A1.

**Table 3.** Interpretation results for the build-up test of well A1.

## Conclusions

In order to describe the flow retardation in complex fractures in a way that considers the SRV region with anomalous diffusion and fractal features, an improved five-region model was established in this work by introducing the time-fractional flux law. Based on the present model, type curves of pressure and pressure derivative without wellbore storage were plotted and five flow stages were identified: bilinear flow, first linear flow, inter-porosity and fractal-anomalous flow, second linear flow, and boundary control flow.

The sensitivity analysis revealed that fractal-anomalous diffusion has a significant impact on pressure-transient behaviors. When the anomalous diffusion exponent decreased from 1 to 0.75, which indicates Darcy flow changing to anomalous diffusion, the pseudo-pressure had less depletion at the early linear flow stages, but this subsequently became greater. When the Hausdorff index changed from 2 to 1.8, greater pressure consumption was needed to achieve the same production. Additionally, stress sensitivity, absorption, and Knudsen diffusion showed non-negligible influences on the pressure-transient response. These effects cannot be ignored. Therefore, the typical five-region flow model which does not take the fractal-anomalous diffusion into account cannot be applied for heterogeneous multi-fractured systems. The present model can be used to provide a more accurate and appropriate interpretation of well-testing data to guide exploration and development.

### Author Contributions

All authors have contributed to this work. Conceptualization, H.T., Q.L., and Y.Z.; software, Q.D.; validation, M.L.; formal analysis, H.T.; investigation, H.T.; resources, L.Z.; data curation, M.L.; writing—original draft preparation, H.T.; writing—review and editing, H.T., Q.L., and Q.D.; supervision, L.Z.

#### Funding

This research was funded by the National Natural Science Foundation of China (Key Program) (Grant No. 51534006), the National Natural Science Foundation of China (Grant No. 51704247), and the National Science and Technology Major Project (2017ZX05009-004).

#### Conflicts of Interest

The authors declare no conflict of interest.

#### Nomenclature

Appendix A. Dimensionless Definitions

Appendix B. Derivations for General Diffusivity Equation in the SRV

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#### Correspondence:

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Tel.: +1-814-852-8648 (H.T.); +86-28-8303-2052 (Q.L.); +86-159-8232-4747 (Y.Z.)