Temperature-dependent thermal transport properties of carbonate minerals and rocks

Carbonate rocks are a significant component of the upper continental crust (Ronov and Yaroshevsky, 1976; Rudnick and Gao, 2003), and thus their thermal transport properties—thermal conductivity (k) and thermal diffusivity (D)—influence important processes such as hydrocarbon maturation and the carbon cycle, in addition to geodynamic processes and geothermal energy potential (e.g., Bickle, 1996; Berner and Caldeira, 1997; Schütz et al., 2012). Because carbonate rocks blanket many platform regions of continental interiors, their role in moderating continental heat flow is potentially important, as thermal conductivity varies more at the low temperatures associated with shallow depths than at the high temperatures commonly found in the deeper interior (Whittington et al., 2009; Nabelek et al., 2010; Liu et al., 2011; Merriman et al., 2013). Despite the ubiquitous occurrence of carbonates, little is known about the effects of mineralogy and texture on their thermal properties above ambient temperatures. The remainder of this section briefly reviews heat transfer and methods for measuring the thermal transport properties of rocks and minerals to high temperature.

All of the above properties vary as functions of temperature (T) and, to a lesser extent, pressure (P) (reviewed by Hofmeister, 2007; Hofmeister et al., 2007). In the range of ∼300–600 K, negative dD/dT dominates over positive dC_P/dT in rocks, and, because changes in ρ are relatively small, k decreases as temperature increases. Variations in k with temperature can be large, because D can decrease by as much as a factor of 10 for crustal rocks (e.g., Merriman et al., 2013; Hofmeister and Branlund, 2015), while C_P typically increases by no more than a factor of two (e.g., Robie and Hemingway, 1995; Waples and Waples, 2004). The decrease in D and k as T increases beyond ambient temperature is a consequence of stimulation of additional lattice vibrations as T increases. More interactions occur as T climbs, and so the mean free path decreases and so does D. In general, the mean free path is considered inversely dependent upon T (see Hofmeister and Branlund [2015] for a review of the relevant literature).

Furthermore, decreases in ρ with increasing T are small because thermal expansivity of crystals is low, typically ∼1–3 × 10^–5 K^–1 (Fei, 1995), and in the crust is offset by increasing P with depth. Elevated P typically increases D, but again the effect is small, on the order of 3%–5% per gigapascal (Hofmeister, 2007; Hofmeister and Pertermann, 2008). This increase in D with increasing P is primarily a result of increasing density, and these effects are negligible compared to those resulting from variations in mineralogy (Hofmeister et al., 2007).

Many experimental methods used to measure k, such as the divided bar method, produce systematic errors (see Hofmeister et al., 2007). Physical contacts between the heat source and the sample are imperfect on a microscopic scale, leading to underestimates of k by ∼10% per contact. In addition, radiative transfer can occur between the heat source and the sensor without interaction with the sample, resulting in measurements of k that are artificially elevated (Mehling et al., 1998). This “ballistic” radiative transfer is an artifact of the experimental method and is not found in natural settings (Hofmeister, 2010).

Contamination of measurements with spurious radiative transfer commonly goes unrecognized because the process is not well understood. Infrared spectroscopic data provide some insights. The effect of ballistic radiative transfer is strongest in minerals that are transparent in the near infrared (near IR) (Fig. 1). Although mineral transparency changes with temperature, many common rock-forming minerals are partially or fully transparent across the range of temperatures at which they are stable. Minerals with high concentrations of water or hydroxyl and high concentrations of Fe²⁺ in large sites are the only types that strongly absorb in this spectral region (Rossman, 1988a, 1988b). Because heat transfer data are typically collected on ∼3–5 mm thicknesses (note in Fig. 1 that absorbance (A) is calculated from A = –log(I_trans/I₀), where I_trans is the intensity of the transmitted light, and I₀ is the intensity of the incident light on the sample). A = 1 describes optically thin conditions for a 1 cm sample. Light crosses typical sedimentary minerals with little attenuation in the near-IR region. Division by the measured thickness provides the absorption coefficient. Both effects reduce throughout. Near room temperature, the effect is small, resulting in scatter in measured values. At high T, however, ballistic transport results in excessive D, and sometimes in an incorrect sign for the temperature derivative, mostly because intensity of blackbody radiation grows as T³. In rocks, scattering along grain boundaries limits this effect, but at the expense of reducing D or k due to thermal grain boundary resistance (Smith et al., 2003). These competing effects create a roughly correct answer for D or k near room T, but with substantial uncertainties.

Because the common sedimentary minerals quartz and feldspar are transparent in the near IR (Fig. 1), experimental methods for the measurement of k of these minerals (and rocks dominated by them) are susceptible to ballistic radiative transfer, particularly above ambient T (Branlund and Hofmeister, 2007). In the laser flash analysis (LFA) method, first developed by Parker et al. (1961), the effects of ballistic radiative transfer can be removed from the time-T curve using the model of Mehling et al. (1998). The model fit is shown in Figure 2. Furthermore, this method avoids physical contact between the heat source and sample, eliminating this error source and thus isolating diffusive heat transfer.

Consequently, the LFA method produces results for thermal diffusivity (D) with errors of ±5% at room temperature and ±<2% at elevated temperatures (Hofmeister, 2006). The effect of removing radiative transfer in quartz and the feldspars has been well documented (Branlund and Hofmeister, 2007, 2008, 2012; Pertermann et al., 2008; Hofmeister et al., 2009). Both show decreasing D as temperature rises above ambient (Branlund and Hofmeister, 2007; Pertermann et al., 2008; Hofmeister et al., 2009), although dD/dT is much higher for quartz than for the feldspars. This pattern extends to quartz- and feldspar-dominated rocks (e.g., Whittington et al., 2009; Miao et al., 2014), although other properties such as porosity (φ) and pore contents become important (Branlund and Hofmeister, 2008).

The dominant minerals in carbonate rocks, calcite (CaCO₃) and dolomite [CaMg(CO₃)₂], have not been examined using LFA. Note that we use the naming convention “dolomite” for the mineral, and “dolomite rock” for the sedimentary rock to avoid confusion, as both the mineral and rock appear in several of the same figures. Measurements of k of both minerals exist in the literature (e.g., Birch and Clark, 1940; Horai, 1971), but these historical studies covered a limited temperature range and used methods that were subject to the experimental errors discussed above. More recently, D of calcite was measured using a photoacoustic spectroscopic method (Ramachandran et al., 2006), but only at room temperature. Thermal diffusivity data for a suite of 11 carbonate minerals were collected across a wide range of temperatures using a modified transient technique (Gratz, 2006), but this method does not fully control radiative transfer and involves one physical contact which artificially reduces D (Hofmeister et al., 2007). Similar to quartz and feldspar, absorption spectra of these minerals in the near IR show a strong potential for spurious radiative transfer in experiments (Fig. 1).

Moreover, there is a paucity of high-temperature D and k data for carbonate minerals and rocks, which can have widely varying textures and porosity depending on their mode of formation and degree of diagenesis or metamorphism. Although the mineralogy of limestones is relatively simple, considerable variety arises from packing of different biological detritus of highly variable shape and size, and, combined with chemical precipitation of calcite, leads to a spectrum of textural variation. Furthermore, recrystallization can occur at various stages of maturity post-diagenesis, with fossils acting as nucleation points or centers of replacement (Lippmann, 1973). Consequently, φ in carbonate rocks varies widely. At near-surface P, calcitic limestone generally has higher φ than dolomite rock, but the reverse is true when P > ∼50 MPa in natural settings (Hantschel and Kauerauf, 2009).

Many workers have proposed models for the prediction of the temperature response of k of rocks using measurements at room T (Clauser and Huenges, 1995; Lee and Deming, 1998). These are based on results from contact methods, however, and do not incorporate mineralogical differences. Lee and Deming (1998) reviewed five such models for T-dependent k based on single room-T measurements of k. However, demonstrably more complex behavior occurs in minerals than can be captured by a single equation for different minerals, an observation that extends to measurements of rocks composed of more than one mineral (Whittington et al., 2009; Merriman et al., 2013), and thus a single equation is unlikely to reproduce the range of observed changes in thermal transport properties with T across all rock types

We present new D data on calcite, dolomite, magnesite (MgCO₃), and rhodochrosite (MnCO₃), and a suite of carbonate rocks, measured using the LFA method. We also measured ambient ρ and the temperature dependence of C_P, from which we calculate k as a function of T. We test existing models and explore the effects of porosity on the T-dependent thermal transport properties of these important rocks.

Four different species of carbonate minerals were studied (Table 1), including two calcite samples (calcite spar and hexagonal calcite [calcite hex], both from the collection of the Department of Earth and Planetary Sciences at Washington University, St. Louis, Missouri, USA), a dolomite sample with both white and grey ends (white dolomite and grey dolomite, from Eugul, Navarre, Spain), magnesite (from Brumado, Bahia, Brazil), and rhodochrosite (from Wudong Mine, Wushuo Prefecture, Guangxi Zhuang, China). Magnesite was included to provide data on the Mg-bearing layers of dolomite. The addition of rhodochrosite permits examination of the effect of cation mass on mineral D. Rock samples include four calcite-dominated limestones from central Missouri: TF-001, a fossiliferous limestone from the Winterset Limestone; TF-002, an oolitic limestone from the Drum Limestone; MO-05, a boundstone from the Burlington Limestone; and MO-07, a micrite from the University of Missouri departmental collection. Additional rocks include: sample KS-002, a sedimentary dolomite rock from the Jefferson City Dolomite (Missouri), four calcite-dominated marbles, samples of Threshold marble (a building stone from a commercial retailer, Lowes, in Maplewood, Missouri, USA), Black marble (also from a commercial retailer, The Tile Store, in St. Anne, Missouri, USA), and Carrara Marble (Italy), including a marble deformed at ambient temperature under triaxial loading (Carrara deformed); sample AW-05, a marble with mixed calcite and dolomite phases (from Ross Lake near North Cascades National Park, Washington, USA); sample 17-4, a dolomitic marble (from the Eureka Valley–Joshua Flat–Beer Creek [EJB] aureole, White Mountains, California, USA; Nabelek and Morgan, 2012); sample 33-1, a forsterite- and chlorite-bearing calcite-dolomite marble (also from the EJB aureole); and a magnesite conglomerate (from the departmental collection at Washington University). The suite of rocks permits exploring effects of (1) depositional environments ranging from intertidal to shallow marine; (2) dolomitization; (3) porosities; and (4) contact and burial metamorphism involving variable mineralogy. We used polarized-light microscopy to describe mineralogy and textures and to measure grain sizes.

Mineral proportions were determined using a Bruker D8 ADVANCE X-ray diffractometer (XRD). Weight-percent proportions were calculated from diffraction patterns with the Bruker software TOPAS (version 5), using the refinement method of Rietveld (1968). Results were converted to volume percent using published values of density for minerals (Deer et al., 1992), and using microprobe data to calculate end-member proportions of solid-solution minerals such as olivine.

Mineral compositions were analyzed by wavelength-dispersive analysis with a JEOL JXA-8200 electron microprobe, and using Probe for Windows software for data reduction (Probe Software, Inc., Eugene, Oregon, USA; http://www.probesoftware.com/). The measured data were corrected with CITZAF software after Armstrong (1995). Oxide and silicate standards were used for calibration: Amelia albite for Na and Si; microcline for K; Gates wollastonite for Ca; Alaska anorthite for Al; synthetic fayalite for Fe; synthetic forsterite for Mg; synthetic TiO₂ for Ti; synthetic Mn-olivine for Mn; and synthetic Cr₂O₃ for Cr. Detection limits, calculated error, and calibration standards are available in Table S1¹.

Chemical analysis of rock samples was performed on limestones TF-001, TF-002, MO-05, MO-07, and KS-002 and marbles 33-1, 17-4, and Black marble by Activation Laboratories, Inc. (Ontario, Canada), using inductively coupled plasma–optical emission spectroscopy (ICP-OES) on lithium metaborate fusions for major oxide analysis, and combustion spectroscopy for total carbon and graphitic carbon analysis (through combustion and binding of carbon with oxygen to form CO and CO₂). Calibration standards and duplicate measurements are available in Table S2 [^{footnote 1}].

We used an evacuated Bomem DA3.02 Fourier transform interferometer with an SiC source, an InSb detector, and a CaF₂ beamsplitter for IR spectroscopy. Instrumental accuracy is 0.01 cm^–1. About 1000 scans were collected at room temperature from ∼1200 to 5000 cm^–1 at a resolution of 2 cm^–1. Data were collected from samples mounted on an ∼1 mm aperture, which served as the reference.

Density was measured using three different methods. (1) Bulk density was determined using a geometric method, where the sample was cut into a regular shape (cube or cylinder) and then measured with a micrometer, and weighed on an analytical balance. Longer cores (1–3 cm) were used to reduce errors associated with imperfections. Uncertainties are estimated at 2% based on variability between different cores of the same sample. (2) Density of the solid and its isolated pore spaces was ascertained using the Archimedean method. Samples were weighed in air and again while immersed in fluid (ethanol). The sample was soaked overnight to enable fluid penetration throughout connected pore space. (3) A Quantachrome multipycnometer was used to measure the bulk density of samples that were powered to <150 µm to allow helium to penetrate previously unconnected pore space. Sample powder was weighed in a sample chamber, which was then placed in a sealed chamber in the multipycnometer. Helium was added to a reference chamber to a measured pressure of ∼117 kPa and opened to the sample chamber. With helium present in both the reference and sample chamber, the new pressure was recorded, and the ratio of reference to sample plus reference chamber pressure was used to calculate the volume of the sample.

These three measurements were combined to calculate total porosity and connected porosity after Avard and Whittington (2012).

For minerals, we used published C_P data (calcite: Jacobs et al., 1981; dolomite: Krupka et al., 1985; magnesite: Robie and Hemingway, 1994; rhodochrosite: Moore, 1943). We combined these data for minerals with the modal mineralogy from XRD results to calculate ideal heat capacity for the rock samples to compare experimental results for powdered rock.

We used a NETZSCH LFA 427, described by Bräuer et al. (1992), to measure thermal diffusivity. For a general descriptions of this method, which is commonly used in industry and materials science, see Maglić and Taylor (1992), Vozár and Hohenauer (2003, 2005), and Criss and Hofmeister (2017). For procedural details regarding our laboratory, see Pertermann and Hofmeister (2006).

In brief, samples were prepared as nearly circular disks between 0.75 and 2 mm thick, with a diameter of ∼10 mm. All rock samples had grain sizes smaller than disk thickness, ensuring that measured D reflects the bulk material rather than individual crystals. Samples were coated with one to two layers of graphite to aid in absorption of the laser pulse and emission of heat from the top side of the sample.

The method is contact free: A Nd:GGG laser is used to heat the bottom of the sample. As heat diffuses through the sample, an IR detector on the opposite side of the sample from the laser records the change in emissions with time, which is directly related to the change in temperature. Thermal diffusivity of the sample at the given temperature is obtained by fitting the time-temperature (t-T) acquisitions using the model of Cowan (1963) at ambient temperature and Mehling et al. (1998) above ∼300 K to remove spurious effects of ballistic radiative transfer.

Maximum temperatures attained depended upon the sample maintaining physical integrity at elevated temperature without the presence of confining pressure. For each sample, D is measured at discrete temperatures along a pre-programmed path. Each reported measurement is the average from t-T curves of between two and six laser shots at that temperature, to ensure reproducibility. Measurement uncertainty is highest near room temperature (up to 5%), decreasing to ∼2% at elevated temperatures, as ascertained by comparison with steel, iron, graphite, and Pyroceram standards reference materials from the National Institute of Standards and Technology and the manufacturer (Henderson et al., 1998a, 1998b). These standards are opaque and also soft, so they do not transmit light ballistically and adhere well to thermocouples, permitting calibration against results obtained from conventional methods, e.g., that of Hust and Lankford (1984). Temperatures were calibrated against the well-established Curie points of iron and cobalt.

Calcite spar is colorless and transparent, and microprobe analysis (Table 2) indicates a nearly pure calcite crystal with trace quantities (<0.1 wt%) of magnesium, iron, and sodium. Calcite hex is a colorless, hexagonal calcite crystal (microprobe data unavailable).

The dolomite samples (white dolomite, grey dolomite) were cut from the same crystal. Both white and grey samples are nearly pure (Table 2) with small amounts of iron (FeO = 0.18−0.42 wt%) and manganese (MnO = 0.1 wt%), but total CaO + MgO is low relative to ideal dolomite. Trace impurities (<0.1 wt% oxide) include Al, Na, and Ti. The color of grey dolomite likely results from a larger proportion of FeO compared to white dolomite.

The translucent magnesite crystal includes substitutions of Mn, Fe, and Ca for magnesium, with Mg composing 98% of cation mass within the crystal (Table 2). Additional trace elements include Na, Al, and K.

The rhodochrosite sample is pink in hand specimen. Microprobe analysis (Table 2) indicates 1.4 wt% FeO, with a smaller amount of Mg and trace quantities of Al, Ca, Na, and K. A solid solution exists between pure rhodochrosite and siderite (FeCO₃), and the mineral formula of this specimen is (Mn_0.97Fe_0.02Mg_0.005)CO₃.

Sample TF-001 is a fossiliferous limestone composed primarily of calcite with ∼3 wt% quartz (Table 1). The sample has a wide range of grain sizes, with the largest grains concentrated in the center of the matrix between fossils (Fig. 3A). Fine-grained material composed mostly of calcite with minor quartz occurs in the center of fossils and along the boundaries between clusters of smaller fossils. Pore space is estimated at 9%, which is the difference between the measured density and the density calculated as the weighted average of the constituent minerals. Bulk rock chemical analysis shows small amounts of Si as well as Fe and Mg (Table 3).

Sample TF-002 is a limestone dominated by ooids and rounded fossils. Ooids are fairly uniform in size, ranging from ∼0.25 to 0.5 mm in diameter, and are composed of fine-grained bands radiating from a central point. Larger grains are concentrated in the matrix (Fig. 3B). This sample has the lowest amount of calcite of the four limestones (93.8 wt%) and the highest φ (13%). Chemical analysis suggests that either XRD analysis overestimates quartz content by several percent (Table 3) or some heterogeneity exists within the sample, and also indicates that the sample has small amounts of Fe, Al, and Mg.

Sample MO-05 (boundstone) is nearly pure calcite (Table 1) with small amounts of dolomite and quartz in a highly variable matrix, which lacks immediately recognizable fossils (Fig. 3C). Porosity (6%) is low compared to that of the above limestones, and not apparent in thin section. Bulk rock chemistry confirms a calcite-dominated mineralogy with small amounts of silica and magnesium (Table 3).

Sample MO-07 (micrite) is nearly 100% calcite (Table 1), with a very small average grain size (∼0.005 mm) and very low φ (∼2%). Most of the sample is composed of indistinguishable small grains; however, clusters of larger grains of calcite occur randomly throughout (Fig. 3D). Chemical analysis confirms the nearly pure calcite makeup of this sample (Table 3).

Sample KS-002 (dolomite rock) is largely dolomite (79%), but also contains ∼11% potassium feldspar and 10% quartz (Table 1). Bulk chemical analysis shows that a small amount of Fe is present, but only trace quantities of Na, suggesting that feldspar is dominated by K. This sample is also typified by a high φ (15%), with an overall fine-grained matrix broken by larger detrital quartz and feldspar grains (Fig. 3E).

White Carrara Marble from Tuscany, Italy, has been quarried since the time of the Roman Empire. XRD revealed calcite as the only mineral phase, and microprobe analysis shows small amounts of Mg and only trace amounts of Fe and Al (Table 2). Two different samples of Carrara were used in this study: a fresh sample and an experimentally deformed sample. The fresh sample shows mostly interlocking grains of calcite with occasional fine-grained interfaces between larger grains (Fig. 3F). In hand specimen, the outer layers of the fresh sample are brittle, but the interior remains relatively coherent. A sample deformed at room temperature by George Dresden was provided by Brian Evans (Xu and Evans, 2010), with deformation resulting in an order-of-magnitude decrease in average grain size (Table 1; Fig. 3G), as well as a fourfold increase in φ (4% fresh, 16% deformed).

Threshold marble is a commercial tile. This sample is also a pure calcite marble with ∼4% porosity, but has a slightly larger average grain size than Carrara Marble (0.08 mm versus 0.06 mm). Threshold marble contains calcite crystals with strong twinning and cleaner grain boundaries than Carrara (Fig. 3H). Microprobe analysis indicates calcite crystals with minor Mg substitution (Table 1).

Black marble is 99% calcite, containing ∼1% of grey quartz veins. Like Carrara Mable and Threshold marble, Black marble contains small amounts of magnesium (Table 3) but lower amounts of total CaO (Table 3), and some quartz (Table 1) which the others lack. In thin section, this sample has what appears to be cloudy veins of graphite (Fig. 3I), but chemical analysis by combustion revealed only small amounts (0.06 wt%). In addition to quartz and (possibly) graphite veins, bands of large, interlocking calcite crystals crisscross the sample (Fig. 3I). Porosity of this marble is relatively low (4%).

Sample 33-1 is a calcitic marble (calcite = 58.5%) with high contents of dolomite (10.3%), forsteritic olivine (16.0%), and chlorite (15.2%) as indicated by XRD (Table 1). In hand specimen, the sample has alternating white and grey layers of thicknesses between 0.5 and 2 cm which are not apparent in thin section (Fig. 3J), but may result from variable amounts of olivine or chlorite. Both olivine and chlorite are nearly pure Mg end members (Table 2). Dolomite crystals are modestly deficient in Mg, whereas calcite contains up to 1.8 wt% MgO. Bulk rock chemical analysis (Table 3) suggests there is insufficient alumina present to support the 15 wt% chlorite indicated by XRD. This may be a scale effect due to compositional heterogeneity between white and grey layers. In thin section (Fig. 3J), large grains of calcite surround rounded to subrounded grains of olivine and dolomite, and no porosity is evident, which is confirmed by density measurements (Table 1). Average grain size is larger than that of all other rock samples except 17-4.

Sample 17-4 is a dolomite-dominated marble from the same metamorphic aureole as sample 33-1, and contains ∼6% calcite. Like 33-1, this sample has white and grey layers in hand specimen, which may result from variable proportions of calcite and dolomite (Fig. 3K). The composition of dolomite is homogeneous throughout (Table 2). Bulk rock chemical analysis found <1% of silica that was not detected by the XRD. 17-4 has the largest grain size of any of our samples (0.4 mm), but no measurable φ (Table 1).

Sample AW-05 is a foliated and lineated calcitic marble (calcite = 56%) accompanied by a substantial proportion of dolomite (∼37%) and some chlorite (∼7%) (Table 1). Like for the other marbles, microprobe analysis indicated that dolomite in this sample is somewhat deficient in Mg. The hand specimen is white with thin foliation planes (1–2 mm). In thin section (Fig. 3L), large calcite crystals form foliation planes of moderately sized interlocking grains surrounded by microcrystalline dolomite. Dolomite grains are highly altered near large calcite crystals, but fresh where large calcite crystals are absent. Average grain size of dolomite matrix is ∼0.1 mm, whereas calcite crystals are as large as 2.0 mm. The bimodal distribution of grain sizes results in an average grain size of ∼0.7 mm. Porosity is minimal (<1%).

The magnesite conglomerate sample is largely magnesite with a small amount of accessory dolomite and ∼8% porosity (Table 1). The magnesite is deficient in Mg and contains large amounts of Ca, with smaller amounts of Si, Al, Na, and K; the dolomite has excess Mg and is deficient in Ca (Table 2).

Experimental C_P results on the rocks studied here agree reasonably well with C_P calculated using literature data on mineral components and modal mineral proportions from XRD analysis (Fig. 4). Measurements of sedimentary samples were limited to <∼600 K. Metamorphic carbonate rocks heated to higher T depart significantly from calculated C_P. Powders for samples heated to high T showed signs of significant alteration: typically, the powder darkened, and mass of the sample dropped by 1%–2%, consistent with the onset of decarbonation reactions. Thus, fits of experimental C_P to Equation 2 (Table 4A) do not include measurements above 600 K.

Because initial C_P measurements on rock samples TF-001, 17-4, and 33-1 and the magnesite conglomerate sample differed from calculated C_P by >3%, duplicate experiments were performed on these samples, and all were within 2% of calculated C_P. All final results were consistent to <2% at temperatures between 360 and 600 K. In the temperature range 280–360 K, all repeat experiments were within 3% of C_P calculated with literature data except that for the magnesite conglomerate, which differed by <4% in this temperature range. This may be due to adsorbed water on the powder used for measurement, as water has a heat capacity >4000 J kg^–1 K^–1 at room T (e.g., He et al., 2015). The powder for the second run (but not the first) was dried in an oven at ∼400 K for 24 h. Variations of 1%–3% are reasonable given the experimental uncertainty and uncertainty in modal mineral proportions.

The thermal diffusivity of calcite spar was measured from ∼300 to 900 K in two orientations: with heat flowing parallel to the c-axis (aa), and with heat flowing perpendicular to the c-axis (ac). At room temperature (296 K), D for calcite was very similar for both orientations: 1.66 mm² s^–1 for aa and 1.63 mm² s^–1 for ac (Fig. 5). D decreased rapidly on heating to ∼500 K, then decreased more slowly at higher temperatures. For the ac orientation, D decreased more rapidly and was ∼3%–10% lower than for aa. By 888 K, D was ∼0.47 mm² s^–1 for aa and ∼0.43 mm² s^–1 (886 K) for ac.

Dolomite had consistently higher D than calcite, varying between 2.82 mm² s^–1 for white dolomite in the ac orientation at 298 K and 2.46 mm² s^–1 for grey dolomite (ac) at 297 K. As with calcite, D again decreased rapidly in the range 300–500 K, and more slowly above ∼700 K. Thermal diffusivity reached ∼0.74 mm² s^–1 at 766 K for grey dolomite in both orientations.

Magnesite and rhodochrosite form a solid solution, and given the lack of anisotropy in heat flow for calcite and low anisotropy in dolomite, an orientation between [001] and [100] was measured for both minerals. The Mg end-member magnesite had the highest D of any carbonate sample across the range of temperatures measured, decreasing from 4.36 mm² s^–1 at 296 K to 1.64 mm² s^–1 at 615 K. D for the Mn end-member rhodochrosite was between that of dolomite and that of calcite, decreasing from 2.16 mm² s^–1 at 296 K to 0.71 mm² s^–1 at 634 K.

Dolomitic limestone KS-002 reflects the higher diffusivity of the mineral dolomite when compared to more calcitic limestones (Fig. 6). KS-002 has a D of 1.48 mm² s^–1 at 293 K, which decreases to 0.55 mm² s^–1 at 597 K. At similar temperatures, micrite MO-07 produced a D of 1.38 mm² s^–1 at 296 K, decreasing to 0.55 mm² s^–1 at 531 K. The samples with large fossil fragments or ooids produce the lowest D for limestones across the range of temperatures. Boundstone MO-05 had a D of 1.30 mm² s^–1 at 295 K, decreasing rapidly to 0.48 mm² s^–1 at 529 K. Sample TF-001, a fossiliferous limestone, was only measured at 294 K, and produced a D of 1.04 mm² s^–1, similar to sample TF-002, an oolitic limestone with a D of 1.03 mm² s^–1 at 293 K. TF-002 was heated to 478 K, however, where D was 0.52 mm² s^–1.

Experimental data for limestones were fit to Equation 3 (Table 4B).

The thermal diffusivity of the marble samples varied more than that of the limestones, depending primarily on dolomite content (Fig. 7). Whereas room-temperature D for the AW-05, Black marble, and Threshold marble samples was comparable to that of limestone, at 1.45 mm² s^–1, 1.37 mm² s^–1, and 1.29 mm² s^–1, respectively, D dropped more rapidly on heating to 0.48 mm² s^–1 at 555 K for the Black marble and 0.60 mm² s^–1 at 561 K for the Threshold marble. Thermal diffusivity of sample AW-05, however, remained elevated at higher temperatures (0.73 mm² s^–1 at 523 K), where its D is similar to that of the dolomitic marbles. The marbles were measured to higher temperatures than the limestones, with D values approaching 0.3 mm² s^–1 for Black marble (0.31 mm² s^–1 at 880 K), calcite marble 33-1 (0.33 mm² s^–1 at 870 K), and Threshold marble (0.35 mm² s^–1 at 877 K). At room temperature (294 K), D for sample 33-1 was 1.49 mm² s^–1 perpendicular to layering and 1.24 mm² s^–1 parallel to layering, but this difference disappeared at high temperature. Dolomitic marble 17-4 had the greatest anisotropy at ambient temperature, from 2.00 mm² s^–1 (parallel, 300 K) to 2.01 mm² s^–1 (perpendicular, 298 K) for the white sample, and from 2.64 mm² s^–1 (parallel, 299 K) to 2.25 mm² s^–1 (perpendicular, 297 K) for the grey sample. These differences were reduced at higher temperatures.

The first disk of Carrara Marble had the lowest D of all samples measured at ambient temperature (0.82 mm² s^–1 at 294 K), falling to 0.33 mm² s^–1 at 702 K. The experimentally deformed sample had a higher D at room temperature (294 K), 1.10 mm² s^–1, but similar high-temperature values of 0.34 mm² s^–1 at 732 K and 0.29 mm² s^–1 at 870 K. A subsequent measurement of the same sample at room temperature (298 K) was only 0.53 mm² s^–1, suggesting that irreversible structural changes occurred in the sample during heating. As the sample is heated, grains within the disk expand differentially, opening new pore space, which remains upon cooling. The result is a more porous, friable sample whose intergrain thermal resistance is much higher.

Three additional disks of fresh Carrara were measured to assess reproducibility and explore the cause of low D in the first disk (Fig. 8, run 2, run 3, and trapezoid). Room-temperature D measurements were moderately higher at 0.91 mm² s^–1 (296 K), 0.94 mm² s^–1 (297 K), and 1.07 mm² s^–1 (297 K). In hand specimen, the sample was brittle to the touch, but when outer layers were removed, the interior was found to be less brittle. Disk 3 was reground twice, each time reducing outer-layer brittleness and increasing D at room T (Fig. 8). Finally, disk 4 (trapezoid) was measured at elevated temperature, and its D dropped rapidly to within error of results for disk 1 by ∼500 K.

Experimental data for some of the Carrara runs (run 1, trapezoid, and deformed), Threshold marble, Black marble, sample 33-1, and magnesite conglomerate were fit to Equation 3. Data for sample 17-4 were fit to Equation 3.

Because C_P and ρ typically vary less than D between carbonate minerals, mineral k mirrors D, with dolomite displaying high k at room T, and a steep negative dk/dT (Fig. 9). Room-temperature k was as high as 6.85 W m^–1 K^–1 (298 K) for grey dolomite (ac orientation) and as low as 5.97 W m^–1 K^–1 (297 K) for white dolomite (ac orientation). As with D, there is no strong dependence on orientation for either grey or white dolomite, as grey dolomite has a higher k at room T in the aa orientation, but white dolomite in the aa orientation has lower room-temperature k compared to the ac orientation. All k values for dolomite converge to a narrow range as temperature increases, and by ∼500 K vary by <4%. The decrease in k as temperature increases appears to level off and approach minimal values near ∼2.3 W m^–1 K^–1 for the highest temperatures at which D was measured (898 K, grey dolomite [aa]).

Calcite shows a similar small dependence of k on orientation of the crystal, with nearly identical k at room T, departing slightly as temperature increases (Fig. 9). Calculated k for calcite remains roughly 60% that of any dolomite sample over the range of temperatures measured. For example, k at room T was ∼3.7 W m^–1 K^–1 in the aa orientation, compared to ∼6.6 W m^–1 K^–1 for grey dolomite (aa). At 620 K, this difference was ∼2.1 W m^–1 K^–1 versus ∼3.3 W m^–1 K^–1 respectively. Both minerals showed similar decreases in k over the range of temperatures in which D was measured. Calcite (aa) k drops by ∼58% between 300 K and 890 K, and grey dolomite (aa) ∼66% over the same temperature range.

For magnesite, the product of high D at all temperatures with higher ρ (3010 kg m^–3) and C_P (Fig. 9) combined to produce k ∼80% higher than for dolomite. Although D of magnesite was measured to only 600 K, the k of magnesite was highly T dependent, dropping by 45% between 300 K and 600 K. Similar percentage decreases in k for both calcite and dolomite occurred across the same temperature range (∼45%–48%).

The k of rhodochrosite was between that of calcite and dolomite. Lower C_P for rhodochrosite (∼700 J kg^–1 K^–1 at 296K) is somewhat offset by higher ρ (3700 kg m^–3) resulting in a room-temperature k of 5.59 W m^–1 K^–1 (296 K), remaining higher than calcite and lower than dolomite to high T (2.50 W m^–1 K^–1 at 634 K). Rhodochrosite k shows a similar dk/dT to that of calcite and dolomite.

As with most of the carbonate minerals, all samples of limestone and dolomite rock show a strong decrease in k with increasing T (Fig. 10A). Samples MO-07 (micrite) and KS-002 (dolomite rock) had very similar k values throughout the range of temperatures calculated, with MO-07 having a k of 3.14 W m^–1 K^–1 at 296 K, decreasing by ∼50% to 1.64 W m^–1 K^–1 at 531 K and similar values for KS-002. Calcitic boundstone MO-05 had a higher dk/dT than any other sample calculated as k dropped by ∼55% from an already low 2.74 W m^–1 K^–1 at 295 K to a very low 1.27 W m^–1 K^–1 at 529 K. Oolitic limestone TF-002 was similarly insulating at high T, but across the entire temperature range calculated (∼2.0 W m^–1 K^–1 at 294 K, and ∼1.2 W m^–1 K^–1 at 478 K).

Thermal conductivity of undeformed Carrara Marble at 294 K (Fig. 10B) was lower than that of any of the limestones at ∼1.8 W m^–1 K^–1, but the deformed sample had a higher k of ∼2.1 W m^–1 K^–1 (294 K). This contrast disappeared at elevated temperatures, with the undeformed sample having a k of 1.29 W m^–1 K^–1 at 484 K and the deformed sample a similar k of 1.19 W m^–1 K^–1 at 513 K. Threshold marble and Black marble both had k in the range of measured limestones as well, but higher than that of the Carrara Marble. Marble 33-1 was the only calcite-dominated marble with room-temperature k higher than that of the limestones at 3.17 W m^–1 K^–1 (average of all orientations), but contains high-D forsterite and dolomite (Table 1).

Unlike sample TF-002 (dolomite rock), dolomite-dominated marble 17-4 strongly reflected the higher k of the mineral dolomite with a room-temperature k of 5.4 W m^–1 K^–1 (300 K, average of four orientations). The decrease in k for sample 17-4 was ∼45% between 300 K and 600 K. This decrease mirrors that of the mineral dolomite across the same temperature range.

Previous LFA measurements have shown that minerals rich in magnesium tend to produce high D when compared to isochemical analogs with heavier cations. For example, Pertermann and Hofmeister (2006) reported a room-temperature D for pure forsterite in [001] of 5.14 mm² s^–1, but substitution of 0.005 mole fraction Co for Mg in the same orientation lowered D to 4.17 mm² s^–1, and 0.08 mole fraction Fe for Mg further reduced D to 2.60 mm² s^–1 also in [001]. This relationship appears to hold for carbonates (Fig. 11). Magnesite, dolomite, and calcite fall on a linear trend of increasing D with decreasing cation mass for group 2 cations. We predict that huntite, a compound carbonate with the formula Mg₃Ca(CO₃)₄, thus would have a room-T thermal diffusivity of ∼3.3 mm² s^–1 if it also follows this trend.

Rhodochrosite does not follow this trend, despite having the same trigonal structure. This suggests that D for the other transition-metal trigonal carbonates (siderite, otavite, smithsonite, sphaerocobaltite, and the compound trigonal carbonate ankerite) cannot be projected from these results using cation mass alone, however we were unable to locate sufficiently pure samples of these minerals to test this hypothesis. A similar pattern is observed when plotting D versus unit cell size (Fig. 11B), with D increasing with decreasing unit cell size from calcite to dolomite to magnesite, and rhodochrosite similarly plotting off of this trend.

Rock thermal diffusivity varies as a function of both mineralogy and texture. Dolomite-dominated rocks reflect the higher D of the mineral dolomite across all temperatures measured for similarly textured rocks. For example, Threshold marble, which is dominated by calcite, had a room-temperature D of 1.29 mm² s^–1, but D of dolomitic marble 17-4 was nearly twice that (2.00–2.64 mm² s^–1 depending on orientation). Both Threshold marble and marble 17-4 have moderately large grains, clean grain boundaries, and low porosities (<5%). This same effect is reflected in sedimentary rocks, where the dolomite rock KS-002 had a D at room T of 1.48 mm² s^–1, compared to 1.03 mm² s^–1 for oolitic limestone (TF-002), despite similar porosities (15% versus 13% respectively).

Several previous studies have explored textural controls on thermal transport properties within carbonate rocks, specifically the effects of porosity (ϕ) on k, resulting in theoretical models based on experimental results, statistical analysis, and numerical models of heat flow on a hand-sample scale (e.g., Carson et al., 2005; Çanakci et al., 2007; Yaşar et al., 2008; Alishaev et al., 2012; Fuchs et al., 2013). Carson et al. (2005) proposed two end-member models of effective k in which heat flow within the material is through either “internal” porosity or “external” porosity (Fig. 12, where Maxwell-Eucken 1 represents the upper bounds of internal porosity and Maxwell-Eucken 2 the lower bounds of external porosity). In “internal porosity materials”, heat flows primarily through the continuous medium surrounding the dispersed phase (in this case, pore spaces), and in “external porosity materials”, heat flows through both the continuous phase and dispersed medium, but preferentially takes the pathway through the dispersed phase. The practical repercussions of this are that k does not necessarily vary linearly with porosity, and consequently neither does D.

Besides porosity, factors affecting D can include grain size, distribution of dispersed phases within the rock, and the presence and nature of fluids within pore space. Grain size varied little for the samples in this study. The LFA method requires rocks to be measured “dry”, allowing us to focus on mineralogy and porosity. Porosities vary from ∼3% to 19% within the suite, and higher porosity correlates with lower bulk-rock D (and consequently, k; see Fig. 12).

Sample selection and preparation may influence measured D (or k). For example, Carrara Marble is dominated by calcite and has a low porosity (ϕ = 4%), but had the lowest D of any sample studied here. The sample is brittle to the touch, but contains a more cohesive core when the outer layers are removed. When D was measured for the internal cohesive core, the sample produced higher a higher D (Fig. 8, Run 3 [reground 2×], “Trapezoid”). This has two implications: First, poorly cohesive rocks may be slow conductors even if little pore space is present. Cement between grains may consist of very fine material, which acts as a thermal barrier to conduction (Branlund and Hofmeister, 2008). Second, poorly consolidated rocks could produce anomalously low D or k even if measured with accurate methods because sample cohesion has been compromised during or following depressurization.

The thermal diffusivity of carbonate minerals is broadly similar to that of many common silicate minerals. For example, directionally averaged forsterite has a room-T thermal diffusivity of ∼2.5 mm² s^–1 (Pertermann and Hofmeister, 2006), compared to directionally averaged dolomite of this study (white + grey) at 2.6 mm² s^–1. This is significantly lower than the D of directionally averaged quartz (4.9 mm² s^–1; Branlund and Hofmeister, 2007), but dD/dT of dolomite is higher than that of forsterite. Near 600 K, dolomite D falls to ∼1.1 mm² s^–1, where D for directionally averaged quartz is ∼1.7 mm² s^–1 and for forsterite is ∼2 mm² s^–1. At the maximum temperature measured for dolomite (898 K), D approaches a limit of ∼0.6 mm² s^–1, while both forsterite and quartz approach ∼1 mm² s^–1. Magnesite displays a D that is similar to that of quartz, with D of magnesite of 4.2 mm² s^–1 at 300 K, and falling to 1.7 mm² s^–1 at 600 K.

Calcite has D values that compare to those of more insulating silicate minerals such as albite (directionally averaged Amelia albite D = 1.36 mm² s^–1 at 293 K; Hofmeister et al., 2009), but significantly lower than that of dolomite. Furthermore, calcite D appears to reach a minimum at a much lower temperature and lower values (∼0.5 mm² s^–1 near 600 K) than that of dolomite. This suggests that, barring textural differences, calcite-dominated rocks should be more insulating than dolomite-, quartz-, or olivine-dominated rocks, and that sedimentary basins dominated by calcite are more insulating than those dominated by dolomite or quartz.

Several models have been published that use single-value low-temperature k measurements to model the temperature dependence of k for the rock. This is an attractive concept, as measuring the temperature dependence of k for every rock from a drill-hole core is impractical and expensive. Figures 13A–13C show the results of using several of these models with low T values for rocks from this study to predict the temperature response of k for the sample compared to the calculated k values using measured D, C_P, and ρ.

Although several models produce similar T versus k trends for some samples (e.g., calculated k for sample 33-1 is reproduced by the Chapman et al. [1984] model with <5% error), no model consistently predicts the T dependence of k for the suite of rocks studied here. Thus, using these models to predict k for a sample across temperatures found in the crust would produce moderate to significant error (∼5%–125%). This is unsurprising, as these models are calibrated using techniques with systematic errors resulting from contacts and/or ballistic radiative transfer, and commonly do not use equations that have a basis in the physics of diffusive transport. Thus, even an accurate low-T measurement used with one of these models will produce an unpredictable error in calculated elevated-T properties.

The results of this study have several implications for basin modeling, and suggest further lines of enquiry. First, closure of pore space with depth in a basin is predicted to lead to an increase in D and k as recrystallization occurs and low-conductivity pore space and amorphous or organic material are replaced with crystalline grains. This is apparent from the contrast in both D and k between porous limestones TF-002 and MO-05, low-porosity MO-07, and calcite-dominated Threshold marble and Black marble (Figs. 6 and 10A), which persists to elevated temperatures. This result is not a new concept (e.g., Carson et al., 2005), but previous methods used to constrain the effects on porosity have an uncontrolled parameter in the form of experimental error associated with imperfect contacts. Note that pressures attained in the crust are on the order of ∼1–1.5 GPa, and because D increases only ∼4% per gigapascal, this effect can be neglected (see introduction and references therein).

Because the results presented here show a stronger temperature dependence and variability for D and k of carbonate rocks than predicted by current models (Fig. 13), the geothermal gradient of crusts with carbonate rocks and the conductive properties that influence it are more strongly coupled. As an example of the interplay of these effects, we present a model of two ad hoc crustal columns with 10 km of carbonate rocks in the upper crust as shown in Figure 14, and using input parameters as listed in Table 6. The two model columns differ by the dominant upper-crustal mineral: one is calcite dominated, and one is dolomite dominated. Inputs for thermal diffusivity were chosen to roughly mirror a reduced porosity with depth, with limestones composing the upper 6 km of the calcite-dominated model, and the calcite marble Threshold marble filling out the lower 4 km of the upper crust. By comparison, dolomite rock KS-002 overlies dolomitic marble 17-4 in the dolomite-dominated upper crust. Both crusts are 30 km thick, with a middle 10 km of a high-modal-quartz tonalite and a lower 10 km of granulite (thermal diffusivity values from Merriman et al., 2013).

The finite-difference software package Lithoheat (Nabelek et al., 2010) was used to calculate geothermal gradients and heat fluxes for both model columns. Lithoheat uses boundary conditions of surface temperature (298 K) and basal temperature (1573 K). Constant basal heat flux was not used because it implies temperatures increase (or decrease) in the asthenosphere in response to elevated (or reduced) temperatures at the base of the lithosphere to keep heat flux into the lithosphere constant (Nabelek and Liu, 2004). Heat production within the lithosphere is thus the major driving force behind the thermal structure of the lithosphere, and in our models was concentrated in the middle tonalite layer, with minor heat production in the granulite layer. Although basins typically contain higher heat production in the upper crust, this is usually concentrated in other rock types such as shales (e.g., Norden and Förster, 2006). Each model column was run to steady state using two different internal heat production (A_rad) values by varying the heat production in the tonalite layer (2.38 mW m^–3 versus 1.19 mW m^–3), for a total of four models.

Figure 14 shows the resulting geothermal gradients, thermal diffusivity, and thermal conductivity profiles. Two columns are shown: one with an upper 10 km of calcite-rich rocks (solid lines), and one with an upper 10 km of dolomitized carbonates (dashed lines). The models predict values of D resulting from computed geothermal gradients that vary strongly depending on the texture and mineralogy of the different layers. For example, D varied by as much as 0.87 mm² s^–1 between the calcite-dominated crust (0.87 mm² s^–1 at 6 km depth) and the dolomitized crust (1.74 mm² s^–1). Even within a single layer for identical crustal configurations, the calculated value of D varied by as much as 13%, and the resulting k as much as 10%, because of variations in temperature between hot and cold models. In other words, the geothermal gradient depends on the ability of the rocks to conduct heat, which in turn influences the geotherm. The resulting temperatures vary by as much as 90 K between calcite- and dolomite-dominated models, even when heat production was the same for both crustal columns. Some of these results would be hidden by surface heat flow measurements, as predicted heat flow at the surface is nearly identical for both low-A_rad models (43.4 mW m^–2 for the calcite-dominated crust versus 45.6 mW m^–2 for the dolomite-dominated crust).

Furthermore, variations in thermal transport properties partially obscure variations in heat production in the crust. This is highlighted by temperatures in the low-A_rad calcite-dominated crust versus those in the high-A_rad dolomitized crust, where average crustal temperatures vary by only <1 K, and peak temperature contrasts are a modest 22 K at 5 km depth (410 K for high-A_rad dolomitized crust, 388 K for low-A_rad calcite-dominated crust).

Multiple properties combine to control thermal diffusivity of both minerals and rocks, the effects of which are more variable at low temperatures. Rock thermal diffusivity is in part controlled by mineralogy, which in carbonate rocks is typically dominated by either generally low-D calcite or higher-D dolomite, with proportions of other common minerals such as quartz or feldspar adding an additional layer of complexity. Mineral D itself is controlled by mineral structure and cation mass, as well as volume of the unit cell. The composite mineral dolomite has D that is the weighted average of that of calcite and magnesite at all temperatures. The data for rhodochrosite suggest that transition-metal carbonates do not follow the same trend as those with alkaline-earth metals (calcite-dolomite-magnesite), and that properties of other carbonate minerals not studied here (e.g., siderite, smithsonite) cannot be predicted from this trend.

For carbonate rocks, texture is a primary control on bulk-rock D, as porosity generally reduces D across all measured temperatures, but in a nonlinear fashion. Additional textural concerns may include dispersed minor phases such as graphite, sample cohesion, and grain size. The size of the suite measured for this work, however, is insufficient to fully characterize these effects. Metamorphism of sedimentary carbonates may increase bulk-rock D, in part by porosity reduction, which occurs in sedimentary rocks under confining pressure or in the presence of fluids, but also by increasing grain size and replacement of lower-D minerals such as calcite with higher-D minerals such as forsterite.

Composition also has the potential to influence the thermal structure of crust that has a large component of carbonate rocks, as a higher contrast exists between the conductive ability of the minerals calcite and dolomite than previously thought. This contrast exists across all temperatures measured (Figs. 5 and 9), and strongly suggests that basins that have undergone dolomitization will conduct heat more efficiently than those that are dominated by calcite (Fig. 14). All other factors being equal, this suggests that geothermal gradients would be lower in dolomitized basins, and petroleum maturation windows or sources of geothermal energy would be deeper.

Thermal conductivity results presented here suggest that published values of k for both carbonate minerals and carbonate rocks contain systematic errors similar to those found in other mineral and rock groups. This typically results in published values for k using contact methods being lower at low temperatures and higher at high temperatures than values calculated through the LFA method. These results show that the conductive ability of carbonate regimes is likely more sensitive to local temperature than previously thought, and models using older k data or temperature-dependent models for k based on older data need re-evaluation. This is particularly true for models that incorporate a large temperature gradient over a small distance. For example, Nabelek et al. (2012) modeled emplacement of a granitic pluton into dolomitic marble using T-dependent D. Compared to a model with a fixed D of 1 mm² s^–1, temperatures in the pluton remained above the solidus more than twice as long because of the feedback between elevated temperatures in the country rock and its ability to diffuse the heat away. An extended residence time provides a longer-lasting thermal engine to drive metamorphic processes such as contact metamorphism or skarn mineralization. Furthermore, models that use a single, low-temperature value of k to predict its change with temperature are based on methods that include significant error, and do not correctly predict the change in k with temperature.

We thank Peter Nabelek for providing samples 17-4 and 33-1, Tom Freeman and Kevin Shelton for samples TF-001 and TF-002, and Brian Evans for samples of Carrara Marble. We also thank Paul Carpenter for feedback on this research and assistance with XRD and microprobe analysis. The comments of two anonymous reviewers significantly improved the manuscript. This research was funded in part by National Science Foundation awards EAR-1524796 to AGW and EAR-1524495 to AMH, American Chemical Society (ACS) Petroleum Research Fund grant 56166-ND8, and a University of Missouri Research Council grant to AGW. We also thank the donors of the ACS Petroleum Research Fund for partial support of this research.