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The fractal geometry, anisotropy, discontinuity, and non-Darcy flow of tight reservoirs exert a significant effect on well production performance. In this study, the reservoir fractal geometry is represented by exponential functions on the basis of microseismic data, while the discontinuity of the fractures is presented as a nonequilibrium effect. The impact of the nonequilibrium effect and the low velocity non-Darcy flow on the temporal scale of the wellbore pressure is predicted herein. Results showed that the time scale analysis accurately simulates gas flow in a tight reservoir. The wellbore pressure gradually increases, whereas the pressure in the matrix lags when the nonequilibrium effect is considered. The wellbore pressure is affected in the early period by the nonequilibrium effect. However, at the later stage, the pressure in the matrix is mainly affected by the non-Darcy flow. When the non-Darcy flow is dominant, the pores without gas flowing through are better presented.

Tight gas reservoirs present an attractive option for energy exploitation due to their abundance, resource richness, and adaptability to the energy situation. Nevertheless, extraction of gas from tight reservoirs is challenging. The most economic and viable approach for tight gas reservoir development is multistage fracturing of horizontal wells. Fracturing produces a multispatial scale reservoir structure [

Microseismic monitoring technology has been widely used in visualization experiments [

During fracturing, the microseismic event distributes unevenly through the heterogeneous rocks [

The discontinuity of fractures promotes different velocities inside natural and hydraulic fractures. However, the flow in the hydraulic fracture is presented by the cubic law and discrete fracture model, which are widely employed in the former literature [

In addition, an assignable effect is exerted by the low gas flow velocity in the unstimulated region [

Last but not the least, during production, the velocity and the pressure reduction rate of each medium vary greatly, which is the multiple time scale fluid flow. However, only the production at each time point is described by the traditional flow model [

In this study, a detailed description of the reservoir heterogeneity extracted from microseismic measurements is provided. Then, a second dimensional heterogeneous, anisotropic, and dynamic coupling gas flow model is proposed. The crossflow coefficients are different from each other, which is dynamic coupling. They are derived by integrating the Navier-Stokes equation. In this way, the microstructure is upscaled more accurately. Due to the great flow velocity difference between mediums, the crossflow is established after the pressure difference between mediums. Therefore, the coefficients also decrease with space in two directions. The nonequilibrium flow is presented based on the wellbore storage effect. Additionally, the non-Darcy flow is captured by a power function, with an _{i}_{i}_{i}

The physical model in this work is a tight gas reservoir, based on the microseismic data and an earlier work. The model is presented in Figure

Schematic of segment of a horizontal well associated with one stage of hydraulic fracturing. The 3-region configuration is used for generating the fractal multiple porosity media model considered in this study.

The production rate of the well is assumed to be constant. Coupling between regions 1 and 2 conforms to the nonequilibrium effect. The intersection between the hydraulic fracture and the natural fracture is shown in Figure

Microstructure of the fracture.

The solution of (

From (

Integrating (

Here,

Region 2 is composed of the inorganic matrix and fracture networks. The natural fracture, the inorganic matrix, and the kerogen pores have a fractal geometry and a multiscale distribution. Hence, the coupling between the flow mechanisms is dynamic. Based on the microseismic data, the heterogeneity in the flow mechanisms is described by exponential functions. The flow equations are as follows:

Region 3 is composed of a homogeneous inorganic matrix. A low velocity non-Darcy flow evolves from this region to region 2. The non-Darcy flow velocity,

The flow equation in region 3 is as follows:

The second dimensional heterogeneity in the model is considered and given by continuous functions,

From equations (

The microseismic data is collected from the field [

Microseismic data: (a) microseismic event distribution along the

Parameters of the tight gas reservoir considered in this study were published previously [

Parameters of the tight gas reservoirs [

Parameter | Value | Parameter | Value |
---|---|---|---|

0.38 | |||

0.25 | 0.2 | ||

0.02 m | |||

0.07 | 200 m | ||

0.04*10^{-14} Pa^{-1}·s^{-1} | |||

10^{-0} Pa^{-1} | |||

400 m | |||

15 m | 200 m | ||

Model validation: (a) wellbore pressure, (b) temporal scale distribution, and (c) error analysis.

At the beginning, there is a fast gas flow out of the hydraulic fracture, which increases the wellbore pressure. Then, the gas starts to flow out of the natural fracture. The natural fracture permeability beside the inner boundary is higher. The rate of wellbore pressure increase slows down. At a later stage, the gas struggles to flow from the matrix. However, more gas flows out of the natural fracture and the hydraulic fracture. Consequently, the wellbore dimensionless gas pressure increases faster.

The temporal scale diagram is presented in Figure ^{-5}. The left peaks in the figure correspond to the fracture. The flow velocity in the hydraulic fracture is higher. Therefore, the peak is quite low but slightly higher than the peak around

The comparison between temporal scale analysis results and field results is shown in Figure

Gas flow from field study compared with the temporal scale analysis.

Figure

The wellbore dimensionless gas pressure at different values of

The temporal scale diagram is shown in Figure ^{-4} in the hydraulic fracture and 10^{-3} to 0.01 in the natural fracture. At a higher ^{-2} become lower. At a later stage, the pressure propagation accelerates. As a result, the time scale of the peak between the time scales 0.01 and 0.05 is less significant. The coupling time between regions 1 and 2 is about 0.1, and the dimensionless coupling time between natural and hydraulic fractures is about 0.1. At higher

Figure

The wellbore dimensionless gas pressure at different heterogeneity conditions: (a) wellbore pressure, (b) temporal scale distribution, and (c) wellbore pressure derivative.

In the second dimensional heterogeneous case, the right side of the peaks around the time scale 10^{-3} to 10^{-2} is higher than that in the first dimensional case. The left part of the peak around 0.1 is steeper. More gas flows to the wellbore. Therefore, a small peak appears around the time scale 0.03. In addition, the gas inside the matrix struggles to flow out at different temporal scales, especially beside the outer boundary. As a consequence, the peak around the time scales 1 to 10 becomes higher and wider, especially to the left of the peak. A small peak appears around the time scale 10. The flowing time in the matrix is 1 to 10. The area with large heterogeneity behaves as a “nonsealing fault.” As the gas pressure propagates through the “nonsealing fault,” the gas in the matrix flows into the “fault” more easily than the gas in the fracture. The dip between the two peaks corresponds to the “nonsealing fault,” while the gas beside the outer boundary also struggles to flow to the wellbore. The dimensionless crossflow time in the matrix beside the outer boundary is 20. Therefore, the peak around 20 becomes lower. Only the peaks to the right of time scale 1 are affected by the heterogeneous crossflow. As shown by the blue line, since the crossflow coefficient beside the inner boundary is higher than that beside the outer boundary, the crossflow rate decreases. The peak around 1 to 10 is consequently lower on its left side. The crossflow mainly occurs beside the inner boundary. A small peak appears around the time scale 1, while the peaks between 10 and 10^{3} become higher, especially to the right of the peaks between 10 and 10^{3}. The flow time in region 3 increases to 10^{3}. At the “nonsealing fault,” the gas in the matrix also has great difficulty in flowing out. Therefore, the dip between 10 and 20 disappears, while the left side of the peak between 20 and 100 becomes flat.

The wellbore gas pressure in the case with low velocity non-Darcy flow is presented in Figure

The wellbore gas pressure for the case with non-Darcy flow: (a) wellbore pressure and (b) temporal scale distribution.

The temporal scale diagram is presented in Figure

This study shows a gas reservoir model of a horizontal well considering a nonequilibrium flow and a non-Darcy flow can be used to describe the reservoir microstructure. A gas flow model considering heterogeneous, anisotropic, and nonequilibrium effects as well as dynamic coupling was established and solved using a temporal scale analysis approach. A case study was analyzed to validate the model, and the effect of heterogeneity of the mass transfer coefficient on the gas temporal scales was discussed. The microseismic data are in good agreement with the field data using an exponential function. The increase in wellbore pressure first slows down, then gradually becomes rapid. The discontinuity is described accurately by the nonequilibrium effect. Pressure and its derivative are affected in the early period by the nonequilibrium effect, whereas they are affected by the non-Darcy flow at a later period. Time of pressure propagation through the matrix lags in the case with the nonequilibrium effect, and the gas flows out of the matrix and natural fracture later. In the case with the heterogeneous crossflow coefficient, the matrix is less developed, especially that beside the outer boundary. Moreover, the heterogeneous flow is better presented in the temporal scale distribution diagram. The non-Darcy flow is well described by a power function for the first time. Pores through which gas does not flow are better handled by the non-Darcy flow exponent. However, at lower non-Darcy flow values, the reservoir is tighter, and the flow velocity further decreases.

Flow regime

Concentration

Gas compressibility coefficient

Nonequilibrium effect coefficient

Total compressibility (Pa^{-1})

Probability of event pairs closer than

Probability of event pairs closer than

Non-Darcy flow coefficient

Anisotropy coefficient

Reservoir height (m)

Permeability (m^{-12})

Non-Darcy flow exponent

Number of hydraulic fractures

Gas pressure (Pa)

Gas production (m^{3}/s)

Conventional gas constant

Temperature

Time (s)

Temporal scale

Hydraulic fracture width (m)

Length along the hydraulic fracture (m)

Length along the wellbore (m)

Compression factor.

Contribution of each temporal scale

Viscosity (Pa·s)

Transfer coefficient (s^{-1}·Pa^{-1})

Pseudopressure function (Pa/s)

Porosity.

Initial

Dimensionless

Natural fracture

From the natural fracture to the matrix

The index of temporal scales

Hydraulic fracture

Matrix

From the matrix to the natural fracture.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This study was supported by the Postdoctoral Application Research Project in Qingdao (qdyy20200066), the Postdoctoral Self-Innovative Project (20CX06094A), and the China Scholarship Council.