Well Interference Analysis of Shale Gas Wells Based on Embedded Discrete Fracture Model

Well interference is commonly observed in shale gas reservoirs due to the small well spacing, and it significantly affects the shale gas production. Effective evaluation of well interference is important to increase the gas production of shale gas wells. Previous researches mainly focus on the well interference phenomenon and production optimization using numerical simulation so that the quantitative analysis of shale gas well interference is rare. Therefore, this paper is aimed at analyzing the well interference of shale gas wells through production type curves. First, the complex fracture networks are described by using the embedded discrete fracture model (EDFM). Second, different cases are designed to characterize different types and degrees of well interference in shale gas reservoirs. Third, numerical modelling is conducted to simulate the well interference and its effect on gas production. Fourth, the type curves are obtained to quantitatively analyze and compare the impact of well interference on shale gas production. Results show that well interference caused by hydraulic fractures mainly reduce the gas production of the parent well while the gas production of child well can be increased owing to the larger equivalent stimulated area. The pressure depletion is obvious when the well communication degree becomes higher. Differences can be found from early to late periods by the combination of log-log and Blasingame type curves. This work provides a method for well interference evaluation, and it can be used to obtain well spacing and adjust fracturing parameter in shale gas reservoirs.


Introduction
The permeability and porosity of shale gas reservoirs are ultralow due to abundant nanopores [1][2][3][4][5]. Stimulation technology is required to obtain commercial productivity [6]. Horizontal well drilling and hydraulic fracturing technologies have been commonly used to generate highpermeability fracture networks [7][8][9][10][11]. Multiwell pad is further introduced to decrease the drilling and fracturing costs, and each well pad is usually composed of six to eight horizontal wells [12,13]. However, well interference is observed in shale gas reservoirs due to the small well spacing (e.g., 300 m to 500 m). How to characterize and analyze the well performance in shale gas reservoirs is important for enhancing the ultimate shale gas production [14].
In recent years, the well interference phenomena in unconventional oil and gas reservoirs have attracted much research interest [15][16][17]. The well interference is modelled using different methods. Lawal et al. simulated and forecasted frac hits in shale gas wells and indicated that the gas production reduction was caused by the flow resistance due to the multiphase flow in the fracture network or the permeability decrease around the wells [18]. Moradi and Angus modelled the frac hits by using dynamic microseismicityconstrained enhanced fracture regions [19]. Guo et al. conducted numerical investigation about the effects of subsequent parent well injection on interwell fracturing interference using reservoir-geomechanics-fracturing modelling approaches [20]. Mohaghegh presented dynamic simulation of frac hit based on artificial intelligence and machine learning methods [21]. The impact of fracturing interference on gas production performance is also analyzed [22]. The mechanisms of well interference and different types of well interference types in shale reservoirs are further investigated [23][24][25]. Various methods are developed to evaluate the well interference caused by hydraulic fracturing in the shale reservoirs. Sardinha et al.
applied frac pressure hits and production interference analysis to estimate the well connectivity [26]. Gupta et al. focused on identifying the well interference by forecasting long-term production and residual analysis [27]. Molina established an analytical model to assess the fraction of frac fits in multiwell pads [28]. Kumar et al. performed integrated analysis of tracer and pressure-interference tests to identify well interference [29]. Arman et al. forecasted the mid and far field frac hits at an Eagle Ford and Wolfcamp well based on shale pressure depletion and well performance using geomechanical constraints [30]. GIS Platform was used to evaluate the risks of frac hits in the Aishwarya Barmer-Hill Field [31]. Magneres et al. developed a workflow to estimate the well interferences and its maximum expected well head pressure using magnitude forecast methodology [32]. To enhance the shale gas production, well spacing needs to be optimized to decrease well interference [33][34][35]. Except for well spacing, the stimulation design also needs to be optimized [36]. Different fracturing technologies are applied to prevent well interference, including fracture geometry control technology [37] and adaptive fracturing [38]. Other prevention methods are also proposed to mitigate the well interference especially the frac hits, including preloading depleted parent wells or fluid injection [39][40][41], chemical or mechanical treatments such as refracturing or solvent/surfactant chemistry blend [42][43][44]. Also, there are some filed case studies about the mitigation measures for well interference [45,46]. However, the remedial costs are huge and the recovering effect is uncertain by the prevention measures. A quick and accurate evaluation of well interference is important for selecting measures to reduce the effect of well interference on shale gas production. Exact characterization of complex fracture networks is required for shale gas wells, especially for the multiple wells with fracture interference [47][48][49]. The embedded discrete fracture model (EDFM) is able to accurately and efficiently deal with both natural fractures and hydraulic fractures, which does not require local grid refinement nearby fractures through nonneighboring connections [50][51][52][53]. PTA and RTA are methods to analyze well interference, which can identify the characteristics of well interference in some specific periods [54][55][56][57][58][59].
The current research mainly focuses on the phenomenon, reason, mechanism, and mitigation suggestions for well interference. The quantitative analysis method for shale gas well interference evaluation needs to be further investigated. Thus, this paper tries to analyze the well interference of shale gas wells through production type curves based on numerical simulation using EDFM technology.

Fracture Characterization and Calculation
Complex fracture networks are generated through largescale hydraulic fracturing in shale gas formations. How to characterize and calculate the formed fracture networks including both hydraulic and natural fractures becomes a crucial issue for efficient development of shale gas resources. The dual porosity and dual permeability models belong to the continuum media, which is unavailable for the fracture description in shale gas reservoirs [60]. A discrete fracture model is more accurate to characterize the hydraulic frac-tures and natural fractures [61]. Although the discrete fracture model based on unstructured grids shows good performance in representing complex fracture geometries, the huge number of grids and big differences among grid scale result in high computation costs and poor convergence as well as difficulty on gridding [62]. Therefore, it is hard to efficiently handle the complex fractures by using the discrete fracture model based on unstructured grids. In recent years, the EDFM was developed to meet the accuracy of discrete fracture models with structured gridding especially in the unconventional oil and gas reservoirs [63]. The grid number is significantly reduced since the refinement near the fractures is not required compare to the discrete fracture model with unstructured gridding. Furthermore, the gridding is easier and the convergence is better based on EDFM.
2.1. Nonneighboring Connections. The nonneighboring connections (NNCs) are proposed to handle the different intersections among natural fractures, hydraulic fractures, and matrices [62]. The fracture can be divided into multiple segments through matrix cells and generate NNCs. The fluid flow between the fracture and matrix can be efficiently modelled using the transmissibility of NNCs [63].
The NNCs can be divided into three categories (see Figure 1), including the connection between the matrix grid and fracture segment, connection between different fracture segments within the same fracture, and connection between different fractures.

Transmissibility of Connection between Matrix
Grid and Fracture Segment. The schematic of fracture-matrix connection can be found in Figure 1(a). Its transmissibility can be calculated by [62] where A f is the area of the fracture plane. ⇉ K denotes the permeability tensor. ⟶ n is the normal vector of the fracture plane. d f −m represents the distance between the matrix and fracture segment. V means the fracture volume. X n denotes the distance from the matrix unit to the fracture segment.

Transmissibility of Connection between Fracture
Segments within the Same Fracture. The connection between fracture segments from the same fracture is shown in Figure 1(b). Its transmissibility can be obtained and expressed as [62] T seg = where k f is the fracture permeability. A c means the common 2 Geofluids plane area of two fracture segments. d seg1 denotes the distance from the center of fracture segment 1 to the common plane. d seg2 denotes the distance from the center of fracture segment 2 to the common plane.

Transmissibility of Connection between Different
Fractures. Figure 1(c) shows the intersection between two different fractures. The transmissibility between them can be calculated by [62] T int = where L int is the length of the intersection line. k f 1 and k f 2 represent the permeability of fracture 1 and fracture 2, respectively. d f 1 and d f 2 denote the weighted average distance from the intersection line to the fracture 1 and fracture 2, respectively. w f 1 and w f 2 are the fracture aperture of fracture 1 and fracture 2, respectively.

Well
Index. The well index in EDFM can be calculated through the effective well index of the fracture segment intersecting with the horizontal wellbore, as shown in [62] WI r e = 0:14 where k f and w f are the permeability and aperture of the fracture. r e and r w represent the supply radius and well radius. L means the fracture length, and W is the fracture height. After the calculation of transmissibility of different NNCs and the well index, the EDFM model can be coupled with commercial reservoir simulators for numerical simulation efficiently [64].

Well Interference Modelling
3.1. Model Design. Based on the well distribution and fracture features in shale gas reservoirs, six physical models are designed to analyze the effect of well interference on production performance. The distribution of horizontal well, hydraulic fractures, and natural fractures of two MFHWs are presented in Figure 2.  3 Geofluids Case 1. Two horizontal wells are not directly connected (see Figure 2(a)). Pressure interference is dominated through matrix.
Case 2. Two horizontal wells are connected through 50% hydraulic fractures (see Figure 2(b)). And well interference is composed of fracturing interference and pressure interference through hydraulic fractures and matrix.
Case 3. Two horizontal wells are connected through all hydraulic fractures (see Figure 2(c)). And well interference is composed of fracturing interference and pressure interference through all hydraulic fractures and matrix.
Case 4. Two horizontal wells are connected through partial natural fractures (see Figure 2(d)). And well interference is composed of fracturing interference and pressure interference through natural fractures and matrix.
Case 5. Two horizontal wells are connected through both hydraulic fractures and natural fractures (see Figure 2(e)). Well interference is caused by fracturing interference and pressure interference through hydraulic, natural fractures, and matrix. Case 6. Two horizontal wells are connected through lots of natural fractures (see Figure 2(f)). The well interference is The basic parameters of shale gas reservoirs, horizontal wells, and fractures can be seen in Table 1.

Pressure Distribution.
To consider the effect of fracturing fluid on gas production, fluids are injected into the par-ent well from Jan 2, 2020 to Jan 20, 2020, and the well was further shut in for 11 days. From Feb 1, 2020, the parent well was put into production. The designed gas production is 250000 m 3 /d and the minimum bottom-hole pressure (BHP) is set as 5 MPa. After about two years, the child well was fractured beginning from Jan 1, 2022 to Jan 20, 2022. The child well began to produce gas and water from Feb 1,  To compare the well interference of different cases, the pressure distribution in different time steps (Mar 2022 and Dec 2025) of two MFHWs considering only hydraulic fractures (Cases 1 to 3) are obtained as shown in Figure 3. In the early period, the difference of pressure distribution between Case 1 and Case 2 is not obvious as shown in Figures 3(a) and 3(b). However, the difference gradually becomes bigger since the well interference becomes stronger when two horizontal wells are connected with partial hydraulic fractures. If the two wells are totally connected through all hydraulic fractures (see Figure 3(c)), the well interference is obvious and the impact of well interference on the pressure distribution is significant. The pressure depletion is obvious when the well communication degree becomes higher, which can provide guidance for well spacing optimization.
Except for the effect of hydraulic fractures on well interference and pressure distribution, the impact of natural fractures also needs to be investigated. The pressure distribution of two MFHWs considering hydraulic fractures and natural fractures under three kinds of well connection conditions are shown in Figure 4. Firstly, the pressure distribution of Case 4 (see Figure 4(a)) is quite different with Case 1 (see Figures 3  (a)). It indicates that well interference is obvious when two horizontal wells are directly connected through hydraulic fractures or natural fractures. In addition, the impact of natural fractures on pressure distribution is relatively weaker than hydraulic fractures especially for the late period (see Figures 4(b) and 4(c)). However, it is hard to quantitatively evaluate the well interference and its impact on gas production only based on the pressure distribution.

Well Interference Analysis
Type cures are developed to further quantitatively analyze the well interference, including the log-log type curve and Blasingame type curve. The feature on log-log type curve and Blasingame type curve under different well connections are identified to evaluate well interference.

Log-Log Type
Curve. The log-log type curve is composed of the rate normalized pressure integral and rate normalized pressure integral derivative. The rate normalized pressure integral can be defined as On the basis of Equation (9), the rate normalized pressure integral derivative can be obtained shown as Equation (10).
where p rni and p rniD represent the rate normalized pressure integral and its derivative, respectively. P i is the initial formation pressure. P wf is the bottom-hole pressure. q is the gas production. t e is the equivalent time, and t e = QðtÞ/ð2qðtÞÞ.

Blasingame Type
Curve. The Blasingame type curves are proposed to handle the production data under changeable pressure and production rate [65]. It is composed of pressure normalized rate, pressure normalized rate integral, and pressure normalized rate integral derivative curves. The pressure normalized rate is defined as The pressure normalized rate integral can be calculated Based on Equation (12), the pressure normalized rate integral be expressed as where R pn , R pni , and R pniD represent the pressure normalized rate, pressure normalized rate integral, and its derivative, respectively. P i is the initial formation pressure. t c means the material balanced time, and t c = QðtÞ/qðtÞ.

Well Interference Type Curve.
To compare the impact of hydraulic fractures and natural fractures, we separate the six cases into two groups. The first group consists of Cases 1, 2, and 3. The second group is composed of Case 1, 4, 5, and 6. Firstly, the production data needs to be analyzed and compared. The production curves of two MFHWs considering only hydraulic fractures under three kinds of well connection conditions (Cases 1 to 3) are shown in Figure 5. After the wells are connected through hydraulic fractures, the gas production of the parent well decreases rapidly and the water production increases quickly. Bigger decline of gas production for the parent well can be identified with the increase of well communication degree (e.g., from Case 2 to Case 3). Although it begins to recover in the later stage, it is still lower than that of parent wells with lower well communication degree. However, the gas production of child well shows the opposite characteristics. Higher communication degree between wells through hydraulic fractures results in the stimulation area or degree for the child well so that the gas production is slightly higher than that with lower communication degree or even disconnection. It indicates that 7 Geofluids well interference caused by hydraulic fractures mainly reduce the gas production of the parent well while the gas production of child well can be increased owing to the larger equivalent stimulation area and degree.
Then, the production and pressure data can be processed by using the Equations (9) and (10). The log-log type curves are generated for the MFHWs considering only hydraulic fractures under three different well connection conditions (Cases 1 to 3), shown in Figure 6. Obvious differences can be found from early to late periods in the log-log type curves when two horizontal wells are connected with hydraulic fractures (Cases 2 and 3) or two wells are not directly connected with hydraulic factures (Case 1). Especially, the distinctions during early period are more obvious. The rate normalized pressure integral derivative drops a lot with the increase of well connectivity, while it becomes larger under late period with the increase of well connectivity. Therefore, well interference can be identified using the log-log type curves.
Furthermore, the Blasingame type curves can also be obtained using the Equations (11) and (13) based on the production and pressure data. Figure 7 shows the Blasingame type   In this part, the impact of natural fractures on the production data and type curves will be analyzed. The production curves of two MFHWs considering both hydraulic fractures and natural fractures under three kinds of well connection conditions (Cases 4 to 6) are shown in Figure 8. Case 1 was included in this part to show the comparison. Compared to Case 1, more fractures are added in to Cases 4 to 6 so that the gas production is improved owing to bigger simulated area. When the fracturing interference     9 Geofluids occurs, the gas production of the parent well decreases due to the impact of fracturing fluids from the child well. Different from Figure 5, the decline degree of gas production is minor when child well is connecting parent well only through natural fractures (Cases 4 and 6). But, if two wells are communicated with natural and hydraulic fractures, the gas production of the parent well decline a lot (Case 5). We can conclude that well interference caused by natural fractures is much weaker compared to hydraulic fractures.
To identify the impact of well interference caused by natural fractures, log-log type curves are generated as shown in Figure 9. On the one hand, differences can be observed between Case 1 and Cases 4 to 6, especially on the rate normalized pressure integral derivative curves. On the other hand, the distinctions are minor if the parent well and child well are communicated with only natural fractures (Cases 4 and 6). Additionally, the impact of fracturing interference caused by hydraulic fractures or numerous natural fractures can be identified using the rate normalized pressure integral derivative curve (Case 5 and 6).
The feature of fracturing interference is not obvious enough on the log-log type curves so that the Blasingame type curves are developed based on the production and pressure data of Cases 1, 4, 5, and 6 (see Figure 10). It is obvious that major differences during early, middle, and late periods can be found on the Blasingame type curves under different well communication degrees. And the signal of fracturing interference caused by the natural fractures is more obvious.

Conclusions
This paper focuses on the well interference analysis of shale gas wells. EDFM technology is introduced to deal with the hydraulic fractures and natural fractures as well as model the well interference caused by hydraulic and natural fractures.
(1) After the wells are communicated through hydraulic fractures, the gas production of the parent well declines a lot due to the fracturing fluids from the child well. However, if two wells are connected only through natural fractures, the decline degree of gas production is much minor. Well interference caused by natural fractures is much weaker compared to hydraulic fractures (2) For wells connected through hydraulic fractures, the differences during the late period on the Blasingame type curves are clearer than that on the log-log type curves, while the features during early period on the log-log type curves are easier to distinguish (3) For wells connected through only natural fractures, the distinctions are minor if the parent well and child well are communicated with only natural fractures, the signal of fracturing interference caused by the natural fractures are more obvious on the Blasingame type curves Thus, the well interference of shale gas reservoirs can be identified by the combination of log-log and Blasingame type curves. This work can be helpful for understanding the well interference feature and its impact on shale gas production.

Data Availability
The data used to support the findings of this study are included in the article.   10 Geofluids

Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this paper.