Shale Gas Transport ModelYou Nan

Professor: Lau Hon Chung

Department of Civil and Environment Engineering

Outline

➢Background

➢Mathematical model

➢Model application

➢Summary

2

Background

(E. Basma, 2016)

1

FE-SEM images of Longmaxi shale(Y. Wang et al., 2014)

Background4

2

SEM of Barnett shale (Ambrose et al., 2010)

Background

SEM of Eagle Ford shale (Naraghi et al., 2015)

c d

3

porosity

Matrix porosity

Organic porosity

Inorganic porosity

Natural fracture

Hydraulic fracture

Background

(Liu et al., 2016)

Na

no-p

ore

• Organic pores

• Inorganic pores

4

Outline

➢Background

➢Mathematical model

➢Model application

➢Summary

7

Bulk gas model

(Song and et al., 2017)

Knudsen number

𝐊𝐧 =𝑴𝒆𝒂𝒏 𝒇𝒓𝒆𝒆 𝒑𝒂𝒕𝒉 𝒐𝒇 𝒈𝒂𝒔𝒎𝒐𝒍𝒆𝒄𝒖𝒍𝒆𝒔

𝒑𝒐𝒓𝒆 𝒅𝒊𝒂𝒎𝒆𝒕𝒆𝒓=

𝛌

𝟐𝒓

Mathematical model

Mean free path of gas molecules

𝝀 =𝝁

𝒑

𝝅𝒛𝑹𝑻

𝟐𝑴

Knudsen diffusionKnudsen diffusion & Slip

8

5

(Song and et al., 2017)

p pressure m Particle mass

μ viscosity r Particle position

v velocity P Momentum

F Force field f(r, P, t) Probability density function of particles

Mathematical model

Darcy’s Law

Navier-Stokes equation

Boltzmann equation

Effect of collisions between particles

Knudsen diffusionKnudsen diffusion & Slip

6

Introduction

(Wu et al., 2016)

• Gas slippage

• Knudsen diffusion

• Surface diffusion

• Molecular diffusion

Mathematical model

Bulk gas

Adsorbed gas

Dissolved gas

10

7

ሶ𝑚𝑣 Mass flux of slip flow (viscous flow) per area ሶ𝑚𝑘 Mass flux of Knudsen diffusion per area

𝜁 Slip coefficient Dk Knudsen diffusion coefficient

ҧ𝑣 Thermodynamic molecular mean velocity 𝐷f Fractal dimension of the pore surface

αTangential momentum accommodation

coefficient at the pore wall (TMAC)δ

Ratio of normalized molecule size to local average

pore diameter

Mathematical modelBulk gas flow = Slip flow + Knudsen diffusion

Knudsen diffusion term

(S. Cruener and P. Huber, 2008)

Modified Hagen-Poiseuille’s equation (Brown et al.,1946)

8

In nano-pores, the number of collisions occurring within

a unit time period can be expressed as

𝑛𝑇 =1

𝑡𝑀+1

𝑡𝑆where

𝑡𝑀 =λ

ҧ𝑣

𝑡𝑆 =2𝑟

ҧ𝑣Therefore

𝜔𝑣 =

1𝑡𝑀

1𝑡𝑀

+1𝑡𝑆

=

ҧ𝑣λ

ҧ𝑣λ+

ҧ𝑣2𝑟

=1

1 +λ2𝑟

=1

1 + 𝐾𝑛

𝜔𝑘 =

1𝑡𝑆

1𝑡𝑀

+1𝑡𝑆

=

ҧ𝑣2𝑟ҧ𝑣λ+

ҧ𝑣2𝑟

=1

2𝑟λ+ 1

=1

1 + Τ1 𝐾𝑛

Mathematical model

Slip flow

𝑴𝒐𝒍𝒆𝒄𝒖𝒍𝒂𝒓 𝒄𝒐𝒍𝒍𝒊𝒔𝒊𝒐𝒏

𝑶𝒗𝒆𝒓𝒂𝒍𝒍 𝒄𝒐𝒍𝒍𝒊𝒔𝒊𝒐𝒏

Knudsen diffusion

𝑾𝒂𝒍𝒍 −𝒎𝒐𝒍𝒆𝒄𝒖𝒍𝒆 𝒄𝒐𝒍𝒍𝒊𝒔𝒊𝒐𝒏

𝑶𝒗𝒆𝒓𝒂𝒍𝒍 𝒄𝒐𝒍𝒍𝒊𝒔𝒊𝒐𝒏

Weighting coefficients?

Mean free path of gas molecules

Mean velocity of gas molecules

Pore diameter

Mean velocity of gas molecules

9

Gas transport modelMathematical model

2. Surface diffusion

Hopping model (Y.D. Chen, R.T. Yang, 1991)

Molecule-molecule collision

1. Bulk gas flow

Slip flow Knudsen diffusion

Molecule-wall collision

3. Total mass flux for one cylindrical capillary

𝑟𝑒𝑓𝑓

𝑟

4. Definition of flow conductance Viscous/slip flow

Knudsen diffusion

Surface diffusion

13

10

Influence factors

Stress dependence The nano-pore radius considering the

stress-sensitive effect can be expressed as

(Dong et al., 2010)

pc Overburden pressure

pe Effective pressure

p pore pressure

sShale permeability

coefficient

qShale porosity

coefficient

r0

Intrinsic pore radius at

atmospheric pressure

po atmospheric pressure

Mathematical modelInfluence factors

14

11

Influence factors

Ideal gas law

pV=nRT

Real gas equation

pV=znRT

z factor

Real gas effect Gas adsorption

Langmuir adsorption model

θ Gas coverage

𝑝𝐿 Langmuir pressure

𝑑𝑚 Gas molecule diameter

Mathematical modelInfluence factors

15

12

Slip flow

Knudsen diffusion

Surface diffusion

Mathematical model

Flow conductance (N. You model)

13

Bulk gas transport model

S.K. Loyalka and S.A.

Hamoodi (1990) simulated

rarefied nitrogen flow in a

cylindrical tube using linearized

Boltzmann molecule model.

Fitting factor: 𝛼 =0.99

Model validation

Mbi: mass flux of bulk gas

Mci: mass flux of Darcy flow (𝜋𝜌𝑟4

8𝜇

𝑑𝑝

𝑑𝑥)

Mki: mass flux of Knudsen diffusion

14

Outline

➢Background

➢Mathematical model

➢Model application

➢Summary

18

Parameter Symbol Unit Value

Shale permeability coefficient s Dimensionless 0.08

Shale porosity coefficient q Dimensionless 0.04

Overburden pressure pc MPa 50

Temperature T K 423

Gas type CH4 - -

Langmuir pressure pL MPa 6.72

TMAC α - 0.5

Fractal dimension of the pore

surface𝐷𝑓 - 2.6

Ratio of normalized molecule size

to local average pore diameterδ - 0.7

Model application19

15

A quick applicationModel application

Contribution of different mechanisms ( Ct=Cv+Ck+Cs )

20

16

Model application

Extrapolation from lab condition to in-situ condition

Lab condition

In-situ condition

0~50MPa

340K

CH4

5MPa

300K

N2• No gas adsorption

• No surface diffusion

• No stress dependence

• Full flow mechanisms

• All influence factors

17

Model application

Extrapolation from lab condition to in-situ condition

Pore size distribution : {r1, r2, … , rn} → {∅1, ∅2, … , ∅𝑛}

𝐶𝑡 =

1

𝑛

∅𝑖𝐶𝑡𝑖

18

Conclusion

Gas transport model

• Account for all flow regimes

• Consider real gas effect, stress dependence and gas adsorption effect

Gas transport model

• When r>10 nm, gas flow is dominated by slip flow.

• When r<5 nm, gas flow is dominated by surface diffusion.

• Knudsen diffusion only makes a relatively significant contribution to the total flow

conductance when p<1000 psi.

19

Conclusion

Relationship between experimental and in-situ data

• Experimental measurement overestimates permeability of shale.

• As reservoir pressure increases, in-situ permeability decreases to a certain point

(more gas adsorption, less surface diffusion, Knudsen diffusion, gas slippage) and

then increases due to decreasing effective pressure.

Future study

• Lab measurement of shale permeability under reservoir condition, aiming to further

validate and possibly improve the mathematical model.

• Application of the proposed mathematical model in reservoir simulation, helping

predict production and optimize exploitation strategy.

20

Thank you!You Nan

Model application

-100

-50

0

50

100

0 1000 2000 3000 4000 5000 6000 7000

I L(%

)

p (psi)

2 nm

real gas stress dependence adsorption layer

-100

-50

0

50

100

0 1000 2000 3000 4000 5000 6000 7000

I L(%

)

p (psi)

10 nm

real gas stress dependence adsorption layer

-100

-50

0

50

100

0 1000 2000 3000 4000 5000 6000 7000

(IL

(%)

p (psi)

5 nm

real gas stress dependence adsorption layer

Influence level of different factors (T=423 K)

y axis: 𝑰𝑳 =𝑪𝒕 (𝒊𝒈𝒏𝒐𝒓𝒆 𝒐𝒏𝒆 𝒊𝒏𝒇𝒍𝒖𝒆𝒏𝒄𝒆 𝒇𝒂𝒄𝒕𝒆𝒓)−𝑪𝒕

𝑪𝒕× 𝟏𝟎𝟎%

26

Model application

-100

-50

0

50

100

0 1000 2000 3000 4000 5000 6000 7000

I L(%

)

p (psi)

100 nm

real gas stress dependence adsorption layer

-100

-50

0

50

100

0 1000 2000 3000 4000 5000 6000 7000

I L(%

)

p (psi)

50 nm

real gas stress dependence adsorption layer

-100

-50

0

50

100

0 1000 2000 3000 4000 5000 6000 7000

I L(%

)

p(psi)

25 nm

real gas stress dependence adsorption layer

Influence level of different factors (T=423 K)

27

Model application

Influence of real gas effect

-4

-2

0

2

4

6

8

10

12

0 10 20 30

Ψ(%

)

p(MPa)

r=5nm (You) r=50nm (You) r=500nm (You)

T=380 K

-4

-2

0

2

4

6

8

10

12

0 10 20 30

Ψ(%

)

p(MPa)

r=5nm (You) r=50nm (You) r=500nm (You)

T=330 K

-4

-2

0

2

4

6

8

10

12

0 10 20 30

Ψ(%

)

p(MPa)

r=5nm (You) r=50nm (You) r=500nm (You)

T=423 K

28