terapaper diffusion and flow mechanisms -1147

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Diffusion and Flow Mechanisms of Shale Gas through Matrix Pores andGas Production ForecastingJuntai Shi, SPE, China University of Petroleum at Beijing; Lei Zhang, Research Institute of Yanchang PetroleumGrouop Co. LTD; Yuansheng Li, China University of Petroleum at Beijing; Wei Yu, The University of Texas at

Austin; Xiangnan He, Ning Liu, Xiangfang Li, China University of Petroleum at Beijing; Tao Wang, ResearchInstitute of Yanchang Petroleum Grouop Co. LTD

Copyright 2013, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Unconventional Resources Conference-Canada held in Calgary, Alberta, Canada, 57 November 2013.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not beenreviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, itsofficers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to

reproduce in print is restricted to an abstract of not more t han 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abst ractThe transport mechanism of gas moving through matrix pores is the bottleneck of conquering the difficulties in shale gas

development. The matrix pores can be divided into organic and inorganic matrix pores. The transport mechanism of shale gas

in organic and inorganic matrix pores is different. However, the present gas transport model only focused on the gas transport

in organic matrix pores, in addition, the impact of organic and inorganic mass ratio has been largely neglected by shale gas

transport models in the literature, leading to an unclear recognition of shale gas production discipline and large derivation

between prediction results by the present models and actual performance of shale gas wells.

In this paper, both the pore size distribution and water distribution in shale matrix pores are investigated. Furthermore, a new

diffusion-slippage-flow model in combination with the gas transport mechanism is proposed. Also, the organic content effect

is considered and the range of Knudsen number is quantified. Finally, a gas production model based on this gas transportmechanism is derived and employed to reveal the discipline of shale gas production.

The preliminary results illustrate that Knudsen diffusion is not suitable for shale gas reservoirs. This is because Knudsen

number is generally less than 10, especially for such shale gas reservoirs with higher initial reservoir pressure. Gas moving

through shale matrix pores to fractures is mainly divided into two forms: in organic matrix pores, both slip effect and

transition diffusion mechanism are dominant; in inorganic matrix pores, the gas-water two-phase flow controls the gas

transport mechanism because of the presence of water in these pores.

The efforts of this work will provide a more accurate technique for forecasting shale gas production, and also give some

insights into scientific evidence to the rational development of shale gas reservoirs.

Keywords: Shale gas, matrix pores, diffusion, slippage, gas production discipline.

Introduction and BackgroundTransport of gas in shale gas reservoirs is a complex multi-scale transport process, which is from macro scale to the

molecular scale (Javadpour 2007). So far, lots of researches on transport mechanisms of shale gas in shale matrix pores and

fractures have been done. For the transport of gas in fractures, nearly all researchers considered there is no doubt in migration

mechanism of fractures, generally believing that gas migration in the fracture system obeys Darcy flow, but the migration

mechanism of gas in matrix system is in the controversial. Zuber et al. (2002),Schepers et al. (2009),Wang and Reed (2009),

Song and Ehlig-Economides (2011) and Song (2013) successively proposed that gas migration from matrix system to the

fracture system in shale gas reservoirs obeys Darcy flow; Rushing et al (1989; 2007), Dahaghi (2010) and Dahaghi and

Mohaghegh (2011) successively proposed gas migration diffuses from matrix system to the fracture system in shale gas

reservoirs; Javadpour (2009) and Ozkan et al. (2009) proposed that flow and diffusion exist at the same time for gas

migration from matrix system to fracture system.


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Recently investigations on apparent gas permeability focused on flow mechanisms of gas in Nano-pores in shale gas

(Clarkson et al. 2011, 2012a, 2012b; Michel et al. 2011; Civan et al. 2011; Sakhaee-Pour and Bryant 2012; Javadpour et al.

2007; Javadpour 2009; Swami and Settari 2012; Swami et al. 2012; Fathi et al. 2012) . Ertekin et al. (1986) introduced a

dynamic slippage factor into apparent gas permeability model by using Knudsen diffusion coefficient. Clarkson et al. (2011,

2012a)extended the dynamic-slippage concept to shale gas reservoirs in order to analyze production characteristics of tight-

gas and shale-gas reservoirs. Numerical simulation demonstrated that dynamic-slippage concept should be incorporated to

accurately forecast the production characteristics of reservoirs with multi-mechanism flow such as shale gas reservoirs.

Michel et al. (2011) also incorporated Knudsen diffusion into the gas slippage factor to develop their own dynamic gas

slippage factor. Civan et al. (2011) and Sakhaee-Pour and Bryant (2012) considered Knudsen diffusion into the slippage

factor and developed an apparent gas permeability model. Javadpour et al. (2007, 2009)considered both Knudsen diffusion

and slippage effect and derived an apparent gas permeability model. Swami and Settari (2012) performed pore scale gas

model for shale gas reservoirs and thought that Knudsen diffusion and slippage should be included in the shale gas

production modeling. Swami et al. (2012)compared various apparent gas permeability models and quantitatively modeled

gas production in shale gas reservoirs by incorporating these models in a numerical simulator. They noted that models

proposed by Civan et al. (2011)and Sakhaee-Pour and Bryant (2012)were a reasonable approach, and the model proposed

by Javadpour et al. (2007, 2009), in which the Knudsen diffusion and slippage effect were both considered, was also

reasonable on a theoretical basis but needed validation against field data.

In this work, first by presenting the pore scale in shale matrix and the pressure characteristics of shale gas reservoirs,

according to Knudsen number, the flow regime in shale gas reservoirs is determined systematically. An apparent gas

permeability model for organic matter in shale matrix is proposed for all flow regimes. Furthermore, an apparent gas

permeability model for inorganic matter in shale matrix considering water effect in inorganic matrix pores is also developed.

Through considering the organic content in shale matrix, the apparent gas permeability model for shale matrix is generated

by combining apparent gas permeability models for organic matter and inorganic matter in shale matrix. Finally, gas

production model of shale gas wells is established and calculated, and gas production decline curve is also obtained. From the

gas production decline curve, it can be clearly seen that gas transport in shale matrix pores significantly affect the gas

production performance.

Transport type of shale gas in matrix pores

Matrix pore distribution of shale gas

At present, a lot of researches have been done for pore scale distribution characteristics of shale matrix. Howard (1991)

proposed that matrix pore diameter of Frio shale is between 5nm and 15nm; Katsub (1992) proposed that the matrix pore

diameter of the shale buried 4400 and 5600m underground is between 2.7nm to 11.5nm; Reed (2007) and Wang et al. (2009)

pointed out that the pore diameter of shale organic matter ranged from 5nm to 1000nm;Nelson (2009) proposed that the

minimum diameter of shale matrix pore throats was detected to be about the size of asphaltene molecular by experiments,

about 10 times of water molecule and methane molecule, about 5nm. And the maximum diameter of shale matrix pores is

about 100nm. Javadpour (2009) revealed that nanometer pores existed in shale matrix by high-pressure mercury injection

experiment and backscatter scanning electron microscopy, and detected Nano-pores and Nano-grooves in shale matrix by

using atomic force microscopy (AFM) at the first time.

According to these researches on the pore scale of shale matrix, it can be concluded that the diameter of shale matrix pores is

mainly distributed in the nanometer level.

Reservoir pressure and temperature of shale gas reservoir

Material for 5 shale gas basins in North America shows that the initial reservoir pressure of shale gas reservoirs is

between 2.76 and 68.95MPa, as shown in Figure 1.


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Figure 1 The initi al formation pressure of 5 shale gas reservoirs in North America

The mean free path of gas molecules in shale gas reservoir

The mean free path of gas molecules is defined as,

2

m2

= B

T

p (1)

where, is the mean free path of gas molecules, m; Bis Boltzmann constant, which is equal to 1.3805 10-23J/K; Tis the

temperature, K;pis the pressure, Pa; m is the molecular collision diameter (effective diameter) of the gas molecules, m.

The mole percentage of gas components for a kind of shale gas is shown in the Table 1, it can be seen that the collision

diameter of each component is different, the mean molecular collision diameter of shale gas weighted according to the mole

fraction is 0.41 nm, and the molar mass is 19.5 kg/kmol.

Table 1 Molecular coll ision diameter of shale gas

Gas Mole fraction Collision diameter Molar mass

component (%) (m, m) (kg/kmol)

CH4 87.4 4.010-10 16

C2H6 0.12 5.210-10 30

CO2 12.48 4.510-10 44

Average 4.110-10 19.5

According to the equation of the mean free path of gas molecules, we can see that is directly proportional to T, but inversely

proportional top. The mean free path of shale gas molecules under different temperature and pressure are calculated, which

are listed in Table 2 and shown in Figure 2.

Table 2 The mean free path of shale gas molecules under dif ferent temperature and pressure

p

Pa

at T=273K

nm

at T=300K

nm

at T=350K

nm

at T=400K

nm

1.00E+05 50.46223 55.45301 64.69517 80.03717

1.00E+06 5.046223 5.545301 6.469517 8.003717

1.00E+07 0.504622 0.55453 0.646952 0.800372

1.00E+08 0.050462 0.055453 0.064695 0.080037

0

5

10

15

20

25

30

Barnett Ohio Antrim New Albany Lewis

InitialReservoirPressure,

MPa

The Minimum

The Maximum


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Figure 2 the mean free path of shale gas molecules under different temperature and pressure

From Table 3 and Figure 2, it can be seen that: (1) temperature has little impact on the mean free path of shale gas

molecules; (2) pressure has great influence on the mean free path of shale gas molecules; (3) when the pressure is less than

10MPa,increases sharply as pressure decreases; (4) when the pressure is greater than 10MPa, is less than 1, and decreasesslightly with the increase of pressure.

Determination of transport mechanism of shale gas in matrix pores

The range of Kn in shale gas matrix pores

Knudsen (1934)defines Knudsen number to describe different gas transport pattern in porous media. The expression of

Knudsen number is,

n

=KD

(2)

whereDis the mean diameter of the pore, m; is the mean free path of gas molecules, m.Roy et al. (2003)put forward that gas transport can be divided into different flow patterns by Knudsen number, for

each flow regime, gas transport followed different control equations, as shown in Figure 3.

Figure 3 Gas flow regimes divided by Kn

When Kn


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When 10-1


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apparent gas permeability equation in the slip flow range can be expressed by Klinkenberg Equation (Klinkenberg 1941),

( ) gg Slip = 1 8 ,

+

k k c

D (4)

where (kg)Slip is apparent gas permeability in slip flow regime, m2; k is the absolute permeability of rock, usually is

approximated by apparent liquid permeability , m2;g is mean free path of gas molecules, m;Dis the pore diameter, m; c is

the proportionality factor, dimensionless.

A similar equation for slip flow was derived by (Roy et al., 2003; Sakhaee-Pour and Bryant, 2012),

( )g Slip gl

2=1 4 ,

+

v

v

k

k D (5)

where v is the tangential momentum-accommodation coefficient, which is close to 0.9 (Roy et al., 2003), so the slip flow

model will be,

( ) ( )g Slip = 1 5 n . +k k K (6)From this equation, we can see that c in Equation 1 is approximately equal to 0.625.

Actually Equation 6 can also be used to calculate the absolute permeability in continuum flow regime, because in

continuum flow regime, Kn 10, gas transport is in free-molecule flow regime. As mentioned above, gas transport in shale matrix pores

may be in the continuum flow regime, slip flow regime, or transition flow regime. Only Knudsen flow cannot be happen in

shale matrix. However, in order to get the gas transport model for the whole transition flow regime and the gas transport

model for all flow regimes (including continuum flow regime, slip flow regime, transition flow regime and Knudsen flow

regime), Knudsen flow model is also presented here.

Ziarani and Aguilera (2012) conducted some experiments to investigate gas flow mechanism in this regime. Sakhaee-Pour and Bryant (2012), Ziarani and Aguilera applied Knudsen diffusion to describe the gas flow characteristics in this

regime. Agarwal et al. (2001) proposed that gas transport mechanism can be investigated by DSMC method and grid

Boltzmann method.

Sakhaee-Pour and Bryant (2012)proposed a flow equation to represent the free-molecule flow,

n n ,= K K iJ D n (9)

whereJKnis mass flux of gas component i, kg/s/m2;DKnis Knudsen diffusivity coefficient, m

2/s; in is the density gradientof gas component i, kg/m

4.

Transmitting equation of the mass flow rate with gas density gradient to that with gas pressre gradient yields,

D Kn m ,= M

J D PRT

(10)


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where M is the molar mass of gas, kg/kmol; R is universal gas constant, which is equal to 8314 Pa.m3/(kmol.K); T is the

temperature, K;Dis the diameter of pores.DKnis expressed by

organic

Kn

8

3 =

D RTD

M (11)

Submitting Equation 11into Equation 9 gives,

organic

D m

8

3 =

DM RTJ PRT M

(12)

Actually,JDcan be expressed by,

D ,=J v (12)

where vis the gas flow velocity, m/s. Equation 12can also be expressed as,

g organic

m

g g

8 1

3

=

M D RTv P

RT M (13)

Comparing the form of Darcy flow equation, we can see that in Knudsen diffusion regime the apparent gas permeability

is,

( ) g organicg Knudseng

8,

3

= M D RT

k

RT M

(14)

where (kg)Knudsen is apparent gas permeability in free-molecule flow regime, m2; is gas viscosity, Pa.s;avgis average gas

density, kg/m3, which expression is,

mg =

Mp

RT (15)

So Equation 14 will be changed to be

( ) g organicg Knudsenm

8

3

=

D RTk

p M (16)

Applying Poiseuilles Equation for the capillary tube model and the expression of the average free path of gas molecule

(see Equation 1), Equation 16can further reorganized as follows,

( )

( )

2 2 2

organic organic g g

g 2Knudsenorganic organic

2 2

g

organicorganic

128

32 3

128= n

3

=

D Rk MT D

Rk K

MT

(17)

The flow-slip-diffusion model of gas in all flow regimes

In order to get the transport model in all flow regimes, we need investigate the transport mechanism in transition flow

regime and get the flow model for this regime first. The flow model in transition flow regime proposed by Sakhaee-Pour and

Bryant (2012)is suitable for 0.1


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higher than 10, it will gradually approach to 0.It will decrease from 1 to 0 with Kn increasing from 0.1 to 10. The largerfis,

the more slip flow effect is, on the contrary, the lowerfis, the more Knudsen diffusion effect is.

In order to get the coefficient Kn0.5and n in Equation 19, Equation proposed by Sakhaee-Pour and Bryant (2012)was

used to generate the fitting data, as shown in Table 3.

Table 3 the apparent gas permeabilit y versus Kn from Sakhaee-Pour and Bryant (2012)

Kn 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(kg)Transiton/ k 1.392693 1.943352 2.497277 3.054468 3.614925 4.178648 4.745637 5.315892

By matching data inTable 3with the proposed model (Equation 18 and 19), Kn0.5and ncan be obtained.Figure 5 and 6

are the comparison between (kg)Transiton/ k/Kn from Sakhaee-Pour and Bryant (2012)and those estimated by the proposed

model in this work. Figure 7 is the comparison between (kg)Transiton/ k from Sakhaee-Pour and Bryant (2012) and those

estimated by the proposed model in this work. From Figure 5, 6 and 7, it can be seen that a good agreement is obtained.

Finally Kn0.5is determined to be 4.5, and nis equal to 5.

Figure 5 Comparison between (kg)Transiton/ k/Kn from Sakhaee-Pour and Bryant (2012)and those estimated by the proposed model in

this work

Figure 6 Comparison between (kg)Transiton/ k/Kn from Sakhaee-Pour and Bryant (2012)and those estimated by the proposed model inthis work

0

4

8

12

16

20

0 0.2 0.4 0.6 0.8 1

kg

/k

/Kn,

dimension

less

Kn, dimensionless

The proposed medel

Equation by Sakhaee-pour and Bryant, 2012

1

10

100

1000

0.1 1 10

kg/k

/Kn,

dimensionless

Kn, dimensionless

The proposed model

Equation by Sakhaee-pour and Bryant, 2012


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Figure 7Comparison between (kg)Transiton/ kfrom Sakhaee-Pour and Bryant (2012)and those estimated by the proposed model in

this work

Figure 8is the plot offversus Kn. As shown in this figure, when Kn is lower than 0.1, fis almost equal to 1, so in this

case it can be considered that there is no Knudsen diffusion effect in gas flow process, which manifests that the proposed

model can be extend to slip flow regime and continuum flow regime. When Kn is larger than 15, fis almost equal to 0, so in

this case it can be considered that there is no slippage effect in gas flow process, which indicates that the proposed model can

also be extend to free-molecule flow regime.

Figure 8Plot of fversus Kn.

Figure 9is the comparison between kg/kfor slip flow regime, transition flow regime, and Knudsen diffusion regime andthose estimated by the proposed model. From this figure, it can be clearly seen that the proposed model can accurately match

data in all flow regimes, which validates the effectiveness of the proposed model for all flow regimes. From this figure, we

can see that kg/krockets for 1


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Figure 9Comparison betweenkg/kfor slip flow regime, transition flow regime, and Knudsen diffusion regime and those estimatedby the proposed model

Finally, the apparent gas permeability model for all flow regimes proposed in this work will be,

( ) 2 24 5gg,organic 4 4 4 4

1284.5 1 5 n n= .

4.5 + n 3 4.5 + n

+ +

K R Kk k

K MT K (20)

For shale gas reservoirs with parameters in Table 4, when the pressure is from 1MPa to 70MPa, the apparent gas

permeabilities for these three pressures are shown in Figure 10. The corresponding Darcy flow models are introduced to

compare the difference between apparent gas permeability for all flow regimes and the absolute permeability.

Figure 10 Plots o f kgversus Kn when T=360K, =0.06, and p= 1MPa, 10 MPa, and 70MPa respectively

From Figure 10, we can see that the apparent gas permeability for Kn close to 1 is lower than that for Kn close to 5,

although the diameter of matrix pores for lower Kn is larger than that for higher Kn at the same average free path of gas

molecules. When the pressure is from 1 MPa to 70 MPa for shale gas reservoirs, the lowest apparent gas permeability appears

in the range of Kn from 1.2 to 0.8. This concludes that when Kn is in the range of 0.8-1.2, the apparent gas permeability is

1

10

100

1000

10000

100000

0.0001 0.001 0.01 0.1 1 10 100

kg/k

Kn

The proposed model

Equation by Sakhaee-pour and Bryant,

2012Knudsen Diffusion Equation

Slip flow model

1.0E-10

1.0E-08

1.0E-06

1.0E-04

1.0E-02

1.0E+00

1.0E+02

0.0001 0.001 0.01 0.1 1 10 100

Kg,mD

Kn

p=1MPa,the proposed model

p=1MPa,Darcy equation






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SPE 167226 11

lowest and the development effect is the worst.

So for shale gas reservoir, we can also see that the apparent gas permeability of shale matrix doesnt always decreases

with decreasing the matrix pore size.

Table 4 Parameters of shale gas reservoirs

,

Pa.s

,

dimensionless

,

m

,

fraction

,

J/K

R,

Pa.m3/Kmol/K

M,

kg/kmol

T,

K

0.000023 1 4.110-10 0.06 1.381023 8314 19.5 360

Transport model of shale gas in inorganic matrix pores

Water distribution in matrix pores (inorganic matrix pores)

The inorganic shale matrix pores distributes in the void of fine sands and clays, as shown in Figure 11. Because the

surface of fine sands and clays is hydrophilic, water adheres to the internal surface of the inorganic shale matrix pores, while

the organic matrix pores is hydrophobic, so water in the matrix pores mainly distributes in the inorganic matrix pores,

generally appears in the inner surface of large inorganic matrix pores, and even occupies the whole small inorganic pores.

Due to the existence of water in the inorganic matrix pores, the effective diameter of inorganic matrix pores will be reduced

or plugged.

Figure 11 Inorganic matrix pores of SEM of Barnett (Ambrose et al. 2010; Wang and Reed 2009)

Gas flow model in inorganic matrix pores

From Figure 11, we can see that the matrix pores in shale gas can be represented by capillary tubes or plates. If the matrix

pores are represented by capillary tubes with same diameters, the liquid water adsorbing at the internal surface of capillary

tubes will reduce the diameter of matrix pores. On the one hand, the apparent gas permeability of the inorganic porous media

decreases with water saturation because of decrease in the pore throat for gas filtration, on the other hand, it will increase

because of the increase of the slippage effect.

Gas transport mechanism in inorganic matrix pores is in the Darcy flow regime or the slip flow regime, so the apparent

gas permeability model in inorganic matrix matter can be expressed by

( ) ginorganic inorganicinorganic

= 1 8 ,

+

k k cD

(21)

where kinorganicis apparent gas permeability in inorganic matrix matter, m2;g is mean free path of gas molecules, m; Dinorganic

is the pore diameter of inorganic matrix pores, m.

For the shale matrix block with Atas the cross sectional area, tas the content of organic matter in shale matrix, Ltas the

length, t as the porosity of matrix, oganicas the porosity of organic matter, andDinorganic,0 as the initial diameter of inorganic

matrix pores, so the relationship between the initial porosity of the inorganic matter and the initial diameter of inorganic

matrix pores can be represented by

( ) ( )

2 2

inorganic 0 t inorganic 0

inorganic 0

t t t

= .4 1 4 1

=

D L D

A L A (22)

The initial permeability of the inorganic matter is,

Inorganic matrix pores


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( )2

inorganic 0 inorganic 0

2inorganic,0 .

32

=

Dk

(23)

Because of the existence of water in inorganic matrix pores, the diameter of inorganic matrix pores becomes Dinorganic, so

the relationship betweenDinorganicand water saturation in the inorganic matterSw,inorganiccan be described as,

( ) ( )2 2 2 2inorganic, 0 inorganic t inorganic, 0 inorganicw,inorganic 2 2

inorganic,0 t inorganic,0

= .

=

D D L D DS

D L D

(24)

Equation 24can also be transformed as,

( )2 2inorganic w,inorganic inorganic, 0= 1 .D S D (25)The porosity of inorganic matter with water saturation of Sw,inorganicwill become

( ) ( )

2 2

inorganic inorganic

inorganic

t t

= .4 1 4 1

=D L D

A L A (26)

Dividing Equation 26by Equation 22and arranging it gives,2

inorganic

inorganic inorganic 02

inorganic 0

.

= D

D (27)

Substituting Equation 25into Equation 27yields,

( )i norganic w,inorgani c inorgani c 01 . = S (28)So the gas permeability of inorganic matter will become,

( )2

inorganic inorganic

2inorganic.

32

=

Dk (29)

Substituting Equation 25 and 28into Equation 29gives,

( ) ( )

22

w,inorganic inorganic,0 inorganic,0

2inorganic

1.

32

=

S Dk (30)

Dividing Equation 30by Equation 23and arranging it gives,

( ) ( )2

w,inorganic ,inorganic,0inorganic 1 . = k S k (31)

Substituting Equation 25 and 31intoEquation 21gives,

( ) ( ) ( )2 1.5 g

inorganic w,inorganic w,inorganicinorganic 0inorganic 0

= 1 8 1

+

k k S c S

D (32)

Assume the water saturation of the matrix block is Sw. As mentioned above, water only appears in inorganic matrix pores,

so water saturation in the inorganic matter will be

ww,inorganic = .

1

SS (33)

Substituting Equation 33into Equation 32gives,

( )2 1.5

w winorganic inorganic 0

= 1 8 1 n1 1

+

S Sk k c K (34)

The coupling transport model of shale gas in organic and inorganic matrix pores

Because the organic matrix pores and Inorganic matrix pores in the parallel in the direction, so the apparent gas permeability

in shale matrix block with organic materials and inorganic materials is the cross surface weighted average apparent gas

permeability in organic matrix pores and gas permeability in inorganic matrix pores. Assume that is the organic matter

content in shale matrix, the total apparent gas permeability model will be the summation of the apparent gas permeability of

organic material and that of inorganic material, which is written as,


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SPE 167226 13

( ) ( )

( ) ( )

2 24 5g

gT 4 4 4 4organicorganic

2 1.5

w w

inorganic,0

1284.5 1 5 n n=

4.5 + n 3 4.5 + n

+ 1 5 1 n 1 ,1 1

+ +

+

K R Kk k

K MT K

S Sk K

(35)

where (k)organicand (k)inorganic,0can be expressed asEquation 20 and Equation 34 for capillary models, respectively. Theinitial porosity of inorganic matter inorganic,0can be calculated by,

( )0 organic

inorganic 0 = ,1

(36)

So the total apparent gas permeability model will changed to be,

( )

( )

2 2 24 5organic organic g

gT 2 4 4 4 4

organic

2 2 1.50 organic inorganic,0 w w

2

1284.5 1 5 n n=

32 4.5 + n 3 4.5 + n

+ 1 5 1 n ,32 1 1

+ +

+

D K R Kk

K MT K

D S SK

(37)

Where kgTis the total apparent gas permeability, m2; is the organic matter content in shale matrix, fraction; 0is the initialporosity of the whole shale matrix block, fraction; organicis the porosity of organic matter, fraction; inorganicis the porosity of

inorganic matter, fraction; inorganic,0 is the initial porosity of inorganic matter, fraction; Dorganic is the diameter of organic

matter pores, m;Dinorganic,0is the initial diameter of inorganic matter pores, m; Swis the water saturation in the whole shale

matrix block, fraction; is the tortuosity of the shale matrix, dimensionless.

Gas produc tion model of shale gas wells

Model assumption

Shale gas reservoir is dual porosity reservoirs. In order to investigate the production characteristic of shale gas wells and the

effect of gas transport in matrix pores on the production. Here, flat model is applied to simplify the dual porosity reservoir

(Bello, 2009), the simplified dual porosity model is shown in Figure 12.

Figure 12 Simplified dual porosi ty model for shale gas reservoirs

The following assumptions are given below for the flat model of shale gas horizontal well with multi-stage fractures,

(1) The formation of shale gas reservoir is horizontal and with the same thickness, in which the gas flow is isothermal

process; (2) Horizontal well with multi-stage fractures is located in the center of reservoir, and no fluid flows at ends of the

horizontal well; (3) Only gas is produced from the well; (4) The effect of gas desorption in matrix pores is considered; (5)

Effects of wellbore storage and skin factor are not considered in the model; (6) Matrix block is in the flat form, with L in the


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14 SPE 167226

characteristic length of matrix block; (7) Matrix piece centerline (dotted line) is the Y axis direction, horizontal well is the X

axis direction; (8) The direction of gas flow in matrix is in the X direction, and the direction of gas flow in fractures is in the

Y direction.

Combining with the characteristics of linear flow in shale gas reservoirs, the flat model contains four directions for flowing:

(1) The linear flow from matrix to fracture, parallel to the X direction; (2) The linear flow from fracture to the horizontal

wellbore, parallel to the Y direction.

Model establishment

Continuity equation for matrix

The continuity equation in shale matrix is expressed in terms of pseudo-pressure and pseudo-time,

( )2 g tm mm i m2

am a

,

=

C

x k t (38)

where tais the pseudo-time, dimensionless; mis the pseudo-pressure in matrix pores, Pa/s; Ctmis the total compressibility in

matrix, Pa-1; Cf is the compressibility of matrix system, Pa-1; Cg is compressibility of gas, Pa

-1; Cd is the desorption

compressibility of gas, Pa-1.The expressions of ta, m, Ctm, and Cdare expressed as follows,

( )t ia

0t( ) ( )

t Ct dt

p C p

= (39)

( )i

m m 2p

p

pp dp

z

= = (40)

tm g f d C C C C = + + (41)

( )2

sc L Ld

m sc sc L

p ZV pC

Z T p p p=

+ (42)

Continuity equation for fracture

The continuity equation in fracture is expressed in terms of pseudo-pressure and pseudo-time,

( )2

g tf f amf m i f

2

/ 2f f a

1 ,/ 2

=

= x L

Ck

x L k x k t (43)

wherefis the pseudo-pressure in fractures, Pa/s; Ctfis the total compressibility in fractures, Pa-1; Cris the compressibility of

fracture system, Pa-1; The expressions of fand Ctfare expressed as follows,

( )t ia

0t( ) ( )

t Ct dt

p C p

= (44)

( )i

f f 2

= = p

p

pp dp

z (45)

tf rgC C C= + (46)

The initial condition and boundary conditions(1) The initial condition and boundary conditions for matrix system

Initial condition: m=0 when t=0;Outer boundary condition:

m/ x=0 whenx=0;

Inner boundary condition: m= fwhenx=L/2

(2) The initial condition and boundary conditions for fracture system

Initial condition: f=0 when t=0;Outer boundary condition:

f/ y=0 wheny=yf;

Inner boundary condition: ( )f f f f cw f2k p k p

q A x hy y

= =

wheny=0.


15/19

SPE 167226 15

Model dimensionless

Dimensionless variable is defined as inTable 5.

Table 5 Definition of dimensionless variable

Variable Equation Variable Equation

Dimensionless pseudo-

pressure

f cw i

D 3

( )

1 291 10

k A

qT

=

.

Dimensionless pseudo-

time ( ) ( )f a

Da

gi ti ti cwm f

k tt

c c A =

+

Dimensionless

production

f cw i wf

3

D

( )1

1 291 10

k A

q qT

=

. Storability

[ ]f tif

m tim f tif

c

c c

=

+

Transmissivity ratioam

cw2

f

112

kA

L k= Dimensionless size of

gas reservoir

fDf

cw

yy

A=

Dimensionless length -

X direction D / 2

xx

L=

Dimensionless length -

Y directionD

cw

yy

A=

Through submitting dimensionless variables in Table 5 to Equation 39 and 43, the dimensionless continuity equations for

matrix and fractures can be obtained.

Dimensionless continuity equation for matrix

The dimensionless continuity equation for matrix is,

( )2

Dm Dm

2

D Da

31

x t

=

(47)

Initial condition: Dm=0 when tDa=0;Outer boundary condition:

Dm/ xD=0 whenxD=0;

Inner boundary condition: Dm= DfwhenxD=1.

Dimensionless continuity equation for fracture

The dimensionless continuity equation for fracture is,

D

2

Df Dm Df 22

D D Da13

xy x t

=

=

(48)

Initial condition: Df=0 when tDa=0;Outer boundary condition:

Df/ xD=0 whenyD=yDf;

Inner boundary condition:

D

Df

D 0

2

yy

=

=

whenyD=0.

Model calculation

Introduce Laplace transformation method, tin real space is transformed to sin Laplace space by the following equation,

0( ) ,

=

sts e dt (49)

wheresis variable in Laplace space.

Through Laplace transformation, the dimensionless pseudo-pressure at the bottom hole of shale gas horizontal well with

multi-stage fractures produced at the constant gas production is derived,

( )

( )

( )

Df

Df

2

Dwf 2

2 1

1

s f s y

s f s y

e

s sf s e

+==

(50)

The relationship between production and pressure in Laplace space is as follows (Van Everdingen and Hurst, 1949; Bello,

2009)


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16 SPE 167226

Dwf D 2

1q

s = (51)

So the dimensionless gas production of shale gas horizontal well with multi-stage fractures produced at the constant bottom

hole flowing pressure is,

( ) ( )

( )

Df

Df

2

D

2

1

2 1

sf s y

s f s y

sf s eq

s e

=

+

(52)

Gas production decline curve

Gas production decline curves can be divided into four stages, including fracture linear flow stage, bilinear flow stage, matrix

linear flow stage, and outer boundary controlling stage. Figure ? is the gas production decline curves with fixed storability

and Dimensionless size of gas reservoir yDf but different transmissivity ratio . As shown the curve with =0.001 in

Figure ?, three vertical dotted lines divides the horizontal axis into four sections. From the left side to the right side, the first

section is the fracture linear flow stage with -1/2 in slope of the curve. The second section is the bilinear flow stage with -3/4

in slope of the curve. The third section the matrix linear flow stage with -1/2 in slope of the curve. The fourth section is the

outer boundary controlling stage with the characteristics that gas production declines rapidly.

Transmissivity ratioreflects thecapacity of fluid flow from matrix system to fracture system, the larger is, the more easily

fluid flows from matrix into fractures. From the expression of transmissivity ratio in Table 5, we can see that the effect of

gas transport in matrix pores on gas production can be revealed in the term , the larger apparent gas permeability kgT

corresponds to the larger transmissivity ratio.

Figure 13 The effect of interporosit y flow coeffic ient on type curves

From Figure 13, it can be seen that the effect ofon gas production mainly appears in bilinear flow stage and matrix linear

flow stage. In bilinear flow stage and matrix linear flow stage, the dimensionless gas production increases with increasing,

the larger is, the less obvious the characteristics in bilinear flow stage is, the shorter matrix linear flow stage is, and the

sooner flow arrives the outer boundary.

ConclusionThrough determination the flow regime in shale gas reservoirs, investigation on gas transport model for shale matrix, and

production forecasting of shale gas wells in this work, the following conclusions are obtained,

(1) Gas transport in shale matrix pores is not unique, there are 3 possibilities: continuum flow, slip flow and transition flow.

For actual shale gas reservoirs, gas transport pattern can be determined by using the decision chart just knowing the pressure,

temperature, and the pore size distribution.

(2) Only Knudsen diffusion is not suitable for shale gas reservoirs. This is because Knudsen number in shale gas reserovirs is

generally less than 10, especially for such shale gas reservoirs with higher initial reservoir pressure.

(3) Agreements between data from flow models for each flow regime and those estimated by the proposed a new diffusion-

slippage-flow model for organic matter in shale matrix validate the effectiveness of the proposed model for all flow regimes.

(4) The apparent gas permeability model for organic matter and inorganic matter in shale matrix can be applied to calculate

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

1.0E+01

1.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03

lgqD

lgtaD

=0.1

=0.01

=0.001

=0.01

yDf=0.5

increasing


17/19

SPE 167226 17

the apparent gas permeability of shale matrix in case that pore size distribution, water saturation in shale matrix, porosity of

shale matrix block, porosity of organic matter are given. Because the organic content and water effect in inorganic shale

matrix pores are considered, the proposed equation is more reasonable to calculate the apparent gas permeability of shale

matrix.

(5) Gas production decline curve of shale gas wells indicates that gas transport in shale matrix pores significantly affect the

gas production performance in the middle stage of production history.

AcknowledgementsThis research was supported by National Natural Science Foundation Project (U1262113) and Science Foundation of China

University of Petroleum, Beijing (YJRC-2013-37). We also recognize the support of MOE Key Laboratory of Petroleum

Engineering in China University of Petroleum (Beijing), The Research Institute of Yanchang Petroleum Group, and the

Center for Petroleum & Geosystems Engineering at The University of Texas at Austin to this paper.

Nomenclature = Mean free path of gas molecules, m;

B = Boltzmann constant, which is equal to 1.3805 10-23J/K;

T = Temperature, K;

p = Pressure, Pa;

m = Molecular collision diameter (effective diameter) of the gas molecules, m;

D = Mean diameter of the pore, m;

vm = Gas velocity in shale matrix pores, m/s;k = Permeability of the shale matrix, m

2;

pm = Pressure in shale matrix pores, Pa;

g = Gas viscosity, Pas.

(kg)Slip = Aapparent gas permeability in slip flow regime, m2;

k = Absolute permeability of rock, m2;

g = Mean free path of gas molecules, m;

D = Pore diameter, m;

c = Proportionality factor, dimensionless.

v = Tangential momentum-accommodation coefficient, dimensionless;

Ja = Slip mass flow rate of gas, kg/s/m2

JD = Knudsen diffusion mass rate of gas, kg/s/m2;

(kg)Transition = Apparent gas permeability in transition flow regime, m2;

kl = Liquid permeability, m

2

;Kn = Knudsen number, dimensionless;

JKn = Mass flux of gas componenti, kg/s/m2;

DKn = Knudsen diffusivity coefficient, m2/s;

in = The density gradient of gas component i, kg/m4.

v = Gas flow velocity, m/s.

(kg)Knudsen = Apparent gas permeability in free-molecule flow regime, m2;

= Gas viscosity, Pa.s;

avg = Average gas density, kg/m3;

f = Weight coefficient, dimensionless;

kinorganic = Apparent gas permeability in inorganic matrix matter, m2;

g = Mean free path of gas molecules, m;

inorganic = Porosity of inorganic matter, fraction;

inorganic,0 = The initial porosity of inorganic matter, fraction;kgT = Total apparent gas permeability, m

2;

= Organic matter content in shale matrix, fraction;

0 = The initial porosity of the whole shale matrix block, fraction;

organic = Porosity of organic matter, fraction;

Dorganic = Diameter of organic matter pores, m;

Dinorganic,0 = The initial diameter of inorganic matter pores, m;

Sw,inorganic = Water saturation in the inorganic matter, fraction;

Sw = Water saturation in the whole shale matrix block, fraction;

= Tortuosity of the shale matrix, dimensionless

ta = Pseudo-time, dimensionless;

m = Pseudo-pressure in matrix pores, Pa/s;

Ctm = Total compressibility in matrix, Pa-1;


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18 SPE 167226

Cf = Compressibility of matrix system, Pa-1;

Cg = Compressibility of gas, Pa-1;

Cd = Desorption compressibility of gas, Pa-1.

f = Pseudo-pressure in fractures, Pa/s;

Ctf = Total compressibility in fractures, Pa-1;

Cr = Compressibility of fracture system, Pa-1;

s = Variable in Laplace space.

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SI Metrc Conversion Factorsft 3.048* E01 = m

scf/D 2.863 640 E02 = standard m3/d

millidarcy 9.869 233 E+14 = m2

lbf/in.2(psi) 6.894 757 E+03 = Pa

cp 1.0* E03 = Pas

lb 2.2046 E+00 = kg

terapaper diffusion and flow mechanisms -1147

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