## Results and Discussion

### Computation of the Proxy Model

#### Modeling of the Unconventional Shale Reservoir

Figure 5a,b shows the matrix permeability and porosity distribution in a heterogeneous shale gas reservoir used in this study. The reservoir size is 899 m × 411 m × 79 m. The total grid number of the model is 49 × 27 × 13. The reservoir properties used for the numerical simulation are listed in Table 1. These properties are similar to those of Marcellus shale. Figure 5c shows a hydraulic fracturing design with four stages with different HF lengths.

**Figure 5.** Shale gas reservoir in 3D grid systems—(a) matrix permeability, (b) matrix porosity, and (c) hydraulic fracturing design.

**Table 1.** Input properties of the synthetic reservoir to computing the proxy modeling.

The natural fractures in the shale gas reservoir affect the connectivity of HF zones. Since they were closely related to the hydraulic fracturing plans, their modeling in the reservoir required many properties such as natural fracture density, direction, size, spacing, and their connectivity (Table 2). In this study, the DFN model was employed to simulate natural fractures for the shale gas reservoir (Figure 6a). Upscaling was used for adjusting the DFN model to the DPDK model (Figure 6). The final DPDK model was applied to replicate the simultaneous matrix-to-matrix and fracture-to-fracture flow in the reservoir.

**Figure 6.** Natural fracture generation and upscaling in 3D grid systems—(a) natural fracture generation, (b) fracture porosity, (c) sigma factor, (d) fracture permeability, x, (e) fracture permeability, y, and (f) facture permeability, z.

**Table 2.** Natural fracture properties of the synthetic reservoir.

#### Proxy Model Based on the Robust Regression

The key parameters used to create the proxy model are HF length and conductivity. Since shale gas reservoirs have very low permeability in the range from 10−8 to 10−6 Darcy, hydraulic fracture length and its conductivity are the most important factors in production [7]. In this research, the proxy model based on the two variables was developed for predicting shale gas productions. When the proxy model was constructed by fitting the robust regression model with the given data, its prediction accuracy was dependent on given data. Samples for the proxy model were 100 training data sets by LHS based on the properties shown in Table 1. We made additional 100 testing data sets using the same conditions to verify the proxy model.

The prediction performance of the robust regression based the proxy model was compared with that of ANN, which typically used a feed forward backpropagation network model for nonlinear prediction. For the ANN, 15 neurons made up the first hidden layer, 10 neurons made up the second hidden layer, and the production was the only parameter in the output layer. Data were divided into three sets—training (70%), validation (15%), and testing (15%). At the same time, the estimation error was computed using the mean absolute percentage error (MAPE) between a commercial simulator (CMG reservoir simulator) and the proxy model. The R2 values showed the goodness of each model’s fit.

Figure 7 is the cross plot between the commercial simulation and the proxy model. It is the average of the productions from the 100 training data. As seen, they provide an excellent match. In the training and testing data, the proxy model showed a better prediction performance than the ANN model. The accuracy of the proposed proxy model was MAPE of less than 3.4%, but that of the ANN model was 6%, especially for the testing sample data. The R2 of the proxy model also appeared higher than that of the ANN model (Table 3).

**Figure 7.** Cross plot of cumulative gas productions at each proxy model.

**Table 3.** Comparison of prediction accuracy between the proxy model and ANN.

Figure 8 shows the average prediction errors of the ANN model and the proposed proxy model using 100 training and 100 testing data. For the prediction error according to the production time, the proposed method provided smaller errors, which decreased with the lapse of time. On the other hand, in the case of the ANN model, it was found that the prediction error moved in a random walk according to time. This was because the model fit of the ANN was different at each prediction point.

**Figure 8.** Prediction accuracy of artificial neural network (ANN) model and the proxy model.

#### Sensitivity Analysis Using the Proxy Model

We can look at the influence of the variables by using the coefficients of the proxy model according to time change in the proposed model. Figure 9 presents the effect of each parameter on the shale gas productions. The ratio of importance was evaluated as the ratio of the coefficients from the proposed model. It showed that the HF length affected the production amount more than the HF conductivity.

**Figure 9.** Influence change of each parameter. (HF#, hydraulic fracture length; HF_Cond#, hydraulic fracture conductivity; # = 1, 2, 3, and 4 denote the location of hydraulic fracture stage.)

Although it can be seen that the influence of HF conductivity was large in the early stage of the production, it decreased with time. This was because the gas flowed along the HF zone of higher permeability, and the volume of gas inside was restricted. Since its length had a large influence on the production of shale gas, in this study, hydraulic fracturing design was carried out in consideration of economy according to the HF length using the developed proxy model.

### Optimization of Hydraulic Fracture Design

#### Assumptions for NPV Calculation

Even if the hydraulic fracturing stage has the same HF length, the gas production will be different due to the reservoir heterogeneity. In other words, the production volume varies depending on each HF length. Therefore, an optimal design was proposed considering the development cost of the shale gas. The stage of HF was four (Figure 10), and the 14,641 designs of the hydraulic fracturing were examined assuming that the HF length varied from 50 to 550 ft for each stage (11 x 11 x 11 x 11) and HF conductivity was 30 md ft.

**Figure 10.** Schematic diagram of the hydraulic fracture design.

For comparing the computation time of the commercial software and the proxy model, it took 475 s to simulate one case by the commercial software (CMG), but it took just 0.000341 s for the proposed proxy model on a desktop with Intel Core 2 Duo CPU 3.30 GHz. The proxy model developed required just infinitesimal fraction of time compared to that of the full model. It was about 1.4 million times faster than the full simulator compared. Since it took a long time to simulate all cases using the commercial software, the proxy model was utilized for the optimization of HF design. Table 4 shows the economic data for the design.

**Table 4.** Economic data for net present value (NPV) calculation [16].

#### Results of Hydraulic Fracture Design

Figure 11 shows the results of HF design based on NPV. It presents the NPV results for HF length as a boxplot. If HF cost was ignored, the longest fracture length would provide the highest NPV. Otherwise, another design would be chosen as the optimal one.

**Figure 11.** Net present values of each hydraulic fracture design.

The highest and lowest NPVs of hydraulic fracturing design were 41.43 and 5.70 million dollars, respectively. The NPV difference between the two was 7.27 times. It was also found that the economic feasibility was different, even if the total length of HF was same, because the reservoir was heterogeneous and the total production varied depending on the degree of fracture network system.

## Conclusions

In this research, we developed a proxy model for the optimization of shale gas hydraulic fracture design. As shale gas production can be quickly evaluated in the various hydraulic fracture designs, the proposed proxy model solves the time-consuming problem of shale gas reservoir simulation. Additionally, an optimum hydraulic fracture design could be selected by considering economic feasibility, even for a heterogeneous shale gas reservoir.

The results of this study are as follows. The shale gas reservoir was modeled by replicating the natural fractures and heterogeneous properties that are characteristic of an actual shale gas reservoir. The proxy model was developed by selecting the two key parameters of hydraulic fracture conductivity and hydraulic fracture length. For improving the prediction performance of the proxy model, LHS was employed to sample good and representative training data sets. It was found that its prediction performances for forecasting shale gas productions were much better than that of ANN model and its computing speed was tremendously faster than that of the commercial simulator. In additional, we could understand and explain the influence parameters in the proxy model.

For the optimization of hydraulic fracturing design with economic feasibility, it was found that its length of 168 m was the most economical. The procedure developed in this paper can be applied for a fast and economic design of hydraulic fracturing of heterogeneous shale gas reservoirs.

### Author Contributions

K.K. and J.C. contributed to the preparation of the whole paper and the proposed model development.

#### Funding

This research was partially funded by projects from MOTIE, Korea.

#### Acknowledgments

The authors thank to research projects supported by the Ministry of Trade, Industry, and Energy (20172510102090, 20142520100440, 2011201030001C). This research was conducted through the Research Institute of Energy and Resources at Seoul National University, Korea.

#### Conflicts of Interest

The authors declare no conflict of interest.

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