Multistage fractured horizontal wells (MFHWs) have become the main technology for shale gas exploration. However, the existing models have neglected the percolation mechanism in nanopores of organic matter and failed to consider the differences among the reservoir properties in different areas. On that account, in this study, a modified apparent permeability model was proposed describing gas flow in shale gas reservoirs by integrating bulk gas flow in nanopores and gas desorption from nanopores. The apparent permeability was introduced into the macroseepage model to establish a dynamic pressure analysis model for MFHWs dualporosity formations. The Laplace transformation and the regular perturbation method were used to obtain an analytical solution. The influences of fracture halflength, fracture permeability, Langmuir volume, matrix radius, matrix permeability, and induced fracture permeability on pressure and production were discussed. Results show that fracture halflength, fracture permeability, and induced fracture permeability exert a significant influence on production. A larger Langmuir volume results in a smaller pressure and pressure derivative. An increase in matrix permeability increases the production rate. Besides, this model fits the actual field data relatively well. It has a reliable theoretical foundation and can preferably describe the dynamic changes of pressure in the exploration process.
Shale gas is known as a key resource to meet the increasing world energy demand because of its rich reserves and extensive distribution [
In recent studies [
Mayerhofer et al. [
In recent years, many general semianalytical models, considering microscopic seepage mechanisms, have been proposed with the rapid development of computer technology [
The total shale gas flux is composed of a viscous flow flux and a Knudsen diffusion flux component. Viscous flow is strongly prominent when the short molecule free path is far less than the pore diameter. In other words, the transport mechanism is governed by viscous flow in macropores (where the pore diameter > 50 nm). The HagenPoiseuille equation can describe the molar flux [
Gas viscosity gradually deviates from the traditional viscosity definition for high Knudsen numbers. Karniadakis et al. [
According to the Darcy equation, the apparent permeability of viscous flow can be expressed as [
When there is shale gas transport through micropores under low pressure, the molecule free path is long and equal to the pore diameter. The Knudsen diffusion is prominent because collisions between molecules and the micropores wall are more frequent than the intermolecular collisions. The flux of the Knudsen diffusion can be expressed as [
By combining (
In the actual shale gas reservoirs, different transport mechanisms exist at the same time. Therefore, the total shale gas flux should be a weighted summation of the viscous flow flux and the Knudsen diffusion flux, based on their different contributions. The weight factors of the viscous flow and the Knudsen diffusion flow are defined as the ratios of intermolecular collisions and molecular/porewall collisions to total collisions, respectively. Based on the definitions for a gas molecule free path [
Therefore, the total apparent permeability in nanopores is
With a decrease in pressure, the gas molecule free path increases immediately. The weight factor of the Knudsen diffusion also increases. The piece of research carried out by Wu et al. [
Organic matter is described by weak strength and strong sensitivity to stress change. With an increase in reservoir pressure, adsorbed gas begins to desorb. Gas desorption results in shrinkage of the organic matrix and to an increased effective hydraulic diameter [
Meanwhile, based on Seidle’s model [
Assume that the pore volumes are proportional to the gas flow channels. In accordance with the capillary model, the effective hydraulic diameter can be obtained as
Because of the large surface area and oilwet characteristic, nanopores have a strong adsorbed gas capacity. The mass balance equation, which considers the adsorbed gas, can be expressed as
Cui et al. [
According to the Langmuir isotherm equation, the effective adsorption porosity can be expressed as
Finally, the apparent permeability model for gas transport in shale nanopores can be expressed as
Figure
Diagram of MFHWs and of the sevenflowregion model.
The main assumptions considered for this model are listed below:
The shale reservoir has been fractured and has a constant reservoir thickness. All hydraulic fractures are symmetric and arranged uniformly along the horizontal well. Length and conductivity of the hydraulic fractures are the same.
Natural fracture permeability is stressdependent and shale gas flow in the matrix is driven by concentration difference.
The initial pressure throughout the reservoir is uniform.
Hydraulic fractures penetrate the entire pay formation. Fracture interference is ignored. After SRV, large fracture networks can be generated in region 2 and it has higher natural fracture permeability compared to the other regions.
The shale gas flow in the reservoir is considered to be an isothermal flow and the effect of gravity and capillary pressure are negligible.
Nonsimulated areas are described by a dualporosity system consisting of a shale matrix and natural fractures.
The only route for shale gas to reach the horizontal wellbore is through hydraulic fractures. The flow rate of MFHW is the sum of the flow rate from every hydraulic fracture.
According to the mass conservation equation, the motion equation, and the state equation, the partial differential formulas for shale gas flow in a matrix system and a fracture system can be obtained as follows [
Pedrosa’s substitution [
To solve (
A series of variables and dimensionless variables are defined in Table
Definitions of variables.
Variable  Expression  Dimensionless expression  

Pseudopressure, psi^{2}/cp 



Permeability modulus factor, psi^{−1} 



Diffusivity, ft^{2}/hr 











Adsorption coefficient, dimensionless 


Time, hr 


Length, ft 








In this upperreservoir region, the governing diffusivity equation for shale gas flow in the
At the top of the reservoir
On the interfaces
Equation (
Consistent with region 7, the governing diffusivity equation for shale gas flow in the
The boundary condition at the top of the reservoir
The boundary conditions on the interfaces of regions 6 and 2 and regions 6 and 4 are
The solution for (
In this region, shale gas flow takes place in the
Along the horizontal well
Flux across the interface of regions 5 and 7
Substituting (
In this region, shale gas flow in the
Pressure continuity condition between regions 5 and 3 (at
By solving (
In a similar way, shale gas flow in this region takes place in the
The same method is also used in region 5, where the governing diffusivity equation can be simplified as
Noflow condition at the outerreservoir boundary
On the interface of regions 4 and 2, the boundary condition is
The solution for (
This region is adjacent to regions 5 and 7, so shale gas flow direction takes place along the
For the
For the
By substituting (
Between two fractures, there is a noflow boundary (at
On the interface of regions 3 and 2, the pressure continuity condition is written as
The solution for (
Similarly, as region 2 is adjacent to hydraulic fracture, the governing diffusivity equation is
Consistent with region 3, the simplified equation can be written as
At the interface of regions 2 and 3, the boundary condition is characterized by flux continuity:
And between the hydraulic fracture level and region 2, the boundary condition is
Solving (
In this region, the diffusivity equation is given by
The solution can be obtained:
Assuming that pressure distribution along the horizontal well is uniform, the dimensionless downhole pressure can be solved by setting
Shale gas flow in hydraulic fractures is not completely linear. The nonlinear streamlines result in an additional pressure drop. Mukherjee and Economides [
Based on the Duhamel principle and the superposition principle, the dimensionless flowing bottom hole pressure (FHBP), which takes into consideration the wellbore storage effect and the skin effect, can be expressed as [
The FHBP for MFHWs, when considering the stress sensitivity effect, can be obtained as
The relationship between the dimensionless rate under the condition of constant pressure and dimensionless pressure caused by the constant rate is described as [
Figure
Basic parameters.
Variable  Value 


0.05 

10^{−8} 

25 

333 

1.38 

2.56 

4.35 

18.66 

4000 

4.3 

1.75 

1 

0.016 

22.4 

0.085 

6.89 

2.72 

2.0 
Variation of
Figures
Model parameters.
Variable  Value 

Reservoir size in 
500 
Distance to noflow boundary, 
250 
Reservoir thickness, 
250 
Fracture width, 
0.01 
Reservoir temperature, 
333 
Gas volumetric factor, 
0.004 
Initial matrix compressibility 
10^{−3} 
Fracture porosity, 
0.45 
Fracture permeability, 
10^{3} 
Hydraulic fracture compressibility, 
10^{−3} 
Hydraulic fracture permeability, 
10^{5} 
Well length, 
1500 
Fracture halflength, 
400 
SRV width, 
250 
Fracture height, 
0.001 
Matrix radius, 
5 
Initial viscosity, 
0.018 
Initial matrix porosity, 
0.05 
Matrix permeability, 
10^{−6} 
Fracture compressibility 
10^{−3} 
Hydraulic fracture porosity, 
0.38 
Effect of hydraulic fracture halflength on pressure.
Effect of hydraulic fracture halflength on production.
Figures
Effect of hydraulic fracture permeability on pressure.
Effect of hydraulic fracture permeability on production.
The effects of the Langmuir volume on pressure and production are plotted in Figures
Effect of Langmuir volume (
Effect of Langmuir volume (
Figures
Effect of matrix radius on pressure.
Effect of matrix radius on production.
Figures
Effect of matrix permeability on pressure.
Effect of matrix permeability on production.
Figures
Effect of induced fracture permeability on pressure.
Effect of induced fracture permeability on production.
The MFHW is located in the transitional zone between the Sichuan Basin and the YunnanGuizhou Plateau (400–1,200 m in altitude) and mainly developed a marine and continental sedimentary facies. The specific stratum parameters are listed in Table
Parameters of the typical well.
Porosity  Viscosity  Compressibility  Temperature 

0.032  1.47 
4.8 
96.8°C 
Wellhead pressure  Depth  Pressure coefficient  Horizontal length 
39 MPa  2900 m  1.96  1438 m 


Fracturing segments  Thickness 
1.62 
1.62 
15  35 m 
As seen in Figure
Production history.
History matching of typical well.
In this study, an analytical model was presented to simulate shale gas flow through MFHW in a shale reservoir. Based on microscale percolation mechanisms in nanopores, a sevenlinearflow model for dualporosity formations was established. This model takes viscous flow, the Knudsen diffusion, the sorptioninduced swelling response, the adsorption/desorption, the matrix shrinkage and the stress sensitivity effect into consideration. The pressure transient analysis was carried out under conditions of constant rate. The following conclusions were deduced:
Temperature, K
Shale tortuosity
Gas molar mass, kg/mol
Universal gas constant, J/(mol
Boltzmann constant, J/K
Shale density, t/m^{3}
Gas density, t/m^{3}
Shale compressibility, MPa^{−1}
Gas compressibility, MPa^{−1}
Mole volume of gas at standard constant, m^{3}/mol
Langmuir volume, m^{3}/t
Langmuir pressure, MPa
Shale matrix Young modulus, MPa
Rarefaction coefficient
Pseudopressure, psi^{2}/cp
Production rate, Mscf/D
Laplace transform parameter
Time, hours
Permeability modulus, psi^{−1}
Hydraulic fracture width, ft
Hydraulic fracture halflength, ft
Reservoir size in
SRV width, ft
Halfwidth between two fractures, ft
Half fracture height, ft
Half reservoir thickness, ft
Radius, ft
Shale viscosity, cp
Porosity
Shape factor
Diffusivity, ft/hour
Flow capacity ratio
Desorption coefficient at average pressure
Storativity ratio
Skin factor
Dimensionless wellbore storage coefficient.
Dimensionless
Natural fracture system
Hydraulic fracture
The
Matrix system
Standard condition
Dimensionless wellbore
Dimensionless length of external boundary.
The authors declare that they have no conflicts of interest.
The authors would like to acknowledge the support from MOE Key Laboratory of Petroleum Engineering in China University of Petroleum (Beijing). The authors would also like to acknowledge the financial support from the National Nature Science Foundation of China (no. 51504265) and Science Foundation of China University of Petroleum, Beijing (no. 2462015YQ0223).