Impact of micro- and macro-scale consistent flows on ...

Journal of Natural Gas Science and Engineering 36 (2016) 1239e1252

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Impact of micro- and macro-scale consistent flows on well performance in fractured shale gas reservoirs Jia Liu a, J.G. Wang b, a, *, Feng Gao b, a, Yang Ju a, c, Furong Tang d a

State Key Laboratory for Geomechanics & Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China c State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology at Beijing, Beijing 100083, China d School of Sciences, China University of Mining and Technology, Xuzhou 221116, China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 October 2015 Received in revised form 30 March 2016 Accepted 4 May 2016 Available online 7 May 2016

Shale gas revolution comes from the skillful combination of horizontal drilling and hydraulic fracturing technology that can create a fractured gas reservoir for gas production. The well performance in the fractured shale gas reservoir is significantly impacted by the complicated gas flow regimes in both fracture network and shale matrix. The consistency between the macro-flow in fracture network and the micro-flow in shale matrix determines the gas production curve, thus being a key issue to gas well design. This paper proposes a numerical model to investigate the impact of micro- and macro-scale consistent gas flows on well performance in fractured shale gas reservoirs. In this numerical model, the macro-scale gas flow follows the Darcy law in the fracture network and the micro-flow in the shale matrix is described by a diffusion-controlled gas transport model. Two apparent diffusion coefficients or models are then obtained. They incorporate viscous flow with slip boundary, molecular diffusion (i.e. molecular self-diffusion), Knudsen diffusion, and surface diffusion in the adsorption layer. The performances of these two diffusion models for the gas transport within shale matrix are investigated and compared with two apparent permeability models proposed by Singh and Javadpour (2013) and Darabi et al. (2012). Furthermore, the pressure-dependent anisotropy of fracture permeability and compressibility is incorporated into the numerical model. This numerical model is verified by an analytical solution and history matching for a Barnett shale gas well. Finally, a fractured gas reservoir with different scenarios is numerically simulated and the shapes of production curves are analyzed through parametric study. It is found that the enhancement of gas recovery efficiency and the life of a shale gas well can be effectively designed if the consistency of micro- and macro-flows can be well designed. © 2016 Elsevier B.V. All rights reserved.

Keywords: Diffusion-controlled gas transport model Fracture-matrix interaction Apparent diffusion coefficient Surface diffusion Anisotropic permeability and compressibility Gas recovery efficiency

1. Introduction Shale gas is playing an increasingly critical role in the world’s natural gas production due to the advancement of enormous reserves, exploration technology, and the demand for cleaner energy such as crude oil and natural gas (Lee and Sohn, 2014; Guo et al., 2015; Kang et al., 2015; Sharma et al., 2015; Zhang et al., 2015). Shale gas reservoirs usually have extremely low permeability, thus the flow behaviors in shale gas reservoirs are different from conventional gas reservoirs. The commercial development of shale gas has been driven by following three key advances: (1) horizontal

* Corresponding author. School of Mechanics and Civil Engineering, China University of Mining and Technology, China. E-mail addresses: [email protected], [email protected] (J.G. Wang). http://dx.doi.org/10.1016/j.jngse.2016.05.005 1875-5100/© 2016 Elsevier B.V. All rights reserved.

well plus multi-staged hydraulic fracturing, (2) understanding of gas storage mechanism in shale matrix, (3) understanding of gas flow mechanism in shale matrix (Li et al., 2014). Therefore, the investigation on the shale gas flow in the shale matrix with rich nanopores is a vital procedure for the commercial development of shale gas reservoirs. The gas flow in the nanopores within shale matrix is considerably complicated (Singh and Javadpour, 2016). Based on Knudsen number,Kn, which is the ratio of molecular mean free path to space characteristic dimension, the gas flow regime can be classified into viscous flow, slip flow, transition flow, and free molecular flow, respectively (Roy et al., 2003; Dongari et al., 2009; Darabi et al., 2012; Hashemifard et al., 2013; Deng et al., 2014; Li et al., 2014; Ren et al., 2015; Zhang et al., 2015). In the viscous flow and slip flow regimes, the Navier-Stokes equation is applicable to the

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description of gas flow. In the transitional flow regime, continuum approximation is not suitable for gas flow. The Knudsen layer, where gas molecules collide with the wall, occupies a significant fraction of the flow domain. In the free molecular flow regime, the Knudsen number is significantly greater than unity. The mean free path is therefore much greater than the length of the flow channel and molecules collide with surfaces bounding the flow wall more frequently than they collide with one another. A lot of mathematical models expressed by apparent permeability have been proposed for the gas flow within shale matrix (Javadpour, 2009; Civan, 2010; Civan et al., 2011; Darabi et al., 2012; Singh and Javadpour, 2013; Wu et al., 2014; Guo et al., 2015). However, no gas transport model is available in expression of apparent diffusion coefficient. The mechanisms of gas flow in shale matrix have been studied through both numerical simulations and experimental measurements. Micro-scale flow simulations have been conducted to investigate the mechanisms of gas desorption and transport within shale matrix (Zhai et al., 2014; Ren et al., 2015; Sharma et al., 2015; Zhang et al., 2015). For example, Zhang et al. (2015) investigated the transport mechanisms and influence factors through Lattice Boltzmann method (LBM). They found that net desorption, diffusion, and slip flow are very sensitive to pore scale. Pore pressure and temperature have also impacts on the mass flux of gas (Teng et al., 2016). Ren et al. (2015) studied the effect of surface diffusion and gas slippage on the velocity of free gas and the mass flux in a kerogen pore via LBM. The simulation results show that the surface diffusion is a more important factor than gas slippage, but it can be negligible in large-size pores. Due to the significant advantages of multi-scale times, LBM became a prevailing numerical simulation approach for the investigation of gas transport mechanisms. In addition, Zhai et al. (2014) used grand canonical Monte Carlo (GCMC) simulation to investigate the adsorption of shale gas in a shale matrix model at different geological depths up to 6 km. They adopted molecular dynamics (MD) simulation to examine the diffusion of shale gas in shale matrix. The GCMC configurations were used as initial input in their MD simulation. Similarly, Sharma et al. (2015) investigated the adsorption and diffusion characteristics of methane and ethane in montmorillonite slit pores via GCMC and MD simulations. Micro-flow simulations can deepen our understandings on the flow mechanisms within shale matrix, but are difficult to be incorporated into a large-scale simulation for a well performance in a fractured gas reservoir. On the other hand, micro-flow experiments on gas flow have also been conducted. Compared to micro-flow simulations, corresponding micro experiment is insufficient due to its difficulty. Guo et al. (2015) presented an experiment for nitrogen flow through nano membranes and derived an apparent permeability based on an advection-diffusion model. In order to investigate the characteristics of nonlinear gas flow in shale matrix, Kang et al. (2015) conducted a core-flooding experiment under the net confining stress at reservoir conditions. They found that the nonlinear flow behavior has a huge difference between nitrogen and methane as a driving fluid. This is because nitrogen and methane have significantly different adsorption properties on organic matter. Therefore, the adsorption property of gas on shale matrix is an important issue to affect the micro-flow of gas within shale matrix. Further, the estimation of gas content in a shale reservoir is a vital issue for gas production. It is known that gas is stored as free gas in tiny spaces of shale matrix or absorbed on the surfaces of organic matter and clays (Sharma et al., 2015). Free gas also exists in fracture network and this free gas is transported to the wellbore driven by the difference between the bottom-hole pressure and the reservoir pressure. The gas pressure difference in matrix pores and fracture network drives the free gas in matrix pores to flow into the

fracture network. This process is slow and usually diffusive. It changes the pressure profile and causes the adsorbed gas on the surface of organic matter to adsorb into pore spaces. This desorption supplies free gas to the pore channel and increases its pore pressure (Zhang et al., 2015). In addition, the adsorbed gas can be migrated in the sorption layer by surface diffusion without any desorption (Sheng et al., 2014; Wu et al., 2015). Therefore, any gas transport model for the gas diffusion in shale matrix should systemically consider these mechanisms. The above-mentioned gas transport mechanisms can be described by two possible types of conceptual models: (1) molecular dynamics model considering the details of gas molecular nature; (2) macro-scale model ignoring the details of gas molecular nature (Wu et al., 2014). The molecular dynamics model can accurately simulate the gas flow in high Knudsen number ranges within nanopores, but its computational time and memory requirement limit its implementation for large-scale problems such as the well performance in fractured shale gas reservoirs. Therefore, molecular dynamics model can only simulate some small-scale problems. Macro-scale model is practical, although it is cannot take all the mechanisms in a shale gas reservoir into considerations. This paper proposes a dual-porosity numerical model to simulate the well performance in a fractured shale gas reservoir. The gas reservoir is regarded as a composite body of fracture network and shale matrix. The formation matrix with low permeability is intersected by fractures with high hydraulic conductivity. In such a fractured shale gas reservoir, the micro- and macro-consistent flows play a crucial role in the performance of shale gas wells. In this paper, the gas flow in fracture network satisfies the Darcy flow and the gas transport in a shale matrix block follows a nonlinear diffusion process. Then, a diffusion-controlled gas transport model is proposed and two new apparent diffusion coefficients are obtained. These two diffusion coefficients incorporate viscous flow considering slip boundary, molecular diffusion (i.e. molecular selfdiffusion), Knudsen diffusion, and surface diffusion together in different ways. They are then applied to the modeling of the gas transport in shale matrix. The superiority and reliability of this diffusion-controlled gas transport model is verified through the comparisons with an analytical solution and the history matching of well production data in Barnett shale reservoirs. Subsequently, the gas production of a horizontal well is numerically simulated and the shapes of production curves are analyzed through parametric study. Finally, the effects of fracture anisotropic permeability and compressibility on gas production rate are explored. This paper is organized as follows. Section 2 presents the diffusion-controlled gas transport model and two apparent diffusion coefficients. Section 3 discusses the gas exchange rate between matrix and fracture network. Section 4 describes the macro-flow in fracture network. The pressure-dependent anisotropy of fracture permeability and compressibility is introduced into the macroflow. Section 5 validates the numerical model with an analytical solution and history matching. The two diffusion models for the gas transport in shale matrix are also compared with two apparent permeability models proposed by other researchers. Section 6 conducts parametric study on the numerical model for a fractured shale gas reservoir. The shapes of production curves and the effect of fracture anisotropy are investigated. The conclusions are given in Section 7. 2. Diffusion-controlled gas transport model In previous studies, seepage-controlled transport models are prevailing for the gas flow in shale matrix. The apparent permeability is the focus (Javadpour, 2009; Civan, 2010). However, no diffusion-controlled gas transport model is available. This section

J. Liu et al. / Journal of Natural Gas Science and Engineering 36 (2016) 1239e1252

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will describe viscous flow, slip flow, molecular diffusion, Knudsen diffusion, and surface diffusion, respectively. Based on these gas flow mechanisms, two apparent diffusion coefficients are obtained. The performance of these two diffusion coefficients is then analyzed and compared with two apparent permeability models. The advantages and shortcomings of these two apparent diffusion coefficients are identified.

slip boundary is valid when the pore diameter is more than 500 nm and the reservoir pressure is more than 10 MPa. In nano-scale pores, gas flow regime is mainly composed of transition flow or slip flow. In addition, free molecular flow is constrained in much smaller diameter pores (that is Knudsen diffusion). Hence, both transition flow and slip flow are more significant in a shale gas reservoir than in a conventional gas reservoir.

2.1. Knudsen number

2.2. Viscous flow with slip boundary

Molecular mean free path is an important molecular dynamic parameter which is frequently used in gas flow dynamics. This free path is defined as the mean path for a molecular free motion in an interval of two consecutive collisions (Loeb, 1934; Civan, 2010):

The gas mass transfer in porous media can be described through Knudsen diffusion, molecular diffusion, and viscous flow when isothermal condition is assumed (Hashemifard et al., 2013). The mass flux for the viscous flow in porous media is obtained according to Hagen-Poiseuille equation, simultaneously considering the slip boundary here. So-called slip flow regime, the NavierStokes equation can be still used with modified boundary conditions to account for rarefaction effects close to the wall. The NavierStokes equation is not applicable in the layer close to the wall, but the flow outside the layer can be described by extrapolating the bulk gas flow towards the wall and applying Maxwell’s slip boundary condition at the wall. The mass transfer rate is finally obtained by the solution of Navier-Stokes equation in a circle crosssection pore channel or pipe. This mass transfer rate can be expressed as below after considering the effect of porosity and tortuosity (Singh and Javadpour, 2013):

k T m l ¼ pffiffiffi B 2 ¼ p m 2pd pm

sffiffiffiffiffiffiffiffiffiffi pRT 2Mg

(1)

where l is the molecular mean free path, kB is the Boltzmann constant, R is the universal gas constant, m is the gas viscosity, Mg is the mole mass, d is the molecular collision diameter, T is the absolute temperature, and pm is the pore pressure in shale matrix. The gas flow in micro- and nano-scale pores is usually constrained, forming different gas flow regimes (Singh and Javadpour, 2016). Knudsen number is used to describe the relative length of mean free path of gas flow in the gas flow channel. Thus, the Knudsen number is defined as the ratio of the molecular mean free path to the characteristic length of the pore channel:

Kn ¼

l dp

(2)

where Kn is the Knudsen number and dp is the mean pore diameter in shale matrix. This dp can be measured through laboratory tests, such as mercury injection and pores imaging using Scanning Electron Microscope (SEM) and Atomic Force Microscopy (AFM). Fig. 1 presents the change of Knudsen number with pressure at the reservoir temperature of 350 K when pore size ranges from 1 nm to 10 mm. Similar figure was drawn by Javadpour et al. (2007) but Fig. 1 was redrawn based on our interested ranges. The gas flow regimes in the pores can be roughly classified according to the Knudsen number. For example, this figure shows that viscous flow without

f M_ C ¼ m x

d2p Mg pm

th

32mRT

Vpm

(3)

·

where MC is the mass flux due to viscous flow, fm is the porosity of shale matrix, th is the tortuosity of pores in shale matrix. The theoretical dimensionless coefficient x considers the correction due to slip velocity. This slip correction coefficient was derived by Brown et al. (1946) as



x¼1þ

8pRT Mg

0:5

  2m 2 1 pm dp av

(4)

where av is the tangential momentum accommodation coefficient (TMAC). It is an empirical parameter and 0  av  1 (Roy et al., 2003). Substituting Eqs. (1) and (2) into Eq. (4) and assuming av ¼ 1, which corresponds to a rough surface that reflects all molecules diffusions (Lunati and Lee, 2014), one obtains a relationship as

x¼1þ8

l



2

dp av

 1

¼ 1 þ 8Kn

(5)

As shown in Fig. 2, the slip correction coefficient increases with the decreases of pore pressure and pore diameter. Therefore, the slip effect should be carefully considered, especially in the later period of production where pore pressure drops to a small value and the slip effect is more prominent. In fact, the slip effect can be treated in many ways such as Langmuir slip condition (Singh and Javadpour, 2016). We do not discuss the treatment method on the slip effect in this paper. 2.3. Molecular diffusion and Knudsen diffusion

Fig. 1. Change of Knudsen number with pore pressure when pore diameter ranges from 1 nm to 10 mm and temperature is T ¼ 350 K (based on Javadpour et al., 2007).

The capability of molecular collisions to transfer mass and momentum is described by molecular diffusion and viscous flow, respectively (Lunati and Lee, 2014). For a pure gas with selfdiffusion, the molecular diffusion coefficient DM is expressed in Eq. (6) through molecular kinetic theory (Dongari et al., 2009;

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Lunati and Lee, 2014). This coefficient for Knudsen diffusion DK can be obtained by replacing l with dp in Eq. (6). That is

60 d=1nm d=5nm d=10nm d=50nm d=100nm

Slip correction coefficient ξ

50

1 DK ¼ um dp 3

d=1μm

40

For porous media, the effects of porosity, tortuosity, and surface roughness (the fractal dimension of the pores surface) on gas flow should be taken into account. The roughness describes the local heterogeneity of pores surface. It is found that Knudsen diffusion decreases with the increase of surface roughness (Darabi et al., 2012). Therefore, the effective Knudsen diffusion coefficient is described by

30

20

10

DeK ¼ 0 0 10

1

2

10

10

Pressure (MPa) Fig. 2. Variation of slip correction coefficient with pore pressure under different pore diameter and T ¼ 350 K.

Hashemifard et al., 2013; Lunati and Lee, 2014). For multicomponent shale gas, the molecular masses and the crosssectional area of collisions are obtained through weighted average (their weights are listed in Table 1). After average, this paper still regards the multi-component gas as a single-component gas. Thus the molecular diffusion is equal to molecular selfdiffusion here (Dongari et al., 2009; Lunati and Lee, 2014).

1 1 l DM ¼ um l ¼ um dp ¼ KnDK 3 3 dp

(6)

where DK is the coefficient for Knudsen diffusion. As molecular velocity obeys the Maxwellian distribution, the arithmetical average of the molecular velocity is

um

sffiffiffiffiffiffiffiffiffiffi 8RT ¼ pMg

(7)

The diffusion is affected by the porosity fm and the tortuosity th of a porous medium. After considering this effect, the effective molecular diffusion coefficient DeM is

DeM ¼

fm

th

KnDK

(9)

(8)

When the molecular mean free path is comparable to or larger than pore diameter, especiallyKn > 10, the wall-molecule collision is more remarkable than intermolecular collisions. Hence, the molecular diffusion should be modified due to the increasing frequency of molecule-wall collision. The modified molecular diffusion is defined as Knudsen diffusion (i.e. free molecular flow regime). In this regime, diffusion is described through the Knudsen diffusion rather than the molecular diffusion (Dongari et al., 2009;

fm  0 Df 2

th

d

(10)

$DK

0

where d is the ratio of normalized molecular size to local average 0 pore diameter d ¼ dm/dp; Df is the fractal dimension of the pores surface, 2 < Df < 3. The tortuosity of pores in shale matrix is one of the most widely studied petrophysical parameter and is defined as the ratio of effective flow path to the macroscopic length, i.e. th ¼ Le/L (Ziarani and Aguilera, 2012). Katsube (2010) proposed a similar model to Pirson’s one (1983) (square root model). He introduced an additional coefficient to account for the shape of connecting pores. Thus, the tortuosity is (Katsube, 2010)

Ffm bf

th ¼

!1=2 (11)

where bf is a constant that depends on the type of connecting pores, bf ¼ 1.5 for sheet-like and bf ¼ 3 for circular/tubular connecting pores. F is the formation resistivity factor which is proposed based on porosity by Archie (1942),

F ¼ fc m

1:2 < c < 3

(12)

Fig. 3 plots the change of tortuosity with porosity when the constant bf ranges between 1.5 and 3. Smaller porosity and constant bf corresponds to larger tortuosity. The value of bf is set as an average of 2.25 in this paper.

2.4. The first apparent diffusion coefficient The effective diffusion coefficient is obtained within the whole range of Knudsen number. It should smoothly vary between the limits of molecular diffusion (Kn / 0) to fully developed Knudsen diffusion (Kn / ∞) (Dongari et al., 2009; Lunati and Lee, 2014).

De ¼

 1=2  2ðDf 2Þ  e 2  e 2 1=2 fm DK þ DM ¼ DK d0 þ Kn2

th

(13)

Corresponding effective diffusion mass flux is

Table 1 Molecular collision diameter of shale gas. Gas component

Mole fraction (%)

Collision diameter (d, nm)

Molar mass (kg/kmol)

CH4 C2H6 CO2 Shale gas

87.4 0.12 12.8 e

0.40 0.52 0.45 0.41

16 30 44 19.5

Data taken from Shi et al. (2013) and Deng et al. (2014).

J. Liu et al. / Journal of Natural Gas Science and Engineering 36 (2016) 1239e1252

is generally smaller than the heat of adsorption. As a result, surface diffusion increases with the decrease of temperature. Therefore, surface diffusion is generally insignificant at temperatures which are high relative to the normal boiling point of the adsorbed gas such as nitrogen, methane, hydrogen and helium (Hashemifard et al., 2013). Only isotherm process is considered. Akkutlu and Fathi (2012) suggest a value of 1  108 ~ 1  106 m2/s as surface diffusion coefficient through history matching. By means of a surface diffusion coefficient which describes the capability of methane diffusion in nanotube of activated carbon (Guo et al., 2008), the same surface diffusion coefficient in shale nano-scale pores is expressed through considering different parameters as (Sheng et al., 2014; Wu et al., 2015)

40 c=2.1 35

bf =1.5 bf =2

30

Tortuosity τh

bf =2.5 bf =3

25 20 15 10

. i h DS ¼ 8:29  107  T 0:5 exp  DH0:8 ðRTÞ

5

0 -3 10

-2

-1

10

10

Porosity Fig. 3. Tortuosity change of matrix pores with porosity.



· e

M ¼

· e

2

MK · e

 · e 2 1=2 þ MM

· e

(14)

· e

where M , M K , and M M denote the mass flux due to effective diffusion, effective Knudsen diffusion, and effective molecular diffusion, respectively. As shown in Fig. 4, the effective diffusion coefficient varies with Knudsen number for2 < 2 < 0.5. In the case of 2 ¼ 1, the effective diffusion becomes Knudsen diffusion and molecular diffusion when Kn > 10 and Kn < 102, respectively. A value of 2 ¼ 1, hence, may be chosen without much loss of accuracy (Dongari et al., 2009). Two limits of effective diffusion mass flux are the Knudsen diffusion mass flux and the molecular diffusion mass flux, respectively. That is,

lim



Kn/∞

· e

M

·

¼ MK

lim



Kn/0

· e

M

·

¼ MM

(15)

For adsorbed gas, in addition to desorption, there is a crucial way of migration, i.e. surface diffusion, which is not well known due to difficulty in measurement (Xiong et al., 2012). Surface diffusion is an activated process, but the diffusion activation energy

Dpe ¼ De þ ð1  fm ÞKT DS =fm

KT ¼ rs rga

e

Diffusion coefficient

VL PL

DM ζ= ζ= ζ= ζ=

e

DK -7

10

-8

-2 -1.5 -1

where rga is the gas density under standard conditions. rgm is the density of gas in shale matrix. rs is the density of shale matrix. VL,PL are the Langmuir volume and pressure constants of shale matrix. crgm is the compressibility coefficient of gas in matrix. For ideal gas, it is defined as

crgm ¼

1 vrgm 1 ¼ rgm vpm pm

-9

·

MD ¼

·

φ m= 0.06, τh = 3.13

-10

D0a

-11 -3

10

-2

10

-1

10

0

10

1

10

2

10

Knudsen number (Kn) Fig. 4. Change of effective diffusion coefficient with Knudsen number.

Mg D0a Vpm RT

(20)

Mg Dpe Vpm RT

(21)

Substituting Eqs. ((3), (13), (17) and (21) into Eq. (20), the diffusion coefficient is expressed as

d= 10nm, Df= 2.5, T= 350K

10

(19)

However, gas transport mechanism in nanopores is not only diffusion, advection (viscous flow including slip boundary) should not be negligible. Thus, the apparent diffusion coefficient for porous media is derived by Fick’s diffusion law as

·

-0.5

10

10

(18)

crgm rgm ðPL þ pm Þ2

where D0a is the first apparent diffusion coefficient, which describes the capability of gas migration incorporating molecular diffusion, Knudsen diffusion, surface diffusion and viscous flow considering · · slip boundary; MT is the total mass flux; M C is the advection mass 0 flux in the surface layer; M_ D is the total diffusion flux as

-5

10

-6

(17)

where KT is the linear slope of the adsorption isotherm. For classical Langmuir isotherm, KT is obtained as

· 0

10

(16)

where DS is the surface diffusion coefficient, DH is the isosteric adsorption heat, which describes the capability of adsorption on shale organic matter. For shale, it is determined as 10~25 kJ$mol1(Yin et al., 2013). Therefore, after the inclusion of surface diffusion, the effective diffusion coefficient of pores is presented as (Ma et al., 1996; Kilislioglu and Bilgin, 2003)

MT ¼ MC þ MD ¼

-4

10

10

1243

¼

fm

th

3

10

þ

" ð1 þ 8KnÞ 1 þ fm KT DS fm

d2p pm 32m

 þ DK

 0 ðDf 2Þ

d

þ Kn

1

1 #

(22)

This is the first apparent diffusion coefficient model (ADC1). This

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J. Liu et al. / Journal of Natural Gas Science and Engineering 36 (2016) 1239e1252

first model combines the viscous flow considering slip boundary with an effective diffusion coefficient (Dongari et al., 2009; Lunati and Lee, 2014). As shown in Fig. 4, the effective diffusion coefficient varies with Knudsen number, which is equal to the effective molecular diffusion coefficient and the effective Knudsen diffusion coefficient in porous media when Knudsen number is smaller and bigger, respectively. One can notice that the Knudsen diffusion coefficient is constant when the pores size and the reservoir temperature are given. However, the molecular diffusion is directly proportional to Knudsen number when Knudsen diffusion coefficient is determined. Therefore, effective diffusion coefficient is constant when Kn > 10 and decreases with the decrease of Knudsen number whenKn < 10. In other words, the apparent diffusion coefficient without considering surface diffusion is equal to the sum of viscous flow and Knudsen diffusion when Knudsen number is bigger. On the contrary, that is equal to the sum of viscous flow and molecular diffusion. It is noted that the slip correction coefficient decreases with the decrease of Knudsen number.

2.5. The second apparent diffusion coefficient

(25)

These two different probability functions vary with Knudsen number as shown in Fig. 5. The domain of individual mechanism associated with the corresponding range of Knudsen number is illustrated. Obviously, the probability function curve of Eq. (24) is smoother in the transition regime and more practical. It is used in this paper. Eq. (20) becomes the following form after considering the probability function of collisions:

· · 00 · · · MT ¼ uK M K þ ð1  uK Þ M C þ M M þ MS

00

Da ¼

fm

th

"

(

ð1  uK Þ ð1 þ 8KnÞ

d2p pm 32m

# )  D 2 þ KnDK þ uK d0 f DK

1 þ fm KT DS fm (27)

This apparent diffusion coefficient includes the probability function of collisions of Eq. (24). This shows that different transport mechanisms have different proportional contributions to the total gas transport (expressed by apparent diffusion coefficient). From Eq. (27), the apparent diffusion is the sum of viscous flow plus

Molecule  Molecule or Molecule  Wall collision frequency Total collision frequency

Particularly, a probability function of collisions with wall is introduced by Wakao et al. (1965):

uK ¼

Kn Kn þ 1

(24)

Molecule-wall collision probability function

Another probability function of collisions with wall is defined by Scott and Dullien (1962):

(23)

molecular diffusion when Kn decreases, however, molecular diffusion (Eq. (6)) becomes smaller and viscous flow (slip correction coefficient is smaller) is predominant. On the contrary, the Knudsen diffusion becomes more significant when Kn increases. At the same time, the molecular diffusion coefficient and the slip correction coefficient become bigger, but their weighted coefficient is smaller. The second diffusion model is the same as the first one when Kn is either sufficiently large or small enough.

1 0.9 0.8

Kn/(1+Kn)

0.7 0.6

Exp[-sinh(1/Kn)]

Slip

Viscous

Transition

Free molecule

0.5 0.4

Molecule-wall collision

Molecule-molecule collision 0.3 0.2 0.1 0 -4 10

-3

10

-2

10

-1

10

(26)

where M_ S is the mass flux due to surface diffusion. Correspondingly, the second apparent diffusion coefficient (ADC2) is obtained as

þ

Molecular dynamics theory shows that the molecule-molecule collision and molecule-wall collision have different contributions to the gas flow in pores. In order to count their different contributions, a probability function of collision is defined as below to describe the proportion of either of the two collisions in total gas collisions (Hashemifard et al., 2013):

u¼

uK ¼ exp½  sinhð1=KnÞ

0

10

1

10

Knudsen number Fig. 5. Distribution of collision probability function with Knudsen number.

2

10

3

10

J. Liu et al. / Journal of Natural Gas Science and Engineering 36 (2016) 1239e1252

2.6. Analysis and comparison of two apparent diffusion models

-4

10

pm=1MPa, ADC1 pm=1MPa, ADC2 pm=10MPa, ADC1 pm=10MPa, ADC2 pm=50MPa, ADC1 pm=50MPa, ADC2

Apparent diffusion coefficient

-5

10

-6

10

-7

10

-8

10

-9

10

0

1

10

2

10

10

Pore diameter (nm) (a) Without surface diffusion -4

10

pm=1MPa, ADC1 pm=1MPa, ADC2 pm=10MPa, ADC1 pm=10MPa, ADC2 pm=50MPa, ADC1 pm=50MPa, ADC2

-5

Apparent diffusion coefficient

The above two diffusion coefficients or models consider the flow mechanisms of viscous flow, slip flow, Knudsen diffusion, molecular diffusion and surface diffusion but they treat the effect of surface diffusion in slightly different ways. In this section, the sensitivity analysis of gas apparent diffusion models is performed in the shale matrix to identify controlling parameters for gas diffusion such as average pore size, reservoir pressure, and surface diffusion coefficient. Eq. (9) and Fig. 4 observe that the Knudsen diffusion coefficient keeps constant when temperature and pore size are fixed. Eq. (6) shows that molecular diffusion is the product of Knudsen number and Knudsen diffusion coefficient. The Knudsen number inversely increases with pore pressure (see Eq. (1)). As reservoir is depleting or pore pressure falls, the Knudsen number increases (see Eq. (2)). At this time, molecular diffusion becomes stronger but Knudsen diffusion keeps constant. Fig. 4 indicates a critical point. When Knudsen number is lower than 0.2, which is obtained under the parameters in this paper, molecular diffusion is weaker than Knudsen diffusion. When Knudsen number is greater than 0.2, molecular diffusion becomes stronger. Therefore, the critical point is at Knudsen number of 0.2 in this paper. Both models show that the apparent diffusion coefficient in shale matrix is a function of gas pressure and pore diameter. Pore size actually exists in shale reservoirs. It may have a complex distribution and vary with pore pressure and adsorbed gas concentration. This section does not consider the distribution and evolution of the nanometer pore size in the process of gas production. The average pore diameter is used in numerical simulations. The curves of apparent diffusion coefficients are drawn at various average pore sizes under different gas pressures. The distribution of collision probability function with Knudsen number is plotted in Fig. 5. Generally, gas flow has four regimes: viscous flow, molecule-molecule collision, molecule-wall collision, and transition between. Both temperature and pore pressure have significant impacts on gas flow regime when the pore size is small. The difference between two models is further identified. The variations of apparent diffusion coefficients with pore size under different pore pressures are shown in Fig. 6. This figure is plotted according to Eqs. (22) and (27). It shows that the trends of both apparent diffusion coefficients without (in Fig. 6(a))/with (in Fig. 6(b)) surface diffusion are almost identical and the increase of pore size corresponds to a bigger apparent diffusion coefficient regardless of pore pressure. Higher gas pressure corresponds to bigger apparent diffusion coefficient. The change trend of apparent diffusion coefficient with pore diameter becomes flatter when gas pressure is smaller. In addition, the surface diffusion strengthens this phenomenon. Fig. 7 presents the evolution of two apparent diffusion coefficients with reservoir pressure and pore size. Particularly, Fig. 7(a) and (c) present the variation of ADC1 without/ with surface diffusion with pore diameter and gas pressure, respectively. Similarly, Fig. 7(b) and (d) present the evolution of ADC2 without/with surface diffusion with pore diameter and gas pressure, respectively. These figures show that the contribution of surface diffusion to total diffusion is bigger when pore size and reservoir pressure are smaller. This contribution increases with the decrease of pore size if the porosity is kept constant. Smaller pore size means bigger specific surface area of pores if the shale matrix has the same porosity. The shale matrix has bigger section area for adsorbed gas. Therefore, the surface diffusion should not be neglected when smaller size pores are dominant within shale gas reservoirs. Langmuir isotherm shows that the adsorbed gas concentration decreases with the decrease of reservoir pressure. This weakens the capacity of surface diffusion. It is further found that the contribution of surface diffusion to total gas transport is

1245

10

-6

10

-7

10

-8

-7

10

2

Ds =10 m /s

-9

10

0

10

1

10

2

10

Pore diameter (nm) (b) With surface diffusion Fig. 6. Comparison between two apparent diffusion coefficients.

negligible under high reservoir pressure. At this time, the viscous flow is dominant although adsorbed gas concentration is higher. We thus conclude that the gas transport capacity within shale matrix is controlled by pore size, gas pressure, and surface diffusion coefficient. In addition, the gas transport model considering a probability function of collision is equivalent to that combining the viscous flow with an effective diffusion in shale matrix. Therefore, the two diffusion models are applicable to the description of gas transport in shale matrix.

3. Gas exchange rate between matrix and fracture network The gas exchange rate depends on the difference between the current gas content and the transient equilibrium of gas content in shale matrix. Thus, a pseudo-steady diffusion model is applied, where diffusion time is introduced to measure their diffusion speed (Wang et al., 2012):

1 qe ¼  ðVm  Ve Þ

t

(28)

where qe is the gas exchange rate between matrix and fracture

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J. Liu et al. / Journal of Natural Gas Science and Engineering 36 (2016) 1239e1252 -4

-4

10

10

pm=1MPa, ADC1 with surface diffusion pm=1MPa, ADC1 without surface diffusion pm=10MPa, ADC1 with surface diffusion pm=10MPa, ADC1 without surface diffusion pm=50MPa, ADC1 with surface diffusion pm=50MPa, ADC1 without surface diffusion

-5

Apparent diffusion coefficient

Apparent diffusion coefficient

-5

10

-6

10

-7

10

-8

-7

10

2

Ds =10 m /s

0

10

1

-6

10

-7

10

-7

2

Ds =10 m /s -8

10

10

2

10

0

1

10

10

2

10

10

Gas pressure (MPa) (c) Effect of surface diffusion on ADC1 under different pore pressures

Pore diameter (nm) (a) Effect of surface diffusion on ADC1 in different pore sizes -4

-4

10

10 pm=1MPa, ADC2 with surface diffusion pm=1MPa, ADC2 without surface diffusion pm=10MPa, ADC2 with surface diffusion pm=10MPa, ADC2 without surface diffusion pm=50MPa, ADC2 with surface diffusion pm=50MPa, ADC2 without surface diffusion

10

-6

10

-7

10

-8

-7

10

ADC2 with surface diffusion,d=1nm ADC2 without surface diffusion,d=1nm ADC2 with surface diffusion,d=10nm ADC2 without surface diffusion,d=10nm ADC2 with surface diffusion,d=100nm ADC2 without surface diffusion,d=100nm

-5

Apparent diffusion coefficient

-5

Apparent diffusion coefficient

10

-9

-9

10

2

Ds =10 m /s

-9

10

ADC1 with surface diffusion, d=1nm ADC1 without surface diffusion,d=1nm ADC1 with surface diffusion,d=10nm ADC1 without surface diffusion,d=10nm ADC1 with surface diffusion,d=100nm ADC1 without surface diffusion,d=100nm

10

-6

10

-7

10

-7

2

Ds =10 m /s -8

10

-9

0

10

1

2

10

10

10

Pore diameter (nm) (b) Effect of surface diffusion on ADC2 in different pore sizes

0

1

10

10

2

10

Gas pressure (MPa) (d) Effect of surface diffusion on ADC2 under different pore pressures

Fig. 7. Evolution of apparent diffusion coefficient under reservoir pressure and pore diameters.

network. t is the diffusion time of shale matrix, Vm is the average remaining gas content in shale matrix, Ve is the gas content in the transient equilibrium within shale matrix. The gas content Vm in shale matrix includes free gas and adsorbed gas and is expressed as (Wang et al., 2012)

Vm

fm rga Va VL þ ð1  fm Þrs rga pm ¼ RT PL þ pm

(29)

In order to understand its physical meaning easily, a storage potential function j(pm) is introduced here for shale matrix. This function describes the capability of shale matrix to store gas under a given gas pressure. Therefore, Vm is expressed in other form as

Vm ¼ jðpm Þ

(30)

If an average of pore pressure within the matrix is equal to the surrounding pore pressure in cleat system which means an equilibrium state, the gas content in transient equilibrium Ve is related to the gas pressure in fracture network as

Ve ¼ jðpÞ The diffusion time is expressed as (Wang et al., 2013)

(31)

t¼

1

(32)

sDa

where Da is the gas apparent diffusion coefficient in shale matrix, s is a shape factor which is a parameter of matrix block (Warren and Root, 1963):

s¼

3p2 L2

(33)

The gas pressure in shale matrix pm is calculated through Eq. (29) when the average remaining gas content in shale matrix is known,

pm ¼

ðlPL þ m  Vm Þ þ

where l ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðlPL þ m  Vm Þ2 þ 4lVm PL 2l

fm rga Va RT ;

m ¼ ð1  fm Þrs rga VL .

(34)

J. Liu et al. / Journal of Natural Gas Science and Engineering 36 (2016) 1239e1252

4. Macro-flow in fracture network

 ff 1  ff 0 ð1 þ nÞ 2E ðp  p0 Þ  ð3 s  3 s0 Þ ¼ exp 3ð1  nÞ 9ð1  nÞ ff 0 Kp

4.1. Pressure-dependent anisotropic fracture permeability The fracture compressibility is defined analogous to pore volume compressibility as follows (Seidle et al., 1992; Chen et al., 2015)

cf ¼ 

1 vff ff vse

(35)

where ff is the fracture porosity and se is the effective stress. However, constant fracture compressibility is not suitable for the match of some experimental data. A stress-dependent fracture compressibility below was presented by McKee et al. (1988) and the compressibility has been discussed by many researchers (Walsh and Grosenbaugh, 1979; Walsh, 1981; Harpalani, 1985; Zimmerman et al., 1986; Chen et al., 2015).

cf ¼

 cf 0 1  eaðse se0 Þ aðse  se0 Þ

(36)

where cf0 is the initial fracture compressibility, se0 is the initial effective stress and a is the declining rate of fracture compressibility with the increase of effective stress. The permeability-stress relationship is directly associated with the fracture compressibility. For example, Chen et al. (2015) presents an exponential form for permeability-stress relationship, see Eq. (37). This has the same form as the permeability model proposed by Seidle et al. (1992) for coal.

k ¼ k0 e3cf ðse se0 Þ

(37)

where k is the fracture permeability and k0 is the initial fracture permeability. Reservoir pressure is more easily measured than stress and shale fracture compressibility may be strongly anisotropic. It is better to express the evolution of fracture permeability with reservoir pressure for either constant mean stress condition (called mean-stress-controlled) or constant horizontal stress condition (called horizontal-stress-controlled) as follows. The former refers to the constant total hydrostatic stress and the later refers to nochange of horizontal constraint on the saturated porous media. Appendix gives the derivation details of these two formulas.

Mean­stress­controlled :

(39) where ff0 is the initial fracture porosity, 3 s is the sorption-induced volumetric strain, E is the Young’s modulus, Kp is the pores volume modulus which is

Kp ¼ ff Kb ¼

ff E 3ð1  2nÞ

(40)

where Kb is the bulk modulus of the fractured porous media. For lower organic matter content in shale, the adsorption capacity should generally not result in significant changes in sorption-induced volumetric strain. The minimal matrix shrinkage or resulting permeability enhancement generally occurs during gas shale production. For shale the change in sorption-induced strain is assumed to be zero (Bustin and Bustin, 2012). The stress-dependent fracture porosity ratio of Eq. (39) becomes

0 1 1  ff 0 ð1 þ nÞ ff @ ðp  p0 ÞA ¼ exp ff 0 3Kp ð1  nÞ

(41)

A constant is defined as



q¼

1  ff 0 ð1  2nÞð1 þ nÞ Eð1  nÞ

(42)

Combining Eqs. (40)e(42), the derivative of the fracture porosity with respect to production time is thus obtained as

vff qff vp ¼ vt ff þ qðp  p0 Þ vt

(43)

4.2. Gas flow equation in fracture network The mass conservation of gas flow in fracture network is expressed by

 v ff rg vt

8 > > > 1þn > > < kx ¼ kx0 ecfy 1naB ðpp0 Þ

 þ V rg $! v gi ¼ qe þ qi

(44)

where rg is the gas density in fracture network, ! v gi is the gas flow velocity in the ith direction, qe is the exchange rate of gas mass between matrix and fracture network, qi is the gas source by injection or other methods, t is the real time. The density of ideal gas is calculated by the equation of state as

> 1þn > > ky ¼ ky0 ecfx 1naB ðpp0 Þ > > :

Horizontal­stress­controlled :

1247

8 > > n > > < kx ¼ kx0 e3cfy 1naB ðpp0 Þ

rg ¼

n > > > k ¼ ky0 e3cfx 1naB ðpp0 Þ > : y

(38) Where kx0, ky0 are the initial permeability in x direction and y direction, respectively. The compressibility is cfx along x direction and cfy along y direction. n is the Poisson ratio, aB is the Biot’s coefficient, p is the gas pressure in fracture and p0 is the initial gas pressure in fracture. Further, following stress-dependent porosity model was proposed by Cui and Bustin (2005) for coal and used by Bustin and Bustin (2012) for shale.

rga Mg p¼ p RT pa

(45)

where pa is the pressure in fracture under standard conditions. The gas flow in fracture network is described by the Darcy flow:

k ! v gi ¼  i Vi p

m

(46)

where ki is the shale fracture permeability in the ith direction (For example, i ¼ x for x direction), Vi p denotes the directional gradient of pressure in the ith direction. Substituting Eqs. (43), (45) and (46) into Eq. (44) obtains the final equation for gas flows:

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ff

J. Liu et al. / Journal of Natural Gas Science and Engineering 36 (2016) 1239e1252

qp 1þ ff þ qðp  p0 Þ

!

  vp k pa þ V$  i pVi p ¼ ðq þ qi Þ vt m rga e

a¼



(47) b¼

5. Model validations 5.1. Verification of numerical model with analytical solution A steady diffusion-controlled gas transport problem is simulated to check the reliability of the numerical model. The continuity equation of gas flow is expressed by

V$J ¼ 0

(48)

The molar flux including viscous flow and diffusion is obtained through Fick’s diffusion law as

  1 km J¼ pm þ D Vpm RT m

(49)

Combining Eq. (49) with Eq. (48), the steady-state gas transport equation in polar coordinates is obtained as

 km

dpm dr

2

 þ ðmD þ km pm Þ

d2 pm 1 dpm þ r dr dr 2

 ¼0

(50)

Eq. (50) is the same as the seepage-controlled transport in steady state flow. For a gas production well in the reservoir with a constant pressure, the boundary conditions are

r ¼ rw ; pm ¼ pw r ¼ re ; pm ¼ pe

(51)

Its analytical solution is then obtained as (Song et al., 2015)

pm ðrÞ ¼

mD þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmDÞ2 þ 2ka ln r þ 2kb

(52)

km

 km p2e  p2w þ 2mDðpe  pw Þ km p2w

2 lnðre =rw Þ

ln re  p2e ln rw þ 2mDðpw ln re  pe ln rw Þ 2 lnðre =rw Þ

where D is a diffusion coefficient and km is the absolute permeability of shale matrix. The following case is used to verify the accuracy of numerical model through this radial flow of shale gas. The basic parameters are as follows: the well radius is 0.1 m, the radius of drainage zone is 10 m, the bottom hole pressure is 1 MPa, the matrix permeability is 1  1021 m2, the external boundary pressure is 10 MPa, and the diffusion coefficient is 1  108 m2/s. Fig. 8 is the comparison of our numerical solution with the analytical solution. Good agreement is observed. Therefore, the current numerical model considering viscous flow and diffusion effect can reliably simulate this radial gas flow in shale matrix. 5.2. History matching of in-situ production data Our ADC2 model is used for history matching of a well performance in the Barnett shale reservoir. The production data are taken from the publication by Yu and Sepehrnoori (2014). The simulation parameters used are listed in Table 2. They have the same values if the parameters have the similar meanings. Tortuosity of shale and molar mass of methane are from Eq. (11) and Table 1, respectively. The last four parameters in Table 2 are obtained from literature or our assumptions. Fig. 9 compares the production rate calculated by our model with in-situ measurements. Generally, the gas production rate is in good agreement with those of field data. The only drawback is that the gas production rate forecasted by our model is little lower than field data in the later period. This may be due to the difference of their influence zones. Our computational model is two-dimensional, where producible area is limited, while the influence zone in site is unknown. Therefore, our numerical model can reasonably predict the in-situ well performance in shale reservoirs. 5.3. Performance comparison between diffusion-controlled transport models with two apparent permeability models

where The performance of our diffusion-controlled transport model for gas flow in shale matrix is compared with that of the apparent permeability models proposed by Singh and Javadpour (2013) and Darabi et al. (2012). The numerical model for this comparison is a

10 9

Gas pressure (MPa)

8

Table 2 Parameters for numerical simulation.

7 6

Analytical Solution by Song et al. (2015)

Parameter

Value

Numerical Solution in this paper

Reservoir depth, h(m) Initial reservoir pressure, pinit(MPa) Reservoir temperature, T(K) Density of shale, rs(kg/m3) Langmuir volume constant, VL(m3/kg) Langmuir pressure constant, PL(MPa) Young’s modulus of shale, E(GPa) Methane dynamic viscosity, m(Pa$s) Initial porosity of shale matrix, 4m0 Bottom hole pressure, pw(MPa) Tortuosity of shale, th Molar mass of methane,Mg(kg/mol) Pore diameter, d(nm) Shale Poisson’s ratio, n The fracture spacing, Lx ¼ Ly(m) Initial porosity of shale fractures, 4f0

1665.1 20.3 338.75 2580 0.00272 4.48 34 2.01  105 0.06 3.45 3.13 0.0195 10 0.28 1 0.01

5 4 3 2 1 0

1

2

3

4

5

6

7

8

9

10

Distance (m) Fig. 8. Verification of numerical model with analytical solution in a two-dimensional steady gas flow problem.

J. Liu et al. / Journal of Natural Gas Science and Engineering 36 (2016) 1239e1252 5

2.5

x 10

Present simulation Field data (Yu and Sepehrnoori (2014))

3

Production rate (m /d)

2

1.5

1

0.5

0

0

1

2

3

4

5

6

7

8

9

10 7

Time (s)

x 10

Fig. 9. History matching of a Barnett Shale well.

1249

problem, only boundary condition is required. The parameters in Table 2 are used for this comparison. The gas pressure profiles are compared in Fig. 10. It is observed that the gas pressure profiles along with radial distance calculated by our model are in good agreement with those of other two apparent permeability models when pore size is bigger. For smaller size pores, our model has faster pressure drops than other two models. This is because our model considers the surface diffusion. Smaller pore size corresponds to stronger surface diffusion. This illustrates that the contribution of surface diffusion increases with a decrease of pore size. The surface diffusion relates to not only the diffusion coefficient but also the adsorbed gas concentration (KT term in Eqs. (22) and (27)). The smaller size pore induces bigger specific surface area under the same porosity. At the same time, the adsorbed gas section area is bigger. Therefore, surface diffusion is not ignorable, especially for shale gas reservoirs with significant portion of small size pores. The above three comparisons indicate that the proposed numerical model based on gas diffusion in shale matrix can be applicable to the prediction of well performance in fractured shale reservoirs. 6. Numerical forecast of gas production curve

10

6.1. Numerical model 9

Gas pressure (MPa)

8

A zone

7

ADC1 (d=1nm)

6

ACD2 (d=1nm) Singh and Javadpour's model (d=1nm)

5

Darabi et al's model (d=1nm) ADC1 (d=10nm) ADC2 (d=10nm) Singh and Javadpour's model (d=10nm)

4

Darabi et al's model (d=10nm)

3

ADC1 (d=100nm) ADC2 (d=100nm) Singh and Javadpour's model (d=100nm)

2

Darabi et al's model (d=100nm)

1

0

1

2

3

4

5

6

7

8

9

10

Distance (m)

(a) Full profile 8.6

This numerical model is used to simulate a horizontal well in a fractured gas reservoir. The transport of single-component gas (shale gas) is computed with the parameters in Table 2 in a fractured gas reservoir. The FEM mesh and corresponding boundary conditions of this numerical model are shown in Fig. 11. The gas is extracted by a horizontal well. The single production well is 0.1 m in radius. The influence region is a rectangle which is 100 m high and 500 m wide. Due to symmetry, a half of the production well is located at the left center of the model. The initial pressure in the shale reservoir isp0 ¼ 20.3 MPa, the horizontal well length is 900 m. ADC2 is used in the numerical simulation. Besides, the model initial-boundary conditions are specified as follows: the bottomhole pressure is pw ¼ 3.45 MPa and other boundaries are specified as no flow boundary conditions for gas flow in fracture network. At the same time, the initial gas pressure in fracture network is p0. For the gas diffusion in matrix, all boundaries have no flow boundary conditions and the initial gas concentration in matrix is Vm0 ¼ [l þ m/(PL þ p0)]p0.

8.4

6.2. Distribution and evolution of nonlinear diffusion coefficient in matrix

Gas pressure (MPa)

8.2 8

The distribution and evolution of diffusion coefficient nearby the wellbore during gas production are discussed. Fig. 12 (a) shows the distribution of nonlinear diffusion coefficient along the horizontal line y ¼ 0,x2[0.1 m,250 m] at different production times.

7.8 7.6

A zone

7.4 7.2 7 6.8 6.6 2

2.5

3

3.5

4

Distance (m)

(b) Local observation Fig. 10. Comparison of gas pressure profiles predicted by our model and other two models.

circle area with a radius of 10 m. A well is set at the center of modeling area. Because this is a steady and single-porosity flow

Fig. 11. Numerical model for shale gas production simulation.

J. Liu et al. / Journal of Natural Gas Science and Engineering 36 (2016) 1239e1252 -7

4

x 10

7

x 10

10

x 10

1.3

Production rate (m /day)

3

1.2

1.1

1

0.9

5

3

Case a Case a Case b Case b Case c Case c Case d Case d Case e Case e

t=0.03 year t=0.32 year t=3.17 year

2

1.5

1

0.5

0.8 0

0

50

100

150

200

250

0

0.5

1

1.5

2

2.5

3

Time (year)

Distance from wellbore (m)

(c) Gas production of different simulated cases

(a) Evolution of apparent diffusion coefficient along the horizontal line 4

1

6

7

x 10

x 10 Mean-stress-controlled (Gas production rate) Mean-stress-controlled (Cumulative gas production) Horizontal-stress-controlled (Gas production rate) Horizontal-stress-controlled (Cumulative gas production)

Production rate (m /day)

0.9

0.8

1.5

4

3

Permeability ratio

0 3.5

3

0.7

0.7 t=0.03 year, t=0.03 year, t=0.32 year, t=0.32 year, t=3.17 year, t=3.17 year,

0.6

0.5

0.4

Parallel to bedding Normal to bedding Parallel to bedding Normal to bedding Parallel to bedding Normal to bedding

1

2 0.5

0

0

50

100

150

200

250

0

0.5

1

2

2.5

3

0 3.5

Time (year)

Distance from wellbore (m)

(b) Evolution of fracture permeability ratio along the horizontal line

1.5

Cumulative gas production (m )

Apparent diffusion coefficient (m /s)

1.4

Cumulative gas production (m )

1250

(d) Gas production under two assumptions for fracture permeability evolution

Fig. 12. Gas production after considering anisotropic fracture permeability and compressibility.

This figure shows that the apparent diffusion coefficient around the wellbore is bigger than the rest area. Meanwhile, the apparent diffusion coefficient increases with production time. This is because the minimum matrix pressure pm is surrounding the wellbore and the maximum value is located along the left and right model boundaries in process of gas production. The apparent diffusion coefficient increases along the horizontal line away from wellbore when the production times are 0.03 and 0.32 years. However, a decrease is also observed when the production time is 3.17 years. This is because reservoir pressure is bigger in the early stage of production and viscous flow dominates the gas transport. In the later stage of production, reservoir pressure is lower, and Knudsen diffusion and surface diffusion dominate the gas transport. Thus, gas diffusion is higher when the reservoir pressure is lower. This causes the distribution and evolution of apparent diffusion coefficient to be complicated near the wellbore. Both gas pressure profile and pore sizes play significant roles in the evolution of diffusion coefficient. 6.3. Effect of anisotropic fracture permeability and compressibility on gas production Shale gas reservoirs have strong anisotropic properties due to

Table 3 Anisotropic parameters for fracture permeability and compressibility. Case

Initial fracture permeability (mD)

Fracture compressibility (MPa1)

kx0

ky0

cfx

cfy

a b c d e

0.5 0.5 0.5 0.5 0.5

0.5 0.1 0.1 0.05 0.05

0.02 0.02 0.02 0.02 0.02

0.02 0.02 0.04 0.04 0.08

different causes of formation. In the simulations, the bedding is in horizontal direction. The permeability ratio along this line in some direction is defined as the ratio of the current permeability to its initial permeability in that direction. For example, the permeability ratio is kx/kx0 for parallel to bedding direction and ky/ky0 for normal to bedding direction. Fig. 12 (b) shows the distribution of fracture anisotropic permeability ratio along the horizontal liney ¼ 0, x2 [0.1 m, 250 m] at different production times. It is clear that the permeability ratio is smaller near wellbore because pressure drawdown is more significant. This figure shows that the

J. Liu et al. / Journal of Natural Gas Science and Engineering 36 (2016) 1239e1252

1251

permeability ratio parallel to bedding direction is bigger than that normal to bedding direction. Besides, the difference of permeability ratios in horizontal and vertical directions is bigger with the increase of production time. The anisotropic fracture permeability and compressibility induce this difference of gas production rate. Table 3 lists the permeability and the compressibility used in five cases of simulations. Fig. 12(c) presents the gas production rate and the cumulative gas production at different cases considering anisotropic fracture permeability and compressibility. The case a and b only consider anisotropic permeability. The difference of gas production rate is bigger. The compressibility normal to bedding is bigger than that parallel to bedding, which induces faster decrease of permeability parallel to bedding. Besides, several remaining cases change both fracture permeability and compressibility. These simulation results show that the effect of anisotropic permeability and compressibility on shale gas production is crucial and cannot be ignored. Further, the gas production rate and the cumulative gas production are plotted in Fig. 12(d) under two assumptions of mean-stress-controlled and horizontal-stress-controlled (see Eq. (38)). The results show that the gas production rate under meanstress-controlled evolution of fracture permeability is slightly lower, thus having a little lower cumulative gas production. When production time is equal to 3.2 years (108 s), the cumulative gas production is 1.34  107 m3 under mean-stress-controlled evolution of fracture permeability and 1.27  107 m3 under horizontalstress-controlled one, respectively. Obviously, the impact of permeability anisotropy on the gas production varies with the evolution of permeability.

that the contribution of surface diffusion to total gas transport is negligible under high reservoir pressure. At this time, the viscous flow is dominant although adsorbed gas concentration is higher. Third, the consistency between the macro-flow in fracture network and the micro-flow in shale matrix determines the gas production curve, thus being a key issue to gas well design. The proposed numerical model in this paper combines the micro-flow in shale matrix and the macro-flow in the fracture network. It can be used for the numerical simulations of gas recovery in fractured shale gas reservoirs. Finally, the anisotropy of fracture permeability and compressibility has a significant impact on gas production. The decline of gas production due to anisotropic permeability can be partially offset through the simultaneous consideration of both anisotropic permeability and compressibility. Mean-stress-controlled and horizontal-stress-controlled evolutions of fracture permeability have a little difference for gas production forecast.

7. Conclusions

The relationship between fracture permeability and reservoir pressure is derived under two assumptions: (1) the reservoir is under uniaxial strain compression (D3 11 ¼ D3 22 ¼ 0), and (2) the overburden stress keeps constant during gas production (Ds33 þ aBDp ¼ 0,Dp ¼ p  p0). These assumptions have been made by researchers (Geertsma, 1957, 1961; Gray, 1987; Seidle et al., 1992; Palmer and Mansoori, 1998; Cui and Bustin, 2005; Chen et al., 2015). For homogeneous isotropic elasticity, the incremental Hooke’s law is

Being different from those models for seepage-controlled gas transport, a diffusion-controlled gas transport model was proposed and two new apparent diffusion coefficients were developed in this paper. This model incorporated viscous flow with slip boundary, molecular diffusion (i.e. molecular self-diffusion), Knudsen diffusion, and surface diffusion together. This model was applied to the gas transport in shale matrix and the effect of surface diffusion under different reservoir pressures and pore sizes was explored. Furthermore, a numerical simulation for a horizontal well in fractured shale gas reservoirs was performed by considering the interaction between matrix and fractures. The model was verified by an analytical solution and the history matching of a shale gas well. Further, the effect of fracture anisotropic permeability and compressibility on gas production rate was briefly investigated. Based on these preliminary investigations, following conclusions can be drawn. First, the diffusion-controlled gas transport model with two apparent diffusion coefficients has the capability to simulate the gas transport within shale matrix. Compared with other two apparent permeability models, these apparent diffusion coefficients can still describe the combined effect of different transport regimes in shale matrix. Surprisingly, the curves of two apparent diffusion coefficients are almost coincident, especially under the consideration of surface diffusion. Second, the surface diffusion is a critical mechanism of gas transport. The contribution of surface diffusion increases with the decrease of pore size if the porosity is kept as a constant. Specific surface area is bigger for smaller size pore if the shale matrix has the same porosity. In this case, shale matrix has bigger section area for adsorbed gas. Therefore, the surface diffusion should be not neglected, especially when small size pores within shale gas reservoirs are dominant. Langmuir isotherm shows that the adsorbed gas concentration decreases with the decrease of reservoir pressure. This weakens the capacity of surface diffusion. It is further found

Acknowledgements The authors are grateful to the financial support from Creative Research and Development Group Program of Jiangsu Province (2014e27) and National Natural Science Foundation of China (Grant No. 51204159, 51374213). Appendix. The evolution of fracture permeability with reservoir pressure

1 E 1 ¼ ½Ds22  nðDs33 þ Ds11 Þ E 1 ¼ ½Ds33  nðDs11 þ Ds22 Þ E

D3 11 ¼ ½Ds11  nðDs22 þ Ds33 Þ D3 22 D3 33

(A1)

For K0-compression and constant overburden stress, following relationship is obtained as

Ds11 ¼ Ds22 ¼

n 1n

Ds33 ¼

n a ðp  p0 Þ 1n B

(A2)

Then, the mean stress is Dsm ¼ 13 ðDs11 þ Ds22 þ Ds33 Þ and the horizontal stress is Dsh ¼ Ds11 ¼ Ds22. The permeability models for constant are obtained as Mean-stress-controlled (total mean stress is constant):

k ¼ k0 e3cf Dsm ¼ k0 ecf 1naB ðpp0 Þ 1þn

Horizontal-stress-controlled constant):

k ¼ k0 e3cf Dsh ¼ k0 e3cf 1naB ðpp0 Þ n

(A3) (total

horizontal

stress

is

(A4)

Further, shale fracture compressibility may be strongly anisotropic (Chen et al., 2015). At this time, the evolutions of fracture permeability with reservoir pressure are

1252

Mean­stress­controlled :

J. Liu et al. / Journal of Natural Gas Science and Engineering 36 (2016) 1239e1252

8 > > > 1þn > > < kx ¼ kx0 ecfy 1naB ðpp0 Þ > 1þn > > ky ¼ ky0 ecfx 1naB ðpp0 Þ > > :

Horizontal­stress­controlled :

8 > > n > > < kx ¼ kx0 e3cfy 1naB ðpp0 Þ n > > > k ¼ ky0 e3cfx 1naB ðpp0 Þ > : y

(A5)

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