## 1. Introduction

In the 2000s, since oil demands in China and India have increased and the oil price continues to rise, unconventional resources such as shale gas, shale oil, oil sand, tight gas, and coal bed methane have been actively developed. In the United States (US), the average production of shale gas in 2010 was 0.43 billion cubic meters per day (bcm/day). With continuous unconventional shale gas developments and technology innovation, it is increased by 3.5 times at 1.50 bcm/day in 2018.

Shale gas is a natural gas trapped in shale formation. Due to low permeability of shale formation, there are many difficulties in its development. Shale gas is actively developed by horizontal drilling and hydraulic fracturing technologies in the US. Its permeability is 10^{−8} to 10^{−6} Darcy, where fluid flow hardly occurs. Therefore, the most important factor in shale gas development and production is the multi-stage hydraulic fracturing to overcome its low flow capacity.

Multi-stage hydraulic fracturing is employed to increase the flow area between hydraulic fracture (HF) networks and shale gas reservoir formation. For its optimization, it is necessary to consider many characteristics of shale gas reservoir such as HF spacing and length, natural fracture networks, matrix permeability, adsorbed gas, and so on. However, the evaluation of productivity and economic feasibility study will require a lot of time and cost.

Many factors influencing shale gas production have been confirmed in some of the previous research. Mayerhofer et al. (2006) [

1] integrated microseismic fracture mapping with a numerical production modeling of fracture networks in the Barnett Shale. The model matched production and pressure history data using fracture and matrix parameters. Cipolla et al. (2009) [

2] used numerical simulations for complex fracture geometry and heterogeneity of shale gas reservoirs. However, it was very time-consuming. Valko and Lee (2010) [

3] estimated reserves using a decline curve model for the production data in a tight gas reservoir. Since the decline curve model only utilized the trend of the production data, it cannot predict the change of production rate with different HF designs.

Biswas (2011) [

4] developed a predictive model of shale gas production using transfer equations and mass balance equations. The model predicted the shale gas production using fluid saturation, fluid compressibility, temperature, storage pressure, and drainage area. However, natural fracture and matrix permeabilities were not taken into account in the properties of the shale reservoir. Yu and Sepehrnoori (2013) [

5] proposed an optimal multiple horizontal well placement considering the influence factors of shale gas production using a response surface method for net present value (NPV). The response surface model was constructed using only 38 sample cases in the NPV prediction. The model with small training data cannot guarantee its predictive performance and may result in inaccurate results.

Xie et al. (2015) [

6] rapidly predicted shale gas production using a fast marching method. However, a single porosity–single permeability model had limitations to simulate the pseudo steady state flow behavior of shale gas production. Kim et al. (2015) [

7] analyzed the main influencing factors affecting the shale gas production, and then constructed a shale gas production prediction model using artificial neural network (ANN). However, in the case of this ANN model, it was difficult to show the weight of key influencing factors on the production. Balan et al. (2016) [

8] conducted an optimization of hydraulic fracturing spacing for tight and shale gas reservoirs. However, only single HF was considered in the analysis.

Kim et al. (2017) [

9] proposed a multi-objective history matching method with a proxy model for the characterization of production performances in the shale gas reservoir. The proxy model was employed for overcoming the time-consuming multi-objective method. However, the proxy model cannot handle the pseudo steady state flow behavior of shale gas productions. Tang et al. (2018) [

10] developed a numerical simulation for multi-scale flow mechanisms in a shale gas reservoir. It can simulate various flow mechanisms such as gas desorption, the Klinkenberg effect, and gas diffusion. Although it may be good to duplicate shale gas behaviors in detail, the simulation cost is too high.

This paper presents an optimal hydraulic fracture design using a developed proxy model. The proxy model used a robust regression scheme and considered the change of the influence of the main variables affecting production at each production time. Its optimum design was selected by feasibility study with the development cost.

## 2. Methodology

Figure 1 shows the proposed workflow for the optimization of hydraulic fracturing design using the proxy model. To improve the accuracy of the proxy model, Latin hypercube sampling (LHS) and a robust regression model were employed.

#### 2.1. Robust Regression for the Proxy Model

A proxy model is a prediction tool to replicate a full simulation model by considering key parameters. It has been used efficiently for the development plan, decision making, evaluation of uncertainty, optimization of operating conditions, and history matching in a hydrocarbon reservoir. Typical proxy models are regression model, response surface method, kriging, and ANN. A linear regression model is one of the oldest statistical methods. It is still employed in today’s world of developing new statistical methods because a linear model is easy to understand while giving least variance among all linear estimation methods.

The linear regression can be fit poorly if the error distribution is not normal, particularly when the errors are heavy-tailed. One scheme is to remove outlier observations. Another method, robust regression, is to use a fitting criterion that is not as vulnerable as least squares to unusual data. Robust regression estimators typically aim to fit a model that describes the majority of a sample. This estimator has the ability to provide accurate parameter estimate values in the presence of outliers or non-normally distributed errors. A few of the popular robust estimators are least absolute value (LAV) estimator, least median of square estimator, M-estimator, and MM-estimator.

In this research, M-estimator is employed for predicting shale gas productions because it is one of the most powerful tools for outlier adjustments, and a heterogeneous shale gas reservoir has many outliers in the data set. It can also identify key parameters on the shale gas production. M-estimator is defined as Equation (1) and minimizes the sum of residual function.

where

$\rho $ is the continuous symmetric function called the objective function with a unique minimum at 0 [

11],

${e}_{j}$ is the residual at time

j,

${y}_{j}$ is the cumulative gas production at time

j,

$x$ are the influence parameters, and

$\beta $ is the regression coefficient of the parameters affecting the shale gas production.

#### 2.2. Latin Hypercube Sampling

Sampling schemes can be typically divided into probabilistic and non-probabilistic methods. The former extracts a sample by assuming the occurrence probability of the target variable as the form of a probability distribution function. In addition, since all possible cases are considered by a probability distribution, the reliability of the result can be secured by many sampling. On the other hand, the other method does not assume a variable as a probability distribution.

Statistically, the more samples there are, the more meaningful the results are. However, fewer samples can reduce the experimental cost and time, if they are properly selected. Latin hypercube sampling is a method that can reduce the number of samples and simulations, and at the same time, it is complementing the disadvantages of the Monte Carlo sampling method, which cannot always perform uniform sampling. It also has the advantage of showing the characteristics of the population better as input variables are smaller [

12]. If there are two variables such as in

Figure 2, samples can be extracted so that each sample interval does not overlap.

Figure 3 shows the comparison of sampling results between random sampling and LHS. LHS gives a uniform distribution than the random sampling. It prevents repeated samples in the sample space. Therefore, it is very useful in the case of high-cost and time-consuming experiments, since it selects relatively small number of samples. In this study, initial reservoir models for developing a proxy model are sampled by LHS.

#### 2.3. Dual Porosity–Dual Permeability Model

The important part of shale gas reservoir modeling is how to model natural fractures existing in the reservoir, because shale gas flows dominantly through the fracture system between the hydraulic and natural fractures. Typical models for replicating natural fractures are discrete fracture network (DFN) model and dual porosity–dual permeability (DPDK) model.

The dual porosity model shown in

Figure 4 converts matrix, fractures, and vugs into homogeneous matrix and fracture. Then, it assumes that the matrix porosity and fracture porosity store the fluid and the fluid flows only through the fractures [

13,

14]. This model is simple, but it is difficult to explain the connection between the matrices [

15], since most fluid flows through both the fracture and the matrix.

The DPDK model can simulate the fluid flow in a matrix and a fracture. The fluid flow between the matrix and the fracture is affected by the matrix fracture coupling factor (

$\mathsf{\sigma}$) in Equation (2), which combines the fracture spacing in the x-, y-, and z-directions to account for the flow of the fluid in connected space in the reservoir. Since there are many fractures, the coupling factor becomes larger resulting in easier fluid flow. In this research, DFN model and DPDK model are used for modeling natural fractures in a shale gas reservoir.

where

${I}_{x},\text{}{I}_{y},\mathrm{and}{I}_{z}$ are the fracture spacing in

x-,

y-, and

z-directions, respectively.

#### 2.4. Net Present Value

NPV is the value of all future cash flows over the entire time of an investment discounted to the present. It is a very important tool for financial decision making and is typically utilized for the development feasibility study. NPV can be compared for different cases to evaluate which one is advantageous in the long-term perspective. In this research, an objective function is NPV for the design of hydraulic fracturing considering the economic feasibility. For this, the shale gas production for each design is computed using the proxy model developed, and NPV is obtained by Equation (3).

where

C_{t} is the net cash flow at time

t,

r is the discount rate, and

t is the period of the cash flow.

## 4. 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.