Numerical Simulation of Well Type Optimization in Tridimensional Development of Multi-Layer Shale Gas Reservoir

Abstract

Aimed at the development of shale gas reservoirs with large reservoir thickness and multiple layers, this paper carried out a numerical simulation study on the optimization of three different well types: horizontal well, deviated well, and vertical well. To make the model more in line with the characteristics of shale gas reservoirs, a two-phase gas–water seepage mathematical model of shale gas reservoirs was established, considering the adsorption and desorption of shale gas, Knudsen diffusion effect, and stress sensitivity effect. The embedded discrete fracture model was used to describe hydraulic fracture and natural fracture. Based on Fortran language, a numerical simulator for multi-layer development of shale gas reservoirs was compiled, and the calculation results were compared with the actual production data of Barnett shale gas reservoirs to verify the reliability of the numerical simulator. The spread range of hydraulic fractures in the reservoir with different natural fracture densities is calculated by the simulation to determine well spacing and fracture spacing. The orthogonal experimental design method is then used to optimize the best combination of well spacing and fracture spacing for different well types. The results show that the well productivity of the high-density (0.012 m/m²) natural fractures reservoir > the well productivity of the medium-density (0.006 m/m²) natural fractures reservoir > the well productivity of the low-density (0.001 m/m²) natural fractures reservoir. According to the design of the orthogonal test, it can be seen that the most significant factor affecting the productivity of horizontal wells is the fracture spacing in the Y direction. For deviated wells and vertical wells, the X-direction well spacing has the greatest impact on its productivity.

1. Introduction

With the sharp decline in reserves and productivity of conventional oil and gas resources, in recent years shale gas has gradually become an important supplement to traditional energy [1]. However, compared with conventional gas reservoirs, shale gas reservoirs generally have ultra-low porosity and permeability. The porosity is usually less than 10%, the permeability varies from micro-Darcy to nano-Darcy, and the reservoir heterogeneity is serious, requiring hydraulic fracturing to enable commercial exploitation [2,3]. Hydraulic fracturing and horizontal drilling are key technologies for shale gas production in most shale reservoirs [4]. Shale gas mainly exists in the two forms of free state and adsorption state in the reservoir. It exists in the matrix pores and natural fractures in the free state, while it is adsorbed on the surface of organic matter in the adsorption state [5]. The existence mode of shale gas and the characteristics of shale gas reservoirs mean that there are multiple migration mechanisms of shale gas in the reservoir, such as Knudsen diffusion, stress sensitivity effect, and adsorption and desorption of adsorbed gas in the production process [6].

In shale gas reservoirs, when the average pore radius is comparable to the mean free path of gas molecules, Knudsen diffusion must be considered in the gas flow matrix. Knudsen diffusion will change the apparent permeability of the matrix to affect shale gas production, and the smaller the shale pore radius, the more obvious the impact on production. Under the effect of surface diffusion and Knudsen diffusion, the apparent permeability of the shale matrix increases, which increases the cumulative gas production of shale gas [7,8,9]. In conventional oil and gas reservoirs, the influence of geomechanics on rock deformation or permeability is usually small, which is mostly ignored in actual production. However, in shale gas reservoirs with nanoscale pores or micro-fractures, geomechanical effects may be relatively large and may have a significant impact on fracture and matrix permeability [10]. Wang et al. [11] showed that the permeability in the Marcellus shale is related to pressure, and the influence of confining pressure on permeability is caused by the decrease of porosity and decreases with the increase in confining pressure. Bustin et al. [12] reported the strong effect of stress (confining pressure) on permeability in Barnett, Muskwa, and Ohio shales, and showed that the degree of permeability reduction with confining pressure is significantly higher in shales than in consolidated sandstone or carbonate. The natural gas in shale gas reservoirs exists in the form of the free phase, and also in the form of adsorbed gas. The adsorbed gas accounts for a large proportion of the total natural gas reserves (20–80%), which is also an important factor affecting recovery [13]. Moreover, adsorption–desorption is an important occurrence form of shale gas, and adsorption gas is a supplement to free gas in the development process, which has a significant impact on the unsteady production capacity in the middle and late stages of gas well production [14]. During the exploitation of shale gas reservoirs, the adsorption and desorption of shale gas will also lead to dynamic changes in the effective pore radius of the shale matrix, thereby affecting the apparent permeability [15]. Therefore, to make the model accurately simulate shale gas productivity, the percolation mechanism of shale gas must be comprehensively considered.

The combination of horizontal drilling and hydraulic fracturing is the key technology for shale gas production. Although hydraulic fracturing can increase the production of shale gas wells, the operating cost is high. Due to the high cost of drilling and completion, the economy of gas wells will be affected by their performance, so the analysis and optimization of well parameters (such as well spacing and fracture spacing) are very important [16,17]. Rafiee et al. [18] proposed two new designs for horizontal wells and improved well performance from both rock mechanics and fluid production, but only a 500-foot fracture spacing was assumed, without optimizing both well location and fracture spacing. Diaz de Souza et al. [19] studied the sensitivity of horizontal well layout in Haynesville shale to obtain the best well spacing. However, the optimization of key parameters of economic development zones, such as fracture spacing and fracture half-length, has not been considered. Wei Yu et al. [20] used the corresponding surface method, based on the net present value, and optimized the fracturing design and the layout of multiple wells at the same time, providing a theoretical reference for the effective exploitation of shale gas reservoirs. Ramanathan et al. [21] used the unconventional fracture model to predict the production performance of multiple well groups with different fracture geometries and points out that the interference between well groups makes the well spacing and the number of fracturing fractures important optimal values. To improve shale gas recovery and reduce engineering costs, it is still necessary to comprehensively optimize the relevant parameters of wells. However, at the same time, for large thickness shale reservoirs, there are few numerical simulation studies on well-type optimization of horizontal wells, deviated wells, and vertical wells.

In this paper, we establish a mathematical model of shale gas reservoir seepage that considers adsorption and desorption, Knudsen diffusion, and shale reservoir stress sensitivity effects and compiles a corresponding numerical simulator based on Fortran language. Firstly, the spread range of hydraulic fractures in the reservoir with different natural fracture densities is calculated by the simulation to determine well spacing and fracture spacing. The orthogonal experimental design method is then used to optimize the best combination of well spacing and fracture spacing for different well types, thereby providing a theoretical basis for well type optimization for the multi-layer development of on-site shale gas reservoirs.

2. Mathematical Model Descriptions

2.1. Fluid-Governing Equations

There are many occurrence modes of shale gas, among which the gas present in the adsorption state on the surface of the pores between the particles is called adsorbed gas. Affected by temperature and pressure, the adsorbed gas will be desorbed into free gas [22]. In this study, it is assumed that only gas and water components exist in shale gas reservoirs, while adsorbed gas exists in the solid phase of rock [23], and its general integral form is:

$$\frac{\partial }{\partial t}{\displaystyle {\int }_{\mathsf{\Omega }}{A}_{\beta }}d\mathsf{\Omega }+{\displaystyle {\int }_{\mathsf{\Gamma }}{F}_{\beta }\cdot nd\mathsf{\Gamma }}={\displaystyle {\int }_{\mathsf{\Omega }}{q}_{\beta }d\mathsf{\Omega }}$$

(1)

where $n$ is the unit normal vector of boundary $\mathsf{\Gamma }$, $A$ is the mass per unit volume, $F$ is mass flux, and $q$ is the source terms on domain $\mathsf{\Omega }$. The mass per unit volume is expressed as:

$$\{\begin{array}{l}{A}_{\beta }=\varphi S_{\beta }{\rho }_{\beta }+m_{\beta },\mathrm{in}\mathrm{matrix}\\ A_{\beta }=\varphi S_{\beta }{\rho }_{\beta },\mathrm{in}\mathrm{fracture}\end{array}$$

(2)

where subscript $\beta$ indicates fluid phases (water or gas), $\varphi$ is the effective porosity of the porous or fractured media, ${\rho }_{\beta }$ is the density of fluid, $m_{\beta }$ is the adsorption/desorption mass term for the gas component per unit volume of formation, and $S_{\beta }$ is the saturation of phase $\beta$.

$$S_g+S_w=1,p_c\left(S_w\right)=p_g-p_w$$

(3)

$${\rho }_w={\rho }_{w0}\exp \left[c_w\left(p_w-p_{w0}\right)\right],{\rho }_g=\frac{p_g{M}_g}{Z_{g}RT}$$

(4)

where the subscript 0 indicates the initial state, $c_w$ is the compression coefficient of water, $p_c$ is the capillary pressure, $p_g$ and $p_w$ are the pressure of gas and water, respectively, $M_g$ is the molar mass of gas, $Z_{\mathrm{g}}$ is the gas compression factor, $R$ is the general gas constant, and $T$ is the reservoir temperature.

The amount of adsorbed gas is determined according to the Langmuir isotherm as a function of gas pressure.

The gas compressibility factor, $Z_g$ can be calculated based on the following equations [24]:

$$Z_g=\left(0.702e^{-2.5T_{pr}}\right)p_{pr}^2-5.524e^{-2.5T_{pr}}{p}_{pr}+0.044T_{pr}^2-0.164T_{pr}$$

(5)

where $p_{pr}$ and $T_{pr}$ indicate the pseudo-reduced pressure and pseudo-reduced temperature, respectively.

The Langmuir isotherm function can determine the adsorption capacity of shale gas:

$$m_g={\rho }_r{\rho }_{gstd}{V}_L\frac{\raisebox{1ex}{$p_g$}\!\left/ \!\raisebox{-1ex}{$Z_g$}\right.}{p_L+\raisebox{1ex}{$p_g$}\!\left/ \!\raisebox{-1ex}{$Z_g$}\right.}$$

(6)

where ${\rho }_r$ is matrix density, ${\rho }_{gstd}$ is gas density at standard condition, and $V_L$ and $p_L$ are the Langmuir volume and Langmuir pressure, respectively. The mass-flux term in Equation (1) is given by [25,26]:

$$\{\begin{array}{l}{F}_w=-{\rho }_w\frac{k_{rw}}{{\mu }_w}k\left(\nabla p_w-{\rho }_{w}g\nabla D\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{in} \mathrm{matrix} \mathrm{and} \mathrm{fracture}\\ F_g=-{\rho }_g\frac{k_{rg}}{u_g}{k}_{\infty }\left(1+\alpha \left(K_n\right)K_n\right)\left(1+\frac{4K_n}{1-b{K}_n}\right)\left(\nabla p_g-{\rho }_{g}g\nabla D\right),\mathrm{in}\mathrm{matrix}\\ F_g=-{\rho }_g\frac{k_{rg}}{u_g}{k}_{\infty }\left(\nabla p_g-{\rho }_{g}g\nabla D\right),\mathrm{in} \mathrm{fracture}\end{array}$$

(7)

where $k$ is the absolute permeability, $k_{\infty }$ is the intrinsic permeability, which, for simplicity, equals absolute permeability in this study, $k_{r\beta }$ and ${\mu }_{\beta }$ are the relative permeability and viscosity of phase $\beta$ (the relative permeabilities are calculated by using the table-lookup approach, in which the correlations between relative permeabilities and water saturation are obtained from laboratory studies), $g$ is gravitational acceleration, $D$ is depth, $b$ is slip coefficient, $K_n$ is Knudsen number and represents the ratio of the average free path of gas molecule to the radius of pore, and $\alpha$ denotes the thinning coefficient, which can be calculated by the following formula [27]:

$$K_n=\frac{\lambda }{r_h},\alpha =\frac{1.358}{1+0.170K_n{}^{-0.4348}}$$

(8)

where $\lambda$ is the average free path of gas molecules and $r_h$ is the average flow radius of gas molecules, and can be calculated by the following formula [27]:

$$\lambda =\frac{k_{b}T{Z}_g}{\sqrt{2}\pi d_m{}^2{p}_g},{\mathrm{r}}_h=2\sqrt{2{\tau }_0}\sqrt{\frac{k_{\infty }}{\varphi }}$$

(9)

where $k_b$ is Boltzmann constant, $d_m$ is the molecular radius, and ${\tau }_0$ is the tortuosity of porous media, dimensionless.

2.2. Stress Sensitivity Effect of Shale Reservoir

The traditional percolation theory generally assumes that the porous media skeleton of fluid flow is completely rigid, and this simplification has defects that are not suitable for actual production. Because the porosity of shale reservoirs is extremely low, it is particularly sensitive to reservoir pressure changes. Shale stress sensitivity is a phenomenon by which the effective stress on the rock skeleton increases due to the decrease of reservoir pressure during shale gas exploitation, resulting in changes in physical parameters such as porosity and permeability of shale reservoirs [28,29].

$$\varphi (p)={\varphi }_0{e}^{{\alpha }_{\varphi }(p-p_0)}$$

(10)

$$k(p)=k_0{e}^{{\alpha }_k(p-p_0)}$$

(11)

where $k(p)$ is reservoir permeability under pressure, $p$, and $p=p_w{S}_w+p_{\mathrm{g}}{S}_g$. $p_0$ is the original pressure of the gas reservoir; $\varphi (p)$ and ${\varphi }_0$ are the porosity corresponding to $p$ and $p_0$, respectively, and $a_k$ and $a_{\varphi }$ are the permeability and porosity variation coefficient, respectively. As the porosity and permeability of the shale matrix changes with reservoir pressure, the average flow radius, $r_h$, is also dynamic, and its value is updated according to Equation (9).

3. Flow Equation Discretization

The structured grids of EDFM are used to explicitly model the fractured shale reservoir. As shown in Figure 1, the matrix region is discretized using an orthogonal structured grid, and the fracture grid is segmented by matrix gridlines. Details are given by Xu et al. [30], Yu et al. [31], and Liu et al. [32].

In this study, using the FVM discrete gas flow governing equations, the time dispersion is approximated by the standard first order [33], and gravity is ignored. Therefore, the residual expression of Equation (1) is:

$$R_{\beta ,i}^{n+1}=\left[{\left(\varphi S_{\beta }{\rho }_{\beta }+m_{\beta }\right)}_i^{n+1}-{\left(\varphi S_{\beta }{\rho }_{\beta }+m_{\beta }\right)}_i^n\right]\frac{V_i}{\Delta t}-{\displaystyle \sum _{j\in G_i}{\left(\rho \lambda \right)}_{ij+1/2}^{n+1}{T}_{ij}\left(p_{\beta ,j}^{n+1}-p_{\beta ,i}^{n+1}\right)-q_{\beta ,i}^{n+1}{V}_i}$$

(12)

where $n$ is the time level, $i$ and $j$ are element numbers, $\Delta t$ is the time step, $V$ is the element volume, $G_i$ is the adjacent element set of element $i$, $ij+1/2$ represents the upstream weight value on the interface between elements $i$ and $j$, $\lambda =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mu $}\right.$ is the gas mobility, and $T_{ij}$ is the transmissibility between element $i$ and $j$, defined as:

$$T_{ij}=\frac{A_{ij}{k}_{ij+1/2}}{d_i+d_j}$$

(13)

where $A_{ij}$ is the interface area, $d_i$ and $d_j$ are vertical distances from the cell center to the interface, and $k_{ij+1/2}$ is an averaged absolute permeability. To include the Knudsen effect on gas flow, the permeability in Equation (11) should be evaluated as ${\left(k\left(1+\alpha K_n\right)\left(1+4K_n/1-b{K}_n\right)\right)}_{ij+1/2}$. In this study, the fluid flow in fractures is modeled explicitly with the embedded-discrete-fracture model. The upstream-weighting is applied for the mobility and transmissibility terms.

In addition, in this paper, the outer boundary conditions of the model are closed boundary conditions:

$${\frac{\partial P_m}{\partial n}|}_{{\mathsf{\Gamma }}_1}={\frac{\partial P_f}{{\partial }_n}|}_{{}_{{\mathsf{\Gamma }}_1}}=0$$

(14)

The internal boundary condition is the constant pressure boundary condition:

$${P|}_{{\mathsf{\Gamma }}_2}=P_w$$

(15)

In the formula, ${\mathsf{\Gamma }}_1$ and ${\mathsf{\Gamma }}_2$ denote the outer boundary and the inner boundary, respectively, $n$ represents the normal unit vector of the outer boundary, and $P_w$ is the bottom hole flowing pressure, MPa.

4. Model Verification

To verify the accuracy of the new model, a well in the Barnett shale gas reservoir was selected, and parameters related to the Barnett shale gas reservoir are shown in

Table 1. Parameters related to the Barnett Shale gas reservoir.

Parameter	Value	Units
3D model size	1100 × 290 × 90	m
Initial reservoir pressure	25	MPa
Bottom hole pressure	2.5	MPa
Langmuir pressure	4.48	MPa
Langmuir volume	2.72 × 10−3	m3/kg
Reservoir temperature	340	K
Well radius	0.1	m
Reservoir porosity	0.06	Dimensionless
Initial permeability	1.4 × 10−19	m2
Hydraulic fracture half-length	40	m
Hydraulic fracture spacing	30	m
Number of hydraulic fractures	28	Dimensionless
Hydraulic fracture height	90	m
Length of horizontal well	900	m
Shale density	2580	kg/m3
Initial gas saturation	0.776	Fraction
Initial water saturation	0.224	Fraction
Gas viscosity	2.01 × 10−5	Pa⋅s
Natural fracture permeability	1 × 10−13	m
Hydraulic fracture permeability	1.17 × 10−11	m
Compression coefficient of water	4.4 × 10−10	Pa−1
Porosity variation coefficient	0.08	Pa−1
Permeability variation coefficient	0.62	Pa−1

. The reservoir parameters for Barnett shale were obtained from the literature [34,35]. The simulation calculated daily gas production for 1600 days and compared it with actual daily gas production from the Barnett shale gas reservoir.

A simulation model was established based on a horizontal well in the Barnett shale gas reservoir, as shown in Figure 2, where the model size (x × y × z) is 1100 m × 290 m × 90 m.

The pressure, water saturation, and gas saturation distributions of the model in different production periods of 1600 days is shown in Figure 3, and the fitting results of daily gas production are shown in Figure 4.

The root mean square method was used to calculate the fitting rate, and the fitting rate between the simulation results and the actual gas production of the Barnett shale gas reservoir reached 86.4%, which verifies the correctness and reliability of the numerical simulator.

5. Results and Discussion

First, the utilization range of hydraulic fractures under different natural fracture densities is calculated, and then the well spacing and fracture spacing in the X, Y, and Z directions of horizontal wells, deviated wells, and vertical wells are determined according to the calculation results. The optimal well spacing and fracture spacing of each well type are optimized by orthogonal experimental design.

5.1. Calculation of Reservoir Utilization Range under Different Natural Fracture Densities

According to the field data, a set of gas reservoir parameters are selected: the initial formation pressure is 38.55 MPa, bottom hole pressure is 10 MPa, wellbore radius is 0.1 m, and formation water viscosity is 0.50 mPa·s. Irreducible water saturation is 0.18. The hydraulic fracture permeability and natural fracture permeability are 1.17 × 10⁻¹¹ m, and 1 × 10⁻¹³ m, respectively. Matrix density is 2800 Kg/m³, Langmuir volume is 8 × 10⁻⁴ m³/kg, and Langmuir pressure is 15 MPa. In the established numerical simulation model, the hydraulic fracture height is 20 m, and the properties of each layer and the development of natural fractures are shown in

Table 2. Properties of each layer and development of natural fractures.

Layer Number	Layer Thickness/m	Porosity	Water Saturation	Gas Saturation	Matrix Permeability/10−3 mD	Natural Fracture Density/m/m2
1	20	0.028	0.25	0.75	0.326	0.012
2	45	0.025	0.23	0.77	0.407	0.012
3	40	0.022	0.22	0.78	0.172	0.006
4	50	0.034	0.24	0.76	0.150	0.006
5	40	0.038	0.21	0.79	0.180	0.006
6	30	0.022	0.30	0.70	0.137	0.001
7	35	0.015	0.21	0.79	0.133	0.001

.

To determine the well spacing and hydraulic fracture spacing under different natural fracture densities, the pressure change range after 15 years of exploitation is calculated, and the reservoir utilization range under high-density (0.012 m/m²), medium-density (0.006 m/m²), and low-density (0.001 m/m²) natural fractures are obtained (Figure 5, Figure 6 and Figure 7), so as to determine the well spacing and fracture spacing in the various directions of different well types.

From the simulation results, it can be seen that, for reservoirs with high-density, medium-density, and low-density natural fracture after 15 years of exploitation, the reservoir utilization ranges in the X direction are 280 m, 260 m, and 240 m, in the Y direction they are 50 m, 45 m, and 40 m, and in the Z direction they are 34 m, 32 m, and 30 m.

Figure 8 shows the production of the well at high-density, medium-density, and low-density natural fractures. As shown above, the density of natural fractures affects the productivity of the well. The well productivity of the high-density (0.012 m/m²) natural fractures reservoir > the well productivity of the medium-density (0.006 m/m²) natural fractures reservoir > the well productivity of the low-density (0.001 m/m²) natural fractures reservoir.

5.2. Productivity Evaluation and Optimization of Different Well Types

A shale gas reservoir model is established to simulate the fracturing production of horizontal wells, deviated wells, and vertical wells. The model size (x × y × z) is 1200 m × 1500 m × 260 m. The properties of each layer and the development of natural fractures are shown in Table 2, and the related parameters of wells and fractures are shown in

Table 3. Well and hydraulic fracture related parameters.

Essential Parameter	Value	Units
Initial reservoir pressure	38.5	MPa
Bottom hole pressure	10	MPa
Shale density	2800	kg/m3
Hydraulic fracture half-length	100	m
Hydraulic fracture height	20	m
Initial gas saturation	0.776	Fraction
Initial water saturation	0.224	Fraction
Gas viscosity	2.0 × 10−5	Pa·s
Natural fracture permeability	1 × 10−13	m
Hydraulic fracture permeability	1.17 × 10−11	m
Porosity variation coefficient	0.08	Pa−1
Permeability variation coefficient	0.62	Pa−1

. The model permeability field diagram, model porosity field diagram, horizontal well layout diagram, deviated well layout diagram, and vertical well layout diagram are shown in Figure 9.

5.2.1. Production Capacity Evaluation and Optimization of Horizontal Wells

According to the reservoir utilization range determined in the previous section, for a horizontal well, the well spacings in the X direction are set as 240, 260, and 280 m; the fracture spacings in the Y direction are set as 40, 45, and 50 m; and the well spacings in the Z direction are set as 30, 32, and 34 m. The reservoir properties and hydraulic fracture-related parameters in the model are from Table 2 and Table 3. The above three factors are combined, and the orthogonal design test method is used to determine the optimization scheme, as shown in

Table 4. Orthogonal design table of horizontal wells.

Level	Factor
	Factor 1 X Direction Well Spacing/m	Factor 2 Y Direction Fracture Spacing/m	Factor 3 Z Direction Well Spacing/m
1	240	40	30
2	260	45	32
3	280	50	34

.

According to the orthogonal design method, nine schemes are simulated respectively. Figure 10 is the pressure distribution field diagram of different development schemes for horizontal wells after 20 years of exploitation, and Figure 11 is the cumulative production of a single well with different development schemes of horizontal wells.

Table 5. Production analysis table for different schemes of horizontal well.

Orthogonal Design Table				Well Production 107 m3
Scheme	Factor-1	Factor-2	Factor-3
1	1	1	1	2.76
2	1	2	2	3.06
3	1	3	3	3.56
4	2	1	2	3.11
5	2	2	3	3.63
6	2	3	1	3.43
7	3	1	3	3.04
8	3	2	1	3.16
9	3	3	2	3.52

is the production analysis table of different schemes of horizontal wells.

According to the production values of the above scheme, the average production values, $K_{\mathrm{avg}}$ (the average value of well production corresponding to different factor levels), of three levels in the three factors are calculated, as shown in Figure 12. The optimal combination of horizontal well development is 260 m in the well spacing in the X direction, 50 m in the fracture spacing in the Y direction, and 34 m in the well spacing in the Z direction.

By calculating the range value, R, of three factors (Maximum average production value minus Minimum average production value), we get R_Factor-2(0.53) > R_Factor-3(0.29) > R_Factor-1(0.26). Therefore, the most significant factor affecting the productivity of horizontal wells is factor-2 (fracture spacing in the Y direction).

5.2.2. Production Capacity Evaluation and Optimization of Deviated Wells

Based on the same gas reservoir model in the above section, a 45-degree deviated well is used for development. X direction well spacing is set at 240, 260, and 280 m; Y direction fracture spacing at 40, 45, and 50 m; and Z direction fracture spacing at 30, 32, and 34 m. The above three factors are combined, and the optimization scheme is determined by the orthogonal design method, as shown in

Table 6. Orthogonal design table of deviated wells.

Level	Factor
	Factor 1 X Direction Well Spacing/m	Factor 2 Y Direction Fracture Spacing/m	Factor 3 Z Direction Fracture Spacing/m
1	240	40	30
2	260	45	32
3	280	50	34

.

According to the orthogonal design, nine schemes are simulated respectively. Figure 13 is the pressure distribution field diagram of different development schemes for deviated wells after 20 years of exploitation, and Figure 14 is the cumulative production of a single well with different development schemes of deviated wells.

Table 7. Yield analysis table for different schemes of deviated well.

Orthogonal Design Table				Well Yield 107 m3
Scheme	Factor-1	Factor-2	Factor-3
1	1	1	1	3.13
2	1	2	2	3.09
3	1	3	3	3.14
4	2	1	2	3.34
5	2	2	3	3.31
6	2	3	1	3.32
7	3	1	3	3.26
8	3	2	1	3.11
9	3	3	2	3.21

is the yield analysis table of the different schemes of deviated wells.

According to the yield values of the above scheme, the average yield values, $K_{\mathrm{avg}}$, of three levels in the three factors are calculated respectively. As shown in Figure 15, the optimal combination of deviated well development is 260 m in the well spacing in the X direction, 40 m in the fracture spacing in the Y direction, and 34 m in the fracture spacing in the Z direction.

By calculating the range value, R, of three factors, for deviated wells, we get R_Factor-1(0.20) > R_Factor-2(0.07) > R_Factor-3(0.05). Therefore, factor-1 (X direction well spacing) has the greatest impact on its productivity.

5.2.3. Production Capacity Evaluation and Optimization of Vertical Well

Based on the same gas reservoir model above, for vertical well development. X direction well spacing is set at 240, 260, and 280 m; Y direction well spacing at 40, 45, and 50 m; and Z direction fracture spacing at 30, 32, and 34 m. The above three factors are combined, and the optimization scheme is determined by the orthogonal design method, as shown in

Table 8. Orthogonal design table for vertical wells.

Level	Factor
	Factor 1 X Direction Well Spacing/m	Factor 2 Y Direction Well Spacing/m	Factor 3 Z Direction Fracture Spacing/m
1	240	40	30
2	260	45	32
3	280	50	34

.

According to the orthogonal design, nine schemes are simulated respectively. Figure 16 is the pressure distribution field of different development schemes after 20 years of exploitation, and Figure 17 is the cumulative production of a single well with different schemes of vertical wells.

Table 9. Yield analysis table for different schemes of vertical wells.

Orthogonal Design Table				Well Yield 107 m3
Scheme	Factor-1	Factor-2	Factor-3
1	1	1	1	1.12
2	1	2	2	1.02
3	1	3	3	0.98
4	2	1	2	1.11
5	2	2	3	1.16
6	2	3	1	1.13
7	3	1	3	1.23
8	3	2	1	1.18
9	3	3	2	1.25

is the yield analysis table of the different schemes of deviated wells.

According to the yield values of the above scheme, the average yield values, $K_{\mathrm{avg}}$, of three levels in three factors are calculated respectively. As shown in Figure 18, the optimal combination of vertical well development is 280 m in the well spacing in the X direction, 40 m in the well spacing in the Y direction, and 30 m in the fracture spacing in the Z direction.

By calculating the range value, R, of three factors, for vertical wells, we get R_Factor-1(0.18) > R_Factor-2(0.03) > R_Factor-3(0.02). Therefore, factor-1 (X direction well spacing) has the greatest impact on its productivity.

6. Conclusions

In this paper, a gas–water two-phase flow model in a shale gas reservoir considering adsorption/desorption, Knudsen diffusion, and stress sensitivity are established. The fluid flow in fractures is modeled explicitly with the embedded discrete fracture model, and the reliability of the numerical model and algorithm is verified by comparing the actual production data of the Barnett shale gas reservoir.

After verification of the simulator, this paper first establishes a shale gas reservoir model with different natural fracture densities and simulates the production of the same horizontal well under different natural fracture densities. The results show that the natural fracture density will affect the utilization range of the well and then greatly affect the production of the well. The orthogonal experimental design method was then used to optimize the best combination of well spacing and fracture spacing for horizontal wells, deviated wells, and vertical wells. The results show that the optimal combination of horizontal well development is 260 m in the well spacing in the X direction, 50 m in the fracture spacing in the Y direction, and 34 m in the well spacing in the Z direction. The optimal combination of deviated well development is 260 m in the well spacing in the X direction, 40 m in the fracture spacing in the Y direction, and 34 m in the fracture spacing in the Z direction. The optimal combination of vertical well development is 280 m in the well spacing in the X direction, 40 m in the well spacing in the Y direction, and 30 m in the fracture spacing in the Z direction.