Estimating effective thermal conductivity of unsaturated bentonites with consideration of coupled thermo-hydro-mechanical effects

International Journal of Heat and Mass Transfer 72 (2014) 656–667

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Estimating effective thermal conductivity of unsaturated bentonites with consideration of coupled thermo-hydro-mechanical effects Yifeng Chen ⇑, Song Zhou, Ran Hu, Chuangbing Zhou State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering (Ministry of Education), Wuhan University, Wuhan 430072, China

a r t i c l e

i n f o

Article history: Received 16 October 2013 Received in revised form 20 January 2014 Accepted 21 January 2014 Available online 18 February 2014 Keywords: Thermal conductivity Compacted bentonite Buffer material THM coupling

a b s t r a c t The thermal conductivity of compacted bentonites is one of the key properties for performance assessment in design of the engineered barrier systems. This study presented an effective thermal conductivity model for compacted bentonites with a consideration of the coupled thermo-hydro-mechanical (THM) phenomena involved in the barrier systems. The model was developed based on the structural connections of pores and the solid phase and the series–parallel arrangements of multiphase fluids (water and air-vapor mixture) in the pore system, and was represented as a function of porosity, the degree of saturation, temperature, and pressures of the fluid phases. The proposed model was comprehensively verified by five sets of laboratory data on the MX-80, FEBEX, Kunigel-V1 and GMZ01 compacted bentonite materials with different dry densities, water content and mineralogical composition, and good agreements were obtained between the model predictions and the laboratory measurements. It is demonstrated that the model predictions strictly fall within the Wiener bounds, and mostly obey the Hashin–Shtrikman bounds over wide ranges of porosity and saturation. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Bentonite has been widely considered as an engineered buffer material for deep geological disposal of radioactive wastes because of its low permeability, good swelling capacity, chemical buffering capability, etc. The thermal conductivity of compacted bentonite is one of the most important properties that influence the transfer of decay heat from waste containers to the host rocks surrounding a geological repository and the temperature distribution within the barrier system [1,2]. The resulting distribution of temperature, as a state variable, significantly affects the mechanical and hydraulic behaviors of the barrier system through other temperature-related state variables and the physical mechanisms such as thermal expansion, phase exchange, and thermal osmosis. The thermal conductivity of the porous buffer materials, therefore, plays a substantial role in the design and performance/safety assessment of the repositories and in understanding the coupled thermo-hydromechanical (THM) phenomena in the barrier system [1,2]. As the coupling processes proceed, this thermal property does not remain a constant, but varies with the properties such as porosity, pore ⇑ Corresponding author at: State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China. Tel./fax: +86 27 68774295. E-mail address: [email protected] (Y. Chen). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.01.053 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

structure and water content as well as the state variables such as temperature and pressures of the three-phase media. Development of thermal conductivity models dependent on porosity, the degree of saturation, temperature and pressure is critical to modeling the coupled heat transfer, multiphase flow and stress/deformation occurring in the barrier system, given that the thermal conductivity measurements [3–5] are always limited and only available in certain well-controlled laboratory conditions [6]. The thermal conductivity models of two phase materials [7–10] have been well developed, but these models are only pplicable to the buffer materials at completely dry or saturated state. Empirical relationships were proposed for unsaturated soils [11– 16], but with limitations of expressing the thermal conductivity as a single function of the degree of saturation. The idea of random phase distribution using the flux concept was applied in models such as Maxwell’s model [17], Fricke’s relation [18], Bruggeman’s theory [19] and Brailsford and Major’s model [20], for predicting the thermal conductivity of two-phase or three-phase materials, but with limited applications to buffer materials. Micromechanical models [21,22] were developed for estimation of the effective thermal conductivity of porous/cracked media in partially saturated conditions, by virtue of homogenization techniques and the analytical solution of the single ellipsoidal inclusion problem. These models have potential advantages of taking into account the

Y. Chen et al. / International Journal of Heat and Mass Transfer 72 (2014) 656–667

microstructure of soils, the heterogeneity and spatial distribution of solid constituents, the interaction between matrix and inclusions and the saturation degree of the porous phase, but their applicability to coupled THM modeling requires further investigations. Statistics models [23–25] were also proposed to examine the effects of aggregate size on soil thermal conductivity, and a comprehensive review can be found in [26] for analytical models for fluid-saturated porous media with periodic structures. The effective thermal conductivity structural models [6,10,27] have been a widely-applied alternative for assembling the contributions of the respective thermal conductivities from the constituents of soils in solid, liquid and gas phases. Compared to the empirical or statistical models, the models of this kind offer a far better understanding about the heat transfer in soils, with improved accuracy, better characterization of soil texture and good extrapolation capability to wide ranges of porosity and saturation [28]; under coupled THM conditions, the effects of mineral chemistry and mechanical effects (such as heat transformation by mechanical work) may also be included [6]. In this type of models, the series and parallel models define the upper and lower bounds (known as the Wiener bounds [29]) for all other estimates. The Hashin–Shtrikman bounds [30] are much narrower than the Wiener bounds, but apply only to isotropic mixtures. Some simple structural models based on the series and parallel models have been developed with different averaging techniques [31,32], but poor performance may be resulted for unsaturated soils because of poor consideration on the soil structures. Tarnawski and Leong [28] extended the two-phase model by Woodside and Messmer [8] for the effective thermal conductivity of unsaturated soils, but with a large set of parameters to be determined. Tong et al. [6] developed an effective thermal conductivity model for unsaturated bentonite that considers the effects of porosity, saturation, temperature and pressure, but this model was only verified against a limited set of laboratory measurements on the MX-80 bentonite and poor performance may exhibit when it applies to other data sets. Microstructure examinations [33–38] demonstrated that within the microstructure of compacted bentonites, there exist three types of pore spaces (i.e. the intra-particle pores, the inter-particle or intra-aggregate pores and the inter-aggregate pores), and the pore size distribution (PSD) curves of compacted bentonites are typically bi-modal [33,34]. Compaction or mechanical loading leads to variation of the PSD curves and porosity of bentonites, but the smaller pore size inside aggregates may not be affected by the magnitude of the compaction pressure [33]. Water saturation results in swelling of the clay materials, which further complicates the microstructure of bentonites. Relating the heat conductivity properties to the microstructure of compacted bentonites, however, remains a difficult task, and the phenomenological structural modeling approach is still commonly applied to make the problem tractable. For this consideration, this study presents a series–parallel structural model for predicting the effective thermal conductivity of unsaturated compacted bentonites, with potential applications for reliable numerical modeling of coupled THM processes involved in the engineered barrier system of radioactive waste repositories [1,2,6]. The proposed model is a function of porosity, the degree of saturation, temperature and pressures of fluid phases, and is comprehensively verified by five sets of laboratory data on the MX-80, FEBEX, Kunigel-V1 and GMZ01 compacted bentonite materials. 2. Model development 2.1. Effective thermal conductivity of dry soils Compacted bentonite materials are commonly characterized as a three-phase mixture, which contains pores and voids surrounded

657

by solid particles and filled with fluids (liquid water and/or gas mixture). In a given representative elementary volume (REV), the volume fraction occupied by phase a (a = s for solid, w for liquid water and g for gas) is given by /a, with /s = 1  /, /w = /Sr and /g = /(1  Sr), where / is porosity and Sr is the degree of saturation. The thermal conductivity of each phase is assumed to be isotropic and denoted by ka (a = s, w, g). Among the structural models, the series and parallel models define respectively the upper and lower bounds (i.e. Wiener bounds) of the effective thermal conductivity in which all components of the mixture are arranged in series or parallel connection. At completely dry state, the Wiener bounds are represented by

" kLdry ¼

X /a a

kUdry ¼

ka

#1 ¼

 1 1/ / þ ks kg

X /a ka ¼ /kg þ ð1  /Þks

ð1aÞ

ð1bÞ

a

where kUdry and kLdry are the upper and lower bounds of thermal conductivity at dry state, respectively. The effective thermal conductivity of dry materials must lie in between the Wiener bounds and could be modeled as a weighted combination of the series and parallel structures [6]

kdry ¼ ð1  gÞkLdry þ gkUdry e

ð2Þ

where kdry is the effective thermal conductivity at dry state, g a e weighting parameter depending solely on the pore structure of soils, which could be represented as a function of /, with 0 < g(/) < 1. Substituting Eq. (1) into Eq. (2) yields

kdry ¼ gð1  /Þks þ g/kg þ ð1  gÞkLdry e

ð3Þ

The first term on the right hand side of Eq. (3) is attributed to the pore structure of soils and the thermal conductivity of solid phase, and different from the second and third terms, is independent of fluid distribution in the pores, as depicted in Fig. 1. This implies that the porosity / can be conceptually decomposed into two components: one component is in series connected to the solid phase (denoted by /1 = (1  g)/) and the remaining component in parallel connected to the effective structure (denoted by /2 = g/). This concept serves as a basis for development of an effective thermal conductivity model for the three-phase mixtures. 2.2. A new series–parallel effective thermal conductivity model For unsaturated soils, the influence of multiphase fluids on the effective thermal conductivity of soils is mainly reproduced by modifying the second and third terms in Eq. (3). At room temperature and under atmospheric pressure, the common values of ks, kw, and kg are, respectively, about 2.0–4.0, 0.6 and 0.026 W/m K. From the effective structure shown in Fig. 1 and the remarkable differences among the thermal conductivities of the three phases, it can be inferred that the effective thermal conductivity of the three-phase mixture is less dependent on the part of pores /2 in parallel connected to the solid phase. This is not the case, however, for the remaining part of pores /1 in series connected to the solid phase, since as water invades into the pores in the imbibition process, the replacement of the low conductivity gas with the higher conductivity water results in a significant decrease of the contact resistance and a rapid increase of the effective thermal conductivity, and vice versa in the dehydration process. Therefore, the effective thermal conductivity of a three phase mixture depends on how the multiphase fluids (water and gas) distribute in the pores and how they are connected to the solid phase. In this study, we consider the following four cases:

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Fig. 1. Sketch of effective structure of two phase soil.

(1) Case 1: Suppose that water and gas in the pores can be arranged in parallel connection, and the second part of pores /2 is preferentially filled with water (or equivalently, /1 is less hydrophilic or larger in size and preferentially filled with gas). If Sr 6 g, then /2 is partially saturated with water while /1 is saturated with gas, as shown in Fig. 2a. In this case, the effective thermal conductivity kg;p reads e L kg;p e ¼ gð1  /Þks þ /ðg  Sr Þkg þ /Sr kw þ ð1  gÞkdry ðif Sr 6 gÞ

ð4aÞ

In Eq. (4), the superscript g indicates that /1 is preferentially filled with gas. (2) Case 2: Suppose that water and gas in the pores can be arranged in parallel connection, and the first part of pores /1 is more hydrophilic or narrower in size and preferentially filled with water. If Sr 6 1  g, then /1 is partially saturated with water while /2 is saturated with gas, as shown in Fig. 3a. In this case, the effective thermal conductivity kw;p can be represented as e

If Sr > g, then /2 is saturated with water while /1 is partially saturated, as shown in Fig. 2b. The corresponding effective thermal conductivity reads

kw;p ¼ gð1  /Þks þ g/kg e  1 1/ /ð1  gÞ þ ð1  gÞ þ ðif Sr 6 1  gÞ ks Sr kw þ ð1  g  Sr Þkg

kg;p e ¼ gð1  /Þks þ g/kw  1 1/ /ð1  gÞ þ ð1  gÞ þ ks ð1  Sr Þkg þ ðSr  gÞkw

If Sr > 1  g, then /1 is saturated with water while /2 is partially saturated, as shown in Fig. 3b. The corresponding effective thermal conductivity can be represented as

ðif Sr > gÞ

ð5aÞ

ð4bÞ

Fig. 2. Sketch of effective structure of three phase soil where water and gas in the pores is arranged in parallel connection and /1 is less hydrophilic or larger in size.

Y. Chen et al. / International Journal of Heat and Mass Transfer 72 (2014) 656–667

kw;p ¼ gð1  /Þks þ /ðSr þ g  1Þkw þ /ð1  Sr Þkg e þ ð1  g kLsat

ÞkLsat

ðif Sr > 1  gÞ

ð5bÞ

þ ð1  gÞkLsat

ks kw /ks þð1/Þkw

where ¼ represents the lower-bound effective thermal conductivity of saturated soils when the solid and liquid phases are arranged in series connection. In Eq. (5), the superscript w indicates /1 is preferentially filled with water. (3) Case 3: Suppose that water and gas in the pores can be arranged in series connection, and the second part of pores /2 is preferentially filled with water. If Sr 6 g, then /2 is partially saturated with water while /1 is saturated with gas. In this case, the effective thermal conductivity kg;s e is given as

kg;s e ¼ gð1  /Þks þ

g2 /kg kw ðg  Sr Þkw þ Sr kg

þ ð1  gÞkLdry ðif Sr 6 gÞ

ð6aÞ

If Sr > g, then /1 is partially saturated while /2 is saturated with water. The corresponding effective thermal conductivity is given as

kg;s e ¼ gð1  /Þks þ g/kw  1 1  / /ð1  Sr Þ /ðSr  gÞ þ ð1  gÞ þ þ ks ð1  gÞkg ð1  gÞkw

ðif Sr > gÞ ð6bÞ

(4) Case 4: Suppose that water and gas in the pores can be arranged in series connection, and the first part of pores /1 is preferentially filled with water. If Sr 6 1  g, then /1 is partially saturated with water while /2 is saturated with gas. In this case, the effective thermal conductivity kw;s reads e

kw;s ¼ gð1  /Þks þ g/kg þ ð1  gÞ e þ

/ð1  g  Sr Þ ð1  gÞkg

kw;s ¼ gð1  /Þks þ e

1



1/ /Sr þ ks ð1  gÞkw

g2 /kw kg ðSr þ g  1Þkg þ Sr kw ðif Sr > 1  gÞ

659

ð7bÞ

For given values of porosity and the degree of saturation, keg;s and kw;p generally yield the lowest and largest predictions in the above e four structural models, respectively, while keg;p and kew;s take values between them. The effective thermal conductivity of soils is influenced by various factors such as the geometry of pores, mineral composition and the wettability of particle surfaces, etc., and may not be accurately represented by any one of the above four structural models. To make the model applicable to different types of soils or a soil with different dry density and water content, a linear combination of Eqs. (4)–(7) is suggested: w;p w;s ke ¼ f1 kg;p þ f3 kg;s e þ f2 ke e þ f 4 ke

ð8Þ

where fi (i = 1, 2, 3, 4) are weighting coefficients, with 0 6 fi 6 1 and

Rfi = 1. It is to be noted that the weighting coefficients could be represented as functions of the degree of saturation, but verification studies against laboratory data (see Section 4) showed that constant values of the weighting coefficients yield acceptable predictive performance of the model, with a smaller set of parameters to be determined. Except for the arithmetic mean, the geometric mean or the harmonic mean of the four structural models could be used to construct Eq. (8), but a comparison study on the FEBEX and GMZ01 data sets implies that the geometric mean or the harmonic mean does not produce better performance than the arithmetic mean. Therefore, Eq. (8) with constant weighting coefficients is adopted in this study, for its simplicity, clear physical meaning and comparable predictive performance.

3. Determination of model parameters

ðif Sr 6 1  gÞ

ð7aÞ

If Sr > 1  g, then /1 is saturated with water while /2 is partially saturated. The corresponding effective thermal conductivity reads

Besides the weighting coefficients fi (i = 1, 2, 3, 4), the proposed model, Eq. (8), contains the following six parameters: ks, kw, kg, g, / and Sr, in which / and Sr are state variables for unsaturated soils.

Fig. 3. Sketch of effective structure of three phase soil where water and gas in the pores is arranged in parallel connection and /1 is more hydrophilic or narrower in size.

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0:135qd þ 64:7 qs  0:947qd

Identification of the parameters is critical for application of the model.

kdry ¼ e

3.1. Thermal conductivities of solid, liquid and gas phases

where qd is the dry density of soils, and qs the density of soil particles. Calibration studies showed that for bentonite, Johansen’s expression may result in unsatisfied prediction because of its insufficient consideration of the thermal conductivities of constituent components. To overcome this limitation, one may adopt the following geometric mean model

The thermal conductivity of solid phase of compacted bentonite ks depends on the mineral constituents and their spatial distribution, and may be a function of temperature. A simple method was suggested by Carson and Sekhon [39] for determination of the thermal conductivity of the solid phase of heterogeneous materials. A widely used estimate of ks is the geometric mean of the thermal conductivities of quartz kq and other minerals ko: 1 11

ks ¼ kq ko

ð9Þ

where 1 is the volume fraction of quartz. The most common values of kq and ko suggested for bentonites are, respectively, 7.7 and 2.0 W/m K [11,28,40]. The dependence of kq on temperature and the value of ko can be found in [24,41,42]. The thermal conductivities of water and gas, kw and kg, depend on their state variables of respective phases, i.e. temperature and pressure (or density). Particularly, the gas phase could be regarded as an ideal mixture gas of dry air and water vapor, with the two constituents completely miscible and occupying the same volume fraction. Therefore, kw and kg could be written in the following general forms:

kw ¼ kw ðT; pw Þ;

kg ¼ kg ðT; pg ; pv Þ

ð10Þ

where T is temperature, pw, pg and pv are pressures of water, gas mixture and vapor, respectively. In this study, the thermal conductivity equations of water and gas mixture given in [6] are adopted. 3.2. Parameter g The parameter g depends on the pore structure of soils, and is a function of porosity /. By Eq. (3), if the thermal conductivity values of dry soils of various porosities are available, then g could be determined by the following expression:

gð/Þ ¼

L kdry e ðks ; kg ; /Þ  kdry

ð11Þ

ð1  /Þks þ /kg  kLdry

Otherwise, if no thermal conductivity data at dry state are available or the laboratory data are limited, Johansen’s empirical expression [11] is commonly used

Experiment data (MX-80) Experiment data (Kunigel-V1) Johansen's model Geometric mean model

0.8

0.6

0.4

0.2 0.3

0.4 Porosity

0.5

0.6

ð13Þ

Eq. (13) seems to be a better alternative because it has rather good predictive capability for porosity at the range between 0 and 1, and strictly lies in between the Weiner bounds or even the Hashin–Shtrikman bounds. Fig. 4 plots the thermal conductivity values of the Kunigel-V1 (Ould-Lahoucine’s data [40]) and MX80 (Madsen’s data [43]) bentonites at dry state. A comparison between the experimental data and the predictions by the geometric mean model, Eq. (13), and Johansen’s model, Eq. (12), implies that the geometric mean model yields much better predictions of the thermal conductivity under dry conditions, while Johansen’s model significantly underestimates the thermal conductivity, especially at low values of porosity. Eqs. (11) and (13) can be approximated by the following simple expression

gð/Þ ¼ ea/þb

ð14Þ

where a and b are fitting parameters. Take China’s GMZ01 bentonite as an example. Under atmospheric pressure and at room temperature (T = 25 °C), the basic model parameters are as follows: ks = 2.342 W/m K, kw = 0.611 W/m K, kg = 0.026 W/m K and qs = 2.66 g/cm3. The values of g estimated by Eqs. (12) and (13) are respectively plotted in Fig. 5, which shows lower g values (and resultant smaller effective thermal conductivities) at low porosity by Johansen’s expression compared to those by the geometric mean model. Similar results are also observed for other bentonites, such as the MX-80 bentonite, and the geometric mean model, Eq. (13), yields better predictions for compacted bentonite materials. Also plotted in Fig. 5 is the best-fitting curve of Eq. (14) for the geometric mean model, which yields the parameters of a = 3.382 and b = 0.063.

b

1.0

0.2

kdry ¼ k/g k1/ e s

Predicted thermal conductivity (W/m K)

Thermal conductivity (W/m K)

a

ð12Þ

1.0 Johansen's model Geometric mean model

0.8

+20% +10% -10% -20%

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Measured thermal conductivity ( W/m K)

Fig. 4. Validation of the geometric mean model for dry bentonites with Madsen’s data on MX-80 and Ould-Lahoucine’s data on Kunigel-V1.

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Y. Chen et al. / International Journal of Heat and Mass Transfer 72 (2014) 656–667

respectively. To ensure the predictive accuracy of the model, the experimental data set should be selected with good reliability and representativeness. The latter requires that the experimental data are uniformly distributed in the full ranges of porosity and the degree of saturation.

1.0

0.8 Geometric mean model Johansen's model Best-fitting model

0.6

4. Model validation

0.4

4.1. Validation of the proposed model

0.2

0.0 0.0

0.2

0.4 0.6 Porosity

0.8

1.0

Fig. 5. Comparison of the values of parameter g determined by Eqs. (12) and (13).

3.3. Weighting coefficients fi When laboratory data of thermal conductivity of a compacted bentonite are available, the weighting coefficients fi (i = 1, 2, 3, 4) could be determined by a constraint optimization problem as follows:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X min RMSE ¼ 1n ðkexp;k  ke;k Þ2 s:t:

P

9 > =

k¼1

fi ¼ 1 and 0 6 fi 6 1 ði ¼ 1; 2; 3; 4Þ

> ;

ð15Þ

The proposed model was comprehensively verified against the laboratory experimental data published in the works of Madsen [43] and Tang et al. [44] on MX-80 bentonite, Ould-Lahoucine et al. [40] on Kunigel-V1 bentonite, Villar [45] on FEBEX bentonite, and Liu et al. [46] on GMZ01 bentonite. Among them, the MX-80, Kunigel-V1 and GMZ bentonites are Na-bentonites whereas the FEBEX bentonite is a Ca-bentonite [47]. Table 1 shows the mineralogical composition of the bentonites, with the montmorillonite content varying in 47–92% and the quartz content in 0.6–15%, respectively. The thermal conductivity testing methods and the physical properties of the clay samples, such as dry density, bulk density, water content, the degree of saturation and porosity, are also listed in Table 1. If the degree of saturation Sr and porosity / are not given in the literature, / is calculated with bulk density qb, gravimetric b or with water content w and specific gravity Gs by / ¼ 1  Gs q qðwþ1Þ w

dry density qd and specific gravity Gs by / ¼ 1  Gqs qd , where qw is w

where RMSE is the root-mean-squared error, n the number of experimental data, kexp,k and ke,k the kth thermal conductivity values of laboratory measurements and the corresponding model predictions,

the density of water. Sr is then calculated by Sr = Gsw(/1  1). Note that if the density of the bound water of the clay materials is not considered, the calculated saturation may be over 100% at saturated

Table 1 Main physical properties of the bentonites. Bentonite

MX-80 [44]

MX-80 [43]

FEBEX [45]

Kunigel-V1 [40]

GMZ01 [46,47]

Quartz content (wt.%) Montmorillonite content (wt.%) Other main mineral composition Specific gravity Gs Available properties

3 92

15 76

2 92

0.6 47–48

11.7 75.4

– 2.76 Dry density, water content 1.45–1.84 –

Feldspar, plagioclase, mica, carbonate 2.76 Bulk density, water content – 1.45–2.35

Plagioclase, K-feldspar, calcite, cristobalite 2.70 Dry density, water content, saturation 1.49–1.76 –

Chalcedony, plagioclase, analcime, dolomite, calcite 2.79 Bulk density, saturation, porosity – 1.35–2.17

Cristobalite, feldspar, kaolinite 2.66 Dry density, water content 0.79–2.04 –

7.0–17.9

0, 7, 14

0.1–27.4

0–34.4

0.92–37.3

27.6–85.8

0–91.4

0–100.0

0–94.7

1.1–100.0

0.34–0.48 Hot wire method

0.20–0.51 –

0.34–0.45 Hot wire method

0.3, 0.4, 0.5 Thermistor probe method

0.23–0.70 Hot wire method

Dry density qd (g/cm3) Bulk density qb (g/ cm3) Water content w (wt.%) Degree of saturation Sr (%) Porosity / Testing method

Table 2 Model parameters of the bentonites. MX-80 [44]

MX-80 [43]

FEBEX [45]

Kunigel-V1 [40]

GMZ01 [46]

Present model

Bentonite a b f1 f2 f3 f4

3.277 0.067 0.3073 0.4010 0.2917 0

3.422 0.061 0.5136 0.2209 0.2655 0

3.265 0.067 0 0.7934 0 0.2066

3.248 0.068 0.1419 0.8309 0 0.0272

3.382 0.063 0.3847 0.6153 0 0

Tong’s model

a b c d e

0.0732 0.7652 0.5648 1.9820 0.2513

0.0626 0.7497 0.5779 1.1130 0.9509

0.0751 0.7634 0.6566 1.1050 0.2224

0.0750 0.7700 0.8017 0.8149 0.1676

0.0668 0.7485 0.8571 1.2987 0.1237

Y. Chen et al. / International Journal of Heat and Mass Transfer 72 (2014) 656–667

state and it should be calibrated to guarantee Sr = 1 at completely saturated state. Given that the measurements were obtained at room temperature (T = 25 °C) and atmospheric pressure (pg = 101 kPa and pv  0), the following values of thermal conductivity of water and gas were taken in the validation: kw = 0.611 W/m K and kg = 0.026 W/m K. The thermal conductivity of solid phase ks was estimated by Eq. (9), with kq = 7.7 W/m K and ko = 2.0 W/m K. The parameter g was calculated by Eq. (14), and the parameters a and b were obtained by best fitting Eqs. (11) and (13), as listed in Table 2. Also listed in Table 2 are the weighting coefficients fi (i = 1, 2, 3, 4) for the bentonites, determined by solving Eq. (15).

Predicted thermal conductivity (W/m K)

a

2.0 1.8

+20% +10% -10% -20%

Tong's model Present model

1.6 1.4 1.2 1.0 0.8 0.6 0.4

water content = 0

0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Fig. 6 shows a comparison between the predicted and measured thermal conductivity values of the MX-80, FEBEX, Kunigel-V1 and GMZ bentonites, respectively. One observes from the plots that the model predictions are generally in good agreements with the experimental data. The model predictions perfectly fit Tang’s data on the MX-80 bentonite (Fig. 6b) and Ould-Lahoucine’s data on the Kunigel-V1 bentonite (Fig. 6d). As for Madsen’s data (Fig. 6a), Villar’s data (Fig. 6c) and Liu’s data (Fig. 6e) on the MX-80, FEBEX and GMZ01 bentonites, respectively, the prediction accuracy is also acceptable. Table 3 lists the prediction errors of the above four typical bentonites, showing that 14–57% of the predictions have a discrepancy lower than 5% and most (75–100%) of the predictions fall

b Predicted thermal conductivity (W/m K)

662

2.0 1.8 1.6

1.8

Tong's model Present model

+20% +10% -10%

1.6

-20%

1.4 1.2 1.0 0.8 0.6 0.4 0.2

water content < 2%

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Measured thermal conductivity (W/m K)

Predicted thermal conductivity (W/m K)

e

-20%

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Measured thermal conductivity (W/m K)

d Predicted thermal conductivity (W/m K)

Predicted thermal conductivity (W/m K)

2.0

+20% +10% -10%

1.4

Measured thermal conductivity (W/m K)

c

Tong's model Present model

2.0 1.8

Tong's model Present model

1.6

+20% +10% -10% -20%

1.4 1.2 1.0 0.8 0.6 0.4 water content = 0

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Measured thermal conductivity (W/m K)

2.0 1.8 1.6 1.4

Tong's model Present model

+20% +10% -10% -20%

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Measured thermal conductivity (W/m K)

Fig. 6. Predicted thermal conductivity by Eq. (8) versus measured thermal conductivity of (a) MX-80 (Madsen’s data), (b) MX-80 (Tang’s data), (c) FEBEX (Villar’s data), (d) Kunigel-V1 (Ould-Lahoucine’s data) and (e) GMZ01 (Liu’s data) bentonites.

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Y. Chen et al. / International Journal of Heat and Mass Transfer 72 (2014) 656–667 Table 3 Relative prediction errors of thermal conductivity of the bentonitesa. Bentonite No. of data points

MX-80 [43]

FEBEX [45]

Kunigel-V1 [40]

GMZ01 [46]

24

29

59

14

105

Percentage of data points with er 6 5%

Present model Tong’s model

54.17% 54.17%

13.79% 27.59%

18.64% 45.76%

57.14% 50.00%

29.52% 19.05%

Percentage of data points with er 6 10%

Present model Tong’s model

58.33% 58.33%

17.24% 31.03%

20.34% 47.46%

64.29% 57.14%

30.48% 20.00%

Percentage of data points with er 6 20%

Present model Tong’s model

100% 100%

89.66% 51.72%

76.27% 84.75%

92.86% 78.57%

75.27% 84.76%

er denotes the relative error at kth data point defined by |kexp,k  ke,k|/kexp,k  100%.

in between the 20% relative error lines. A larger discrepancy between the predicted and measured thermal conductivity values only occurs for the experimental data on FEBEX bentonite with water content lower than 1.9%. For practical use, however, this prediction error is of secondary importance, because the compacted bentonite blocks in a barrier system generally have much higher moisture content. Based on the experimental observations, Tang et al. [44] presented a linear correlation between the effective thermal conductivity of bentonites and the volume fraction of air as follows:

ke ¼ a/g þ ksat

ð16Þ

where a is the slope of the curve, ksat the thermal conductivity at saturated state, and /g the volume fraction of pore air in the three-phase mixture.

Thermal conductivity (W/m K)

a

b

1.6 Laboratory data (Tang's data) Laboratory data (Madsen's data) Model predictions (Tang's data) Model predictions (Madsen's data)

1.4 1.2 1.0

Linear correlation for Madsen's data = 2.34 +1.34

0.8 0.6

Fig. 7 plots the thermal conductivity values of the above four bentonites against the volume fraction of air, showing an approximately linear decrease of the thermal conductivity with increasing air volume fraction. The best-fitted values of a and ksat are listed in Table 4, with a high correlation coefficient R2 varying in between 0.87 and 0.98. Using the linear correlation, however, there is no guarantee that the predicted thermal conductivity values lie within the Wiener bounds, and are always positive and meaningful for increasing values of the air volume fraction, as evidenced by the plot in Fig. 7d where the predicted values of thermal conductivity become negative as the volume fraction of air, /g, takes values over 0.61 for the GMZ01 bentonite. It can be inferred from Fig. 7a–c and Table 4 that for the MX-80, FEBEX and Kunigel-V1 bentonites, similar phenomenon occurs as long as the volume fraction of air increases up to 0.57, 0.65, and 0.59, respectively, implying that

Linear correlation for Tang's data

0.4

Thermal conductivity (W/m K)

a

MX-80 [44]

0.1

0.2 0.3 Volume fraction of air

0.4

d

1.4

1.2 1.0 0.8 0.6 Linear correlation

0.4

1.0 0.8 0.6 0.4

0.1

Linear correlation

0.2

0.2 0.3 Volume fraction of air

0.4

0.5

1.8 1.6

Laboratory data Model predictions

1.2

0.0 0.0

Laboratory data Model predictions

0.0 0.0

0.5

Thermal conductivity (W/m K)

Thermal conductivity (W/m K)

c

1.4

0.2

0.2 0.0 0.0

1.6

Laboratory data Model predictions

1.4 1.2 1.0

Linear correlation

0.8 0.6 0.4 0.2

0.1

0.2 0.3 0.4 Volume fraction of air

0.5

0.6

0.0 0.0

0.1

0.2 0.3 0.4 0.5 Volume fraction of air

0.6

0.7

Fig. 7. Variation of the effective thermal conductivity of bentonites with the volume fraction of pore air: (a) MX-80; (b) FEBEX; (c) Kunigel-V1 and (d) GMZ01.

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Table 4 Parameters in Tang’s linear correlation. Bentonite

MX-80 [44]

MX-80 [43]

FEBEX [45]

Kunigel-V1 [40]

GMZ01 [46,47]

a (W/m K)

1.98 1.13 0.87

2.34 1.34 0.98

1.96 1.28 0.87

2.23 1.32 0.96

2.36 1.45 0.92

ksat (W/m K) R2

Tang’s linear correlation is only valid for /g >  ksat/a. The proposed model overcomes this limitation and well reproduces the relationship for various values of dry density and water content, as shown in Fig. 7.

Tong’s model, the parameters, g and g2, are approximated by the following empirical expressions:

4.2. Comparison with Tong’s model

where a–e are fitting parameters. For the above five data sets, the best fitting parameters of Tong’s model are listed in Table 2. Fig. 6 and Table 3 show that both models have similar performance for Tang’s data on the MX-80 bentonite and Liu’s data on the GMZ01 bentonite. But for Madsen’s data on the MX-80 bentonite and Ould-Lahoucine’s data on the Kunigel-V1 bentonite, the proposed model exhibits better performance, especially for the experimental data of low water content (or dry bentonite samples). For Villar’s data (Fig. 6c) on the FEBEX bentonite, if the fitting parameters given by Tong et al. [6] was used (a = 0.0644, b = 0.8613, c = 0.5943, d = 1.4207, and e = 0.02377), then the performance of Tong’s model is much worse than that of the proposed model. In this comparison study, the parameters involved in g and g2 were re-determined by best fitting the experimental data, as listed in Table 2. Fig. 6c shows that Tong’s model seems to have better performance for the experimental data of lower water content, while the proposed model has better predictive

A comparison is also made between the performance of the proposed model and that of Tong’s model [6]. Both models were established based on the series–parallel effective structure of soils and were intended to capture the influences of porosity, moisture content and temperature on the effective thermal conductivity of buffer materials. The major difference is that in Tong’s model, the effective thermal conductivity is estimated by the following linear combination of kL and kU:

ke ¼ ð1  g2 ÞkL þ g2 kU

ð17aÞ

where kL and kU are the thermal conductivities when the fluid phases in pores (water and gas) are, respectively, in series and parallel connected to the part of solid phase (1  g)(1  /) (see Fig. 1), and g2 a parameter dependent on pore structure and saturation. In

b

2.5 e

Effective thermal conductivity (W/m K)

Effective thermal conductivity (W/m K)

a

H-S upper bound H-S lower bound Wiener upper bound Wiener lower bound

2.0

1.5

1.0

0.5

0.0 0.0

g ¼ a/b ; g2 ¼ cSd/þe r

0.2

0.4

0.6

0.8

1.0

ð17bÞ

2.5 e H-S upper bound H-S lower bound Wiener upper bound Wiener lower bound

2.0

1.5

1.0

0.5

0.0 0.0

0.2

0.4

Effective thermal conductivity (W/m K)

c

0.6

0.8

1.0

Porosity

Porosity

2.5 e H-S upper bound H-S lower bound Wiener upper bound Wiener lower bound

2.0

1.5

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Porosity Fig. 8. Evolution of the effective thermal conductivity against porosity with various constant saturations: (a) Sr = 10%; (b) Sr = 50% and (c) Sr = 90%.

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Y. Chen et al. / International Journal of Heat and Mass Transfer 72 (2014) 656–667

2.5

2.0

H-S upper bound H-S lower bound

b

Wiener upper bound Wiener lower bound

e

1.5

1.0

0.5

0.0 0.0

0.2

0.4

0.6

2.5

Effective thermal conductivity (W/m K)

Effective thermal conductivity (W/m K)

a

0.8

1.0

2.0

e

1.5

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Saturation

Saturation

c 2.5 Effective thermal conductivity (W/m K)

Wiener upper bound Wiener lower bound

H-S upper bound H-S lower bound

2.0

Wiener upper bound Wiener lower bound

H-S upper bound H-S lower bound e

1.5

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Saturation Fig. 9. Evolution of the effective thermal conductivity against saturation with various constant porosities: (a) / = 30%; (b) / = 40% and (c) / = 50%.

2.5

Sr=50%

Sr=10%

Sr=60%

Sr=20%

Sr=70%

Sr=30%

2.0

Sr=80%

Sr=40%

Sr=90%

1.5

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Porosity

b Effective thermal conductivity (W/m K)

Effective thermal conductivity (W/m K)

a

2.5

2.0

=10% =50% =90%

=20% =60%

=30% =70%

=40% =80%

1.5

1.0

0.5

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Saturation

Fig. 10. Evolution of the effective thermal conductivity against (a) porosity with various constant saturations, and (b) saturation with various constant porosities.

accuracy for the experimental data with water content higher than 1.9%. 4.3. Responses of the proposed model with respect to porosity and saturation Without loss of generality, the GMZ01 bentonite was taken for discussion of the responses of the proposed model to variations of porosity and the degree of saturation. The material parameters are the same with those for GMZ01 bentontie given in Section 4.1. Figs. 8 and 9 plot, respectively, the behaviors of the effective thermal conductivity with porosity when the degree of saturation is fixed to 10%, 50%, and 90%, and with the degree of saturation under

different porosity values of 30%, 40%, and 50%. Also plotted in the figures are the Wiener bounds and the Hashin–Shtrikman bounds (interested readers may refer to e.g. [6] for the expressions of the bounds for three-phase mixtures). The results show that the model predictions strictly fall in between the Wiener bounds for any values of porosity and saturation. It is also demonstrated that even the compacted bentonite is not an isotropic material in nature, the predicted values of the effective thermal conductivity fall mostly within the Hashin–Shtrikman bounds, in wide ranges of / 2 [0, 0.98] and Sr 2 [0, 1]. Fig. 10 shows the responses of the proposed model with respect to porosity and the degree of saturation. As depicted in Fig. 10a, the effective thermal conductivity reduces with increasing porosity at

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a decreasing rate at any given degree of saturation. For small values of porosity (e.g. <5–10%), the effective thermal conductivity is mainly determined by the thermal conductivity of the solid phase, and the influence of water content is rather weak. For high values of porosity (e.g. >90–95%), on the other hand, the effective thermal conductivity is mainly determined by the thermal conductivities of water and gas, with the lowest values bounded by the thermal conductivity of gas when the water content tends to zero. From Fig. 10b, one observes that the effective thermal conductivity increases with increasing degree of saturation. For porosity values higher than 60%, the effective thermal conductivity increases almost linearly with the degree of saturation, while for porosity values lower than 50%, remarkable nonlinearity is presented in the curves. As a typical example, for / = 40% (which is a rather common value for buffer materials), the variation in the effective thermal conductivity with the degree of saturation exhibits different increasing rates for Sr = 0–25%, 25–75% and 75–100%, in which a larger increasing rate appears for Sr = 25–75%. It can be inferred from Eqs. (4)–(7)that the predicted thermal conductivity against the degree of saturation may not be smooth at Sr = g or Sr = 1  g (where g is dependent on porosity), as a result of the consideration of the connections of water and gas in pore spaces and the assumption of different pore size associated with /1 and /2. For instance, the parameters listed in Table 2 indicate that for / = 10%, the thermal conductivity versus saturation curve loses smoothness at Sr = g  0.7, as depicted in Fig. 10b, but this effect does not significantly influence the performance of the model.

5. Conclusions A new effective thermal conductivity model was proposed for compacted bentonites in which, as buffer materials of radioactive waste repositories, coupled thermo-hydro-mechanical processes are involved. The model was developed based on the structural connections of pores and the solid phase and the series–parallel arrangements of multiphase fluids (water and air-vapor mixture) in the pore system. In the model, the effective thermal conductivity of bentonites is represented as a function of porosity, the degree of saturation, temperature, and pressures of the fluid phases. Therefore, the influences of deformation, heat transfer and multiphase flow in the engineered barrier system on the heat conductivity property are considered. The proposed model was comprehensively verified by five sets of laboratory data on the MX-80, FEBEX, Kunigel-V1 and GMZ01 compacted bentonite materials with different dry densities, water content and mineralogical composition, and good agreements were obtained between the model predictions and the laboratory measurements. The model predictions strictly fall within the Wiener bounds, and mostly obey the Hashin–Shtrikman bounds in wide ranges of porosity and saturation, e.g. / 2 [0, 0.98] and Sr 2 [0, 1] for the GMZ01 bentonite. The performance of the proposed model was compared with that of Tong’s model [6], which similarly considered the variations of thermal conductivity in a coupled THM system with multiphase flow, but with different considerations of the structural arrangements of the solid, liquid and gas phases. The comparison studies show that the proposed model exhibits slightly better performance than Tong’s model. As a major shortcoming, the proposed model is phenomenological in nature, and is not related to the microstructure of compacted bentonites. The pore size distribution and its change, as well as the swelling behavior induced by water saturation, during the coupled THM processes are expected to have a non-negligible influence on the heat transport behaviors of the barrier system. This issue could be potentially addressed with a multi-scale micromechanical modeling approach, and we will leave it for our future study.

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