Mass Transport in Porous Media With Variable Mass

Abstract

We present a theoretical and numerical study of mass transport in a porous medium saturated with a fluid and characterised by an evolving internal structure. The dynamics of the porous medium and the fluid as well as their reciprocal interactions are described at a coarse scale, so that the fundamental tools of Mixture Theory and Continuum Mechanics can be used. The evolution of the internal structure of the porous medium, which is here primarily imputed either to growth or to mass exchange with the fluid, is investigated by enriching the space of kinematic variables of the mixture with a set of structural descriptors, each of which is power-conjugate to generalised forces satisfying a balance law. Establishing the influence of the structural change of the porous medium on the transport properties of the mixture and, thus, on the quantities characterising fluid flow is the crux of our contribution.

Appendix

The following relations are used in the computations above:

$$ {{\partial J}\over {\partial {\mathbf{C}}}} = \tfrac{1}{2}J {\mathbf{C}}^{-1}, \quad {{\partial {\mathbf{C}}}\over {\partial {\mathbf{F}}}} = {\mathbf{\delta}}\;\overline{\otimes}\;{\mathbf{F}}^{T} + {\mathbf{F}}^{T}\underline {\otimes}\;{\mathbf{\delta}}, $$

(130)

$$ {{\partial {\mathbf{C}}^{-1}}\over{\partial {\mathbf{C}}}} = - {\mathbb{I}} = - \tfrac{1}{2}\big\lbrack {\mathbf{C}}^{-1} \underline {\otimes}\;{\mathbf{C}}^{-1} + {\mathbf{C}}^{-1} \overline{\otimes}\;{\mathbf{C}}^{-1} \big\rbrack. $$

(131)

The tensor ${\mathbb{I}}_{n}$ is obtained from (131) by substituting ${\mathbf{C}}$ with ${\mathbf{C}}_{e}$.

Given two second-order tensors ${\mathbf{A}}$ and ${\mathbf{B}}$, the products ${\mathbf{A}}\;\overline{\otimes}\;{\mathbf{B}}$ and ${\mathbf{A}}\;\underline {\otimes}\;{\mathbf{B}}$ have the following index representation [18]

$$ \lbrack{\mathbf{A}}\;\overline{\otimes}\;{\mathbf{B}}\rbrack_{IJMN} = A_{IN}B_{JM}, \quad \lbrack{\mathbf{A}}\;\underline {\otimes}\;{\mathbf{B}}\rbrack_{IJMN} = A_{IM}B_{JN}. $$

(132)

To perform the linearisation procedure, we set $\varvec{u} = \varvec{u}_{0} + \varvec{h}$, and use the following Gateaux-derivatives:

$$ \begin{aligned} {\mathfrak{D}}(J\pi{\mathbf{C}}^{-1})(\varvec{u}_{0},\pi_{0})[\varvec{h},\theta] &= (J_{0}({\mathbf{C}}_{0})^{-T}\!\!: {\mathfrak{D}}{\mathbf{E}}(\varvec{u}_{0})\lbrack\varvec{h}\rbrack) \pi_{0}({\mathbf{C}}_{0})^{-1} \\ &+ J_{0} \theta ({\mathbf{C}}_{0})^{-1} - J_{0}\pi_{0} 2 ({\mathbf{C}}_{0})^{-1} \lbrace {\mathfrak{D}}{\mathbf{E}}(\varvec{u}_{0})[\varvec{h}]\rbrace({\mathbf{C}}_{0})^{-1}, \end{aligned} $$

(133)

$$ {\mathfrak{D}}\dot{{\mathbf{E}}}_{v}(\varvec{u}_{0},\varvec{v}_{v})[\varvec{h}] = \tfrac{1}{2}\big\lbrack ({\mathbf{H}})^{T}\dot{{\mathbf{F}}}_{v} + (\dot{{\mathbf{F}}}_{v})^{T}{\mathbf{H}}\big\rbrack, $$

(134)

$$ {\mathfrak{D}}{\mathbf{S}}(\varvec{u}_{0})[\varvec{h}] = {\mathbb{C}}_{r}(\varvec{u}_{0})\!\!:{\mathfrak{D}}{\mathbf{E}}(\varvec{u}_{0})[\varvec{h}], $$

(135)

$$ {\mathbb{C}}_{r} = J_{a} ({\mathbf{F}}_{a})^{-1}\underline {\otimes}({\mathbf{F}}_{a})^{-1}\!\!:{\mathbb{C}}_{n}\!\!:({\mathbf{F}}_{a})^{-T}\underline {\otimes}({\mathbf{F}}_{a})^{-T}. $$

(136)

The formulae in Sect. 3.1 are retrieved by setting $\varvec{u}\equiv\varvec{u}^{m,k},\; \varvec{u}_{0}\equiv\varvec{u}^{m,k-1}$ and $\varvec{h}\;\equiv\;\varvec{h}^{m,k}$.