Hierarchical porous media with multiple separate spatial scales are ubiquitous in substructures, unconventional strata, chemical engineering systems, energy storage, etc. The development of a highly accurate and highly efficient prediction model for gas transport in these multiscale porous media is of great importance for structural optimization design but remains an open research topic. In this study, we conduct a detailed multi-scale analysis of steady-state gas transport in a multiscale microporous matrix using the iterated asymptotic homogenization method. The upscaled permeability models and the governing equations for gas flow in the continuum and slip flow regimes in multiscale porous media with more than three separate spatial scales are derived accordingly. The accuracy and reliability of the developed models are confirmed and verified by the results of high-fidelity direct numerical simulation. The contribution of the first-order auxiliary cell functions to the upscaled permeability of porous media with different configurations is analyzed in detail, which provides a fundamental understanding of how these heterogeneities at each scale affect the macroscopic flow resistance and equivalent permeability. We also analyze the differences between the low-order and high-order multiscale models and the contrast between the gas transport processes in a steady and transient state. This work guides establishing highly efficient prediction models for gaseous microflows in complex porous media with arbitrary multiscale heterogeneities.
REFERENCES
1.
H.
Zuo
,
C.
Zhai
,
F.
Javadpour
,
S.
Deng
, and
H.
Li
, “
Multiple upscaling procedures for gas transfer in tight shale matrix-fracture systems
,” Geoenergy Sci. Eng.
226
, 211764
(2023
).2.
S.
Deng
,
H.
Li
,
G.
Ma
,
H.
Huang
, and
X.
Li
, “
Simulation of shale–proppant interaction in hydraulic fracturing by the discrete element method
,” Int. J. Rock Mech. Min. Sci.
70
, 10
(2014
).3.
C.
Montgomery
and
M.
Smith
, “
Hydraulic fracturing: History of an enduring technology
,” J. Petroleum Technol.
62
(12
), 26
(2010
).4.
X.
Jiang
,
S.
Deng
,
H.
Li
, and
H.
Zuo
, “
3D pore nanostructures characterization and stress-dependent permeability of organic-rich shales in northern Guizhou Depression, China
,” J. Rock Mech. Geotech. Eng.
14
(2
), 407
(2022
).5.
P.
Tahmasebi
,
F.
Javadpour
,
M.
Sahimi
, and
M.
Piri
, “
Multiscale study for stochastic characterization of shale samples
,” Adv. Water Resour.
89
, 91
(2016
).6.
R. G.
Loucks
,
R. M.
Reed
,
S. C.
Ruppel
, and
D. M.
Jarvie
, “
Morphology, genesis, and distribution of nanometer-scale pores in siliceous mudstones of the Mississippian Barnett Shale
,” J. Sediment. Res.
79
, 848
(2009
).7.
C. C.
Mei
and
B.
Vernescu
, Homogenization Methods for Multiscale Mechanics
(
World Scientific Publishing Co. Pte. Ltd
,
Singapore
, 2010
).8.
Z.
Huang
,
J.
Yao
,
Y.
Li
,
C.
Wang
, and
X.
Lv
, “
Numerical calculation of equivalent permeability tensor for fractured vuggy porous media based on homogenization theory
,” Comput. Phys. Commun.
9
, 180
–204
(2011
).9.
S.
Korneev
and
I.
Battiato
, “
Sequential homogenization of reactive transport in polydisperse porous media
,” Multiscale Model. Simul.
14
, 1301
(2016
).10.
T.
Zaouter
,
D.
Lasseux
, and
M.
Prat
, “
Gas slip flow in a fracture: Local Reynolds equation and upscaled macroscopic model
,” J. Fluid Mech.
837
, 413
(2018
).11.
D.
Lasseux
,
F. J. V.
Parada
, and
M. L.
Porter
, “
An improved macroscale model for gas slip flow in porous media
,” J. Fluid Mech.
805
, 118
(2016
).12.
D.
Lasseux
,
F. J. V.
Parada
,
J. A. O.
Tapia
, and
B.
Goyeau
, “
A macroscopic model for slightly compressible gas slip-flow in homogeneous porous media
,” Phys. Fluids
26
, 053102
(2014
).13.
J.
Warren
and
P.
Root
, “
The behavior of naturally fractured reservoirs
,” Soc. Petroleum Eng. J.
3
, 245
(1963
).14.
C. C.
Mei
and
J. L.
Auriault
, “
Mechanics of heterogeneous porous media with several spatial scales
,” Proc. R. Soc. A
426
, 34
(1989
).15.
T.
Wu
,
X.
Li
,
J.
Zhao
, and
D.
Zhang
, “
Multiscale pore structure and its effect on gas transport in organic-rich shale
,” Water Resour. Res.
53
, 5438
, https://doi.org/10.1002/2017WR020780 (2017
).16.
J.-L.
Auriault
,
C.
Boutin
, and
C.
Geindreau
, Homogenization of Coupled Phenomena in Heterogenous Media
(
John Wiley & Sons, Inc
,
Great Britain and USA
, 2010
).17.
E.
Skjetne
and
J.-L.
Auriault
, “
Homogenization of wall-slip gas flow through porous media
,” Transp. Porous Media
36
, 293
–306
(1999
).18.
P.
Royer
and
J.-L.
Auriault
, “
Transient quasi-static gas flow through a rigid porous medium with double porosity
,” Transp. Porous Med.
17
, 33
(1994
).19.
T.
Zhang
,
F.
Javadpour
,
Y.
Yin
, and
X. F.
Li
, “
Upscaling water flow in composite nanoporous shale matrix using lattice Boltzmann method
,” Water Resour. Res.
56
, e2019WR026007
, https://doi.org/10.1029/2019WR026007 (2020
).20.
T.
Zhang
,
S. G.
Luo
,
H.
Zhou
,
H. R.
Hu
,
L. H.
Zhang
,
Y. L.
Zhao
,
J.
Li
, and
F.
Javadpour
, “
Pore-scale modelling of water sorption in nanopore systems of shale
,” Int. J. Coal Geol.
273
, 104266
(2023
).21.
T.
Zhang
,
L. H.
Zhang
,
Y. L.
Zhao
,
R. H.
Zhang
,
D. X.
Zhang
,
X.
He
,
F.
Ge
,
J. F.
Wu
, and
F.
Javadpour
, “
Ganglia dynamics during imbibition and drainage processes in nanoporous systems
,” Phys. fluids
34
, 042016
(2022
).22.
S.
Li
,
F.
Jiang
,
B.
Wei
,
J.
Hou
, and
H. H.
Liu
, “
Prediction of three-phase relative permeabilities of Berea sandstone using lattice Boltzmann method
,” Phys. Fluids
33
, 063302
(2021
).23.
Z. H.
Chai
,
B. C.
Shi
,
J. H.
Lu
, and
Z. L.
Guo
, “
Non-Darcy flow in disordered porous media: A lattice Boltzmann study
,” Comput. Fluids
39
, 2069
(2010
).24.
Z. H.
Chai
,
J. H.
Lu
,
B. C.
Shi
, and
Z. L.
Guo
, “
Gas slippage effect on the permeability of circular cylinders in a square array
,” Int. J. Heat Mass Transfer
54
, 3009
(2011
).25.
H.
Huang
and
X. Y.
Lu
, “
Relative permeabilities and coupling effects in steady-state gas-liquid flow in porous media: A lattice Boltzmann study
,” Phys. Fluids
21
, 092104
(2009
).26.
H.
Zuo
,
S. C.
Deng
, and
H. B.
Li
, “
Simulation of permeability of hydraulic fractures by computational fluid dynamics
,” J. Nat. Gas Sci. Eng.
67
, 122
(2019
).27.
X.
Miao
,
M.
Gan
,
Y.
Wang
,
L.
Zhang
, and
H.
Jing
, “
Permeability evolution of defective wellbore cement under geologic carbon sequestration conditions by means of varying-radius pipeline modeling
,” J. Nat. Gas Sci. Eng.
98
, 104393
(2022
).28.
C. C.
Mei
and
B.
Vernescu
, Homogenization Methods for Multiscale Mechanics
(
World Scientific Publishing Co. Pte. Ltd
.,
USA
, 2010
).29.
30.
K.
Wu
,
Z.
Chen
,
X.
Li
,
C.
Guo
, and
M.
Wei
, “
A model for multiple transport mechanisms through nanopores of shalegas reservoirs with real gas effects-adsorption-mechanic coupling
,” Int. J. Heat Mass Transfer
93
, 408
(2016
).31.
H.
Zuo
,
C.
Zhai
,
S.
Deng
,
X.
Jiang
, and
F.
Javadpour
, “
Lattice Boltzmann modeling of gaseous microflow in shale nanoporous media
,” Fuel
337
, 127087
(2023
).32.
P.
Royer
and
C.
Boutin
, “
Time analysis of the three characteristic behaviours of dual-porosity media. I: Fluid flow and solute transport
,” Transp. Porous Med.
95
, 603
(2012
).33.
H.
Zuo
,
F.
Javadpour
,
C.
Zhai
,
S.
Deng
, and
H.
Li
, “
On gas transport modes in matrix-fracture systems with arbitrary multiscale configurations
,” J. Nat. Gas Sci. Eng.
108
, 104839
(2022
).34.
H.
Zuo
,
Z.
Yang
,
S.
Deng
, and
H.
Li
, “
High-order asymptotic solutions for gas transport in heterogeneous media with multiple spatial scales
,” Phys. Fluids
35
, 013106
(2023
).35.
H.
Zuo
,
S. C.
Deng
, and
H. B.
Li
, “
Limitations of lattice Boltzmann modeling of micro-flows in complex nanopores
,” Acta Geol. Sin. (Engl. Ed.)
93
(6
), 1808
–1822
(2019
).36.
L.
Wu
,
M. T.
Ho
,
L.
Germanou
,
X. J.
Gu
,
C.
Liu
,
K.
Xu
, and
Y. H.
Zhang
, “
On the apparent permeability of porous media in rarefied gas flows
,” J. Fluid Mech.
822
, 398
(2017
).37.
W.
Su
,
S.
Lindsay
,
H. H.
Liu
, and
L.
Wu
, “
Comparative study of the discrete velocity and lattice Boltzmann methods for rarefied gas flows through irregular channels
,” Phys. Rev. E
96
, 023309
(2017
).38.
J. J.
Yu
,
N. Y.
Li
,
B.
Hui
,
W.
Zhao
,
Y.
Li
,
J.
Kang
,
P.
Hu
, and
Y.
Chen
, “
Experimental simulation of fracture propagation and extension in hydraulic fracturing: A state-of-the-art review
,” Fuel
363
, 131021
(2024
).39.
A.
Bensoussan
,
J. L.
Lions
, and
G.
Papanicolaou
, Asymptotic Analysis for Periodic Structures
(
North Holland
,
Amsterdam
, 1978
).40.
G.
Allaire
and
M.
Briane
, “
Multiscale convergence and reiterated homogenization
,” Proc. R. Soc. Edinburgh Sect. A
126
(2
), 297
–342
(1996
).41.
Z.
Yang
,
Y.
Sun
,
J.
Cui
,
Z.
Yang
, and
T.
Guan
, “
A three-scale homogenization algorithm for coupled conductionradiation problems in porous materials with multiple configurations
,” Int. J. Heat Mass Transfer
125
, 1196
(2018
).42.
A.
Ramírez-Torres
,
R.
Penta
,
R.
Rodríguez-Ramos
,
J.
Merodio
,
F. J.
Sabina
,
J.
Bravo-Castillero
,
R.
Guinovart-Díaz
,
L.
Preziosi
, and
A.
Grillo
, “
Three scales asymptotic homogenization and its application to layered hierarchical hard tissues
,” Int. J. Solids Struct.
130–131
, 190
(2018
).43.
X. F.
Guan
,
X.
Liu
,
X.
Jia
,
Y.
Yuan
,
J. Z.
Cui
, and
H. A.
Mang
, “
A stochastic multiscale model for predicting mechanical properties of fiber reinforced concrete
,” Int. J. Solids Struct.
56–57
, 280
(2015
).44.
X. L.
Liu
,
E.
Chung
, and
L.
Zhang
, “
Iterated numerical homogenization for multiscale elliptic equations with monotone nonlinearity
,” Multiscale Model. Simul.
19
(4
), 1601
(2021
).45.
H.
Zuo
,
F.
Javadpour
,
S.
Deng
, and
H.
Li
, “
Liquid slippage on rough hydrophobic surfaces with and without entrapped bubbles
,” Phys. Fluids
32
, 082003
(2020
).46.
D.
Einzel
,
P.
Panzer
, and
M.
Liu
, “
Boundary condition for fluid flow: Curved or rough surfaces
,” Phys. Rev. Lett.
64
, 2269
(1990
).47.
P.
Panzer
,
M.
Liu
, and
D.
Einzel
, “
The effects of boundary curvature on hydrodynamic fluid flow: Calculation of slip lengths
,” Int. J. Mod. Phys. B
06
, 3251
(1992
).48.
Z. Q.
Yang
,
Y. Z.
Liu
,
Y.
Sun
, and
Q.
Ma
, “
A second-order reduced multiscale method for nonlinear shell structures with orthogonal periodic configurations
,” Numer. Meth. Eng.
123
, 128
(2022
).49.
F. J.
Valdés-Parada
and
D.
Lasseux
, “
Flow near porous media boundaries including inertia and slip: A one-domain approach
,” Phys. Fluids
33
(7
), 073612
(2021
).50.
F. J.
Valdés-Parada
and
D.
Lasseux
, “
A novel one-domain approach for modeling flow in a fluid-porous system including inertia and slip effects
,” Phys. Fluids
33
(2
), 022106
(2021
).51.
D.
Lasseux
,
F. J.
Valdés-Parada
, and
A.
Bottaro
, “
Upscaled model for unsteady slip flow in porous media
,” J. Fluid Mech.
923
, A37
(2021
).52.
Y.
Yin
,
Z. G.
Qu
,
M.
Prodanović
, and
C. J.
Landry
, “
Identifying the dominant transport mechanism in single nanoscale pores and 3D nanoporous media
,” Fundam. Res.
3
(3
), 409
(2023
).53.
H.
Wang
,
Y.
Yin
,
X. Y.
Hui
,
J. Q.
Bai
, and
Z. G.
Qu
, “
Prediction of effective diffusivity of porous media using deep learning method based on sample structure information self-amplification
,” Energy AI
2
, 100035
(2020
).54.
L.
Chen
,
Q. J.
Kang
,
Z. X.
Dai
,
H. S.
Viswanathan
, and
W. Q.
Tao
, “
Permeability prediction of shale matrix reconstructed using the elementary building block model
,” Fuel
160
, 346
(2015
).55.
H.
Zuo
,
Y.
Yin
,
Z. Q.
Yang
,
S. C.
Deng
,
H. B.
Li
, and
H. M.
Liu
, “
High-order models for convection–diffusion-reaction transport in multiscale porous media
,” Chem. Eng. Sci.
286
, 119663
(2024
).56.
H.
Zuo
,
Z. Q.
Yang
,
J. Z.
Cui
,
S. C.
Deng
,
H. B.
Li
, and
Z. T.
Guo
, “
High‐order models for hydro‐mechanical coupling problems in multiscale porous media
,” Numer. Meth. Eng.
125
(11
), e7456
(2024
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
2024
Author(s)
You do not currently have access to this content.
