Estimation of ground thermal conductivity through indicator kriging: Nation-scale application and vertical profile analysis in Japan

Highlights

• Ground thermal conductivity was estimated by indicator kriging with borehole data.

• The estimation adopted probability-weighted averaging of individual soil conductivity.

• The nation-scale estimation was performed from 46 thousand borehole data in Japan.

• Dependence of estimates on topography was shown, but variation among cells was large.

• All vertical profiles were fitted by a logarithmic curve without a peak or with a peak.

Abstract

The estimation of ground thermal conductivity using indicator kriging and surrounding borehole data at 5-m intervals to a depth of 200 m on regular 0.5-km cells of Japan was demonstrated. The probabilities of occurrence of eight soil/rock types were estimated for their most likely thicknesses and used as weights for estimation of ground thermal conductivity. Individual thermal conductivity of each soil/rock type is determined to obtain sufficient agreements of ground thermal conductivities between estimates and in-situ measurements. The estimation was performed over the land with 46 thousand borehole data. Local maps of Kanto showed the relation of soil/rock types and ground thermal conductivities with five topographic categories. Averaged 10-km-cell maps of Japan indicated the sudden changes of ground thermal conductivity even among adjacent cells due to geologic complexity. Vertical increases of ground thermal conductivity were statistically observed in all topographic categories, and the vertical profile of each location followed a logarithmic curve without or with a peak.

Introduction

Ground thermal conductivity is the overall effective thermal conductivity of layered soils and rocks comprising a geologic formation between the ground surface and any target depth for research in the fields of thermal science and engineering. The effective thermal conductivity of each type of soil or rock, referred to as individual conductivity in this work, varies depending on the thermal conductivity, volume content, and arrangement of the particles, internal water, and air. Many texts (e.g., Ingersoll et al., 1954; Jumkis, 1977; Japan Society of Thermophysical Properties, 1990; Chiasson, 2016; ASHRAE, 2019) have summarized the individual conductivities of various geologic materials, ranging between less than 1 Wm⁻¹ K⁻¹ for unsaturated clayey sediments and more than 5 Wm⁻¹ K⁻¹ for crystallized rocks. The relations of individual conductivity versus density and water content have been experimentally obtained for sand and clay (e.g., Farouki, 1981). As the average of different individual conductivities of the soils and rocks constituting a formation, the ground thermal conductivity potentially varies in the range of individual conductivities, and there is not only horizontal variations for target locations but also vertical variations for target depths. In terms of topography, the ground thermal conductivity is higher in steep mountains where consolidated rocks of high individual conductivity appear at a shallower depth than in lowlands where fine soils of low individual conductivity are dominant. Additionally, at each location, the ground thermal conductivity often has a depth-increasing trend, where the geology changes from unconsolidated soils to consolidated rocks and the density of rocks increases with depth (Ingebritsen et al., 2006).

The ground thermal conductivity is required to analyze heat transfer by conduction horizontally or radially along a formation having a bottom coinciding with the target depth for estimation. As a representative case, ground thermal conductivity is essential in the design and planning of borehole heat exchangers (BHEs) in ground-source heat pump (GSHP) systems. GSHP systems are widely used in heating/cooling buildings, melting snow, supplying hot water, and meeting other heat demands and achieve high efficiency on the basis of the stability of ground temperatures. The systems potentially reduce energy consumption and CO₂ emissions relative to the performances of conventional systems, such as gas boilers and air-source heat pump systems, and their implementation has been increasing in many countries, as summarized by Lund and Boyd (2016). GSHP systems need one or multiple BHEs to extract the heat energy equivalent to the heat demands from the underground. Among various ground heat exchangers, the BHE is constructed by installing one or multiple U-tubes in a borehole with grouting and is applied the most frequently due to its flexibility in terms of installation and heat demands. One reason for the popularity is that the BHE can be constructed in a limited area of high-density buildings, as seen in most developed cities. Another reason is that the heat extraction rate can increase as customers require by deep or multiple constructions of the BHEs. The required total length of the BHEs directly affects the capital costs of GSHP systems and thus should be reduced to a minimum considering various factors of the heat demand, heat pump performance, and geology. In particular, the ground thermal conductivity is linearly related to heat conduction around the BHE, as the most critical geologic property alongside the volumetric heat capacity and initial temperature. As an example, in the practical method of ASHRAE, the length of a single BHE in soft clay having individual conductivity of 1.2 Wm⁻¹ K⁻¹ would be about twice that in consolidated limestone having individual conductivity of 3.1 Wm⁻¹ K⁻¹ (Bernier, 2006). This means that the ground thermal conductivity was not purely linearly related to but rather significantly positively related to the required length

Despite its importance, ground thermal conductivity is often problematic to determine practically in the design and planning of GSHP systems (Schillereff et al., 2008). Most texts and guides used for the design and planning recommend in situ measurements, referred to as thermal response tests (TRTs), to obtain the ground thermal conductivity at each site. TRTs have been developed over several decades, as reviewed by Spitler and Gehlin (2015). The measurement and analysis methodology is simple in practice; i.e., the heat transfer fluids are circulated with a constant heating rate in a testing BHE, and the temperatures of the fluids are measured at the shanks of U-tubes within several days. The temperature increases are analyzed by fitting a straight line against logarithmic time according to the theory of the infinite line heat source. However, an actual BHE is needed even for testing, and the in-situ measurement is thus often expensive for a small GSHP system, especially in the initial stage of planning. In addition, the actual value is only obtained for the same depth as the test borehole, not for multiple depths, even if the planner considers that the ground thermal conductivity might vary vertically. In other words, it is almost impossible for economic reasons to estimate the vertical profile of ground thermal conductivity by TRTs at different depths. It has thus been long desired to establish a methodology considering the depth-dependence of ground thermal conductivity (Spitler, 2011).

Alternatively to adopting TRTs, ground thermal conductivity is approximately estimated as a thickness-weighted average of the individual thermal conductivity of each soil and rock type constituting the formation along a BHE (Banks, 2008). Taking the example of Japan, a simple method is used instead of conducting TRTs to estimate the ground thermal conductivity when the borehole data regarding geological information are obtained. Although the simple method provides only estimates (and not actual values) of the ground thermal conductivity, it is potentially applicable in estimating the vertical profile of the ground thermal conductivity when information about the thickness of each soil and rock type is obtainable. In fact, the methodology has been applied to estimate regional distributions of the ground thermal conductivity in a target area for the evaluation of possible uses of GSHP systems (Fujii et al., 2007; Nam and Ooka, 2011; Russo et al., 2014; Fuchs and Balling, 2016; Casasso and Sethi, 2017; Viesi et al., 2018). Meanwhile, previously obtained drilling records are limited and are not necessarily close to the target location. The interpolation method for estimating the thickness of soils and rocks should be discussed in applying the simple method. However, the above researchers commonly assumed that a formation comprises horizontally stratified layers, and that the ground thermal conductivity is constant in each layer but different among layers. To the best of our knowledge, no research has also shown the 3D distribution of ground thermal conductivity on a larger scale, such as a nationwide-scale, and discussed the vertical profiles in contributing to the optimal design of BHE lengths.

The present study applied indicator kriging with surrounding borehole data as a probability-concept method of interpolating the thicknesses of soil/rock types for estimation of ground thermal conductivity. Kriging is a fundamental interpolation method in geostatistics, and various kriging methods have been developed (Deutsch and Journel, 1994). Kriging has an advantage over other deterministic methods, such as the inverse-distance method, in that it considers uncertainty included in any borehole data, especially in non-core sampling borehole data for the construction of water wells. The present study used not only core-sampling borehole data but also non-core sampling borehole data, providing better coverage of deep zones for the installation of BHEs. Indicator kriging is a geostatistical interpolation procedure for categorical variables; e.g., it is adopted in finding geologic descriptions in borehole data (Deutsch, 2002). Another, more popular, simple or ordinary kriging method is used for continuous variables but is not appropriate for limited measurements of ground thermal conductivity. The present study defined eight rock/soil types for indicator kriging and estimated the probability of occurrence of each soil/rock type at that type’s most likely thickness. The ground thermal conductivity was estimated as a probability-weighted average of individual conductivities of eight soil/rock types. Individual conductivities of the eight soil/rock types were determined to match estimates made from in-situ measurements in our previous study (Sakata et al., 2018). The ground thermal conductivity was estimated at 5-m vertical intervals to a depth of 200 m in regular 0.5-km cells over land in Japan. Local maps of the ground thermal conductivity in the Kanto area, including Tokyo, and averaged 10-km-cell maps for all Japan were obtained at multiple depths. The present study also summarized estimates among five topographic categories statistically and analyzed a vertical profile in each cell by fitting a regression curve without and with a peak. On the basis of these results, this study discusses the variability of ground thermal conductivity, the dependence of ground thermal conductivity on the topography, and the simple modeling of the vertical profile at a location for practical use.