Rev I.Gavriliev

Melnikov Permafrost Institute,Siberian Branch of the Russian Academy of Sciences,Yakutsk 677010,Russia

1 Introduction

Soils are the product of rock destruction by physical and chemical weathering over long geological time periods.Schematically,this can be represented as follows.First,joints develop in a mass of rock by the action of climatic factors.After many repeated cycles of freezing and thawing of water in fractures the rock is broken into blocks,normally angular in shape.Further weathering results in smaller fragments which remain in place or move downhill to form residual soils and slope deposits.Part of the destructed material is transported by water and deposited at a distance,sometime far away from the parent rock.Particles are abraded during the transportation process,so alluvial deposits normally consist of rounded grains.Clay particles are largely produced by the chemical weathering of mineral formations and have complex configurations with rough,often corroded (pitted) surfaces.The bulk of clay particles have irregular forms,such as needle,starred,clustered,and lath like.

An opposite picture is observed when sediments are converted into rock.Lithification begins with compaction of sediments due to the weight of the overlying material and the effect of tectonic forces associated with crust folding.During different stages of metamorphism (diagenesis,epigenesist and catagenesis),the rocks undergo changes in original composition and texture.At high pressures and temperatures,melting or dissolution of minerals occurs at grain-to-grain contacts,and the excess material is bulged out or crystallizes in regions of lower stress in the rock.Thus,mineral particles of sedimentary rocks undergo some kind of plastic deformation through geologic time,gradually filling the entire space.Particles bind together at the contacts ("the contact spot") and rigid crystal bindings develop between the particles.

A model for estimating the thermal conductivity of soils and rocks of different origins should therefore take into account the changes in particle shape over the entire range of porosity from 0 to 1 in order to consider the entire cycle of transitions of geological materials.

2 Theoretical model accounting for microstructural changes

A large number of theoretical models and methods were developed for estimating the thermal conductivity of various particulate materials (Chudnovsky,1962;Ivanov,1962;Vasiliev and Tanaeva,1971;Dulnev and Zarichnyak,1974;Farouki,1986).However,most of the methods do not address the structural changes of the model and their validity is limited to a narrow range of material’s density.In permafrost investigations,it is essential that properties of snow,soils and rocks are studied in relation to the history of sediment formation through geologic time.Therefore,a universal theoretical model with changing particle shapes was proposed by the present author to describe the processes of sedimentary rock formation from sediments and snow compaction with further glacierization which account for diagenetic and post-diagenetic changes in microstructure,as well as the processes of rock weathering and soil formation (Gavriliev,1992;Gavrilyev,1996;Gavrilyev,2011).The scheme of the model (Figure 1) presents the solid component in a cubic cell by three intersecting ellipsoids of revolution.

Figure 1 Particle shapes in the soil thermal conductivity model at different semi-axes ratios of ellipsoids δ=a/R.1:faceted (δ >1);2:spherical (δ=1);3:worn (δ <1);4:cruciate (δ <1)

In this scheme,depending on the semi-axes ratio of the ellipsoidsa/R,the porosity of the system varies from 0 to 1 and the particle attains a variety of shapes,such as cubical,faceted,spherical,worn,and cruciate.This logically represents real change in particle shape through the sedimentary history,i.e.,the key requirement to the model – adequate representation of the real system – is met.In this scheme,the particles always maintain contact with each other and the system remains stable and isotropic.The coordinate number is constant and equal to six;the relation between thermal conductivity and porosity is realized by changing the particle shape at various size ratios of the ellipsoids of revolution.Ata/R≥1,a contact spot appears automatically in the model,which represents rigid bonding between the particles that provides hard,monolithic rock structure (Gavriliev,1996).

All calculations are made in terms of the parameterδ=a/R,which is a unique function of the porositym2(dry densityγd):

whereλmodis the resulting thermal conductivity of the model andφscis the correction for heat transfer across the contact spot,W/(m·K).

The thermal conductivity of the model,λad,is given by the following equations.Atδ≤1,there is:

atδ≥ 1,there is:

whereδ = a/R;

The correction factorφscis given by

wherercis the radius of the contact spot between the particles.

It is assumed in equation(5)that the spot contact between particles is formed of the same material as the particle by its flattening at high pressure or by its squeezing (solution and crystallization) due to selective growth of cement in sandstones (quartz cement grows on quartz particles and feldspar on feldspar particles).In a general case,however,the contact spot may consist of a foreign material resulting,for example,from precipitation of salts from solution at the particle contacts.In this case,the correction factorφscis given by

whereK2=1-λ3/λ1;λ1,λ2andλ3are the thermal conductivities of the solid,medium and contact spot(contact cement),respectively.

The relative size of the contact spot is expressed in terms of the system’s porosity as:

The soil porositym2or the volume fraction of the mineral particlem1is a unique function of the parameterδand is given by the following equations,atδ≤1,there is:

atδ≥1,there is:

The increase in the volume fraction of the solids due to the contact spot is expressed by

The dry density of the soil is

whereρsis the solids unit weight.

Figure 2 shows the relationship between soil porositym2and parameterδ=a/R.

Figure 2 Porosity of the model vs parameter δ=a/R.1:without contact spot;2:with contact spot

The above equations can be used to calculate the thermal conductivity of soils and sedimentary rocks in the saturated frozen and unfrozen states,as well as in the air-dry state in relation to porositym2and thermal conductivityλ1of the solid particles (a two-component system).The predictions obtained are presented as nomograms in figure 3.It should be noted that in this case,porositym2refers to the entire volume fraction of the soil or rock which is completely filled either with ice,water,or air.This porosity is related to volume fractionmsand dry densityγdby

The proposed model assumes that the material consists of mineral particles of the same composition.However,naturally occurring soils always contain particles of various compositions and they can be treated in modeling as multi-component heterogeneous systems with a statistical particle distribution.

Figure 3 Nomograms for calculating the thermal conductivity of soils and rocks in dry (a),saturated unfrozen (b) and frozen (c)states in terms of total porosity m2 and solids thermal conductivity λ1 (W/(m·K)).1:0.5;2:1.0;3:1.5;4:2.0;5:2.5;6:3.0;7:3.5;8:4.0;9:4.5;10:5.0;11:6.0;12:7.0

3 Accounting for mineral composition of soils and rocks

Mineralogy of solid particles can be taken into account in the proposed prediction method by three ways.In the first case,the statistical-probabilistic distribution of cubic elementary cells of the universal model with components of different mineralogical compositions is estimated based on the generalized conductivity theory for heterogeneous systems developed by Odelevsky (1951) and Zarichnyak and Novikov (1978).The second case is when the fill prevails over the solid skeleton,and one can proceed from the matrix scheme and analyze perturbation to the homogeneous temperature field in the matrix by inclusions of different composition.The third way is to estimate the mean value of thermal conductivity for the complete mineralogical composition of soils or rocks and then to substitute it into parameters of the solid component of the elementary cell in the universal model.

3.1 Statistical probability scheme

Odelevsky (1951),studying the behavior of dielectric inclusions of spherical shape in a homogeneous electrical field,obtained a relationship for the generalized conductivity (including thermal conductivity) of a multiphase statistical mixture with non-elongated particles in the form:

whereλiandmiare respectively the thermal conductivity and volume concentration of the particles of theith component;λis the mixture-averaged thermal conductivity;0≤mi≤1.

Finding the value ofλfrom equation(13)is associated with the solution of a higher-order equation.The degree of an equation depends on the number of components in a mixture.When there are three or more components,the use of equation(13)becomes problematic.

In the scheme of Zarichnyak and Novikov (1978),the multi-component system is represented by cubic elements arranged in random (probabilistic) fashion.A two-layer set of heterogeneous cubes in the direction of heat flow is separated in the volume of the system;in two other orthogonal directions the geometrical model is unlimited.The thermal conductivity of such a system is determined as a sum of parallel and serial connections of conductors when cube pairs of homogeneous (nii,njj) and heterogeneous (nij) components are formed.The probability (W) of the formation of such pairs is uniquely determined by the volume concentration (m) of the components:Wii=nii/n=mimi=mi2;Wjj=njj/n=mj2;Wij=ninj/n= mimj(heremi=ni/n;mj=nj/n;niandnjare the number of particles of theith andjth components,respectively;andnis the total number of particles).

Thus,the following expression was derived for the mechanical mixture with any number of non-interacting components (Zarichnyak and Novikov,1978):

or in a more compact form

whereiandjare numbers of the component combinations under the sum sign.

Equation(15)is well suited for estimating the effect of the mineral composition of soils and rocks on their thermal conductivity.To do this,we arrange particles in a cubic array and divide the entire space around particles into uniform cubes.Each cube is a two-component system:particle + fill (air,water,and ice).Particles in the cubes are similar in shape,but differ in composition and combine into thesnumber of components.Since the cubic cells form a poreless structure,the porosity of one cell reflects the porosity of the whole systemms,which is expressed by the semi-axes ratio of the ellipsoids of revolutiona/R.

Thermal conductivities of cubical components in equation(15)can be specified with account for contact heat transfer,using the model proposed by the author,in terms of thermal conductivities of particles (λ1) and fill (λ2),and their volume fractionsm1andm2by equations(1)–(12)(Gavril’ev,1992).

3.2 Matrix scheme

At the same time,a simpler (though less correct)method can be proposed to directly account for the heterogeneous composition of particles in thermal conductivity calculations,which is based on the analysis of perturbation to the temperature field of a medium by inclusions of different composition.For generality,we consider an ellipsoidal particle shape,since with the change in the ratio of semi-axes the particles transform into other figures,such as a sphere,plate,or cylinder.

We have a matrix medium (fill) of thermal conductivityλ2with heterogeneous inclusions (particles)of ellipsoidal shape embedded in a sufficiently regular arrangement.Letm1be the volume fraction of all particles,of which themipart is characterized by conductivityλi.From this medium we cut an ellipsoid with semi-axesa＞b＞c,assuming that it hasnheterogeneous inclusionsn=∑ni(niis the number of particles of theith material) and place it into the homogeneous medium of conductivityλ2with a linear temperature distribution in the OZ axis direction.Then,assuming the semi-minor axis of the ellipsoid matched with the OZ axis,a disturbance of the steady-state temperature distribution will be observed,which accordings to the field theory (Ovchinnikov,1971),has the form

whereλis the effective thermal conductivity of the ellipsoid in the OZ axis direction;C(0) andC(λ) are the integrals of the form

f(t)=(a2+t)(b2+t)(с2+t);tis the integration variable,andGis the temperature gradient in the homogeneous matrix medium.

If the same ellipsoid is completely filled with inclusions of theith composition,the temperature perturbation will be

whereλiis the thermal conductivity of the inclusions of theith material.

However,because the inclusions only occupy part of the ellipsoid volumem1=∑mi,their contribution to the total temperature perturbation ΔTis estimated proportionally to their volume fractionmi/(miΔTi′).

Hence,we write:

Considering equations(16)and(19),we obtain

Let us introduce the notation:2Kf=abcС(0) which is the particle shape factor.From equation(10),we finally obtain the following relationship for the effective thermal conductivity of soils and rocks which takes into account the mineralogical composition of particles

Let us examine the particle shape factorKf.C(0) is the integral of the form (Ovchinnikov,1971)

E(ψ,p) is the elliptic integral of the second kind,is the amplitude andis the modulus of the integral.

The elliptic integralE(ψ,p) is tabulated,and the shape factor of inclusions can be readily found from the ratio of the particle dimensionsa,b,andc.For practical purposes,calculations can be limited to the more simple case of ellipsoids of revolution.Then,the integralC(0) is expressed in terms of elementary functions (Carslaw and Jaeger,1959).Let us consider two examples.

1) The particles have a shape of an oblate ellipsoid of revolution (a=b>c).Then,along the semi-axes we have

2) The inclusions have a shape of a prolate ellipsoid of revolution (b2=c

Figure 4 shows graphically the shape factorsKffor oblate and prolate ellipsoids of revolution calculated with equations(23)–(26)in relation to the ratio of the ellipse’s semi-minor (c) and semi-major (a) axes at different directions.

Figure 4 Shape factor Kf of soil particles in the form of oblate(1 and 1') and prolate (2 and 2') ellipsoids of revolution versus parameter c/a for different directions.1 and 2 refer to along the axis of revolution;1' and 2' refer to perpendicular to the axis of revolution

It is seen from figure 4 that for spherical inclusions(с/а=1),Kf=1/3;for the plane inclusions (с/a=0):perpendicular to the heat flowKf⊥=1 and parallel to the flowKf||=0.In the case of cylindrical inclusions (с/а=0)we have:Kf||=0 (along the flow) andKf⊥=1/2 (perpendicular to the flow).

3.3 Estimating mean thermal conductivity of solid particles

Besides the methods presented above,the mineralogical composition of soils and rocks can also be taken into account by estimating the mean value of thermal conductivity of the particles of various compositions in the solid skeleton of the model.When the mineral composition of soil or rock is dominated by one mineral,it is proposed to use equation(22)using the thermal conductivity of predominant mineralλpminstead ofλ2and replacing the parametermibymi'·m1(heremi'=Vi'/V1refers to the fraction of thei-th mineral in the solid particles volumeV1,i.e.,soil mineral composition;m1=V1/Vrefers to the fraction of the solid particles in the total soil volumeV).This is based on the thought experiment,in which all mineral particles are grouped together to form a single monolithic system.

As a rough approximation,the mean value of thermal conductivity of mineral particles for the universal model can be found by averaging the thermal conductivities of minerals in series and parallel to heat flow using the following equation (Gavriliev,1989):

whereλiandmi'are the thermal conductivity and volume fraction of thei-th group of minerals in the soil or rock solids.

One can also use the geometric mean method,in which the mean thermal conductivity of minerals is represented as the product of the thermal conductivity of mineralsλibeing raised to the power of their volumetric proportionmi'in the total volume of the minerals equal to 1 and is expressed by Woodside and Messmer (1961) and Sasset al.(1971):

Equation(29)is recommended when the thermal conductivity of each mineral does not contrast by more than one order of magnitude.Data from Horai(1971) indicate that most rock-forming minerals fall within this range.The geometric mean method was widely used by C?te and Konrad (2005) to calculate the thermal conductivity of soils and construction materials.

4 Discussion of results

1) Now let us examine the consequences of equation(22).Assume that soil particles consist of the same material,i.e.,a single-component skeleton is considered (i=1).Then,from equation(22)we obtain

This equation was first obtained by the author in relation to angular and rounded gravels (Gavriliev,1986).This paper also specified the values ofKffor gravel inclusions having the shape of:

(1) oblate ellipsoids of revolution

(2) prolate ellipsoids of revolution

2) In the case of spherical inclusions (Kf=1/3),equation(28)leads to the equations of Maxwell (1873)and Odelevsky (1951).The latter examined cubic inclusions and proved the validity of equation(28)over the entire range of medium’s porosity.The same is true for equation(22).

The volumetric heat capacity of the soil or rock is calculated by the additive equation

whereсdis specific heat of the dry soil or rock;γdis dry density of the soil or rock;Wis gravimetric total moisture content expressed as a fraction;сwis specific heat of water (liquid water and ice),4.2×103J/(kg·K)at positive temperatures and 2.1×103J/(kg·K) at negative temperatures.

The specific heat capacity of the dry soil or rock is computed,taking into account its mineral composition,by

whereсiis specific heat of thei-th mineral in the soil or rock;piis mass fraction of thei-th mineral in the soil or rock.

The specific heat capacities for minerals can be found readily in reference books (e.g.,Birchet al.,1942;Clark,1966).

The thermal diffusivity is given by the relation:

The methodology presented above allows one to calculate the thermal conductivity of soils and rocks in dry and saturated frozen and unfrozen states,i.e.,in the two-component state.The effect of mineral composition can be taken into account by estimating the statistical-probabilistic distribution of cubic elementary cells with heterogeneous particles of the solid skeleton or by averaging the thermal conductivities of minerals in the solid component of the model.To illustrate estimation of the mean thermal conductivity of the solid skeleton of the universal model for different mineralogical compositions,we compared the results of thermal conductivity computations using the described methods for three rock types:quartzite,gabbro and granite.In these calculations,we used the mineralogical composition and thermal conductivity of rock-forming minerals as given in C?te and Konrad(2005) (Table 1).Computed results are presented in table 2.As is seen,the computed and measured values for gabbro and crushed granite agree quite well.The values for quartzite show a greater difference.

Table 1 Mineralogical composition of rocks and thermal conductivity of minerals (C?te and Konrad,2005)

Table 2 Computed thermal conductivity of rocks

First,the matrix method gives a higher thermal conductivity than the other prediction methods.Second,the measured thermal conductivity is considerably lower than all the computed values.For the prediction methods we can note the following.In theoretical terms,the matrix scheme approach is preferable.When one of the components (minerals) prevails,the theoretical scheme in which the homogeneous temperature field is perturbed by foreign inclusions works perfectly,with no assumptions.The matter is different when it comes to whether the thermal conductivity value is correct for quartz,the commonest rock-forming mineral.Quartz is a complex anisotropic mineral.Its thermal conductivity varies greatly depending on the optic axis orientation relative to the direction of heat flow.From data compiled by Birchet al.(1942) and Clark (1966),the average thermal conductivity of quartz parallel and perpendicular to the optic axis is:=11.0 W/(m·K) and=6.40 W/(m·K).It is not clear what average thermal conductivity should be used in computations for quartz.There is also a question concerning the measured thermal conductivity for quartzite.Theλvalue is strongly dependent on sampling location.Data of numerous authors for quartzite samples from different locations in Japan,Germany,South Africa (Transvaal)and USA (South Dakota) show that the thermal conductivity of quartzite varies over a wide range,from 3.64 to 6.54 W/(m·K),with most common values between 5.15 and 6.54 W/(m·K) (Birchet al.,1942;Clark,1966).

5 Conclusion

In permafrost investigations,it is essential that properties of soils and rocks be studied in relation to the history of sediment formation through geologic time.In order to account for the diagenetic and post-diagenetic alterations in sediment microstructure during lithification of unconsolidated sediments into sedimentary rocks or conversely during rock weathering and erosion,a universal theoretical model with changing particle shapes has been proposed.The model consists of three intersecting ellipsoids of revolution in a cubic cell.Depending on the semi-axes ratio of the ellipsoidsa/R,the porosity of the system varies from 0 to 1 and the particle attains a variety of shapes,such as cubical,faceted,spherical,worn,and cruciate.This logically represents the real changes in particle shape through the sedimentary history,i.e.,the key requirement to the model – adequate representation of the real system – is met.All calculations are made in terms of the parametera/R,which is a unique function of porosity (or dry density).

This paper has presented a methodological scheme for estimating the thermal properties of soils and rocks,which takes into account their origin and mineralogical composition.Three approaches have been developed.The first is founded on structural modeling of contact heat interaction of particles with a fill and estimates the statistically probable distribution of particles in the volume of the medium.The second approach is based on the analysis of perturbation of the temperature field of the matrix medium by heterogeneous inclusions of ellipsoidal shape.The third approach involves finding the mean thermal conductivity of the solid skeleton in the universal model at different composition of rock-forming minerals.

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