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Numerical Modelling of Swelling Pressure in Unsaturated Expansive Elasto-Plastic Porous Media

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Abstract

The focus of this work is to provide a new concept for accessing the swelling stress in expansive porous media, especially in highly compacted bentonite. The key to the new approach is the simulation with a chemical swelling model of an infinitesimal volume change followed by a back compaction Process. Free extension is allowed in the first step, to calculate the interlayer porosity change (micro) and the induced volume change potential (macro). The object-oriented FEM simulator GeoSys/RockFlow allows the combination of different processes. The hydro-mechanic/chemical (H2M/C) model takes into consideration two phase flow and deformation, as well as chemical swelling effects. The negative displacements on each boundary after the free extension simulation are taken as Dirichlet boundary conditions of the back compaction problem. The deformation step is simulated in the context of elasto-plasticity using the modified Cam-Clay model. The stresses obtained by back compaction represent the swelling pressure. A 2D example of compacted bentonite is analyzed with the new H2M/C model.

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Abbreviations

A :

Particle surface area (m2)

B :

Strain-displacement matrix (−)

c i :

Concentration of the ith ion (M)

C e :

Fourth order tensor of elasticity (Pa)

d :

Mean diameter of smectite particles (m)

D :

Diffusion coefficient (m2 s−1)

e :

Void ration in Cam-Clay model (−)

E :

Yong’s modulus (Pa)

\({\mathcal{F}}\) :

Yield function (Pa)

F :

Faraday coefficient (=96485.309 C mol−1)

\({\mathcal{G}}\) :

Plastic potential (Pa)

g :

Gravity Vector (m s−2)

J :

Flux (kg m−1 s−1)

I :

Ionic strength (mol m−3)

k :

Permeability (m2)

k :

Permeability tensor (m2)

k rel :

Relative permeability (−)

k sw rel :

Swelling relative permeability(−)

M :

Slope of critical state line (−)

m :

Effective layer number (−)

m :

Unit projection vector (−)

N :

Finite element shape function (−)

n :

Porosity (−)

n tot :

Total porosity (−)

n IL :

Interlayer porosity (−)

n ILmax :

Maximum interlayer porosity (−)

n IP :

Interparticle porosity (−)

n IPmin :

Minimum interparticle porosity (−)

n sw :

Porosity change potential (−)

p csl :

Preconsolidation pressure (Pa)

p :

Pressure or hydrostatic pressure (Pa)

p :

Pressure vector (Pa)

p c :

Suction (Pa)

p c :

Suction vector (Pa)

P S :

Mean stress (Pa)

\({\mathcal{Q}}\) :

Source term (m3 s−1)

q :

Norm of deviatoric stress tensor (Pa)

R :

Gas constant (J mol−1 K−1)

S total :

Total specific surface (m2 g−1)

S 0 :

External specific surface (m2 g−1)

S :

Saturation (−)

S eff :

Effective saturation (−)

S :

Saturation vector (−)

T :

Temperature (K)

u :

Displacement vector (−)

v :

Velocity vector (m s−1)

v s :

Solid velocity vector (m s−1)

X :

Mass fraction (−)

z i :

Ionic charge of the ith ion (−)

β:

Volume fraction of expansive minerals (−)

γ:

Phase (−)

λ:

Virgin compression index (−)

Λ:

Plastic multiplier (−)

κ:

Swelling/re-compression index (−)

μ:

Fluid viscosity (Pa s)

ργ :

Density of phase γ (kg m−3)

ρd :

Dry density of compacted bentonite (g m−3)

σ :

stress tensor ()

σ:

Stress ()

ε :

Strain tensor ()

ε:

Strain ()

τ:

Mean thickness of the effective sheets (m)

ν:

Poisson ratio (−)

ε:

Dielectric constant (−)

ε0 :

Permittivity of free space (8.854 × 10−12 C V −1 m −1)

η:

Dimensionless coefficient (−)

θ:

Volumetric water content (−)

θ s :

Saturated volumetric water content (−)

θ r :

Residual volumetric water content (−)

δ:

Thickness of diffuse double layer (DDL) (m)

ξ:

van Genuchten parameter (−)

Ω:

Domain (−)

ϕ:

Shape function (−)

ω:

Test function (−)

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Authors and Affiliations

  1. Center for Applied Geoscience, GeoSystemsResearch, University of Tübingen, Sigwartstr. 10, D-72076, Tübingen, Germany

    Mingliang Xie, Wenqing Wang, Joëlle De Jonge & Olaf Kolditz

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  1. Mingliang Xie
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  2. Wenqing Wang
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  3. Joëlle De Jonge
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  4. Olaf Kolditz
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Correspondence to Mingliang Xie.

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Xie, M., Wang, W., De Jonge, J. et al. Numerical Modelling of Swelling Pressure in Unsaturated Expansive Elasto-Plastic Porous Media. Transp Porous Med 66, 311–339 (2007). https://doi.org/10.1007/s11242-006-0013-0

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