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Hyperelastic Multiphase Porous Media with Strain-Dependent Retention Laws

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Abstract

This article presents poroelastic laws accounting for a retention behavior dependent also on porosity, as suggested by experimental evidence. Motivated by the numerical formulation of the corresponding boundary-value problem presented in a companion article, these constitutive equations employ displacements and fluid pressures as primary variables. The thermodynamic admissibility of the proposed rate laws for stress and fluid contents is assessed by means of symmetry and Maxwell conditions obtained from the Biot theory. In the case of strain-dependent saturation, the two elasticity tensors describing the drained response in saturated and unsaturated conditions, respectively, are proven to be in general not coincident, with their difference depending on capillary pressure and porosity. Furthermore, it is shown that besides the stress decomposition proposed by Coussy, also the stress split proposed by Lewis and Schrefler is consistent with the Biot framework. The former decomposition is obtained for retention laws depending only on capillary pressure, as expected. The Lewis–Schrefler split is proven to be consistent with retention models depending also on porosity. In these developments, the compressibility of all the phases is taken into account, in order to assess the thermodynamic consistency of an extension of the Biot’s coefficient to partially saturated anisotropic porous media.

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Abbreviations

b :

Biot’s tensor

b α :

Coupling tensors (α = w, g)

C αβ :

Tangent storage moduli (α, β = w, g)

\({\mathbb{C}_{\rm sk}}\) :

Solid-skeleton elasticity tensor (full saturation)

\({\tilde{\mathbb{C}}_{\rm sk}}\) :

Solid-skeleton elasticity tensor (variable saturation)

dv :

Trace of spatial strain rate of the solid skeleton

e α :

Logarithmic volumetric strains of fluids (α = w, g) and solid phase (α = s)

J :

Jacobian of solid-skeleton deformation

J s :

Jacobian of solid-phase deformation

M α :

Fluid (α = w, g) and solid (α = s) mass contents (per unit reference volume of porous medium)

n :

Porosity (current void volume per unit volume of deformed porous medium)

p α :

Fluid pressures (α = w, g)

p c :

Capillary pressure

p f :

Average pore pressure

\({\check{p}_{\rm f}}\) :

Approximated average pore pressure (given by integration of (53))

p s :

Mean stress in the solid phase

q α :

Mass flow of fluids (α = w, g)

s α :

Saturation degree of fluids (α = w, g) (current fluid volume per unit deformed void volume)

\({\check{s}'_\alpha}\) :

Total derivative of function \({\check{s}_\alpha(p_{\rm c})}\) (α = w, g)

\({\breve{s}\,'_\alpha}\) :

Total derivative of function \({\breve{s}_\alpha(n p_{\rm c})}\) (α = w, g)

\({\varepsilon}\) :

Infinitesimal strain tensor of the solid skeleton

\({\varepsilon_{\rm v}}\) :

Trace of \({\varepsilon}\)

\({\varepsilon_\alpha}\) :

Trace of infinitesimal strain tensors of fluids (α = w, g) and solid phase (α = s)

κ α :

Tangent bulk moduli of fluids (α = w, g)

κ s :

Bulk modulus of the solid phase

μ α :

Free enthalpy of fluid (α = w, g) (per unit mass of fluid)

ξ α :

Coupling functions (α = w, g) defined by (32) and related to retention law

\({\check{\xi}_\alpha}\) :

Particular form (49)3 of ξ α (α = w, g) obtained for s α = \({\check{s}_\alpha(p_{\rm c})}\)

\({\breve{\xi}_\alpha}\) :

Particular form (64) of ξ α (α = w, g) obtained for \({s_\alpha = \breve{s}_\alpha(n p_{\rm c})}\)

ρ α :

Intrinsic densities of fluids (α = w, g) and solid phase (α = s)

σ :

Total Cauchy stress tensor in the porous medium

σ':

Effective Cauchy stress in the porous medium

ψ :

Helmholtz free energy of the porous medium (per unit reference volume of porous medium)

ψ α :

Helmholtz free ener. of fluids (α = w, g) and solid phase (α = s) (per unit mass of phase)

ω :

Function defined by (59) (related to porosity-dependent retention)

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Authors and Affiliations

  1. Facoltà di Ingegneria, Università del Molise, via Duca degli Abruzzi, 86039, Termoli, CB, Italy

    Carlo Callari

  2. Dipartimento di Ingegneria Civile, Università di Roma “Tor Vergata”, via Politecnico 1, 00133, Roma, Italy

    Andrea Abati

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  1. Carlo Callari
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  2. Andrea Abati
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Correspondence to Carlo Callari.

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Callari, C., Abati, A. Hyperelastic Multiphase Porous Media with Strain-Dependent Retention Laws. Transp Porous Med 86, 155–176 (2011). https://doi.org/10.1007/s11242-010-9614-8

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