Effective thermal conductivity of fluid-saturated rocks

Engineering Geology 135–136 (2012) 24–39

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Effective thermal conductivity of fluid-saturated rocks Experiment and modeling M.G. Alishaev, I.M. Abdulagatov ⁎, Z.Z. Abdulagatova Geothermal Research Institute of the Dagestan Scientific Center of the Russian Academy of Sciences, 367003 Makhachkala, Shamilya Str. 39-A, Dagestan, Russia

a r t i c l e

i n f o

Article history: Received 14 June 2011 Received in revised form 1 February 2012 Accepted 5 March 2012 Available online 16 March 2012 Keywords: Density Heat capacity Heat transfer Porous rocks Thermal conductivity Sandstone

a b s t r a c t Effective thermal conductivity (ETC) of dry, gas-, oil-, and water-saturated rocks with various porosities has been measured over a temperature range from 273 K to 523 K at atmospheric pressure with a steady-state guarded parallel-plate apparatus. The expanded uncertainty of thermal conductivity and temperature measurements at the 95% confidence level with a coverage factor of k = 2 were estimated to be 4% and 30 mK, respectively. This uncertainty in ETC measurement does not include the uncertainty due to contact thermal resistance and radiative conductivity. The temperature coefficients, (∂ ln λ/∂ T)P, for fluid-saturated rocks were calculated by using the measured ETC. We interpreted measured ETC data for fluid-saturated rocks using various theoretical models in order to check their accuracy, predictive capability, and applicability. The effect of saturating fluids, structure (size, shape, and distribution of the pores), porosity, and mineralogical composition on temperature and porosity dependences of the ETC of fluid-saturated rocks was discussed. A new simple equation for ETC of fluid-saturated rocks which takes into account structure of porous media has been proposed. Using the Hofmiester model and measured thermal conductivities of dry rock materials, the values of thermodynamic properties (density, thermal expansion coefficient, enthalpy, and heat capacity) were predicted. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Effective thermal conductivity (ETC) studies of the fluid-saturated porous materials are useful in a number of applications such as petroleum and natural gas geology; utilization of hydrothermal energy and underground thermal energy storage; applications to geothermal problems, hydro-geological studies, drilling and drilling fluids, well logging, fluid mechanics in porous media and multi-phase flow. All of these processes are requiring accurate knowledge of the ETC and other high-temperature and high-pressure thermal behavior of dry and fluid saturated porous rock materials. ETC measurements are also a research interest. Unfortunately, ETC of fluid-saturated porous rocks cannot be accurately predicted theoretically, due to their complicated physical and chemical structures (irregularity of the microstructure). There are a number of models for the prediction of ETC which is based on theoretical studies (see review in Abdulagatova et al., 2009). Any model requires reliable ETC measurements for validation. All heat transfer problems require for their solution knowledge of the ETC of porous materials. Physically it is very clear that ETC of such system is a function of thermal conductivities of solid phase, λS, fluid phase, λf, and conductivity of the ⁎ Corresponding author at present address: Thermophysical Properties Division, National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80303, USA. Tel.: + 1 303 497 4027; fax: + 1 303 497 5224. E-mail address: [email protected] (I.M. Abdulagatov). 0013-7952/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2012.03.001

interferences (contact region, grain–grain, λSS, and grain–liquid, λSf, interface). The conductivities of the solid and fluid phases are well known, while the nature and physical mechanism of the contact ther−1 −1 mal resistances, λSS (solid–solid) and λSf (solid–fluid) interferences is still less understandable. Basically, the measurements of the ETC of rock materials were performed by the following methods: conventional contact methods: (1) need-probe (uncertainty 2–3%); (2) divided-bar (uncertainty 4%); (3) guarded hot plate (4%); hot wire (uncertainty 4–5%); and contact-free method: (5) optical scanning, laser-flash analysis (LFA) (uncertainty 2%). A comprehensive literature review of the experimental ETC measurement techniques of dry and fluid-saturated porous materials at high temperatures and high pressures is reported in our previous publication Abdulagatova et al. (2009) (see also, Somerton, 1992; Hammerschmidt and Sabuga, 2000; Aichlmayr and Kulacki, 2006; Hofmeister, 2007; Hofmeister et al., 2007). The main purposes of this study are: (1) to provide an accurate (with estimate uncertainty of 4%) experimental ETC data for geological porous materials (rocks) with various porosities at temperatures from 275 K to 523 K and at atmospheric pressure using a guarded parallel-plate method, which has been used previously for accurate measurements on other solids and rocks (dry and fluid-saturated) materials (Abdulagatov et al., 1998, 2000, 2002, 2006; Abdulagatova et al., 2009, 2010); (2) to study the effect of temperature on the ETC behavior of dry and fluid-saturated porous rocks; (3) to test the validity (applicability), accuracy, and predictive capability of the various

M.G. Alishaev et al. / Engineering Geology 135–136 (2012) 24–39

25

Fig. 1. Thermal conductivity cell (a) and compensation heater (b).

theoretical and semi-empirical models for the ETC of rocks as a function of temperature and porosity; (4) to develop simple model to predict the ETC of fluid-saturated rocks materials; (5) to study of the

relation between ETC and other thermodynamic properties such as density, heat capacity, enthalpy etc. 2. Experimental

1.60 Fused quartz

1.55

λ, W·m-1·K-1

P=0.1 MPa

1.50

1.45

1.40

1.35

1.30 270

300

330

360

390

420

T, K Fig. 2. Comparison of measured thermal conductivities of fused quartz with reported data. ● — Abdulagatov et al. (2000); ○ — Touloukian et al. (1989) (recommended); × — Devyatkov et al. (1960); □ — Sugawara (1968); ▲ — Camirand (2004); ■ — Clauser and Huenges (1995); (- · - · - · -) — Kaye and Higgins (1926); (············) — Ratcliffe (1959);▼ — Xia Cheng et al. (1990), Δ — Incropera and Dewitt (1981) (recommended); ♦ — Horai and Susaki (1989).

ETC of dry and fluid (gas-, oil-, and water)-saturated rocks has been measured by a guarded parallel-plate apparatus. It is an absolute, steady-state measurement device with an operational temperature range of 270 K to 600 K and hydrostatic pressures up to 1000 MPa. The method (apparatus, procedure of measurements, and detailed uncertainty assessment) has been described fully in our previous several publications (Abdulagatov et al., 1998, 2000, 2002, 2006; Abdulagatova et al., 2009, 2010, 2011), thus only a brief review will be given here. Schematic diagrams of the construction of thermal conductivity cell and the compensation heater are shown in Fig. 1. Thermal conductivity apparatus consists of a high-pressure chamber, a thermal-conductivity measuring cell, an air thermostat, a high precision temperature regulator, and a high-pressure liquid and gas compressors. The temperature in the air thermostat was controlled automatically to within ±5 mK. In this method, thermal conductivity is obtained from simultaneous measurements of the steady-state heat flux Q and temperature gradient in the sample placed between the heating and cooling plates. The good thermal contact between the heater and the sample was assured using thermal interface materials (wetting agent such as vaseline or glycerin). To reduce the effect of contact resistance (imperfect thermal contact between the sample and the adjacent bronze disk) the sample and heater surfaces were

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Table 1 Measured values of ETC of dry, gas-, water-, and oil-saturated rocks at atmospheric pressure as a function of temperature. T, K

Sandstone-1

Limestone

Amphibolites

Granulate

Pyroxene-granulate

273 323 373 423

2.01 1.93 1.88 1.82

1.94 1.78 1.65 1.55

3.52 3.02 2.63 2.35

2.06 1.96 1.85 1.77

2.40 2.42 2.44 2.46

T, K

Siltstone

Dolomite

Sandstone-2 (argon-saturated)

Sandstone-2 (oil-saturated)

Sandstone-2 (water-saturated)

275 323 373 423 473 523

1.93 2.06 2.20 2.34 2.48 2.62

3.14 3.14 3.14 3.14 3.14 3.14

0.62 0.64 0.67 0.70 – –

1.64 1.69 1.70 1.73 – –

2.25 2.26 2.27 2.30 – –

T, K

Sandstone-3 (air-saturated)

T,K

Sandstone-3 (water-saturated)

T, K

Sandstone-3 (oil-saturated)

273 323 373 423 473 523 – – – – – – – –

2.71 2.63 2.56 2.49 2.42 2.34 – – – – – – – –

281 289 297 305 325 347 368 391 398 417 454 472 498 518

3.81 3.81 3.78 3.77 3.75 3.72 3.66 3.65 3.62 3.57 3.53 3.49 3.44 3.41

288 318 344 356 388 428 470 500 515 – – – – –

3.18 3.15 3.11 3.09 3.02 2.96 2.91 2.87 2.83 – – – – –

T, K

Andesite

273 323 373 423

2.90 2.93 2.95 2.99

polished flat to within 0.05 mm and smooth. The spring was also used to create contact pressure (pressure of 1 to 2 MPa was applied axially) to improve the thermal contact between the sample and heater. Two thermocouples were embedded in the center of the inner surface of the bronze disk (see Figure 1b). The heater was located between

these thermocouples. Two other thermocouples were soldered to the body of the heater. The temperature difference (temperature gradient) and temperature of the chamber were measured with four copper–constantan thermocouples (see Figure 1a, b).

4.0 3.5 3.6

2.5

λ (W·m-1·K-1)

λ (mW·m-1·K-1)

3.0

2.0 1.5

3.1

2.6

1.0 2.1 0.5 0.0 263

313

363

413

463

513

T (K)

1.6 273

323

373

423

473

523

T (K) Fig. 3. Measured ETC of dry, gas-, oil-, and water-saturated rock materials as a function of temperature. □ — Limestone; ■ — Sandstone-1; ○ — Granulate; ● — Amphibolites;▲ — Dolomite; Δ — Siltstone;× — Pyroxene-Granulate; ♦ — Sandstone-3; ◊ — Sandstone-2; ∇ — Andesite.

Fig. 4. Temperature dependence of the measured ETC of different type fluid-saturated sandstones. ● — Sandstone-3 (water-saturated); ▲ — Sandstone-3 (oil-saturated); ○ — Sandstone-2 (water-saturated); Δ — Sandstone-2 (oil-saturated).

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0.88 Water-saturated

0.74

Oil-saturated 0.81

0.69

λETC/λS

0.74

0.67

0.64

0.60 0.59 0.53

0.46 0.025

0.030

0.035

0.040

0.045

0.54 0.075

0.095

λoil/λS

0.115

0.135

λw/λS

Fig. 5. Normalized ETC of rock-oil and rock-water systems (λETC/λS) as a function of normalized thermal conductivity of saturating oil (λoil/λS) and water (λW/λS)

The thermal conductivity λ of the specimen was deduced from the relation λ¼S

1

h1

Q−Q los ΔT 1 þ

S2 h2

ΔT 2

;

ð1Þ

where Q = Q1 + Q2 is the heat flow transferred from the heater to the upper and lower specimens; Q1 and Q2 are the heat flows transferred by conduction through the lower and upper specimens, respectively; Qlos is the heat losses through the lateral surface of the samples; S1 and S2 are the cross-sectional areas of the specimens that heat flows through; h1 and h2 are the height of the samples; and ΔT1 and ΔT2 are the temperature differences across the samples thickness. The thermal conductivity was obtained from the measured quantities: Q, Qlos, ΔT1, ΔT2, S1, S2, h1, and h2. The heat flow Q from the heater was distributed between the two samples Q1 and Q2. The values of Q were corrected by a specimens lateral loss factor Qlos. The lateral heat-losses were measured by using samples of pyrex glass of well known conductivity, Qlos = λpyrΔT ln2πh . Two thermocouples were ðd=DÞ embedded in the center of the inner surface of the bronze disk. From the uncertainty of the measured quantities and the corrections,

Table 2 Temperature coefficient of the ETC, βT, of dry and fluid saturated rocks at 0.1 MPa. Water-saturated (Sandstone-3)

Oil-saturated (Sandstone-3)

Dry (Sandstone-3)

T, K

βT × 103, K− 1

T, K

βT × 103, K− 1

T, K

βT × 103, K− 1

281 289 297 305 325 347 368 391 398 417 454 472 498 518

− 0.4984 − 0.4964 − 0.4945 − 0.4925 − 0.4877 − 0.4826 − 0.4777 − 0.4725 − 0.4710 − 0.4668 − 0.4589 − 0.4551 − 0.4498 − 0.4458

288 318 344 356 388 428 470 500 515

− 0.5494 − 0.5405 − 0.5330 − 0.5296 − 0.5208 − 0.5102 − 0.4995 − 0.4921 − 0.4885

273 323 373 423 473 523

− 0.6206 − 0.6020 − 0.5844 − 0.5678 − 0.5521 − 0.5373

βT/10− 4 K− 1 P/MPa

Sandstone

Limestone

Amphibolites

Granulate

Pyroxenegranulate

0.1

6.2–6.8

14–17

23–34

9–11

1.6–1.7

-0.63

-0.60 Gas-saturated

Table 3 Values of coefficients A and B in Eq. (2) for ETC.

βTx103 (K-1)

-0.57

-0.54 Oil- saturated

-0.51 Water- saturated

-0.48

-0.45 275

315

355

395

435

475

515

T (K) Fig. 6. Derived temperature coefficient of the ETC, βT, of gas, oil-, and water-saturated sandstone-3 (porosity ϕ = 13%) as a function of temperature.

Rocks

A

B × 103

Sandstone-1 Limestone Amphibolites Granulate Pyroxene-granulate Siltstone Dolomite Sandstone-2 (argon-saturated) Sandstone-2 (oil-saturated) Sandstone-2 (water-saturated) Sandstone-3 (air) Sandstone-3 (water-saturated) Sandstone-3 (oil-saturated) Andesite

0.406 0.279 0.025 0.338 0.435 0.661 0.318 1.965 0.661 0.462 0.306 0.225 0.263 0.363

0.33982 0.87108 0.94901 0.53819 − 0.06737 − 0.54202 0.00000 − 1.26503 − 0.19813 − 0.06156 0.22866 0.13015 0.17311 − 0.06643

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λ (W·m-1·K-1)

3.3

2.0 1.9

3.0

Amphibolites

Limstone

1.8 2.7 1.7 2.4 2.1 273

1.6

323

373

423

473

1.5 270

523

305

340

T (K)

2.1

λ (W·m-1·K-1)

375

410

T (K)

Granulate

2.0 1.9 1.8 1.7 270

300

330

360

390

420

T (K) Fig. 7. Comparison the present ETC data with the values predicted by correlation by Vosteen and Schellschmidt (2003).

conductivity measurements were made with standard (reference) material (fused quartz) using the present apparatus. Fused quartz has been recommended as a standard material for the test and calibration of the thermal conductivity apparatus (see for example, Devyatkov et al., 1960). The thermal conductivity of fused quartz has been used previously by many authors to calibrate apparatus for measuring ETC of rock specimens using various versions of the contact methods. Excellent agreement within 1.0–2.5% was found between the present data and the majority of the reported values for the reference sample (fused quartz) by other authors (see Figure 2). Unfortunately, to our knowledge, there are no reported thermal conductivity data

the total combined expanded uncertainty in the thermal conductivity measurement at the 95% confidence level with a coverage factor of k = 2 less than 4%. The reproducibility of the measurements is about 1.0%. This value of the uncertainty does not include the uncertainty due to contact resistance and radiative heat transfer. Therefore, the uncertainty of the ETC data obtained with the method is probably higher than 4%. According to Hofmeister et al. (2007) contact resistance with heaters and thermocouples, and possibly among constituent grains, leads to systematic and substantial underestimation of the lattice thermal conductivity by 20%. To check and confirm the validity of the method and procedure of the measurements, the thermal

λ (W·m-1·K-1)

3.8 3.7

b

3.13

2

8

3

3.6

4

δλ

5

3.5

7

3.06 =9 %

max

6

6

7

3.4 3.3 275

3.20

a

1

2

2.99

3 4

5

1

295

315

335

355

2.92 280

375

310

340

T (K) 2.8

λ (W·m-1·K-1)

370

400

430

T (K)

c

2.6 7 3

2.4 1

2.2 270

320

370

4

2

420

470

6 5

520

T (K) Fig. 8. (a, b, c). Measured and predicted values of ETC of water- (a), oil- (b), and gas- (c) saturated Sandstone-3 as a function of temperature at atmospheric pressure. (a): 1 — Zimmerman (1989); 2 — Hsu et al. (1994, 1995); 3 — Kunii and Smith (1960); 4 — Hadley (1986); 5 — Zehner and Schlünder (1970); 6 — Keller et al. (1999); 7 — Buntebarth and Rueff (1987). (b): 1 — Chapman et al. (1984); 2 — Tikhomirov (1968); 3 — Sass et al. (1971); 4 — Funnell et al. (1996); 5 — Eq. (3); 6 — Anand (1971) and Anand et al. (1973); 7 — Asaad (1955); 8 — Zimmerman (1989). (c): 4 — this work Eq. (2); 1 — Tikhomirov (1968); 2 — Sekiguchi (1984); 6 — Funnell et al. (1996); 7 — Rzhevsky and Novik (1971); 5 — Fricke (1924); 3 — Sass et al. (1971).

M.G. Alishaev et al. / Engineering Geology 135–136 (2012) 24–39

29

22 31

Granulate

Limstone

a=0.1

19 26

a=0.1

a=0.2

16

α x 105 (K-1)

21

a=0.2 13

16

a=0.3

11

10 a=0.4

a=0.4

7

6

1 270

a=0.3

4

320

370

420

470

520

1 270

330

390

T (K)

450

510

570

T (K)

Fig. 9. Thermal expansion coefficients for limestone and granulate derived from the present ETC measurements as a function of temperature at atmospheric pressure.

for fused quartz obtained using the contact-free methods to compare with the present data. This excellent agreement for fused quartz demonstrates the reliability and accuracy of the present measurements for porous rocks and correct operation of the instrument. 2.1. Materials descriptions 2.1.1. Physical and mineralogical characteristics of the samples and locations Sandstone-1 Aktash, Dagestan, Russia, borehole #1, 2977 m, porosity ϕ = 5%, density 2180 kg·m − 3 weakly cemented, gray color, moderate grained, and slightly aerated; Sandstone-2 Buinaks, Dagestan, Russia, open porosity ϕ = 16.2%, density 2700 kg·m − 3; Sandstone-3 Solonchak, Dagestan, Russia, borehole #34, weakly cemented, weakly carbonated, gray color, moderate grained, about 80% to 90% forming material basically has crystalline structure, open and interconnected

pores with random orientation, ϕ = 13%, depth 3941 m density 2180 kg·m − 3; Limestone Soltangasha, Dagestan, Russia, borehole #96, 201 m, ϕ = 5%, density 2380 kg·m − 3; Amphibolites Kola ultra-deep borehole, 10,000 m, ϕ = 1%, density 2610 kg·m − 3; Granulate Saxonian Granulate Mountains, Germany, quartz 38%, plagioclase 9%, K-feldspar 47%, biotite 1%, 4% granite, mixture of crystalline and amorphous structures, ϕ = 1%, density 2060 kg·m − 3; Pyroxene-granulate Saxonian Granulate Mountains, Germany, ϕ=1.2%, granite 6%, clinopyroxene 39%, plagioclase 34%, non-transparent minerals 11%, orthopyroxene 9%, ambibolb 1%, basically amorphous structure, density 3200 kg·m− 3; Siltstone Dmitrievskoe, from oil-gas field, borehole #D44, Dagestan, Russia, 4570–4574 m, ϕ = 1–2%; Dolomite Dmitrievskoe, from oil-gas field, borehole D44, Dagestan, Russia, 4247–4248 m; ϕ = 1–2%; 2070 Granulate

Limstone

2380

2060 a=0.4

2350

a=0.3

2320

ρ (kg·m-3)

a=0.3

2050

2040 a=0.2

2290

2030

2260 a=0.1

2230

2200 273

a=0.2

2020 a=0.1

2010

323

373

423

T (K)

473

523

2000 270

a=0.05

320

370

420

470

520

T (K)

Fig. 10. Density of limestone and granulate derived from the present ETC measurements as a function of temperature at atmospheric pressure.

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M.G. Alishaev et al. / Engineering Geology 135–136 (2012) 24–39

3.0 4.8 Limstone 4.2

(dCP/dP)T x105 (kJ·kg-1·K-1·MPa-1

Granulate

a=0.1

a=0.05

2.5

3.6 2.0 3.0 a=0.2 1.5

2.4 1.8

a=0.1

1.0

1.2 0.5 0.6 0.0 270

320

370

420

470

520

0.0 270

330

390

T (K)

450

510

570

T (K)

Fig. 11. Derived values of pressure derivative of the heat capacity,

Andesite Saatlinskii ultra deep borehole, Dagestan, Russia, quartzite andesite-basalt, 6240 m, ϕ = 1.0%, density 2540 kg·m − 3. 3. Results and discussion The results of ETC measurements for dry, gas-, oil-, and watersaturated rocks with various porosities are presented in Table 1 and shown in Figs. 3 and 4 as a function of temperature at atmospheric pressure. Temperature dependence was measured from 273 K to 523 K and at atmospheric pressure. As one can see from Fig. 3, the temperature dependence of the ETC for various types of rocks is different. Fig. 5 shows the plot of (λeff/λS) versus of (λoil/λS) and (λW/λS), where λoil, λW, and λS are the thermal conductivity of saturated fluid (oil and water) and solid matrix, respectively. This figure demonstrates the effect of conductivity of the saturated fluid on the ETC of solid– fluid system. As one can see, the plots of (λeff/λS) versus of (λoil/λS) and (λW/λS) are almost linear. The temperature changes of ETC (see Figures






∂C P , ∂P T

as a function of temperature.

3 and 4) are nearly linear and decreasing or increasing with temperature increase depending on the type of rock. For example, for Sandstone-1, Sandstone-3, Limestone, Amphibolite, and Granulate, the ETC monotonically decreases with temperature, while for Sandstone-2, Siltstone, Pyroxene-granulate, and Andesite, the ETC monotonically increases with temperature with a different rate. For dolomite the ETC is almost independent of temperature. For amorphous materials thermal conductivity is found to increase with increasing temperatures, while for crystalline rocks the ETC decreases with temperature. More complicated behavior of ETC is found for rocks with mixture of amorphous and crystalline structures. Pyroxene-granulite, for example, is a mixed crystalline and amorphous structure. Since ETC for pyroxene-granulite increase with temperature (see Table 1 and Figure 3); therefore, in this sample the amorphous structure dominates crystalline components. This can be readily explained on the bases of mineralogical contents of the rocks and their structure. However, almost all rocks are composed of mixtures of crystalline and amorphous structures,

0.450

0.52

a=0.1

Limstone

Granulate

0.445

(dH/dP)T x103 (m3·kg-1)

a=0.05 a=0.2

0.51 a=0.1

0.440

a=0.2

0.435

a=0.3

0.50

a=0.3

0.430

a=0.4

0.49 0.425 a=0.4

0.48 270

320

370

420

470

520

0.420 270

335

Fig. 12. Derived values of pressure derivative of the enthalpy,

400

465

530

T (K)

T (K)


∂H , ∂P T

as a function of temperature.

595

M.G. Alishaev et al. / Engineering Geology 135–136 (2012) 24–39

31

0.055 0.047 Limstone

Granulate

(CP-CV) (dp/dT)-1 (cm3·g-1)

0.050 a=0.05

0.042 0.045

a=0.1 0.037

0.040

a=0.1 a=0.2 0.032

0.035

0.027

0.030

0.025 270

320

370

420

470

520

0.022 270

330

390

T (K)

450

510

T (K)


−1 ∂P Fig. 13. Derived values of ðC P −C V Þ ∂T , as a function of temperature. T

therefore, the ETC behavior of rocks is more complicated (see below). An increase of ETC with increasing temperature was found also by several authors (Vulis and Putseluiko, 1956; Sugawara and Yoshizawa, 1961; Roy et al., 1981; Pribnow et al., 2000). Results of ETC measurements from Somerton (1958), Khan (1964), and Anand (1971) for various type rocks showed that in general, ETC decreases linearly with increasing temperature. In order to quantitatively estimate the effect of temperature on ETC of air-, water-, and
 ∂λ oil-saturated rocks the temperature coefficient of ETC, βT ¼ λ1eff ∂Teff , P

was calculated using the present experimental ETC data. The temperature in the range from 288 K to 520 K affected the ETC of oil-saturated sandstone by up to 12%. The derived values of βT as a function of temperature are shown in Fig. 6 and given in Table 2. As one can see from Table 2, the values of βT changed within (0.48–0.60) × 10− 3 K − 1, in the temperature range from 288 K to 520 K. The derived values of βTalmost linearly decrease with increasing temperature. For limestone, amphibolites, granulate, and Pyroxene-granulate the derived

values of temperature coefficient are within (14–17), (23–34), (9–11), and (1.6–1.7 ) × 10− 4 K − 1, respectively. Theoretically, lattice conductivity tends to be inversely proportional to temperature (λlat ∝ T− 1 for three-phonon mode, Peierls, 1956), while radiative (heat transfer by phonons) conductivity tends to be proportional to the cube of temperature (λrad ∝ T 3) (Clauser, 1988), if it neglects the temperature dependency of absorbance. For four-phonon processes and T > θ (Debye temperature) λlat ∝ (BT + B1T 2)− 1 (Klemens, 1969) and λlat ∝ (BT) − 1 + (B1T 2)− 1 by Ziman (1962). Because the mean free path of the phonons cannot become less than the unit cell, the law λlat ∝ T − 1 is no longer valid at high temperatures (T > θ) and λlat = (B/T)[(2/3)(B1/T)1/2 + (1/3)(T/B1)] (Roufosse and Klemens, 1974). The relationship λlat ∝ T − 1 is valid only for structurally perfect isotropic single crystals. However, many rocks are composed of mixtures of highly disordered crystals of different compositions, therefore, the thermal conductivity of rocks tends to decrease more slowly than λlat ∝ T − 1 (λlat ∝ T − n, where 0 b n b 1) and may tend to actually increase, in some case, with increasing temperature (Somerton, 1992).

6.5 4.0 5.5

T = 297 K

P=0.1 MPa

P=0.1 MPa

3.5

λ (W·m-1·K-1)

λ (W·m-1·K-1)

3.2

T=473 K

4.5

5 4 3

2.5

2

β=1

2.4

β=1.5

1.6 β=1.67

1

1.5

0.5 0.0

β=3.2

0.8

0.2

0.4

φ

0.6

0.8

1.0

Fig. 14. Maximum and minimum of the ETC of water-saturated sandstone as a function of porosity predicted from various models. 1 — Geometric mean (λmin); 2 — Walsh and Decker (1966) (λmin); 3 — Hashin and Shtrikman (1962) (λmin); 4 — Walsh and Decker (1966) (λmax); 5 — Arithmetic mean (λmax).

0.0 0.0

0.2

0.4

φ

0.6

0.8

1.0

Fig. 15. ETC of gas-saturated sandstone as a function of porosity predicted from Fricke's (1924) model for various values of structural parameter β together with our experimental value for the porosity of 13% at selected temperature of 473 K and pressure of 0.1 MPa.

32

M.G. Alishaev et al. / Engineering Geology 135–136 (2012) 24–39

7 6.5 6 5.5

λ (W·m-1·K-1)

5 T=297 K

4.5

T=294 K

P=0.1 MPa

P=0.1 MPa

4 3.5

3

2.5

2 1

1 2

1.5

2

1

3

3

0.5 0.0

0.2

0.4

φ

0.6

0 1.0 0.0

0.8

0.2

0.4

φ

0.6

0.8

1.0

Fig. 16. ETC of water- (left) and oil- (right) saturated sandstones (porosity of 13%) as a function of porosity predicted from Keller et al.'s (1999) model for various values of the ratio of contact and cross-section areas. 1 — AC/AS = 0.2; 2 — AC/AS = 0.1; 3 — AC/AS = 0.05.

The present experimental results for the ETC of different types of rock materials as a function of temperature were fitted to the linearEucken law (Eucken, 1940) (see also, Buntebarth, 1991a,b; Sass et al., 1992; Pribnow et al., 1996)

Some authors suggested that variation of the lattice conductivity with temperature is λlat ∝ T − 5/4, for the four-phonon mode approach (Beck et al., 1978). Thermal conductivity of feldspar aggregates, glasses, and vitreous materials increase with temperature, n b 0 (Birch and Clark, 1940a,b; Ratcliffe, 1959; Somerton, 1992; Abdulagatov et al., 2000). Our measured ETC data for fluid-saturated sandstones as a function of temperature were fitted to the power law, λ = AT− n. We found that for gas-saturated sandstone the values of exponent n varied from 0.3 to 0.4, therefore, the sandstone under study is mixed crystals (see also Ziman, 1962). For water-saturated sandstones the values of exponent n varied from 0.17 to 0.19, while for oil-saturated sandstones the values of n are changed from 0.20 to 0.22. For example, for water-saturated sandstones A = 10.584 and n = 0.18, while for oil-saturated sandstone A = 10.191 and n = 0.204.

T=297 K

−1

¼ A þ BT;

ð2Þ

where A (the thermal resistance at zero T = 0) is related to the scattering of phonons by impurities and imperfections, and B (rate of increase in the thermal resistance) is related to phonon–phonon scattering, approximately proportional to an inverse power of sound velocity (Ziman, 1962) and is caused by the increase of atomic oscillations. As was shown by Lee and Kingery (1960) and Clark (1969), at high temperatures (above the Debye temperature, T > Θ) the thermal conductive (lattice or phonon conductivity) of many electrically nonconducting

a

6.5

λ (W·m-1·K-1)

λ

6.1

P=0.1 MPa

4.6

5.0 n=2.5

3.5

T=294 K

b

P=0.1 MPa

λ max

3.1 n=1

2.0

n=3

1.6

n=6

λ min

exp. n=4.5

0.5 0.0

0.2

0.4

φ

0.6

0.8

1.0

0.1 0.0

0.2

φ

0.6

0.8

1.0

c

6.5 T=297 K

λ (W·m-1·K-1)

0.4

P=0.1 MPa

5.0 1

3.5 2 3 4

2.0 0.5 0.0

0.2

0.4

φ

0.6

0.8

1.0

Fig. 17. ETC of oil- and water-saturated sandstone as a function of porosity predicted from Sugawara and Yoshizawa (1961, 1968) (a and b) and Buntebarth and Rueff's (1987) models for various values of empirical parameters n and α together with the present measured value for the porosity of 13% at selected temperatures. (a) — oil-saturated; (b and c) water-saturated; (c): 1 − α = 0.0; 2 − α = 0.1; 3 − α = 3; and 4 − α = 8.

M.G. Alishaev et al. / Engineering Geology 135–136 (2012) 24–39

33

7 6.5

T=297 K

T=294 K

P=0.1 MPa

P=0.1 MPa

6 5.5

Oil-saturated sandstone

Water-saturated sandstone

5

λ (W·m-1·K-1)

4.5 4

λ spher

3.5

λ spher

3 λ need

λ need

2.5

2

λcrack

λcrack

1.5

1

0.5 0.0

0.2

0.4

0.6

φ

0.8

1.0

0 0.0

0.2

0.4

0.6

φ

0.8

1.0

Fig. 18. ETC of water- and oil-saturated sandstones as a function of porosity predicted from Zimmerman's (1989) model for the various types of pores (thin cracks, spherical and needle-like pores), ϕ = 13%.

(dielectrics) solids, including the majority of geophysically interesting materials, is given by Eq. (2). The derived values of coefficients A and B for rocks under study are presented in Table 3. As one can see from Table 3, for rocks with ETC decreasing with temperature (crystalline type rocks) the value of parameter B in Eq. (2) is positive, while for rocks with ETC increasing with temperature (amorphous type rocks) the value of B is negative. Vosteen and Schellschmidt (2003) proposed general equation for thermal conductivity of rocks. This equation is excellent predicting the present measured ETC data for crystalline rocks (average absolute deviation, AAD, is within 2–3 %, see Fig. 7). Fig. 8a, b, c shows the temperature dependence of measured ETC of fluid-saturated sandstones and the values predicted from various models. As these figures demonstrate, the models by Funnell et al.

(1996); Fricke (1924); Rzhevsky and Novik (1971) show good agreement (deviations within 1–2%) with our ETC data for gas-saturated sandstone, while some empirical predictive models by Chapman et al. (1984); Sekiguchi (1984); Tikhomirov (1968); Sass et al. (1971) are systematically lower at high temperatures by (4% to 11%). These models do not take into account the mineralogical composition of porous rocks, for example, the quartz content which considerably affects the slope (rate) of ETC changes. Excellent agreement within 1.2% and 2.7% was found between Funnell et al. (1996), geometric mean models (see Figure 8b), and the present experimental data, respectively. Fig. 8c demonstrate the temperature dependence of the predicted ETC of water-saturated sandstone from various porosity models with the temperature dependence of λS(T), λf(T), and ϕ(T).

Water - saturated sandstone 6.5

6.5

λ(W·m-1·K-1)

P=0.1 MPa

T=297 K

5.5

5.5

4.5

4.5

3 1

2 6 4

3.5

3.5

3 9

5

8 5

2.5

2.5

7

2

1.5

0.5 0.0

4

1 1.5

0.2

0.4

0.6

φ

0.8

0.5 1.0 0.0

0.2

0.4

0.6

0.8

1.0

φ

Fig. 19. ETC of water-saturated sandstone as a function of porosity predicted from various theoretical models together with the present experimental value. Left: 1 — Odelevskii (1951); 2 — Waff (1974); 3 — Brown (1955), first order; 4 — Bruggeman (1935); 5 — Asaad (1955); 6 — Chan and Tien (1973); 7 — Huang (1971); 8 — Ziman (1962); 9 — Chan and Jeffrey (1983); Right: 1 — Russell (1935); 2 — Rzhevsky and Novik (1971); 3 — Ribaud (1937); 4 — Mendel (1997); 5 — Intermediate model.

34

M.G. Alishaev et al. / Engineering Geology 135–136 (2012) 24–39

P=0.1 MPa

T=297 K

4.7

Φ =0626 (n=6.928)

4.0

P=0.1 MPa

6.1 T=297 K

5.1

λ (W·m-1·K-1)

3.3 4.1

Φ =0.078 (exp.)

1 4 5

3.1

2.6

2

6 3

1.9

2.1 7

Φ =0.1589 (n=3/2)

8

1.1

0.1 0.0

0.2

1.2

0.4

φ

0.6

0.8

1.0

0.5 0.0

0.2

0.4

φ

0.6

0.8

1.0

Fig. 20. ETC of water-saturated sandstone as a function of porosity predicted from various mixing-law models (left) and Kunii and Smith (1960) model (right) together with the present measurements. Left: 1 — arithmetic mean, first order; 2 — Maxwell model, second order; 3 — arithmetic mean, second order; 4 — geometric mean, second order; 5 — geometric mean, first order; 6 — Maxwell model, first order; 7 — harmonic mean, second order; 8 — harmonic mean, first order.

As one can see, the agreement is good enough (maximum AAD is 9%). Excellent agreement within 0.2% was found with Zimmerman's (1989) model, while the values of ETC predicted with Hsu et al.'s (1995) model were systematically lower than the present measurements by 0.5%. Kunii and Smith's (1960) prediction deviates from our ETC data by 1.3% (prediction values systematically lower). A deviation of about 3.5% was observed for the values of ETC predicted with Hadley's (1986) model.

derived equation for thermal conductivity of solid materials is (Hofmeister, 1999) " # !   T 298 a χ′ λðT Þ ¼ λð298Þ exp −ð4γ Th þ 1=3Þ ∫ α ðθÞdθ 1 þ 0 P ; ð3Þ T χ0 298 where χ0' = dχT/dP≈ 4 to 5 is constant, χ0≈135 GPa (Honda and Yuen, 2004). According to Hofmeister model (3), the temperature coefficient

3.1. ETC measurements and equation of state

of ETC, βT, is related to the thermal expansivity coefficient α as
1 a β þ : α¼− ð4γ þ 1=3Þ T T

Hofmeister (1999) developed a new theory which can explain the effect of temperature and pressure on ETC of solids. Her model for the lattice thermal conductivity,λlat, is based on phonon lifetimes. She studied the pressure and temperature dependence of transport properties from the Grüneisen parameter, γ = αVKT2/CV, isothermal bulk

 ∂P modulus, χ T ¼ −V ∂V , and thermal expansivity, α ¼ V1 ∂V . The ∂T T

Therefore, the measured values of the temperature coefficient of ETC, βT (derived from Eq. (2), see also Table 2 and Figure 6), can be used to calculate the values of thermal expansivity coefficient, α.

P

6.5

P=0.1 MPa

ð4Þ

6.5

T=297 K

5.5

P=0.1 MPa

T=297 K

5.5

λ (W·m-1·K-1)

γc =0

4.5

4.5 γ c=0 for 3-dimensional unit cell

3.5

3.5

1

γ c =0.01 for 3-dimensional unit cell

2.5

2

2.5 γ c=0.01

1.5

0.5 0.0

1.5

0.2

0.4

φ

0.6

0.8

1.0

0.5 0.0

0.2

0.4

φ

0.6

0.8

1.0

Fig. 21. ETC of water-saturated sandstone as a function of porosity predicted from Hsu et al.'s (1994, 1995) model (left) for various values of the touching parameter, γc = c/a, and Maxwell's models (1 — for spherical fluid-filled voids; 2 — for spherical particles suspended in a continued fluid).

M.G. Alishaev et al. / Engineering Geology 135–136 (2012) 24–39

6.5 P=0.1 MPa

6.5

T=297 K

P=0.1 MPa

5.5

λ (W·m-1·K-1)

35

T=297 K

5.5

4.5

4.5

C=1.25 m c =10/9

3.5

3.5

2.5

2.5

C=1.15 m c =1.0

1.5

α p=0.02

f 0 =0.8

1.5 α p=0.02 f 0=0.6

0.5 0.0

0.2

0.4

φ

0.6

0.8

1.0

0.5 0.0

0.2

0.4

φ

0.6

0.8

1.0

Fig. 22. ETC of water-saturated sandstone as a function of porosity predicted from the Zehner and Schlünder (1970) (left) and Hadley's (1986) (right) models for various values of the model's parameters. Left: (- - - -) — Zehner and Schlünder (1970); () — this work (experiment); Right: (- - - -) — Hadley (1986); () — this work (experiment).

Then the derived values of thermal expansivity coefficient α can be used to calculate the specific volume of the rocks as a function of temperature at atmospheric pressure, i.e. to develop equation of state V(T, P = 0.1). The values of thermal expansivity coefficient α derived from the present ETC measurements using the relation (4) for two selected dry rock materials (as example) at atmospheric pressure are shown in Fig. 9 for various values of parameter a. Temperature coefficient of ETC, βT, can be expressed as a function of temperature from Eq. (2) as βT ¼ −

value of ρ0(T0) at room temperature, T0 = 293 K are 2380 kg·m− 3 and 2060 kg·m− 3, respectively. The values of density calculated from Eq. (6) are presented in Fig. 10. Eqs. (4) to (6) together with well-known thermodynamic relations were used to calculate some derived thermodynamic properties of limestone and granulate as a function of temperature. Figs. 11 to 13 show derived thermodynamic properties of limestone and granulate from the present ETC measurements as a function of temperature for different values of parameter a. 3.2. Simplified porosity dependence model

B or βT ¼ −Bλ; ðA þ BT Þ

ð5Þ

By integrating of Eq. (4), one obtains the equation for density of dry rocks at atmospheric pressure as a function of temperature   a C λ T ρðT Þ ¼ ρ0 ðT 0 Þ ; ð6Þ λ0 T0 1 where C ¼ − ð4γþ1=3 Þ; ρ0(T0) and λ0(T0) are the density and ETC of the rock at reference temperature T0. For limestone and granulate the

In this section we considered the effect of structural properties of rocks on porosity dependence of ETC. Measurements of the ETC of porous rock materials by using various techniques are time consuming and costly. Therefore, predictive models are very important. Modeling is the only practical approach to explore the dependency of ETC of porous media on morphology (size, shape, their distribution and orientation, form and structure of rocks) of the porous media. For many engineering and scientific applications the relationship between physical bulk properties and porosity is needed. Porosity of a material is the most important parameter

1.0

6.5

β B=-0.057

0.8

4.5 P=0.1 MPa

0.6

T=297 K

solid phase

3.5

y

λ (W·m-1·K-1)

5.5

0.4

2.5

1.5

0.2 B=-0.15 (exp.)

0.5 0.0

0.2

pore space (fluid) 0.4

φ

0.6

0.8

1.0

0.0 0.0

0.2

β2 0.4

0.6

0.8

1.0

x Fig. 23. ETC of water-saturated sandstone as a function of porosity predicted from the Krupiczka's (1967) model. (- - - -) — Krupiczka (1967); () — this work (experiment).

Fig. 24. The model of rock pores. Pore-structure model used to develop ETC calculation.

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M.G. Alishaev et al. / Engineering Geology 135–136 (2012) 24–39

Table 4 Measured and predicted from Eq. (10) values of ETC of gas-, water-, and oil-saturated rocks. ϕ0 (%)

λS, W·m− 1 ·K− 1

λf, W·m− 1 ·K− 1

λexp, W·m− 1 ·K− 1

λcal, W·m− 1 ·K− 1

References

0.28 0.28 0.28

3.00 3.00 3.00

0.018 0.141 0.607

0.63 1.62 2.25

0.96 1.24 2.30

This work This work This work

Sandstone-3 (T = 298 K) Air 13.0 Oil 13.0 Water 13.0

0.28 0.28 0.28

6.24 6.24 6.24

0.026 0.170 0.607

2.67 3.17 3.78

2.61 3.20 3.80

This work This work This work

Sandstone (T = 298 K) Argon 10.5 Oil 10.5 Water 10.5

0.28 0.28 0.28

3.85 3.85 3.85

0.018 0.141 0.607

1.95 2.56 3.79

1.94 2.24 3.36

Kurbanov (2000) Kurbanov (2000) Kurbanov (2000)

Sandstone (T = 298 K) Argon 12.0 Oil 12.0 Water 12.0

0.28 0.28 0.28

4.54 4.54 4.54

0.018 0.141 0.607

1.28 2.40 3.50

2.50 2.38 3.76

Kurbanov (2000) Kurbanov (2000) Kurbanov (2000)

Sandstone (T = 298 K) Argon 6.9 Oil 6.9 Water 6.9

0.28 0.28 0.28

3.95 3.95 3.95

0.018 0.141 0.607

2.20 3.21 4.35

2.37 2.67 3.59

Kurbanov (2000) Kurbanov (2000) Kurbanov (2000)

Sandstone (T = 298 K) Argon 13.9 Oil 13.9 Water 13.9

0.28 0.28 0.28

4.56 4.56 4.56

0.018 0.141 0.607

1.43 2.70 3.91

1.76 2.15 3.65

Kurbanov (2000) Kurbanov (2000) Kurbanov (2000)

Sandstone (T = 298 K) Argon 7.0 Oil 7.0 Water 7.0

0.28 0.28 0.28

3.05 3.05 3.05

0.018 0.141 0.607

1.99 2.12 2.33

1.87 2.10 2.77

Kurbanov (2000) Kurbanov (2000) Kurbanov (2000)

Sandstone-1 (T = 305.4 K) Air 19.6 Oil 19.6 Water 19.6

0.205 0.205 0.205

9.86 9.86 9.86

0.026 0.133 0.611

0.877 1.362 2.755

0.787 1.419 2.700

Somerton (1958) Somerton (1958) Somerton (1958)

Sandstone-2 (T = 305.4 K) Air 0.4 Oil 0.4 Water 0.4

0.5 0.5 0.5

3.81 3.81 3.81

0.026 0.133 0.611

0.493 1.000 1.817

0.810 1.160 1.890

Somerton (1958) Somerton (1958) Somerton (1958)

Limestone (T = 305.4 K) Air 18.6 Oil 18.6 Water 18.6

0.24 0.24 0.24

8.31 8.31 8.31

0.026 0.133 0.611

1.701 2.155 3.548

1.712 2.224 3.299

Somerton (1958) Somerton (1958) Somerton (1958)

Siltstone (T = 305.4 K) Air 36.0 Oil 36.0 Water 36.0

0.405 0.405 0.405

3.81 3.81 3.81

0.026 0.133 0.611

0.585 0.957 1.793

0.587 0.959 1.726

Somerton (1958) Somerton (1958) Somerton (1958)

Sand (fine) (T = 305.4 K) Air 38.0 Oil 38.0 Water 38.0

0.39 0.39 0.39

9.35 9.35 9.35

0.026 0.133 0.611

0.627 1.385 2.752

0.651 1.263 2.520

Somerton (1958) Somerton (1958) Somerton (1958)

Sand (coarse) (T = 305.4 K) Air 34.0 Oil 34.0 Water 34.0

0.39 0.39 0.39

9.35 9.35 9.35

0.026 0.133 0.611

0.557 1.644 3.072

1.281 1.859 3.048

Somerton (1958) Somerton (1958) Somerton (1958)

Silty sand (T = 305.4 K) Air 43.0 Oil 43.0 Water 43.0

0.459 0.459 0.459

3.98 3.98 3.98

0.026 0.133 0.611

0.452 1.082 1.921

0.483 0.874 1.679

Somerton (1958) Somerton (1958) Somerton (1958)

Fluid Sandstone − 2 Argon Oil Water

ϕ (%) (T = 298 K) 16.2 16.2 16.2

affecting its mechanical, thermal, and electrical properties, but their relationship to porosity is poorly understood. Increasing porosity leads in general to a decrease in any bulk property, but in detail, the microstructure of the granular material is important, i.e. the shape and spatial

arrangement of the grains (texture) as well as the conditions in the contact area of the connecting grains (contact stiffness). Textural– structural peculiarities of rocks and oriented fracturing are some of the main factors governing the ETC behavior of porous rock materials.

M.G. Alishaev et al. / Engineering Geology 135–136 (2012) 24–39

6

a

λ (W·m-1·K-1)

6

b

5

5

1

1

4 3

4 3

2

2

5 3

4

1

2

4

3

2

0 0.00

37

1

5

0.06

0.12

φ

0.18

0 0.30 0.00

0.24

0.06

0.12

6

0.18

0.24

0.30

c

5

λ (W·m-1·K-1)

φ

1

4 3 2

2 3

1 0 0.00

0.06

4

0.12

5

0.18

φ

0.24

0.30

Fig. 25. Porosity dependence of the ETC fluid-saturated sandstones predicted by the present model at T = 305.37 K and 0.1 MPa. (a) — water-saturated; (b) — oil-saturated; (c) — airsaturated; 1 — Odelevskii (1951); 2 — Geometric mean; 3 — Maxwell model; 4 — Somerton (1958); 5 — this work (Eq. (11)).

The physical limits for the ETC fluid-saturated rocks are very well established (see Figure 14). ETC of small to modest solid–fluid conductivity ratio (λS/λf) media can be predicted with reasonable accuracy, while our understanding of heat conduction in large solid–fluid conductivity ratio, (λS/λf)→∞, still is incomplete. In order to estimate the ETC of porous fluid-saturated rocks, it is essential to define precisely the contribution of each component (mineral, λS, and fluids, λf) to bulk conductivity. The results of the comparison of the present ETC data for fluidsaturated sandstones with some of the frequently used theoretical models reported by other authors are depicted in Figs. 15 to 23. In these figures the ETC of fluid-saturated sandstones are presented as a function of porosity for different values of structural model parameters. These figures also demonstrate how porosity behavior of the ETC depends on structural parameter which defined the size, shape, distribution, orientation, and touching area of the grains. Unfortunately, in majority of the cases theoretically estimated values of the structural parameter do not agree well with the fitted values. This makes it difficult to accurately predict the porosity behavior of the ETC fluid-saturated rock materials using theoretical models (Figure 7). Unfortunately, most developed models have their disadvantages: some overestimate while others underestimate systematically the true (measured) bulk thermal conductivity. Most of them are valid only for a specific range of volume ratios (porosities), and yield completely unreasonable results outside this range. Parallel and series models are easy to understand, but have the disadvantage of being a rather special case, applicable mostly to bedded sediments. They lead to the well known arithmetic and harmonic means, respectively, and define upper and lower limits for all other models. Most predictive models work to within 10–15% accuracy (for typical conductivity ratios, i.e. λS/λf b 10). The empirical models have their shortcomings in that the resulting models may be applicable only to the particular suite of rocks being investigated. In the present work we developed a new simple ETC model for fluidsaturated rocks based on the pore structure. As was mentioned above, in general, thermal conductivity of fluid-saturated porous rock materials can be consider in the following functional form λ = f(λS, λf, ϕ, ϕ0), where λS and λf are the thermal conductivities of the pure phases (solid matrix and fluid, respectively); ϕ is the porosity; and ϕ0 is the structural parameter (shape factor) which depend on the pore shape, their distribution and orientation. Let us consider that the rock sample

part of the solid matrix is continuously connected to media, and other part contacted with the fluid. Each void is connected only to one other pore or not connected to any pore (isolated). Void shape is regular and uniform in their size and in their distribution throughout the matrix. It is natural to consider that the connected part of the sample is (1 − β 2), and other part belongs to the pores and the solid matrix. Also, the micro-size of the alternation is distributed according some law, y(x). Fig. 24 shows one of the likely (realistic) distribution in the case when heat flow is directed along the axis y. The curve y = y(x), where 0 ≤ x ≤ β 2, shown in Fig. 24, have to envelop the area which is equal to the porosity of the sample, ϕ β

2

∫ yðxÞdx ¼ ϕ:

ð7Þ

0

The parameter β is the characteristic total linear size of the pores. Square in the β means that along the axis x we have a pore crosssection area. The effective thermal conductivity of the sample defined as (Rubinshtein, 1972)
 β2 2 λ ¼ λS 1−β þ ∫ 0

dx : yðxÞ=λf þ ½1−yðxÞ=λS

ð8Þ

It is very clear that selection of the analytical form of the distribution curve y(x) and the parameter β depends on the structure of the porous media (rock sample), their orientation and distribution. In this work, as an example, we used the following structural function 2

yðxÞ ¼ β cos



 πx 2 ; where 0≤x≤β : 2 2β

ð9Þ

Taking the integrals (7) and (8) we have λ 2 2 ¼ 1−β þ β λS

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi λf ; where β ¼ 3 2ϕ: ð1−βÞλf þ βλS

ð10Þ

Eq. (10) allows estimating the ETC of fluid-saturated porous media (rock materials) using the matrix thermal conductivity of the λS, fluid thermal conductivity λf, and structural parameter β. At ϕ > 0.5 the

38

M.G. Alishaev et al. / Engineering Geology 135–136 (2012) 24–39

phases (solid and liquid) change roles, i.e. we have the case, when the solid particles are suspended in the continuing liquid phase. At pffiffiffiffiffiffiffiffiffiffi ϕ = 0.5, Eq. (10) reduced to the geometric mean model λ ¼ λS λf as a particular case. At λf = 0 (vacuum in the pores), we have λ = λS(1 − β 2), therefore, the condition β = 1 is defined as the thermal conductivity limit. For the structural function (9) the limited value of the porosity is ϕ = 0.5. Therefore, because the ETC cannot become less than zero the condition β = 1 is defined as the physical limit of the thermal conductivity for this model. In order to take into account the geometric structure of the porous media and underground condition let us consider pffiffiffiffiffiffiffiffiffiffiffiffi some modifications of Eq. (10). Let us consider that β ¼ 3 ϕ=ϕ0 , where ϕ0 is the maximum possible value of the porosity of sample, 0 ≤ ϕ ≤ ϕ0. It is obvious that for fluid saturated rocks the values of ϕ0 is less than 1. Also, we consider the case, when λf ≪ λS (the ratio λf/λS is small, especially for gas saturated porous media). Then, Eq. (10) for ETC can be simplified as

λ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1− ðϕ=ϕ0 Þ2 ÞλS þ λS λf ðϕ=ϕ0 Þ:

ð11Þ

pffiffiffiffiffiffiffiffiffiffi At ϕ = ϕ0 we have a geometric mean model λ ¼ λS λf . We applied Eq. (11) for various types fluid saturated rocks (see Table 4). Fig. 25 shows the comparison between the present measured and predicted with the present model (10) values of ETC for various fluid-saturated sandstones. This figure also contains the values of ETC predicted with various theoretical models reported by other authors. The value of the structural parameter ϕ0 for various types of rocks changes from 0.2 to 0.5 (see Table 4). As one can see from Table 4, for all sandstones with various porosities and at the same temperature the value of structural parameter is the same ϕ0 = 0.28. 4. Conclusions The ETC of dry, gas-, oil-, and water-saturated rock materials (Sandstone-1, Sandstone-3, Limestone, Amphibolite, Granulate, Sandstone-2, Siltstone, Pyroxene-granulate, and Andesite) in the temperature range from 273 K to 523 K has been measured using a steady-state guarded parallel-plate apparatus with an estimated uncertainty of 4%. The effect of saturating fluids, structure (size, shape, and distribution of the pores), and mineralogical composition on temperature and porosity dependences of the ETC of fluid-saturated rocks was studied. The temperature changes of the ETC for Sandstone-1, Sandstone-3, Limestone, Amphibolite, and Granulate are nearly linear and decreasing with temperature increase, while for Sandstone-2, Siltstone, Pyroxene-granulate, and Andesite ETC monotonically increases with temperature with a different rate. For dolomite ETC is almost independent of temperature. A new simple model for ETC of fluid-saturated rocks which is taking into account structure of porous media has been proposed. Using Hofmiester's model and measured thermal conductivities for dry rock materials the values of thermodynamic properties (density, thermal expansion coefficient, enthalpy, and heat capacity) were predicted. The measured ETC of fluidsaturated rocks was used to examine the accuracy, predictive capability, and applicability of various theoretical and semi-empirical models. Acknowledgments One of us (I.M.A.) thanks the Thermophysical Properties Division at the National Institute of Standards and technology (NIST) for the opportunity to work as a Guest Researcher at NIST during the course of this research.

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