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Pore-Scale Model for Thermal Conductivity of Unsaturated Sand

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Abstract

A new framework is proposed for predicting thermal conductivity dry out curves (TCDC) quantifying the relationship between soil thermal conductivity and degree of saturation at non-elevated temperatures. Pore-scale expressions are derived to quantify water retention and corresponding conductive heat transport for an idealized contacting-sphere geometry approximating that of granular (sand-sized) porous media. Measured water retention behavior is used to constrain a simulated soil–water characteristic curve and corresponding TCDC by differentiating pores containing water in the form of inter particle liquid bridges from pores containing water in the form of saturated pockets. Modeled TCDCs compare well with experimental measurements for representative fine-, medium-, and coarse-grained sands. Predicted thermal conductivity is within 5–10 % of the full scale range for saturations greater than 20 %. Model predictions perform as well as or better than predictions made using many existing empirical approaches. Because the model directly incorporates basic soil properties and process variables in its formulation, including grain size, grain size distribution, and wetting direction, it becomes a potentially useful framework to improve understanding of fundamental controls on soil thermal conductivity.

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Abbreviations

C c :

Coefficient of curvature

C u :

Coefficient of uniformity

d 50 :

Median grain size by mass

e :

Void ratio

g :

Campbell et al. (1994) TCDC modeling function

G s :

Specific gravity of soil solids

m c :

Clay content

n :

Porosity; mixing model exponent

q t :

Heat flux

r 1 :

Radius in toroidal approximation for liquid bridge

r 2 :

Radius in toroidal approximation for liquid bridge

R :

Spherical particle radius

R min :

Minimum particle radius in f(R)

R max :

Maximum particle radius in f(R)

R fill :

Particle radius in f(R) corresponding to snap off

R 2 :

Coefficient of determination

S :

Degree of saturation

T s :

Air-water interfacial tension

V a :

Volume of air

V s :

Volume of soil solids

V t :

Volume total

V w :

Volume of water

w :

Gravimetric water content

x f :

Fractional area of liquid bridge

α :

Fitting parameter in Hu et al. (2001) TCDC model; contact angle

κ :

Kersten number

λ :

Thermal conductivity

λ a :

Thermal conductivity of air

λ p :

Pore-scale thermal conductivity

λ q :

Thermal conductivity of quartz

λ s :

Thermal conductivity of mineral solids

λ w :

Thermal conductivity of water

λ dry :

Thermal conductivity at dry conditions

λ sat :

Thermal conductivity at full saturation

θ :

Volumetric water content

θ crit :

Volumetric water content at snap-off

θ o :

Cut-off water content for liquid return flow, Campbell et al. (1994)

ρ b :

Dry bulk soil density

ρ s :

Density of soil solids

ψ :

Soil suction

μ :

Mean particle radius in f(R)

σ :

Standard deviation of particle radius in f(R)

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Acknowledgments

This material is based upon work supported by the National Science Foundation (NSF) under Grant CMMI 0968768. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF.

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  1. Department of Civil and Environmental Engineering, University of Wisconsin-Madison, Madison, WI, 53706, USA

    William J. Likos

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Likos, W.J. Pore-Scale Model for Thermal Conductivity of Unsaturated Sand. Geotech Geol Eng 33, 179–192 (2015). https://doi.org/10.1007/s10706-014-9744-9

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