Transport in Porous Media

, Volume 89, Issue 3, pp 475–486

# Poroelasticity-II: On the Equilibrium State of the Fluid-Filled Penetrable Poroelastic Body

Article

## Abstract

Part-II of the paper titled “Poroelasticity-II” is devoted to defining the joint equilibrium state of a fluid and elastic penetrable material, encapsulated in a rigid volume, Ω. One can ask different questions in respect to such an equilibrium state, such as: (1) In what physical state will the fluid be, if the porous volume of the poroelastic material is filled with a given mass of fluid? (2) What is the capacity of a volume filled with a penetrable sponge in respect to the fluid being in specific state? etc. The point is that, in the frame of the most popular theory of poroelastic materials, Biot’s theory, such a question cannot be addressed because this theory is formulated as a linear theory in the neighborhood of some equilibrium state. In this article, we continue to describe our theory that can be applied to solve such problems. We consider a simplified case of a linear-elastic matrix and a fluid having a Van der Waals equation of state and address the problem of the equilibrium state.

## Keywords

Poroelastic materials Fluid Van der Waals Equilibrium

## List of Symbols

$${\varsigma}$$

Biot’s “fluid content” parameter

u

Vector of matrix displacement in Biot’s nomenclature

U

Vector of fluid displacement in Biot’s nomenclature

ξS

Vector of matrix displacement

ξf

Vector of the fluid displacement

Vf

Fluid particle velocity

$${\varphi}$$

Porosity as the argument of the equation of state

$${\varphi_0}$$

Equilibrium porosity of the poroelastic material body filled with a fluid

$${\varphi_{00}}$$

Porosity of the “empty” poroelastic material body (non-filled with a fluid)

Ω

Volume of the container

Mf

Mass of fluid

ρf

Density of the fluid as the argument of equation of state

ρf0

Density of the fluid in the equilibrium state

ρS00

Density of matrix material in the “empy” state (not filled with a fluid)

a and b

Van der Waals constants

pf

Pressure of the fluid

T

Temperature

$${\bar{{\varepsilon}}_{ij}}$$

Tensor of the internal strain (See Part I)

$${\bar{{\varepsilon}}}$$

Bulk part of the tensor of the internal strain

$${\bar{{\varepsilon}}_0}$$

Bulk part of the tensor of the internal strain in an equilibrium state of fluid-filled poroelastic material material

$${I_2 = \bar{{\varepsilon}}_{ij}\bar{{\varepsilon }}_{ij}}$$

Second invariant of the strain tensor

eS

Internal energy of the matrix, which we consider (in neglection of heat effects) a function of $${\bar{{\varepsilon}}_{ij}}$$ and $${\varphi }$$

δij

Kronecker delta

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