Thermophysical properties of the Po Basin rocks

Geophys. J. Int. (2011) 186, 69–81 doi: 10.1111/j.1365-246X.2011.05040.x

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Thermophysical properties of the Po Basin rocks

V. Pasquale, G. Gola, P. Chiozzi and M. VerdoyaDipartimento per lo Studio del Territorio e delle sue Risorse, Settore di Geofisica, Universita di Genova, Viale Benedetto XV 5, I-16132 Genova, Italy.E-mail: [email protected]

Accepted 2011 April 6. Received 2011 April 6; in original form 2010 November 8

S U M M A R YWe present an analysis on thermal properties, density and porosity of clastic, chemi-cal/biochemical and intrasedimentary volcanic rocks collected from petroleum explorationwells of the Po Basin (Northern Italy). Moreover, we investigate the applicability of theHashin–Shtrikman’s model for a mineral aggregate in combination with the Zimmerman’smodel that takes pore shape into account, to calculate the bulk thermal conductivity. In caseof macroscopically isotropic rocks, deviations between predicted and measured values rangefrom −2.2 per cent to 6.9 per cent, and significantly decrease if a proper pore aspect ratiois chosen. Regarding the volumetric heat capacity, approximate estimates were obtained bymeans of a weighted average of the volumetric heat capacity of the mineral grains and that ofthe pore-filling water. The differences between the computed and measured values range from–6.2 per cent to 4.9 per cent and, on average, the computed volumetric heat capacity is lowerby 1.6 per cent. The water loss during compaction and the temperature increase with depth aremain factors controlling thermal properties. An anisotropy effect occurs in the case of rocksrich in sheet silicates. Due to rotation of these minerals, the vertical thermal conductivity ofsheet silicates decreases exponentially with burial depth from 2.13 W m−1 K−1 at the surfaceto 0.52 W m−1 K−1 at 4.5 km. The laboratory data allow the formulation of compaction curvesfor the different sedimentary rock types. Examples of estimations of in situ vertical thermalconductivity and heat flow are finally given for two petroleum wells at which lithostratigraphicinformation is known in good detail.

Key words: Downhole methods; Heat flow; Microstructures; Sedimentary basin processes;Heat generation and transport.

1 I N T RO D U C T I O N

The Po Basin is a several hundred-kilometre wide sedimentarybasin, mainly filled with clastic and chemical/biochemical deposits,enclosed between the Alps and Apennines orogenic belts (Fig. 1).Pasquale & Verdoya (1990) gave a first insight into the basin surfaceheat flow, which is a basic constraint for studying the undergroundthermal regime. Their study was based on the assumption that heatconduction is the dominant heat transfer mechanism in the basin.However, the presence of aquifers at different depths, like thoseoccurring in most sedimentary basins, implies caution in the heat-flow interpretation, as the actual thermal regime can be a result ofthe complex superposition of heat and mass transfer processes re-lated to groundwater flow (e.g. Jessop 1990; Ingebritsen & Sanford1998).

A possible approach to unravel the thermal regime is the compari-son of the thermal data available from wells with a purely conductivethermal model. Deviations from this model may allow the detec-tion of convective effects, the subsequent modelling of the type offlow in the deep regional aquifer and, ultimately, the geothermal

resource assessment (Pasquale et al. 2008a). However, this kindof approach is applicable only provided that the thermophysicalproperties of the sedimentary sequence are known in detail, as vari-ation in these properties can lead to significant misinterpretation ofthermal anomalies (Deming 1994; Clauser & Huenges 1995; Daviset al. 2007).

This work falls within a research program aimed at the betterunderstanding of the geothermal potential of the Po Basin deepsedimentary sequences. We present results of laboratory measure-ments of thermal properties, density and porosity of rock samplesrecovered from several petroleum wells drilled in the basin. Theresults are then tested with mixing models for the prediction of thethermal conductivity and the volumetric heat capacity, based on theknowledge of volume fractions of the rock-forming minerals. Todescribe the basin rock properties under any possible condition ofburial depth, temperature and anisotropy, we suggest an approachthat allows the inference of in situ thermal parameters on the ba-sis of the mineral composition or lithostratigraphic data. Finally,examples of calculation of thermal conductivity and surface heatflow are given for two deep wells.

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Geophysical Journal International

70 V. Pasquale et al.

Figure 1. Location of wells (full circles) whose core specimens were used for laboratory measurements.

2 RO C K T Y P E S

More than 100 core samples from 25 petroleum exploration wells,scattered in the Po Basin (Fig. 1), were made available by the Italiannational oil company (Eni E&P Division San Donato Milanese,Milan). The cores provide a broad collection of the basin mainlithologies up to 6500 m depth. Most of them are sedimentaryand include clastic and chemical/biochemical rocks. A few samplesare effusive rocks belonging to intrasedimentary volcanic bodies.The sampled lithotypes are macroscopically isotropic, except somesiltstones, shales and silty shales exhibiting horizontal bedding ofsheet silicates.

Powder X-ray diffraction analyses, carried out for all sam-ples, gave information about the rock mineral composition. Thin

section analyses of selected representative samples, impregnatedwith methylene blue, provided a visual estimation of grain size,sorting, fabric and pore shape. Table 1 lists the investigated rocksaccording to their origin and composition together with the numberof available samples for each lithotype.

Clastic sediments are predominantly consolidated rocks and con-sist of framework silicates and carbonate grains scattered in anargillaceous matrix or calcareous cement. Most samples are marlsand silty marls of marine origin, formed by a calcium carbonatemud containing variable amount of clays. In addition to clay andcalcium carbonate, commonly silt-sized grains of quartz, plagio-clase and K-feldspar are present. Shales are mainly composed of amix of clay minerals and tiny fragments of other minerals, especiallymuscovite, quartz and, to a lesser extent, feldspars and calcite. Thesampled argillaceous sandstones are lithic and feldspatic arenites,

Table 1. Laboratory results of physical properties. kr is the thermal conductivity of water-saturated isotropic (code 1−5 and 9−18) samples, φ is the porosity,ρrcr and ρr are the volumetric heat capacity and the density, respectively, of both isotropic and anisotropic (code 6−8) dry samples. The standard deviation (inbrackets) and the number n of samples are listed.

kr (W m−1 K−1) ρrcr (kJ m−3 K−1) φ (per cent) ρr (kg m−3)

Rock Code/Lithotype n Range Mean Range Mean Range Mean Range Mean

Clastic 1−Marl 19 2.15−3.08 2.77 (0.23) 1310−2038 1808 (176) 6.0−37.0 15.1 (8.4) 1787−2530 2278 (240)2−Silty marl 18 2.85−3.66 3.16 (0.26) 1790−2150 1937 (125) 2.0−20.0 12.8 (5.5) 2150−2670 2359 (156)3−Calcareous

marl6 1.99−2.37 2.17 (0.13) 1406−1617 1495 (72) 22.0−35.0 30.8 (5.0) 1693−2008 1801 (123)

4−Argillaceouslimestone

3 3.58−3.63 3.60 (0.03) 1977−2094 2036 (59) 7.5−12.0 9.3 (2.4) 2477−2588 2520 (80)

5−Argillaceoussandstone

6 2.60−3.40 3.00 (0.29) 1630−2059 1884 (155) 8.0−25.0 15.1 (6.2) 1990−2560 2330 (222)

6−Siltstone 4 − − 1853−2145 2003 (119) 6.0−18.0 11.1 (5.4) 2368−2560 2492 (107)7−Shale 6 − − 1780−1970 1854 (63) 4.8−22.0 15.9 (6.3) 2120−2400 2220 (106)8−Silty shale 6 − − 1680−1830 1739 (55) 5.0−21.0 13.2 (6.3) 2200−2570 2340 (145)9−Calcarenite 3 2.18−2.50 2.34 (0.16) 1370−1810 1590 (220) 25.0−32.0 29.0 (3.6) 1834−1997 1917 (82)

Chemical−biochemical

Carbonate 10−Mudstone 5 3.04−3.48 3.30 (0.16) 2090−2188 2148(36) 0.5−6.0 2.7(2.1) 2550−2695 2630 (59)

11−Wackestone 5 3.10−3.20 3.16 (0.04) 1980−2190 2108 (80) 3.0−10.0 6.0 (3.0) 2500−2670 2590 (75)12−Packstone 4 3.00−3.45 3.23 (0.18) 2014−2109 2058 (39) 3.0−6.0 4.3 (1.3) 2550−2655 2620 (59)13−Grainstone 5 2.95−3.36 3.12 (0.16) 1950−2070 2010 (55) 6.5−12.0 8.8 (2.5) 2400−2540 2480 (73)14−Dolostone 5 4.25−5.45 4.60 (0.49) 2240−2400 2331 (58) 1.5−7.5 3.8 (2.4) 2800−2630 2735 (73)

Siliceous 15−Radiolarite 4 3.16−3.46 3.37 (0.14) 1953−2107 2021 (69) 0.5−5.5 2.4 (2.2) 2550−2650 2600 (48)Evaporitic 16−Anhydrite 5 3.15−3.65 3.39 (0.22) 1930−1970 1945 (17) 0.5−5.0 2.7 (1.8) 2680−2780 2730 (52)

17−Gypsum 5 1.40−1.64 1.54 (0.09) 2325−2500 2445 (71) 0.5−7.0 2.4 (2.6) 2260−2400 2350 (61)Igneous Effusive 18−Dacite 4 3.56−3.91 3.73 (0.18) 2140−2160 2151 (8) 1.5−7.5 3.3 (2.8) 2690−2500 2610 (102)

C© 2011 The Authors, GJI, 186, 69–81

Geophysical Journal International C© 2011 RAS

Properties of the Po Basin rocks 71

Figure 2. Photomicrograph, under plane-polarized light, of some clastic rocks of Table 1. (a) 2-Silty marl: scattered silt (quartz and plagioclase) with planctonicand bentonic foraminifera partly pyritized and locally pyrite blades in marly matrix; porosity is localized in layers along pyrite blades; (b) 3-Calcareous marl:silt (prevalently quartz) and planctonic foraminifera scattered in a marly matrix with intragranular and chalky porosity; (c) 6-Siltstone: silt (quartz, plagioclase,K-feldspar and mica) and micrite grains in argillaceous matrix with scarce porosity; (d) 4-Argillaceous sandstone: medium to fine sandstone formed by quartz,plagioclase, K-feldspar and mica grains in argillaceous matrix, intragranular, intergranular with fracture porosity.

primarily composed by cemented sandy sediment, in many casesdominated by sand-sized rock fragments and quartz. The grain sizeranges from medium to fine, with argillaceous matrix and/or cal-careous cement. Both clasts and matrix of calcarenites are generallycalcareous, and only a low percentage of grains is formed by quartzand clay minerals. Fig. 2 shows a few examples of the microscopicstructure of some samples of clastic sediments.

Chemical/biochemical sediments include carbonatic, evaporiticand siliceous rocks. Carbonates are classified on the basis of thedepositional texture according to Dunham (1962). The most abun-dant mineral is calcium–magnesium carbonate, which is presentboth in calcareous bioclastic fragments and micrite particles. Evap-oritic rocks (anhydrite and gypsum) are fine grained. In anhydrites,gypsum and calcite minerals are associated. Siliceous rocks arerepresented by radiolarites. Examples of the microscopic structureof samples belonging to chemical/biochemical rocks are given inFig. 3.

The volcanic rock samples are lavas of dacitic composition. Theyhave a porphyritic texture and contain from 25 to 35 per cent of phe-nocrystals, mainly of plagioclase + quartz + biotite. Phenocrystsare set in cryptocrystalline groundmasses consisting of plagioclase,quartz, oxides and minor K-feldspar. Alteration processes have pro-duced kaolinite and illite–smectite clays minerals through chemicalweathering of aluminium silicate minerals like feldspar.

3 L A B O R AT O RY M E A S U R E M E N T S

Measurements were made at standard laboratory conditions on sam-ples both water saturated and dehydrated with a fan-forced drying

oven at 105 ◦C for 24 hr. Rocks with water-sensitive or hydratedphases (like clay, gypsum and anhydrite) were dried at lower temper-ature (65 ◦C) for preserving the mineral assemblage and preventingthe increase of effective porosity, due to hydration-water removaland alteration of the pore fabric.

Both bulk density and grain density were determined. Mass wasmeasured with a high precision balance and the bulk volume wasdetermined by means of immersion in distilled water. Porosity wascomputed as the ratio of the difference of grain and bulk densitiesto the grain density.

The solid volume was inferred with two different approaches:(i) for permeable and consolidated rocks, the mass change betweenthe dehydrated and water-saturated conditions was accounted forby the influx of water into the pore spaces (this procedure there-fore yielded a direct measurement of porosity), (ii) for poorly ce-mented, argillaceous and evaporitic rocks, a helium pycnometer wasused.

Thermal properties were measured with the device ISOMET (Ap-plied Precision, Ltd., Bratislava, Slovakia), which can be equippedwith needle and plane probes suitable for simultaneous determina-tion of thermal conductivity, thermal diffusivity and volumetric heatcapacity. The measurement was based on the analysis of the tem-perature response of the analysed material to heat impulses inducedby electrical heating. A thin layer heater in thermal contact with thesurface of the sample was used (see Carslaw & Jaeger 1959, formathematical formulation). Since the three thermal properties arelinked together, we considered only the thermal conductivity k (inW m−1 K−1) and the volumetric heat capacity (given by the productbetween density ρ [in kg m−3] and specific heat c [J kg−1 K−1]).

C© 2011 The Authors, GJI, 186, 69–81



Figure 3. Photomicrograph, under plane-polarized light, of some chemical/biochemical rocks of Table 1. (a) 15-Radiolarite: radiolarian interbedded withsiliceous shale, much of the matrix is silicified, porosity is scarce; (b) 10-Mudstone: calcite occurs as very fine crystalline grains to micrite-size grains,locally microcrystalline quartz replaces radiolarian and chert nodules, porosity is scarce or absent; (c) 12-Packstone: silt to micrite-size non-skeletal grains,mollusca and crinoid fragments arranged in a self-supporting framework, yet also contains matrix of calcareous mud and argillaceous layers, scarce porosity;(d) 14-Dolostone: very fine crystalline dolomite, crystals are typically subhedral to euhedral and locally surrounded by clays minerals, intercrystalline porositylittle developed or absent.

Reproducibility is 3 per cent for both parameters, and accuracy is 5per cent for k and 15 per cent for ρc.

A series of preliminary measurements on standard materials withknown thermal properties was carried out for the calibration of thedevice. Moreover, to verify the device performances several trialswith another apparatus implemented at our laboratory (Pasquale1982; Pasquale 1983) were run on the same standard materialsand a selection of rock samples. The results obtained with the twodifferent equipments were in substantial agreement.

Thermal conductivity was measured parallel and perpendicular tothe core axis, which always coincides with the vertical. Anisotropyratios, that is, the ratio between the maximum and minimum thermalconductivity, turned out to be negligible (<1.05) in many samples,so that most of the rocks can be regarded as isotropic. Table 1summarizes the thermal conductivity results of the water-saturatedsamples that showed negligible anisotropy, together with porosity,density and dry volumetric heat capacity of all samples. The meanthermal conductivity ranges from 1.54 to 4.60 W m−1 K−1, cor-responding to gypsum and dolostone, respectively. Besides dolo-stones, larger values of conductivity characterize dacites and an-hydrites, whereas lower conductivity is typical of calcareous marls.Carbonate rocks and argillaceous sandstones show intermediate val-ues. The mean values of dry volumetric heat capacity range from1495 (calcareous marls) to 2445 kJ m−3 K−1 (gypsum). Porosityvaries from 2.4–2.7 per cent (radiolarite, gypsum, mudstones andanhydrite) to 29.0–30.8 per cent (calcarenite and calcareous marl).

Albeit macroscopically homogeneous, a few samples of shales,silty shales and siltstones had bedding on a scale <1 mm, so that

their anisotropy ratio was not negligible. Table 2 details the com-position of these samples together with the measured horizontal(kx) and vertical (kz) thermal conductivity, porosity and the depth atwhich they were recovered. Fig. 4 shows kx, kz and the sheet silicatevolume fraction vss against depth z. Their correlation is in the form

kx = 0.117z + 2.334, (1)

kz = −0.054z + 2.037, (2)

vss = −0.015z + 0.426, (3)

where z is in km. The horizontal thermal conductivity increasesappreciably with depth, whereas the vertical thermal conductiv-ity and the sheet silicate fraction decrease. Anisotropy factorvaries from 1.09 for the highly porous siltstone coming from1261 m depth to 1.68 for the compact shale recovered at 3980 mdepth.

4 M O D E L L I N G O F T H E R M A LPA R A M E T E R S

Many mixing models have been developed to estimate thermalconductivity and volumetric heat capacity of rocks from otherparameters (e.g. Scharli & Rybach 2001; Wang et al. 2006;Abdulagatova et al. 2009). The inference of the thermal con-ductivity from information on the volume fraction and the con-ductivity of rock-forming minerals requires the modelling of the

C© 2011 The Authors, GJI, 186, 69–81



Table 2. Depth z, porosity φ, horizontal kx and vertical kz measured thermal conductivity, mineral composition (per cent), and computed thermal conductivitykss of the sheet silicate fraction of Table 1. Cc, Calcite; Dol, Dolomite; Qtz, Quartz; Kf, Potassium feldspar; Pl, Plagioclase; SS, Sheet silicate.

Mineral

Code/Lithotype z (m) φ (per cent) kx (W m−1 K−1) kz (W m−1 K−1) Cc Dol Qtz Kf Pl SS kss (W m−1 K−1)

6−Siltstone 1261 18.0 2.97 2.72 33.6 1.1 39.0 4.2 10.9 11.2 0.852330 12.5 2.58 1.98 15.4 1.5 31.5 2.6 19.6 29.4 0.573730 7.8 2.91 1.98 29.0 1.5 22.0 3.8 15.1 28.6 0.504396 6.0 2.72 1.89 10.3 2.5 29.0 6.0 19.4 32.8 0.48

7−Shale 660 21.0 2.31 1.85 8.6 2.1 30.0 − 5.0 54.3 1.301190 18.0 2.34 1.71 7.9 − 35.0 − 7.1 50.0 0.881350 16.0 2.20 1.57 12.4 − 25.5 1.4 11.5 49.2 0.832232 12.0 2.35 1.47 10.5 1.5 30.2 − 4.8 53.0 0.632757 7.0 2.54 1.52 5.5 − 35.6 1.3 3.8 53.8 0.603980 5.0 2.99 1.78 11.3 − 36.0 1.1 7.9 43.7 0.49

8−Silty shale 806 22.0 2.45 2.04 6.4 1.4 38.2 2.5 10.4 41.1 1.171018 20.0 2.81 2.34 5.3 1.1 46.4 1.6 10.3 35.3 1.101721 19.0 2.27 1.77 12.1 1.8 28.0 1.2 15.9 41.0 0.782028 17.0 2.62 2.11 10.0 − 34.7 5.0 20.3 30.0 0.722846 12.5 2.71 2.01 3.5 − 33.5 8.5 17.5 37.0 0.703520 4.8 2.77 1.92 11.3 − 33.0 1.5 16.0 38.2 0.55

Figure 4. Horizontal kx and vertical kz measured thermal conductivity, and sheet silicate fraction versus depth of the samples in Table 2. Solid lines are the fitcurves.

distribution of the various constituents. The reliability of the resultsdepends on the accuracy of the available information and on howthe rock characteristics affect the thermal conductivity. The prob-lem is to compute the disturbance to a linear heat flow througha uniform medium due to the presence of a region of differentconductivity.

Literature values of thermal conductivity, density and specificheat of the minerals recognized in our samples are summarizedin Table 3. There is substantial agreement in the thermal conduc-tivity values, albeit derived from different sources. However, wenoted some significant discrepancies, which generally may reflectvariations in the type of determination. In case of measurements onsingle crystals or monomineralic aggregates, we preferred the latter.The values of thermal parameters for potassium feldspar, plagio-

clase and sheet silicates correspond to the most abundant minerals,that is, microcline, oligoclase and the smectite–illite mineral group,respectively.

4.1 Volumetric heat capacity

Because the specific heat of the air is comparable with those of therock-forming minerals, we focused on the volumetric heat capacitymeasurements carried out on dry rocks. Provided that the mineralcomposition and physical properties of minerals and air are known,the volumetric heat capacity (in J m−3 K–1) of a dry rock ρrcr can becomputed as the weighted average of the volumetric heat capacityof the matrix ρmcm and that of the air ρaca in the voids

ρr cr = (1 − φ) ρm cm + φ ρa ca, (4)

C© 2011 The Authors, GJI, 186, 69–81



Table 3. Thermal conductivity, density and specific heat of rock-formingminerals, air and water at standard laboratory conditions.

Material

Thermalconductivity

(W m−1 K−1)Density(kg m−3)

Specific heat(J kg−1 K−1)

Quartz−α 7.69a 2647a 740b

Quartz 3.71a 2618a 735b

microcrystallinePlagioclase 1.97a 2642a 837b

K−feldspar 2.40a 2562a 700b

Calcite 3.59a 2721a 815b

Dolomite 5.51a 2857a 870b

Sheet silicates 1.88c 2630c 832d

Anhydrite 4.76a 2978a 585b

Gypsum 1.30e 2320b 1070b

Air 0.026 1.225 1005Water 0.60 1000 4186

Notes: aHorai (1971). bWaples & Waples (2004a). cBrigaud & Vasseur(1989). dHadgu et al. (2007). eClauser & Huenges (1995).

Figure 5. Comparison of average values of computed and measured dryvolumetric heat capacity ρrcr. Broken lines are ±7 per cent deviation fromone to one correspondence (continuous line). The code number next to eachdata point refers to samples listed in Table 1.

where ρmcm = ∑nj=1 v jρ j c j ,

∑nj=1 v j = 1, φ is the porosity, vj, ρ j

and cj are the volume fraction, density and specific heat of the jthmineral, respectively and n is the number of mineral components.A comparison of the results of average volumetric heat capacity,estimated for each lithology with the laboratory values is shown inFig. 5. The differences are relatively small (from –6.2 per cent to4.9 per cent) and the computed ρrcr is lower by 1.6 per cent.

4.2 Thermal conductivity

4.2.1 Isotropic rocks

If the conductivity of the mineral phases and the mineral compo-sition are known, the matrix thermal conductivity of isotropic rockcan be computed. The conductivity of the water-saturated rock can

be then estimated by considering also the effect of the pore-fillingwater.

Among the several models involving the application of mixinglaws for a mineral aggregate (e.g. Jessop 1990), we used the modelby Hashin & Shtrikman (1962), which was originally proposed forthe magnetic permeability of macroscopically homogeneous andisotropic materials. Due to mathematical analogies, it can be ex-tended to thermal conductivity calculations. The matrix conductiv-ity of the rock kmo (in W m–1 K–1) is given by

kmo = (kU + kL )

2, (5)

where kU is the conductivity upper bound defined as

kU = kmax + Amax

1 − amax Amax(5a)

with kmax is the maximum thermal conductivity of the mineralphases, amax = 1

3 kmax and

Amax =n∑

j=1

v j

1(k j − kmax

) + amax

for k j �= kmax, (5b)

where kj the thermal conductivity of the jth mineral, and vj and nare as in eq. (4). By replacing the minimum thermal conductivity ofthe mineral phases and the index ‘max’ with ‘min’ in eqs (5a) and(5b), a similar expression can be obtained for the lower conductivitybound kL.

To determine the bulk conductivity, we have taken into accountthe porosity according to the model by Zimmerman (1989). Poresare assumed to be isolated spheroids and their shape is defined by theaspect ratio, a, which is the ratio of the length of the unequal axis tothe length of one of the equal axes. Thus, pores have spherical, oblateand prolate shape, for a = 1, a < 1 and a > 1, respectively. In extremecases, that is, when pores consist of thin cracks, spheroids assumeneedle like, tubular shape (a → ∞) or thin coin like shape (a → 0).For an isotropic rock, an average orientation of the unequal axis ofthe spheroid with respect to the temperature gradient is considered.If the pores are randomly oriented and distributed spheroids, thecomputed thermal conductivity kHZ is

kH Z = kmo[(1 − φ) (1 − r ) + r β φ]

[(1 − φ) (1 − r ) + β φ], (6)

where kmo is the matrix thermal conductivity of eq. (5), φ the poros-ity and r the ratio of the thermal conductivity of the pore-filling waterand the matrix thermal conductivity. The parameter β is given by

β = (1 − r )

3

[4

2 + M (r − 1)+ 1

1 + (r − 1) (1 − M)

],

where for a < 1

M = 2θ − sin 2θ

2 tan θ sin2 θ, with θ = arcos (a)

and for a > 1

M = 1

sin2 θ− cos2 θ

2 sin3 θln

(1 + sin θ

1 − sin θ

), with θ = arcos

(1/

a).

For a = 1 the parameter M = 3(1 − r)/(2 + r).A comparison of the results for an aspect ratio a = 1 with mea-

sured values of thermal conductivity is shown in Fig. 6. The dif-ferences between the computed and measured values range from–2.2 per cent to 6.9 per cent and, on average, the computedthermal conductivity is larger by 3.5 per cent. A model that as-sumes spherical pores thus seems to be consistent with laboratoryresults.

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https://www.researchgate.net/publication/248253857_Thermal_conductivity_of_fluid-saturated_rocks?el=1_x_8&enrichId=rgreq-240b0a77-7c5d-4811-81a8-fbd4293319d2&enrichSource=Y292ZXJQYWdlOzIzMDMxMTk3ODtBUzoxMDIxNjMxNDEzNjU3NzFAMTQwMTM2OTAwMDAyOA==


Figure 6. Comparison of thermal conductivity kr measured on water-saturated samples with thermal conductivity kHZ computed with the Hashin–Shtrikman’smodel and conductivity kG computed with the geometric mixing model. kHZ takes account of porosity according to the Zimmerman’s model. Details as inFig. 5.

Figure 7. Photomicrograph, under plane-polarized light, of some clastic and chemical/biochemical rocks of Table 1 exhibiting porosity shape enhancedby methylene blue. (a) 5-Argillaceous sandstone: high porosity related to interparticle pores and intraparticle pores within planktonic foraminifera shell; (b)3-Calcareous marl: high porosity related to chalky porosity of the matrix, moldic and intraparticle pores within planktonic foraminifera shells; (c) 14-Dolostone:vugs and dissolution-enlarged fractures; (d) 12-Packstone: well-cemented limestone with open fractures that cross rock fabric.

Fig. 7 shows, however, that in some samples pores differ from aspherical shape. Well-cemented clastic rocks and crystalline rockshave a brittle behaviour, thus they may tend to split. In addition,fractures can be enlarged by dissolution, particularly in carbonaterocks. The best agreement between computed and measured thermalconductivity of mudstone, packstone, dolostone and dacite samplesis thus obtained for a more appropriate pore aspect ratio, that is,for thin coin-like shaped cracks. Moldic and intraparticle porositywithin planktonic foraminifera shells was observed in argillaceoussandstones and calcareous marls. For these rocks, deviations be-tween computed and measured thermal conductivity can signifi-

cantly decrease (on average 1.3 per cent) if a proper aspect ratio(a = 0.1) is chosen.

We also applied another technique to compute the thermal con-ductivity, that is, the geometric mixing model (see Jessop 1990)commonly used for porous rocks

kG =n∏

j=1

kν jj , (7)

where symbols are as in eq. (5b). The obtained values are plot-ted against the measured ones in Fig. 6. This method provides a

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Figure 8. Regression plot of inferred vertical thermal conductivity of sheet silicates kss and vertical matrix thermal conductivity kma as function of depth z ofthe rocks of Table 2. Solid lines are the fit curves.

less satisfactory estimation as the differences between computedand measured values range from –13.8 per cent to 0.8 per centand, on average, the computed conductivity is underestimated by4.7 per cent.

4.2.2 Anisotropic rocks

Anisotropy of shales, silty shales and siltstones is mainly due to theeffect of orientation of the clay and mica platelets during burial.Then the decrease of the vertical thermal conductivity kz with depth(Fig. 4 and eq. 2) is due to the fact that the thermal conductivityof the sheet silicates, kss, varies with depth. As the sample mineralcomposition and thermal conductivity of carbonate and frameworksilicate minerals are known, from eq. (7) we find

kss =(

kz

kνCcCc kνDol

Dol kνQtzQtz k

νK fK f kνPl

Pl

)1/νSS

, (8)

where the subscripts of the conductivity and the volume fractionindicate the minerals of Table 2. This table also reports the estimatedkss values ranging from 0.48 to 1.30 W m−1 K−1. Fig. 8 shows kss

versus depth z and the best fitting curve which is of the form

kss = 1.622 exp (−1.12 z) + 0.507, (9)

where the depth is expressed in kilometres. kss is expected to de-crease exponentially with depth from 2.13 W m−1 K−1 at the surfaceto 0.52 W m−1 K−1 at 4.5 km.

Consequently, we expect that also the vertical matrix conductivitykma (in W m–1 K–1) of anisotropic rocks decreases with depth. kma

can be determined by means of the relation

kz = k1−φ

makφ

w, (10)

where kw is thermal conductivity of water. The computed valuesof kma against depth (in kilometres) are well fitted by the linearexpression (Fig. 8)

kma = −0.251z + 2.899. (11)

At the depth of 4.5 km, kma of shales, silty shales and siltstonesis 1.77 W m−1 K−1.

4.3 Burial depth and temperature effects

Thermal conductivity and volumetric heat capacity measured orcomputed at standard pressure and temperature do not reflect therock properties at depth. Thus corrections must be applied to ex-trapolate thermal parameters to in situ conditions (Beardsmore &Cull 2001 and references therein).

The rock thermal conductivity can be regarded as a result ofthree main overlapping effects depending on burial depth and tem-perature: (i) the relative abundance of pore-filling water comparedto the solid part, which decreases with compaction, (ii) the pore-filling water conductivity, which increases with temperature and(iii) the matrix thermal conductivity, which decreases with depth asa function of temperature (pressure has only minor influence withinsedimentary basins).

In igneous, siliceous and evaporitic rocks, generally porosityvaries little with depth. Significant changes occur, instead, in clas-tic and carbonate rocks, during progressive burial, which involvescompaction caused by the weight of the overlying sediments, the ex-pulsion of intergranular fluids and the volume reorganization of thesediment grains. Compared with most minerals, pore-filling waterhas much lower conductivity and higher volumetric heat capacity,so that the rock thermal properties are very sensitive to the porosityfraction. The porosity decays exponentially with depth according to(Sclater & Christie 1980)

φ = φo exp (−b z) , (12)

where φo is the porosity at the surface (z = 0) and b (km−1) isthe compaction factor. The sedimentary rocks of the Po Basinwere grouped into four categories to infer appropriate parame-ters of the compaction curves. Fig. 9 depicts the porosity–depthcurves together with the computed values of φo and b for each rockcategory.

C© 2011 The Authors, GJI, 186, 69–81



Figure 9. Porosity–depth compaction curves for the clastic and carbonate rocks of Table 1. Values of φo and b of eq. (11) are shown.

Deming & Chapman (1988) provided an estimation of the thermalconductivity of water kw as function of temperature, T (◦C) in theform

kw = 0.5648 + 1.878 10−3T − 7.231 10−6T 2 for T ≤ 137 ◦C(13a)

kw = 0.6020 + 1.309 10−3T − 5.140 10−6T 2 for T > 137◦C.(13b)

Concerning the matrix conductivity, it is worth to remind that heatconduction in solids can occur either via lattice vibrations or viaradiation processes. As thermal radiation becomes important onlyat very high temperatures, in our analysis we took into account onlythe lattice contribution. Nevertheless, the T−1 relationship for thelattice conductivity agrees with experimental data only for isotropic,structurally perfect, single crystals (Roy et al. 1981; Buntebarth1991; Lee & Deming 1998). The thermal conductivity of rocksdecreases more slowly than T−1, and low conductivity rocks mayeven show a slight increase in thermal conductivity with temperature(Tikhomirov 1968; Cermak & Rybach 1982; Somerton 1992).

Lee & Deming (1998) evaluated the accuracy of different thermalconductivity temperature corrections over a large set of experimen-tal data of several rock types. The lowest mean relative error is givenby the expression (Sekiguchi 1984)

km = kM +[

To TM

TM − To(kmo − kM )

(1

T− 1

TM

)], (14)

where km is the matrix thermal conductivity at in situ temperature T(in K), kmo is the matrix conductivity at surface temperature To, kM

and TM are the thermal conductivity and the absolute temperatureat which Sekiguchi refers to as the assumed point of 1.8418 W m−1

K−1 and 1473 K, respectively. The thermal conductivity increaseswith burial because of rapidly declining porosity and increasingproportion of the conductive mineral matrix. At large depths, asporosity decreases more slowly and temperature rises, the effect ofdeclining matrix conductivity takes over and the conductivity showsa reversed trend.

The specific heat of both water and matrix increases with tem-perature. Somerton (1992) proposed an expression for the specificheat of pure water as a function of temperature, which holds in the20–290 ◦C range

cw = (4245 − 1.841 T ) 103

ρw

, (15)

where ρw is the density of the liquid water, whose temperaturedependence is

ρw = ρw20

1 + (T − 20) αw

, (16)

where ρw20 is the water density at 20 ◦C (in kg m–3), and αw (inK–1) is the thermal expansion coefficient of water given by

αw = 0.0002115 + 1.32 10−6T + 1.09 10−8T 2. (17)

As long as the pressure is high enough to keep the water in a liquidphase, the volumetric heat capacity of water ρwcw, under subsurface(high pressure) conditions, can be estimated with good accuracyby using eqs (15 − 17), without including pressure dependence(Somerton 1992; Waples & Waples 2004b).

Since the volumetric thermal expansion coefficient of the rocksis very small (about 10−5 K−1), density was considered as constantover the temperature range expected within the sedimentary basin,so that the volumetric heat capacity of the matrix increases in accor-dance with the rise of the specific heat as a function of temperature.The temperature dependence of volumetric heat capacity for anymineral matrix can be computed using the equation (Hantschel &Kauerauf 2009)

ρmcm = ρm cm20

(0.953 + 2.29 10−3 T − 2.835 10−6 T 2

+ 1.191 10−9 T 3),

(18)

where ρm and cm20 are the grain density, taken as a constant, andthe specific heat of the rock matrix at 20 ◦C, respectively.

5 A P P L I C AT I O N T O W E L L DATA

The modelling techniques and corrections presented in the fore-going sections permit the estimation of in situ thermal parameterson the basis of the rock mineral composition or, alternatively, fromlithostratigraphic data. The latter information is more frequentlyavailable from drilling reports. As a pratical example, we inferredin situ thermal conductivity at the Mortara and Turbigo wells ofthe Po Basin (Fig. 1) whose lithostratigraphic sequence is known ingood detail.

The Mortara well is 5905 m deep and lies above a structuralhigh corresponding to a Tertiary volcanic edifice buried belowthe Miocene–Quaternary clastic cover. The Turbigo well reached6631 m depth, where it encountered a Mesozoic structural high

C© 2011 The Authors, GJI, 186, 69–81



buried beneath an Eocene–Quaternary clastic cover. Detailed lithos-tratigraphic information and formation temperature data for thesewells are available from Eni E&P Division.

The in situ vertical thermal conductivity was estimated at themiddle point of 20 m intervals by the combination of rock matrixand water thermal conductivities obtained with eq. (7) and the pa-rameters of Fig. 9. The matrix thermal conductivity for isotropicrocks was computed with values of kr and φ as in Table 1. Forthe rocks rich in sheet silicates, eq. (11) was used. The tempera-ture effect on the water was corrected by means of eqs (13a) and(13b), whereas eq. (14) was used for the rock matrix. A regionalgeothermal gradient of 25.2 mK m−1, as estimated by Pasqualeet al. (2008b), was assumed.

Fig. 10 shows the lithostratigraphic column and in situ thermalconductivity for the two wells. In the first kilometres, the com-paction effect is larger than that due to the temperature and, forthe same lithotype, this results in an increase of conductivity withdepth. Both wells show that the maximum values of conductivityoccur in the deepest formations (dacites and dolomites). Horizons ofshales, silty shales and siltstones are present at different depths andexhibit minima. Due to the presence of thermally anisotropic sheetsilicates, note that the conductivity is constant or decreases withdepth.

The classical approach of the thermal resistance method (Bullard1939) was then applied to estimate the terrestrial heat flow in thetwo wells. The thermal resistance R (in m2 K W–1) along the verticalbetween the surface and the depth d is

R = � zd∑

z=0

(1

kin

), (19)

where �z is constant and equals to 20 m and kin is the estimate in situvertical thermal conductivity (in W m–1 K–1). Thus, the subsurfacetemperatures in a horizontally layered, isotropic medium is relatedto the thermal resistance as

Td = To + qo R, (20)

where Td is the temperature at depth z = d, To is the temperatureat depth z = 0 and qo is the surface heat flow (in W m–2). Thelinear fit of data, constrained with a surface temperature of 12.5 ◦C,allows the estimate of the surface heat flow. Figs 11 and 12 depictthe formation temperature as a function of the thermal resistance inthe two analysed wells and the inferred values of surface heat flow.The goodness of the linear fit suggests that internal heat sources(e.g. fluid convection, radioactive heat generation) are of negligibleimportance. The geotherms computed by means of eq. (20) are alsodisplayed.

6 D I S C U S S I O N

Accurate determinations of physical properties of a wide vari-ety of sedimentary and intrasedimentary volcanic rocks of the PoBasin allow the calculation of the thermal conductivity as functionof depth, by taking into account the effects due to burial, tem-perature and anisotropy. We obtained information on the thermalconductivity anisotropy from measurements on oriented samples.Chemical/biochemical, volcanic and most of clastic rocks analysedare thermally isotropic, in agreement with literature data (Deming1994; Clauser & Huenges 1995; Davis et al. 2007). We noted thatanisotropy is enhanced in fine-grained and sheet silicate-rich rocksand it increases with burial depth, in relation to a preferential orien-tation of the sheet silicates during compaction (Bennet et al. 1981;

Pribnow & Sass 1995; Waples & Tirsgaard 2002; Davis et al. 2007).This information was used to improve the accuracy and reliabilityof the vertical component of the thermal conductivity.

Cores decompression upon recovery from a well and the con-sequent increase in pore fraction when the overburden pressureis removed can be a problem in porosity determination (Pasqualeet al. 2006). On the other hand, porosity obtained in the laboratoryby saturating samples with water or with the helium pycnometercan be underestimated, because it measures only the interconnectedpores. However, errors due to laboratory procedures and the poros-ity rebound mechanism are of opposite sign and approximately ofthe same magnitude (Hamilton 1976). Therefore, one can reason-ably assume that the laboratory measurements reflect the porosityat the depth at which the sample was recovered.

Laboratory results of thermal properties were used to anal-yse the applicability of two techniques for the thermal conduc-tivity estimation from mineral composition of macroscopicallyisotropic and homogeneous rocks. The first technique combinesthe Hashin–Shtrikman’s model with the Zimmerman’s model. Theformer model provides accurate estimations of the matrix ther-mal conductivity whereas the latter takes into account the struc-ture influence, that is, the pore shape effects. Photomicrographanalysis can help to distinguish flat fractures or oblate poros-ity from a spherical shape and to include a proper aspect ra-tio in the model. The second technique is based on the geo-metric mixing model and yields underestimation of the thermalconductivity.

As the thermal conductivity anisotropy is not negligible in shales,silty shales and siltstones, we developed a relationship to infer thevertical thermal conductivity of the sheet silicates at different depthof burial. The results of the computed conductivity at large depthor high degree of compaction are well consistent with thermal con-ductivity values of sheet silicates perpendicular to main cleavageplane obtained by Diment & Pratt (1988) and Williams & Ander-son (1990). The estimated vertical thermal conductivity of the sheetsilicates significantly decreases with depth. This is consistent withthe process of sheet silicate mineral deposition. As soon as theydeposit, mineral are randomly oriented. Subsequent burial is asso-ciated with porosity decrease and during compaction the clay andmica platelets rotate to a preferred horizontal orientation, whichdecreases the vertical thermal conductivity.

The volumetric heat capacity is a scalar value and an isotropicphysical property. Given that the specific heat of a solid elementis the same whether it is free or part of a solid compound, thevolumetric heat capacity was estimated as a weighted average ofthe volumetric heat capacity of the mineral grains and water. Whencalculating the volumetric heat capacity, since the trends of specificheat as function of temperature of minerals and non-porous rocksare similar and the thermal expansion coefficient is negligible, weapplied the same correction for any mineral matrix. The differencesbetween non-porous rocks and minerals are small, usually less than1 per cent at temperatures below 500 ◦C, that is, those of majorinterest in sedimentary basins (Waples & Waples 2004a).

The temperature dependence of the matrix conductivity wastaken into account according the empirical correction suggestedby Sekiguchi (1984). This correction is independent on mineralogyand porosity. The error involved in applying the Sekiguchi’s correc-tion increases with temperature, but it tends to be relatively smallover a wide range of rock thermal conductivity (Lee & Deming1998).

The inference of thermal conductivity and volumetric heat ca-pacity has been usually approached by correlating geophysical logs

C© 2011 The Authors, GJI, 186, 69–81



Figure 10. Vertical thermal conductivity kin of the Mortara and Turbigo wells as inferred from lithostratigraphic information. Formation names are indicated.

directly with the thermal properties or by estimating mineralogyfrom log data and then using mixing laws for thermal properties es-timates (e.g. Vasseur et al. 1995; Hartmann et al. 2005). A completeset of geophysical logs is only seldom available, whereas informa-tion on the mineral composition of the rocks or the lithostratigraphiccolumn are often accessible.

The thermal conductivity versus depth was estimated at the Mor-tara and Turbigo wells from lithostratigraphic information. If theestimated thermal conductivity is correct and thermal regime ispurely steady-state conductive, the heat flow should be constantwith depth. This seems to occur for both wells, showing only smallfluctuations due to errors in estimating the degree of sediment com-paction and minor heat sources.

7 C O N C LU S I O N S

Laboratory results of thermal properties of rocks from the Po Basinwere compared to results obtained by applying mixture theories. Forisotropic rocks, experimental values of thermal conductivity are sat-isfactorily predicted by models in which randomly sized spheroidalpores are randomly dispersed in a homogeneous, isotropic andmultimineral matrix. The thermal conductivity anisotropy issignificant in some clastic rocks, and it is explained with the thermalconductivity variation of the sheet silicates as function of the burialdepth. The volumetric heat capacity was estimated as a weightedaverage of the volumetric heat capacity of the minerals and thepore-filling water. Based on this analysis, we propose an approachto calculate the thermal properties of the basin rocks at in situ

C© 2011 The Authors, GJI, 186, 69–81



Figure 11. Bullard plot and temperature profile of the Mortara well. Formation temperatures are indicated with dots. The surface heat flow (qo) and thecoefficient of determination (R-square) are shown.

Figure 12. Bullard plot and temperature profile of the Turbigo well. Details as in Fig. 11.

C© 2011 The Authors, GJI, 186, 69–81



conditions. The methodology takes into account changes in thethermal properties due to temperature, burial depth and anisotropyeffects, and it is applicable to high accuracy estimations of ther-mal properties versus depth in the Po Basin. If information on themineral content or lithostratigraphic data is available, the approachcan be used for studies of heat flow and thermal regime in anysedimentary basin.

A C K N OW L E D G M E N T S

This work was carried out within the framework of the MIUR−2008project ‘Geothermal resources of the Mesozoic basement of thePo Basin: groundwater flow and heat transport’. The authors areindebted to the Eni E&P Division (San Donato Milanese, Milan),which made available the core samples for laboratory investigations,A. Frixa, E. Vitagliano and two anonymous reviewers for the usefuldiscussions and comments.

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Thermophysical properties of the Po Basin rocks