### 3.5 Solution with the finite element method

The DFM can be used to simplify the description of the hydraulic fracture (Yao et al. 2010; Huang et al. 2011). Three-dimensional fractures are converted into two-dimensional surface elements (as shown in Fig. 2). In order to guarantee equal integral value, the fracture aperture should be multiplied before the surface integral.

DFM greatly decreases the number of grids and improves the numerical computation efficiency.

**Fig. 2** Schematic diagram of the discrete-fracture model.

In Eqs. (14) and (15), the DFM is applied to simplify the fracture surface. The FEM is used to solve it, in which tetrahedron elements are used in the matrix, and two-dimensional triangular elements are used in the fracture surface (Li et al. 2010; Yao et al. 2013c). The equations are nonlinear and hard to solve directly; therefore, iteration is used to solve them at an arbitrary time point. The pressure of *n*th time step is applied to obtain the pressure of the (*n* + 1)th time step. To guarantee algorithm stability, an implicit backward difference method about time is utilized.

## 4 Numerical example and analysis of factors affecting production

In this paper, a multi-stage fractured horizontal well in the middle of a box-shaped sealed reservoir is simulated, with its coordinate ranging from (*x*_{1}, *y*_{0}, *z*_{0}) to (*x*_{2}, *y*_{0}, *z*_{0}). Vertical hydraulic fractures are in the shape of rectangles symmetrically distributed around the horizontal well (as shown in Fig. 3).

**Fig. 3** 3D schematic map of the fractured horizontal well.

The gas reservoir is penetrated up and down by hydraulic fractures. Meanwhile, the gas reservoir is assumed to be homogeneous, and the influence of gravity is ignored. The basic parameters of the gas reservoir and fractured horizontal well are shown in Table 1.

**Table 1** Basic data used in the simulation of the multi-staged fractured horizontal well.

### 4.1 Influence of transport mechanisms

Because of the nanopores in shale reservoirs, the matrix porosity and permeability are extremely low. Due to different pore sizes in porous media, the intrinsic matrix permeability differs, and thus, the influence of viscous flow, Knudsen diffusion, adsorbed layer, and surface diffusion on transport mechanisms varies substantially. Figure 4 shows simulated cumulative production versus intrinsic permeability under different transport mechanisms over 20 years.

As illustrated in Fig. 4, when *k _{∞}* > 10

^{−5}μm

^{2}, because of the relatively large pore size, gas transport in the porous matrix is dominated by viscous flow, Knudsen diffusion and surface diffusion have little impact on the productivity of fractured horizontal wells and can be ignored. When 10

^{−7}μm

^{2}<

*k*< 10

_{∞}^{−5}μm

^{2}, with a decrease in the pore size, the influence of Knudsen diffusion becomes bigger and gradually affects the productivity of fractured horizontal wells and cannot be ignored, but the influence of the surface diffusion still can be ignored. When

*k*

_{∞}< 10

^{−7}μm

^{2}, because of the extremely small pore size, gas transport in a porous matrix is dominated by Knudsen diffusion, which has the biggest contribution to the production of fractured horizontal wells. Meanwhile, the impact of surface diffusion on the production of fractured horizontal wells becomes bigger and thus cannot be ignored.

**Fig. 4** Cumulative production predicted by different transport models.

### 4.2 Influence of intrinsic matrix permeability

For a single-porosity shale gas reservoir, the intrinsic matrix permeability is one of the main factors that affect the productivity of fractured horizontal wells. From Fig. 5, with an increase in the intrinsic matrix permeability, daily gas production and cumulative gas production increase dramatically. With a decrease in the intrinsic matrix permeability, gas transport in porous media is gradually dominated by Knudsen diffusion and surface diffusion, and the impact of viscous flow is reduced, which slows the production decline rate and guarantees long-term stable production.

**Fig. 5** Effects of intrinsic matrix permeability on production and cumulative production.

### 4.3 Influence of an adsorption layer

As shown in Fig. 6, because of the existence of adsorbed gas, the effective pore size and effective porosity decrease. With a decrease in the pore radius, the fractured horizontal well production gradually diminishes for the same production time. The production that involves the adsorption layer is lower than production to which the adsorption layer does not contribute. Furthermore, with a decrease in the pore radius, the ratio of the adsorption layer thickness divided by the pore radius gradually increases, and the impact of adsorption layer on production becomes bigger.

**Fig. 6** Impact of the adsorption layer on fractured horizontal well productivity with different pore radii.

The reduction in cumulative production considering the adsorption layer is a percentage that defined as the difference of the cumulative production between without and with the adsorption layer divided by the cumulative production without the adsorption layer. As shown in Fig. 7, when *r* > 10 nm, whether the adsorption layer thickness is considered or not, the production does not change much, with the reduction in cumulative production all below 10 %. When *r* < 10 nm, the influence of the adsorption layer thickness on production cannot be ignored. When *r* decreases to 1 nm, compared with the cumulative production of 10,000 days without the adsorption layer, the reduction in cumulative production considering the adsorption layer increased up to 52.3 %.

**Fig. 7** Impact of adsorption layer on the reduction in the cumulative production under different pore radii.

As can be seen in Fig. 8, when *r* > 10 nm, gas transport in a porous medium is dominated by viscous flow. Thus, with a decrease in the pore radius, considering the adsorption layer thickness, the decline rate of the fractured horizontal well production is relatively small. When *r* < 10 nm, Knudsen diffusion and surface diffusion become the main transport mechanisms. In this condition, with a decrease in the pore radius, considering the adsorption layer thickness, the fractured horizontal well production decrease degree gradually becomes bigger. Therefore, the influence of adsorption layer on production performance and productivity should not be ignored for nano-scale shale gas reservoirs.

**Fig. 8** Contributions to cumulative production by viscous flow, Knudsen diffusion, and surface diffusion under different pore radii.

### 4.4 Influence of gas reservoir pressure

Figure 9 shows that formation pore pressure dramatically impacts on the gas transport mechanisms in nanopores. When *r* > 10 nm, gas transport in matrix is dominated by viscous flow while the adsorption layer thickness has little impact on gas transport. Under the same drawdown pressure, with an increase in the formation pore pressure, the amount of absorbed gas increases correspondingly, and the production of a fractured horizontal well gradually increases. When *r* < 10 nm, gas transport is dominated by Knudsen diffusion and surface diffusion, and the influence of the adsorption layer thickness on gas transport can no longer be ignored.

**Fig. 9** Cumulative production under different pore pressures and pore radii.

In this condition, with a decrease in the formation pore pressure, the amount of absorbed gas decreases correspondingly and the thickness of adsorption layer gradually decreases. The increase in well production caused by Knudsen diffusion and surface diffusion exceeds the production loss caused by viscous flow, which makes the production of the fractured horizontal well increase with a decrease in the pore pressure. Surface diffusion varies inversely with the pore pressure, and the variation trend of its contribution to production with pressure change is the opposite to that of viscous flow.

When the pore radius is extremely small (e.g., 1 nm), with an increase in the pore pressure, the amount of absorbed gas and the thickness of the adsorption layer increase correspondingly, resulting in a decrease in the effective flow radius and effective porosity. Meanwhile, the cumulative production decreases under the same drawdown pressure (as shown in Fig. 10). When the pore pressure is relatively small, with an increase in the pore pressure, the amount of absorbed gas and the adsorption layer thickness increase significantly and the cumulative production drops dramatically.

**Fig. 10** Cumulative production under different pore pressures at the same drawdown pressure with r = 1 nm.

When the pore pressure is higher than 30 MPa, the increasing tendency of the amount of gas and the adsorption layer thickness slow down and tend to constant values. Meanwhile, the effective pore radius and the effective porosity gradually become stable, and the decrease in the cumulative production becomes smaller. The higher the pore pressure is, the smaller the influence of Knudsen diffusion and surface diffusion on production is. When the pore pressure is high enough, the influence of surface diffusion on production is hard to observe and thus can be ignored.