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Modeling Unsaturated Flow in Absorbent Swelling Porous Media: Part 1. Theory

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Abstract

The flow and deformation processes in swelling porous media are modeled for absorbent hygiene products (e.g., diapers, wipes, papers etc.). The first part of the article derives the fundamental equations for the hysteretic unsaturated flow, liquid absorption, and large deformation. The final set of model equations consists of balance equations of mobile and absorbed (immobile) liquid combined with a series of constitutive relationships. The resulting equation system is strongly nonlinear and requires advanced numerical strategies for solving. The second part of the article focuses on numerical solution and presents simulation results for 2D and 3D applications.

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Abbreviations

A p :

Solid–liquid interface area per REV, L2

A p, total :

Maximum solid–liquid interface area per REV, L2

asl(sl):

Saturation-dependent fraction of the solid–liquid interface area, 1

a :

Solid displacement direction vector, 1

a i :

Direction vector at node i, 1

a i :

Spatial components of a, 1

B :

Thickness, L

C :

Stiffness tensor, ML−1 T−2

C :

Intrinsic concentration, ML−3

\({\bar C}\) :

Bulk concentration, M L−3

C a :

Pore constant, L−1

D/Dt:

Material derivative, T−1

d :

Strain vector, 1

d s :

Volumetric solid strain, 1

e :

 = −g/|g|, Gravitational unit vector, 1

f :

External supply or function

G :

Geometry constant, 1

g :

Gravity vector, LT−2

g :

 = |g|, Gravitational acceleration, LT−2

H :

Surface tension head, L2

I :

Identity tensor, 1

J s :

Jacobian of solid domain, volume dilatation function, 1

j :

Diffusive (nonadvective) flux vector, ML−2 T−1

K :

Hydraulic conductivity tensor, L T−1

k :

Permeability tensor, L2

k r :

Relative permeability, 1

k + :

Reaction rate constant, M−1L T−1

L :

Gradient operator, L−1

L :

Differential operator, ML−3 T−1

\({l_{i}^{\rm s}}\) :

Side lengths of a small solid cuboid, L

M :

Molar mass, M

m :

Unit tensor, 1

m :

Mass, M

m :

VG curve fitting parameter, 1

\({m_{2}^{\rm s}}\) :

AGM x-load, 1

\({\hat m_{2}^{\rm s}}\) :

\({=(m_{2\max}^{\rm s} - m_{2}^{\rm s})/m_{2\max}^{\rm s}}\) , Normalized AGM x-load, 1

\({m_{2\max}^{\rm s}}\) :

Maximum AGM x-load, 1

n :

Pore size distribution index, 1

p :

Pressure, ML−1 T−2

Q :

Mass supply, ML−3 T−1

q :

Volumetric Darcy flux, LT−1

R :

Chemical reaction term, ML−3 T−1

R :

Radius, L

r :

Pore radius or distance, L

s :

Saturation, 1

u :

Solid displacement vector, placement transformation function, L

u :

Scalar solid displacement norm, L

V :

REV volume, L3

V p :

Pore volume, L3

v :

Velocity vector, LT−1

x :

Eulerian spatial coordinates, L

x i :

Components of x, L

α :

VG curve fitting parameter, L−1

Γs :

Closed boundary of solid control space Ωs, L2

γ :

Liquid compressibility, M−1 LT2

\({\bar\gamma}\) :

\({=\gamma \rho_{0}^{\rm l} g}\) , Specific liquid compressibility, L−1

γ ij :

Shear strain component, 1

Δz :

Vertical extent, L

δ :

Exponential fitting parameter, 1

δ ij :

\({=\left\{\begin{array}{ll}1,\, i=j\\ 0, i\neq j \end{array}\right.}\) Kronecker delta

\({\varepsilon}\) :

Porosity, void space, 1

\({\varepsilon^{\alpha}}\) :

Volume fraction of α-phase, 1

μ :

Dynamic viscosity, ML−1 T−1

ρ :

Density or intrinsic concentration, ML−3

σ :

Solid stress tensor, ML−1 T−2

σ * :

Liquid surface tension, ML−2 T−2

τ :

AGM reaction (speed) rate constant, T−1

\({\phi^{\rm l}}\) , \({\phi^{\rm s}}\):

Deformation (sink/source) terms for liquid and solid, respectively, T−1, ML−1 T−1

ψ l :

Pressure head of liquid phase l, L

Ωs :

Control space of porous solid or domain, L3

ω k :

Mass fraction of species k, 1

ω :

Reaction rate modifier, 1

∇:

Nabla (vector) operator (= grad), L−1

∇ i :

 = ∂/∂ x i , Partial differentiation with respect to x i

AGMraw :

Available AGM in reaction

AGMconsumed :

Consumed AGM in reaction

AGM:

AGM

c :

Capillary

e :

Effective or elemental

H2O:

Water

I :

Material Lagrangian coordinate, ranging from 1 to 3

i, j:

Spatial Eulerian coordinate, ranging from 1 to 3, or nodal indices

k :

Species indicator

L → S:

AGM absorbed liquid

0:

Reference, initial or dry

p :

Pore

r :

Residual, reactive or relative

α:

Phase indicator

D:

Number of space dimension

g:

Gas phase

l:

Liquid phase

s:

Solid phase

T:

Transpose

AGM:

Absorbent gelling material

CM:

Inert carrier material

REV:

Representative elementary volume

RHS:

Right-hand side

SAP:

Superabsorbent polymer

VG:

van Genuchten

[. . .]:

Chemical activity, molar bulk concentration

() · ():

Vector dot (scalar) product

() ⊗ ():

Tensor (dyadic) product

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Author information

Authors and Affiliations

  1. DHI-Wasy GmbH, Waltersdorfer Str. 105, Berlin, 12526, Germany

    Hans-Jörg G. Diersch, Volker Clausnitzer & Volodymyr Myrnyy

  2. Procter & Gamble Service GmbH, Sulzbacher Str. 40, Schwalbach am Taunus, 65824, Germany

    Rodrigo Rosati, Mattias Schmidt, Holger Beruda & Bruno J. Ehrnsperger

  3. Procter & Gamble Services Company SA, Temselaan 100, Strombeek-Bever, 1853, Belgium

    Raffaele Virgilio

Authors
  1. Hans-Jörg G. Diersch
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  2. Volker Clausnitzer
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  3. Volodymyr Myrnyy
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  4. Rodrigo Rosati
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  5. Mattias Schmidt
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  6. Holger Beruda
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  7. Bruno J. Ehrnsperger
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  8. Raffaele Virgilio
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Corresponding author

Correspondence to Hans-Jörg G. Diersch.

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Diersch, HJ.G., Clausnitzer, V., Myrnyy, V. et al. Modeling Unsaturated Flow in Absorbent Swelling Porous Media: Part 1. Theory. Transp Porous Med 83, 437–464 (2010). https://doi.org/10.1007/s11242-009-9454-6

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  • DOI: https://doi.org/10.1007/s11242-009-9454-6

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