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Method for Evaluating the Equivalent Thermal Conductivity of a Freezing Rock Mass Containing Systematic Fractures

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Abstract

Rock deterioration is caused by freeze–thaw cycles, and this deterioration or loss of strength induces failure when the rock thaws. To accurately analyze freeze–thaw cycles, a proper method for evaluating the thermal conductivity of a fractured rock mass is needed. In this study, a simplified model of a fracture was derived from knowledge of actual fractures in rock masses, and the possible thermal resistance types of a fracture were analyzed. Then, according to the definition of thermal resistance, models were established for the normal and tangential thermal resistances of a fracture by considering the ice-water phase transition. Based on the thermal resistance models, a method for evaluating the equivalent thermal conductivity of a rock mass with a set of systematic fractures was established. The effect of the convective heat transfer of the water in the fractures was considered based on energy contributions. According to the superposition principle and the introduction of fracture parameters (strike angle, dip direction angle, and dip angle), the proposed method was extended to rock masses with multiple sets of fractures, and a sensitivity analysis was conducted. The thermal conductivity of a rock mass with connected fractures was more sensitive to the fracture parameters than that of a rock mass with intermittent fractures. Finally, the proposed method was verified by comparison with an experiment using a block of cement with artificial fissures under freezing conditions; the calculated temperature agreed well with the experimental results.

Highlights

  • A method for evaluating the thermal conductivity of a freezing rock mass is proposed.

  • The method considered the phase transition, convection and fracture parameters.

  • The convective heat transfer was considered based on energy contributions.

  • The thermal conductivity is more sensitive to the rock mass with connected fractures.

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Notes

  1. Equivalent thermal conductivity refers to the thermal conductivity, but it is the equivalent value of a continuum having the same volume as REV.

  2. Fracture ratio refers to the ratio of the volume of fractures in RVE to the total volume of RVE.

Abbreviations

REV:

Representative elementary volume

ETC:

Equivalent thermal conductivity

A :

Flow area

A as :

Actual contact area

A bs :

Nominal area

N s :

Number of contact points

Q :

Heat transmission rate

R :

Thermal resistance

R c :

Thermal conduction resistance

R f :

Thermal convection resistance

R fa :

Thermal convection resistance of air

R fw :

Thermal convection resistance of water

R m :

Thermal conduction resistance of the filling medium

R n :

Normal thermal resistance of fractured media

R r :

Thermal radiation resistance

R s :

Thermal conduction resistance of the rock at the contact point

R t :

Tangential thermal resistance of fractured media

R tr :

Convective heat transfer resistance

S :

Saturation

T :

Temperature

ΔT :

Temperature difference

a :

Length of the fracture

\(\overline{a}_{{\text{s}}}\) :

Average contact radius

a s :

Actual contact radius

b :

Center distance of the fractures

b s :

Nominal radius

\(\overline{c}_{{\text{s}}}\) :

Average interface contraction factor

c s :

Interface contraction factor

c a :

Specific heat of air

c w :

Specific heat of water

h :

Convective heat transfer coefficient

k :

Thermal conductivity

k a :

Thermal conductivity of air

k fm :

Thermal conductivity of other solid fillers in the fracture

k i :

Thermal conductivity of ice

k m :

Normal equivalent thermal conductivity of the filling medium

k mt :

Tangential equivalent thermal conductivity of the filling medium

k n :

Normal equivalent thermal conductivity

k t :

Tangential equivalent thermal conductivity

k w :

Thermal conductivity of unfrozen water

l :

Row spacing of the fractures

l m :

Length of the fracture

v a :

Velocity of air

v w :

Velocity of water flow

ΩT :

Temperature effect coefficient

α :

Angle between the normal direction of the fracture and the horizontal coordinate axis

β :

Dip angle of the fracture

δ' :

Thickness of the single-layer plate

δ :

Aperture of the fracture

ζ :

Fracture ratio of the rock mass

θ :

Volume content

θ fm :

Content of solid fillers

κ :

Boundary temperature factor

ρ a :

Density of air

ρ w :

Density of water

φ :

Porosity

χ :

Content of unfrozen water in the fracture fillings

m :

Order number of dominant fractures

w, i, fmt, r and a:

(Properties associated with) water, ice, solid fill, contact bodies and air

g(c s) = (1 − ω)3/2 :

Contact influence function

ω = A as /A bs :

Contact area ratio

ψ = ωδ + (1 − ω)5/2 k m /k r :

Normal heat transfer factor

ξ = aφSχ(1 − ω)ρ w c w ν w  + k mt :

Tangential heat transfer factor

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Acknowledgements

We express our gratitude to the anonymous reviewers for their constructive comments and suggestions. This research was supported by the Talent-Technology Foundation (No. RC1804) and Special Funds of the Natural Science Foundation (No. ZR18017) of Xi’an University of Architecture and Technology, the Natural Science Basic Research Program of Shaanxi (Nos. 2019JQ-756, 2022JM-190), the Special Scientific Research Project of Shaanxi Education Department (No. 19JK0452), the China Postdoctoral Science Foundation (No. 2019M663648), and the Open Fund of State Key Laboratory of Road Engineering Safety and Health in Cold and High-Altitude Regions (No. YGY2020KYPT-03).

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Authors and Affiliations

  1. School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an, 710055, China

    Naifei Liu & Zhanping Song

  2. Institute of Geotechnical Engineering, Xi’an University of Technology, Xi’an, 710048, China

    Ning Li & Guofeng Li

  3. Post-Doctoral Research Center, CCCC First Highway Consultants Co., Ltd, Xi’an, 710075, China

    Naifei Liu & Shuangjie Wang

  4. Shaanxi Key Laboratory of Geotechnical and Underground Space Engineering, XAUAT, Xi’an, 710055, China

    Naifei Liu & Zhanping Song

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  1. Naifei Liu
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  2. Ning Li
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  3. Guofeng Li
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Correspondence to Naifei Liu.

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Liu, N., Li, N., Li, G. et al. Method for Evaluating the Equivalent Thermal Conductivity of a Freezing Rock Mass Containing Systematic Fractures. Rock Mech Rock Eng 55, 7333–7355 (2022). https://doi.org/10.1007/s00603-022-03038-9

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