Lattice Boltzmann Method simulations of transport
properties of partially water saturated clay.
Magdalena Dymitrowska, S.Gueddani, S. ben hadj Hassine, IRSN V. Pot, INRA
A. Genty, CEA
Journées Diphasiques MoMaS, 5-8 octobre 2015, Nice
Plan
Motivations
Lattice Boltzman Method presentation
Opalinus clay rough fractures
Relative permeability versus saturation
Effective diffusion coefficient
Conclusions and perspectives
Gas migration within repository
hydraulic gradient -> quicker advective transport of water and RN
mechanical dammage to EBs and the host rock
slower resaturation of bentonite plugs and chemical perturbation
explosion risques
● mostly corrosion H2 ( also CH4, CO2..)
● very low permeability k~10-21 m2
● high saturation Sw>0.9
● strong capilarity Pc > 20 MPa
● discontinuities (interfaces)
● THM problems
2-j flow Darcy model
many conceptual problems
lack of reliable experimental data when Sw~1 – very senstitive !
no model for dilatant pathways (localized gas flow)
simulations on repository scale possible
scattering of results obtained with the same simulation tool
Need to go down to the pore scale:
no Darcy assumption
insight into local phenomena
extraction of homogenised Pc(Sw), Kr,I(Sw), De(Sw)
possible tackling of contact line motion problem
Opalinus : Vilard et al. 2009
Andra, FORGE D1.3
phase-in icomponent offlux diffusive
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Lattice Boltzmann Method (1/3)
Kinetic theory of gases : Boltzmann equation (1872): - binary collisions - molecular chaos Boltzmann H-theorem: Collision Interval Theory:
- state close to equilibrium - BGK version
Simplification of forcing terme: Link to hydrodynamics:
𝜕𝑡 + 𝑒 ∙ 𝛻𝑟 + 𝑎 ∙ 𝛻𝑒 f 𝑟 , 𝑒 , 𝑡 = 𝜕𝑡𝑓 𝑐𝑜𝑙𝑙 where f – one particle probability function e – particle velocity a – external force
𝜕𝑡𝑓 𝑐𝑜𝑙𝑙 = 𝑑Ω 𝑑3𝑒 0 𝜎(Ω) 𝑒 − 𝑒 (0) 𝑓′𝑓′ 0 − 𝑓𝑓 0
𝑑𝐻 𝑡
𝑑𝑡≤ 0 𝑤ℎ𝑒𝑟𝑒 𝐻 = 𝑓𝑙𝑛𝑓𝑑3𝑒
𝜕𝑡𝑓 𝑐𝑜𝑙𝑙 = −𝑓 − 𝑓𝑒𝑞
𝜏 𝑤ℎ𝑒𝑟𝑒 𝑓𝑒𝑞 =
𝜚
2𝜋𝑅𝑇𝐷02
𝑒𝑥𝑝 −𝑒 − 𝑢 2
2𝑅𝑇
𝛻𝑒𝑓 ≈ 𝛻𝑒𝑓𝑒𝑞 = −
𝑎∙ 𝑒−𝑢
𝑅𝑇𝑓𝑒𝑞
𝜌 = 𝑓𝑑𝑒 𝜌𝑢 = 𝑓𝑒 𝑑𝑒 𝜌ℇ = 𝑓 𝑒 − 𝑢 2𝑑𝑒
Lattice Boltzmann Method (2/3)
Continuous equation : Discretisation methods heuristic (Frisch et al.’86, Wolfram ‘86)
-constant temperature and low Mach number
finite difference approx. of Boltzmann equation (He and Luo ‘97)
Final LBM equation (BGK model)
to be solved by an appropriate scheme
𝜕𝑡 + 𝑒 ∙ 𝛻𝑟 f 𝑟 , 𝑒 , 𝑡 = −𝑓−𝑓𝑒𝑞
𝜏+ 𝑎∙ 𝑒−𝑢
𝑅𝑇𝑓𝑒𝑞
𝑓𝑒𝑞 =𝜚
2𝜋𝑅𝑇𝐷02
𝑒𝑥𝑝 −𝑒 2
2𝑅𝑇1 +
𝑒 ∙ 𝑢
𝑅𝑇+(𝑒 ∙ 𝑢)2
2(𝑅𝑇)2−𝑢2
2𝑅𝑇+ ℴ(𝑢3)
𝑒 → 𝑒𝑎 where 𝑎 = 0,… , 𝑁 lattice definition 𝑓 → 𝑓𝑎 𝑓𝑒𝑞 → 𝑓𝑎
𝑒𝑞 = 𝐴𝑎 + 𝐵𝑎𝑒𝑎𝑖𝑢𝑖 + 𝐶𝑎𝑢2 + 𝐷𝑎𝑒𝑎𝑖𝑒𝑎𝑗𝑢𝑖𝑢𝑗
𝜕𝑡 + 𝑒 𝑎𝑗 ∙ 𝜕𝑗 𝑓𝑎 = −𝑓𝑎−𝑓𝑎
𝑒𝑞
𝜏+ 𝑎∙ 𝑒𝑎−𝑢
𝑅𝑇𝑓𝑎𝑒𝑞
Lattice Boltzmann Method (3/3)
Two phase flow LBM code:
• Gustensen model : non-ideal gases in the nearly incompressible limit
• 2 LB equations: distribution function for each fluid
• 2 relaxation times algorithm in 3D and 19 velocities (D3Q19)
• wetting + interfacial tension
• bounce-back condition at solid walls
• parallelised on GPU with CUDA
Opalinus clay fractures
96-1 96-2 96-3
96-4
103-4A
103-2C 103-1
103-3C
103-4B
• Ventilation Experiment II at Mont Terri • -tomography with voxel size of 0.7 µm • total porosity : 18% • percolating porosity : 0.1 à 2.4 % • micro-fractures with aperture of 2.1 to 29 µm
sample <b> - / + init size in pixels
96-2_1 3.3 42/43 3 0.69 1171*1130*201
96-2_2 3.1 48/48 3 0.69 1171*1166*1024
96-4 3.1 24/24 2.9 2.4 1163*1050*1024
103-1_1 2.1 20/20 3.8 0.68 500*325*198
103-1_2 29.6 18/26 3.8 0.68 324*325*198
103-4B 20 15/5 6 1.8 339*450*115
Selected percolating fractures
Sample 103-3-C
Sample 96-2-B
Voxel = 0,7 m Size : (48,50,24) Porosity = 5,9%
Voxel = 0,7 m Size : (34,70,50) Porosity = 2,2%
Maximal opening of flow sections
X X
Two-phase flow in fractures
Two-phase flow in fractures – characteristic numbers
Under relevant conditions ∆h = 1m/m and L = 7 m
● V=𝐿2
12𝜇𝛻𝑃 ≈ 10−5𝑚/𝑠
● 𝑅𝑒 =𝜌𝑉𝐿
𝜇= 3. 10−8
● 𝐶𝑎 =𝜇𝑉
𝜎= 5. 10−7
● 𝑀 =𝜇𝑤
𝜇𝑔= 111
● 𝐵𝑜 =∆𝜌𝑔𝐿2
𝜎= 10−7
Capillary fingering regime in Lenormand diagram
LBM numerical stability :
• Ca=10-4 M=102 Bo=0 Re=10-2
Percolation gradient theory (Lefort PhD 2014)
● stabilisation of capillary flow by viscous effects
𝐶𝑎
Σ𝑀2,19<<1
● for 𝐶𝑎 = 5.10−7 critical distance 𝑋𝑐 = 20 𝜇𝑚
Physical system:
Ca=5.10-7; M=102
Relative permeability of plane fractures - Poisseuille
Generalized Darcy 𝒒𝒊 = −𝐾𝑘𝑟,𝑖
𝜇𝑖𝜵𝑝𝑖 − 𝒈𝜌𝑖
3-layers Poisseuille flow :
𝑘𝑟,𝑤 𝑆𝑤 =1
2𝑆𝑤2 3 − 𝑆𝑤
𝑘𝑟,𝑔 = 1 − 𝑆𝑤1 − 3𝛾
2−1 + 3𝛾
21 − 𝑆𝑤
2
LBM results for sufficient / minimal number of points:
Relative permeability of plane fractures – initial conditions
percolating phases
gas phase initialised as cylinder
spontaneous jump to lower energy interface
strong effect on permeability!
Relative permeability of plane fractures – initial conditions 2/2
segmented flow
discontinuous gas phase : cylinders or bubbles
collapse of kr,g for segmented flow
! strong effect of IC !
Relative permeability in Venturi conditions
narrowing segments L =20lu
bubble of R=10
at Ca=10-4 no deformation when 2R >= L
at Ca=10-2 deformation when 2R = L
at Ca=0.1 deformation possible for 2R > L
Gas entrappement possible in clays !
Relative permeability of rough clay fracture
narrowest aperture L = 28lu
Conclusions for LBM permeability calculations
I. strong influence of initial condition on planar fracture flow
II. spontaneous segmentation of flow in rough fracture
III. undeformable gas bubbles in in-situ conditions
IV. snap of effect at relatively low water saturation Kr,g=0 for Sw > 0.6
Is this effect realistic ?
LBM model lacks:
gas compressibility
transport of dissolved gas
pore space evolution with pressure
lattice pinning for bubbles radii < 3 lu – influence on the results ?
Effective diffusion in partially saturated clay
Effective diffusion in model media
Fick law 𝜕𝑆𝑤𝐶
𝜕𝑡= −𝛻𝐷𝑒𝛻𝐶
objective : De(Sw), Archie law :𝐷𝑒 = 𝐷𝑜𝜃𝑚𝑆𝑤
𝑛
Step 1 : steady state distribution of water et Sw<1 (LBM)
Step 2 : diffusion of tracer in liquid phase (LBM)
Numerical Calculation of Effective Diffusion in Unsaturated Porous Media by the TRT Lattice Boltzmann Method, Genty et al., TPM 2014
Fracture of 100x152x18 voxels of 0,7m
Application to a clay fracture
Phase separation
24%
24%
45% 35%
29% 55%
95% 72%
60%
Diffusion in percolating liquid phase
Fitting LBM profiles with analytical solution
Declusterisation
Quality of fit at different water saturations
Effective diffusion coefficient De= f(Sw)
De/De(Sw=1)
Saturation []
milde discretisation effect non-monotonous at low saturation speed up effects at high water saturation
Discussion of diffusion results
De(Sw) linear for intermediate Sw (Archie law with n<1 ? not known in litterature)
De >1 close to full saturation :
Last bubbles clogging pore intersections declusterisation errases dead end tortuosity τ lowered
De increased
De non monotonous at low water saturations
wetting fluid structure lice-like tortuosity higher De lowered
more fractures to be studied
adding one more discretisation level
including matrix diffusion and discontinuous pores
Perspectives
Convection - diffusion in single rough fracture by LBM
Coupling of 2-j flow with deformable/fracturable solid matrix
Smooth Particle Hydrodynamics
Dissipative Particle Dynamics
Lattice-Spring + LBM
Testing pore-scale methods against nanofluidic experiments
This work was conducted with the financial support of Needs-Mipor
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