Temperature dependence of the relationship of thermal diffusivity versus thermal conductivity for crystalline rocks

Thermal diffusivity governs the transient heat transport equation. Thus, a realistic characterisation of this parameter and its temperature dependence is crucial for geothermal modelling. Due to sparse information from boreholes, lack of samples, and elaborate measurement procedures, there is often insufficient data on thermal diffusivity at elevated temperatures. We make use of existing data on crystalline (metamorphic and magmatic) rock samples from the Kola Peninsula and the Eastern Alps and develop a general relationship for the temperature dependence of thermal diffusivity up to 300°C. The temperature dependence of thermal conductivity is parameterised itself, using an empirical relationship which is set up for both data sets as well. Hence, only thermal conductivity at ambient temperatures is required for determining the temperature dependence of thermal diffusivity. We obtain different coefficients for both data sets which can be explained by different geological settings, and therefore different mineral compositions and internal structures. Comparisons with other expressions for these rock physical parameters show a good agreement at ambient conditions. General relations for thermal diffusivity at elevated temperatures are rare. A comparison of our results with data from two crystalline samples from the KTB and data from the Southern Indian Granulite Province highlights the need for further data, which will help to quantify uncertainties.     

Soil thermal properties at Kalpakkam in coastal South India

Time series of soil surface and subsurface temperatures, soil heat flux, net radiation, air temperature and wind speed were measured at two locations in Kalpakkam, coastal southeast India. The data were analysed to estimate soil thermal diffusivity, thermal conductivity, volumetric heat capacity and soil heat flux. This paper describes the results and discusses their implications.     

An analysis of MONTBLEX data on heat and momentum flux at Jodhpur

Parameterization of sensible heat and momentum fluxes as inferred from an analysis of tower observations archived during MONTBLEX-90 at Jodhpur is proposed, both in terms of standard exchange coefficientsC _H andC _D respectively and also according to free convection scaling. Both coefficients increase rapidly at low winds (the latter more strongly) and with increasing instability. All the sensible heat flux data at Jodhpur (wind speed at 10 m Ū₁₀ < 8 ms⁻¹) also obey free convection scaling, with the flux proportional to the ‘4/3’ power of an appropriate temperature difference such as that between 1 and 30 m. Furthermore, for Ū₁₀ < 4 ms⁻¹ the momentum flux displays a linear dependence on wind speed.     

Estimating apparent thermal diffusivity of soil using field temperature time series

Reliable estimates of soil thermal properties such as heat capacity, thermal conductivity, and diffusivity are important in analysis of heat transmission through soils in applications such as shallow geothermal applications, buried electrical conduits, and in general heat/fluid flow analyses. A number of analytical, numerical and experimental methods are available to determine the soil thermal properties. In this paper, the analytical and numerical methods developed on the basis of one-dimensional heat conduction equation are used to estimate the apparent thermal diffusivity of soil. Three of the four analytical methods, Amplitude, Phase, and Arctangent provide explicit equations for the apparent thermal diffusivity. Two methods, Harmonic and Numerical, make use of large number of temperature measurements to implicitly solve for the apparent thermal diffusivity. The temperature time series data monitored at different depths in two field sites in Melbourne, Australia for more than 2 year period were used to estimate the apparent thermal diffusivity of soil down to 2 m depth. The apparent thermal diffusivity was calculated using all five methods and compared with laboratory experimental results. The effectiveness of each method in predicting the thermal diffusivity was compared and observed discrepancies were discussed. Finally, the observed soil temperature data for a 12 month period are used to model the temperature variation in the ground analytically using Harmonic method and the model prediction for the following 12 month was compared independently with the field measurements. The analytical model prediction is found to be in good agreement with the field monitored data.     

Thermal diffusivity of garnets at high temperature

Thermal diffusivity (D) of garnets with diverse chemical compositions was measured using the laser-flash technique, which is accurate (±2%) and isolates the lattice component from direct radiative transfer. Temperatures ranged from ~290 to ~1,600 K (unless limited by melting). Seven synthetic (e.g., YAG, GGG) and 15 natural garnets with two types of ionic substitution [Ca₃(Fe,Al)₂Si₃O₁₂ and (Mg,Fe,Ca)₃Al₂Si₃O₁₂] and varying amounts of OH^- were examined. Cation substitution or hydroxyl incorporation lowers D from end-member values. Thermal diffusivity is constant once the temperature (T) exceeds a critical value (T _sat) of ~1,100 to 1,500 K. From ~290 K to T _sat, the measurements are best represented by 1/D=A+BT+CT ² where A, B, and C are constants. These constants vary little among diverse chemical compositions, suggesting that the oxygen sublattice controls heat transport. Higher order terms are needed only when T _sat is low, such as Ant Hill garnet wherein 1/D=0.049403+0.0032299T−2.3992T ²×10⁻⁶+6.0168T ³×10⁻¹⁰(1/D in s/mm²; T in K). The mean free path (λ, computed from D and sound velocities) is slightly larger than the lattice parameter above T _sat, in accord with phonon–phonon interactions requiring non-localized modes. At most temperatures, λ is nm-sized. Large values of λ are obtained by extrapolation to a few Kelvins, suggesting that boundary scattering can only be important at extremely cold temperatures. The observed behavior with T and chemical composition is consistent with the damped harmonic oscillator model. Phonon transport is best represented by inverse thermal diffusivity wherein 1/D goes as T ⁿ where n is between 1 and 3 up to ~200 K, depends on a quadratic or cubic polynomial at moderate T, but is constant above T _sat. The predicted and observed temperature response of 1/D mimics the well-known form for heat capacity, in that acoustic modes control heat transport near cryogenic temperatures, optic phonons dominate above ambient temperature, and a limit analogous to that of Dulong and Petit is reached at very high temperature, due to full population of discrete phonon states.     

The behavior of a multilayered porous thermo-elastic medium with anisotropic thermal diffusivity and permeability

With the aid of integral transform techniques, this paper presents an extended precise integration solution for thermal consolidation problems of a multilayered porous thermo-elastic medium with anisotropic thermal diffusivity and permeability due to a heat source. From the fundamental governing equations, ordinary differential equations are derived by employing Laplace–Hankel transforms. By applying the extended precise integration method, equations in the transformed domain can be solved, and the actual solutions are further obtained by adopting a numerical inverse transformation. The accuracy and feasibility of the proposed theory is demonstrated by contrastive analysis with existing studies. Finally, several examples are carried out to investigate the influence of heat source’s type, axial distance, burial depth of heat source, ratio of thermo-permeability, permeability anisotropy, thermal diffusivity anisotropy and stratification on the thermal consolidation process.     

青藏公路沿线多年冻土区土壤热通量参数化方案的优化和检验

土壤热通量是地表能量平衡的重要分量,其估算方案在研究地表能量平衡研究中必不可少。利用青藏公路沿线5个站点0~20 cm的实测土壤层温、湿度及5 cm土壤热通量资料,以翁笃鸣气候学计算方案为基础建立了优化的5 cm土壤热通量计算方案。通过唐古拉和西大滩两个独立站点的检验结果表明,优化方案的结果相对于原方案有较大的改善,唐古拉和西大滩5 cm土壤热通量均方根误差值分别减小了3.2 W·m^-2和4.8 W·m^-2,而相对误差分别减小了61.9%和36.1%,即新方案能够较好地估算出青藏公路沿线多年冻土区5 cm土壤热通量。使用优化方案模拟了青藏公路沿线11个站点5 cm土壤热通量变化,结果显示,近十年青藏公路沿线土壤热通量呈现出增大的趋势,其中,5 cm土壤热通量增大了近1.0 W·m^-2,而且各观测场的年平均土壤热通量值均大于0.0 W·m^-2,表明就年尺度而言,热量有盈余,盈余热量用于加热下层土壤,引起活动层厚度增加,平均状况下土壤热通量每增大1.0 W·m^-2,活动层厚度增大约21.0 cm。     

A methodology to determine thermal conductivity of soils from flux measurement

Determination of thermal properties of soils (viz., thermal resistivity, thermal conductivity, thermal diffusivity and heat capacity), which primarily influence heat migration through the soil mass, is essential in situations where geomaterials are relentlessly subjected to higher temperatures and temperature variations. These properties of the soil mainly depend upon its type, mineralogy, particle size and gradational characteristics, density and water content. In this context, earlier researchers have determined thermal conductivity of soils by employing a thermal probe (a line heat source), which works on the principle of transient method (TM) of heat conduction. However, this methodology cannot be employed for establishing the heat flow (read thermal regime) through the soil. Hence, development of an alternate technique, which facilitates quantification of temporal and spatial variation of the heat flux and temperature in the soil mass, becomes essential. With this in mind, a methodology to determine thermal conductivity of soils by employing the concept of thermal flux measurement (TFM) has been developed and its details are presented in this paper. Results obtained from the TM and TFM have also been critically evaluated for the sake of validation and generating more confidence in the proposed methodology.     

Inversion of multiple thermal indicators: Quantitative methods of determining paleoheat flux and geological parameters IV. Case histories using thermal indicator tomography

A quantitative tomographic method to determine simultaneously several geological, geochemical, and geothermal parameters associated with reconstruction of the geohistory and thermal history of sediments in a well is presented. Using vitrinite reflectance data from the well Inigok-1, National Petroleum Reserve of Alaska, the numerical algorithm was tested and found to be effective in delineating the variation of heat flux with time. In addition, the size and timing of a major unconformity also were bracketed. Application of tomography using apatite fission track distributions with depth as a thermal indicator enabled not only the thermal history of two wells in the NW Canning Basin of Australia to be determined, but also the chemical parameters associated with fission track annealing to be constrained. Results of both the Alaska study and the Australian study were consistent with the qualitative behavior inferred from current geological models.     

土壤中热量传输计算的研究进展与展望

土壤中热量传输计算的研究分为土壤热通量计算方法和土壤导温率计算方法两方面的成果. 目前已有土壤热通量的计算方法, 在土壤表层可以得到较好结果, 但在土壤深层的适用性无法确定. 多孔介质模型和考虑水的渗流造成的影响, 是多年冻土区土壤导温率的计算方法的研究方面新的进展. 为研究青藏铁路沿线冻土演化趋势找到合适的土壤热量传输计算方案, 应是今后工作的一个重点.     

Soil heat flux and day time surface energy balance closure at astronomical observatory,Thiruvananthapuram, south Kerala

Soil heat flux is an important input component of surface energy balance. Estimates of soil heat flux were made in the year 2008 using soil temperature data at Astronomical Observatory, Thiruvananthapuram, south Kerala. Hourly values of soil heat flux from 00 to 24 LST are presented for selected days typical of the winter, pre-monsoon, SW monsoon and NE monsoon seasons. The diurnal variation is characterized by a cross-over from negative to positive values at 0700 h, occurrence of maximum around noon and return to negative values in the late evening. The energy storage term for the soil layer 0–0.05 m is calculated and the ground heat flux G ^? is estimated in all seasons. Daytime surface energy balance at the surface on wet and dry seasons is investigated. The average Bowen’s ratio during the wet and dry seasons were 0.541 and 0.515, respectively indicating that considerable evaporation takes place at the surface. The separate energy balance components were examined and the mean surface energy balance closure was found to be 0.742 and 0.795 for wet and dry seasons, respectively. When a new method that accounts for both soil thermal conduction and soil thermal convection was adopted to calculate the surface heat flux, the energy balance closure was found to be improved. Thus on the land surface under study, the soil vertical water movement is significant.     

Thermal diffusivity of natural and synthetic garnet solid solution series

Knowledge of heat transport properties as a function of mineral- and rock-composition and temperature is of major relevance to understand and model heat transfer in the Earth’s interior. A systematic study on 13 natural and 4 synthetic garnets was carried out in an attempt to obtain a better systematic understanding of the processes that affect the heat transport in minerals, especially the effect of chemical substitution in solid solution series. It is found that substitution significantly lowers the thermal diffusivity from end-member values for both synthetic and natural garnets with a minimum of thermal diffusivity at an intermediate composition. The thermal diffusivity as a function of the degree of substitution can be described by the approach of Padture and Klemens (J Am Ceram Soc 80 (4):1018–1020, 1997). With increasing temperature the thermal diffusivity decreases due to phonon-phonon-scattering effects. A quantitative analysis of the high-temperature behaviour was carried out by using the model of Roufosse and Klemens (J Geophys Res 79 (5):703–705, 1974), which takes a lower limit of thermal diffusivity at elevated temperatures into account. The model allows for an extrapolation of the deduced room temperature thermal diffusivities to higher temperatures. Furthermore, the model was modified to determine the high temperature limit of the thermal diffusivity for all investigated natural garnets D _min to be 0.64 ± 0.03 mm²/s.     

青藏高原那曲地区非均质土壤导温率的变化及土壤温度数值模拟

土壤中的热传输是地-气能量交换的组成部分,而非均质土壤导温率/导热率的获得是研究土壤热传输过程中的一个难点,是至今仍未很好解决的问题.分析了一维热传导方程在非均质土壤条件下的适用形式,利用青藏高原那曲站实测土壤温度资料,计算了非均质土壤导温率,计算结果表明,土壤导温率有明显的随深度和季节的变化.利用考虑非均质土壤导温率参数的土壤温度数值模式,对那曲地区各层土壤温度的年变化进行了模拟试验,模拟结果显示,按冷季和暖季分别采用两组不同的土壤导温率,对土壤温度的年变化已有较好的模拟效果.     

Variability of soil moisture and its relationship with surface albedo and soil thermal diffusivity at Astronomical Observatory,Thiruvananthapuram, south Kerala

Continuous observation data collected over the year 2008 at Astronomical Observatory, Thiruvananthapuram in south Kerala (76°59′E longitude and 8°30′N latitude) are used to study the diurnal, monthly and seasonal soil moisture variations. The effect of rainfall on diurnal and seasonal soil moisture is discussed. We have investigated relationships of soil moisture with surface albedo and soil thermal diffusivity. The diurnal variation of surface albedo appears as a U-shaped curve on sunny days. Surface albedo decreases with the increase of solar elevation angle, and it tends to be a constant when solar elevation angle is greater than 40°. So the daily average surface albedo was calculated using the data when solar elevation angle is greater than 40°. The results indicate that the mean daily surface albedo decreases with increases in soil moisture content, showing a typical exponential relation between the surface albedo and the soil moisture. Soil thermal diffusivity increases firstly and then decreases with the increase of soil moisture.     

潮间带滩涂沉积物与水体热通量模拟

为了揭示滩涂对近岸水温影响,开展潮间带滩涂沉积物与海水之间热量交换研究。以韩国西南海岸的滩涂为例,建立滩涂沉积物温度模型,模拟不同潮汐状态下沉积物垂向剖面温度以及沉积物与水体间的热通量,并分析了季节、滩涂位置、潮位-太阳辐射相位对热通量的影响。研究表明:模拟出的沉积物温度与实测值吻合较好。沉积物与水体存在大量热量交换,且集中在淹没后的前3 h,最大热通量可达398.7 W/m2。冬季月份海水向滩涂净传热。夏季月份滩涂向水体净传热,且当滩涂淹没时刻发生在当地正午或正午过后3 h内,滩涂传递给水体的热量相对较大,达2.0 MJ/（m2·d）;累积热通量随滩涂干滩率的减小而减小。研究成果为进一步深入研究滩涂影响下近岸水温变化提供了技术支撑。     

Estimation of surface heat flux and inversion height with a Doppler acoustic sounder

In this paper, acoustic sounder (sodar) derived vertical velocity variance (σ _w ² ) and inversion height (Z _i) are used to compute the surface heat flux during the convective activity in the morning hours. The surface heat flux computed by these methods is found to be of the same order of magnitude as that obtained from tower measurements. Inversion heights derived from sodar reflectivity profiles averaged for an hour are compared with those obtained from the σ _w ² /Z profile. Variation of σ _w ² in the mixed layer is discussed. The data were collected during the Monsoon Trough Boundary Layer Experiment 1990 at Kharagpur. The analysis is made for four days which represent the pre-monsoon, onset, active and relatively weak phases of the summer monsoon 1990. The interaction of the ABL with the monsoon activity is studied in terms of the variation of inversion height, vertical velocity variance and surface heat flux as monsoon progresses from June to August.     

南海区域海气热通量的变化特征分析

应用由卫星SSM/I(Special Sensor Microwave/Imager)和AVHRR(Advanced Very High Resolution Radiometer)遥感资料,使用先进的海气通量计算方法(TAGA COARE3.0),计算出南海1987年7月至2004年12月共200个月的海气界面的感热和潜热通量(0.25°×0.25°),其结果与实测结果比较发现,由卫星反演的海气热通量与实测结果非常一致.与GSSTF2的结果相比,其时空分布变化特征基本一致.由此说明,利用卫星遥感获得的热通量可以用来进行中国近海海气相互作用的研究以及作为我国气候预测研究的重要依据.由多年南海海气热通量的分析表明,南海区域热通量的变化具有显著的年变化和年际变化特征,其周期分别是0.5a、1a、准3a和6～11a.其中准3a和6～11a周期与中国旱涝的周期一致.因此,可以认为南海区域热通量的年际变化对中国的旱涝分布将起着不可低估的作用.     

利用土壤温度估算土壤有效热导系数和水流通量

土壤有效热导系数和水流通量是研究大气-土壤系统水分转化的重要变量。采用动态回归模型（DHR）获取土壤温度的振幅和相位变化,联合土壤水热耦合方程的解析解估算土壤有效热导系数和土壤水流通量,并将该方法应用于毛乌素沙地的土壤水流通量估算中。结果显示,实例研究估算获得的有效热导系数在10-7m2/s数量级变化,且随振幅比的增加呈指数增加,随相位差的增加呈指数衰减。当土壤含水率小于0.08时,有效热导系数呈线性增加;当土壤含水率大于0.08时,接近恒定值（定量0.08）,土壤水流通量随土壤含水率的变化无明显的线性关系。     

Surface sensible heat flux over a deep moist convective region and its interplay with synoptic and mesoscale features

The surface parameters are being evaluated using a methodology which considers the vertical temperature structure of MONTBLEX ’90. For this 30 metre micrometeorological tower data are utilised. In this process, a concept of isolated layers has been introduced. The parameters have been evaluated following two successive iterative processes to give a consistent value both in the flux-profile relation and the related similarity relation. The heat flux obtained using the present methodology, is then utilised to examine its interplay with the synoptic as well as mesoscale features. The existence of such an interplay is observed.