Biological Driven Phase Transitions in Fully or Partly Saturated Porous Media: A Multi-Component FEM Simulation Based on the Theory of Porous Media

Abstract

Bacterial methane oxidation in landfill cover soils, which turns the emitting methane caused by waste degradation into carbon dioxide, reduces the climate impact of landfill gas emissions significantly, since methane is estimated to have a global warming potential (GWP) of 25 over 100 years (GWP of CO₂ = 1). To understand and forecast the biological processes, a Finite-Element Model (FEM) is developed to simulate the behavior of methanotrophic layers. A multiphasic continuum mechanical approach based on the extended Theory of Porous Media (eTPM) is chosen, providing a macroscopic, multi-component view on the bacterial progress including the relevant gas transport processes of diffusion and advection in porous media as well as bacterial driven conversion processes. The presented thermodynamically consistent model also considers energy production within the gas phase resulting from exothermic reactions. An experimental setup was developed to validate the model also in terms of temperature development via the thermal imaging technique.

Appendices

Appendix 1: Kinematics of Porous Media

The homogenized phases φ ^α are composed of particles X_α. At any time t each spatial point of the current placement is simultaneously occupied by every particle X_S, X_L, X_G of the solid, liquid, and gas phase, whereby each particle proceeds from a different reference position X _α at the time t = t₀. Thus, each constituent is assigned to its own independent motion function χ _α with

(68)

Equation (68)₁ is called the Lagrange description of motion, Eq. (68)₂ is the inverse map of the motion for a fixed time t, which represents the Euler description. The Euler description develops from the postulation that χ _α is unique and uniquely invertible at any given time t. A mathematically necessary and sufficient condition for the existence of Eq. (68)₂ is given, if the Jacobian J_α = detF _α differs from zero with F _α = Grad_αχ _α and its inverse $\mathbf F_{\pmb {\alpha }}^{\mathrm {-1}}=\mathrm {grad}\,{\mathbf {x}}_{{\pmb {\alpha }}}$ as the deformation gradient. “Grad_α” denotes the differential operator referring to the reference position X _α of the constituent φ ^α, while “grad” refers to the actual position x. During the deformation process, F _α is restricted to .

The velocity and acceleration vector, cf. (68)₃₋₄, are defined with the material time derivatives of the Lagrange description. The material time derivative for scalar fields depending on x and t with respect to the trajectory of φ ^α is given with $(\ldots )^{\prime }_{{\pmb {\alpha }}}=\partial (\ldots )\,/\,\partial \mathrm {t}+[\mathrm{grad}(\ldots )]\cdot {\mathbf {x}}^{\prime }_{\pmb {\alpha }}$.

With (68)₃ the material and spatial velocity gradient are defined with

$$\displaystyle \begin{aligned} ({\mathbf{F}}_{{\pmb{\alpha}}})^{\prime}_{{\pmb{\alpha}}} = \mathrm{Grad}_{{\pmb{\alpha}}}{\mathbf{x}}^{\prime}_{\pmb{\alpha}}, \quad {\mathbf{L}}_{{\pmb{\alpha}}} = (\mathrm{Grad}_{{\pmb{\alpha}}}{\mathbf{x}}^{\prime}_{\pmb{\alpha}})\,\mathbf F_{\pmb{\alpha}}^{\mathrm{-1}} = \mathrm{grad}\,{\mathbf{x}}^{\prime}_{\pmb{\alpha}} \end{aligned} $$

(69)

The tensor ${\mathbf {D}}_{{\pmb {\alpha }}}=\text{1/2}\,({\mathbf {L}}_{{\pmb {\alpha }}}+{\mathbf {L}}_{{\pmb {\alpha }}}^{\mathrm {T}})$ is defined as the symmetric part of the spatial velocity gradient. An extended explanation of the kinematics of porous media is given in [6] and [9].

Appendix 2: Evaluation of Entropy Inequality

The resultant system of field equations is set up by 29 equations, namely Eqs. (22), (24), (25), (27), (28), (29), (30), as well as the physical constraint $\hat {\mathbf {p}}^{\mathbf S} + \textstyle \sum _{\upgamma }\,\hat {\mathbf {p}}^{\mathrm {G}\upgamma } = \mathbf {o}$, the partial densities ρ ^S = n^Sρ ^SR and ρ ^G = n^Gρ ^GR, respectively. The field equations contain overall 91 scalar field quantities, namely

$$\displaystyle \begin{aligned} \mathcal{F}=\{&\rho^{\mathbf{S}},\rho^{\mathbf{G}},\rho^{\mathbf{S}\mathrm{R}},\rho^{\mathbf{G}\mathrm{R}},\mathrm{n}^{\mathbf{S}},\mathrm{n}^{\mathbf{G}},\mathrm{c}^{\mathbf{G}},{\mathbf{x}}^{\prime}_{\mathbf{S}},{\mathbf{x}}^{\prime}_{\mathrm{G}\upgamma},\\ &{\mathbf{T}}^{\mathbf{S}},{\mathbf{T}}^{\mathbf{G}\upgamma},\mathbf{b},\hat{\mathbf{p}}^{\mathbf S},\hat{\mathbf{p}}^{\mathrm{G}\upgamma},\hat{\rho}^{\mathrm G\upgamma},\psi^{\mathbf{S}},\psi^{\mathrm G\upgamma},\eta^{\mathbf{S}},\eta^{\mathrm{G}\upgamma},\mathbf{q},\,\theta\}\,, \end{aligned} $$

(70)

where the Cauchy stresses T ^S and T ^Gγ are symmetric. Since the solid phase is assumed to be incompressible and not involved into mass transfer, the real density of the solid is known and stays constant. The gravitational acceleration b is known as well and nitrogen is not involved into gas exchange, which leads to five known field quantities

$$\displaystyle \begin{aligned} \mathcal{K}=\{\rho^{\mathbf{S}\mathrm{R}}\,,\mathbf{b}\,,\hat{\rho}^{\mathrm{GN}}\}. \end{aligned} $$

(71)

In consequence, 57 field quantities remain for which constitutive relations have to be found. The constitutive quantities are chosen with

$$\displaystyle \begin{aligned} \mathcal{C}=\{{\mathbf{T}}^{\mathbf{S}},{\mathbf{T}}^{\mathrm{G}\upgamma},\hat{\mathbf{p}}^{\mathrm{G}\upgamma},\hat{\rho}^{\mathrm{GO}},\hat{\rho}^{\mathrm{GC}}, \psi^{\mathbf{S}},\psi^{\mathrm G\upgamma},\eta^{\mathbf{S}},\eta^{\mathrm{G}\upgamma},\mathbf{q}\}. {} \end{aligned} $$

(72)

In order to provide a thermodynamic consistent description of the material behavior, the entropy inequality for the mixture is evaluated in accordance to [5], which yields restrictions for the constitutive relations. Following the principle of equipresence postulated by [35], the set of constitutive variables (72) can depend on the following process variables

$$\displaystyle \begin{aligned} \mathcal{P}=\{{\mathbf{C}}_{\mathbf{S}},\mathrm{c}^{\mathrm{G}\upgamma}_{\mathrm{mol}},\hat{\rho}^{\mathrm G\upgamma},\theta,\mathrm{grad}\>\theta,{\mathbf{w}}_{\mathrm{G}\upgamma\mathbf{S}}\}. {} \end{aligned} $$

(73)

Here, ${\mathbf {C}}_{\mathbf {S}} = {\mathbf {F}}_{\mathbf {S}}^{\mathrm {T}}{\mathbf {F}}_{\mathbf {S}}$ denotes the right Cauchy–Green deformation tensor. The entropy inequality for the binary model (solid and gas mixture) reads

$$\displaystyle \begin{aligned} &{\mathbf{T}}^{\mathbf{S}}\cdot{\mathbf{D}}_{\mathbf{S}} - \rho^{\mathbf{S}}(\psi^{\mathbf{S}})^{\prime}_{\mathbf{S}} - \rho^{\mathbf{S}}(\theta)^{\prime}_{\mathbf S}\,\eta^{\mathbf{S}} - \hat{\mathbf{p}}^{\mathbf S}\cdot{\mathbf{x}}^{\prime}_{\mathbf{S}}\\ {} -&\textstyle\sum_\upgamma[{\mathbf{T}}^{\mathrm{G}\upgamma}\cdot{\mathbf{D}}_{\mathrm{G}\upgamma} - \rho^{\mathrm{G}\upgamma}(\psi^{\mathrm{G}\upgamma})^{\prime}_{\mathrm{G}\upgamma}-\rho^{\mathrm{G}\upgamma}(\theta)^{\prime}_{\mathrm G \upgamma}\,\eta^{\mathrm{G}\upgamma} -\hat{\mathbf{p}}^{\mathrm{G}\upgamma}\cdot{\mathbf{x}}^{\prime}_{\mathrm{G}\upgamma}] \\ {} -&\textstyle\sum_\upgamma\,\hat{\rho}^{\mathrm G\upgamma}\psi^{\mathrm G\upgamma}-\dfrac{1}{\theta}\mathbf{q}\,\mathrm{grad}\,\theta\geq 0 \end{aligned} $$

(74)

It will be enhanced by the constraining saturation condition with the Lagrange multiplicator λ with

$$\displaystyle \begin{aligned} \lambda\left((\mathrm{n}^{\mathbf{S}})^{\prime}_{\mathbf{S}}+(\mathrm{n}^{\mathbf{G}})^{\prime}_{\mathbf{S}}\right)\,. \end{aligned} $$

(75)

The overall gas balance in terms of molar concentration solved for $(\mathrm {n}^{\mathbf {G}})^{\prime }_{\mathbf {S}}$ reads

$$\displaystyle \begin{aligned} (\mathrm{n}^{\mathbf{G}})^{\prime}_{\mathbf{S}} = &-\dfrac{\mathrm{n}^{\mathbf{G}}}{\mathrm{c}^{\mathbf{G}}_{\mathrm{mol}}}\textstyle\sum_\upgamma(\mathrm{c}^{\mathrm{G}\upgamma}_{\mathrm{mol}})^{\prime}_{\mathrm{G}\upgamma}-\mathrm{n}^{\mathbf{G}}\textstyle\sum_\upgamma\mathrm{x}^{\mathrm{G}\upgamma}_{\mathrm{mol}}{\mathbf{D}}_{\mathrm{G}\upgamma}\cdot\mathbf I\\ {} &-\mathrm{grad}\,\mathrm{n}^{\mathbf{G}}\textstyle\sum_\upgamma\mathrm{x}^{\mathrm{G}\upgamma}_{\mathrm{mol}}\cdot{\mathbf{w}}_{\mathrm{G}\upgamma\mathrm{S}} +\textstyle\sum_\upgamma\dfrac{\hat{\rho}^{\mathrm G\upgamma}_{\mathrm{mol}}}{\mathrm{c}^{\mathbf{G}}_{\mathrm{mol}}}\,. \end{aligned} $$

(76)

A mass specific Helmholtz free energy ψ ^Gγ for the partial gas components is introduced with

$$\displaystyle \begin{aligned} \rho^{\mathrm{G}\upgamma}\psi^{\mathrm G\upgamma} = \mathrm{n}^{\mathbf{G}}\rho^{\mathrm{G}\upgamma\mathrm x}\psi^{\mathrm G\upgamma}=\mathrm{n}^{\mathbf{G}}\psi^{\mathrm G\upgamma\mathrm x}.\end{aligned} $$

(77)

For simplification we postulate that the Helmholtz free energy functions ψ ^α hold the following dependencies

$$\displaystyle \begin{aligned} \psi^{\mathbf{S}} &=\psi^{\mathbf{S}}({\mathbf{C}}_{\mathbf{S}},\theta)\\ {} \psi^{\mathrm G\upgamma\mathrm x} &= \psi^{\mathrm G\upgamma\mathrm x}(\rho^{\mathrm{G}\upgamma\mathrm x},\theta). \end{aligned} $$

(78)

With the chosen dependencies of the Helmholtz free energy functions on the process variables, the material time derivatives $(\psi ^{{\pmb {\alpha }}})^{\prime }_{{\pmb {\alpha }}}$ read

$$\displaystyle \begin{aligned} \psi^{\mathbf{S}}&:=\psi^{\mathbf{S}}({\mathbf{C}}_{\mathbf{S}},\theta) \quad \Rightarrow \quad (\psi^{\mathbf{S}})^{\prime}_{\mathbf{S}} = \mathrm 2\,\rho^{\mathbf{S}}{\mathbf{F}}_{\mathbf{S}}\displaystyle{\frac{\partial\psi^{\mathbf{S}}}{\partial{\mathbf{C}}_{\mathbf{S}}}}{\mathbf{F}}_{\mathbf{S}}^{\mathrm{T}}\cdot{\mathbf{D}}_{\mathbf{S}}+\frac{\partial\psi^{\mathbf{S}}}{\partial\theta}(\theta)^{\prime}_{\mathbf S}\\ {} \psi^{\mathrm G\upgamma\mathrm x}&:=\psi^{\mathrm G\upgamma\mathrm x}(\rho^{\mathrm{G}\upgamma\mathrm x},\theta) \quad \Rightarrow \quad (\psi^{\mathrm{G}\upgamma\mathrm x})^{\prime}_{\mathrm{G}\upgamma} = \frac{\partial\psi^{\mathrm G\upgamma\mathrm x}}{\partial\rho^{\mathrm{G}\upgamma\mathrm x}}(\rho^{\mathrm{G}\upgamma\mathrm x})^{\prime}_{\mathrm{G}\upgamma}+\frac{\partial\psi^{\mathrm G\upgamma\mathrm x}}{\partial\theta}(\theta)^{\prime}_{\mathrm G \upgamma}, \end{aligned} $$

(79)

the entropy inequality finally reads (sorted by the process variables and their derivatives):

$$\displaystyle \begin{aligned} &[{\mathbf{T}}^{\mathbf{S}}+\mathrm{n}^{\mathbf{S}}\lambda\mathbf I\,-2\,\rho^{\mathbf{S}}{\mathbf{F}}_{\mathbf{S}}\displaystyle{\frac{\partial\psi^{\mathbf{S}}}{\partial{\mathbf{C}}_{\mathbf{S}}}}{\mathbf{F}}_{\mathbf{S}}^{\mathrm{T}}]\cdot{\mathbf{D}}_{\mathbf{S}} -\rho^{\mathbf{S}}[\eta^{\mathbf{S}}+\frac{\partial\psi^{\mathbf{S}}}{\partial\theta}](\theta)^{\prime}_{\mathbf S}+ \\ {} &\textstyle\sum_\upgamma([{\mathbf{T}}^{\mathrm{G}\upgamma}+\mathrm{n}^{\mathbf{G}}\mathrm{x}^{\mathrm{G}\upgamma}\lambda\,\mathbf I]\cdot{\mathbf{D}}_{\mathrm{G}\upgamma} - \rho^{\mathrm{G}\upgamma}[\eta^{\mathrm{G}\upgamma}+\displaystyle{\frac{1}{\rho^{\mathrm{G}\upgamma\mathrm x}}}\frac{\partial\psi^{\mathrm G\upgamma\mathrm x}}{\partial\theta}](\theta)^{\prime}_{\mathrm G \upgamma}+\\ {} &[\mathrm{x}^{\mathrm{G}\upgamma}\lambda\displaystyle{\frac{\rho^{\mathrm{G}\upgamma}}{(\rho^{\mathrm{G}\upgamma\mathrm x})^{2}}}+\frac{\rho^{\mathrm{G}\upgamma}}{(\rho^{\mathrm{G}\upgamma\mathrm x})^{2}}\psi^{\mathrm G\upgamma\mathrm x}- \frac{\rho^{\mathrm{G}\upgamma}}{\rho^{\mathrm{G}\upgamma\mathrm x}}\frac{\partial\psi^{\mathrm G\upgamma\mathrm x}}{\partial\rho^{\mathrm{G}\upgamma\mathrm x}}](\rho^{\mathrm{G}\upgamma\mathrm x})^{\prime}_{\mathrm{G}\upgamma}-\\ {} &[\hat{\mathbf{p}}^{\mathrm{G}\upgamma}-\mathrm{grad}\>\mathrm{n}^{\mathbf{G}}\mathrm{x}^{\mathrm{G}\upgamma}\lambda]\cdot{\mathbf{w}}_{\mathrm{G}\upgamma\mathbf{S}}- \hat{\rho}^{\mathrm G\upgamma}[\psi^{\mathrm G\upgamma}+\displaystyle{\frac{\mathrm{x}^{\mathrm{G}\upgamma}\lambda}{\rho^{\mathrm{G}\upgamma\mathrm x}}}])- \displaystyle{\frac{1}{\theta}\,\mathbf{q}\,\mathrm{grad}\>\theta}\geq 0. \end{aligned} $$

(80)

The evaluation of the restrictions for the constitutive relations is presented in Sect. 4.