The effects of storativity ratio ω on pressure and rate are shown in Fig. 7. The storativity ratio mainly affects the width and depth of the “dip” in the pseudo-pressure derivative curve. The smaller the value of ω, the deeper and wider the “dip” in the derivative curve. This is because ω reflects the gas capacity in the fracture network. A smaller value of ω means less gas in the fracture network; thus, the gas in the matrix must transfer into the fracture network earlier. The smaller the value of ω is, the smaller the gas rate at the early stage.

**Fig. 7** Effect of storativity ratio on pseudo-pressure and rate curves. a Dimensionless pseudo-pressure and pseudo-pressure derivative curve, b dimensionless rate curve

The effects of fractal dimension D_{f} on pressure and rate are shown in Fig. 8. The fractal dimension significantly affects the curves throughout entire stages, except for the early stage when wellbore storage is the predominant mechanism. D_{f} reflects the development of the fracture network. The greater the value of D_{f}, the more the induced fractures produced by the fracturing process. The fractal permeability of the fracture network reduces slowly further away from the hydraulic fracture, which leads to a lower seepage resistance. Therefore, the position of pseudo-pressure and pseudo-pressure derivative curves becomes lower, and the value of gas rate becomes higher. Therefore, D_{f} is a satisfactory parameter that can be used to describe and evaluate the effects of multistage fracturing.

**Fig. 8** Effect of fractal dimension on pseudo-pressure and rate curves. a Dimensionless pseudo-pressure and pseudo-pressure derivative curve, b dimensionless rate curve

The effects of conductivity index θ on pressure and rate are shown in Fig. 9. The positions of the pseudo-pressure and pseudo-pressure derivative curves are lower when the value of θ is smaller. The gas rate decreases as the value of θ increases. This is because θ is related to the topology of the fracture network and reflects the connectivity of the fractal fracture network. In general, the more complex the fracture network is, the worse the connectivity of the fracture network, and the larger the value of θ. Therefore, θ can be also used to evaluate the quality of the fracture network.

**Fig. 9** Effect of conductivity index on pseudo-pressure and rate curves. a Dimensionless pseudo-pressure and pseudo-pressure derivative curve, b dimensionless rate curve

The effects of fracture conductivity w_{f}k_{1ref }on pressure and rate are shown in Fig. 10. The fracture conductivity mainly affects the pressure and rate at the early and middle stages. The larger the value of w_{f}k_{1ref}, the easier the gas flow in the hydraulic fracture. Therefore, the position of the pseudo-pressure and pseudo-pressure derivative becomes lower, and the position of gas rate becomes higher. However, the larger the fracture conductivity, the earlier the gas rate begins to decrease. Hence, fracture conductivity for a specific tight gas reservoir has an optimal value.

**Fig. 10** Effect of fracture conductivity on pseudo-pressure and rate curves. a Dimensionless pseudo-pressure and pseudo-pressure derivative curve, b dimensionless rate curve

The effects of dimensionless permeability modulus γ_{D} on pressure and rate are shown in Fig. 11. The definition of dimensionless permeability modulus varies under different working systems. Therefore, the values of γ_{D} in Fig. 11a and b are different. The stress sensitivity of fractures also significantly affects the curves throughout the different stages, except for the early time when wellbore storage is the predominant mechanism. The larger the value of γ_{D}, the faster the decrease in fracture permeability, which leads to a higher position of pseudo-pressure and pseudo-pressure derivative curves and a lower gas rate. When the gas rate is constant, a large pressure drop is required to maintain the constant rate as the value of γ_{D} is large, which leads to a short seepage period.

**Fig. 11** Effect of dimensionless permeability modulus on pseudo-pressure and rate curves. a Dimensionless pseudo-pressure and pseudo-pressure derivative curve, b dimensionless rate curve

## Conclusions

In this paper, we established an analytical model for an MFHW in tight gas reservoirs based on the trilinear flow model. Fractal porosity and permeability were applied to consider the heterogeneous distribution of the complex fracture network. Moreover, the stress sensitivity of fractures was considered in the model. Pedrosa substitution, perturbation method, and Laplace transformation were employed to solve this model. The transient pressure and rate type curves were obtained, and a sensitivity analysis was performed. The following conclusions can be obtained:

1. The transient pressure and rate type curves contain five flow regimes, including the wellbore storage stage, transitional flow, inter-porosity flow, compound linear flow, and boundary dominated flow.

2. Inter-porosity flow coefficient is related to the density of the fracture network. The larger the inter-porosity flow coefficient is, the lower the position of pressure curves and the larger the gas rate at the inter-porosity flow stage. Storativity ratio reflects the storage capacity of the fracture network. The larger the storativity ratio is, the lower the position of pressure curves and the larger the gas rate at the early stage.

3. Fractal dimension and conductivity index can fully reflect the development and connectivity of the complex fracture network. When the fractal dimension is larger and the connectivity index is smaller, more fractures are produced by fracturing process, and the connectivity between fractures is better, which leads to a lower position of pressure curves and a larger gas rate.

4. Fracture conductivity considerably affects the pressure and rate at the early and middle stages. A larger fracture conductivity leads to a lower position of the pressure curves and a larger gas rate at early and middle stages.

5. The effect of the stress sensitivity of the fracture is obvious and cannot be neglected. The larger the dimensionless permeability modulus, the higher the position of pressure curves and the lower the gas rate.

6. The model presented here can be utilized to recognize formation properties and forecast the pressure and rate dynamics of tight gas reservoirs. In addition, the new model is recommended as an evaluation model for screening attractive tight gas reservoirs and evaluating the effect of fracturing.

#### Acknowledgements

This research was supported by the National Basic Research Program of China (2015CB250900).

#### List of symbols

### Subscript

## Appendix 1: Analytical solutions in the Laplace domain

### Outer reservoir (region 3)

By taking the Laplace transformation, the model of region 3 in Laplace domain can be obtained:

The general solution of the model in the Laplace domain is as follows:

By taking the derivative of Eq. (A.2) with respect to x_{D}, we obtain:

Substituting inner and outer boundary conditions into Eqs. (A.2) and (A.3), we can obtain:

Substituting Eq. (A.4) into Eq. (A.3), we obtain:

Stimulated reservoir (region 2)

Obviously, there is a strong nonlinearity in the model of region 2. Hence, the Pedrosa substitution and the perturbation method are applied to linearize the equations (Pedrosa 1986; Wang 2014):

Similarly, dimensionless pseudo-pressure of region 1 can be also substituted to the following form:

Considering that the value of γD is always small, usually γDζ < 1. The zero-order perturbation solution can satisfy accuracy requirement. Therefore, the model of region 2 can be transformed to the following form:

By taking the Laplace transformation, the model of region 2 in Laplace domain can be obtained:

Equation (A.13) can be transformed to the following form:

Substituting Eqs. (A.5) and (A.14) into Eq. (A.12), we obtain:

The generalized Bessel function can be described as follows:

When 2+m−n > 0, the general solution of the function is as follows:

Obviously, 2+m−n=θ+2 > 0. Therefore, the general solution of the model is as follows:

By taking the derivative of Eq. (A.20) with respect to y_{D}, we obtain:

Substituting inner and outer boundary conditions into Eqs. (A.20) and (A.21), we can obtain:

Substituting Eq. (A.22) into Eq. (A.21), we obtain:

Hydraulic fracture (region 1)

Similarly, the model of region 1 in Laplace domain is as follows:

The general solution of the governing equation is as follows:

By taking the derivative of Eq. (A.25) with respect to x_{D}, we obtain:

**1. Constant gas rate**

Substituting constant gas rate inner boundary condition and the outer boundary condition into Eqs. (A.25) and (A.26), we can obtain:

Substituting Eq. (A.27) into Eq. (A.25), the solution of hydraulic fracture can be obtained as follows:

By setting x_{D}=0 in Eq. (A.28), we can obtain the bottom hole pressure of the fractured horizontal well in Laplace domain as follows:

**2. Constant bottom hole pressure**

Substituting the constant bottom hole pressure inner boundary condition and the outer boundary condition into Eq. (A.25) and Eq. (A.26), we can obtain:

Substituting Eq. (A.30) into Eq. (A.26), and setting x_{D}=0, we can obtain:

Finally, the dimensionless gas rate in Laplace domain can be obtained: