Geofluids
Volume 2018 (2018), Article ID 1690102, 9 pages
https://doi.org/10.1155/2018/1690102

Research Article

Numerical Simulation Study on Fracture Parameter Optimization in Developing Low-Permeability Anisotropic Reservoirs

Jie Liu,1,2 Zhenhua Xu,³ Zhe Yuan,1,2 Hanyu Bie,1,2 and Pengcheng Liu1,2

¹School of Energy Resources, China University of Geosciences, Beijing 100083, China
²Key Laboratory of Marine Reservoir Evolution and Hydrocarbon Enrichment Mechanism, Ministry of Education, Beijing 100083, China
³China Southern Petroleum Exploration & Development Corporation, Haikou 570216, China

Correspondence should be addressed to Pengcheng Liu; lpc@cugb.edu.cn

Received 12 April 2018; Revised 31 July 2018; Accepted 28 August 2018; Published 24 December 2018

Academic Editor: Marco Petitta

Copyright © 2018 Jie Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The diamond-shape inverted nine-spot well pattern is widely used in developing low-permeability reservoirs with fractures. However, production wells with equal fracture lengths will lead to nonuniform displacement, especially in anisotropic reservoir. Previous researches mainly focused on equal-length fractures, while studies on the unequal-length fractures which can dramatically improve the development efficiency were little. In this paper, a corresponding numerical model with unequal length of fracture designed in the edge and the corner wells was built in a low-permeability anisotropic reservoir. The main objective was to examine and evaluate the effects of anisotropic permeability and fracture parameter on the waterflooding in the diamond-shape inverted nine-spot well pattern. The results indicate that different fractures penetration ratio and anisotropic permeability both result in different development efficiency. Fracture of the edge well are more easily to be water breakthrough, while the increase of penetration ratio of injection well effectively enhance oil recovery. Moreover, the most optimal penetration ratios of production well fractures under different kx : ky are determined. With the increase of kx : ky, the optimized penetration ratio of corner wells fracture decrease, while that of the edge wells increase. Setting unequal length fractures in low-permeability anisotropic reservoirs can effectively improve the oil displacement efficiency in the waterflooding process.

1. Introduction

Waterflooding technique is one of the most common methods for enhancing oil recovery in developing oil reservoirs [1]. Different forms of water injection well pattern were used for the development of reservoirs, including five-point method, seven-point method, and nine-point method. The diamond-shape inverted nine-spot well pattern has been proven to be superior to others in developing low-permeability anisotropic hydrocarbon reservoir [2]. Hydraulic fracturing technique is also widely applied to further enhance oil recovery of low permeability reservoirs [3, 4]. The fractures can improve the development efficiency of anisotropic hydrocarbon reservoirs [5]. However, it can also lead to nonuniform waterflooding, affecting the ultimate oil recovery of waterflood areal well pattern. The optimized fracture length, azimuth, diversion capability, and other parameters are related to the reservoir parameters. The well spacing of the injection well pattern and the selection of the well pattern are also related to the reservoir properties [6–9].

Reservoirs are generally found to be anisotropic in developing oil reservoirs. Due to the heterogeneity of deposition, fingering in porous media is a common phenomenon which affects the oil recovery as the anisotropy of the reservoir permeability [10–12]. Reservoir anisotropy has a significant influence on the effect of waterflooding, which generally leads to nonuniform displacement [13–17]. Well with fractures can increase the amount of water injection and liquid production. It can also improve the swept area to further displace the residual oil. However, too short fracture lengths result in a worse oil recovery, while too long fracture lengths will cause water breakthroughs in the displacement front, shortening the water-free oil production period and failing to meet the economic benefits [18, 19]. Numerical simulation streamline analysis, tracer tracking, and other methods can be used to evaluate the effects of waterflood process and analyze the cumulative oil production curve. Previous researches have present that waterflood development of low permeability hydrocarbon reservoir should select smaller well spacing and increased differential pressure between injection wells and production well [20–22]. In the recent years, with the continuous improvement of horizontal well technology, horizontal well-vertical well combined well pattern has also been used in developing low-permeability anisotropic reservoirs. In order to better cope with the physical properties of the low-permeability anisotropic reservoir, many attempts have been tested to investigate several new methods and innovations of waterflooding well patterns [23–27].

At present, most of the fracture length in development well pattern is equal. Previous studies mainly focused on the effects of anisotropic permeability on the fracture parameters such as fracture length and fracture azimuth. Researches on the influence of penetration ratio of well fractures in the diamond-shape inverted nine-spot well pattern were little. Therefore, it is necessary for the optimization of fracture parameter in the diamond-shape inverted nine-spot well pattern. The main objective of this paper is to investigate the effect of the anisotropic permeability on the development of waterflood in different fractures parameters of the diamond-shape inverted nine-spot well pattern. Streamline simulation is conducted to explain the waterflood process with unequal length fractures. Observation is mainly focused on two basic conditions: injection wells with fractures or no fractures. The effects of different fracture lengths in the edge and the corner production wells under different low-permeability anisotropic reservoir were investigated, and a series of unequal fracture length optimization results were obtained. The research results provide reasonable suggestions for the development in low-permeability anisotropic reservoir.

2. Reservoir Modeling

2.1. Simulation Model

A typical low-permeability anisotropic reservoir data of a block of Changqing oilfield (China) was selected to build the numerical model. The component model of Eclipse software was used for the simulation of waterflooding process. The key reservoir properties used in the model are presented in Table 1.

Table 1: Key simulation parameters used in the model.

Items
Values
Items
Values

Depth to reservoir top, m
3150
Average permeability, mD
5

Initial formation pressure, MPa
28.5
Average porosity, f
0.124

Oil viscosity, mPa·s
3
Initial oil saturation, f
0.646

Oil density, kg/m³
869.5
Oil volume coefficient, f
1.14

Formation water viscosity, mPa·s
0.5
Water compression coefficient, 1/MPa
0.52 × 10⁻⁴

Formation water density, kg/m³
1000
Rock compression coefficient, 1/MPa
0.32 × 10⁻⁴

2.2. Well Pattern and Grid Modeling

The grid model for the low-permeability anisotropic reservoir is shown in Figure 1. The depth of the top layer of the reservoir is 2950 m, the grid number is 61 × 97 × 4, and each grid volume is 15 m × 15 m × 3.75 m. The well spacing of the diamond-shape inverted nine-spot well pattern which is the distance of two edge well along the horizontal direction is 450 m in this model, and the well array distance is 145 m. The local grid refinement (LGR) method was used to build the fracture model. The fracture width is 0.25 m, the fracture permeability is 5000 mD, and the height of the fracture is equal to the thickness of the reservoir.

Figure 1: Grid model and well pattern illustration.

3. Effects of Reservoir Anisotropy on Waterflooding

The formation anisotropy in low permeability reservoirs is mainly influenced by the physical properties of deposit sediment and the fracture direction. The fracture can be divided into two categories: natural fractures and artificial fractures. As the characteristics of low permeability, artificial fractures of wells are generally carried out in the waterflooding process, and anisotropy caused by fractures is an important factor influencing the displacement efficiency of waterflooding. The effects of anisotropic permeability on the displacement efficiency are analyzed by streamline simulation and diagram of residual oil distribution.

3.1. Effects of Penetration Ratio of Facture

In the diamond-shape inverted nine-spot well pattern, the fracture length of the production well can significantly affect the flow of the reservoir fluid. An overlong fracture of corner well lead to an earlier water breakthrough time of corner well, while an overlong fracture of edge well result in a worse swept area of injected water. In order to study the displacement efficiency of different fracture geometries through streamline simulation, we set the ratio of x-direction and y-direction permeability (kx : ky) of the reservoir model as 1 : 1. Four different length fracture cases were designed (Figure 2). A penetration ratio is used to characterize the fracture length in the fractured wells. The penetration ratio is defined as the ratio of the fracture half-length to the half of well spacing. Three penetration ratios of fractured well are selected to discuss their impacts on the oil recovery, penetration ratio of injection well fracture (named Pi in this article), penetration ratio of edge production well fracture (named Pe in this article), and penetration ratio of corner production well fracture (named Pc in this article).

Figure 2: Streamline simulation results corresponding to different cases in Table 1. (a) Case no. 1. (b) Case no. 2. (c) Case no. 3. (d) Case no. 4.

The penetration ratio of injection well, edge well, and corner well in case no. 1 is all 0.23. This is a control case. Case no. 2 is carried out to investigate the effect of Pe which is 0.63. Cases no. 3 and no. 4 are conducted to study the effects of Pc and Pi with a value of 0.63, respectively. Table 2 summarizes the initial conditions and results of different simulations.

Table 2: Schemes parameters of all streamline simulations.

No.
Pi, fraction
Pc, fraction
Pe, fraction
Water cut after 2 years production, %
Oil recovery after 2 years production, %

1
0.23
0.23
0.23
39.96
38.71

2
0.23
0.23
0.63
64.71
36.63

3
0.23
0.63
0.23
35.16
40.44

4
0.63
0.23
0.23
69.11
43.55

From Figure 2(a), the streamlines are converged to the fractures in the waterflooding process. The fracture length of the injection wells and the production wells are equal. Injected water preferentially flow towards to the fracture of the edge production wells, causing a dramatically increasing of the water cut of the edge wells. Moreover, more residual oil is remained around the corner well, which means the displacement efficiency near the edge well is better. The nonuniform displacement is caused by the equal fracture lengths of the corner well and the edge well. From Figure 2(b), Pe is greater than Pc, resulting in an earlier water breakthrough time in the edge well and nonuniform displacement.

For Figure 2(c), the fracture length of the edge well is shorter than the fracture lengths of the corner well. Compared with Figures 2(a) and 2(b), the distribution of the streamline is more homogeneous, and the displacement is relatively evenly distributed. Table 2 and Figure 2(c) has the lowest water cut of 35.16% and higher ultimate oil recovery of 40.44%, indicating that when the fracture length of injection wells is relatively short (0.23), the combination pattern of longer length of the corner well fractures and shorter length of the edge well fractures can improve the displacement efficiency of waterflooding process.

From Figure 2(d), the fracture length of the injection well is increased. Compared with other cases, the breakthrough time of waterflooding in the production well decreases. With the increase amount of injected water, the amount of liquid production increase, resulting in an improvement of the water cut of the corner and edge wells and the oil recovery. In this case, the water cut of 69.11% and the ultimate oil recovery of 43.55% are both the highest. The results of streamline simulation show that the edge well fractures are more easily to be water breakthrough than the corner well fractures.

3.2. Effects of Anisotropic Permeability

In Figures 3(a) and 3(b), the injection wells are not hydraulically fractured, and Pe and Pc are both 0.37. In Figures 3(c) and 3(d), the injection well and the production well have the same penetration ratio of 0.37. From Figure 3(a), the shape of waterflood front of the homogeneous reservoir is regular hexagon, and the rate of waterflood in the x-direction and y-direction is uniform, indicating a homogeneous displacement. From Figure 3(b), due to the anisotropic permeability of the reservoir, the formation fluid flows faster in the x-direction, forming an elliptical shape displacement front. However, it can be seen that the red area in Figures 2(a) and 2(b) is relative larger, especially near the corner well, indicating a worse displacement efficiency.

Figure 3: Plane distribution of remaining oil saturation at different kx : ky and fracture parameters combination. (a) Reservoir permeability kx : ky = 1 : 1 with no fractured injection well; (b) reservoir permeability kx : ky = 3 : 1 with no fractured injection well; (c) injection well has fractures and reservoir permeability kx : ky is 1 : 1; (d) injection well have fractures and reservoir permeability kx : ky is 3 : 1.

From Figures 3(c) and 3(d), when the injection well is fractured, the displacement front is more elliptical. Compare to Figures 2(a) and 2(c) and Figures 2(b) and 2(d), respectively, under the same kx : ky condition, the injection well fracture change the shape of the waterflood displacement front and cause a faster liquid flow along the fracture direction (i.e., the x-direction). The residual oil near the corner well reduce significantly, indicating a better oil recovery. Therefore, a fractured injection well is recommended to further enhance the oil recovery.

In the development of anisotropic reservoirs in the diamond-shape inverted nine-spot well pattern, when the fracture length of the corner well and the edge well is equal, the displacement process is nonuniform. Nonuniform displacement phenomenon is more serious when an anisotropic permeability direction existed in the reservoir, causing the earlier water breakthrough of the edge well. Therefore, it is necessary to further study the fracture parameter optimization of corner well and edge well to improve the displacement efficiency and enhance the oil recovery.

4. Fracture Parameter Optimization

Through the above studies, it can be found that when the injection wells are free of hydraulic fractures, the waterflood is relatively uniform and the water-free oil production period is longer. The water cut of the well group increases slowly. When the injection wells are hydraulically fractured, the amount of injected water and the liquid production can significantly increase, and finally enhance the oil recovery. However, the water cut of well group increases dramatically as well. Therefore, the designed fracture length in the diamond-shape inverted nine-spot well pattern should be optimized.

In order to study the effect of anisotropic permeability on the fractures structure of the well pattern, the kx : ky of the reservoir permeability is set to 1 : 1, 3 : 1, 6 : 1, and 10 : 1. Based on this setting, two groups of cases with no fractures and a fracture penetration ratio of 0.37 of injection wells are set. Then, 12 sets of penetration ratio of corner and edge wells fractures are, respectively, set as 0.17, 0.23, 0.30, 0.37, 0.43, 0.50, 0.57, 0.63, 0.70, 0.77, 0.83, and 0.90, and the specific parameters of the schemes are shown in Figure 4.

Figure 4: Schemes parameters settings block diagram.

4.1. Injection Well with No Fractures

Under the condition of no fractures in the injection wells and different anisotropy of permeability, the relationship curves between the oil recovery and the fracture penetration ratio were obtained by numerical simulation (Figure 5). The x-axis is the penetration ratio of the edge well fracture, y-axis is the penetration ratio of the corner well fracture, and z-axis is the oil recovery.

(a)

(b)

(c)

(d)

Figure 5: Three-dimensional color map surface of oil recovery with no fractured injection well. (a) Reservoir permeability kx : ky is 1 : 1; (b) reservoir permeability kx : ky is 3 : 1; (c) reservoir permeability kx : ky is 6 : 1; (d) reservoir permeability kx : ky is 10 : 1.

From Figure 5(a), the reservoir is homogeneous with reservoir permeability kx : ky of 1 : 1. As the existence of fractures in both edge and corner well, the distance from the bottom of the injection well to the corner well fractures is smaller than that to the edge well. With the increase of Pe, the water breakthrough time in the edge well becomes earlier, resulting in more nonuniform displacement, increased liquid production, and water cut. Therefore, the oil recovery decreases as Pe increases.

From Figures 5(b)–5(d), the oil recovery first increases and then decreases. As reservoir permeability kx : ky is greater than 1, an advantage flowing channel is formed in the x-direction, leading to an earlier water breakthrough time in the corner well than that of kx : ky = 1. With the increase of Pe, balancing the effect of reservoir anisotropy, the displacement process is more uniform, resulting in an increasing oil recovery. When Pe continues to increase, the highest oil recovery point will be obtained. The waterflood displacement efficiency is the best under this condition. However, after continue increasing Pe, the effect of increasing liquid production is weaker than that of increasing water cut, resulting in the reduction of oil recovery. Moreover, with the enhancement of reservoir anisotropy, the liquid production and water cut of the edge well are both increased. Longer fracture length of the edge well is required for balancing the effect of reservoir anisotropy, resulting in the consistent increase of Pe corresponding to the highest oil recovery. On the other hand, for the case of constant Pe, the effect of Pc on liquid production, water cut, and oil recovery is generally consistent with that of the edge well fractures.

Under the anisotropic reservoir and nonfractured injection well condition, the optimal penetration ratio of production well fracture is shown in Table 3. It can be seen that with the increase of reservoir permeability kx : ky, the water cut increases greatly from 26.3% to 94.4%, while the oil recovery has a slight increase from 46.9% to 54.6%. The optimized penetration ratio of the edge and corner well fractures is significantly affected by reservoir anisotropy. The optimal Pc decreases from 0.83 to 0.43 with the increase of kx : ky, while the optimal Pe increases from 0.17 to 0.63.

Table 3: Optimal penetration ratio of production well fracture when the injection well is not fractured.

No.
kx : ky
Pc, fraction
Pe, fraction
Water cut after 6 years production, %
Oil recovery after 6 years production, %

(a)
1 : 1
0.83
0.17
26.3
46.9

(b)
3 : 1
0.57
0.37
80.7
51.3

(c)
6 : 1
0.5
0.5
91.4
53.3

(d)
10 : 1
0.43
0.63
94.4
54.6

4.2. Injection Well with Fractures

Under the condition of injection wells with fractures and different degrees of permeability anisotropy, the relationship curves between the oil recovery percent of reserves and the fracture length were obtained by numerical simulation (Figure 6). The x-axis is the penetration ratio of the edge well fracture, the y-axis is the penetration ratio of the corner well fracture, and the z-axis is the oil recovery of the reservoir.

(a)

(b)

(c)

(d)

Figure 6: Three-dimensional color map surface of oil recovery percent of reserves when the water injection well has fracture. (a) Reservoir permeability kx : ky is 1 : 1; (b) reservoir permeability kx : ky is 3 : 1; (c) reservoir permeability kx : ky is 6 : 1; (d) reservoir permeability kx : ky is 10 : 1.

As shown in Figure 6(a), for the case of the homogeneous reservoir, due to the existence of fractures in the injection well, when Pc is constant, with the increase of Pe, the oil recovery first increases, then decreases, and finally increase slightly. In the early stage, the water cut is low as the fracture length of the corner well is relatively small. The effect of increasing liquid production is stronger than that of increasing water cut, resulting in an increasing period of oil recovery. In the middle stage, as the relatively high water cut, the effect of increasing liquid production is weaker, resulting in a falling period of oil recovery. In the later stage, the fracture length is quite long (fracture penetration ratio is greater than 0.77); the water cut maintains at a high level. The amount of liquid production increases with the increase of Pe, while the water cut increases little, leading to a slight increase of oil recovery.

For Figures 6(b) and 6(c), the reservoir permeability kx : ky is 3 : 1 and 6 : 1. With the enhancement of the reservoir anisotropy, Pe corresponding to the highest oil recovery is also increasing. For constant kx : ky, with the increase of Pe, balancing the effect of reservoir anisotropy, the waterflood displacement becomes more uniform. The oil recovery increases as well until reaching the maximum; the penetration ratio at this time is the most optimal. When Pe continues to increase, the water breakthrough of the edge well is much earlier. The effect of increasing water cut is stronger than that of increasing liquid production, resulting in a descent of oil recovery.

As shown in Figure 6(d), for the case of the high anisotropic reservoir permeability of 10 : 1, the oil recovery increases monotonously with the increase of the penetration ratio. It indicates that in serious heterogeneity reservoir, as long as the fracture length in the edge wells increase, the nonuniform displacement can be further improved. However, the water cut increases dramatically at the same time, and the water breakthrough time becomes earlier. The stable production period is too short, which results in a poor development efficiency. In addition, under constant Pe condition, the effect of Pc on liquid production, water cut, and oil recovery is similar with the case of no fracturing in the injection well.

In the anisotropic reservoirs, the maximum of oil recovery corresponds to the optimized penetration ratio of the edge well fracture under the injection well with fracture condition. The optimal penetration ratio results of production well fractures are shown in Table 4. With the increase of reservoir permeability kx : ky, the water cut increases from 87.8% to 97.1%, while the oil recovery increases from 52.3% to 59.1%. The optimal Pc decreases from 0.77 to 0.30, while the optimal Pe increases from 0.43 to 0.90. Compared with the nonfractured injection wells, the ultimate oil recovery increased. However, the water cut of the same production period also relatively increased.

Table 4: Optimal penetration ratio of production well fracture when the injection well is fractured.

No.
kx : ky
Pe, fraction
Pe, fraction
Water cut after 6 years production, %
Oil recovery after 6 years production, %

(a)
1 : 1
0.77
0.43
87.8
52.3

(b)
3 : 1
0.63
0.63
94.6
55.8

(c)
6 : 1
0.57
0.70
96.5
57.8

(d)
10 : 1
0.30
0.90
97.1
59.1

5. Conclusions

Based on the presented work above, the following conclusions can be drawn.

(1)Anisotropic permeability and fracture penetration ratio of fractured well are the key factors of displacement efficiency and oil recovery in developing a low-permeability anisotropic reservoir by diamond-shape inverted nine-spot well pattern. The streamline simulation results shows that the edge well are more easily to be water breakthrough than that of the corner well, causing a greater impact on the water cut of the reservoir. The increase of penetration ratio of injection well may significantly enhance the oil recovery(2)The displacement area of a homogeneous reservoir with no fractured injection well is a regular hexagon. The residual oil is mainly distributed near the corner well, indicating a poor displacement efficiency. Under constant penetration ratio of production well condition, with the increase of anisotropic permeability, an elliptical shape nonuniform displacement is formed. When the injection well is fractured, the displacement front is more elliptical with a great reduction of the residual oil, indicating a better oil recovery(3)When the injection well is not hydraulically fractured, with the increase of kx : ky from 1 : 1 to 10 : 1, the optimized penetration ratio of the corner well fracture decreases from 0.83 to 0.43, and the optimized penetration ratio of the edge well fracture increases from 0.17 to 0.63. When the injection well is hydraulically fractured, for the kx : ky of 1 : 1, 3 : 1, 6 : 1, and 10 : 1, the optimized penetration ratio of the corner well fracture is 0.77, 0.63, 0.57, and 0.30, respectively; while the optimized penetration ratio of the edge well fracture is 0.43, 0.63, 0.70, and 0.90, respectively. Under the equal kx : ky conditions, the injection well fracture has little impact on the penetration ratio optimization of the corner well fracture, while it has a greater influence on the penetration ratio optimization of the edge well fracture

Nomenclature

kx: | Reservoir permeability along the x-direction
ky: | Reservoir permeability along the y-direction
Pi: | Penetration ratio of injection well fracture
Pe: | Penetration ratio of edge production well fracture
Pc: | Penetration ratio of corner production well fracture.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was financially supported from the National Natural Science Foundation of China (Grant no. 51774256), the Science and Technology Special Funds of China for 2016ZX05015-002, and the Fundamental Research Funds for the Central Universities of China (Grant no. 2-9-2017-310, 53200759268).

References

P. B. Crawford, “Factors affecting waterflood pattern performance and selection,” Journal of Petroleum Technology, vol. 12, no. 12, pp. 11–15, 1960.

X. Tong, “A comparative study of the characteristics and susceptibility of pattern-type water-injection well networks from the viewpoint of balanced waterfloods,” Society of Petroleum Engineers Journal, vol. 23, no. 06, pp. 892–900, 1983.

A. Settari and H. S. Price, “Simulation of hydraulic fracturing in low-permeability reservoirs,” Society of Petroleum Engineers Journal, vol. 24, no. 2, pp. 141–152, 1984.

F. Liu, Z. Luo, Y. Sang, L. Zhao, and C. Zhou, “Deformation behavior between hydraulic and natural fractures using fully coupled hydromechanical model with XFEM,” Mathematical Problems in Engineering, vol. 2017, Article ID 6373957, 12 pages, 2017.

M. A. Aghighi and S. S. Rahman, “Horizontal permeability anisotropy: effect upon the evaluation and design of primary and secondary hydraulic fracture treatments in tight gas reservoirs,” Journal of Petroleum Science and Engineering, vol. 74, no. 1-2, pp. 4–13, 2010.

R. F. Lemon, H. J. Patel, and J. R. Dempsey, “Effects of fracture and reservoir parameters on recovery from low permeability gas reservoirs,” in Proceedings of Fall Meeting of the Society of Petroleum Engineers of AIME, pp. 1–13, Houston, Texas, USA, October 1974.

S. A. Holditch, J. W. Jennings, S. H. Neuse, and R. E. Wyman, “The optimization of well spacing and fracture length in low permeability gas reservoirs,” in Proceedings of SPE Annual Fall Technical Conference and Exhibition, pp. 1–12, Houston, Texas, USA, October 1978.

D. N. Meehan, R. N. Horne, and K. Aziz, “Effects of reservoir heterogeneity and fracture azimuth on optimization of fracture length and well spacing,” in Proceedings of International Meeting on Petroleum Engineering, pp. 567–585, Tianjin, China, November 1988.

Z. Wu, L. Fan, and S. Zhao, “Effects of hydraulic gradient, intersecting angle, aperture, and fracture length on the nonlinearity of fluid flow in smooth intersecting fractures: an experimental investigation,” Geofluids, vol. 2018, Article ID 9352608, 14 pages, 2018.

N. A. Berruin and R. A. Morse, “Waterflood performance of heterogeneous systems,” Journal of Petroleum Technology, vol. 31, no. 07, pp. 829–836, 1979.

L. W. Lake, “The origins of anisotropy (includes associated papers 18394 and 18458),” Journal of Petroleum Technology, vol. 40, no. 04, pp. 395-396, 1988.

A. K. Permadi, I. P. Yuwono, and A. J. S. Simanjuntak, “Effects of vertical heterogeneity on waterflood performance in stratified reservoirs: a case study in Bangko Field, Indonesia,” in Proceedings of SPE Asia Pacific Conference on Integrated Modelling for Asset Management, pp. 1–12, Kuala Lumpur, Malaysia, March 2004.

C. L. Bargas and J. L. Yanosik, “The effects of vertical fractures on areal sweep efficiency in adverse mobility ratio floods,” in Proceedings of International Meeting on Petroleum Engineering, pp. 611–620, Tianjin, China, November 1988.

E. O. Mazo, J. Moreno, and D. Schiozer, “Effects of directional permeability anisotropy on sweep efficiency of water injection under fracturing conditions process,” in Canadian International Petroleum Conference, Calgary, AB, Canada, June 2007.

N. Henderson and L. Pena, “Simulating effects of the permeability anisotropy on the formation of viscous fingers during waterflood operations,” Journal of Petroleum Science and Engineering, vol. 153, pp. 178–186, 2017.

D. N. Meehan, “Optimization of fracture length and well spacing in heterogeneous reservoirs,” SPE Production & Facilities, vol. 10, no. 2, pp. 82–88, 1995.

J. M. Gatens, W. J. Lee, C. W. Hopkins, and D. E. Lancaster, “The effect of permeability anisotropy on the evaluation and design of hydraulic fracture treatments and well performance,” in Proceedings of SPE Gas Technology Symposium, Houston, Texas, USA, January 1991.

X. Bian, S. Zhang, J. Zhang, and D. Wang, “Well spacing design for low and ultra-low permeability reservoirs developed by hydraulic fracturing,” Petroleum Exploration and Development, vol. 42, no. 5, pp. 705–711, 2015.

J. Wang, X. Zheng, Z. Wang, and C. Tian, “An approach to improve sweep efficiency of extra low permeability reservoir by vertical injection wells and production well with large scale hydrofracture,” in SPE Kingdom of Saudi Arabia Annual Technical Symposium and Exhibition, pp. 1–9, Dammam, Saudi Arabia, April 2017.

O. Izgec, “Optimizing volumetric sweep efficiency in waterfloods by integrating streamlines, design of experiments, and hydrocarbon F-Φ curves,” in Proceedings of SPE Western Regional Meeting, Anaheim, CA, USA, May 2010.

W.-j. Luo, J.-l. Wang, X.-d. Wang, and Y.-f. Zhou, “A streamline approach for identification of the flowing and stagnant zones for five-spot well patterns in low permeability reservoirs,” Journal of Hydrodynamics, Ser. B, vol. 25, no. 5, pp. 710–717, 2013.

T. Fan, X. Song, S. Wu et al., “A mathematical model and numerical simulation of waterflood induced dynamic fractures of low permeability reservoirs,” Petroleum Exploration and Development, vol. 42, no. 4, pp. 541–−547, 2015.

Y. T. Liu, “Methodology for horizontal well pattern design in anisotropic oil reservoirs,” Petroleum Exploration and Development, vol. 35, no. 5, pp. 619–624, 2008.

J. E. Onwunalu and L. Durlofsky, “A new well-pattern-optimization procedure for large-scale field development,” SPE Journal, vol. 16, no. 3, pp. 594–607, 2011.

K. Zhang, Y. Chen, L. Zhang et al., “Well pattern optimization using NEWUOA algorithm,” Journal of Petroleum Science and Engineering, vol. 134, pp. 257–272, 2015.

J. Zhao, J. Fan, Y. He, Z. Yang, W. Gao, and W. Gao, “Optimization of horizontal well injection-production parameters for ultra-low permeable–tight oil production: a case from Changqing Oilfield, Ordos Basin, NW China,” Petroleum Exploration and Development, vol. 42, no. 1, pp. 74–82, 2015.

D. Wang, Y. Li, B. Chen et al., “Ensemble-based optimization of interwell connectivity in heterogeneous waterflooding reservoirs,” Journal of Natural Gas Science and Engineering, vol. 38, pp. 245–256, 2017.