The document presents a fractal model to predict the effective thermal conductivity (λ) of unsaturated soils. Existing empirical and physical models are reviewed. A theoretical fractal model is developed using parameters like fractal dimensions for pore area (Dp) and tortuosity (Dt). The model accounts for effects of saturation and void ratio on λ. Model predictions agree reasonably well with experimental data, improving upon existing fractal models. Further research on particle contact area and testing different soil types is suggested.

1. Evaluation of Thermal
Conductivity of Soils Using
Fractal Approach
By Daniel Diamond

2. Thesis Overview
 Analysis of existing fractal models on the effective thermal
conductivity of unsaturated soils.
 Development of a theoretical model using fractal theory to
predict the effective thermal conductivity (λ) of unsaturated
soils as a function of void ratio (e).
 Model was verified by comparing predictions to existing
empirical models and experimental data.
 Predictions of λ are generally found to be in good agreement
with experimental data but overestimates λef at low porosity
levels.

3. Why Thermal Conductivity?
 Importance of λ in various areas of engineering,
specifically relating to earth-contact facilities.
 Ground heat exchangers;
 High voltage cables;
 Nuclear waste vaults;
 Ability to transfer heat away from underground facilities.

4. Empirical Models
 Kersten (1949)
 Kersten Function
 Completely empirical
 Limited to degrees of saturation greater than 0.3
 Johansen (1975)
 Cote and Konrad (2005)

5. Physical-Based Models
 Usowicz (1992)
 Model considers a soil elementary
unit cell containing a number of
overlapping layers of spheres.
 Modeled as a network of thermal
resistors arranged in series and in
parallel

6. Fractal Theory
 Fractal Theory
 Objects in nature: irregular and disordered such as rough
surfaces, coastlines and islands.
 Fractal object measured by M(L)~L^(Dp)
(Mandelbrot, 1982)
 Dp is the fractal dimension of the object
 M is the measure of an object e.g. the length of a line.

7. Fractal Models
 Huai et al. (2007)
 Generated self-similar fractal
to model the structure of
porous media.
 Sierpinski Carpet and the
Ben Avraham and Havlin
carpet
 However, porous media is not
exacty self-similar.

9. Fractal Models
 Kou et al. (2009)
 Thermal-electrical
analogy and statistical
self-similarity of porous
media

10. Issues with Current Fractal Model
 Current fractal models assume the soil (fractal porous
media) is equal to statistically self-similar fractals.
 Implies only one pore with
maximum diameter, rmax
 Saturation effects has not been taken
into account appropriately.
 Current fractal models use an arbitrary representative length
L0 and assume At=L0
2, which is incorrect.
 Assume constant value for Dp, Dt and the ratio rmin/rmax.

11. Pore Area Fractal Dimension, Dp
 Current fractal models determine Dp using the formula
derived from Katz and Thompson (1985)
 However, they assumes a constant ratio rmin/rmax, which
is incorrect since the ratio decreases significantly with
decreasing void ratio (e).
 In my current model, Dp was derived as a function of e
from Khoshghalb et al. (2014) based on works from
Eshelby (1957).

12. Determination of Dp

13. Tortuosity Fractal Dimension, Dt
 Fractal dimension, Dt characterizes the convolutedness of
pore capillaries through heterogeneous media.
 Wheatcraft and Tyler (1988) performed a fractal-monte carlo
simulation to produce Lt(r)=r1-DtL0
Dt where Lt(r) and r are the
tortuous length and pore diameter, respectively.
 L0 is some representative length. Current fractal models
assume At=L0
2, which is incorrect.
 I have rearranged tortuous length as Lt(r)= L0(Lth/r)Dt-1 where
Lth is the threshold length of the soil sample. Lth/r represents
a representative elementary volume (REV).

14. Representative Elementary Volume
(REV)
 Roberts (1994) illustrates this notion using a 1D
continuum model for density

15. Determination of Dt

16. Coordination Number (CN)
and Contact Ratio (CR)
 Both the coordination number (CN) and contact ratio (CR)
are essential parameters in the flow capabilities of the solid
phase.
 The coordination number (CN) represents the number of
particle contact points of a specific particle.
 Hasan and Alshibli (2010) determined a relationship between
CN and void ratio (e) expressed as:
e=2.23e-0.13CN
 Very limited research exists on CR , however existing
literature suggests a value of approximately 4%.

17. 3D Representation
L0 Pore area, with diameter r2
filled with water
Pore area, with diameter r1
filled with air
Lt(r)
rmin rmax

18. Current Fractal Model

19. Results & Findings
 The λef is normalized with respect to λs and compared
to normalized experimental data.
 At all degrees of saturation, the current fractal model
predicts λef with reasonable accuracy.
 The effects of saturation have been sufficiently
accounted for, which demonstrates its superiority over
existing fractal models.

24. Comparison With Existing
Fractal Models
 Of the existing fractal models, many assumptions have
been made to account for a lack of research
surrounding various parameters.
 Constant values for fractal dimensions Dp and Dt.
 At=L0
2
 My current model presented rectifies these
assumptions that over-simply the theory behind the
effective thermal conductivity of unsaturated soils.

25. Further Research
 Further research should be performed on the function
for the real contact area of particles.
 The current model is very intuitive and flexible, which
allows for an array of soil types to be tested. This may
highlight correlations with various parameters in the
model such as REV, CN etc.
 The model presents further insight into heat flow
parameters of porous media and establishes a solid
foundation to further develop this fractal model in other
areas of engineering.