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