Abstract
Trilinear flow model is an effective method to reproduce the flow behavior for horizontal wells with multistage hydraulic fracture treatments in unconventional reservoirs. However, models developed so far for transient analysis have rarely considered the inflow performance of wells. This paper introduces a new composite dual-porosity trilinear flow model for the multiple-fractured horizontal well in the naturally fractured reservoirs. The analytical solution is derived under a constant rate condition for analyzing transient pressure behaviors and generating the transient inflow performance relationships (IPRs). The plots of pressure profiles with time could provide insightful information about various flow regimes that develop throughout the entire production cycle. Sensitivity analysis of pressure and pressure derivative response was also performed by varying different parameters (such as hydraulic fracture width and permeability, reservoir configurations, etc.) and by which the impacts of different parameters on the durations of regimes as well as the productivity index can be confirmed. The main outcomes obtained from this study are as follows: (1) the ability to characterize naturally fractured reservoirs using a new composite dual-porosity trilinear flow model; (2) the application of analytical solutions of transient analysis to generate transient IPR curves for different flow regimes; (3) understanding the effect of reservoir configurations, fractures, and matrix characteristics on pressure distribution, flow regime duration, and transient IPR. More specifically, the pressure drop increases and the productivity index decreases with the decrease of the hydraulic fracture conductivity and the increase of matrix permeability and the skin factor. Also, the larger hydraulic fracture spacing and drainage area result in the later onset of the pseudo-steady-state regime. (4) A comprehensive study on transient pressure behaviors and transient inflow performance can provide valuable information to characterize the multifractured complex systems as well as some insights into the production.
1. Introduction
Hydraulic fracturing technology has been extensively used in the exploitation of unconventional reservoirs. Due to the rise of flow conductivity near wellbore, the seepage process in various porous media becomes more complex than that in conventional reservoirs. Over the past few decades, to solve the pressure or rate problem, there have been many well-developed models that can characterize the flow in the fractured formation.
Since the porous media of the fracture network in the enhanced fracture region consist of two independent systems, namely, induced fractures and the surrounding matrix, such region is usually characterized by dual-porosity models. Warren and Root1 first introduced the concept of dual porosity, in which natural fractures provide the main flow path, while the matrix provides the main storage space. The transfer of fluids from the matrix to the fracture was considered as pseudo-steady-state (PSS) flow by assuming a PSS matrix-fracture shape factor. Given that the fracture network is superimposed in the primary porosity,2 the pressure drops at any point of the matrix are the same.3
For the hydraulically fractured horizontal wells, the fluid movement in the formation is generally described by multilinear flow. Brown et al.4 introduced the classical trilinear model, whose analytical solution provides the information of the pressure transient response of 1-D flow in three regions: hydraulic fractures, the stimulated reservoir region surrounding the hydraulic fractures (SRV), and original low-permeability reservoir region (USRV). Since then, even though lots of multilinear models were presented with different reservoir configurations,5−7 most existing linear flow models regard the USRV as single-porosity media. Since natural fractures exist not only in induced fracture network but also in the unstimulated reservoir region, such naturally fractured reservoirs should be idealized and mathematically treated as composite double-porosity regions. In this paper, the coupled dual-porosity model in both SRV and USRV was developed for naturally fractured reservoirs using the PSS approach.
Moreover, there have been numerous works analyzing the performance of fractured wells through the transient pressure behavior and production decline analysis.8−10 Zhang et al.11 developed a semianalytical model to investigate the transient behavior of multiwing fractured vertical wells in coal-bed methane gas reservoirs considering the effects of adsorption–desorption, diffusion, and viscosity flow. Wang et al.12 and Kou et al.13 provided novel mathematical models of vertical wells with multiple radial hydraulic fractures in the stress-sensitive coal-bed methane reservoir to simulate the transient pressure responses. The effects of the key parameters, such as permeability modulus and the radius of stimulated reservoir volume were also analyzed by solving the semianalytical solutions. Kou et al.14 proposed a transient analysis model of a depleted well with multiple radial hydraulic fractures in a stress-sensitive coal seam to evaluate the CO2 storage capacity in coal seams.
However, very few studies relate the transient behaviors and flow regimes to the inflow performance prediction. Al-Rbeawi15 generated the time-variant inflow performance relationship (IPR) based on the traditional trilinear-flow model and classified the IPR curves with distinct flow regimes by using wellbore pressure decline and its derivative profiles. This paper follows a similar workflow and establishes a new trilinear flow model to investigate the pressure behaviors, flow regimes, as well as IPR of multiple-fractured horizontal wells in naturally fractured reservoirs. Sensitivity analysis is also performed to illustrate various controls on flow-regime durations, transient behaviors, and inflow performance.
2. Mathematical Model
2.1. Trilinear Flow Model Description
As shown in Figure 1, the symmetrical pattern is one-quarter of a rectangular drainage region at an individual hydraulic fracture. The pattern is divided into three parts (Figure 2): the outer region (USRV), the inner region (SRV), and hydraulic fractures. In both outer and inner regions, the naturally fractured reservoir is characterized by a dual-porosity model, thus forming two porosity–permeability systems for the matrix and natural fractures. The analytical solutions are derived by using the continuity conditions for flux and pressure, as well as the boundary conditions for each flow process, respectively.
Figure 1.
Idealization and simplification of the physical model.
Figure 2.
Dimensions of the physical model.
The assumptions using for the derivation of the trilinear flow model are made as follows
The matrix permeability in the outer and inner regions is identical.
The hydraulic fractures are assumed to have the same half-length, width, and spacing in the formation.
Single-phase 1-D flow is developed in the three regions.
The pseudo-steady dual-porosity model (sugar cube model1) is applied in the outer and inner regions.
2.2. Analytical Model Development
To simplify the derivation process, the governing diffusivity equations and related initial/boundary conditions are developed in the Laplace domain.
Starting with the 1-D linear flow in the outer region, the diffusivity equation and the associated boundary conditions can be represented in the following dimensionless form
For a naturally fractured system
 |
1 |
For a matrix system
 |
2 |
Substituting eq 2 to 1, the differential equation is solved to give
 |
3 |
where
 |
Also, the outer and inner boundary conditions are
 |
4 |
 |
5 |
The general solution for dimensionless pressure in the outer region is given by
 |
6 |
Flow in the inner region adjacent to the outer region is assumed as 1-D linear flow with the direction perpendicular to the hydraulic fracture. Following steps similar to those of the outer region, the diffusivity equation and related boundary conditions are given by the following equation
 |
7 |
where
 |
 |
 |
and
 |
8 |
 |
9 |
Then, the dimensionless pressure solution for the inner region is obtained as
 |
10 |
This dimensionless pressure solution of 1-D linear flow represents the pressure distribution inside the hydraulic fracture. The pressure behavior can be described by
 |
11 |
where
 |
The boundary conditions are given as follows
 |
12 |
 |
13 |
The dimensionless pressure solution at the hydraulic fracture face can be obtained by
 |
14 |
Considering the effect of non-Darcy flow and the mechanical skin factor, a total skin factor (SF) is introduced, so eq 14 can be rewritten as
 |
15 |
The solutions above in the Laplace domain can then be inverted into the time domain numerically using the Stehfest16 algorithm.
According to Van Everdignen and Hurst,17 the dimensionless production rate with a constant bottom hole pressure is given by
 |
16 |
and the dimensionless cumulated production is given by
 |
17 |
Using the above equations, the transient production behavior of fractured horizontal wells can be analyzed. Details of analytical model development are given in Appendix B.
3. Transient Behaviors and Flow Regimes
Based on the model presented in the previous sections, the transient pressure behaviors can be obtained for constant rate wells, while transient rate behaviors can be obtained for constant bottom-hole pressure wells, which helps to distinguish between distinct flow regimes that develop throughout the production process.
These plots can provide information to estimate the starting and ending times of all developed flow regimes, as shown in Figures 3 and 4. These figures describe the pressure and pressure derivative curves, and the production rate decline curve and cumulative production profile obtained by using the reservoir parameters are listed in Table 1. Three flow stages developed within a fractured horizontal well in a fractured reservoir can be easily identified from the plots, including the bilinear flow, linear flow, and PSS flow. It is worth mentioning that the linear flow in the hydraulic fracture may hardly be captured due to its extremely limited duration.
Figure 3.
Dimensionless pressure and pressure derivative curves.
Figure 4.
Dimensionless rate and cumulative production profile.
Table 1. Reservoir and Fluid Data.
| reservoir and fluid data | values |
|---|---|
| reservoir pressure (MPa) | 50 |
| reservoir thickness (m) | 10 |
| hydraulic fracture half-length (m) | 180 |
| hydraulic fracture width (m) | 0.0018 |
| reservoir size in x-direction, XeD | 2 |
| distance between hydraulic fracture, YeD | 2 |
| porosity of the matrix | 0.1 |
| porosity of fracture in SRV | 0.3 |
| porosity of fracture in USRV | 0.25 |
| porosity of hydraulic fracture | 0.1 |
| compressibility of the matrix (MPa–1) | 2.30 × 10–4 |
| compressibility of fracture in USRV (MPa–1) | 3.00 × 10–4 |
| compressibility of fracture in SRV (MPa–1) | 3.00 × 10–4 |
| compressibility of hydraulic fracture (MPa–1) | 2.00 × 10–4 |
| permeability of the matrix (mD) | 0.1 |
| permeability of fracture in USRV (mD) | 1 |
| permeability of fracture in SRV (mD) | 10 |
| permeability of hydraulic fracture (mD) | 10,000 |
| oil viscosity (mPa s) | 2 |
| oil formation volume factor | 1.05 |
| shape factor | 20 |
Stage 1: the linear flow in hydraulic fractures usually lasts for a very short time and sometimes cannot even be observed due to the presence of the wellbore storage effect.
Stage 2: the bilinear flow is the first easily identifiable flow regime in the transient pressure and rate curves, and it flows linearly from both the hydraulic fractures and the inner region to the wellbore. This phase corresponds to a slope of 0.25 on the pressure derivative curves.
Stage 3: after the bilinear flow, the linear flow represents the flow from the inner region and can be identified by a 0.5 slope on the pressure derivative curves.
Stage 4: the PSS flow is the last flow regime observed during the whole process, indicating that the pressure pulse reaches the boundary of the reservoir. It can be characterized by a slope of one for pressure derivative curves. Meanwhile, both the pressure curve and the pressure derivative curve overlap with each other. Depending on the reservoir configurations and properties, this flow period usually takes a long time to arrive, sometimes even decades.
4. Transient IPR Curves
Based on the above studies, the time when each flow regime occurs can be verified by transient behavior analysis. Therefore, the precondition of obtaining the time-variant IPR of each flow period can be implemented.
A procedure presented in previous studies18,19 for generating transient IPR under different production conditions is used here. According to the constant-rate pressure results obtained by using eq 14 established by the model above, at any time of interest, the flow rates with corresponding bottom hole pressures are known from the curves as shown in Figure 5, illustrating the IPR curves as a function of time as shown in Figure 6. Similarly, pairing the constant-Pwf flow rate results obtained from eq 15 with the corresponding pressures can also lead to transient IPR curves, as shown in Figures 7 and 8.
Figure 5.
Flowing pressure profile at different flow rates (Q1 < Q2 < Q3).
Figure 6.
Transient IPR curves at constant flow rates (t1 < t2 < t3).
Figure 7.
Production rate profile at different flowing pressures (Pwf 1 < Pwf 2 < Pwf 3).
Figure 8.
Transient IPR curves under constant flowing bottom hole pressures (t1 < t2 < t3).
5. Results and Discussion
Based on the above theory, the transient behaviors and transient IPR curves of different flow regimes of a horizontal well with multiple hydraulic fractures in naturally fractured reservoirs were plotted and analyzed. In this part, sensitivity analysis is conducted to specify how and when the key parameters affect transient behaviors and IPR curves. The well is assumed to be under constant-rate conditions in this part.
5.1. Permeability of Hydraulic Fractures
It is known that fracture permeability has a significant effect on the conductivity of fractures. With the increasing permeability, the fluid flows more easily in the porous medium toward the wellbore. Figure 9 shows the effect of different permeability on transient pressure and derivative pressure curve. The proposed model mainly yields different responses in the early period and the same responses at the late times. Knowing that hydraulic fracture permeability is a critical property related to flow in hydraulic fractures, it is reasonable to exhibit divergence in the bilinear flow regime on the curves. On the contrary, the shapes and trends of the later pressure behavior plots show no change with different values of hydraulic fracture permeability. Thus, it does not influence the linear flow and PSS flow which developed in the inner and outer regions.
Figure 9.
Pressure behaviors for different hydraulic fracture permeability.
To compare in the same real-time domain, select a specified time point to generate a transient IPR in the early flow regime, and the result is shown in Figure 10. The larger permeability, the larger the productivity index (the reciprocal of the slope).
Figure 10.
Transient IPR curves for different hydraulic fracture permeabilities.
5.2. Width of Hydraulic Fractures
Another parameter that dominates the fracture conductivity is the fracture width. The increased width means the strong storage capacity of hydraulic fractures and will result in more time for the fluid to flow within the hydraulic fractures. Similarly, the impact of the hydraulic fracture width on the pressure profile mainly concentrates in the early period, as shown in Figure 11. Meanwhile, according to the corresponding IPR curves in Figure 12, in the case of wider fractures, the reservoir could offer more flow rates at the same flowing bottom hole pressure. Based on the above results, the conductivity proportional to permeability and dimensionless hydraulic fracture width is proved to be conversely proportional with the dimensionless pressure drop and directly proportional to the productivity index.
Figure 11.
Pressure behaviors for different hydraulic fracture widths.
Figure 12.
Transient IPR curves for different hydraulic fracture widths.
5.3. Drainage Distance (Xe)
Because the hydraulic fractures are not fully extended to the formation, the length of the drainage region in the X-direction is one of the important parameters of reservoir configuration. Figure 13 indicates the effect of different sizes of the unstimulated region on the wellbore pressure behavior. It can be seen that in reservoirs with a small drainage area, the dimensionless time of the PSS flow starts earlier, while in those with a large drainage area, the pressure pulse needs more time to reach the boundary, which contributes to the late onset of the PSS period. Moreover, the pressure drops required for reservoir fluid to move in the outer region are higher in the small drainage area. The starting moment of the PSS regime to generate transient IPR are selected, and the curves are shown in Figure 14. This figure demonstrates that the productivity index of the small outer region is higher than that of the large outer region when the PSS regime starts.
Figure 13.
Pressure behaviors for different drainage distances.
Figure 14.
Transient IPR curves for different drainage distances.
5.4. Spacing between Hydraulic Factures (Ye)
Another parameter that describes the dimensions of the drainage area is the hydraulic fracture spacing in the Y direction. Figure 15 indicates the effect of the different sizes of the stimulated region on the wellbore pressure behavior. Similarly, there is no difference in the beginning period until the pressure pulse reaches the no-flow boundary in the Y direction. The long-distance between fractures results in a long time for the whole inner region to participate in the flow and also leads to a delay in the PSS flow. The transient IPR at the start time of the PSS regime is given in Figure 16, where a narrower spacing corresponds to a relatively higher productivity index.
Figure 15.
Pressure behaviors for different hydraulic facture spacings.
Figure 16.
Transient IPR curves for different hydraulic facture spacings.
5.5. Permeability of the Matrix
Figure 17 shows the effect of different matrix permeabilities on the wellbore pressure. It can be observed that all curves show the same trend and shape but the absolute values differ, indicating that the permeability of the matrix makes no impact on the duration distribution of each flow regime. The pressure behavior and IPR curves in Figure 18 illuminate that an increase in the matrix permeability raises the pressure drop and deteriorates the production rate.
Figure 17.
Pressure behaviors for different permeabilities of the matrix.
Figure 18.
Transient IPR curves for different permeabilities of the matrix.
5.6. Skin Factor
As depicted in Figure 19, as the skin factor reduces, the dimensionless pressure drops decreases. However, the curves converge to a stabilized value during the PSS period, meaning that this effect cannot reach the late time pressure profile. Meanwhile, different cases share the same pressure derivative plot. Thus, one can conclude that the skin factor does not affect the arrival time of each flow regime. Figure 20 indicates that the increasing skin factor contributes to a sharp IPR curve with a small productivity index.
Figure 19.
Pressure behaviors for different skin factors.
Figure 20.
Transient IPR curves for different skin factors.
6. Conclusions
In this paper, an analytical trilinear model was developed to investigate the complex dual-porosity system of a naturally fractured reservoir. The solution of the model was derived to analyze the transient pressure behavior, flow regimes, and IPR of multiple fractured horizontal wells. The following conclusions can be drawn
-
1.
The complex dual-porosity system model takes account both inner and outer natural fractures, thus providing more accurate results when characterizing the naturally fractured reservoirs.
-
2.
The plots of the dimensionless transient pressure and pressure derivative can be helpful to identify different flow regimes that developed sequentially during the whole production period. Also, by inverting the constant rate results of the transient analysis to real-time units, a series of flow rates can be used in each flow regime to obtain the transient IPR curves.
-
3.
The sensitivity analysis of different parameters is conducted to examine the effects on the pressure behavior as well as the productivity index of the wells.
-
4.
The hydraulic fracture conductivity, mainly determined by hydraulic fracture permeability and width, plays a more pronounced role in the early period. The multiple-fractured horizontal wells with larger conductivity result in smaller pressure depletion and a larger productivity index.
-
5.
The effects of reservoir configurations on the transient behavior are significant. The drainage distance (Xe) is investigated to influence the flow duration in the outer region, and the PSS flow occurs earlier with a decrease in the drainage distance. The hydraulic fracture spacing (Ye) controls the inner region, and the larger spacing results in a longer time for the whole inner region to participate in the flow and finally leads to a delay in the PSS flow.
-
6.
The matrix permeability and skin factor are confirmed to have no impact on the duration of each flow regime. As the matrix permeability and skin factor increase, the pressure drop increases and the productivity index decreases.
Acknowledgments
The authors of this article would like to express their gratitude to the National Major Projects of Oil and Gas (973 program) 2017ZX05072006-002, “Study on initial productivity of fractured well in tight oil”. The authors are grateful to the editor and the reviewers for their critical comments, which were helpful in the preparation of this article.
Glossary
Nomenclatures
- P
pressure, MPa
- P̅D
pressure in Laplace domain, dimensionless
- Q
flow rate, m3/D
- q̅D
flow rate in the Laplace domain, dimensionless
- x
x-coordinate, m
- y
y-coordinate, m
- xf
fracture half-length, m
- wF
hydraulic-fracture width, m
- Xe
reservoir boundary, m
- Ye
fracture half spacing, m
- λ
matrix-fracture transfer parameter
- s
Laplace operator
- η
reservoir diffusivity ratio
- B
oil formation volume factor, m3/m3
- c
compressibility, MPa–1
- h
formation thickness, m
- k
permeability, mD
- rw
wellbore radius, m
- t
time, s
- μ
viscosity, cp
- α
matrix shape factor, m–2
- ϕ
porosity, dimensionless
- ω
storage ratio, dimensionless
Glossary
Subscripts
- D
dimensionless
- i
inner region (SRV)
- o
outer region (USRV)
- e
external boundary
- f,F
hydraulic fracture
- m
matrix
- nf
natural fracture
- wf
wellbore flow
- t
total
Appendix A
Dimensionless Parameters
 |
A1 |
 |
A2 |
 |
A3 |
 |
A4 |
 |
A5 |
 |
A6 |
 |
A7 |
 |
A8 |
 |
A9 |
 |
A10 |
 |
A11 |
 |
A12 |
 |
A13 |
 |
A14 |
 |
A15 |
 |
A16 |
 |
A17 |
 |
A18 |
 |
A19 |
 |
A20 |
 |
A21 |
 |
A22 |
Appendix B
Derivation of Solution for Pressure Drop in the Composite Dual-Porosity System
In the following part, the pressure solutions in the Laplace domain with the constant rate for the trilinear flow in a rectangular composite reservoir are derived. The multiple hydraulic fractured horizontal well extends in the naturally fractured formation, where dual-porosity media originally exist.
Outer-Region System
The flow in the outer-region system is in the direction perpendicular to the inner-region system (X-direction). It is assumed that there is no flow beyond the outer drainage area.
The mathematical model in the fracture system
 |
B1 |
In the dimensionless form, the equation is converted to
 |
B2 |
In the Laplace domain, the equation is
 |
B3 |
The mathematical model in the matrix system
 |
B4 |
In the dimensionless form, the equation is converted to
 |
B5 |
In the Laplace domain, the equation is
 |
B6 |
Substituting eq B6 into B3, we can obtain
 |
B7 |
where
 |
The no-flow outer-boundary condition is given by
 |
B8 |
The inner-boundary condition is given by
 |
B9 |
The pressure solution of the outer drainage area in the Laplace domain can be obtained
 |
B10 |
Inner-Region System
The flow in the inner-region system is in the direction perpendicular to the hydraulic fracture (Y-direction). It is assumed there is a no-flow boundary between two hydraulic fractures.
The mathematical model in the fracture system
 |
B11 |
Flow rate continuity between the outer and inner regions is given by
 |
B12 |
 |
B13 |
In the dimensionless form, the equation is converted to
 |
B14 |
In the Laplace domain, the equation is
 |
B15 |
 |
B16 |
 |
B17 |
where
 |
The pressure of the inner region is not a function of the distance in the x-direction; eq B17 is simplified to
 |
B18 |
where
 |
The mathematical model in the matrix system
 |
B19 |
In the dimensionless form, the equation is converted to
 |
B20 |
In the Laplace domain, the equation is
 |
B21 |
Substituting eq. B21 into B18, we can obtain
 |
B22 |
where
 |
The no-flow boundary condition is given by
 |
B23 |
The inner-boundary condition is given by
 |
B24 |
The pressure solution of the inner drainage area in the Laplace domain can be obtained
 |
B25 |
Hydraulic fracture system
The flow in the outer-region system is in the direction perpendicular to the inner-region system (X-direction). It is assumed that there is no flow beyond the outer drainage area.
The mathematical model
 |
B26 |
Integrating both sides of eq B26 as follows
 |
B27 |
using the following assumptions
 |
B28 |
 |
B29 |
results in
 |
B30 |
Flux continuity between the inner region and hydraulic fracture is given by
 |
B31 |
 |
B32 |
In the dimensionless form, the equation is converted to
 |
B33 |
In the Laplace domain, the equation is
 |
B34 |
 |
B35 |
 |
B36 |
where
 |
The pressure of the hydraulic fracture is not a function of the distance in the y-direction; eq B36 is simplified to
 |
B37 |
where
 |
The no-flow outer boundary condition is given by
 |
B38 |
The inner-wellbore condition is given by
 |
B39 |
Finally, we obtain the bottom hole wellbore pressure in the Laplace domain:
 |
B40 |
The authors declare no competing financial interest.
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