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The Impacts of Pore Structure and Relative Humidity on Gas Transport in Shale: A Numerical Study by the Image-Based Multi-scale Pore Network Model

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Abstract

Due to the multi-scale pore size and complex gas-bound water distribution, it is challenging to accurately predict gas transport property in shale. Given the known heterogeneities, single-resolution pore-scale imaging is not reliable for representative pore structure characterization. In this study, the image-based shale multi-scale pore network model (MPNM) is proposed and the impacts of pore structure and relative humidity (RH) on gas transport are analyzed in detail. 3D binary images are constructed by the multiple-point statistics method from a section of low-resolution SEM image which covers the large-scale pore structure and fine-scale SEM images with the same physical size at high resolution. The maximal ball fitting method is applied to extract large-scale pore network model (LPNM) and fine-scale pore network models (FPNMs) from the 3D binary images, respectively. MPNM is obtained by merging the LPNM and FPNMs based on the proposed procedure. The confined gas-bound water distribution at different RH is calculated considering the disjoining pressure resulting from van der Waals force, electric double-layer interactions and structural force. Gas slippage in irregular pores is considered for gas transport. Pore structure parameters and gas permeabilities are calculated based on the MPNM, LPNM and FPNMs. Study results indicate that the gas permeability of MPNM is more close to the laboratory pressure pulse decay measured gas permeability of studied sample. Gas permeability decreases with the increasing RH and drops to zero at average pore radius less than 12 nm and RH larger than 0.7.

Article Highlights

  • The image-based shale multi scale pore network model (MPNM) is proposed based on low resolution and high resolution SEM images.

  • Permeability of MPNM is more close to the laboratory measured permeability compared with that of fine scale and large scale pore network.

  • Gas permeability drops to zero at average pore radius less than 12 nm and relative humidity larger than 0.7.

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The data used in figures can be found at https://doi.org/10.6084/m9.figshare.14502897.v1.

Abbreviations

A :

Cross section area given pore or throat (m2)

A 11 :

Hamaker constant for solid phase

A 22 :

Hamaker constant for gas phase

A 33 :

Hamaker constant for water phase

A 132 :

Nonretarded Hamaker constant representing solid and gas interacting across water film

A pnm :

PNM cross section area

\(A_{tri\_eff}^{{}}\) :

Effective transport area considering the water film thickness (m2)

e :

Electron charge

f cir(Kn cir):

Flow condition function

f tri(G,Kn tri):

Flow condition function in irregular triangle pore

g :

Gas flow conductance (m3 s−1 Pa−1)

g Darcy :

Gas flow conductance at Darcy flow case (m3 s−1 Pa−1)

g ij :

Gas flow conductance between pore i and pore j (m3 s−1 Pa−1)

g i, g j, g t :

Gas flow conductance of pore i, pore j and its connected throat (m3 s−1 Pa−1)

G :

Shape factor, dimensionless

h :

Water film thickness (m)

h p :

Planck’s constants (J/s)

h c :

Critical water film thickness (m)

k B :

Boltzmann constant (J K−1)

k :

Gas permeability at certain RH (m2)

k dry :

Gas permeability at dry condition (m2)

k Darcy :

Gas permeability at non-slip flow condition (m2)

Kn cir :

Knudsen number in circular pore

Kn squ :

Knudsen number in square pore

K str :

Magnitude of the structural forces

l :

Pore length (m)

L i, L j, L t :

Length of pore i, pore j and connected throat (m)

L thi :

Maximum throat length in the group of Ts of each region (Ω1, Ω1,…, Ω9) (m)

L t :

The throat length of the resultant throat connecting Pli and Ps (m)

L vt :

Virtual throat length (m)

M w :

Gas molar mass (g mol−1)

n ∞ :

Ion concentration (mol/m3)

N inlet :

Total number of inlet pores

N i (r t ) :

Throat radius of FPNM i (m)

N i(L t):

Throat length distribution of FPNM i

P e :

Excess pressure (Pa)

P s :

Pores on FPNMs that are connected with Pol

p s :

Saturated vapor pressure (Pa)

P d :

Perimeter of given pore or throat (m)

P li :

Corresponding large-scale pores in each region Ωi

P ol :

The pores with pore size larger than the corresponding rthi in each FPNM

Q:

Gas flux in single pore (m3 s−1)

r :

Pore radius (m)

R :

Universal gas constant, dimensionless

RH:

Relative humidity, dimensionless

RHc :

Critical relative humidity, dimensionless

r t :

The throat radius of the resultant throat connecting Pli and Ps (m)

r eff :

Effective gas transport radius considering the water film thickness (m)

r tri_eff :

Effective inscribed radius in triangle pore considering the water film thickness (m)

r thi :

Large-scale threshold pore size of each region (m)

r vt :

Virtual throat radius (m)

T :

Temperature (K)

T c :

Water critical temperature (K)

T s :

Throats on FPNMs that are connected with Pol

w eff :

Effective side length of the square considering the water film thickness (m)

Y e :

Dimensionless electrostatic potential

z :

Ion valence, dimensionless

αcir :

Rarefaction coefficient, dimensionless

α duct :

Rarefaction coefficient in square pore, dimensionless

β 1,2,3 :

Corner half angle of each corner

β :

Slip coefficient, dimensionless

μ g :

Gas viscosity (Pa s)

ν e :

Primary electronic absorption frequency in the ultraviolet region

v m :

Water molar volume (m3/mol)

Ωi :

Region number according to spatial range of each FPNM, dimensionless

γ :

Interfacial tension (N/m)

Π(h):

Disjoining pressure (Pa)

Πvd(h):

Van der Waals force (Pa)

Πel(h):

Electrostatic force (Pa)

Πstr(h):

Structural force (Pa)

Ψ e :

Electrostatic potential (V)

ξ :

Gas mean free path (m)

\(\chi\) :

Inverse of Debye length (m−1)

ε :

Dielectric constants

λ str :

Characteristic length of the structural force (m)

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Acknowledgements

Wenhui Song acknowledges the Fundamental Research Funds for the Central Universities (No. 20CX06088A) and Qingdao Postdoctoral Applied Research Project (qdyy20200083).

Funding

This study was funded by National Natural Science Foundation of China (No. 52034010), Fundamental Research Funds for the Central Universities (No. 20CX06088A), Qingdao Postdoctoral Applied Research Project (qdyy20200083) and National Natural Science Foundation (52034010).

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  1. Research Centre of Multiphase Flow in Porous Media, China University of Petroleum (East China), Changjiang West Road 66, Economic Technical Development Zone, Qingdao, 266580, Shandong Province, People’s Republic of China

    Wenhui Song, Jun Yao, Kai Zhang, Hai Sun & Yongfei Yang

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Song, W., Yao, J., Zhang, K. et al. The Impacts of Pore Structure and Relative Humidity on Gas Transport in Shale: A Numerical Study by the Image-Based Multi-scale Pore Network Model. Transp Porous Med 144, 229–253 (2022). https://doi.org/10.1007/s11242-021-01663-6

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