Pressure and rate transient modeling of multi fractured horizontal wells in shale gas condensate reservoirs

Journal Pre-proof Pressure and rate transient modeling of multi fractured horizontal wells in shale gas condensate reservoirs Sadegh Dahim, Amin Taghavinejad, Milad Razghandi, Hamed Rahimi, Kianoosh Moeini, Saeid Jamshidi, Mohammad Sharifi PII:

S0920-4105(19)30987-8

DOI:

https://doi.org/10.1016/j.petrol.2019.106566

Reference:

PETROL 106566

To appear in:

Journal of Petroleum Science and Engineering

Received Date: 10 September 2018 Revised Date:

28 August 2019

Accepted Date: 6 October 2019

Please cite this article as: Dahim, S., Taghavinejad, A., Razghandi, M., Rahimi, H., Moeini, K., Jamshidi, S., Sharifi, M., Pressure and rate transient modeling of multi fractured horizontal wells in shale gas condensate reservoirs, Journal of Petroleum Science and Engineering (2019), doi: https:// doi.org/10.1016/j.petrol.2019.106566. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

Pressure and Rate Transient Modeling of Multi Fractured Horizontal Wells in Shale Gas Condensate Reservoirs Sadegh Dahim1, Amin Taghavinejad2, Milad Razghandi1, Hamed Rahimi2, Kianoosh Moeini3, Saeid Jamshidi1, Mohammad Sharifi2, 1. Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran 2. Department of Petroleum Engineering, Amirkabir University of Technology, Tehran, Iran 3. Department of Chemical Engineering, Technical University of Denmark, Lyngby, Denmark

Abstract Gas condensate production using technology of multi-stage hydraulically fracturing in shale gas condensate reservoirs’ horizontal wells is a new topic of unconventional resources studies. So, shale gas condensate as a new source of energy can be considered as an important issue for development and further studies. In this work, a semi-analytical solution of gas and oil two-phase flow is presented for pressure transient analysis (PTA) and rate transient analysis (RTA) of a shale gas condensate reservoir’s production data. Fluid flow assumption here is flow in a pseudo triple-porosity porous media, which are matrix, natural fractures and adsorbed gas. Adsorbed gas is a form of gas in porous rock pores like free gas, but unlike that, it is not free to flow initially and requires pressure drop to desorb from micro- (or even nano-) pores of rock to flow in matrix and then in fracture. In this study, desorption of the adsorbed gas is modeled with Langmuir isotherm theory. Based on the results, type curves of two-phase pseudo-pressure and also twophase rate decline are plotted. These type curves are analyzed with different sensitivities to Langmuir volume, interporosity flow parameters, horizontal well length, number of multiplefractures and Corey model relative permeability exponents. Τhe strength of this work is to plot type curves of pressure and rate transient analysis of multi-stage hydraulically fractured well (MHFW) in shale gas condensate reservoir in terms of rich gas production of the well. Keywords: Pseudo triple-porosity – Pressure transient analysis – Multi-stage hydraulically fractured well – Shale gas condensate – Rate transient analysis

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

Pressure and Rate Transient Modeling of Multi Fractured Horizontal Wells in Shale Gas Condensate Reservoirs Sadegh Dahim1, Amin Taghavinejad2, Milad Razghandi1, Hamed Rahimi2, Kianoosh Moeini3, Saeid Jamshidi1, Mohammad Sharifi2, 1. Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran 2. Department of Petroleum Engineering, Amirkabir University of Technology, Tehran, Iran 3. Department of Chemical Engineering, Technical University of Denmark, Lyngby, Denmark

1. Introduction Due to the reduction of conventional hydrocarbon reserves in the world, exploration and development of the unconventional reservoirs such as shale oil, gas hydrate, and shale gas resources have been increased. There are adequate amounts of gas shale reservoirs around the world; in which, gas is freely flowing not only in the fractures and pores of shale matrix, but also in the form of adsorbed gas on the matrix grains which result in extremely high difference in the flow equations and pressure investigations used for obtaining reservoir characteristics in the case of not considering this quota of storage. The cited reservoirs have extremely low permeability and porosity (low reservoir quality), which result in a different approach of production and development in comparison with conventional oil and gas reservoirs (Hill and Nelson, 2000). As a common practice, they highly require the utilization of hydraulic fracturing and horizontal wells technology for the sake of economic hydrocarbon production. Gas condensate reservoirs could be stated as a specific type of hydrocarbon reservoirs, which can be found either in conventional or unconventional basins. Gas condensate reservoirs demonstrate a complex thermodynamic behavior at pressure conditions below dew point pressure, leading to the formation of condensate and the consequent alteration of reservoir gas. The formation of condensates near the wellbore, diminishes the reservoir flow efficiency. For the first time, (O'Dell and Miller, 1967) suggested gas equations by using pseudo-pressure concept to be as a solution for condensate blockage. (Fussel, 1973) found the (O'Dell and Miller, 1967) approach pessimistic to model the gas condensate reservoirs productivity. After that, Raghavan worked on several models estimating reservoir flow capacity using the concept of steady-state pseudopressure (Jones and Raghavan, 1988; Jones et al., 1989; Raghavan et al., 1995). After the research conducted by (Fevang, 1995) and later (Xu and Lee, 1999), common models for well test analysis of gas condensate reservoirs amended based on radial composite regions with different mobility values (Gringarten et al., 2000; Haji Seyedi et al., 2014; Yousefi et al., 2014).

1

1 2 3 4 5 6 7 8 9 10 11 12 13

For having a properly developed production from unconventional reservoirs, drilling horizontal wells and implementing hydraulic fracturing methods are usually implemented. It must be noted that the mathematical models of these wells and fractures for conventional reservoirs have been investigated through various researchers (Belyadi et al., 2010; Crosby et al., 2002; Larsen and Hegre, 1994; Medeiros et al., 2007; Raghavan et al., 1997; Restrepo, 2008; Wan and Aziz, 2002; Wei and Economides, 2005). (Medeiros et al., 2007) studied the pressure transient analysis of hydraulically fractured horizontal wells considering different fracture conductivities in longitudinal and transverse fractures, (Restrepo, 2008) worked on pressure behavior of multiple fractures of wells with different type of conductivities, and (Wan and Aziz, 2002) presented a new semi-analytical well model which relates well rate to the well pressure in multiple fractured horizontal wells (MFHWs). Moreover, studies in hydraulic fracturing indicate the capability of improving the productivity of wells in gas condensate reservoirs (Indriati and Wang, 2002; Langedijk et al., 2000; Sedarat et al., 2014).

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

Production analysis of MFHWs in shale gas reservoirs have been studied by several researchers. As the most outstanding samples of these studies, various analytical (Bello, 2009; Belyadi et al., 2010; Fan and Ettehadtavakol, 2017; Liang et al., 2017; Nobakht and Clarkson, 2011a; Nobakht and Clarkson, 2011b; Nobakht and Clarkson, 2012a; Nobakht and Clarkson, 2012b; Nobakht et al., 2011; Wei and Economides, 2005, Mahmoodi et al. 2019) semi-analytical (He et al., 2018; Li et al., 2018; Wan and Aziz, 2002; Wang et al., 2017) and numerical models (Al-Twaijri, 2017; Mayerhofer et al., 2006; Miao et al., 2018) have been presented for the analysis and evaluation of reservoir and well production and also rate transient analysis (RTA) in shale gas reservoirs. (Wei and Economides, 2005) studied on fracture performance and well productivity of horizontal wells with transverse hydraulic fractures. (Mayerhofer et al., 2006) presented the results of integrating microseismic fracture mapping with numerical production modeling of fracture networks in the Barnett shale. (Bello, 2009) introduced a new method for RTA of fractured horizontal wells in shale gas reservoirs with transient linear behavior. (Nobakht and Clarkson, 2011a; Nobakht and Clarkson, 2011b) presented new analytical models for the production data analysis of shale gas reservoirs with linear flow considering constant pressure and constant rate production conditions, respectively. (Nobakht and Clarkson, 2012b) developed their analytical production data analysis method for constant-rate boundary condition tight and shale gas reservoirs. (Nobakht and Clarkson, 2012a) presented the multiwell production data analysis of MFHWs in tight and shale gas reservoirs. (Al-Twaijri, 2017) introduced a comprehensive numerical model for simulating two-phase flow in shale gas reservoirs. (He et al., 2018) studied on a new improved RTA model of MFHWs with non-uniform hydraulic fracture properties. (Miao et al., 2018) implemented a new RTA model for shale gas reservoirs coupled the effect of slip flow and surface diffusion. Furthermore, in some researches, Arps-based experimental decline curve analysis has been studied (Baihly et al., 2010; Duong, 2011; Guo et al., 2016; Kenomore et al., 2018). (Baihly et al., 2010) compared the shale gas production decline over time and different basins. (Kenomore et al., 2018) studied shale gas production decline trend

2

1 2

over time in the Barnett shale for the purpose of estimated ultimate recovery (EUR) forecast in this type of reservoirs.

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Although most of the researches in the area of flow in unconventional reservoirs concentrates on analyzing the performance of the well production and production forecast, various analytical (Brown et al., 2011; Ozkan et al., 2011; Sheng et al., 2015), semi-analytical (Gu et al., 2017; He et al., 2016; Medeiros et al., 2007; Tian et al., 2014; Xiao et al., 2016; Xiao et al., 2017; Zeng et al., 2015; Zeng et al., 2018; Zhao et al., 2016; Zhao et al., 2013) and numerical (Cheng, 2011; Gao et al., 2019; Wu et al., 2018) models have been developed for analyzing the pressure behavior and well test analysis of flow via the MFHWs in shale gas reservoirs. (Ozkan et al., 2011) compared the fractured horizontal wells performance in conventional and unconventional reservoirs using PTA techniques. (Zhao et al., 2013) presented a new type-curve analysis method for PTA and RTA of shale gas reservoirs with MFHWs. (Tian et al., 2014) provided a new well test model for MFHWs in shale gas reservoirs considering dual diffusion in matrix medium which divide the flow into five regimes; early linear flow; pseudo-steady state inter-porosity flow; the diffusion from matrix into micro-fractures; the diffusion from matrix into hydraulic fractures and boundary-dominated flow. (Sheng et al., 2015) presented a multiple porosity media model to describe gas flow from kerogen to multi-fractured horizontal wells in shale gas reservoirs. (Zhao et al., 2016) investigated the effects of Knudsen diffusion and gas slippage on pressure response and production performance of MFHWs with complex seepage mechanism in box-shaped shale gas reservoirs. (Xiao et al., 2017) presented a semi-analytical PTA methodology for multiwell-pad-production in sale gas reservoirs. (Zeng et al., 2018) introduced a PTA model for shale gas reservoirs’ MFHWs proposing a modified apparent permeability which describes gas flow in shale gas reservoirs by integrating bulk gas flow in nanopores and gas desorption from nanopores.

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

Various studies focusing on the production evaluation of tight/shale gas condensate reservoirs have also been implemented through different research based projects (Behmanesh et al., 2013b; Behmanesh et al., 2015b; Clarkson and Qanbari, 2015; Khanal et al., 2017; Labed, 2016; Orangi et al., 2011). (Behmanesh et al., 2013b) implemented analytical production data analysis of constant flowing borehole pressure wells producing from liquid rich shale gas condensate reservoirs which only accounts for transient linear flow period. (Behmanesh et al., 2015b) updated their previous model for production data analysis of shale gas condensate reservoirs and presented it with more theoretical bases. (Clarkson and Qanbari, 2015) provided a history-match and forecasted tight gas condensate and oil wells using the dynamic drainage area concept. (Khanal et al., 2017) used a compositional shale gas condensate reservoir model and implemented their data driven approach on it using component analysis for the purpose of production forecast. (Labed, 2016) considered the effect of Knudsen flow and pore size distribution affected phase behavior of gas condensate in his numerical simulation model for gas condensate flow in shale gas reservoirs. (Orangi et al., 2011) compared the impact of rock, fluid and hydraulic fractures in unconventional shale oil and gas condensate reservoirs production 3

1 2 3 4 5 6

using a simulation model. Furthermore, (Ibrahim et al., 2017) worked on an analytical PTA model (build-up test) for a shale gas condensate well integrating with completion data, petrophysical data, fluid sample analysis, mini-frac analysis, and flowback data. In comparison with previous works, the novelty of this study is considering the effect of adsorbed gas in two phase flow of shale gas condensate reservoir producing with a MFHW by a new semi-analytical method.

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

In most of the studies implementing analytical or semi-analytical RTA/PTA techniques in tight/shale gas reservoirs or tight/shale gas condensate reservoirs, average reservoir pressure is calculated using material balance methods. These studies have been presented for both singlephase (Behmanesh et al., 2015a; Clarkson et al., 2011; Clarkson and Williams-Kovacs, 2013; Mattar et al., 2006; Mattar and McNeil, 1998; McNeil, 1995) and two-phase flow (Behmanesh et al., 2013a; Behmanesh et al., 2015b; Behmanesh et al., 2017; Clarkson and Qanbari, 2015; Sureshjani et al., 2014) studies; through which, by the determination of average reservoir pressure, the pressure-dependent parameters of flow would be updated in each time step of the solution. The usual technique to consider the average reservoir pressure for linearizing the flow equations in gas condensates is coupling the material balance equations with the analytical or semi-analytical fluid flow models. In (Clarkson and Qanbari, 2015), transient and boundary dominated linear flow have been considered and material balance equations assist the determination of average pressure in each time-step of solution. Material balance in transient linear flow utilizes the concept of dynamic drainage area and considers the reservoir as increasing distances of investigation by elapsing time. Same approach of transient linear flow is considered by (Behmanesh et al., 2013a; Behmanesh et al., 2015b), however a self-illustrated definition of two-phase pseudo-pressure and pseudo-time function was used to linearize the governing equations. Furthermore, (Sureshjani et al., 2014) semi-analytically modeled gas condensate boundary-dominated flow coupled with modified material balance equation to obtain gas-in-place. Later studies by (Behmanesh et al., 2017) developed a semi-analytical rate transient and pressure transient boundary-dominated flow model. They utilized the single-phase gas flow theory alongside with the two-phase viscosity and two-phase compressibility factor to develop a modified two-phase (gas and oil) pseudo-time formulation. Moreover, they combined the superposition principle and constant rate solution was combined to consider dynamic variations in pressure and rate. Moreover, material balance was applied to calculate the average pressure implicitly. As one of the other novelties of this work is to consider all possible flow regimes in the life of a producing MFHW in shale gas condensate reservoir, the simple approach of transient linear flow is not the key for determination of average reservoir pressure in each time-step of solution. So, a simple approach has to be applied in such condition of solution.

36 37 38 39

In this study, the governing flow equations of gas reservoirs (flow in dual porosity system of natural fractures and then in infinite-conductivity hydraulic fractures) have been combined with adsorption equations (similar to (Zhao et al., 2013)). The novelty of this work is due to the consideration of the effect of existence of two phase in such shale gas system. Taking into 4

1 2 3 4 5 6 7 8 9 10 11

account the existence of condensate and its equation in a fractured horizontal well, a reliable model has been developed for by investigating the pressure and pseudo-pressure changes. The conducted semi-analytical approach by considering the linearized flow-equations to solve them in Laplace space, demonstrates some similarities with the approach of utilizing material balance method for obtaining average pressure; however, differences exist. It must be stated that due to complexity of equations to consider various flow regimes in the flowing period and also infiniteacting behavior of reservoir, exact average pressure is not used. In each time-step of this solution a simplified approach is considered to use a proper average reservoir pressure for the calculation of non-linearity coefficients specially for two-phase flow condition of gas condensates. This simplified approach uses the concept of arithmetic average between reservoir initial pressure and wellbore flowing pressure in each time-step.

12 13 14 15 16 17 18 19 20 21

2. Model description The schematic of shale gas reservoirs has been illustrated in Figure 1. As it is demonstrated, the majority of the gas is the gas stored in matrix. Owing to the low permeability of the matrix, it can be assumed that the fluid is flowing to the well mainly via the fractures and the matrix is deprived of flowing capacity. Furthermore, after the fracture and matrix fluid storages, this reservoir has a third type of flow capacity on matrix grains, known as adsorbed. By considering this pseudo triple-porosity performance which is dual porosity with adsorbed gas storage, this new type of gas can be considered as the third porous medium after matrix and fracture’s free gas. (Song, 2010)

5

Natural Fracture Medium

Fracture free gas (2nd Porosity)

Dual porosity modeled natural fractures Natural fractures

Matrix Medium

Induced fracture

Matrix free gas (1st Porosity) Adsorbed gas (on shale grains) (3rd Porosity)

1 2

Figure 1: The schema of a shale gas reservoir with pseudo triple-porosity

3 4 5 6 7 8

The procedure of adsorbed gas flow to the hydraulic fractures and then producing well has been illustrated in the Figure 2. Initially, the adsorbed gas on the matrix particles is desorbed and performs as a free gas in the matrix. Afterwards, the fluid enters the natural fractures as a result of the pressure difference between matrix and fracture, which is then conducted to the hydraulic fractures and then toward producing well.

9

6

1 2

Figure 2: The mechanism of fluid flow in shale gas reservoir with pseudo triple-porosity

3

7

1 2

Figure 3: Top view schema of the fractured horizontal well with discrete elements

3 4 5 6

The schema of the fractured horizontal well has been shown in Figure 3. As it is depicted, some points are considered on each fracture for the purpose of increasing the precision of point source analytical solution at each segment of hydraulic fracture stages.

7

The used assumptions made in this modeling could be stated as:

8 9 10 11 12 13 14 15

1. The reservoir is horizontal and naturally fractured with two sets of fractures. Also, its upper and lower boundaries are impermeable (no-flow). 2. The flow conductivity in hydraulic fracture stages is infinite which results in the equality of pressure drop in all sectors of hydraulic fractures. 3. The producing fluid is merely flowing via fractures. In other words, matrix deliverability is considered zero. 4. The desorbed fluid from matrix grains surface is only in the form of gas. Furthermore, the adsorbed gas only enters the matrix and does not enter the fracture directly.

16

Table 1: Base case; reservoir and well data

General data 24×106 350 0. 101325×106

Initial reservoir pressure, Pi [Pa] Initial reservoir temperature, T [K] Standard condition pressure [Pa] 8

Standard condition temperature [K] Horizontal well length, 2L [m] Reservoir thickness, h [m] Fracture half-length, Lf [m] Absolute permeability of fracture medium, kf [m2] Absolute permeability of matrix medium, km [m2] Porosity of fracture medium, φf [Dimensionless]

288.15 1000 80 40 1.0×10-14 1.0×10-19 0.08

Porosity of matrix medium, φm [Dimensionless] Number of natural fracture sets, n Characteristic length, lc [m] Compressibility of fracture medium rock, crf [Pa-1] Compressibility of matrix medium rock, crm [Pa-1] Langmuir pressure, PL [Pa] Langmuir volume, VL [sm3/m3] Skin factor, S [Dimensionless] Number of fractures, M Discrete number of each half-fracture Dry gas system data Gas production rate (at standard condition), qsc [m3/day] Gas specific gravity, γ [Dimensionless] Condensate gas system data Rich gas production rate (at standard condition), qsc [m3/day] Dew point pressure, Pdew [Pa]

0.035 2 40

10-8 10-10 1.5×106 4 0.5 4 5 85,000 1.05 85,000 28×106

1 2 3 4 5 6 7 8 9 10

3. Development of the model In this section, governing equations and model are developed for the single-phase (dry gas) at first (Zhao et al., 2013). Afterwards, the method is extended for two-phase condition (condensate gas). A new set of type curves for pressure transient analysis and also rate transient analysis are presented in the form of dimensionless pseudo-pressure and dimensionless rate versus dimensionless time, respectively. Then the effect of different parameters especially those with higher significance that are more important in two-phase flow, is investigated on type curves.

11 12

3.1. Single-phase condition

13 14 15

Single-phase flow equations for MFHW in shale gas reservoir were proposed by (Zhao et al., 2013). Equations (1) and (2) are acquired for fracture and matrix media respectively, via utilizing mass balance. (The fully explained derivation is provided in Appendix A). 9

∂m ( Pf ) 1 ∂  2 ∂m ( Pf )  r = ω − λ ( m ( Pm ) − m ( Pf ) )   D f rD2 ∂rD  ∂rD  ∂t D

(1)

−λ ( m ( Pm ) − m ( Pf ) ) = (1 − ωf )

(2)

∂m ( Pm ) ∂m ( Pm ) + ωad ∂t D ∂t D

1 2

For solving the above equations, the Laplace transform is used as shown below:

1 ∂  2 ∂∆m ( Pf )   rD  = ωf s∆m ( Pf ) − λ ( ∆m ( Pm ) − ∆m ( Pf ) ) rD2 ∂rD  ∂rD 

−λ ( ∆m ( Pm ) − ∆m ( Pf ) ) = (1 − ωf ) s∆m ( Pm ) + ωad s∆m ( Pm )

3 4 5

(3) (4)

Where; ∆m ( P ) = m ( Pi ) − m ( P )

Pursuing equation (4), the pseudo-pressure difference of matrix can be obtained from that of the fracture:

∆m ( Pm ) = 6

λ ∆m ( Pf ) λ + (1 − ωf + ωad ) s

(5)

7

By substituting the equation (5) into equation (3) the below equation can be achieved:

8

1 ∂  2 ∂∆m(pf )   rD  = f ( s ) ∆m ( pf ) rD2 ∂rD  ∂rD 

(6)

9

In which, f(s) can be computed from the below equation: f (s ) =

λ (1 + ωad ) + ωf (1 − ωf + ωad ) s λ + (1 − ωf + ωad ) s

s

(7)

10 11 12

Table 2: Introducing the dimensionless and pseudo triple-porosity flow parameters

rD =

r L

Dimensionless distance

tD =

kf t ( φf ctfg + φm ctmg ) µfg L2

Dimensionless time

10

m D ( P ) = 86, 400

πkhTsc ∆m ( P ) Psc q sc T

c t = Sg cg + So co + cr * λ ( m −f ) = α

ωf = ωad = 1

4n ( n + 2 ) km 2 L , α= l2c kf

Dimensionless pseudo-pressure Total compressibility of fracture/matrix Interporosity flow coefficient

ctfgφf ctfg φf + ctmgφm

Storability ratio of the fracture system

cgpφm ctfgφf + ctmg φm

Storability ratio of adsorption gas

* So equals to zero for single-phase flow condition. 2

3 4 5 6 7 8

It should be noted that the term ωad, which is defined in Table 2, is pressure dependent because of the term adsorption compressibility ( cgp ) and has to be updated in each time-step of solution by new average matrix pressure values. The general solution of single and two-phase flow condition is discussed in the following sections. Also, further information about calculation of adsorption compressibility is debated in Appendix A.

9

3.2. Two-phase condition 10 11

In terms of gas condensate, two-phase flow condition will appear in the pressures below dew

12

point, whose governing equations are different from the ones mentioned before. In this

13

condition, the fluid saturations in the shale matrix and fracture media are also included in the equations as unknown parameters.

14 15

Mass balance for gas phase in both fracture and matrix media is provided below (the fully explained derivation is written in Appendix B).

∂m ( Pfg ) ωf 2Pfg ∂Sfg 1 ∂  2 ∂m ( Pfg )  Sfg  rD = ωf + 2 rD ∂rD  ∂rD  k rfg ∂t D k rfg µ fg Zfg c tfg ∂t D   k M  k µ −λ  rmg mg m ( Pmg ) − rmg fg m ( Pfg )  k M  k rfg µ mg fg  rfg 

11

(8)

k k rmg M fg  µ fg rmg µ fg −λ  m ( Pmg ) −   k rfg µ mg k rfg M mg  µ mg 

2   Smg ∂m ( Pmg ) m P  ( fg )  = (1 − ωf )  k rfg ∂t D  

(1 − ωf ) 2Pmg ∂Smg + ωad ∂m ( Pmg ) + k rfg µ mg Zmg c tmg ∂t D k rfg ∂t D

(9)

1 2 3

Using mass balance for oil phase in fracture and matrix media leads to equation (10) and (11), respectively. ∂m ( Pfg ) ωf 2Pfg ∂Sfg 1 ∂  2 ∂m ( Pfg )  Sfo c tfo µ fo   r = ω − D f 2 rD ∂rD  ∂rD  k rfo c tfg µ fg ∂t D k rfo µ fo Z fo c tfg ∂t D  

(10)

k M  k µ −λ  rmo mo m ( Pmg ) − rmo fo m ( Pfg )  k rfo µ mo  k rfo M fo 

k µ  k M µ µ S c ∂m ( Pmg ) −λ  rmo fg m ( Pmg ) − rmo fo fo 2fg m ( Pfg )  = (1 − ωf ) mo tmo ∂t k rfo M mo µ mo k rfo c tmg  k rfo µ mo  (1 − ωf ) 2Pmg ∂Smg − k rfo µ mo Zmo c tmg ∂t

(11)

4 5 6

For eliminating the saturation derivative terms from the equations, the equations of oil and

7

gas in fracture and matrix are combined together, distinctly, which results in the equations

8

provided below. Equation (12) is related to flow in fracture and equation (13) is related to flow in matrix.

(1 + α )

Sfo c tfo µ fo  ∂m ( Pfg ) 1 ∂  2 ∂m ( Pfg )   Sfg   r = ω + α ωf   D f rD 2 ∂rD  ∂rD   k rfg k rfo c tfg µ fg  ∂t D  

  k rmg M mg k M −λ   + α rmo mo   k rfg M fg k rfo M fo 

   k rmg µ fg k µ  + α rmo fo  m ( Pfg )   m ( Pmg ) −   k rfo µ mo    k rfg µ mg 

12

(12)

  k rmg µ fg µ  k + β rmo fg  m ( Pmg ) ...   k rfo µ mo    k rfg µ mg −λ  2   k rmg M fg  µ fg  k M µ fo µ fg −   + β rmo fo k rfo M mo µ mo 2   k rfg M mg  µ mg     Smg S c + β (1 − ωf ) mo tmo  (1 − ωf ) k rfg k rfo c tmg 

   =    m ( Pfg )     

(13)

 ∂m ( Pmg ) ωad ∂m ( Pmg ) +  ∂t k rfg ∂t D 

1 2

α and β coefficients are calculated as defined below:

3

α(g −o) =

k rfoµfo Zfo k rfg µ fg Zfg

4

β( g − o ) =

k rfo µ mo Z mo k rfg µ mg Zmg

5 6

It should be noted that pseudo-pressure, which has been used in equations (12) and (13), is a

7

two-phase pseudo-pressure based on gas pressure that will be applied in the next equations of

8

gas condensate system model as well. The cited pseudo-pressure is defined by the notation of equation (14). P  k  k   m ( P ) = 2∫   r  +  r   PdP P0  µz    o  µz g 

(14)

9 10 11

Also, it is notable that all of the dimensionless variables in two-phase condition are in terms

12

of gas phase in fracture medium. Furthermore, for making the equations completely linear in

13

each time-step of solution, they have to be solved with known values of pressure dependent

14

parameters such as; fluid saturations, relative permeabilities, compressibility coefficients and

15

so on. Using the non-linearity coefficients of flow in matrix and fracture media in their

16

average pressures can linearize the equations and make them tractable for further calculations and solution.

13

1 2

In the following, developing the Laplace space diffusivity equations for shale gas condensate

3

reservoirs has been illustrated. Laplace transform for the equations (12) and (13) can be expressed as following form:

(1 + α )

Sfo c tfo µ fo  1 ∂  2 ∂∆m ( Pfg )   Sfg   r = ω + α ωf  s∆m ( Pfg )  D   k rfg f rD 2 ∂rD  ∂rD k rfo c tfg µ fg   

 M k µ µ k M  k −λ   mg + α rmo mo  ∆m ( Pmg ) −  rmg fg + α rmo fo     M fg k rfo M fo  k rfo µ mo  k rfg µ mg    k rmg µ fg  µ fg  k + β rmo ∆m ( Pmg )    k rfo µ mo    k rfg µ mg  = −λ  2    k rmg M fg  µ fg   µ µ k M fo fo fg  ∆m ( Pfg )  −   + β rmo 2 k rfo M mo µ mo    k M mg  µ mg      rfg   Smg S c + β (1 − ωf ) mo tmo  (1 − ωf ) k rfg k rfo c tmg 

4

Where; ∆m ( P ) = m ( Pi ) − m ( P )

   ∆m ( Pfg )    

(15)

(16)

 ω  s∆m ( Pmg ) + ad s∆m ( Pmg ) k rfg 

5 6 7

Pseudo-pressure difference of matrix can be expressed in terms of fracture pseudo-pressure difference according to the equation (16).

∆m ( Pmg ) =

k λ  rmg k  rfg

2 k  µfg  k M µ foµ fg  rmg M fg λ   + β rmo fo k rfo M mo µ mo 2   k rfg M mg  µ mg    ∆m ( Pfg ) µfg Smg k rmo µ fg   Smo c tmo ωad  +β + β (1 − ωf ) +  +  (1 − ωf ) s µ mg k rfo µ mo   k rfg k rfo c tmg k rfg 

(17)

8 9 10

By substituting the equation (17) into equation (15), the results can be shown as equation (18) which is similar to single-phase one, equation (6):

1 ∂  2 ∂∆m(Pf )   rD  = f ( s ) ∆m ( Pf ) rD2 ∂rD  ∂rD 

(18)

11 12

In which, f(s) is calculated as shown below:

14

  Sfg  M S c µ k M ωf + α fo tfo fo ωf  s − λ  mg + α rmo mo     k rfo c tfg µ fg k rfo M fo 1   k rfg   M fg f (s ) =  1 + α   k rmg µ fg k rmo µ fo   +λ  k µ + α k µ  rfo mo    rfg mg

   F       

(19)

1

F=

2

3 4

k λ  rmg k  rfg

 k M  µ 2 k M µ µ  λ  rmg fg  fg  + β rmo fo fo 2fg  k rfo M mo µ mo   k rfg M mg  µ mg    µfg S k µ   S c ω  + β rmo fg  +  (1 − ωf ) mg + β (1 − ωf ) mo tmo + ad  s µmg k rfo µmo   k rfg k rfo c tmg k rfg 

It is worth mentioning that by eliminating oil phase related terms in above f(s) formulation, the single-phase f(s) can be captured (Equation 7).

5 6

3.3. General solution of the governing equations 7 8

As it was mentioned, the ultimate-form flow equation in both single- and two-phase

9

conditions, equations (6) and (18), has a same governing equation in Laplace space, except

10

for the definition of function f(s). The solution and the procedure for solving this equation have been explained precisely by Ozkan (Ozkan et al., 1988).

11

Related initial and boundary conditions are as follows. 12

Initial condition: 13

∆m ( Pf )

tD =0

=0

(20)

Outer boundary condition: 14

∆m ( Pf )

rD → ∞

=0

(21)

Inner boundary condition:



lim 2πk f L  rD 2

ε→ 0

+



∂∆m ( Pf )  ∂rD

q% δ(t)  =− 86, 400  r =ε D

15 15

(22)

1 2

Inner boundary condition is related to the surface of a vanishingly small sphere with a pulsewise flux of the amount of q% at t=0.

3 4

The solution for the mentioned equation for the horizontal well with impermeable boundaries (Zhao et al., 2013) is provided below (Ozkan et al., 1988).

∆m ( Pf ) = 5

PscT q(s)  K0 f ( s )rD    Tsc kf h

(23)

In which, the dimensionless radius can be obtained via the below equation:

rD =

6

( x D − x wD ) + ( yD − ywD ) 2

2

7 8

The pressure-drop related to element “i” on point (x, y) can be obtained from equation (24). ∆Lfi 2 ∆L xˆ i − fi 2

Psc T qi (s)  2 2 K 0 f (s) ( x D − x wD ) + ( yD − y wD )  dx w    Tsc k f h Where; ∆Lfi is the length of hydraulic fracture i-th element. ∆mi (Pf ) = ∫

9

xˆ i +

∆Lfi =

10 11 12

14

Lf N

According to the definition of dimensionless pseudo-pressure and rate, the dimensionless pseudo-pressure of element “i” can be expressed as following form:

mDi (Pf ) = 13

(24)

q Di s∆LfDi

∆LfDi 2 ∆L − fDi 2

∫

K 0  f (s) 

( x D − xˆ Di − ξ ) + ( yD − ywDi ) 2

2

 dξ 

(25)

Where;

∆LfDi =

∆Lfi L

15 16

It should be said that qDi, which is dimensionless rate of i-th element, is a constant term due

17

to constant terminal rate assumption so its Laplace inverse can be obtained by just producing a “1/s” in the equation and its original value. Its value is defined below:

18

q Di ( s ) =

16

q i ( s ) ∆L fi q sc

1 2

Owing to the superposition theorem, the pressure-drop in each point is obtained from the

3

combination of the effect of the pressure-drop in all elements on that point, so the pressure drop in element “j” will be computed via following equation. m D ( xˆ Dj , yˆ Dj ) =

4 5 6

7

∑

11 12 13

i =1

m Di ( xˆ Dj , yˆ Dj )

(26)

MF*2 N

∑ i =1

m Di ( xˆ Dj , yˆ Dj )

(27)

It is clear that the combination of dimensionless rates equates to 1, which is referred as the below equation. MF*2 N

10

∑

The conductivity of fractures is assumed to be infinite, resulting the pressure drop in each point to be equal with that of the well. m wD =

8 9

MF*2 N

i =1

q Di = 1

(28)

With combining the pressure drop equation for each point and equation (28), MF*2N+1 equations will be obtained, which are utilized for calculating pressure drop in the well and the flow rate in each nodal point.

 A1,1  ...   A k,1   ...  A 2 N*MF,1   1

.. .. .. ..

A1,k ... A k,k ...

.. .. .. ..

A1,2N*MF ... A k,2N*MF ...

.. A 2 N*MF,k .. A 2N*MF,2N*MF .. 1 .. 1

−1  q D1  0 −1  q D1  0 −1  .   .  =   −1  .   .  −1  q DMF*2N  0     0   m wD  1 

14 15

16

In the mentioned matrix, the Ai,j-th entry can be calculated as follows:

1 Ai, j = s∆LfDi

∫

∆LfDj

−

2 ∆LfDj 2

 K0  f (s) 

( xˆ

17

17

Di

2 2 − xˆ Dj − ξ ) + ( yˆ Di − ywDj )  dξ 

(29)

1 2 3 4 5

Numerical integration and Laplace inversion with Stehfest algorithm (Stehfest, 1970) are applied for calculating the coefficients of the above equation and its ultimate solution respectively. Ultimate solution can be achieved by Gaussian elimination method. Moreover, the below equation must be applied preliminary to the inverse Laplace transform, for showing the effect of skin. (van Everdingen and Hurst, 1949)

mwD = mwDN +

Skin s

(30)

6 7 8 9

10

Also, dimensionless rate in Laplace domain (in constant terminal pressure case) can be achieved by using following expression (van Everdingen and Hurst, 1949). This is used for plotting dimensionless rate decline curve for the MFHW.

qD .mwD =

1 s2

11 12 13 14 15

The mentioned procedure of solution for single and two-phase flow conditions, must be applied for a time duration to be able to plot a curve of pseudo-pressure (or rate) values versus time. In the following, Figure 4 and 5 illustrates the workflow of solution to plot type curves within a range of time values for single and two-phase flow conditions, respectively.

16

18

1 2 3

Figure 4: Workflow of solution for plotting PTA type curves in shale gas reservoirs with MFHW (single-phase flow)

4

19

1 2 3

Figure 5: Workflow of solution for plotting PTA type curves in shale gas condensate reservoirs with MFHW (twophase flow)

4 5

20

1

4. Results

2

4.1. Results for single-phase flow

3 4 5 6

The results for the single-phase flow (dry gas) are shown below. In Figure 6, a comparison between this study and numerical model built by a commercial numerical simulator is shown. As can be observed, there is a match between these two different models capture the same physique of flow for a single-phase shale gas reservoir under a 30-year constant flow-rate production.

7

8 9 10

Figure 6: Comparison of semi-analytical and numerical results of shale gas reservoir PTA type curves (single-phase flow)

11

21

1 2 Figure 7: Effect of Langmuir adsorption model volume (VL) on PTA type curves in single-phase condition

3 4

The effect of Langmuir volume on pseudo-pressure derivative curve can be captured by a dip

5

in the curve, which can be deeper with the increase in Langmuir volume (VL), in terms of existence.

6 7

Rate decline curves also can be generated for constant wellbore flowing pressure condition

8

and transient period rate decline curves are provided in this work. The formulation of the

9

dimensionless rate is exactly in the diverse form of the dimensionless pseudo-pressure

10

(1/mD). Decline curves of constant wellbore pressure condition generated with the sensitivity of Langmuir volume are provided below, Figure 8.

22

1 2 Figure 8: Effect of Langmuir adsorption model volume (VL) on RTA type curves in single-phase condition

3 4 5 6 7

4.2. Results for two-phase flow In the case of two-phase flow, a synthetic compound of a gas condensate with the given components in Table 3 is used. The results are recorded at temperature of 350 Kelvin.

8

9

10

Table 3: Condensate gas fluid components; mole fraction

Component

Mole Fraction

Methane (C1)

0.850

Ethane (C2)

0.025

Octane (C8)

0.025

Decane (C10)

0.100

23

1 2

Similar to single-phase flow condition, the comparison between this study and numerical

3

model built by commercial numerical simulator is shown in Figure 9. In this plot, the match

4

between the two semi-analytical and numerical models can be seen; in which, both of them

5

indicate the same physique of flow for a two-phase shale gas condensate reservoir under a 30-year constant flow-rate production.

6

7 8 9 10

Figure 9: Comparison of semi-analytical and numerical results of shale gas condensate reservoir PTA type curves (two-phase flow)

11

Following figure (Figure 10) depicts the effect of the Langmuir volume on type curves could

12

be illustrated. Similar to single-phase condition, the Langmuir volume has the same sensitivity impact on the type curves in terms of two-phase pseudo-pressure concept.

13

24

1 2 3

Figure 10: Effect of Langmuir volume on PTA type curves in two-phase condition

4

As it can be observed from the above figure, there are several flow regimes in a multi-

5

fractured horizontal well (single- or two-phase flow condition). Flow regimes “A” to “E” as depicted in Figure 10, are illustrated below in Figure 11.

25

A

B

C

D

E

1 2

Figure 11: Schematic of different flow regimes in a multi-stage hydraulically fractured horizontal well

3 4

-

Flow regime A: First linear flow; This flow regime is the linear flow from matrix to each half fracture, which can be diagnosed by a “1/2” slope of pseudo-pressure derivative

5

curve. 6 7 8

-

Flow regime B: First elliptical flow; By propagating pressure reduction pulse, an elliptical flow emerges around each whole-fracture. This regime signature is a “1/3” slope of pseudo-pressure derivative curve.

26

1 2

-

Flow regime C: Second linear flow; By more propagating pressure reduction pulse, another linear flow regime takes place from both lateral sides of system into fractures and

3

the well. 4 5

-

Flow regime D: Second elliptical flow; In the next propagation effect, a late elliptical flow can be seen toward the whole system of fractures and the horizontal well.

6 7

-

Flow regime E: Infinite acting radial flow (IARF); Finally, the pressure variations

8

resulted in a fully radial flow, which is formed as a response of the homogenous region of

9

reservoir (matrix) to the whole system of fractures and well (stimulated reservoir volume, SRV).

10 11

Second linear and elliptical flows also can be detected by the signatures mentioned at “A” and “B” regions.

12 13

The flow regime between “D” and “E” is the interporosity flow which is controlled by both matrix free and adsorbed gas flow into natural fractures.

14

27

1 2

Figure 12: Effect of horizontal well length on PTA type curves in two-phase condition

3 4 5 6 7 8

The effect of horizontal well length on type curves has been shown in Figure 12. Horizontal well length is capable to control the intensity of the linear and elliptical flows of the horizontal well flow regimes. It is obvious that less well length leads to longer first and second linear and elliptical flows, later inter-porosity and fully radial flow and totally lower dimensionless pseudopressure values (at least in our model).

9

28

1 2

Figure 13: Effect of number of multi-fractures on PTA type curves in two-phase condition

3 4 5 6 7

The effect of quantity of the multiple hydraulic fractures has been shown in the provided graph, Figure 13. In terms of two-phase flow condition; the number of hydraulic fractures, affects the duration of the first linear and elliptical flows. The longer first linear and elliptical flows are witnessed by the decrease in the fracture numbers.

8 9 10 11

Figure 14 shows the impact of Langmuir volume on rate decline and rate decline derivative curves. In terms of two-phase flow dimensionless pseudo-pressure curve, a dip is formed which can be affected by the Langmuir volume, in terms of existence. The more Langmuir volume is, the deeper dip will be.

29

1 2

Figure 14: Effect Langmuir volume on RTA type curves in two-phase condition

3 4 5 6

More scrutiny on the multiple-porosity dip in pseudo-pressure derivative yields the three below figures, which show the effect of storability ratio of fracture system (ωf), storability ratio of adsorbed gas system (ωad) and interporosity flow coefficient (λ) on this dip.

7 8 9 10

Sensitivity of ωf controls the intensity of dip of the derivative plots and also the linear and elliptical flow regions. The higher the value of ωf is, the shallower multiple-porosity dip will be. Moreover, the linear and elliptical flow region on the derivative plots will be deeper (Lower dimensionless pseudo-pressure values, or in other words lower draw-down).

30

1 2

Figure 15: Effect of storability ratio of fracture system on PTA type curves in two-phase condition

3 4 5

The sensitivity to ωad is very similar to Langmuir volume (VL) sensitivity on type curves in interporosity dip but more gently.

6 7

Figure 16: Effect of storability ratio of adsorbed gas on PTA type curves in two-phase condition

8

31

1 2

Interporosity flow coefficient (λ), controls the initiation point of the dip on derivative plots. The higher the value of λ is, the sooner the start of dip will be.

3 4

Figure 17: Effect of interporosity flow coefficient on PTA type curves in two-phase condition

5 6 7 8 9 10 11

The special effects of two-phase flow condition in a lower dew point pressure gas condensate system, investigated below with sensitivity on relative permeability of gas and oil phases Corey exponents in both matrix and fracture media. Corey model (Corey, 1954) exponent of each medium-fluid decrease with its relative permeability increment, so high exponents imply low corresponding relative permeability values. The Corey model relative permeability calculation can be performed by (Corey, 1954):

12

S*o = 13

So − Soc 1 − Soc − Sgc

K ro = ( S*o ) K rg = (1 − S*o )

ng

14

32

no

(1 − (S ) ) * ng o

1 2

Figure 18: Effect of matrix gas relative permeability Corey exponent on PTA type curves in two-phase condition

3

4 5

Figure 19: Effect of matrix oil relative permeability Corey exponent on PTA type curves in two-phase condition

6 7 8

Figures 18 and 19 depict the effect of matrix oil and gas relative permeability on type curves in terms of Corey exponents. These exponent values have an effect similar to λ on the dip of the

33

1 2

derivative plots. Their lower values (higher matrix gas and oil relative permeability contents) provide sooner start of the dip and also the lower amounts of dimensionless pseudo-pressure.

3

4 5

Figure 20: Effect of fracture gas relative permeability Corey exponent on PTA type curves in two-phase condition

6

7 8

Figure 21: Effect of fracture oil relative permeability Corey exponent on PTA type curves in two-phase condition

9 34

1 2 3 4 5

The effect of fracture oil relative permeability values has resulted in lower values (higher fracture oil relative permeability contents) providing the later initiation of the dip and also the higher amounts of dimensionless pseudo-pressure. Furthermore, fracture gas relative permeability values affect linear and elliptical flow regions; the lower nfg (higher krfg) is, the higher dimensionless pseudo-pressure values are.

6 7

4.3. Comparison between single-phase and two-phase flows

8 9 10 11

In this study, two different models are considered for single-phase flow (shale gas reservoir) and two-phase flow (shale gas condensate reservoir). The two phase flowing model is extension of single dry gas model in single-phase but considering the effect of condensate drop out and changing relative permeability.

12 13 14

In this section, a comparison approach is considered between these two flow conditions and it is shown in Figure 22. In this figure, wellbore flowing pressure of gas is plotted versus production time for both flow conditions.

15

16 17 18

Figure 22: Comparison of wellbore flowing pressure reduction in shale gas reservoir (single-phase) and shale gas condensate reservoir (two-phase)

19 20 21

Regarding to the figure above, trend of wellbore flowing pressure reduction in shale gas reservoir and shale gas condensate reservoir base case models are in a reasonable match with 35

1 2 3 4 5 6 7 8

their corresponding numerically derived trend using commercial reservoir simulator results. These trends for single-phase and two-phase flows express that the amount of wellbore flowing pressure values in a shale gas condensate reservoir is expected to be lower than in a shale gas reservoir (with exactly the same reservoir properties) due to difference in fluid model and consequently the effect of changing the relative permeability. It is worth mentioning that the current study is developed for a relatively simple fracture geometry and without considering phenomena like non-darcy flow and positive inertia for modeling gas condensate. These can be done in future works.

9 10 11

5. Conclusions

12 13 14 15 16

In this study, a new semi-analytical model was proposed for modeling the multistage hydraulically fractured wells in shale gas condensate reservoirs. This solution is general and can work for both single- (Zhao et al., 2013) and two-phase flow. This can be used for analyzing pressure and rate data in shale gas condensate reservoirs. Following conclusions are derived from this study:

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

1- A semi-analytical mathematical model developed for fluid flow through a multi-stage hydraulically fractured well in a shale gas condensate reservoir for the purpose of pressure transient or rate transient analysis using type curves. The result of the current study is compared with the result from the numerical simulator and consistent match is observed. 2- Using rich gas production data (rate and pressure) and proposed model for shale gas condensate reservoir, PTA and RTA curves can be plotted with a shape similar to the curves for shale gas reservoirs. 3- The effect of different parameters on pseudo-pressure and production rate data are investigated in this study. These parameters are; A) Horizontal well length (2L); Controls linear and elliptical flows. B) The number of multi-fractures (MF); Controls the early linear and elliptical flows. The lower values of A and B provide longer linear and elliptical flow duration in type curves. C) Adsorbed gas storability (ωad); Its bigger values yield more desorption of adsorbed gas. The higher ωad values are, the deeper dip of the derivative curves will be. D) The relative permeability saturation exponents of Corey model; The more the values of krmo and krmg or the less the values of krfo are, the earlier the start of the dip in type curves will be. Besides, higher krfg provides more mwD and qD values.

36

36

1 2 3 4 5 6

Nomenclature

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Latin Letters c: Isothermal compressibility coefficient [Pa-1] c : Average isothermal compressibility coefficient [Pa-1] cgp: Pseudo compressibility of adsorbed gas [Pa-1] k: Absolute permeability [m2] kr: Relative permeability lc: Characteristic length [m] Mg: Gas molecular weight [kg/kmol] Mo: Oil molecular weight [kg/kmol] n: Number of normal set of fractures PL: Langmuir pressure [Pa] R: Universal gas constant, R=8.3144598 [J.mol-1.K-1] s: Laplace variable S: Skin factor Sfluid: Fluid saturation V: Average matrix gas volume concentration [sm3/m3] VE: Volume of gas adsorbed per unit of shale gas reservoir in equilibrium at the pressure Pm [sm3/m3] VL: Langmuir volume [sm3/m3] x, y and z: Distance coordinates [m] Z: Deviation factor P: Pressure [Pa] q: Flow rate [m3/day] t: Time [s] h: Formation thickness [m] Lf: Fracture half-length [m] r: Radial distance [m] T: Temperature [K]

36 37

Greek Letters α: Warren-Root model shape factor 37

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

∆Lfi: Length of hydraulic fracture i-th element (m) µ: Fluid viscosity [Pa.s] ρ: Fluid density [kg/m3] φ : Porosity δ: Dirac delta function symbol Subscripts f: Fracture m: Matrix o: Oil g: Gas sc: Standard condition t: Total r: Rock D: Dimensionless i: Initial w: Well L: Langmuir

20

Acknowledgment 21 22 23 24

The authors appreciate KAPPA Engineering for providing an academic license of KAPPA Workstation 5.12.04 – Rubis module for Amirkabir University of Technology. Also, authors would like to appreciate reviewers whose creative comments helped this study to be more valuable.

25

26

Appendix

27 28

Appendix A: Derivation of the equations for single-phase condition (dry gas case) (Zhao et al., 2013)

29 30

By using mass balance equation in the fracture medium for gas and considering Darcy flow for fluid velocity (Darcy, 1856) in spherical coordinates, the result is followed:

1 ∂ 2 ∂ 1 ∂  P M k f ∂Pf  ∂ r ρf u ) = ( ρf φf ) − q m ⇒ 2  r 2 f (  = ( ρf φf ) − q m 2 r ∂r ∂t r ∂r  ZRT µ ∂r  ∂t

(A.1)

31 32 33

The right term of the equation can be rearranged by using derivatives chain rule and definition of the fluid and rock compressibility. 38

cg = 1

1 ∂ρ ρ ∂Pf

cr,f =

1 ∂φf φf ∂Pf

∂ ( φf ρf ) ∂P = φf ρf ctf f ∂t ∂t

(A.2)

2 3 4 5

In equation (A.1), the term “qmg” is the amount of transferred gas from matrix to the fracture as a result of the pressure difference between the fracture and matrix media, which is calculated via the following equation (Warren and Root, 1963):

6

km ( ρm Pm − ρf Pf ) µ Where; α is the shape factor and can be defined as follows.

7

α=

8

lc is considered equals to average Lf value of all hydraulic fractures.

qm = α

(A.3)

4n ( n + 2 ) lc 2

9 10 11

By substitution of equations (A.2) and (A.3) into (A.1) and using the definition of gas density, the new equation is: ∂P 1 ∂  2 Pf M k f ∂Pf  Pf M k P M PM  c tf f − α m  m Pm − f Pf  r  = φf 2 r ∂r  ZRT µ ∂r  ZRT ∂t µ  ZRT ZRT 

(A.4)

12 13

Above equation is rearranged with introducing pseudo-pressure as:

14

m ( P ) = 2∫

15

∂m ( P ) 2P = ∂P µz

P dP P0 µz P

∂m ( Pf ) 1 ∂  2 ∂m ( Pf )  µ k ctf − α m ( m ( Pm ) − m ( Pf ) ) r  = φf 2 r ∂r  ∂r  kf ∂t kf

39

(A.5)

1 2

In addition to pseudo-pressure definition, also the average compressibility value is used to make the equation linear.

3

On the other side, mass balance equation in matrix medium is:

4

− ( q m + q* ) =

∂ ( ρ m φm ) ∂t

(A.6)

5 6

7

In which q* is the amount of the diffusion rate of gas from rock particles and can be calculated via below equation:

∂ (1 − φm − φf ) V  MPsc (1 − φm − φf ) ∂V q* = ρsc  = ∂t RTsc ∂t

(A.7)

8

Variations of the volume in above equation is calculated as follows:

9

VL Pmg ∂Pm ∂VE VL ∂Pm VL PL ∂Pm = − = 2 2 ∂t PL + Pm ∂t ( PL + Pm ) ∂t ( PL + Pm ) ∂t

(A.8)

10

The right term of matrix mass balance equation (similar to fracture’s) can be simplified as:

11

∂ ( φmρm ) ∂P = φmρmctm m ∂t ∂t

(A.9)

12

By substituting the equations (A.7), (A.8) and (A.9) into equation (A.6), the results are:

−α 13

MP (1 − φm − φf ) VL Pm ∂Pm km ∂P ( ρm Pm − ρf Pf ) = φmρmc tm m + sc 2 µ ∂t RTsc ( PL + Pm ) ∂t

(A.10)

14 15

16

The mentioned equation is rephrased by using the definition of density and pseudo-pressure in it. Also, average compressibility values used to make the equation linear. ∂m ( Pm ) ∂m ( Pm ) + φ m µ m cgp ∂t ∂t Where, the average pseudo compressibility of adsorbed gas is:

(

)

−αk m m ( Pmg ) − m ( Pfg ) = φ m µ m ctm

40

(A.11)

1

cgp =

Psc T (1 − φ m − φ f ) Z mg φ m Tsc

(P

L

VL

+ Pmg )

2

2

It is noticeable that average matrix pressure ( Pmg ) is the average pressure of reservoir matrix

3 4 5 6

over the entire reservoir dimension. This matrix average pressure can be calculated via arithmetic average of wellbore flowing pressure and initial pressure in each time step of solution as shown in Equation (A.12). Furthermore, average Z factor of matrix gas is calculated from the (Dranchuk and Abou-Kassem, 1975) method at average matrix pressure.

Pmg = 7

Pi + Pwf 2

(A.12)

8 9

10

Equations (A.5) and (A.11) are rearranged by using dimensionless parameters which are brought in the Table 2.

∂m ( Pf ) 1 ∂  2 ∂m ( Pf )  − λ ( m ( Pm ) − m ( Pf ) )  rD  = ωf 2 rD ∂rD  ∂rD  ∂t D ∂m ( Pm ) ∂m ( Pm ) −λ ( m ( Pm ) − m ( Pf ) ) = (1 − ωf ) + ωad ∂t D ∂t D

(A.13) (A.14)

11

Appendix B: Derivation of the equations for two-phase flow condition (condensate gas case) 12 13 14

The liquid phase will be appeared if the pressure is dropped below dew point and the condition will be altered to two-phase flow. In which, derivation of the equations is similar to single-phase flow in some features. The procedure of proofing the equations is brought in the following steps.

15 16 17

Gas/oil flow is infinite in all directions from matrix medium into fracture medium. Therefore, mass balance in spherical coordinates must be used. Mass balance in this coordinate system for fracture medium can be written as follows: 1 ∂ 2 ∂ r ρ f,p u f,p ) ±∑ q g/d = ( ρ f,p φf Sf,p ) ( 2 r ∂r ∂t

(B.1)

18 19 20

In the above equation, “qg/d” is rate of generation (and/or depletion) of mass per unit volume and subscripts of f and p refer to fracture and phase, respectively. By using mass balance equation in the fracture medium for gas phase, the result is as follows:

41

1 ∂ 2 ∂ r ρ fg u fg ) = ( ρ fg φf Sfg ) − q mg ( 2 r ∂r ∂t

(B.2)

1 2 3

The term “qmg” is the amount of transferred gas from matrix to the fracture as a result of the pressure difference between the fracture and matrix. Replacing Darcy's law (Darcy, 1856) with the gas flow rate term (ufg) will result in the following equation: 1 ∂  2 Pfg M fg k f k rfg ∂Pfg  ∂ r  = ( ρfg φf Sfg ) − q mg r 2 ∂r  Zfg RT µ fg ∂r  ∂t

(B.3)

4

5 6

The right side of the equation can be rephrased by using derivatives chain rule and the definition of the fluid and rock compressibility. ∂ ( φf ρfg Sfg )

7

= Sfg

+ φf ρ fg

∂Sfg

∂t ∂t ∂t 1 ∂ρfg  cg = ∂ ( φf ρfg ) ρ fg ∂Pfg  ∂P = φf ρfg c tfg fg ⇒ ∂t ∂t 1 ∂φf  c r,f = φf ∂Pfg  ∂ ( ρfg φf Sfg )

8

∂ ( φf ρfg )

∂t

∂P ∂S   = φf ρfg  Sfg c tfg fg + fg  ∂t ∂t  

(B.4)

9

The term “qmg” can be calculated via the below equation (Warren and Root, 1963): q mg = α

k m k rmg µ mg

(ρ

P − ρfg Pfg )

(B.5)

mg mg

10 11 12

By substitution of equations (B.4) and (B.5) into (B.3) and using definition of gas density (ρ=PM/ZRT), the new equation is followed as: Pfg M fg ∂P P M ∂S 1 ∂  2 Pfg M fg k f k rfg ∂Pfg  Sfg c tfg fg + φf fg fg fg  r  = φf 2 r ∂r  Zfg RT µ fg ∂r  Zfg RT ∂t Zfg RT ∂t  k k P M P M −α m rmg  mg mg Pmg − fg fg Pfg   µ mg  Zmg RT Zfg RT  42

(B.6)

1 2

Above equation can be reformed by using the pseudo-pressure definition, which was mentioned in single-phase condition.

3

∂m ( Pfg ) P ∂Sfg  µ fg  1 ∂  2 ∂m ( Pfg )  r  = φf  Sfg c tfg  + 2 fg 2 r ∂r  ∂r  k f k rfg  ∂t µ fg Zfg ∂t       k k M µ −α m rmg  mg m ( Pmg ) − fg m ( Pfg )   k f k rfg  M fg µ mg 

(B.7)

4

Equation (B.8) below defines the pseudo-pressure function (Al-Hussainy et al., 1966).

m ( p ) =2∫ 5

P

P0

P dP µZ

(B.8)

6 7

As the deliverability in matrix medium is considered zero, generation of mass, depletion of mass

8

and mass accumulation terms will form mass balance in matrix medium. Mass balance equation for matrix medium is:

9

− ( q mg + q* ) =

∂ ( ρmgφmSmg ) ∂t

(B.9)

10 11

q* is the amount of the diffusion rate of gas from rock particles that can be calculated by the following equation:

12

∂ (1 − φm − φf ) V  Mmg Psc (1 − φm − φf ) ∂V = q* = ρsc  ∂t RTsc ∂t

(B.10)

13 14

15

Variations of the volume in the above equation, which is calculated via Langmuir adsorption model, is shown below (Langmuir, 1916): ∂Pmg VL Pmg ∂Pmg ∂VE VL ∂Pmg VL PL = − = 2 2 ∂t PL + Pmg ∂t ( PL + Pmg ) ∂t ( PL + Pmg ) ∂t

(B.11)

Therefore, with respect to the equation (B.11), equation (B.10) can be redefined as follows:

43

∂ (1 − φm − φf ) V  M mg Psc (1 − φm − φf ) VL Pmg q* = ρsc  = ∂t RTsc (P + P L

1

mg

)

∂Pmg 2

∂t

(B.12)

2

The right term of mass balance equation (such as fracture) can be simplified as: ∂ ( φmρ mgSmg )

∂P ∂S   = φmρ mg  Smg c tmg mg + mg  ∂t ∂t  

∂t

3

(B.13)

By substituting the above equation, the result is: ∂P ∂S   P − ρfg Pfg ) = φm ρmg  Smg c tmg mg + mg  µ mg ∂t ∂t   M mg Psc (1 − φm − φf ) VL Pmg ∂Pmg

−α + 4

k m k rmg

(ρ

mg mg

RTsc

(P

L

+ Pmg )

2

(B.14)

∂t

5 6

The mentioned equation is rephrased by using the definition of density and pseudo-pressure in it. (As it was applied on equations B.6 and B.7 for the fracture).

 ∂m ( Pmg )   µfg Pmg ∂Smg   M fg m ( Pfg )  = φmµmg M mg  Smg c tmg −αk m k rmg  M mg m ( Pmg ) − +2    µ mg ∂t µmg Zmg ∂t      +

M mg Psc T (1 − φm − φf ) µmg Zmg Tsc

(P

L

∂m ( Pmg )

VL

+ Pmg )

2

(B.15)

∂t

7 8

In which, pseudo compressibility of adsorbed gas is: c gp = 9

Psc T (1 − φ m − φ f ) Z mg φ m Tsc

(P

L

VL

+ Pmg )

(B.16)

2

10 11

The ultimate equations for fracture and matrix (B.7 and B.15) are followed below using dimensionless parameters provided in Table 2.

44

∂m ( Pfg ) ωf 2Pfg ∂Sfg 1 ∂  2 ∂m ( Pfg )  Sfg  rD = ωf + 2 rD ∂rD  ∂rD  k rfg ∂t D k rfg µ fg Zfg c tfg ∂t D   k M  µ k −λ  rmg mg m ( Pmg ) − rmg fg m ( Pfg )  k M  k rfg µ mg fg  rfg  2 k  k rmg M fg  µ fg  Smg ∂m ( Pmg ) rmg µ fg −λ  m ( Pmg ) −   m ( Pfg )  = (1 − ωf )  k rfg µ mg k rfg M mg  µ mg   k rfg ∂t D   + 1

(1 − ωf ) k rfg

2Pmg

∂Smg

µ mg Z mg c tmg ∂t D

+

ωad ∂m ( Pmg ) k rfg ∂t D

(B.17)

(B.18)

2 3 4

Similar to the proved equations for the gas phase in the fracture and matrix (equations B.17 and B.18), the equations for the oil phase can also be proved. The capillary pressure is ignored in the model, so the pressures of oil and gas phases are equal.

5 6

The same procedure of equations (B.3) to (B.4), (B.5) and (B.7) is as below for the equations (B.19) to (B.22):

1 ∂  2 Pfg M fo k f k rfo ∂Pfg  ∂ r  = ( ρ fo φf Sfo ) − q mo r 2 ∂r  Zfo RT µ fo ∂r  ∂t

(B.19)

7

∂ ( φ f ρfoSfo )

8

= Sfo

∂ ( φf ρfo )

− φ f ρfo

∂Sfg

∂t ∂t ∂t 1 ∂ρfo  co = ρfo ∂Pfg  ∂Pfg ∂ ( φf ρfo ) = φf ρ fo c tfo ⇒ ∂t ∂t 1 ∂φf  c r ,f =  φf ∂Pfg 

∂P ∂S  ∂ ( ρfoφf Sfo )  = φf ρfo  Sfoctfo fg − fg  ∂t ∂t ∂t   q mo = α

(B.20)

k m k rmo ( ρ mo Pmg − ρ fo Pfg ) µ mo

(B.21)

9

45

1

Pfg M fo ∂Pfg Pfg M fo ∂Sfg 1 ∂  2 Pfg M fo k f k rfo ∂Pfg  Sfo c tfo − φf r  = φf 2 r ∂r  Z fo RT µ fo ∂r  Z fo RT ∂t Z fo RT ∂t Pfg M fo  k k  Pmg M mo Pmg − Pfg  −α m rmo  Z fo RT µ mo  Z mo RT 

∂m ( Pfg ) 2Pfg ∂Sfg  µ fo  1 ∂  2 ∂m ( Pfg )      = φ − r S c f fo tfo r 2 ∂r  ∂r  k f k rfo  ∂t µ fo Zfo ∂t     

2

 k k M µ −α m rmo  mo m ( Pmg ) − fo m ( Pfg )  µ mo k f k rfo  M fo 

(B.22)

3 4

Furthermore, the alternatives of matrix equations (B.9, B.13, B.14 and B.15) for oil phase are provided below;

∂ ( ρmoφmSmo ) ∂t ∂P ∂S  ∂ ( φmρmoSmo )  = φmρmo  Smo ctmo mg − mg  ∂t ∂t ∂t   ∂P ∂S   k k −α m rmo ( ρmo Pmg − ρfo Pfg ) = φmρmo  Smo ctmo mg − mg  µmo ∂t ∂t     µ −αk m k rmo  M mo m ( Pmg ) − fo M fo m ( Pfg )  = µ mo   −q mo =

 ∂m ( Pmg ) P ∂Smg φmµ mo M mo  Smo c tmo − 2 mg  ∂t µ mo Z mo ∂t 

   

(B.23) (B.24) (B.25)

(B.26)

5 6

Finally, the alternatives of (B.17) and (B.18) for oil phase are as following: ∂m ( Pfg ) ωf 2Pfg ∂Sfg 1 ∂  2 ∂m ( Pfg )  Sfo c tfo µ fo   r = ω − D f 2 rD ∂rD  ∂rD  k rfo c tfg µ fg ∂t D k rfo µ fo Z fo c tfg ∂t D   k M  k µ −λ  rmo mo m ( Pmg ) − rmo fo m ( Pfg )  k rfo µ mo  k rfo M fo 

46

(B.27)

1 2 3 4 5 6 7

k µ  k M µ µ S c ∂m ( Pmg ) −λ  rmo fg m ( Pmg ) − rmo fo fo 2fg m ( Pfg )  = (1 − ωf ) mo tmo k rfo M mo µ mo k rfo c tmg ∂t  k rfo µ mo  (1 − ωf ) 2Pmg ∂Smg − k rfo µ mo Zmo c tmg ∂t

(B.28)

It should be said that all the equations derived in the Appendix B are still non-linear. For making the differential equations linear, the non-linearity coefficients of matrix and fracture media (mentioned in Figure 5) have to be evaluated in average matrix pressure and average fracture pressure, respectively. These average pressure values, same as the simplified approach in singlephase flow condition, are determined by arithmetic averaging the wellbore flowing pressure and initial pressure as shown in equation (B.29).

Pm or f = 8

Pi + Pwf 2

(B.29)

Appendix C: Related equations in two-phase flow solution 9

Computing saturation of the phases by flash calculation 10

Saturation of gas and liquid phases can be computed via following equations:

vg =

Zg RT

P Zo RT vo = P

Sg = Sg = 11

(C.1)

n v vg

n v vg + (1 − n v ) vo

(1 − n v ) vo n v vg + (1 − n v ) vo

12

Calculating the viscosity of the equations 13 14 15

Lee et al. (Lee et al., 1966) equation is used to calculate the viscosity for gas phase. Similarly; this equation can be used for the condensate phase viscosity, because of the high ratio of methane, which provide a proper precision.

µg = K1 exp ( XρY )

(C.2) 47

1 2

In which the relative parameters are:

ρ=

PM g Zg RT

( 0.00094 + 2 ×10 M ) T = ( 209 + 19M + T ) −6

K1

3

1.5

g

g

986 + 0.01M g T Y = 2.4 − 0.2X

X = 3.5 +

4 5

Calculating the relative permeability 6

Relative permeabilities are calculated from Corey model. (Corey, 1954) S*o =

So − Soc 1 − Soc − Sgc

K ro = (S*o )

4

(C.3) 2

7

(

K rg = (1 − S*o ) 1 − (S*o )

2

)

8

Calculating gas and oil phases compressibility 9 10

Isothermal compressibility coefficient of gas and oil is calculated from the equation provided below (Ahmed, 1989):

cg = 11

1 1  ∂Z  −   P Z  ∂P T

(C.4)

12 13

Also, this formulation can be used for oil phase due to the reason mentioned in the gas viscosity calculation section.

14

15 16

References 17

Ahmed, T., 1989. Hydrocarbon Phase Behavior. Gulf Publishing Company, Houston, Texas. 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Al-Hussainy, R., Ramey Jr., H.J. and Crawford, P.B., 1966. The Flow of Real Gases through Porous Media. Journal of Petroleum Technology: 625-636. Al-Twaijri, M.H., 2017. A comprehensive numerical model for simulating two-phase flow in shale gas reservoirs with complex hydraulic and natural fractures, The University of Texas at Austin. Baihly, J.D., Altman, R.M., Malpani, R. and Luo, F., 2010. Shale Gas Production Decline Trend Comparison over Time and Basins SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, Florence, Italy. Behmanesh, H., Hamdi, H. and Clarkson, C.R., 2013a. Production data analysis of liquid rich shale gas condensate reservoirs, SPE Unconventional Resources Conference Canada. Society of Petroleum Engineers. Behmanesh, H., Hamdi, H. and Clarkson, C.R., 2013b. Production Data Analysis of Liquid Rich Shale Gas Condensate Reservoirs, SPE Unconventional Resources Conference. Soicety of Petroleum Engineers, Calgary, Alberta, Canada. Behmanesh, H., Hamdi, H. and Clarkson, C.R., 2015a. Analysis of transient linear flow associated with hydraulically-fractured tight oil wells exhibiting multi-phase flow, SPE Middle East Unconventional Resources Conference and Exhibition. Society of Petroleum Engineers, Muscat, Oman. Behmanesh, H., Hamdi, H. and Clarkson, C.R., 2015b. Production data analysis of tight gas condensate reservoirs. Journal of Natural Gas Science and Engineering, 22: 22-34. Behmanesh, H., Hamdi, H. and Clarkson, C.R., 2017. Production data analysis of gas condensate reservoirs using two-phase viscosity and two-phase compressibility. Journal of Natural Gas Science and Engineering, 47: 47-58. Bello, R.O., 2009. RATE TRANSIENT ANALYSIS IN SHALE GAS RESERVOIRS WITH TRANSIENT LINEAR BEHAVIOR, Texas A&M University, 174 pp. Belyadi, A.M., Aminian, K., Ameri, S. and Boston, A., 2010. Performance of the hydraulically fractured horizontal wells in low permeability formation, SPE Eastern Regional Meeting. Brown, M., Ozkan, E., Raghavan, R. and Kazemi, H., 2011. Practical Solutions for Pressure Transient Responses of Fractured Horizontal Wells in Unconventional Reservoirs. SPE Reservoir Evaluation & Engineering, 14(06): 663-676. Cheng, Y., 2011. Pressure transient characteristics of hydraulically fractured horizontal shale gas wells, SPE Eastern Regional Meeting. Clarkson, C.R., Nobakht, M., Kaviani, D. and Ertekin, T., 2011. Production Analysis of Tight Gas and Shale Gas Reservoirs Using the Dynamic-Slippage Concept North American Unconventional Gas Conference and Exhibition. Society of Petroleum Engineers, The Woodlands, Texas, USA. Clarkson, C.R. and Qanbari, F., 2015. History-Matching and Forecasting Tight Gas Condensate and Oil Wells Using the Dynamic Drainage Area Concept, SPE/CSUR Unconventional Resources Conference. Society of Petroleum Engineers, Calgary, Alberta, Canada. Clarkson, C.R. and Williams-Kovacs, J.D., 2013. A new method for modeling multi-phase flowback of multi-fractured horizontal tight oil wells to determine hydraulic fracture properties, SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, New Orleans, Louisiana, USA. Corey, A.T., 1954. The interrelation between gas and oil relative permeabilities. Producers Monthly, 19: 38–41. Crosby, D., Rahman, M.M., Rahman, M. and Rahman, S., 2002. Single and multiple transverse fracture initiation from horizontal wells. Journal of Petroleum Science and Engineering, 35: 191-204. Darcy, H., 1856. Les fontaines publiques de la ville de Dijon. Victor Dalmont, Paris. Dranchuk, P.M. and Abou-Kassem, H., 1975. Calculation of Z Factors For Natural Gases Using Equations of State. Journal of Canadian Petroleum Technology, 14(3). Duong, A.N., 2011. Rate-Decline Analysis for Fracture-Dominated Shale Reservoirs. SPE Reservoir Evaluation & Engineering, 14(03): 377-387.

49

1 2

Fan, D. and Ettehadtavakol, A., 2017. Analytical model of gas transport in heterogeneous hydraulicallyfractured organic-rich shale media

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Fuel, 207: 625-640. Fevang, O., 1995. Gas Condensate Flow Behavior and Sampling, Norges Tekniske Hogskole. Fussel, D.D., 1973. Single well performance for gas condensate reservoirs. Journal of Petroleum Technology, 25(07): 860-870. Gao, D., Liu, Y., Wang, D. and Han, G., 2019. Numerical Analysis of Transient Pressure Behaviors with Shale Gas MFHWs Interference. Energies, 12(2). Gringarten, A.C., Al-Lamki, A. and Daungkaew, S., 2000. Well Test Analysis in Gas-Condensate Reservoirs, SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, Dallas, Texas. Gu, D. et al., 2017. Pressure transient analysis of multiple fractured horizontal wells in naturally fractured unconventional reservoirs based on fractal theory and fractional calculus. Petroleum, 3(3): 326339. Guo, K., Zhang, B., Aleklett, K. and Hook, M., 2016. Production Patterns of Eagle Ford Shale Gas: DeclineCurve Analysis Using 1084 Wells. Sustainability, 8(10). Haji Seyedi, S.H., Jamshidi, S. and Masihi, M., 2014. A novel method for prediction of parameters of naturally fractured condensate reservoirs using pressure response analysis. Journal of Natural Gas Science and Engineering, 19: 13-22. He, Y. et al., 2016. A Semianalytical Methodology To Diagnose the Locations of Underperforming Hydraulic Fractures Through Pressure-Transient Analysis in Tight Gas Reservoir. SPE Journal, 22(03). He, Y. et al., 2018. An Improved Rate-Transient Analysis Model of Multi-Fractured Horizontal Wells with Non-Uniform Hydraulic Fracture Properties. Energies, 11. Hill, D.G. and Nelson, C., 2000. Gas productive fractured shales: an overview and update. Gas Tips, 6: 413. Ibrahim, M., Vosburgh, E., Darabel, A., Olayinka, O. and Larsen, S., 2017. Pressure Transient Analysis for a Unique Shale Gas Condensate Well, Actual Field Case, SPE Annual Technical Conference and Exhibition Society of Petroleum Engineers, San Antonio, Texas, USA. Indriati, S. and Wang, X., 2002. International Symposium and Exhibition on Formation Damage Control Society of Petroleum Engineers, Lafayette, Louisiana Jones, J.R. and Raghavan, R., 1988. Interpretation of flowing well response in gascondensate wells. SPE Formation Evaluation, 3(03). Jones, J.R., Vo, D.T. and Raghavan, R., 1989. Interpretation of pressure-buildup responses in gas condensate wells. SPE Formation Evaluation, 4(01). Kenomore, M., Hassan, M., Malakooti, R., Dhakal, H. and Shah, A., 2018. Shale gas production decline trend over time in the Barnett Shale. Journal of Petroleum Science and Engineering, 165: 691710. Khanal, A., Khoshghadam, M., Lee, J.W. and Nikolaoua, M., 2017. New forecasting method for liquid rich shale gas condensate reservoirs with data driven approach using principal component analysis. Journal of Natural Gas Science and Engineering, 38: 621-637. Labed, I., 2016. Gas-condensate flow modelling for shale gas reservoirs, Robert Gordon University, 166 pp. Langedijk, R.A. et al., 2000. Optimization of Hydraulic Fracturing in a Deep, Multilayered, GasCondensate Reservoir, SPE Annual Technical Conference and Exhibition Society of Petroleum Engineers, Dallas, Texas. Langmuir, L., 1916. The constitution of fundamental properties of solids and liquids Journal of American Chemical Society, 38: 2221-2295. Larsen, L. and Hegre, T., 1994. Pressure transient analysis of multifractured horizontal wells, SPE Annual Technical Conference and Exhibition. 50

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

Lee, A.L., Gonzalez, M.H. and Eakin, B.E., 1966. The Viscosity of Natural Gases. Journal of Petroleum Technology, 18(08): 997–1000. Li, Y., Zuo, L., Yu, W. and Chen, Y., 2018. A Fully Three Dimensional Semianalytical Model for Shale Gas Reservoirs with Hydraulic Fractures. Energies 11. Liang, P., Aguilera, R. and Mattar, L., 2017. A New Method for Production-Data Analysis and Well Testing by Use of Superposition Rate, Unconventional Resources Technology Conference. Society of Petroleum Engineers, San Antonio, Texas, USA. Mahmoodi, S., Abasi,M and Sharifi,M., New fluid flow model for hydraulic fractured wells with nonuniform fracture geometry and permeability, Journal of Natural Gas Science and Engineering, 2019,vol 68, p 102914. Mattar, L., Anderson, D. and Stotts, G., 2006. Dynamic Material Balance-Oil-or Gas-in-Place Without Shut-Ins Journal of Canadian Petroleum Technology, 45(11). Mattar, L. and McNeil, R., 1998. The "Flowing" Gas Material Balance. Journal of Canadian Petroleum Technology, 37(02). Mayerhofer, M.J., Lolon, E.P., Youngblood, J.E. and Heinze, J.R., 2006. Integration of microseismicfracture-mapping results with numerical fracture network production modeling in the Barnett Shale, SPE Annual Technical Conference and Exhibition. McNeil, R., 1995. The "Flowing" Material Balance Procedure Annual Technical Meeting. Petroleum Society of Canada, Alberta Medeiros, F., Kurtoglu, B., Ozkan, E. and Kazemi, H., 2007. Pressure-transient performances of hydraulically fractured horizontal wells in locally and globally naturally fractured formations, International Petroleum Technology Conference. Miao, Y. et al., 2018. A new rate-transient analysis model for shale gas reservoirs coupled the effect of slip flow and surface diffusion. International Journal of Heat and Mass Transfer, 124: 1-10. Nobakht, M. and Clarkson, C.R., 2011a. A New Analytical Method for Analyzing Production Data from Shale Gas Reservoirs Exhibiting Linear Flow: Constant Pressure Production North American Unconventional Gas Conference and Exhibition. Society of Petroleum Engineers, The Woodlands, Texas, USA Nobakht, M. and Clarkson, C.R., 2011b. A New Analytical Method for Analyzing Production Data from Shale Gas Reservoirs Exhibiting Linear Flow: Constant Rate Production North American Unconventional Gas Conference and Exhibition. Society of Petroleum Engineers, The Woodlands, Texas, USA Nobakht, M. and Clarkson, C.R., 2012a. Multiwell analysis of multifractured horizontal wells in tight/shale gas reservoirs, SPE Canadian Unconventional Resources Conference. Nobakht, M. and Clarkson, C.R., 2012b. A New Analytical Method for Analyzing Linear Flow in Tight/Shale Gas Reservoirs: Constant-Rate Boundary Condition SPE Reservoir Evaluation & Engineering, 15(01). Nobakht, M., Clarkson, C.R. and Kaviani, D., 2011. New Type Curves for Analyzing Horizontal Well With Multiple Fractures in Shale Gas Reservoirs Canadian Unconventional Resources Conference. Society of Petroleum Engineers, Calgary, Alberta, Canada. O'Dell, H.G. and Miller, R.N., 1967. Successfully cycling a low-permeability highyield gas condensate reservoir. Journal of Petroleum Technology, 19(01): 41-47. Orangi, A., Nagarajan, N.R., Honarpour, M.M. and Rosenzweig , J.J., 2011. Unconventional Shale Oil and Gas-Condensate Reservoir Production, Impact of Rock, Fluid, and Hydraulic Fractures, SPE Hydraulic Fracturing Technology Conference. Society of Petroleum Engineers, The Woodlands, Texas, USA. Ozkan, E., Brown, M.L., Raghavan, R. and Kazemi, H., 1988. Performance of horizontal wells, The University of Tulsa. Ozkan, E., Brown, M.L., Raghavan, R. and Kazemi, H., 2011. Comparison of fractured-horizontal-well performance in tight sand and shale reservoirs. SPE Reservoir Evaluation & Engineering, 14: 248-259. 51

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Raghavan, R., Chen, C. and Agarwal, B., 1997. An Analysis of Horizontal Wells Intercepted by Multiple Fractures. SPE Journal, 2(03): 235-245. Raghavan, R., Chu, W.-C. and Jones, J.R., 1995. Practical considerations in the analysis of gascondensate well tests, SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, Dallas, Texas. Restrepo, D.P., 2008. Pressure behavior of a system containing multiple vertical fractures, University of Oklahoma. Sedarat, E., Taheri, A. and Ataei, A.R., 2014. The Effect of Hydraulic Fracturing on the Performance of Gas Condensate Systems. Petroleum Science and Technology, 32: 912–921. Sheng, G. et al., 2015. A multiple porosity media model for multi-fractured horizontal wells in shale gas reservoirs. Journal of Natural Gas Science and Engineering, 27(3): 1562-1573. Song, B., 2010. Pressure transient analysis and production analysis for New Albany shale gas wells, Texas A&M University. Stehfest, H., 1970. Numerical inversion of Laplace transform. Communications of the ACM, 13: 47–49. Sureshjani, M.H., Behmanesh, H. and Clarkson, C.R., 2014. A new semi-analytical method for analyzing production data from constant flowing pressure wells in gas condensate reservoirs during boundary-dominated flow, SPE Western North American and Rocky Mountain Joint Meeting. Society of Petroleum Engineers. Tian, L. et al., 2014. Well testing model for multi-fractured horizontal well for shale gas reservoirs with consideration of dual diffusion in matrix. Journal of Natural Gas Science and Engineering, 21: 283-295. van Everdingen, A.F. and Hurst, W., 1949. The application of the laplace transformation to flow problems in reservoirs. Trans. AIME 186: 305-327. Wan, J. and Aziz, K., 2002. Semi-Analytical Well Model of Horizontal Wells With Multiple Hydraulic Fractures. SPE Journal, 7(04): 437-445. Wang, J., Wang, X. and Dong, W., 2017. Rate decline curves analysis of multiple-fractured horizontal wells in heterogeneous reservoirs. Journal of Hydrology, 553: 527-539. Warren, J.E. and Root, P.J., 1963. The behavior of naturally fractured reservoirs. SPE Journal, 3: 245– 255. Wei, Y. and Economides, M.J., 2005. Transverse hydraulic fractures from a horizontal well, SPE Annual Technical Conference and Exhibition. Wu, M. et al., 2018. Pressure transient analysis of multiple fractured horizontal well in composite shale gas reservoirs by boundary element method. Journal of Petroleum Science and Engineering, 162: 84-101. Xiao, C. et al., 2016. Comprehensive application of semi-analytical PTA and RTA to quantitatively determine abandonment pressure for CO2 storage in depleted shale gas reservoirs. Journal of Petroleum Science and Engineering, 146: 813-831. Xiao, C. et al., 2017. A Semianalytical Methodology for Pressure-Transient Analysis of Multiwell-PadProduction Scheme in Shale Gas Reservoirs, Part 1: New Insights Into Flow Regimes and Multiwell Interference. SPE Journal, 23(03). Xu, S. and Lee, J.W., 1999. Two-phase well test analysis of gas condensate reservoirs, SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers, Houston, Texas Yousefi, S.H., Eslamian, A. and Rashidi, F., 2014. Investigation of well test behavior in gas condensate reservoir using single-phase pseudo-pressure function. Korean Journal of Chemical Engineering, 33(1): 20-28. Zeng, H., Fan, D., Yao, J. and Sun, H., 2015. Pressure and rate transient analysis of composite shale gas reservoirs considering multiple mechanisms. Journal of Natural Gas Science and Engineering, 27(2): 914-925. Zeng, Y., Wang, Q., Ning, Z. and Sun, H., 2018. A Mathematical Pressure Transient Analysis Model for Multiple Fractured Horizontal Wells in Shale Gas Reservoirs. Geofluids, 2018.

52

1 2 3 4 5 6

Zhao, Y. et al., 2016. Pressure response and production performance for multi-fractured horizontal wells with complex seepage mechanism in box-shaped shale gas reservoir. Journal of Natural Gas Science and Engineering, 32: 66-80. Zhao, Y., Zhang, L., Zhao, J., Luo, J. and Zhang, B., 2013. “Triple porosity” modeling of transient well test and rate decline analysis for multi-fractured horizontal well in shale gas reservoirs. Journal of Petroleum Science and Engineering, 110: 253-262.

7

53

Highlights 1- A new semi-analytical model for shale gas condensate reservoirs fluid flow is developed. 2- Effect of two-phase flow on pressure and rate transient analysis is investigated by introducing a new definition of pseudo-pressure. 3- Fluid flow modeling coupled with flash calculation are used to generate the new PTA and RTA type curves. 4- Special two-phase flow type curves for different relative permeability values are generated.