# A Semianalytical Model for Analyzing the Infill Well-Caused Fracture Interference from Shale Gas Reservoirs

Authors: Sidong Fang; Yonghui Wu; Cheng Dai; Liqiang Ma; Hua Liu
Journal: Geofluids
 (2021)

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## Abstract

Drilling infill well has been widely used in many plays to enhance the recovery of shale gas, but the infill well-caused fracture interference is a very important issue that should be taken into consideration. The well interference makes it difficult for the conventional models to make production predictions, fracture characterization, and production data analysis. In this paper, a semianalytical model is proposed for this purpose by discretizing the whole control volume of the parent and infill wells into several linear flow zones. In this way, three important issues can be further handled very naturally, including fracture connection between the parent and infill wells, different SRV properties for zones with different distances to the wellbore, and different production times for adjacent wellbores. The approximate expressions for different flow regimes are used in making production predictions in the time domain, and a flowing material balance method and a simple iteration are used to update the model parameters step by step. The proposed model is shown to be reasonable and accurate for handling multiwell interference problems after comparing with the commercial numerical simulator tNavigator. The synthetical cases show that the fracture parameters, SRV properties, and well infill time have a significant influence on the production performance of both the parent and infill wells. The results show that the production of the parent well will be dramatically enhanced when it is connected with the infill well via high-conductive hydraulic fractures. Longer unconnected fractures and more fracturing stages/clusters for the infill well will result in higher production for the infill well, but a negative effect is observed for the parent well. The permeability of the distant well SRV has a similar influence on the parent and infill wells. The results also show that late time well interference will result in a more significant increase in production rate on the log-log plots for the severe depletion around the parent well. Finally, the proposed model is used to analyze the production data of a field case from Fuling shale in Southwestern China. After analyzing the production data, several parameters can be obtained for both parent and infill wells, including the fracture lengths and conductivities, numbers of connected fractures, and the near and distant well permeabilities of the SRV. This gives a basic and practical technique for production prediction, formation and fracture evaluation, and well connectivity analysis from shale gas wells with fracture connection.


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## Body

## 1. Introduction

With the development of the technology in drilling long horizontal wells and multiple-stage hydraulic fracturing, unconventional shale gas resources have been economically developed in many counties around the world, such as the USA, China, and Canada [1]. However, the recovery of shale gas is quite low at present because of the high decline rate in shale gas production. At present, several techniques are presented for this problem, including CO2 sequestration in the shale formation [2–4] and refracturing and drilling infill wells [5, 6]. In particular, drilling infill wells are widely used in many plays and the well spaces have been tightened to 200~300 meters. A major concern of drilling an infill well is the infill well-caused well interference, which makes it difficult for making production predictions and interpreting the fracture parameter and evaluating the well connectivity.Many researchers have reported the well interference phenomenon from several aspects, including fracture prorogation simulation, field microseismic monitoring, and production and pressure tests. Fracture prorogation studies show that fractures growing from adjacent wells tend to attract each other and result in fracture connections [7]. Field microseismic data also show that the phenomenon of fracture connection between adjacent wells is quite common, especially for infill well cases, where there is pressure sink in the formation after the long-time production [8, 9]. Production and pressure tests show that the pressure, and gas, and water production of a parent well will dramatically change after the hydraulic fracturing of an adjacent infill well [10–12]. Therefore, it is of great importance to analyze the effects of infill well-caused fracture interference for shale gas reservoirs.Numerical models are quite comprehensive and robust for fracture interference analysis for their ability in handling complex fracture networks, nonlinearities, heterogeneities, and other complex problems. The present numerical models mainly include discrete fracture models (DFM) and embedded discrete fracture models (EDFM) [13–19]. The EDFM is quite efficient in analyzing the effects of infill well-caused fracture interference because the time-consuming grid system is not needed to be discretized two times after adding the infill well. Green element method- (GEM-) based discrete fracture model is another efficient method for unconventional reservoir simulation, which borrows the advantages of EDFM in discretization and the boundary element method (BEM) in precision [20, 21]. By using numerical models, several studies have studied the effects of well interference on the production and pressure performance [22–25]. In these studies, the wells are assumed to be connected with low-conductivity reactivated natural fractures and high-conductivity hydraulic fractures.Because numerical models are often time-consuming in matching the production and pressure data when there are many fractures in the model, analytical models are still often used in reservoir engineering. The analytical models for unconventional oil and gas reservoirs are often based on linear flow assumptions for the long-time linear flow regimes observed in the field. In the linear flow models, the reactivated natural fractures are treated using the SRV concept and the analytical solutions can be derived by discretizing the whole control volume of the wellbore with several linear flow zones [26–28]. Several studies also enriched the analytical models to more complex problems, such as the two-phase flow in retrograde gas and condensate oil reservoirs, and early-time two-phase flow back data analysis [29, 30]. Some studies also use this concept to approximately analyze the production data for wells with fracture interference [12, 31–33]. The slopes exhibited by the pressure and rate curves on the log-log plots are used to diagnose the well interference. It is reported that well interference will increase the decline rate on the type curves. However, the present analytical model can only be used for production decline analysis qualitatively. In addition, the wells are assumed to start producing at the same time, so the problem of infill well-caused fracture interference is not considered in these models.In this paper, we will present a semianalytical model for production prediction and production data analysis based on the linear flow assumptions. This model mainly handled three important problems concerning the well interference, including different production times for different wells, well connections with part of the fractures, and different formation properties for near and distant well SRV regions. In the following, we first presented the mathematical fundamentals of the semianalytical model. Then, the proposed model is benchmarked with a commercial simulator and several synthetical cases are used to analyze the effects of different parameters. Finally, a field case is used to show the application of the proposed model.


## 2. Methodology

### 2.1. Physical Model

For shale gas reservoirs, as shown in Figure1, multiple high-conductive fractures and an SRV will be generated after hydraulic fracturing. Before adding the infill well and there is no well interference, the SRV is near the parent well, just like SRV1 in Figure 1(a) and SRV1 and SRV5 in Figure 1(b). After the hydraulic fracturing of the infill well, SRV can be generated near the infill well, just like SRV3 in Figure 1(a) and SRV3 in Figure 1(b). In addition, the distant well area can also be stimulated because the depletion around the parent well and the hydraulic fractures are tending to propagate into the SRV of the parent well. Therefore, some fractures will be connected between the parent and infill wells and the distant well SRV can be generated, just like SRV2 in Figure 1(a) and SRV2 and SRV4 in Figure 1(b).Figure 1
Schematic of a well pad with fracture connection between the parent and infill wells.
(a)
Two-well interference(b)
Three-well interferenceIn this paper, we assume that part of the distant well SRV will contribute to the parent well and the other part will contribute to the infill well. The distant well SRV, including SRV2 in Figure1(a) and SRV2 and SRV4 in Figure 1(b), contribute to both the parent and infill wells, so there should be a contribution ratio to the parent well, which can be defined using the fracture length. Based on this assumption, the representative zones can be obtained in Figure 1 for both parent and infill wells. For the two-well case shown in Figure 1(a), zones A and B are the representative zones of the parent well and the whole control volume of the parent well can be formed with the combinations of A and B. For the infill well shown in Figure 1(a), zones C and D can be regarded as the representative zones. One should note that zones A and C are different models because the near and distant well SRVs are not stating produce at the same time. In the same way, we can analyze the three-well case shown in Figure 1(b). Another assumption is that the unstimulated reservoir volume is not considered in this model because the production contribution from the unstimulated reservoir is not significant. In addition, this contribution cannot be observed on log-log plots in late flow regimes.Taking half of the representative zones shown in Figure1, as shown in Figure 2, we can further extract the basic linear flow models, including the two-region linear flow model, convergence linear flow model, and single-linear flow model. In this way, we can simplify the complex multiwell interference model to basic linear flow models. In Section 2.2, we will present the mathematical fundamentals for the physical model.Figure 2
Schematic of the typical flow models for analyzing the infill well-caused fracture interference.
(a)
Two-region linear(b)
Convergence linear(c)
Single linear


### 2.2. Mathematical Model

#### 2.2.1. Analytical Solution in Linear Systems

For the single-linear flow, taking the schematic shown in Figure2(c) for example, the governing equation can be given by
(1)∂2ψD∂yD2=1ηD∂ψD∂tD.It should be noted that the gas desorption from the tight matrix pores is considered using desorption compressibility, which is defined bycd=pscZT/ϕZscTscVLpL/ppL+p2 in our previous work [34]. Here, we should state that the geomechanics and complex transport mechanisms in the shale nanopores are not seriously considered in this analytical model [35–42]. In this model, these transport mechanisms can be characterized using apparent permeability and handled with the pseudopressure approximately in the analytical model. The pseudopressure can be defined with ψ=2∫0pfappp/μZdp, where fapp is a permeability modifier.The definitions of the dimensionless variables are shown in Table1.Table 1
Definitions of the dimensionless parameters used in the model derivation.
VariablesDefinitionVariablesDefinitionDimensionless pressure for gas (at constant rate)ψD=krHψi−ψ/1.291×10−3qscTDimensionless pressure for gas (at constant BHP)ψD=ψi−ψ/ψi−ψwfDimensionless rate for oil (at constant rate)ψD=krHpi−p/1.842qscBμDimensionless pressure for oil (at constant BHP)ψD=pi−p/pi−pwfReference diffusivity (106 mD/s)ηr=0.0864kr/ϕctμrDimensionless timetD=ηr/Lr2tDimensionless length:x directionxD=x/LrDimensionless length:y directionyD=y/LrDimensionless diffusivityηD=η/ηr=1/ηr0.0864k/ϕctμDimensionless permeabilitykD=k/krThe initial condition is given by(2)ψDtD=0=0.The inner boundary condition is(3)∂ψD∂yDyD=yfD=0.The outer boundary condition is coupled with the hydraulic fractures. Since infinite-conductivity fracture is assumed in this model, the inner boundary condition can be written as(4)ψDyD=0=1+sc.Laplace transformation can be used to obtain the analytical solution of the single-linear flow model shown in equations (1)–(3). According to our previous work [43], the production solution can be directly given by
(5)q¯D=1+scskDxfDπsηDtanhsηD⋅yfD.It should be noted that the numerical algorithm proposed by Stehfest should be used to obtain the solution of equation (5) in the time domain [44]. Because there are mainly two flow regimes for the infinite conductive fracture model, linear flow regime, and boundary dominated flow regime, the regional approximate solutions [45] can be used to simplify the solution.For the linear flow regime, the solution for the constant bottom hole pressure condition case is given by(6)1qD=11+scπ2kDxfDπηDtD.Taking the dimensionless variables into equation (6), we can obtain
(7)qsc=2π1+sckHψi−ψwfxf1.291×10−3Tπη¯t.For the long-term boundary-dominated flow regime, the solution can be written as(8)qsc=αgπ2kHψ¯−ψwf1.291×10−3Txfyf.αg is a factor, which makes the production rate curves continuous at the connection of the two flow regimes. The value of αg can be determined using the following expression
(9)qsclinearflow,telf=qscboundarydominatedflow,telf,where telf is the time of deviation from the linear flow regime to the boundary-dominated flow regime and it can be estimated as follows:
(10)telf=ϕμictikyf0.58362.The bilinear flow may occur when the permeability of the SRV is high and the fracture cannot be regarded as infinitely conductive. In this case, the approximation of the production [43] can be given as
(11)1qD=11+sc1.2254πkDkfDwfDηDtD4.Taking the dimensionless variables into equation (11), we can obtain
(12)qsc=201.4kkfwf1+scHψi−ψwfTη¯t4.


#### 2.2.2. Average Pressure Calculation

We can find that some parameters in equations (7) and (8) are average pressure dependent, such as η¯j and ψ¯. Therefore, to obtain the production solution of the system, another step is to obtain the average pressure. In this section, the flowing material balance equations are used for this purpose. According to equation (10), the distance of investigation can be written as
(13)yinv=0.5836ktϕμicti.Therefore, for the single-linear flow system, the control volume of the fracture can be given by(14)Vinv=4xfyinvHϕ.Using the material balance method, we can obtain(15)Gp=IGIP−RGIP.In which,Gp is the cumulative gas production, IGIP is the initial gas reserve, RGIP is the remaining gas reserve, and they can be calculated with
(16)Gp=∫0tqgscdt,(17)IGIP=VinvSgiBgi+VLpipL+pi,(18)RGIP=VinvSgiB¯g+VLp¯pL+p¯.It should be noted that the water in the formation is irreducible, so the initial gas saturation is used in equation (18).Taking equations (16)–(18) into equation (15), we can obtain
(19)SgiB¯g+VLp¯pL+p¯=SgiBgi+VLpipL+pi−GpVinv.Equation (19) can be rewritten as
(20)fp¯=SgiB¯g+VLp¯pL+p¯+GpVinv−SgiBgi−VLpipL+pi=0.Taking the derivation of Eq. (20), we obtain
(21)f′p¯=−SgiB¯g2dBgdp+VLpLpL+p¯2.We can use the Newton-Raphson iteration to solve equations (20) and (21). The iteration equation can be written as
(22)p¯k+1=p¯k−ωfp¯kf′p¯k,where ω depends on the iteration steps n, ω=1/2n−1.


#### 2.2.3. Production Prediction for Different Regions

For the cases shown in Figure1, the whole reservoir can be characterized with representative regions, including A–G. In addition, each region is combined with the typical flow models shown in Figure 2.For the model shown in Figure1(c), the solution can be obtained using the mathematical derivations shown in Sections 2.3.1 and 2.3.2. The detailed procedure can be concluded in Figure 3.Figure 3
The procedure to predict the production for a single-linear flow model.Using the procedure for a single-linear flow, we can obtain the production of the typical regions and the horizontal wells. Table2 shows the composition of different regions shown in Figure 1. For the parent and infill wells, the total gas production is the summation of different typical regions. It should be noted that the production times for the parent and infill wells are different, so this should be considered in predicting the production of the parent wells.Table 2
The composition of the typical sections shown in Figure1.
Typical regionCompositionProduction timeALinear (Figure2(c))+two−region linear (Figure 2(a))At a different timeBLinear (Figure2(c))+convergence linear (Figure 2(b))At a different timeCLinear (Figure2(c))+two−region linear (Figure 2(a))At the same timeDLinear (Figure2(c))+convergence linear (Figure 2(b))At the same timeETwo-region linear (Figure2(a))+two−region linear (Figure 2(a))At a different timeFConvergence linear (Figure2(b))+convergence linear (Figure 2(b))At a different timeGTwo-region linear (Figure2(a))+convergence linear (Figure 2(b))At a different time


## 2.1. Physical Model

For shale gas reservoirs, as shown in Figure1, multiple high-conductive fractures and an SRV will be generated after hydraulic fracturing. Before adding the infill well and there is no well interference, the SRV is near the parent well, just like SRV1 in Figure 1(a) and SRV1 and SRV5 in Figure 1(b). After the hydraulic fracturing of the infill well, SRV can be generated near the infill well, just like SRV3 in Figure 1(a) and SRV3 in Figure 1(b). In addition, the distant well area can also be stimulated because the depletion around the parent well and the hydraulic fractures are tending to propagate into the SRV of the parent well. Therefore, some fractures will be connected between the parent and infill wells and the distant well SRV can be generated, just like SRV2 in Figure 1(a) and SRV2 and SRV4 in Figure 1(b).Figure 1
Schematic of a well pad with fracture connection between the parent and infill wells.
(a)
Two-well interference(b)
Three-well interferenceIn this paper, we assume that part of the distant well SRV will contribute to the parent well and the other part will contribute to the infill well. The distant well SRV, including SRV2 in Figure1(a) and SRV2 and SRV4 in Figure 1(b), contribute to both the parent and infill wells, so there should be a contribution ratio to the parent well, which can be defined using the fracture length. Based on this assumption, the representative zones can be obtained in Figure 1 for both parent and infill wells. For the two-well case shown in Figure 1(a), zones A and B are the representative zones of the parent well and the whole control volume of the parent well can be formed with the combinations of A and B. For the infill well shown in Figure 1(a), zones C and D can be regarded as the representative zones. One should note that zones A and C are different models because the near and distant well SRVs are not stating produce at the same time. In the same way, we can analyze the three-well case shown in Figure 1(b). Another assumption is that the unstimulated reservoir volume is not considered in this model because the production contribution from the unstimulated reservoir is not significant. In addition, this contribution cannot be observed on log-log plots in late flow regimes.Taking half of the representative zones shown in Figure1, as shown in Figure 2, we can further extract the basic linear flow models, including the two-region linear flow model, convergence linear flow model, and single-linear flow model. In this way, we can simplify the complex multiwell interference model to basic linear flow models. In Section 2.2, we will present the mathematical fundamentals for the physical model.Figure 2
Schematic of the typical flow models for analyzing the infill well-caused fracture interference.
(a)
Two-region linear(b)
Convergence linear(c)
Single linear


## 2.2. Mathematical Model

### 2.2.1. Analytical Solution in Linear Systems

For the single-linear flow, taking the schematic shown in Figure2(c) for example, the governing equation can be given by
(1)∂2ψD∂yD2=1ηD∂ψD∂tD.It should be noted that the gas desorption from the tight matrix pores is considered using desorption compressibility, which is defined bycd=pscZT/ϕZscTscVLpL/ppL+p2 in our previous work [34]. Here, we should state that the geomechanics and complex transport mechanisms in the shale nanopores are not seriously considered in this analytical model [35–42]. In this model, these transport mechanisms can be characterized using apparent permeability and handled with the pseudopressure approximately in the analytical model. The pseudopressure can be defined with ψ=2∫0pfappp/μZdp, where fapp is a permeability modifier.The definitions of the dimensionless variables are shown in Table1.Table 1
Definitions of the dimensionless parameters used in the model derivation.
VariablesDefinitionVariablesDefinitionDimensionless pressure for gas (at constant rate)ψD=krHψi−ψ/1.291×10−3qscTDimensionless pressure for gas (at constant BHP)ψD=ψi−ψ/ψi−ψwfDimensionless rate for oil (at constant rate)ψD=krHpi−p/1.842qscBμDimensionless pressure for oil (at constant BHP)ψD=pi−p/pi−pwfReference diffusivity (106 mD/s)ηr=0.0864kr/ϕctμrDimensionless timetD=ηr/Lr2tDimensionless length:x directionxD=x/LrDimensionless length:y directionyD=y/LrDimensionless diffusivityηD=η/ηr=1/ηr0.0864k/ϕctμDimensionless permeabilitykD=k/krThe initial condition is given by(2)ψDtD=0=0.The inner boundary condition is(3)∂ψD∂yDyD=yfD=0.The outer boundary condition is coupled with the hydraulic fractures. Since infinite-conductivity fracture is assumed in this model, the inner boundary condition can be written as(4)ψDyD=0=1+sc.Laplace transformation can be used to obtain the analytical solution of the single-linear flow model shown in equations (1)–(3). According to our previous work [43], the production solution can be directly given by
(5)q¯D=1+scskDxfDπsηDtanhsηD⋅yfD.It should be noted that the numerical algorithm proposed by Stehfest should be used to obtain the solution of equation (5) in the time domain [44]. Because there are mainly two flow regimes for the infinite conductive fracture model, linear flow regime, and boundary dominated flow regime, the regional approximate solutions [45] can be used to simplify the solution.For the linear flow regime, the solution for the constant bottom hole pressure condition case is given by(6)1qD=11+scπ2kDxfDπηDtD.Taking the dimensionless variables into equation (6), we can obtain
(7)qsc=2π1+sckHψi−ψwfxf1.291×10−3Tπη¯t.For the long-term boundary-dominated flow regime, the solution can be written as(8)qsc=αgπ2kHψ¯−ψwf1.291×10−3Txfyf.αg is a factor, which makes the production rate curves continuous at the connection of the two flow regimes. The value of αg can be determined using the following expression
(9)qsclinearflow,telf=qscboundarydominatedflow,telf,where telf is the time of deviation from the linear flow regime to the boundary-dominated flow regime and it can be estimated as follows:
(10)telf=ϕμictikyf0.58362.The bilinear flow may occur when the permeability of the SRV is high and the fracture cannot be regarded as infinitely conductive. In this case, the approximation of the production [43] can be given as
(11)1qD=11+sc1.2254πkDkfDwfDηDtD4.Taking the dimensionless variables into equation (11), we can obtain
(12)qsc=201.4kkfwf1+scHψi−ψwfTη¯t4.


### 2.2.2. Average Pressure Calculation

We can find that some parameters in equations (7) and (8) are average pressure dependent, such as η¯j and ψ¯. Therefore, to obtain the production solution of the system, another step is to obtain the average pressure. In this section, the flowing material balance equations are used for this purpose. According to equation (10), the distance of investigation can be written as
(13)yinv=0.5836ktϕμicti.Therefore, for the single-linear flow system, the control volume of the fracture can be given by(14)Vinv=4xfyinvHϕ.Using the material balance method, we can obtain(15)Gp=IGIP−RGIP.In which,Gp is the cumulative gas production, IGIP is the initial gas reserve, RGIP is the remaining gas reserve, and they can be calculated with
(16)Gp=∫0tqgscdt,(17)IGIP=VinvSgiBgi+VLpipL+pi,(18)RGIP=VinvSgiB¯g+VLp¯pL+p¯.It should be noted that the water in the formation is irreducible, so the initial gas saturation is used in equation (18).Taking equations (16)–(18) into equation (15), we can obtain
(19)SgiB¯g+VLp¯pL+p¯=SgiBgi+VLpipL+pi−GpVinv.Equation (19) can be rewritten as
(20)fp¯=SgiB¯g+VLp¯pL+p¯+GpVinv−SgiBgi−VLpipL+pi=0.Taking the derivation of Eq. (20), we obtain
(21)f′p¯=−SgiB¯g2dBgdp+VLpLpL+p¯2.We can use the Newton-Raphson iteration to solve equations (20) and (21). The iteration equation can be written as
(22)p¯k+1=p¯k−ωfp¯kf′p¯k,where ω depends on the iteration steps n, ω=1/2n−1.


### 2.2.3. Production Prediction for Different Regions

For the cases shown in Figure1, the whole reservoir can be characterized with representative regions, including A–G. In addition, each region is combined with the typical flow models shown in Figure 2.For the model shown in Figure1(c), the solution can be obtained using the mathematical derivations shown in Sections 2.3.1 and 2.3.2. The detailed procedure can be concluded in Figure 3.Figure 3
The procedure to predict the production for a single-linear flow model.Using the procedure for a single-linear flow, we can obtain the production of the typical regions and the horizontal wells. Table2 shows the composition of different regions shown in Figure 1. For the parent and infill wells, the total gas production is the summation of different typical regions. It should be noted that the production times for the parent and infill wells are different, so this should be considered in predicting the production of the parent wells.Table 2
The composition of the typical sections shown in Figure1.
Typical regionCompositionProduction timeALinear (Figure2(c))+two−region linear (Figure 2(a))At a different timeBLinear (Figure2(c))+convergence linear (Figure 2(b))At a different timeCLinear (Figure2(c))+two−region linear (Figure 2(a))At the same timeDLinear (Figure2(c))+convergence linear (Figure 2(b))At the same timeETwo-region linear (Figure2(a))+two−region linear (Figure 2(a))At a different timeFConvergence linear (Figure2(b))+convergence linear (Figure 2(b))At a different timeGTwo-region linear (Figure2(a))+convergence linear (Figure 2(b))At a different time


## 2.2.1. Analytical Solution in Linear Systems

For the single-linear flow, taking the schematic shown in Figure2(c) for example, the governing equation can be given by
(1)∂2ψD∂yD2=1ηD∂ψD∂tD.It should be noted that the gas desorption from the tight matrix pores is considered using desorption compressibility, which is defined bycd=pscZT/ϕZscTscVLpL/ppL+p2 in our previous work [34]. Here, we should state that the geomechanics and complex transport mechanisms in the shale nanopores are not seriously considered in this analytical model [35–42]. In this model, these transport mechanisms can be characterized using apparent permeability and handled with the pseudopressure approximately in the analytical model. The pseudopressure can be defined with ψ=2∫0pfappp/μZdp, where fapp is a permeability modifier.The definitions of the dimensionless variables are shown in Table1.Table 1
Definitions of the dimensionless parameters used in the model derivation.
VariablesDefinitionVariablesDefinitionDimensionless pressure for gas (at constant rate)ψD=krHψi−ψ/1.291×10−3qscTDimensionless pressure for gas (at constant BHP)ψD=ψi−ψ/ψi−ψwfDimensionless rate for oil (at constant rate)ψD=krHpi−p/1.842qscBμDimensionless pressure for oil (at constant BHP)ψD=pi−p/pi−pwfReference diffusivity (106 mD/s)ηr=0.0864kr/ϕctμrDimensionless timetD=ηr/Lr2tDimensionless length:x directionxD=x/LrDimensionless length:y directionyD=y/LrDimensionless diffusivityηD=η/ηr=1/ηr0.0864k/ϕctμDimensionless permeabilitykD=k/krThe initial condition is given by(2)ψDtD=0=0.The inner boundary condition is(3)∂ψD∂yDyD=yfD=0.The outer boundary condition is coupled with the hydraulic fractures. Since infinite-conductivity fracture is assumed in this model, the inner boundary condition can be written as(4)ψDyD=0=1+sc.Laplace transformation can be used to obtain the analytical solution of the single-linear flow model shown in equations (1)–(3). According to our previous work [43], the production solution can be directly given by
(5)q¯D=1+scskDxfDπsηDtanhsηD⋅yfD.It should be noted that the numerical algorithm proposed by Stehfest should be used to obtain the solution of equation (5) in the time domain [44]. Because there are mainly two flow regimes for the infinite conductive fracture model, linear flow regime, and boundary dominated flow regime, the regional approximate solutions [45] can be used to simplify the solution.For the linear flow regime, the solution for the constant bottom hole pressure condition case is given by(6)1qD=11+scπ2kDxfDπηDtD.Taking the dimensionless variables into equation (6), we can obtain
(7)qsc=2π1+sckHψi−ψwfxf1.291×10−3Tπη¯t.For the long-term boundary-dominated flow regime, the solution can be written as(8)qsc=αgπ2kHψ¯−ψwf1.291×10−3Txfyf.αg is a factor, which makes the production rate curves continuous at the connection of the two flow regimes. The value of αg can be determined using the following expression
(9)qsclinearflow,telf=qscboundarydominatedflow,telf,where telf is the time of deviation from the linear flow regime to the boundary-dominated flow regime and it can be estimated as follows:
(10)telf=ϕμictikyf0.58362.The bilinear flow may occur when the permeability of the SRV is high and the fracture cannot be regarded as infinitely conductive. In this case, the approximation of the production [43] can be given as
(11)1qD=11+sc1.2254πkDkfDwfDηDtD4.Taking the dimensionless variables into equation (11), we can obtain
(12)qsc=201.4kkfwf1+scHψi−ψwfTη¯t4.


## 2.2.2. Average Pressure Calculation

We can find that some parameters in equations (7) and (8) are average pressure dependent, such as η¯j and ψ¯. Therefore, to obtain the production solution of the system, another step is to obtain the average pressure. In this section, the flowing material balance equations are used for this purpose. According to equation (10), the distance of investigation can be written as
(13)yinv=0.5836ktϕμicti.Therefore, for the single-linear flow system, the control volume of the fracture can be given by(14)Vinv=4xfyinvHϕ.Using the material balance method, we can obtain(15)Gp=IGIP−RGIP.In which,Gp is the cumulative gas production, IGIP is the initial gas reserve, RGIP is the remaining gas reserve, and they can be calculated with
(16)Gp=∫0tqgscdt,(17)IGIP=VinvSgiBgi+VLpipL+pi,(18)RGIP=VinvSgiB¯g+VLp¯pL+p¯.It should be noted that the water in the formation is irreducible, so the initial gas saturation is used in equation (18).Taking equations (16)–(18) into equation (15), we can obtain
(19)SgiB¯g+VLp¯pL+p¯=SgiBgi+VLpipL+pi−GpVinv.Equation (19) can be rewritten as
(20)fp¯=SgiB¯g+VLp¯pL+p¯+GpVinv−SgiBgi−VLpipL+pi=0.Taking the derivation of Eq. (20), we obtain
(21)f′p¯=−SgiB¯g2dBgdp+VLpLpL+p¯2.We can use the Newton-Raphson iteration to solve equations (20) and (21). The iteration equation can be written as
(22)p¯k+1=p¯k−ωfp¯kf′p¯k,where ω depends on the iteration steps n, ω=1/2n−1.


## 2.2.3. Production Prediction for Different Regions

For the cases shown in Figure1, the whole reservoir can be characterized with representative regions, including A–G. In addition, each region is combined with the typical flow models shown in Figure 2.For the model shown in Figure1(c), the solution can be obtained using the mathematical derivations shown in Sections 2.3.1 and 2.3.2. The detailed procedure can be concluded in Figure 3.Figure 3
The procedure to predict the production for a single-linear flow model.Using the procedure for a single-linear flow, we can obtain the production of the typical regions and the horizontal wells. Table2 shows the composition of different regions shown in Figure 1. For the parent and infill wells, the total gas production is the summation of different typical regions. It should be noted that the production times for the parent and infill wells are different, so this should be considered in predicting the production of the parent wells.Table 2
The composition of the typical sections shown in Figure1.
Typical regionCompositionProduction timeALinear (Figure2(c))+two−region linear (Figure 2(a))At a different timeBLinear (Figure2(c))+convergence linear (Figure 2(b))At a different timeCLinear (Figure2(c))+two−region linear (Figure 2(a))At the same timeDLinear (Figure2(c))+convergence linear (Figure 2(b))At the same timeETwo-region linear (Figure2(a))+two−region linear (Figure 2(a))At a different timeFConvergence linear (Figure2(b))+convergence linear (Figure 2(b))At a different timeGTwo-region linear (Figure2(a))+convergence linear (Figure 2(b))At a different time


## 3. Results and Discussion

In this section, we will first validate the proposed semianalytical model by using the commercial numerical simulator tNavigator. Then, we will analyze the effects of the connection condition between the parent and infill wells on the production performance. Finally, a field case from the Fuling shale gas field in Southwestern China is provided to show the application of the proposed model. Because only a two-well interference field case is accessible at present, the case shown in Figure1(a) is analyzed in this section. However, the same method can be used for cases shown in Figure 1(b).


### 3.1. Model Validation

In this section, two cases are used to validate the proposed semianalytical model, the single-linear flow case and two-well interference case. In the following, we first use the single-linear flow case shown in Figure2(c) to validate the model in the calculation, and then, a two-well interference case shown in Figure 1(a) is used to validate the model in model assumptions. For the two cases, the reservoir and fracture properties are shown in Table 3 and the gas PVT properties are calculated using the method proposed by Lee et al. [46] and shown in Figure 4(a).Table 3
Parameters for the validation case.
ParameterValueParameterValueInitial pressure (MPa)75Formation temperature (K)408Initial water saturation (irreducible)0.56Formation thickness (m)25Matrix permeability (mD)1×10−5Porosity of the matrix0.051Rock compressibility (MPa−1)8×10−5Hydraulic fracture porosity0.3Hydraulic fracture width (m)0.01Bottom hole pressure (MPa)5Hydraulic fracture permeability (mD)10000Fracture half-length of the infill well (m)82.5Fracture half-length of the parent well (m)122.5Number of fractures for the infill well11Number of fractures for the parent well16Number of connected fractures6Permeability of the region between the two wells (mD)3×10−5Well space (m)300Permeability of the SRV for the parent well (mD)1×10−4Permeability of the SRV for the infill well (mD)6×10−5Figure 4
The procedure to predict the production for a single-linear flow model.
(a)
Gas PVT properties(b)
Validation results for the single-linear flowThe validation result of the single-linear flow case is shown in Figure4(b). We can find that the proposed semianalytical model has a good match with the numerical simulation results. In addition, for this case, there are mainly two flow regimes exhibited by the gas production curves on the log-log plot—the formation linear flow and boundary-dominated flow regimes. This shows that the computation of the semianalytical model is reliable.We further analyzed the validity of the assumptions of the regional linear flow model in handling infill well-caused fracture interference. In this case, the infill well is assumed to start to produce 720 days later than the parent well. Figure5 shows the validation result, which shows that good matches are obtained for both the parent and infill wells. Figure 5(a) shows that the parent well has been in the boundary-dominated flow regime when it gets interfered by the infill well. It should be noted that the flow regimes after the infill time are composite for the parent well. The flow for the old SRV, the xf1 part shown in Figures 2(a) and 2(b), is under the boundary-dominated flow regime, while the new SRV generated by the fracturing of the infill well is under a linear flow regime. Two methods can be used to analyze this combined flow regime. The first method is by history matching, which will be presented in Section 3.3. The other method is by straight line analysis after processing the production data by deducting the contribution from the near-well SRV, but this will be our future work. One should also notice that the time shown in Figure 5(b) is the production time of the infill well, which is 720 days later than the production time of the parent well. The flow regimes cannot be exhibited by the log-log plots if the production time of the parent well is used, just like the late flow regimes shown in Figure 5(a).Figure 5
The gas production comparison results for the two-well interference case.
(a)
Results for the parent well(b)
Results for the infill well


### 3.2. Synthetic Case Study

In this section, we will analyze the infill well-caused well interference on the production performance of both the parent and infill wells. The analyzed parameters include the numbers of the connected fractures between the parent and infill wells, the length of the unconnected fractures for the infill well, the fracture numbers of the infill well, the permeability of the new SRV region between the parent and infill wells, and the well infill time. In this section, the fracture and SRV properties of the parent well are assumed to be the same for all the studied cases. Table4 shows the analyzed parameters, and other parameters are shown in Table 3. One should note that the values in bold font in Table 4 are the default value for the parameters.Table 4
The analyzed parameters for the synthetic cases.
ParameterValueParameterValueFracture half-length of the parent well (m)120Number of fractures for the parent well40Permeability of the near-well SRV (mD)1×10−4Infill time (day)360/540/720/900Number of connected fractures10/20/30/40Number of fractures for the infill well20/30/40/50Permeability of the distant well SRV (×10−5 mD)1,3, 6, 10Fracture half-length of the infill well (m)40/80/120/160


#### 3.2.1. The Effect of the Connected Fracture Numbers

Figure6 shows the effects of connected fracture numbers on the production of the parent and infill wells. The results show that the production of the parent well will be significantly enhanced when it gets fracture interference from the infill well. This is because of the gas contribution from the far well SRV, which is generated after the hydraulic fracturing of the infill well. The results also show that the production of both parent and infill wells will be higher with more connected fractures at early flow regimes, while lower production rates are observed for late flow regimes. This shows that the remaining gas in the formation will be extracted by the wells earlier with more connected fractures.Figure 6
The effects of connected fracture numbers on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


#### 3.2.2. The Effect of the Length of Unconnected Fractures

In this case, the length of the parent well is given a constant value and the length of the unconnected fractures for the infill well is given as 40, 80, 120, and 160 m to analyze the effects of well interference.As shown in Figure7, the length of the unconnected fractures has a significant influence on the production of both the parent and infill wells. For the infill well, much larger SRV and control volume can be obtained with longer fracture, so the production of the infill well is significantly enhanced. For the parent well, fracture connection can increase the gas production for all the cases but higher production will be obtained with shorter unconnected fractures. This is because a longer fracture of the infill well will increase the near SRV of the infill well and reduce the distant well SRV, which causes the production increase for the parent well.Figure 7
The effects of the unconnected fracture length on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


#### 3.2.3. The Effect of the Fracture Stages/Clusters

Figure8 shows the effects of the fracture stages/clusters of the infill well on shale gas production. The results show that the fracture stages of the infill well have limited influence on the production of the parent well. This is because the connected fracture number is given as 20 for all the scenarios, while this may be not true in the field. Because it is more possible to have more fractures connected with the parent well if more stages/clusters are used in the hydraulic fracturing of the infill well. For the infill well, more fracture stages/clusters will result in higher gas production in the early flow regimes and lower in the late flow regimes. This is quite like the influence of the fracture length shown in Figure 7(b), but the influence of the fracture length is more significant with the studied parameters.Figure 8
The effects of fracture stages/clusters on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


#### 3.2.4. The Effect of the SRV Permeability

In this section, we are focused on analyzing the effects of the permeability of the distant well SRV generated during the stimulation of the infill well. For comparison, the permeability of the near well SRV for both the parent and infill wells is given as1×10−4mD. Figure 9 shows that the influence of the SRV permeability on the parent and infill well is quite similar. In the early flow regimes, larger production rates are obtained with higher SRV permeabilities, while an opposite influence is observed on the curves for the late flow regimes. This is because more gas reserves are produced from the wells with higher SRV permeabilities and left limited remain reserves for the late flow regimes.Figure 9
The effects of the SRV permeability on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


#### 3.2.5. The Effect of the Well Infill Time

In this section, we assume that the infill well starts to fracture and produce at 360, 540, 720, and 900 days after the parent well.Figure10(a) shows that well interference in later flow regimes will result in higher production “pick” when the parent well gets interfered by the infill well. In addition, the amount of production increase will be larger with later fracture interference. This is because the depletion in the control volume of the parent well is more severe and a larger proportion of the distant well SRV will contribute to the parent well. Therefore, in Figure 10(b), we can find that the production of the infill well will be lower when the infill well is added into the well pad at a later time.Figure 10
The effects of the well infill time on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


### 3.3. Field Case Study

To further benchmark the proposed semianalytical model, a field case from Southwestern China is used in the analysis. The parent well is 1450 meters long and hydraulically fractured with 15 stages. After producing for 1484 days, an infill well is fractured and starts to produce. The distance between the parent and infill wells is 300 meters. Figure11 shows the production performance of both the parent and infill wells. Significantly, there is a boom in the gas production for the parent well when the infill well is added to the well pad. Therefore, in this case, we will use the proposed semianalytical model to analyze the production data and inverse the fracture properties and evaluate the inter-well connectivity by using history matching. Some of the formation properties are shown in Table 5.Figure 11
The production performance of the parent and infill wells for the field case.
(a)
Performance of the parent well(b)
Performance of the infill wellTable 5
The formation properties and fitted parameters using the semianalytical model.
ParameterValueParameterValueInitial pressure (MPa)75Formation temperature (K)408Initial water saturation (irreducible)0.56Formation thickness (m)25Matrix permeability (mD)1×10−5Porosity of the matrix0.051Rock compressibility (MPa−1)8×10−5Hydraulic fracture porosity0.3Well space (m)300Permeability of the distant well SRV (mD)5×10−5Langmuir pressure (MPa)5Langmuir volume (m3/t)3∗Hydraulic fracture conductivity (D·cm)0.05∗Number of connected fractures15∗Number of fractures for the parent well15∗Number of fractures for the infill well25∗Half-length of the fractures for the parent well (m)140∗Half-length of the fractures for the infill well (m)90∗Permeability of the near-well SRV (mD)5×10−4∗Contribution ratio to the parent well0.7Because the production data is under variable BHP and rate conditions, we should use the normalized production rate and material balance time to process the production data. The production data obtained using the proposed semianalytical model is also handled using this concept because of the nonlinearity of the semianalytical model. In this paper, both the production data and the semianalytical results are handled in this way. The history matching results are shown in Figure12. For both the parent and infill wells, there are mainly two flow regimes exhibited in the log-log plots. The first flow regime is a bilinear flow regime, which is characterized by a 1/4 slope on the log-log plots. The second flow regime is a boundary-dominated flow regime, which has a slope of 1. Therefore, the approximated expressions, equations (8) and (12), are used in the semianalytical model. The history matching results show that a good match to the field data is obtained. The fitted parameters are shown in Table 5, in which character “∗” is used to mark the fitted parameters. We should admit that this may be not the only one to match the field data because of the uncertainties in the model. It should be noted that the dimensionless fracture conductivity is about 2π, so the early flow regime is bilinear flow. This is following the assumption for using the bilinear flow model. One should also note that the contribution ratio to the parent well, shown in Table 5, is obtained by history matching for field cases.Figure 12
The production analysis results for the parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


## 3.1. Model Validation

In this section, two cases are used to validate the proposed semianalytical model, the single-linear flow case and two-well interference case. In the following, we first use the single-linear flow case shown in Figure2(c) to validate the model in the calculation, and then, a two-well interference case shown in Figure 1(a) is used to validate the model in model assumptions. For the two cases, the reservoir and fracture properties are shown in Table 3 and the gas PVT properties are calculated using the method proposed by Lee et al. [46] and shown in Figure 4(a).Table 3
Parameters for the validation case.
ParameterValueParameterValueInitial pressure (MPa)75Formation temperature (K)408Initial water saturation (irreducible)0.56Formation thickness (m)25Matrix permeability (mD)1×10−5Porosity of the matrix0.051Rock compressibility (MPa−1)8×10−5Hydraulic fracture porosity0.3Hydraulic fracture width (m)0.01Bottom hole pressure (MPa)5Hydraulic fracture permeability (mD)10000Fracture half-length of the infill well (m)82.5Fracture half-length of the parent well (m)122.5Number of fractures for the infill well11Number of fractures for the parent well16Number of connected fractures6Permeability of the region between the two wells (mD)3×10−5Well space (m)300Permeability of the SRV for the parent well (mD)1×10−4Permeability of the SRV for the infill well (mD)6×10−5Figure 4
The procedure to predict the production for a single-linear flow model.
(a)
Gas PVT properties(b)
Validation results for the single-linear flowThe validation result of the single-linear flow case is shown in Figure4(b). We can find that the proposed semianalytical model has a good match with the numerical simulation results. In addition, for this case, there are mainly two flow regimes exhibited by the gas production curves on the log-log plot—the formation linear flow and boundary-dominated flow regimes. This shows that the computation of the semianalytical model is reliable.We further analyzed the validity of the assumptions of the regional linear flow model in handling infill well-caused fracture interference. In this case, the infill well is assumed to start to produce 720 days later than the parent well. Figure5 shows the validation result, which shows that good matches are obtained for both the parent and infill wells. Figure 5(a) shows that the parent well has been in the boundary-dominated flow regime when it gets interfered by the infill well. It should be noted that the flow regimes after the infill time are composite for the parent well. The flow for the old SRV, the xf1 part shown in Figures 2(a) and 2(b), is under the boundary-dominated flow regime, while the new SRV generated by the fracturing of the infill well is under a linear flow regime. Two methods can be used to analyze this combined flow regime. The first method is by history matching, which will be presented in Section 3.3. The other method is by straight line analysis after processing the production data by deducting the contribution from the near-well SRV, but this will be our future work. One should also notice that the time shown in Figure 5(b) is the production time of the infill well, which is 720 days later than the production time of the parent well. The flow regimes cannot be exhibited by the log-log plots if the production time of the parent well is used, just like the late flow regimes shown in Figure 5(a).Figure 5
The gas production comparison results for the two-well interference case.
(a)
Results for the parent well(b)
Results for the infill well


## 3.2. Synthetic Case Study

In this section, we will analyze the infill well-caused well interference on the production performance of both the parent and infill wells. The analyzed parameters include the numbers of the connected fractures between the parent and infill wells, the length of the unconnected fractures for the infill well, the fracture numbers of the infill well, the permeability of the new SRV region between the parent and infill wells, and the well infill time. In this section, the fracture and SRV properties of the parent well are assumed to be the same for all the studied cases. Table4 shows the analyzed parameters, and other parameters are shown in Table 3. One should note that the values in bold font in Table 4 are the default value for the parameters.Table 4
The analyzed parameters for the synthetic cases.
ParameterValueParameterValueFracture half-length of the parent well (m)120Number of fractures for the parent well40Permeability of the near-well SRV (mD)1×10−4Infill time (day)360/540/720/900Number of connected fractures10/20/30/40Number of fractures for the infill well20/30/40/50Permeability of the distant well SRV (×10−5 mD)1,3, 6, 10Fracture half-length of the infill well (m)40/80/120/160


### 3.2.1. The Effect of the Connected Fracture Numbers

Figure6 shows the effects of connected fracture numbers on the production of the parent and infill wells. The results show that the production of the parent well will be significantly enhanced when it gets fracture interference from the infill well. This is because of the gas contribution from the far well SRV, which is generated after the hydraulic fracturing of the infill well. The results also show that the production of both parent and infill wells will be higher with more connected fractures at early flow regimes, while lower production rates are observed for late flow regimes. This shows that the remaining gas in the formation will be extracted by the wells earlier with more connected fractures.Figure 6
The effects of connected fracture numbers on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


### 3.2.2. The Effect of the Length of Unconnected Fractures

In this case, the length of the parent well is given a constant value and the length of the unconnected fractures for the infill well is given as 40, 80, 120, and 160 m to analyze the effects of well interference.As shown in Figure7, the length of the unconnected fractures has a significant influence on the production of both the parent and infill wells. For the infill well, much larger SRV and control volume can be obtained with longer fracture, so the production of the infill well is significantly enhanced. For the parent well, fracture connection can increase the gas production for all the cases but higher production will be obtained with shorter unconnected fractures. This is because a longer fracture of the infill well will increase the near SRV of the infill well and reduce the distant well SRV, which causes the production increase for the parent well.Figure 7
The effects of the unconnected fracture length on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


### 3.2.3. The Effect of the Fracture Stages/Clusters

Figure8 shows the effects of the fracture stages/clusters of the infill well on shale gas production. The results show that the fracture stages of the infill well have limited influence on the production of the parent well. This is because the connected fracture number is given as 20 for all the scenarios, while this may be not true in the field. Because it is more possible to have more fractures connected with the parent well if more stages/clusters are used in the hydraulic fracturing of the infill well. For the infill well, more fracture stages/clusters will result in higher gas production in the early flow regimes and lower in the late flow regimes. This is quite like the influence of the fracture length shown in Figure 7(b), but the influence of the fracture length is more significant with the studied parameters.Figure 8
The effects of fracture stages/clusters on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


### 3.2.4. The Effect of the SRV Permeability

In this section, we are focused on analyzing the effects of the permeability of the distant well SRV generated during the stimulation of the infill well. For comparison, the permeability of the near well SRV for both the parent and infill wells is given as1×10−4mD. Figure 9 shows that the influence of the SRV permeability on the parent and infill well is quite similar. In the early flow regimes, larger production rates are obtained with higher SRV permeabilities, while an opposite influence is observed on the curves for the late flow regimes. This is because more gas reserves are produced from the wells with higher SRV permeabilities and left limited remain reserves for the late flow regimes.Figure 9
The effects of the SRV permeability on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


### 3.2.5. The Effect of the Well Infill Time

In this section, we assume that the infill well starts to fracture and produce at 360, 540, 720, and 900 days after the parent well.Figure10(a) shows that well interference in later flow regimes will result in higher production “pick” when the parent well gets interfered by the infill well. In addition, the amount of production increase will be larger with later fracture interference. This is because the depletion in the control volume of the parent well is more severe and a larger proportion of the distant well SRV will contribute to the parent well. Therefore, in Figure 10(b), we can find that the production of the infill well will be lower when the infill well is added into the well pad at a later time.Figure 10
The effects of the well infill time on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


## 3.2.1. The Effect of the Connected Fracture Numbers

Figure6 shows the effects of connected fracture numbers on the production of the parent and infill wells. The results show that the production of the parent well will be significantly enhanced when it gets fracture interference from the infill well. This is because of the gas contribution from the far well SRV, which is generated after the hydraulic fracturing of the infill well. The results also show that the production of both parent and infill wells will be higher with more connected fractures at early flow regimes, while lower production rates are observed for late flow regimes. This shows that the remaining gas in the formation will be extracted by the wells earlier with more connected fractures.Figure 6
The effects of connected fracture numbers on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


## 3.2.2. The Effect of the Length of Unconnected Fractures

In this case, the length of the parent well is given a constant value and the length of the unconnected fractures for the infill well is given as 40, 80, 120, and 160 m to analyze the effects of well interference.As shown in Figure7, the length of the unconnected fractures has a significant influence on the production of both the parent and infill wells. For the infill well, much larger SRV and control volume can be obtained with longer fracture, so the production of the infill well is significantly enhanced. For the parent well, fracture connection can increase the gas production for all the cases but higher production will be obtained with shorter unconnected fractures. This is because a longer fracture of the infill well will increase the near SRV of the infill well and reduce the distant well SRV, which causes the production increase for the parent well.Figure 7
The effects of the unconnected fracture length on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


## 3.2.3. The Effect of the Fracture Stages/Clusters

Figure8 shows the effects of the fracture stages/clusters of the infill well on shale gas production. The results show that the fracture stages of the infill well have limited influence on the production of the parent well. This is because the connected fracture number is given as 20 for all the scenarios, while this may be not true in the field. Because it is more possible to have more fractures connected with the parent well if more stages/clusters are used in the hydraulic fracturing of the infill well. For the infill well, more fracture stages/clusters will result in higher gas production in the early flow regimes and lower in the late flow regimes. This is quite like the influence of the fracture length shown in Figure 7(b), but the influence of the fracture length is more significant with the studied parameters.Figure 8
The effects of fracture stages/clusters on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


## 3.2.4. The Effect of the SRV Permeability

In this section, we are focused on analyzing the effects of the permeability of the distant well SRV generated during the stimulation of the infill well. For comparison, the permeability of the near well SRV for both the parent and infill wells is given as1×10−4mD. Figure 9 shows that the influence of the SRV permeability on the parent and infill well is quite similar. In the early flow regimes, larger production rates are obtained with higher SRV permeabilities, while an opposite influence is observed on the curves for the late flow regimes. This is because more gas reserves are produced from the wells with higher SRV permeabilities and left limited remain reserves for the late flow regimes.Figure 9
The effects of the SRV permeability on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


## 3.2.5. The Effect of the Well Infill Time

In this section, we assume that the infill well starts to fracture and produce at 360, 540, 720, and 900 days after the parent well.Figure10(a) shows that well interference in later flow regimes will result in higher production “pick” when the parent well gets interfered by the infill well. In addition, the amount of production increase will be larger with later fracture interference. This is because the depletion in the control volume of the parent well is more severe and a larger proportion of the distant well SRV will contribute to the parent well. Therefore, in Figure 10(b), we can find that the production of the infill well will be lower when the infill well is added into the well pad at a later time.Figure 10
The effects of the well infill time on the gas production of both parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


## 3.3. Field Case Study

To further benchmark the proposed semianalytical model, a field case from Southwestern China is used in the analysis. The parent well is 1450 meters long and hydraulically fractured with 15 stages. After producing for 1484 days, an infill well is fractured and starts to produce. The distance between the parent and infill wells is 300 meters. Figure11 shows the production performance of both the parent and infill wells. Significantly, there is a boom in the gas production for the parent well when the infill well is added to the well pad. Therefore, in this case, we will use the proposed semianalytical model to analyze the production data and inverse the fracture properties and evaluate the inter-well connectivity by using history matching. Some of the formation properties are shown in Table 5.Figure 11
The production performance of the parent and infill wells for the field case.
(a)
Performance of the parent well(b)
Performance of the infill wellTable 5
The formation properties and fitted parameters using the semianalytical model.
ParameterValueParameterValueInitial pressure (MPa)75Formation temperature (K)408Initial water saturation (irreducible)0.56Formation thickness (m)25Matrix permeability (mD)1×10−5Porosity of the matrix0.051Rock compressibility (MPa−1)8×10−5Hydraulic fracture porosity0.3Well space (m)300Permeability of the distant well SRV (mD)5×10−5Langmuir pressure (MPa)5Langmuir volume (m3/t)3∗Hydraulic fracture conductivity (D·cm)0.05∗Number of connected fractures15∗Number of fractures for the parent well15∗Number of fractures for the infill well25∗Half-length of the fractures for the parent well (m)140∗Half-length of the fractures for the infill well (m)90∗Permeability of the near-well SRV (mD)5×10−4∗Contribution ratio to the parent well0.7Because the production data is under variable BHP and rate conditions, we should use the normalized production rate and material balance time to process the production data. The production data obtained using the proposed semianalytical model is also handled using this concept because of the nonlinearity of the semianalytical model. In this paper, both the production data and the semianalytical results are handled in this way. The history matching results are shown in Figure12. For both the parent and infill wells, there are mainly two flow regimes exhibited in the log-log plots. The first flow regime is a bilinear flow regime, which is characterized by a 1/4 slope on the log-log plots. The second flow regime is a boundary-dominated flow regime, which has a slope of 1. Therefore, the approximated expressions, equations (8) and (12), are used in the semianalytical model. The history matching results show that a good match to the field data is obtained. The fitted parameters are shown in Table 5, in which character “∗” is used to mark the fitted parameters. We should admit that this may be not the only one to match the field data because of the uncertainties in the model. It should be noted that the dimensionless fracture conductivity is about 2π, so the early flow regime is bilinear flow. This is following the assumption for using the bilinear flow model. One should also note that the contribution ratio to the parent well, shown in Table 5, is obtained by history matching for field cases.Figure 12
The production analysis results for the parent and infill wells.
(a)
Results for the parent well(b)
Results for the infill well


## 4. Conclusions

In this paper, a practical semianalytical model is proposed to analyze the infill well-caused fracture interference on the gas production performance. Based on the studies in this paper, the following conclusions are guaranteed:(1)
Strict analytical models cannot handle the problem of infill well-caused fracture interference because the parent and infill wells are not in production at the same time, which makes the analytical model nonlinear and nonhomogeneous. The validation over the numerical simulator shows that this problem can be handled by dividing the whole model into several linear flow regions and the production of the wellbore can be obtained by adding the production contribution from all the parts(2)
When two wells are connected with high-conductivity fractures, the production of the parent well will significantly increase when the parent well gets interfered. The production rates of the wells are influenced by the fracture parameters, including the numbers of connected fractures, the length of the unconnected fractures, and fracturing stages/clusters for the infill well. Although similar effects are observed on the production rates curves, the length of the fractures and fracturing stages/clusters are particularly significant(3)
The production is also affected by the SRV permeability and well infill time, which influence the production contribution from the distant well SRV by changing the properties and the production contribution ratio to the parent well. The production increase of the parent well upon getting fracture interference will be more significant in late time infill well production(4)
The normalized production rate and material balance time can be used to analyze the production data for multiwell cases with fracture connections. The interwell connection conditions and the fracture properties of the parent and infill wells can be interpreted with the proposed semianalytical model



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*Source: 1877778-2021-11-01.xml*