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A Model for Flow and Deformation in Unsaturated Swelling Porous Media

Abstract

A thermomechanical theory for multiphase transport in unsaturated swelling porous media is developed on the basis of Hybrid Mixture Theory (saturated systems can also be modeled as a special case of this general theory). The aim is to comprehensively and non-empirically describe the effect of viscoelastic deformation on fluid transport (and vice versa) for swelling porous materials. Three phases are considered in the system: the swelling solid matrix s, liquid l, and air a. The Coleman–Noll procedure is used to obtain the restrictions on the form of the constitutive equations. The form of Darcy’s law for the fluid phase, which takes into account both Fickian and non-Fickian transport, is slightly different from the forms obtained by other researchers though all the terms have been included. When the fluid phases interact with the swelling solid porous matrix, deformation occurs. Viscoelastic large deformation of the solid matrix is investigated. A simple form of differential-integral equation is obtained for the fluid transport under isothermal conditions, which can be coupled with the deformation of the solid matrix to solve for transport in an unsaturated system. The modeling theory thus developed, which involves two-way coupling of the viscoelastic solid deformation and fluid transport, can be applied to study the processing of biopolymers, for example, soaking of foodstuffs and stress-crack predictions. Moreover, extension and modification of this modeling theory can be applied to study a vast variety of problems, such as drying of gels, consolidation of clays, drug delivery, and absorption of liquids in diapers.

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Abbreviations

A n :

Fourth order material coefficient tensor

\({A^{\alpha_j}}\) :

Helmholtz free energy of the jth component in α phase

A α :

Helmholtz free energy of the α phase

\({b^{\alpha_j}}\) :

External entropy source for the jth component in α phase

b α :

External entropy source for the α phase

B :

Material coefficient related to the bulk relaxation function

B :

Material coefficient

B α :

Fourth order viscous dissipation tensor

\({C^{\alpha_j}}\) :

Mass fraction of the jth component in α phase

C s :

Right Cauchy-Green strain tensor of the solid phase

\({\bar{\bf C}^{\rm s}}\) :

Right Cauchy-Green strain tensor associated with \({{\bar{\bf F}}^{s}}\)

d α :

Rate of deformation tensor of the α phase

d α :

Material constant related to the initial bulk modulus of α phase

\({e^{\alpha_j}}\) :

Energy density of the jth component in α phase

e α :

Energy density of the α phase

\({\hat{e}_{\alpha_j}^{\beta}}\) :

Rate of mass transfer from phase β to the jth component in α phase

\({\hat{e}_{\alpha}^{\beta}}\) :

Rate of mass transfer from β phase to the α phase

E s :

Lagrangian strain tensor of the solid phase

F s :

Deformation gradient of the solid phase

\({\bar{\bf F}^{\rm s}}\) :

The multiplicative decomposition of the deformation gradient

\({{\bf g}^{\alpha_j}}\) :

Gravitational force on the jth component in α phase

g α :

Gravitational force on the α phase

G(t) :

Relaxation function in shear

\({h^{\alpha_j}}\) :

External supply of energy to the jth component in α phase

h α :

External supply of energy to α phase

H α :

Third order material coefficient tensor

\({\hat{{\bf i}}^{\alpha_j}}\) :

Rate of momentum gain to the jth component in α phase due to interaction with other species in the same phase

I :

Identity tensor

I k :

Principal invariants of the right Cauchy-Green tensor C s

\({\bar{I}_{k}}\) :

Principal invariants of \({{\bar{\bf C}}^{s}}\)

J s :

Determinant of the deformation gradient

K α :

Initial bulk modulus of α phase

K α :

Second rank coefficient tensor from linearization

L :

Laplace transform operator

M α :

Material coefficient

p α :

Physical pressure in the α phase

\({{\bf q}^{\alpha_j}}\) :

Heat flux vector for the jth component in α phase

q α :

Heat flux vector for α phase

\({\hat{Q}^{\alpha_j}}\) :

Rate of energy gain to the jth component in α phase due to interaction with other species in the same phase

\({\hat{Q}_{\alpha_j}^{\beta}}\) :

Rate of energy transfer from phase β to the jth component in α phase

\({\hat{Q}_{\alpha}^{\beta}}\) :

Rate of energy transfer from phase β to α phase

\({\hat{r}^{\alpha_j}}\) :

Rate of mass gain to the jth component in α phase due to interaction with other species in the same phase

R a :

Resistivity tensor

R l :

Resistivity tensor

S :

Material coefficient related to the shear relaxation function

S s :

Second Piola-Kirchhoff stress tensor

t :

Time

\({{\bf t}^{\alpha_j}}\) :

Stress tensor of the jth component in α phase

t α :

Stress tensor of the α phase

t se :

Terzaghi stress for the solid phase

t sh :

Hydration stress for the solid phase

T :

Temperature

\({\hat{{\bf T}}_{\alpha_j}^{\beta}}\) :

Rate of momentum transfer to the jth component in α phase from β phase

\({\hat{{\bf T}}_{\alpha}^{\beta}}\) :

Rate of momentum transfer to α phase from β phase

u s :

Displacement of the solid phase

\({{\bf v}^{\alpha_j}}\) :

Velocity of the jth component in α phase

v α :

Velocity of the α phase

v α,β :

Relative velocity between phase α and phase β

δ :

Dirac’s delta function

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

Volume fraction of the α phase

\({\eta^{\alpha_j}}\) :

Entropy of the jth component in α phase

η α :

Entropy of the α phase

\({\hat{\eta^{\alpha_j}}}\) :

Entropy gain by the jth component in α phase due to interaction with other species in the same phase

λα :

Lagrange multiplier for the continuity equation of α phase

\({\Lambda^{\alpha_j}}\) :

Entropy production per unit mass density for the jth component in α phase

Λα :

Entropy production per unit mass density for α phase

μ :

Viscosity of the liquid phase

\({\mu^{\alpha_j}}\) :

Chemical potential of the jth component in α phase

μ α :

Chemical potential of α phase

μ α :

Initial shear modulus of α phase

π l :

Swelling pressure due to interaction of the solid phase and the bulk fluid

\({\rho^{\alpha_j}}\) :

Density of the jth component in the α phase

ρ α :

Density of the α phase

\({{\boldsymbol{\phi}}^{\alpha_j}}\) :

Entropy flux vector for the jth component in α phase

\({{\boldsymbol{\phi}}^{\alpha}}\) :

Entropy flux vector for α phase

\({\hat{{\Phi}}_{\alpha_j}^{\beta}}\) :

Entropy transfer to the jth component in α phase from β phase

\({\hat{{\Phi}}_{\alpha}^{\beta}}\) :

Entropy transfer to α phase from β phase

a:

Air phase

A:

Particle composed of solid phase and the vicinal fluid

wA:

Vicinal water in the particle

sA:

Solid phase in the particle

f:

Fluid composed of vicinal water and the bulk phase fluid

j:

jth component of species

l:

Liquid phase

s:

Solid phase

vs:

The viscoelastic part

(·)0 :

Initial value of (·)

α :

α phase

\({\hat{(\cdot)}_{\alpha}^{\beta}}\) :

Exchange from β phase to α phase

(·)α,s :

Difference of two quantities ((·)α − (·)s)

\({\frac{D^{\alpha_j}(\cdot)}{Dt}}\) :

Material time derivative of a variable with respect to velocity of jth component in the α phase

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Correspondence to Subrata Mukherjee.

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Zhu, H., Dhall, A., Mukherjee, S. et al. A Model for Flow and Deformation in Unsaturated Swelling Porous Media. Transp Porous Med 84, 335–369 (2010). https://doi.org/10.1007/s11242-009-9505-z

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Keywords

  • Unsaturated
  • Swelling
  • Porous
  • Thermodynamics
  • Large deformation
  • Viscoelasticity