A SemiAnalytical PressureTransient Model to Detect Interwell Interference of MultiWellPadProduction Scheme in Shale Gas Reservoirs
^{1}
College of Petroleum Engineering, China University of Petroleum,
Beijing
102249 China
^{2}
Research Institute of Petroleum Exploration and Development, CNPC,
Beijing
100083 China
^{*} Corresponding author email: C.Xiao@tudelft.nl
Received:
30
October
2016
Accepted:
26
September
2017
Recently, MultiWellPadProduction (MWPP) scheme has been in the center of attention as a promising technology to improve Shale Gas (SG) recovery. However, InterWell Pressure Interference (IWPI) induced by MWPP scheme severely distorts flow regimes, which strongly challenges the traditional pressuretransient analysis methods, which focus on Single MultiFractured Horizontal Wells (SMFHW) without IWPI. Therefore, a methodology to identify pressuretransient response of MWPP scheme without and with IWPI is urgent. To fill this gap, by utilizing superposition theory, Gauss elimination and Stehfest numerical algorithm, the pressuretransient solution of MWPP scheme was established, as a result, type flow regimes can be identified by considering MWIP. Our results show that our proposed model demonstrates promising calculation speed and acceptable accuracy compared to numerical simulation. Part of flow regimes are significantly distorted by IWPI. In addition, well rate mainly determines the distortion of pressure curves, while fracture length, well spacing, fracture spacing mainly determine when the IWPI occurs. The smaller the gas rate, the more severely flow regimes are distorted. As the well spacing increases, fracture length decreases, fracture spacing decreases, occurrence of IWPI becomes later. Stress sensitivity coefficient approximately has no influences on distortion of pressure curves and occurrence of IWPI. This work gains some additional insights on pressuretransient response for MWPP scheme in SG reservoir, which can provide considerable guidance on fracture properties estimation as well as well pattern optimization for MWPP scheme.
© L. Tian et al., published by IFP Energies nouvelles, 2018
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Introduction
Developing unconventional resource has challenged conventional methodologies and engineers for the past few decades. Recently, advances in technology and new analysis methods have made Single MultiFractured Horizontal Wells (SMFHW) to become more economical. One of the major challenges is the determination of optimal stage and well spacing in a specific drilling area. Rate transient analysis is commonly used to assess the effectiveness of fracture stimulation of production wells, which is used to determine additional optimization of well/stage spacing. Beside rate transient analysis method, pressuretransient analysis of horizontal well with multistage fractures is of importance as well in reservoir engineering because it provides valuable information concerning well completion, well placement, well spacing, calculation of dynamic reserve, and most importantly, different flow regimes characterized by pressuretransient behavior provide estimations of the insitu reservoir properties and fracture geometry and conductivity.
The presence of complex fracture topology has significant impacts on establishment of pressuretransient model for unconventional reservoirs (Khvoenkova and Delorme, 2011; Baroni et al., 2015; Noetinger, 2015). Originally, some analytical approaches have been used to model the transient flow behavior in such systems. With assuming uniform distribution of identical HF along horizontal well, Ozkan and Raghavan (1991) and Ozkan et al. (2011) utilized the concept of Lee and Brockenbrough (1986) of trilinear model with inner reservoir of naturally fractured to represent the SMFHW performance in unconventional reservoirs. Brown et al. (2011) presented an analytical trilinear flow model that incorporates transient fluid transfer from matrix to fracture to simulate the pressure transient and production behavior of fractured horizontal wells in unconventional reservoirs. However, those proposed analytical models are unable to explicitly represent the Hydraulic Fracture Networks (HFNW) system induced by interaction between Main Hydraulic Fractures (MHF) and Secondary Hydraulic Fractures (SHF). To fill this gap, Jones et al. (2013), Mirzaei and Cipolla, 2012, Cipolla and Wallace, 2014, Farah and Ding (2016) used an unstructuredgrid simulator to analyze the type curves of HFNW system. This latest unstructuredgrid technology can make the simulation of fracture complexity more accurate by refining the vicinity of the high conductivity fractures network with fine grids. However, it is inevitable to increase the complication and economical consumption of computation. Subsequently, several ingenious semianalytical methods have been proposed to overcome the shortage of inaccurate analytical methods and timeconsuming numerical methods. Zhou et al. (2014) and Yu et al. (2015) combined analytical reservoir solutions with numerical fracture network solution to characterize fracture network complexity. Similarly, Jia et al. (2015) and Chen et al. (2016) utilized startransformation (KarimiFard et al., 2003) and boundary element integration methods to characterize fluid seepage within complex fracture network (induced hydraulic fractures and discrete natural fractures). These semianalytical models provided quick insights into fracturenetwork performance and formed foundations to efficiently and accurately analyze transient pressure response.
Recently, MultiWellPadProduction (MWPP) scheme has been in the center of attention as a promising technology to economically improve SG recovery (Awada et al., 2015; Guindon, 2015). Microseismic fracturing mapping shows hydraulic fractures extending between wells, gaining the existence of InterWell Pressure Interference (IWPI) (Farley and Hutchinson, 2014; Sardinha et al., 2014). Although recognizing IWPI within MWPP scheme can provide valuable insights to gain a better understanding of the fracture design (Soroush et al., 2013; Kaviani et al., 2010), enhancement of the possibilities of IWPI in MWPP scheme severely distorts flow regimes, which increases the burden of parameter estimations (Awada et al., 2015). Consequently, this technology strongly challenges the traditional pressuretransient analysis methods mentioned above, which focus on SMFHW without pressure interference. Therefore, a methodology to identify pressuretransient response of MWPP scheme without and with IWPI is of significance for reservoir engineers.
In this work, our objectives are to obtain better understanding of IWPI in MWPP scheme by addressing the following questions:

What kinds of flow regimes for MWPP scheme can we obtain? And what is the difference between MWPP scheme and SMFHW in flow regimes?

How to identify IWPI for MWPP scheme based on obtained flow regimes?
To answer questions outlined above, following three issues should be solved sequentially and systematically:

Developing an efficient semianalytical mathematical model for MWPP scheme;

Based on proposed semianalytical pressuretransient model, establishing methodology to identify different flow regimes in MWPP scheme;

Based on proposed semianalytical pressuretransient model, analyzing influences of key parameters on flow regimes.
Described below are the attributes of our methodology framework. Section 2 describes the conceptual model of HFNW system in MWPP scheme and development of semianalytical pressuretransient model in detail. Section 3 systematically implemented model validation, methodology to identify the IWPI and propose some new insights into flow regimes in MWPP scheme. Finally, Section 4 summarizes our contribution and promising work in the future. More information on model derivation can be found in Appendix section.
1 Development of SemiAnalytical PressureTransient Model
Single phase gas is assumed to derive the semianalytical model. We envisage two distinct flow regimes governed by different physics: SG reservoir flow and HFNW system flow. In the following sections, we first describe the conceptual model of SG formation and MWPP scheme used for presentation of our approach and then mathematically model these two flow processes, respectively. Finally, SG reservoir flow model and HFNW system flow model are coupled dynamically.
1.1 Conceptual Model
1.1.1 MultiWellPadProduction (MWPP) Scheme
Figure 1a illustrates the placement of MWPP schemes in Shale Gas (SG) reservoir. We mainly focus on single MWPP scheme at the earlyintermediate production period. Each MWPP scheme contains several SMFHW. Figures 1b and c illustrate the microseismic surveillance within one MWPP scheme. As Figure 1b, the hydraulic fractures system can be idealized to several regular fractures. As Figure 1c, the complex hydraulic fracture network system has been formed. Therefore, a general HFNW is idealized to characterize both of these situations (as shown in Fig. 1d), and the corresponding interference can be classified into following two types (Awada et al., 2015):

Interference through HFNW. Interference directly through connected HFNW refers to communication in SG reservoir when HFNW connection is created between two wells, as illustrated in Figure 1d(2);

Interference through reservoir. Interference through the SG reservoir would occur when HFNW are not directly connected between wells but are in close proximity. In Figure 1d, when the fracture conductivity of green zone tends to be zero, it shows the fracture configuration where communication through SG reservoir rock may be observed as illustrated in Figure 1d(1).
Fig. 1
The schematic illustration of a MWPP scheme in SG reservoir, a) is layout of MWPP scheme in a SG field, b) and c) are two possible scenarios of MWPP scheme, d) is idealizations of HFNW system. 
Fig. 2
The schematic illustration of discretized HFNW system in the conceptual model. 
1.1.2 SG Reservoir Model
The conceptual model of a SG formation can be described as follows: The reservoir, within infinite boundary, is treated as a 2D flat. SG reservoir is assumed to be isotropic, including Natural Fractures (NF) system and matrix system, and bounded by upper and lower impermeable strata. Three media exist in the SG reservoir: (1) the lowest permeable shale matrix, (2) the moderate permeable NF, and (3) the highest permeable HFNW connecting to wellbore.
To conveniently describe our methodology, we chose two wells to be our research objective. Two wells produced at constant gas rate q_{1} and q_{2}. Fluid in SG reservoir flows into HFNW at varying flowrate strength q_{f} along fractures. In addition, some other assumptions are made as follows:

Two horizontal wells are intercepted by HFNW. The HFNW are assumed to fully penetrate SG reservoir;

The HFNW of two wells has different fracture conductivity;

SG reservoir has uniform thickness h. The initial pressure is P_{i}; and the initial temperature is T;

Gas seepage within NF system meets Darcy's law. Gas unsteadystate diffusion within matrix is assumed to obey Second Fick's law. NF system is stressdependent with initial permeability k_{ri};

Compressibility coefficient of the slightly compressible SG is constant;

Impacts of gravity and capillary pressure are neglected;

Gas absorption and adsorption meets Langmuir isotherm equation;

Wellbore storage and skin factor are considered;

No frictional pressure loss inside the wellbore is considered.
1.2 Mathematical Model
1.2.1 Hydraulic Fracture Networks Discrete
As Figure 2, we further classify the HFNW into MHF and SHF. The properties of MHF and SHF in HFNW for well1 include: permeability, k_{mf}_{1}, k_{sf}_{1}; fracture width, w_{mf}_{1}, w_{sf}_{1}; width of HFNW, W_{f}_{1}, halflength of HFNW, L_{f}_{1}. The properties of MHF and SHF in HFNW for well2 include: permeability, k_{mf}_{2}, k_{sf}_{2}; fracture width, w_{mf}_{2}, w_{sf}_{2}; width of HFNW, W_{f}_{2}, halflength of HFNW, L_{f}_{2}. The distance between two wells is L_{w}. The vertical distance of HFNW for two wells is L_{vf}_{12}. The horizontal distance of HFNW for two wells is L_{hf}_{12}. To establish a mathematical model, we first subdivide the HFNW systems. HFNW of well1 is divided into N_{1} subfracture segments, NFNW of well2 is divided into N_{2} subfracture segments. The hydraulic fracture number of well1 is M_{1}. The hydraulic fracture number of well2 is M_{2}. The length of subfracture segment of well1 and well2 can be presented as ΔL_{f1}, ΔL_{f2}, respectively. We can summarize fracture properties as follows:

Well1: N_{1}, M_{1}, L_{f}_{1}, ΔL_{f1}, W_{f}_{1}, k_{mf}_{1}, k_{sf}_{1}, w_{mf}_{1}, w_{sf}_{1}

Well2: N_{2}, M_{2}, L_{f}_{2}, ΔL_{f2}, W_{f}_{2}, k_{mf}_{2}, k_{sf}_{2}, w_{mf}_{2}, w_{sf}_{2}
1.2.2 Model of SG Flow in SG Reservoir System
To develop mathematical models in SG reservoir, the formula of NF system and matrix system can be established separately and then dynamically coupled (Tian et al., 2014, 2016; Wang, 2014; Liu et al., 2015). By applying the principle of integration, the pressure distribution of a random position (x_{D}, y_{D}) caused by one fracture segment (x_{WD}, y_{WD}) is given as follows (more information on dimensionless definition and model derivation can be found in Appendix A and Appendix B), (1) where
As one can seen from Figure 2, the HFNW system has been divided subfracture segments. According to Equation (1), we can obtain transient pressure response at the center of each segment of HFNW system in MWPP scheme by the superposition principle: (2) where (3) (4)
is the pressure response at the oth fracture segment, caused by the flux of bth fracture segment in ath hydraulic fracture for well1; is the pressure response at the oth fracture segment, caused by the flux of jth fracture segment in ith hydraulic fracture for well2.
1.2.3 Model of SG Flow in Hydraulic Fracture Networks System
Currently, modeling of fluid flow in HFNW system mainly focus on two issues: flow state (compressible and incompressible flow) within independent fracture segment and fluid transfer at connecting point of two crossing fracture segments. For former issue about flow state, Jia et al. (2015), Zeng et al. (2012) analyzed the unsteady state flow with considering fluid compressibility, while Chen et al. (2016) and Zhou et al. (2014) analyzed the pseudosteady state flow without considering fluid compressibility. In our paper, we establish a generic model by considering the fluid compressibility.
For the later issue about fluid transfer at interacting point of HFNW system. Zhou et al. (2014) and Chen et al. (2016) artificially determined the flow direction, while Jia et al. (2015) employed startransformation to automatically determine flow direction. In this paper, the orthogonal hydraulic fractures system is idealized to characterize the geometry of HFNW, therefore, the startransformation is utilized to automatically determine flow direction and solve the issue of fluid transfer at interacting point of HFNW system.
Independent Fracture Segment. Here, semianalytical method (Zeng et al., 2012; Zhou et al., 2014; Chen et al., 2016) is applied to develop the model of independent fracture segment considering finite hydraulic fracture conductivity. Zeng et al. (2012) proposed a semianalytical method which divided fracture system into several segments and each segment was still solved analytically. As one can see in Figure 3, we can chose ith fracture segment to analyze the flow equation. Fluid flow from position l_{D1} to l_{D2}, because of the fluid supplement of segment flux q_{fi}, rate increases from q_{ci}_{1} to q_{ci}_{2} along l.
We take ith fracture segment as an example, the solutions of other fracture segment are similar. Deriving from the semianalytical method proposed by Zeng et al. (2012), the governing equation of the ith independent fracture segment in Laplace space can be established as follows: (5)
Boundary conditions at ϵ_{i1} and ϵ_{i1} can be given by (6) (7)
Combining with Equations (5) and (6), we can obtain the pressure distribution within the ith.
Independent Fracture Segment. The solution in Laplace space for ith fracture segment can be mathematically characterized by flow rate q_{ci}_{1}, q_{ci}_{2} at both sides of the ith fracture segment and the fluid influx q_{fi} from SG reservoir into this fracture segment. (8)
_{Therefore, pressure at the center of the ith fracture segment} (9) _{where} (10) (11) (12)
Connecting Fracture Segment. Here, the StarDelta transformation is adopted to solve the fluid transfer at the connecting fracture segment. Figure 4 depicts the transformation for four interconnected fracture segments. KarimiFard et al. (2003) used this transformation to eliminate intermediate control volume in discrete fracture network simulation. Taking the StarDelta transformation shown in Figure 4 and eliminate intersection of cell 0, we make the four segments connecting directly. The transmissibility between two adjacent segments can be written as, (13) where, T_{Di}_{,0} is the dimensionless transmissibility between ith fracture segment and intersection 0. (14)
Fracture segment 4 in Figure 4 is taken as an example to illustrate the application of StarDelta transformation. The flow equation of segment 4 can be given by (15)
Furthermore, Equation (15) can change as the following form after the transformation (16)
Fig. 3
Illustration of gas flow within ith fracture segment. 
Fig. 4
Illustration of StarDelta transformation. 
1.3 Solution of TransientPressure for MWPP Scheme
Considering the proposed equations, there are three unknowns, , for each fracture segment, additional two unknowns are bottomhole pressure of two wells. Therefore, the total number of unknowns is equal to [3(M_{1} × N_{1} + M_{2} × N_{2}) + 2]. A closed [3(M_{1} × N_{1} + M_{2} × N_{2}) + 2]order matrix from the following conditions can be obtained.
Combing Equations (2) and (9), pressure continuity between HFNW system and SG reservoir can be satisfied at the center of ith fracture segment, (17)
After rewriting Equation (17) at each fracture segment, we can obtain totally (M_{1} × N_{1} + M_{2} × N_{2})equations.
For two connected fracture segments, ith segment and (i+1)th segment, the pressure and fluid rate at connecting point respectively are equal to each other, (18) (19)
After rewriting Equations (18) and (19) for each fracture segment, we can obtain another totally 2(M_{1} × N_{1} + M_{2} × N_{2})equations.
Another two equations are required to form a closed matrix. Well1 and well2 are producing at constant rate q_{1} and q_{2}, respectively. Here, we define a new variable ε, represents the ratio between q_{1} and (q_{1} + q_{2}), namely, ϵ = q_{1}/(q_{1} + q_{2}). Then, (20) (21)
Finally, a closed [3(M_{1} × N_{1} + M_{2} × N_{2}) + 2]order matrix is formed. By applying Gauss elimination and Stehfest numerical algorithm (Stehfest, 1970), the bottomhole pressure solution, and , can be solved. In Laplace domain, the wellbore storage effects can be easily added into the solution with Duhamel's theorem. For the definition of dimensionless pseudopressure (Eq. A1), the solution can be formulated as follows: (22) (23) where, CD1, CD2 are dimensionless wellbore storage coefficient for well1 and well2 which are defined as
By the Stehfest numerical invention algorithm (Stehfest, 1970), the solution in real space can be obtained. One can get the bottomhole pressure for MWPP scheme in SG reservoir by taking the stress sensitivity of NF into consideration. (24) (25)
2 Results and Discussion
In this section, four aspects will be systematically analyzed: (1) influence of fracture discretization level on pressure response; (2) model validation by comparison between semianalytical model and fully numerical simulation; (3) identification of flow regimes based on special pressuretransient characteristics; and (4) sensitivity analysis of pressure response to fracture parameters. The relevant parameters are shown in Table 1.
The basic input parameters in numerical simulation.
2.1 Discretization Level of HFNW System
Due to the inappropriate discretization of HFNW system, early transient pressure response can be distorted by artifacts of fracture subdivision (Chen et al., 2016). Our approach is partially dominated by the discretization level of HFNW system. The basic dimensionless parameters are as follows: C_{1D} = C_{2D} = 0, ζ_{D} = 0, λ = 0.002, γ = 0.15, ω = 0.0035, C_{ηD} = 10^{5}, L_{f}_{1D} = 250, L_{f}_{2D} = 250, W_{f}_{1D} = 500, W_{f}_{2D} = 500, L_{wD} = 5000, L_{vf}_{12D} = 4500, L_{hf}_{12D} = 0, C_{f}_{1D} = 125, C_{f}_{2D} = 125. In Figure 5, ψ_{wD} represents the Dimensionless PseudoPressure (DPP), dψ_{wD}/d(ln t_{D}) represents the Dimensionless PseudoPressure Derivative (DPPD). As Figure 5, when the number of fracture segments, M, is larger than 12 for each hydraulic fracture, the calculating results do not change appreciably. Therefore, each fracture is divided into 12 segments in our work, including for the model validation and sensitivity analysis.
Fig. 5
Sensitivity analysis on the number of divided fracture segments. 
2.2 Model Validation
To our best knowledge, so far, no semianalytical pressuretransient models for multiwells have been developed. Therefore, a fully numerical simulation, modeled by a commercial simulator CMGGEM module, is utilized to validate our proposed model for MWPP scheme. The top view is shown in Figure 6. Three representative HFNW system are modeled: Figure 6a models the nonconnection between HFNW system of the two wells, Figure 6b models the direct connection between HFNW system of the two wells. Figure 6c models the regular HFNW with transverse MHF and without SHF. The input data are listed in Table 1. Figure 6d illustrates one representative segment to represent one part of the reservoir volume around a hydraulic fracture. Four MHF are orthogonal to the horizontal well at 250 m fracture spacing. Other SHF are orthogonal to main fractures. The model is a 2D model with 100 grid cells in the xdirection, 50 grid cells in ydirection and only one grid cell in the zdirection. The multiporosity model and Multiple INteracting Continua (MINC) method are applied to subdivide the matrix so that the transient diffusion in matrix can be simulated. Desorption phenomenon is characterized to be instant desorption model. DKLSLGR technology is employed to characterize HFNW system. Each hydraulic fracture is represented by a 3 × 3 locally refined grid with 0.025mwide. Then, the pressure solution is calculated under a constant production rate. We assume that ratio between q_{1} and q_{2} is 1:4, and the fracture properties of two wells are consistent. After that, the numerical solutions are compared with semianalytical results calculated in this paper. As we can see in Figure 7, there is a good agreement between our results and CMG's, which indicates that our model is reliable.
Fig. 6
Top view of the numerical model of Case II in CMGGEM module: a) nonconnection between HFNW system, b) direct connection between HFNW system, c) regular HFNW with transverse MHF and without SHF, d) Grid refinement for hydraulic fractures. 
Fig. 7
Comparison of our results of model with that of CMG simulator: a) HFNW system as Figure 6(a), b) HFNW system as Figure 6(b), c) HFNW system as Figure 6(c). 
2.3 Identification of Flow Regime
The main goal of our research is identifying flow regimes of MWPP scheme in SG reservoir. As the numerical validation section, we model three different kinds of HFNW system. For the previous two HFNW system, currently, it is still not very common to comprehensively describe the flow regimes for complex HFNW system although several contributions have been made. Chen et al. (2016) and Jia et al. (2015) separately developed mathematical models to identify the flow regimes in complex Hens system. Besides wellknown classic flow regimes (linearflow, bilinear flow regimes, etc.), the authors also added some new flow regimes by themselves. For example, “fluid feed” flow regime is induced by fluid transfer between MHF and SHF. Pseudo Boundary Dominated Flow (PBDF) is induced the permeability contrast between ultralow permeable SG reservoir and highpermeable Hens system. Figure 8 shows the pressure distribution within MWPI at different production time. As Figure 8, it is apparently to observe that the IWPI indeed occurs for these three kinds of Hens system. To obtain some practical and generic analysis, the flow regimes of Hens with transverse MHF and without SHF are identified in our work as Figure 6c.
We can set some common parameters for those two wells: M_{1} = 4, M_{2} = 4, ζ_{D} = 0.05, λ = 0.002, γ = 0.15, ω = 0.0035, C_{ηD} = 10^{5}. Setting consistent MHF properties (conductivity and halflength) L_{f}_{1D} = L_{fD}_{2} = 200, C_{f}_{1D} = C_{f}_{2D} = 10. Another two parameters, L_{hf}_{12D} and L_{vf}_{12D}, determine the different interference type illustrated as Figure 9, which can be mathematically described as follows:

Figure 9(1): L_{vf}_{12D} = L_{f}_{1D} + L_{fD}_{2} + 100 = 500, and L_{hf}_{12D} = 0;

Figure 9(2): L_{vf}_{12D} = L_{f}_{1D} + L_{fD}_{2} + 100 = 500, and L_{hf}_{12D} = 500.
DPP and the DPPD of WIPS scheme are shown in Figure 10. To clearly describe the type curves, we will compare the characteristics of pressure curves between WIPS and SMFHW. By comparing between SMFHWs and WIPS scheme, we can add some additional information and better explanation into these distorted flow regimes, which can be described in detail as follows:
Regime I: The pure wellbore storage period regime. DP curve and DPD curve align, and the slope of curves are equal to 1. This stage is mainly controlled by wellbore storage effect and difficult to be impacted by the MWPI. Thus, the type curve of WIPS and SMFHW overlap with each other.
Regime II: The transition flow regime. The early stage of this regime gradually derives from the straight line which has unit slope. This stage is mainly controlled by fluid properties and also difficult to be impacted by MWPI. Thus, the type curve of WIPS and SMFHW also overlap with each other.
Regime III: The linear flow regime within HFs. This stage is mainly dominated by fracture conductivity. At this linear flow regime, and also difficult to be impacted by MWPI. Thus, the type curve of WIPS and SMFHW also overlap with each other.
Regime IV: The bilinear flow regime. This stage is mainly controlled by fracture length. At this bilinear flow regime, we can start to detect the MWPI for type Figure 1(b1). The slope of the DPD curves is actually bigger than 0.25. The distortion degree of pressure curve for small gas rate is also more significant than that of big gas rate. But, it is still difficult to be impacted by the MWPI for type Figure 1(b2), Thus, the type curve of WIPS and SMFHWs still overlap with each other.
Regime V: The early pseudoradial flow regime. This stage is mainly dominated by fracture spacing. At this pseudoradial flow regime, we can start to detect the MWPI for type Figure 1(b2). The slope of the DPPD curves is actually bigger than 0. The distortion degree of pressure curve for small gas rate is more significant than that of big gas rate.
Regime VI: The intermediatetime linear flow regime. This stage is mainly dominated by wellbore length. At this intermediatetime linear flow regime, we also can detect the MWPI for type Figure 1(b1) and Figure 1(b2). The slope of the DPD curves is actually bigger than 0.5. The distortion degree of pressure curve for small gas rate is more significant than that of big gas rate.
Regime VII: The latetime pseudoradial flow regime. The shape of DPD curve is a horizontal line. The value of this horizontal well is equal to 0.5.
Fig. 8
Pressure distribution of MWPP scheme at different production time. 
Fig. 9
I dealizations of four possible illustrations of hydraulic fractures. 
Fig. 10
Comparison of pressure curves between WIPS and SMFHW (q_{1}:q_{2} = 1:4): (a) Figure 1b, case1, (b) Figure 1b, case2. 
2.4 Sensitivity Analysis
In this section, we conduct some sensitivity analysis on pressuretransient response for MWPP scheme in SG reservoir. The key factors that influence the transient pressure response for MWPP scheme include hydraulic fracture halflength L_{fD}, hydraulic fracture conductivity C_{fD}, hydraulic fracture spacing L_{f}_{12D}, well spacing L_{wD}, ratio of well rate ε, stress sensitivity ζ_{D}. Here, we just take the Figure 2d(2) and traditional definition of DPP as an example. Our analysis is also on basis of assumption that fracture properties of two wells are consistent. Other situations can be analyzed similarly. Some dimensionless parameters can be: S = 0.2,C_{D} = 10, ζ_{D} = 0.05, λ = 0.002, γ = 0.15, ω = 0.0035, C_{ηD} = 10^{5}, L_{f}_{1D} = 2000, L_{f}_{2D} = 2000, L_{wD} = 3000, L_{f}_{12D} = 1000. C_{f}_{1D} = 50,C_{f}_{1D} = 50. The results are discussed in detail as follows:
Ratio of well rate, ε. We set ε to be 0.1, 0.3, 0.5 respectively. Figure 11 shows effects of ratio of well rate on pressure performance for MWPP scheme. We can judge the occurrence of MWPI by whether pressure curves of two wells overlap together. MWPI starts form first radialflow regimes. Subsequently, first radialflow regime and second linearflow regime are distorted severely. When the shape of pressure curves are distorted by the MWPI, the smaller the well rate, the more severely the pressure curves are distorted. Moreover, the smaller the well rate, the bigger the DPP and DPPD. Therefore, we can judge the well rate of two wells based on the relative position of the DPP curves and DPPD curves. We also can summarize that the ratio of well rate ε approximately has no any influence on the time when the MWPI occurs. This phenomenon can be strictly explained as follows: the time when the MWPI occurs is mainly relied on pressure wave propagation within reservoir and fracture system, which is dominated by fracture and rock properties, such as rock compressibility, fracture conductivity, halflength and well spacing. Thus, if we want to adjust the occurrence of MWPI, changing the production rate is useless.
Well spacing, L_{wD}. We set L_{wD} to be 450, 650, 900 respectively, and we also set ε to be 1:4. Figure 12 illustrates the impacts of well spacing L_{wD} on pressure performance for MWPP scheme. Similarly, we also can judge the occurrence of MWPI by whether pressure curves of two wells overlap together. On condition of different well spacing L_{wD}, MWPI basically starts form first radialflow regimes. As the well spacing L_{wD} increases, the occurrence of MWPI becomes later. Besides, pressure curves will be split at part of flow regimes (such as first radialflow regime and second linearflow regime), the pressure curves will overlap again subsequently. We also can clearly observe another phenomenon that well spacing almost does not distort the shape of pressure curves, the pressure curves just move upward or downward (the slope of pressure curves keeps constant). When the gas rate is big enough, the impacts of well spacing on multiwell interference can hardly be identified (the dot line). Therefore, if we can find that the pressure curves are not distorted apparently, we can make an original judgment that the well rate is relative big.
Hydraulic Fracture spacing, L_{f}_{12D}. We set L_{f}_{12D} to be 100, 500, 1000 respectively, and we also set ε to be 1:4. Figure 13 illustrates the impacts of hydraulic fracture spacing L_{f}_{12D} on pressure performance for MWPP scheme. Similar to the effects of well spacing on pressure curves, hydraulic fracture spacing mainly impacts the occurrence of MWPI and has no any influence on the distortion of flow regimes, the pressure curves just move upward or downward (the slope of pressure curves keeps constant). Different from the effects of well spacing on pressure curves, the impact of fracture spacing is more significant than that of well spacing. When L_{f}_{12D} = 100, MWPI basically starts from first linearflow regime when L_{f}_{12D} = 1000, MWPI basically starts from first radialflow regimes. As the fracture spacing L_{f}_{12D} increases, the occurrence of MWPI becomes later. Therefore, if we can find that the pressure curves are distorted at the early flow regime, we can make an original judgment that the stimulated HF may be closed to each other. Similarly, fracture spacing also just splits part of flow regimes (such as first linearflow regime, bilinear flow regime, first radialflow regime and second linearflow regime), the pressure curves will overlap again subsequently. Similarly, when the gas rate is big enough, the impacts of fracture spacing on MWPI also cannot be observed from pressure curves.
Hydraulic fracture length, L_{f}_{1D}, L_{f}_{2D}. We set L_{f}_{1D} = L_{f}_{2D} = L_{fD} to be 1000, 1500, 2000 respectively and ε to be 1:4. Figure 14 illustrates the impacts of hydraulic fracture length L_{fD} on pressure performance for MWPP scheme. We can systematically analyze the impacts of L_{fD} on pressure performance from three aspects: (1) before the occurrence of MWPI, for a certain fracture length, the pressure curves of two wells will overlap together. However, when fracture length is varying, the pressure curves will paralleled move upward or downward. As the fracture length increases, the pressure curves will paralleled move downward; (2) when the MWPI occurs, as the fracture length L_{fD} increases, the occurrence of MWPI becomes earlier. For example, when L_{fD} = 2000, MWPI basically starts form first linearflow regime. when L_{fD} = 1000, MWPI basically starts form first bilinearflow regime; (3) when MWPI reaches certain degree, the pressure curves will overlap again subsequently. The bigger the gas rate, the more lately the pressure curves overlap. For example, for well1, the pressure curves will overlap at second linearflow regime, for well2, the pressure curves will overlap at pseudosteady diffusion regime. In conclusion, fracture length impacts the whole flow regimes for MWPP scheme.
Stress sensitivity coefficient, ζ. We set ζ_{D} to be 0, 0.03, 0.05 respectively and ε to be 1:4. Figure 15 illustrates the impacts of stress sensitivity coefficient ζ_{D} on pressure performance for MWPP scheme. Different from the previous factors, including well spacing, ratio of gas rate, fracture spacing and fracture length, it is almost impossible to find that stress sensitivity coefficient can induce influences on the occurrence of MWPI. However, stress sensitivity coefficient can distort flow regimes at an inverse direction. For example, when ζ = 0.03, radialflow regimes and pseudosteady diffusion regimes are distorted. When ζ = 0.05, radialflow regimes, pseudosteady diffusion regimes and second linearflow regime are distorted. As the ζ increases, the distortion of pressure curves becomes severe, and more flow regimes will be distorted. It is also found that the pressure curves do not overlap again due to the existence of stress sensitivity. This meaningful finding can assist us to make a preliminary judgment that the gas rate of two wells is different or the SG reservoir is stress sensitivity, although these two factors distort the flow regimes due to different mechanisms.
Fig. 11
Effects of ratio of gas rate on pressure curves for MWPP scheme. 
Fig. 12
Effects of well spacing on pressure curves for MWPP scheme. 
Fig. 13
Effects of fracture spacing on pressure curves for MWPP scheme. 
Fig. 14
Effects of fracture halflength on pressure curves for MWPP scheme. 
Fig. 15
Effects of stress sensitivity coefficient on pressure curves for MWPP scheme. 
Conclusion
To gain better understanding about well performance of MWPP scheme, in this paper, we develop a new semianalytical pressure transient model in Laplace domain to identify flow regimes without and with IWPI. Model validation is implemented using CMG numerical simulator, and sensitivity analysis is also conducted. Some meaningful conclusions are summarized as following:

There is good agreement between our model and numerical simulation, moreover, and our approach also gives a much faster calculation speed compared to numerical simulation, both of which demonstrate the accuracy and efficiency of our method;

Some expected flow regimes are apparently distorted by IWPI. The slope of type curves which characterizes the linear or bilinear flow regime is no longer equal to 0.5 or 0.25. The horizontal line which characterize radial flow regime is no longer equal to 0.5. For different interference type, IWPI can distort different flow regimes. Interference directly through HF is more rapid than interference through reservoir;

Well rate and stress sensitivity coefficient mainly determine the distortion of pressure curves. As the well rate decreases or stress sensitivity coefficient increases, the distortion of pressure curves will become severe. Well rate will distort pressure curves when IWPI occurs, on the contrary, stress sensitivity coefficient can distort pressure curves at an inverse direction which is from radial flow to diffusion flow regime, the bigger the stress sensitivity coefficient, the more flow regimes will be distorted;

Fracture length, well spacing, fracture spacing mainly determine when the IWPI occurs. As the well spacing increases, fracture length decreases, fracture spacing decreases, the occurrence of IWPI becomes later. For well spacing, fracture spacing, when IWPI occurs, pressure curves split, and then overlap again. For fracture length, pressure curves will always split until IWPI reaches certain degree, which is the ending of IWPI flow regime.
Nomenclature
MWPP: MultiWellPadProduction
IWPI: InterWell Pressure Interference
SMFHW: Single MultiFractured Horizontal Wells
Hens: Hydraulic Fracture Networks; DPP Dimensionless Pseudo Pressure
DPPD: Dimensionless Pseudo Pressure Derivation
MHFs: Main Hydraulic Fractures
SHFs: Secondary Hydraulic Fractures
P_{i}: initial formation pressure, MPa
ψ_{i}: initial pseudopressure, MPa^{2}/(mPa · s)
T_{sc}: temperature under standard condition, K
P_{sc}: pressure at standard condition, MPa
C_{t}: total compressibility, MPa^{−1}
ψ: pseudopressure, MPa^{2}/(mPa · s)
ψ_{f}: fracture pseudopressure, MPa^{2}/(mPa · s)
L_{f}_{1}: hydraulic fracture half length of well1, m
L_{f}: hydraulic fracture half length of well2, m
L_{hf}_{12}: horizontal distance of the Hens between two wells, m
L_{vf}: vertical distance of the Hens between two wells, m
r: radial coordinate in NF system, m
r: radial coordinate in matrix system, m
l: coordination of hydraulic fracture, m
q_{1}: well production rate of well1, m^{3}/d
q_{2}: well production rate of well2, m^{3}/d
q_{sc}: well production rate under standard condition, m^{3}/d
k_{ri}: initial permeability of NF system, D
k_{f}: permeability of hydraulic fractures for well1, D
k_{f}_{2}: permeability of hydraulic fractures for well2, D
V: gas concentration, sm^{3}/m^{3}
w_{f}: width of hydraulic fractures for well1, m
w_{f}_{2}: width of hydraulic fractures for well2, m
W_{f}_{1}: width of Hens system for well1, m
W_{f}_{2}: width of Hens system for well2, m
ζ: stress sensitivity coefficient, (mPa · s)/MPa^{2}
C_{g}: gas compressibility, MPa^{−1}
M_{1}: total number of hydraulic fracture for well1, integer
M_{2}: total number of hydraulic fracture for well1, integer
M_{w}: total number of wells, integer
q_{f}_{D}: dimensionless flux rate
q_{c}_{D}: dimensionless fracture rate
x_{D}, y_{D}: dimensionless space
r_{D}: dimensionless radial coordinate in NF system, m
r_{mD}: dimensionless radial coordinate in matrix system, m
L_{f}_{1D}: dimensionless hydraulic fracture half length of well1, m
L_{f}_{2D}: dimensionless hydraulic fracture half length of well2, m
L_{f}_{12D}: dimensionless fracture distance between two wells, m
L_{wD}: dimensionless distance between two wells, m
C_{f}_{1D}: dimensionless hydraulic fracture conductivity for well1
C_{f}_{2D}: dimensionless hydraulic fracture conductivity for well2
C_{1D}: dimensionless wellbore storage coefficient for well1
C_{2D}: dimensionless wellbore storage coefficient for well2
Subscript
Superscript
Acknowledgments
The authors are grateful to the anonymous reviewers for their insightful and constructive comments for improving the manuscript. The authors acknowledge a fund from the National Union Foundation (NUF) of China (No. U1562102) and supports from the MOE Key Laboratory of Petroleum Engineering.
Appendix A Dimensionless definitions
In our research, we introduce a new definition of dimensionless pseudopressure based on the total rate of all wells. Therefore, dimensionless pseudopressure for SG system and HFNW system can be given as follows, (A1)
For the dimensionless time (A2) where:
For the dimensionless spacing and fracture length (A3) (A4)
For the dimensionless gas rate influx, gas concentration and gas flow rate can be defined, respectively: (A5)
For the fractureflow model, the hydraulic fracture conductivity can be assumed to be uniform for the same well, respectively, at the same time, the fracture conductivity can be varying for every well. Therefore, dimensionless fracture conductivity can be defined as (A6)
Dimensionless transmissibility coefficient of HF can be presented as: (A7)
Dimensionless storage ratio of SG formation can be presented as: (A8)
Dimensionless adsorption index which denotes the SG desorption ability can be defined as, (A9)
Dimensionless diffusion coefficient which denotes the SG transferring from matrix into natural fracture can be defined as, (A10)
Dimensionless stress sensitivity coefficient can be defined as, (A11)
Appendix B Derivation of line source solution
To develop the mathematical models in SG reservoir, the mathematical formula for NF system and matrix system can be established respectively and then be dynamically coupled. The model development can be given as follows:
Seepage model for SG in NF system. To begin with, flow in natural fractures is assumed to be single phase by obeying Darcy's law. Combining with gas state equation and motion equation, the SG flow in natural fracture system can be described as following diffusion equation with consideration of gas adsorption behavior: (B1)
The pseudopressure function was used to account for the pressure dependent gas properties, the governing Equation (B1) with the formula of pseudopressure is as follows: (B2)
The stressdependent permeability can be described by introducing a permeability modular ζ, the relationship between permeability and pseudo pressure can be (Pedrosa, 1986): (B3) where, k_{fi} is initial permeability of natural fracture under initial pressure condition, Darcy unit; ζ is permeability modular, (mPa · s)/MPa^{2}; ψ_{i} is the initial pseudopressure, MPa^{2}/(mPa · s).
Submitting Equation (B3) into Equation (B2), the final formation of governing formula for natural fracture system can be transformed as follows: (B4)
Based on the theory of line sink, the inner condition is presented as follows: (B6)
The reservoir is assumed to be infinite, and the outer boundary is as follows: (B7) where, k_{r} is the permeability of SG formation, D; ψ is the pseudopressure in fracture system, MPa^{2}/(mPa · s); µ is the viscosity of shale gas, mPa · s; Φ is porosity of fracture system, fraction; C_{t} is the total compressibility coefficient, 1/MPa; t is the time, h; r is radial distance, m. P_{sc} is the pressure at standard condition, MPa; T_{sc} is the temperature at standard condition, K; q is production rate of line sink, sm^{3}/d; h is the thickness of the reservoir, m.
For the convenience of solution, some dimensionless variables are defined previously. With the definition of these dimensionless variables, Equations (B1)–(B7) with the dimensionless formation can be presented as follows: (B8) (B9) (B10) (B11)
Equation (B8) shows that the seepage model is strongly nonlinear and some addition method needs to be applied to obtain analytical solution. Here, the perturbation technology and the Presoda transformation are applied to linearize the equations (Pedrosa, 1986): (B12)
According to the theory implemented by Pedrosa (1986), performing a parameter perturbation in ζ_{D} by defining the following series: (B13a) (B13b) (B13c)
Considering the facts that the ζ_{D}, dimensionless stress sensitivity coefficient, is always small, thus the zeroorder perturbation solution can greatly meets the requirements. The final formation of Equations (B8)–(B11) are as follow: (B14) (B15) (B16)
The Laplace transformation with respect to t_{D} is then used to deal with Equations (B14)–(B16), the Laplace transform is based on t_{D} and functions as follows: (B17)
And we can obtain the formation of governing equation of SG reservoir system in the Laplace domain, (B18) (B19) (B20)
Seepage model for SG in matrix system. Due to the ultralow permeability of matrix, the fluid flowing in the matrix system is treated as unsteadysteady state flow (de Swaan, 1990; Noetinger et al., 2001; Landereau et al., 2001). Thus, we assume that SG flow within shale matrix obeys Second Fick’s law. The shale matrix can be treated as spherical geometry, the SG flowing in the matrix system can be described as follows (Wang, 2014): (B21)
The diffusive flow in the spherical SG matrix blocks is symmetric, thus the center of matrix blocks can be treated as a noflow boundary, which gives the following inner boundary condition, (B22)
Adsorptive SG concentration on the external surface of the matrix blocks can be evaluated at the gas pressure in the natural fracture system, so the outer boundary condition can be described as follows (B23)
Furthermore, adsorption behavior of SG can be described by Langmuir isotherm equation (Langmuir, 1918). Thus, (B24)
And the initial condition (B25) where, D is SG diffusion coefficient, m^{2}/s; R_{m} is the radius of spherical matrix, m; V_{i} is the initial concentration in matrix, sm^{3}/m^{3}; V_{E} is the SG concentration at the surface of matrix, sm^{3}/m^{3}.
With the dimensionless definitions given in Appendix A, Equations (B21)–(B22) can be written in the following dimensionless formula in Laplace space, (B26) (B27) (B28)
The solution of Equations (B26)–(B28) can be given by (Wang, 2014) (B29)
Coupling seepage model for SG system. The SG flow in NFs system is represented by Equation (B14), the internal source item representing the effects of desorption of SG is given by the following equation based on the spherical matrix blocks assumption, (B30)
Combination of Equation (B29) and Equation (B30) in Laplace domain gives (B31)
Finally, substituting Equation (B31) into Equation (B18), one can get the following governing equation coupling NFs system and matrix system, (B32) where,
The general solution of Equation (B32) for the infiniteacting SG reservoir can be given by (Ozkan and Raghavan 1991, Xiao et al., 2016), (B33) By the requirement that P_{D} vanish at infinity, we must have A=0 in equation. From the condition given by Equation (B20), we can obtain (B34)By applying the principle of integration, the pressure distribution of a random position (x_{D}, y_{D}) caused by one fracture segment (x_{WD}, y_{WD}) is given by the following equation.
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All Tables
All Figures
Fig. 1
The schematic illustration of a MWPP scheme in SG reservoir, a) is layout of MWPP scheme in a SG field, b) and c) are two possible scenarios of MWPP scheme, d) is idealizations of HFNW system. 

In the text 
Fig. 2
The schematic illustration of discretized HFNW system in the conceptual model. 

In the text 
Fig. 3
Illustration of gas flow within ith fracture segment. 

In the text 
Fig. 4
Illustration of StarDelta transformation. 

In the text 
Fig. 5
Sensitivity analysis on the number of divided fracture segments. 

In the text 
Fig. 6
Top view of the numerical model of Case II in CMGGEM module: a) nonconnection between HFNW system, b) direct connection between HFNW system, c) regular HFNW with transverse MHF and without SHF, d) Grid refinement for hydraulic fractures. 

In the text 
Fig. 7
Comparison of our results of model with that of CMG simulator: a) HFNW system as Figure 6(a), b) HFNW system as Figure 6(b), c) HFNW system as Figure 6(c). 

In the text 
Fig. 8
Pressure distribution of MWPP scheme at different production time. 

In the text 
Fig. 9
I dealizations of four possible illustrations of hydraulic fractures. 

In the text 
Fig. 10
Comparison of pressure curves between WIPS and SMFHW (q_{1}:q_{2} = 1:4): (a) Figure 1b, case1, (b) Figure 1b, case2. 

In the text 
Fig. 11
Effects of ratio of gas rate on pressure curves for MWPP scheme. 

In the text 
Fig. 12
Effects of well spacing on pressure curves for MWPP scheme. 

In the text 
Fig. 13
Effects of fracture spacing on pressure curves for MWPP scheme. 

In the text 
Fig. 14
Effects of fracture halflength on pressure curves for MWPP scheme. 

In the text 
Fig. 15
Effects of stress sensitivity coefficient on pressure curves for MWPP scheme. 

In the text 