Effective heat conductivity in porous media

Laws for calculating effective heat conductivity coefficients. More...

Description

In porous media, the effective heat conductivity depends on the solid-fluid conductivity ratio, the volume fractions of the constituent phases, and the geometry of the solid-fluid interface [5]. The following laws are implemented:

Johansen (two fluid phases)

ThermalConductivityJohansen [47] computes the thermal conductivity of dry and the wet soil material and interpolates using the Kersten number. The effective wet conductivity is based on a geometric average and the effective dry conductivity is based on a semi-emprical relation and fitted to medium quartz sand.

The effective thermal conductivity is given by

$${\lambda }_{\text{eff}}={\lambda }_{\text{dry}}+\text{Ke}({\lambda }_{\text{wet}}-{\lambda }_{\text{dry}}),{\lambda }_{\text{wet}}={\lambda }_{\text{s}}^{(1-\varphi )}{\lambda }_{\text{w}}^{\varphi },{\lambda }_{\text{dry}}=\frac{0.135{\rho }_{\text{s}}\varphi +64.7}{{\rho }_{\text{s}}-0.947{\rho }_{\text{s}}\varphi },$$

where $\varphi$ is the porosity, ${\lambda }_{\alpha }$ is the thermal conductivity of phase $\alpha$, ${\rho }_{\text{s}}$ denotes the density of the solid phase, and the Kersten number is given by $\text{Ke}=(\kappa S_{\text{w}})/(1+(1-\kappa )S_{\text{w}})$, with the wetting phase saturation $S_w$ and a fitting parameter $\kappa =15.6$ for medium quartz sand.

Somerton (two fluid phases)

The Somerton method [81] computes the thermal conductivity of dry and the wet soil material. It uses a root function of the water saturation to compute the effective thermal conductivity for a two-phase fluidsystem. The individual thermal conductivities are calculated as geometric mean of the thermal conductivity of the porous material and of the respective fluid phase.

The effective thermal conductivity of ThermalConductivitySomertonTwoP is given by

$${\lambda }_{\text{eff}}={\lambda }_{\text{g,eff}}+\sqrt{S_{\text{w}}}({\lambda }_{\text{w,eff}}-{\lambda }_{\text{g,eff}})$$

with $S_{\text{w}}$ the water saturation, $S_{\text{n}}$ the NAPL saturation, the effective phase saturations given by ${\lambda }_{\alpha ,\text{eff}}=({\lambda }_{\text{s}}{)}^{(1-\varphi )}({\lambda }_{\alpha }{)}^{\varphi },\alpha \in \{\text{w,n,g}\}$ (geometric mean) and ${\lambda }_{\text{s}}$ is the thermal conductivity of the solid phase. The effective conductivity ${\lambda }_{\text{g,eff}}$ corresponds to dry conditions, whereas the effective conductivity ${\lambda }_{\text{g,eff}}$ corresponds to wet conditions.

Somerton (three fluid phases)

The Somerton method [81] computes the thermal conductivity of dry and the wet soil material. It is extended here to a three phase system of water (w), NAPL (n) and gas (g). It uses a root function of the water and NAPL saturation to compute the effective thermal conductivity for a three-phase fluidsystem. The individual thermal conductivities are calculated as geometric mean of the thermal conductivity of the porous material and of the respective fluid phase.

The effective thermal conductivity of ThermalConductivitySomertonThreeP is given by

$${\lambda }_{\text{eff}}={\lambda }_{\text{g,eff}}+\sqrt{S_{\text{w}}}({\lambda }_{\text{w,eff}}-{\lambda }_{\text{g,eff}})+\sqrt{S_{\text{n}}}({\lambda }_{\text{n,eff}}-{\lambda }_{\text{g,eff}})$$

with $S_{\text{w}}$ the water saturation, $S_{\text{n}}$ the NAPL saturation, the effective phase saturations given by ${\lambda }_{\alpha ,\text{eff}}=({\lambda }_{\text{s}}{)}^{(1-\varphi )}({\lambda }_{\alpha }{)}^{\varphi },\alpha \in \{\text{w,n,g}\}$ (geometric mean) and ${\lambda }_{\text{s}}$ is the thermal conductivity of the solid phase.

Average (multiple fluid phases, one solid phase)

The effective thermal conductivity of ThermalConductivityAverage is calculated as a weighted arithmetic average of the thermal conductivities of the solid and the fluid phases. The weights are determined by the volume fraction the phase occupies. Denoting the volume fractions by $n_{\alpha }$, we have

$${\lambda }_{\text{eff}}=\sum _{\alpha }{\lambda }_{\alpha }{n}_{\alpha }/\sum _{\alpha }{n}_{\alpha },$$

summing over both fluid and solid phases. With the porosity $\varphi$ as the sum of all fluid volume fractions, we can equivalently write

$${\lambda }_{\text{eff}}={\lambda }_{\text{s}}(1-\varphi )+{\lambda }_{\text{f}}\varphi ,$$

where ${\lambda }_{\text{s}}$ is the thermal conductivity of the solid phase, and the effective thermal conductivity of the liquid phases is computed as an arithmetic average weighted with the fluid saturations.

Files
file	johansen.hh
	Relation for the saturation-dependent effective thermal conductivity.
file	somerton.hh
	Effective thermal conductivity after Somerton.
file	thermalconductivitysomerton3p.hh
	Effective thermal conductivity after Somerton.
file	thermalconductivityaverage.hh
	Effective thermal conductivity based on weighted arithmetic average.

Classes
class	Dumux::ThermalConductivityJohansen< Scalar >
	Relation for the saturation-dependent effective thermal conductivity. More...
class	Dumux::ThermalConductivitySomertonTwoP< Scalar >
	Effective thermal conductivity after Somerton. More...
class	Dumux::ThermalConductivitySomertonThreeP< Scalar >
	Effective thermal conductivity after Somerton. More...
class	Dumux::ThermalConductivityAverage< Scalar >
	Effective thermal conductivity based on weighted arithmetic average. More...