Modeling Unsaturated Flow in Absorbent Swelling Porous Media: Part 2. Numerical Simulation


For solving flow, absorption and deformation processes in porous media containing absorbent gelling materials a finite element method is developed and applied. Adaptive techniques are preferred where spatial, temporal, and residual errors are controlled. Mesh-movement and mesh-refinement strategies are incorporated. Spline approximations are used for better and more flexible descriptions of experimental data and measured relations. Applied to hysteretic material behavior an extended method is described. Comparisons to analytical results and experimental findings are given. A basic sensitivity analysis of two selected key parameters is presented. Practical applications refer to diaper-core flow simulations.

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A :

Saturation difference in drying curves (1)

a :

Solid displacement direction vector (1)

a i :

Direction vector at node i (1)

B :

Saturation difference in wetting curves (1)

b :

Generalized sink/source vector

b :

Height (L)

C :

Solution vector of concentration (ML −3)

C :

Intrinsic concentration (ML −3)

\({\bar C}\) :

Bulk concentration (ML −3)

c :

Scaling factor (1)

d :

Error vector

d :

Volumetric solid strain (1)

E :

Error vector

e :

Gravitational unit vector (1)

f :

Generalized flux vector

h :

 = z + ψ, Hydraulic head of liquid (L)

K :

Hydraulic conductivity tensor (LT −1)

K :

Scalar hydraulic conductivity (LT −1)

k r :

Relative permeability (1)

J s :

Jacobian of solid domain, volume dilatation function (1)

L :

Partial differential equation operator

l :

Characteristic element length (L)

m :

Generalized storage vector

\({m_2^s}\) :

AGM x-load (1)

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

Maximum AGM x-load (1)

N :

Base sample size

n :

Outward pointing unit normal vector (1)

n :

VG pore size distribution index (1)

p :

Parameter vector

Q :

Volumetric flow rate (\({L^3T^{-1}}\))

q n :

Normal flux (LT −1)

R :

Residual vector (\({L^3T^{-1}}\))

R c :

Kinetic reaction term for immobile liquid (\({ML^{-3}T^{-1}}\))

R ψ :

Kinetic reaction term for mobile liquid (T −1)

S :

Solution vector of saturation (1)

S i :

Sensitivity index of parameter i (1)

s :

Saturation (1)

T :

Pulse-infiltration period (T)

t :

Time (T)

U :

Solution vector of solid displacement (L)

u :

Scalar solid displacement norm (L)

w :

Width (L)

x :

Spatial coordinate vector (L)

z :

Vertical coordinate (L)

α :

VG curve fitting parameter (L −1)

β upwind :

Numerical dispersivity (L)

Γ :

Closed boundary (L 2)

γ :

Specific liquid compressibility (L −1)


Increment or difference

δ :

Temporal error tolerance (1)

ε :

Porosity, void space (1)

η :

Residual error tolerance (\({L^3T^{-1}}\))

κ :

Artificial compression of solid (L −1 T)

ξ :

Mesh refinement error criterion (1)

\({\sigma^{\circ}}\) :

Artificial (dampening) ‘diffusive’ stress of solid (L)

τ :

AGM reaction (speed) rate constant (T −1)

ϕ :

State variable vector

Ψ :

Solution vector of pressure head (L)

ψ :

Pressure head of liquid (L)

Ω :

Domain (L 3)


Nabla (vector) operator (= grad) (L −1)



AGM0 :

AGM at initial time

c :


D :

Dirichlet-type BC

d :


e :




i, j:

Nodal or parameter index

L :






N :

Number of nodes

N :

Neumann-type BC

n :

Time plane



R :




w :


c :


D :

Number of space dimension

d :


L :


P :


R :


s :

Solid phase

T :



Iteration counter


Absorbent gelling material


Adaptive mesh refinement


Boundary condition


Initial condition


Interface manager


Right-hand side


Root-mean square


Sequential iterative approach


van Genuchten


Two dimensions or two-dimensional


Three dimensions or three-dimensional

\({()\cdot()}\) :

Vector dot (scalar) product

\({()\otimes()}\) :

Tensor (dyadic) product

\({\dot{()}}\) :

Differentiation with respect to time t

\({\tilde{()}}\) :

Approximate solution


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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 2. Numerical Simulation. Transp Porous Med 86, 753–776 (2011).

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  • Unsaturated flow
  • Absorbent gelling material
  • Swelling porous media
  • Finite element analysis
  • Diaper flow modeling