A Semianalytical Mathematical Model and Seepage Characteristics of Off-Center Fractured Vertical Wells with Multiwing Asymmetrical Fractures

Ji, Anzhao; Wang, Yufeng; Xu, Youjie; Zhang, Guangsheng; Liu, Xuefen

doi:https://doi.org/10.1155/2023/5226318

Mathematical Problems in Engineering

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Abstract Introduction Conclusions Abbreviations Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Review Article | Open Access

Volume 2023 | Article ID 5226318 | https://doi.org/10.1155/2023/5226318

A Semianalytical Mathematical Model and Seepage Characteristics of Off-Center Fractured Vertical Wells with Multiwing Asymmetrical Fractures

Anzhao Ji,¹Yufeng Wang,¹Youjie Xu,²Guangsheng Zhang,¹and Xuefen Liu¹

Academic Editor: Muhammad Afzal Rana

Received22 Dec 2022

Revised10 Feb 2023

Accepted09 Apr 2023

Published21 Apr 2023

Abstract

The complexity of formation conditions leads to multiwing asymmetric fractures after large-scale fracturing. According to a well pattern model and reservoir characteristics, a testing well is away from the center of the reservoir, and the existing well-test mathematical model cannot meet the field data analysis demand. Therefore, the mathematical model for the fractured wells with multiwing asymmetrical fractures is established. The model solution of the Laplace domain is obtained by the Laplace transform, nonuniform fracture discretization, and pressure drop superposition principle. The model is compared with the numerical model in this paper, and the result shows that the presented semianalytical model and the calculation method are correct. The eight main flow stages are divided according to pressure derivative characteristics and the seepage process of off-center fractured vertical wells with multiwing asymmetrical fractures. The larger off-center distance will lead to higher pressures and derivative curves. Larger fracture asymmetry factors are higher pressures and derivative curves during bilinear and linear flow regimes. The field data show that the model is applicable and that the model can give the guidance of fractured well formation evaluation.

1. Introduction

The seepage mechanism is complex for low-permeability oil reservoirs. It is difficult to exploit low-permeability reservoirs because of low-permeability characteristics [1, 2]. Since low-permeability characteristics have complex pore structures, some scholars have studied the seepage mechanism of complex pore throat structures using finite element simulation [3, 4]. Therefore, hydraulic fracturing technology is used to enhance oil well productivity by adding seepage channels [5, 6]. There are a large number of low-permeability resources in China, so the development of low-permeability resources is of great significance to increase the total production of oil and gas wells in China [7, 8]. The complexity of formation conditions leads to the asymmetry between fractures. The wellbore pressure analysis is an important method for fracture evaluation [9, 10]. Therefore, the establishment of a good mathematical model for off-center multiwing fractured wells is urgent.

Many scholars have studied the unsteady seepage theory of hydraulic fractured vertical wells [11–13]. Cinco-Ley et al. [11] first proposed the coupling method for the hydraulic fractured model and surface source solution, and then, Huang et al. [12] and Wang et al. [13] simulated the relationship between the wellbore pressure and production time by a numerical simulation method and proposed an analytical model for vertically fractured wells. The abovementioned model assumed that fractures are symmetrical about the wellbore, which is different from the actual situation of fracturing wells.

An infinite outer boundary is an ideal case; a number of oil wells are located in oil reservoirs with different boundaries (the closed boundary, constant pressure boundary, and mixed boundary). Although some scholars presented the semianalytical model and discussed the pressure transient curve characteristic, their research aim focused on the infinite outer boundary with different formation models [14–16]. For composite reservoirs, Ren et al. [17, 18] presented multiwing fractured wells in linear composite reservoirs. Xu et al. [19, 20] presented the fractured well in radial composite reservoirs. A boundary element can calculate the wellbore pressure solution of multiwing fractured wells in complex and mixed boundaries. However, the abovementioned model assumed that the well is located at the reservoir center.

In fact, the complexity of the formation stress field distribution makes the fracture lengths of hydraulic fractures unequal and fracture stretch from different directions. Therefore, it is possible to form multiple fractures of unequal length around the wellbore. Some scholars have simulated the wellbore pressure of asymmetric fractured wells by a numerical simulation method and a semianalytical solution. Fracture asymmetry was first introduced by Crawford and Landrum [21]. Bennett et al. [22] discussed the effect of asymmetric fractures on a rate decline curve through numerical simulations. Berumen et al. [23, 24] used numerical simulation methods to analyze wellbore pressure of fractured wells with asymmetric fractures. The disadvantage of the numerical solution is low efficiency and low precision, so some scholars have introduced an analytical and semianalytical solution to calculate the wellbore pressure. Rodriguez et al. [25, 26] first introduced the analytical method of asymmetric fractured vertical wells. However, the abovementioned models only give the pressure solution of fractured wells with biwing symmetrical fractures. In order to couple the hydraulic fracture model and the surface source solution of reservoirs, Wang et al. [13, 27–29] introduced Green’s function to calculate the wellbore pressure of the fractured well with asymmetric fractures. Mahmood et al. [4, 30] combined the boundary element method with the source function to analyze the wellbore pressure of fractured wells with asymmetrical fractures.

Although these scholars have conducted extensive research on the dynamic characteristics of the wellbore pressure of fractured wells with asymmetric fractures, these models assumed that the well is located in the center of reservoirs, which is the idealized model. Peaceman [31] proposed a simulation method for the off-center wells based on the concept of “equivalent well block radius” using numerical simulation technology. Rosa et al. [32] proposed a vertical off-center well analytical model for radial composite reservoirs. Deng et al. [33] proposed a semianalytical model for an off-center well with two-wing symmetrical fractures in multiregion heterogeneous reservoirs and analyzed the wellbore pressure characteristics. Zhao et al. [15] combined the boundary element method with the source function to give a semianalytical solution of off-center wells with two-wing symmetrical fractures. The abovementioned models assume that the fracture is symmetrical about the wellbore. Ji et al. [34] studied the seepage of asymmetric fractures of off-center fractured wells, but the study does not involve multiwing asymmetrical fractures.

In fact, most of wells are deviated from the reservoir center. At the same time, after large-scale volume fracturing of vertical wells, multiwing fractures around the wellbore will be formed because of uneven stress. Unfortunately, there is no corresponding mathematical model describing off-center multiwing fractured wells. Therefore, the major aim of this paper is to present a semianalytical mathematical model for off-center fractured wells with multiwing asymmetrical fractures. The mathematical model is solved by the Laplace transform, coupling and Green’s function. The numerical solution results verify the correctness of the semianalytical model. Wellbore pressure characteristic curves are discussed. The influence of key parameters on the wellbore pressure and rate distribution is analyzed. The result can help readers and engineers understand the seepage characteristics of off-center fractured wells with multiwing asymmetrical fractures.

2. Physical Model and Basic Assumptions

The physical conceptual model is shown in Figure 1. The wellbore location is not in the center of reservoirs. The lengths of hydraulic fractures on both sides of the wellbore are unequal. The basic assumptions in the process of establishing the model are given as follows: (1) The rate of the well is q_sc. (2) The distance from the wellbore to the center of reservoirs is r_o. (3) The angle between the fracture and the x-axis is θ_F. (4) The length of each fracture is not equal, and the distance from the wellbore to the fracture center is x_asym. (5) The seepage in the fracture and the seepage in reservoirs obey the isothermal Darcy’s law. (6) The outer boundary condition of reservoirs is closed.

3. Mathematics and Solutions of Multiwing Fractured Off-Center Wells

3.1. Off-Center Line Source Solution

The point source function method is the main method to solve the pressure distribution of complex hydraulic fractures. When the wellbore is located in the center of reservoirs, the solution is integrated along the fracture direction, and the surface source solution of the fractured well with infinite conductivity can be obtained, which only considers the radial flow. If the wellbore is not in the center of the reservoirs, the fluid flow is not just radial. Therefore, before establishing the mathematical model, it is very important to deduce the off-center linear source function.

For the convenience of mathematical model solving, the following dimensionless variable definition is given in Table 1.

In the process of model establishment and solution, two basic concepts need to be used, which are the global coordinate system and the local coordinate system. The global coordinate system is centered on reservoirs. The local coordinate system is centered on the wellbore.

In the global coordinate system r-θ-z, it is assumed that the line source is located in (r′, θ′, z′). According to the work of Guo et al. [35] and Xu et al. [36], the two-dimensional governing equation in the Laplace domain [33, 37] is

The solution of governing equation (1) can be expressed aswhere is the line source solution that the well is located in the reservoir center, which is given by Van Everdingen and Hurst [38]. When E is chosen, + E should satisfy the outer boundary conditions of the governing equations, and the contribution of E should approach zero when the linear source is a reservoir center. According to the research results of Ozkan and Raghavan [39], the central line source solution is expressed aswhere is the dimensionless distance between the pressure drop calculation point and the point source in the global coordinate system.

According to the Bessel function addition theorem [39], the Bessel function addition theorem can be expressed as

Substituting equation (5) into equation (3) can obtain the expression of the central line source solution:

According to equations (3) and (6), when the wellbore is not in the reservoir center, the coefficient D in equation (6) is changed. When the calculation point is close to the point source, the coefficient D should be close to 0, which not only satisfies the constant bottom-hole production conditions but also satisfies the outer boundary conditions. Therefore, the off-center linear source solution can be expressed by the following equation:

For circular closed boundaries,

For circular constant-pressure boundaries,

Substituting equation (7) into equations (8) and (9), the corresponding coefficients D_n can be obtained for the circular closed boundary and the constant pressure boundary:

Considering the derivative relation of Bessel function equations (11) and (12), equation (10) can be expressed as equation (13).

3.2. The Formation Discrete Model

Substituting equation (13) into equation (7) and considering the rate of each fracture as a nonuniform flow rate, the surface source solution of the nonuniform rate of the i-th fractures was obtained to integrate the linear source solution along the i-th fracture direction:

However, it is very difficult to directly integrate equation (14). According to the geometric model of multiwing asymmetrical fractures in Figure 1, the positional relationship between the calculation point and the line source can be obtained, as shown in Figure 2. Once the location of the well and the geometry of the fracture are determined, the calculation point is only related to the pressure drop production point on the fracture surface.

According to the relationship between the well and the fracture in Figure 2, and with the help of the sine theorem relationship, the following mathematical relationships exist for the dimensionless radius, the dimensionless off-center distance, and the angle between the fracture and the horizontal direction, which can be expressed by

The integral equation of equation (15) contains the Bessel function, and its analytical expression is very complicated. According to the method of dealing with symmetrical fractures in the literature [36, 40], the fracture unit is discretized (Figure 3). Each fracture segment can be regarded as a calculation object, and its flow rate is constant. Combined with the mass conservation equation, the pressure equation and flow equation can be solved simultaneously. Each hydraulic fracture is divided into 2N segments; the fracture grids are on the left side of the wellbore, and the fracture grid number on the right side of the wellbore is N. In the local coordinate system, the dimensionless radius of the endpoint can be expressed by

In the local coordinate system, the radius of the midpoint of the discrete fracture grid can be expressed by

The radius and angle of the discrete fracture grid endpoints in the global coordinate system can be expressed by

The radius and angle of the discrete fracture grid midpoints in the global coordinate system can be expressed by

According to the discrete scheme of the fracture segments in Figure 3, combining with equation (14), the dimensionless pressure drop of the j-th grid of the i-th fracture is represented by equation (21):

Equation (20) expresses the calculation of the pressure drop of the j-th grid of the i-th fracture. The fracture segment midpoint of the local coordinate system can be determined by equation (17). The integration upper and lower limits of equation (20) are represented, respectively, by the starting radius and the ending radius of the fracture segment in the local coordinate system. The starting radius and ending radius of the fracture unit can be determined by equation (16). The coefficient of equation (20) contains the dimensionless outer boundary radius in the global coordinate system. Therefore, when equation (20) is solved, there are both the parameters of the local coordinate system and the parameters of the global coordinate system. To unify the parameters, the parameters in the local coordinate system can be transferred to the global coordinate system through equations (19) and (20). However, when the variable of the local coordinate system is converted into the integral variable of the global coordinate system, the integral variable can be expressed either by the radius of the global coordinate system or by an angle. In this paper, the integral variable of the local coordinate system is first converted into the integral variable of the global coordinate system, and then, the integral variable is converted into the integral of the angle according to the differential relationship between the radius and the angle in the polar coordinate system. Through the above analysis, the integral variable of the local coordinate system in equation (20) was transformed into the integral variable of the global coordinate system, which can be expressed by

Equation (21) shows the integration form towards the radius in the global coordinate system. If the angle between the fracture and the x-axis is 0 or 180 degrees, equation (21) can be used to directly integrate about the radius in the global coordinate system. Otherwise, the integral form of the radius is converted into the integral of the angle in the global coordinate system. It needs to be considered that there exists a differential relationship between the arc length and angle in a polar coordinate system, which can be expressed bywhere .

By the pressure drop superposition, the bottom-hole pressure drop of the off-center vertical well with infinite conductivity asymmetric fractures can be obtained, which can be expressed by

3.3. Coupling of Fracture and Reservoirs

The fluid flow of the fracture is only considered a linear flow. The wing length of the hydraulic fracture is unequal. There is no fluid flow at both the fracture ends. According to the research results of the literature [13], the flow coupling relationship between reservoirs and the i-th fracture can be expressed by

The equation on the left hand side of equation (24) represents the fluid flow in the fracture, and the right represents the fluid flow in reservoirs. Equation (24) is discretized, which can be expressed by the linear equation system of the following equation:where .

According to the mass conservation theorem,

The areal flow rate per unit length of the fracture segment and the average fracture pressure can be obtained by the simultaneous solution of linear equations (25) and (26). The wellbore pressure cannot be calculated directly. According to the calculated fracture segment length surface flow and the average fracture pressure , these two parameters are substituted into equation (24), where . The dimensionless wellbore pressure of a vertical well with finite conductivity asymmetric fracture off-center fracturing can be obtained.

Considering the skin coefficient and wellbore storage coefficient, the calculation expression of the wellbore pressure can be expressed by the following equation [38]:

The bottom-hole pressure in the Laplace domain can be obtained by using equation (27), and the bottom-hole pressure in the time domain can be obtained by the formula of Stehfest numerical inversion [41].

4. Model Validation and Wellbore Pressure Characterization

4.1. Model Validation

The accuracy of the model needs to be verified. With the aid of the Saphir numerical well-testing analysis software, a numerical model for an off-center well with asymmetric multiwing fractures is established for the closed boundary in cylindrical homogeneous reservoirs (Figure 4). The basic parameters of the model are as follows: Reservoir thickness is 10 m. Reservoir permeability is 18.42 mD. The outer boundary radius of reservoirs is 2000 m. The off-center distance is 1200 m. Reservoir porosity is 10%. The comprehensive compressibility of the reservoir is 0.0001 MPa⁻¹. The temperature of reservoirs is 100°C. The volume coefficient of crude oil is 1.02. The production rate of the well is 100 m³/d. The conductivity of fractures is 9210 mD.m; that is, the dimensionless conductivity of each fracture is 5. The angle between the first fracture and the horizontal direction is 0 degrees. The length of the two wings of the fracture is equal, both 100 m. The angle between the second fracture and the horizontal direction is 90 degrees. The length of the two wings of the fracture is not equal. The length of the fracture on the north side is 50 m, and the length of the fracture on the south side is 150 m. In the analytical calculation, the dimensionless wellbore storage coefficient is 0.0001 and the well skin coefficient is 0.01. The geometric models of reservoirs and wellbore location are shown in the left panel of Figure 4, and the geometry model of fractures is shown in the right panel of Figure 4.

According to the above parameters, the dimensionless wellbore pressure and the dimensionless time are calculated by numerical well testing of the Saphir module of KAPPA-Workstation. The theoretical curve of the semianalytical solution is also calculated according to equation (27). It can be seen that the numerical well test calculation results are consistent with the semianalytical solution results (Figure 4). Then, the characteristics of the wellbore pressure are analyzed, which are mainly divided into eight flow stages. The wellbore pressure derivative curve characteristics of each stage are shown in Figure 5. (1) The first stage is the wellbore storage effect. The wellbore pressure and the pressure derivative curve overlap, and the curve is a 1-slope straight line. (2) The second stage is the skin effect response stage. The wellbore pressure derivative is a “hump” type. (3) The third stage is the bilinear flow between reservoirs and fractures, and the wellbore pressure derivative curve is a 1/4-slope straight line. (4) The fourth stage is the linear flow of reservoirs, and the wellbore pressure derivative curve is a 1/2-slope straight line. (5) The fifth stage is the elliptical flow of fluid around the fracture. The wellbore pressure derivative curve is a 0.36-slope straight line. (6) The sixth stage is the radial flow stage, and the wellbore pressure derivative curve is a 0.5-value horizontal line. (7) The seventh stage is the response to the wellbore pressure stage of the closed boundary close to the fracture, and the wellbore pressure derivative curve is upturned. (8) The eighth stage is the response stage of the circular closed outer boundary to the wellbore pressure, and the wellbore pressure derivative curve is a 1-slope straight line.

4.2. Result Analysis

The dimensionless wellbore storage coefficient C_D is 10⁻⁴. The skin coefficient S is 0.01. The asymmetry factor x_asymD is 0.5. The average half-length of fractures is 100 m. The first fracture coincides with the horizontal direction, and the angle between the second fracture and the horizontal direction is 90 degrees. The outer boundary radius Re is 2000 m. The off-center distance r_o is 1200 m.

Figure 6 displays the relationship among the dimensionless wellbore pressure, pressure derivative, and dimensionless time under different fracture conductivity. The dimensionless fracture conductivity has a significant influence on the wellbore pressure during bilinear and linear flow regimes. The larger the dimensionless fracture conductivity is, the smaller the seepage resistance is. Therefore, the larger the dimensionless fracture conductivity, the smaller the dimensionless wellbore pressure during bilinear and linear flow regimes.

Figure 7 displays the effect of dimensionless fracture conductivity on the dimensionless production distribution. The length of the No. 1 and No. 3 fractures are 100 m in the horizontal direction, the length of the No. 2 fracture is 50 m in the vertical direction, and the length of the No. 4 fracture is 100 m in the vertical direction. It can be seen in Figure 7 that the dimensionless fractures conductivity has a great influence on the rate distribution of asymmetric fractures in off-center wells. The solid line of Figure 7 shows the relationship between the dimensionless production and the dimensionless time of the four fractures with C_FD = 5, and the dotted line shows the relationship between the dimensionless production and the dimensionless time of the four fractures with C_FD = 20. It can be seen in Figure 7 that the No. 3 fracture has the greatest contribution to the production under the condition of different fracture conductivity, because the No. 3 fracture is located on the right side of the center of reservoirs, and the fluid supply range is large. The No. 2 fracture contributes the least to the production because the length of the No. 2 fracture is only half the length of the other fractures. Considering the permeability of the homogeneous matrix, the supply area of reservoirs connected to the No. 2 fracture is small, so the production of the No. 2 fracture is low. The contribution of No. 1 and No. 4 fractures to production is not much different, but the dimensionless production of the No. 4 fracture is higher than that of the No. 1 fracture. Because the No. 1 fracture is close to the boundary of reservoirs, the range of fluid replenishment from reservoirs to the No. 1 fracture is smaller than that of the No. 4 fracture for the closed boundary. With an increase in the production time, when a certain production time is reached, the dimensionless production of each fracture remains stable and the contribution of dimensionless production remains unchanged. Therefore, in the whole flow, the greater the dimensionless conductivity of the fracture and the longer the fracture, the higher the contribution to the production. The greater the dimensionless conductivity of the fracture and the longer the fracture, the greater the contribution of the fracture to the dimensionless production.

Figure 8 displays the effect of different off-center distances on the curve of the dimensionless wellbore pressure and pressure derivative. The off-center distance parameter in the calculation is shown in Figure 8. When the pressure wave spreads to the boundary, the off-center distance has a significant influence on the dimensionless wellbore pressure and pressure derivative curve. The farther the well is from the center of the reservoir, the closer the well and fracture are to the reservoir boundary. Therefore, the earlier the boundary response occurs. If the off-center distance is larger, the area corresponding to the radial flow of fluid around the well and the fracture becomes smaller and the duration of the radial flow becomes shorter. When the radial flow area is reduced, a larger pressure drop is required to maintain constant production, so the dimensionless pressure and pressure derivative values of the boundary response section become larger.

Figure 9 displays the effect of the asymmetry factor on the bottom-hole dimensionless pressure and pressure derivative curves. The value of the asymmetry factor in the calculation is shown in Figure 9. The larger the asymmetry factor is, the more obvious the asymmetry of the fracture is. The larger the fracture asymmetry factor is, the smaller the early pressure drop is. Because the fracture asymmetry increases, the length difference between the two wings of the fracture is greater. Under the same production, the contribution of the long wing side of the fracture to the production plays a leading role as the length of the long wing side of the fracture increases. As the fracture length increases, the seepage resistance decreases. In the bilinear flow stage of reservoirs and fractures, the pressure difference consumed in the production process decreases. As the production process progresses, this trend continues to diminish. When the flow reaches the plane radial flow stage, the dimensionless pressure and pressure derivative curves of different symmetry factors overlap.

5. Field Data Analysis

The field data are used to verify and apply the research results of this paper. The basic parameters are as follows: The wellbore radius is 0.1397 m. The height of the reservoir is 5.56 m. Reservoir porosity is 0.126. The average oil rate is 10.56 m³/d. The viscosity is 5.3 MPa.s. The relative density is 0.83. The volume factor is 1.02. The total compressibility coefficient is 3.51 × 10⁻⁴ MPa⁻¹.

According to the pressure derivative curve and the well test data, the six flow regimes can be diagnosed, which include the bilinear flow (③), the linear flow of reservoirs (④), the elliptical flow of fluid around the fracture (⑤), the radial flow stage (⑥), the response to the wellbore pressure stage of the closed boundary close to the fracture (⑦), and the response stage of the circular closed outer boundary to the wellbore pressure the bilinear flow (⑧). The fitting is shown in Figure 10. The fitting reservoir permeability is 15.6 mD, the off-center distance is 1220 m, and the control radius of a single well is 2215.4 m. The distance from the center of hydraulic fractures to the wellbore is 54 m, and the half-length of hydraulic fractures is 98 m.

6. Conclusions

(1)Considering the seepage problem of off-center well fracturing with asymmetric multiwing fractures, the corresponding physical and mathematical models are established. According to the point source theory and the Laplace transform, the pressure distribution of each fracture is obtained. According to the coupling relationship of pressure between the fracture and reservoirs, the semianalytical solution of the Laplace domain for asymmetric multiwing fracture fracturing off-center wells is obtained.(2)According to the treatment method of nonuniform flow, each fracturing fracture is discretized into 2N segments. Combined with the mass conservation equation of flow, the flow of each discrete fracture unit and the average pressure of the fracture are solved simultaneously, and the analytical solution of the pressure in time and space is obtained by the Stehfest numerical inversion method. The numerical solution of the Saphir module of numerical well testing analysis of KAPPA-Workstation is used to verify the correctness of the analytical solution.(3)According to the response characteristics of the wellbore pressure and the pressure derivative curve, the seepage process of the model can be divided into 8 stages: the wellbore storage effect response stage, the skin effect response stage, the bilinear flow response stage of reservoirs and fractures, the linear flow response stage of reservoirs, the elliptical flow response stage of the fluid around the fracture, the radial flow response stage, the response stage of the closed boundary close to the fracture to the wellbore pressure, and the response stage of the circular closed outer boundary to the wellbore pressure. According to the characteristics of these stages, the parameters of reservoirs can be interpreted.(4)The influence of the dimensionless conductivity coefficient, off-center distance, and fracture asymmetry factor on the wellbore pressure and production distribution curve was analyzed. The results shows that the larger the dimensionless conductivity coefficient in the linear and bilinear stages, the lower the value of the wellbore pressure and the pressure derivative curve; the larger the dimensionless conductivity coefficient and the longer the length of the fracture, the greater the fracture production. After reaching the plane radial flow stage, the dimensionless conductivity coefficient of the fracture and the length of the fracture have little effect on the relative distribution of production, and the production of each fracture tends to be stable. Before the plane radial flow, the off-center distance has no effect on the wellbore pressure and the pressure derivative curve. When the pressure wave spreads to the closed boundary on the side closer to the boundary, the curve of the bottom-hole pressure and pressure derivative is upward warped, and the larger the off-center distance is, the earlier the upward warping time of the curve is; the asymmetry factor of the fracture only affects the linear flow and bilinear flow between the fractures and reservoirs. The larger the asymmetry factor of the fracture is, the smaller the pressure drop in the linear and bilinear flow stages is.

As shown in Figure 5, the analytical model in this paper is consistent with the numerical model. However, future works will be focused on the optimization of the model for this case. The model is a useful tool to investigate the flow behavior of off-center fractured vertical wells with multiwing asymmetrical fractures. With this essential knowledge, we can evaluate off-center fractured vertical well performance and stimulation effectiveness in reservoirs.

Abbreviations

Symbol

:	Volume factor of reservoir fluid, dimensionless
:	Dimensionless well storage coefficient, dimensionless
:	Dimensionless conductivity of the fracture, dimensionless
:	Comprehensive compressibility of oil reservoirs, 10 MPa⁻¹
:	Green’s function
:	Reservoirs thickness, cm
:	The derivative of first kind n-order modified Bessel function to x
:	Modified Bessel function of the first kind of n-order
:	Reservoir permeability, D
:	Modified Bessel function of the second kind of n-order
:	The derivative of the second kind n-order modified Bessel function to x
:	Fracture right wing length, cm
:	Fracture left length, cm
:	Fracture wing length, cm
:	Reference length, cm
:	The total number of fractures
:	The number of grids per fracture
:	The initial formation pressure of the reservoir, 0.1 MPa
:	The pressure of hydraulic fracture, 0.1 MPa
:	Average pressure of the fracture, 0.1 MPa
:	Fracture surface flow per unit length, cm²/s
:	Production of wells under standard conditions, m³/s
:	The position in the direction of r in the cylindrical coordinate system, cm
:	The position of the point source in the r-direction in the cylindrical coordinate system, cm
:	The radius of the fracture in the local coordinate system, cm
:	The distance from any position of the fracture to any position of the reservoirs in the global coordinate system, cm
:	The distance from the pressure calculation point to the center of the reservoirs in the global coordinate system, cm
:	Off-center distance, cm
:	Outer boundary radius of reservoirs, cm
:	Skin factor, dimensionless
s:	Laplace variables
:	Production time, s
:	Fracture width, cm
, :	Any position in reservoirs, cm
:	The distance from the wellbore position to the fracture center, cm
, :	The location of the well in reservoirs, cm
:	The position of the direction in the cylindrical coordinate system, rad
:	The position of the point source in the direction in the cylindrical coordinate system, rad
:	The angle between the fracture and the x-axis in the local coordinate system, rad
:	The angle between any position of the fracture and the x-axis direction in the global coordinate system, rad
:	The angle between the j-th grid midpoint of the i-th fracture and the x-axis in the global coordinate system, rad
:	Porosity, dimensionless
:	Reservoir fluid viscosity, MPa.s

Subscript

D:	Dimensionless
F:	Hydraulic fracture.

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This article was supported by the Natural Science Foundation of Gansu Province (22JR11RM169), the National Natural Science Foundation of China Regional Science Foundation Project (42162015), the Innovation Foundation of Colleges and Universities in Gansu Province (2021A-127), and the Science and Technology Foundation of Qingyang City (QY2021B-F002).

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