1. Introduction

The water content has great influence on the distribution of temperature field for soils. Since Gardner et al. (1955; 1959) found that it was difficult to quantitatively analyze the influence, lots of researchers began to pay attention to the relationship between soil temperature field and soil water content (Cray,1966; Haridasan & Jensen, 1972; Nakano et al.,1984). Hopmans & Dane (1986) added different amounts of water to the soil, and found that the depth of soils affected by the same ambient temperature changed with the initial water content. Wang & Li (2018) pointed out that the influence depth was about 40cm with the top cryogenic boundary at 0℃ in the silty soil, and the temperature field of the soil had a certain relation with the initial water content of the soil. Using the measured data collected by the observatory, the relationship between soil water content and soil thermal diffusivity had also been studied (Roxy et al., 2010; Roxy et al.,2014). Their research showed that soil thermal diffusivity increased first and then decreased with the increase of water content.

Many models for simulating soil thermal characteristics had been developed. Parikh et al. (1979) used an unstable method to detect the thermal diffusivity of silty soil. Côté & Konrad (2005) modified the thermal conductivity model to calculate the dry thermal conductivity of soil; Balland & Arp (2005) considered the influence of the soil organic matter based on the Johansen model; Lu et al. (2007) modified the model of Côté& Konrad (2005) to suit the finegrained soils at low-saturation. Barryet al. (2015) evaluated most of soil thermal models. Lukiashchenko& Arkhangelskaya (2018) considered the influence of soil texture in simulating thermal characteristics of soils.

The soil in Beijing is seasonally frozen, which is harmful to urban engineering construction. However, there is still a lack of research on the thermal conduction characteristics of soils in Beijing area. Therefore, in this paper, we take three common soil types in Beijing area as research samples: sandy soil, silty clay and clay. We measure the temperature field of soil samples in a test barrel under 0℃ temperature boundary. Based on the related theory and test results of soil temperature field, we back calculate the thermal diffusivity of soil with different volumetric water content, and analyze the relationship between the thermal diffusivity and the volumetric water content of soils.

2. Tests

2.1 Soil samples in test

Three kinds of soil samples are selected in Beijing area for the tests. As shown in Fig.1, the sandy soil sample is from Mentougou District in Beijing (115°50′N, 40°3′E); the silty clay sample is from Daxing District in Beijing (116°25′N, 39°30′E); the clay sample is from Haidian District in Beijing (116°21′N, 40°0′E).Fig.2 shows the photographs of the soil samples of sandy soil, silty clay and clay, respectively. The basic characteristics of the soil samples are given in Table 1.

2.2 Test instrument

The schematic drawing of instrument is shown in Fig. 3. The soil sample is laid in a test barrel. The size of the test barrel is 100cm in height, 18cm in inner diameter and 20cm in outer diameter. The material of test barrel is polymethyl methacrylate. The test barrel is wrapped with a thermal insulation layer to avoid the influence of external temperature. The temperature control devices are at the top and bottom of the barrel, respectively. The top temperature control device keeps 0℃, and the bottom temperature control device keeps 22℃ which is the same as the room temperature.

The probes of the temperature sensor are placed at 1cm, 2cm, 5cm, 9cm, 15cm, 25cm and 40cm in depth respectively. The probes are placed in the middle of the soil column to reduce the boundary influence. The work range of temperature sensor is -50℃~50℃; the accuracy is 0.1℃; the sampling interval is two seconds.

2.3 Procedure of test

The room temperature is kept at 22℃ during the experiment. The soils are prepared under a constant dry density with different volumetric water content. Then, the soils and temperature sensors are laid into the test barrel. For the three kinds of soils, each one has five sets of tests with different volumetric water content. The volumetric water content and the dry density of the laid soil samples are shown in Table 2.

Apply the temperature control devices at the top and the bottom of the soil sample respectively, and the data of temperature sensors is recorded in the meantime. With the working of the temperature control devices and the recording of the temperature sensors, the test lasts 24 hours.

3. Test results

3.1 Sandy soil

For the sandy soil with different volumetric water content, the temperature curves changing with time of each sensor in soil samples are obtained as shown in Fig. 4.

It can be seen from Fig. 4 that, the temperature of the soil in depth of 25cm drops rapidly in five hours, while the temperature of the soil deeper than 25cm almost do not change. From Fig. 4(c) ~ (e), we can see the temperature of soil deeper than 25cm start to slowly drop after five hours. From Fig. 4 (a) and (b), it can be seen that the temperature of the soil at the depth of 40 cm nearly do not change in 24 hours.

3.2 Silty clay

The temperature curves changing with time for silty clay are obtained as shown in Fig. 5.

Similar to the sandy soil, it can be seen from Fig. 5 that the temperature of the silty clay in depth of 25cm drops rapidly in four hours, while the temperature of the sample deeper than 25cm almost do not change. From Fig. 5 (c) ~ (e), we can see the temperature of the silty clay deeper than 25cm start to slowly drop after four hours. From Fig. 5 (a) and (b), it can be seen that the temperature of the sample at the depth of 40 cm nearly do not change in 24 hours.

Table 1: Basic characteristics of three kinds of soils

Soil	Particle-size distribution, %			Plasticity index	Specific gravity	Mass water content,%	Bulk density, g/cm³
	2mm~0.5mm	0.5mm~0.075mm	≤0.075mm
Sandy soil	43.79	53.72	2.49	\	2.67	\	1.69
Silty clay	\			15.41	2.69	9.21	1.66
Clay	\			18.15	2.70	12.41	1.58

Table 2: The volumetric water content and dry density of three kinds of soils

Volumetric Water content, m3m-3	Number	Sandy soil	Silty clay	Clay
	(a)	0.008	0.025	0.030
	(b)	0.012	0.052	0.060
	(c)	0.020	0.084	0.091
	(d)	0.054	0.123	0.150
	(e)	0.091	0.170	0.203
Dry density, g/cm³		1.70	1.52	1.41

3.3 Clay

The temperature curves changing with time for clay are obtained as shown in Fig.6.

It can be seen from Fig. 6 that the temperature of the clay in depth of 25cm drops rapidly in three hours, while the temperature of the sample deeper than 25cm almost do not change. From Fig. 6 (c) ~ (e), we can see the temperature of the clay deeper than 25cm start to slowly drop after four hours. From Fig. 6 (a) and (b), it can be seen that the temperature of the sample at the depth of 40 cm nearly do not change in 24 hours.

4. Analysis

4.1 Temperature field

To research the influence of volumetric water content to the temperature field for soils in Beijing, the final (t=24 h) temperature field is monitored from tests of three kinds of soils in depth of 40 cm are drawn in Fig. 7.

As can be seen from Fig. 7, it can be found that no matter sandy soil, silty clay or clay, the temperature of the same embedment depth of soil is lower than that of soil with higher value of water content.

It indicates that in range of volumetric water content for sandy soil (0.008m³ m^-3~0.091m³ m^-3), silty clay (0.025m³ m^-3~0.170m³ m^-3) and clay (0.030m³ m^-3~0.203m³ m^-3) as given in tests, the heat transfer capability of soil will be higher when the volumetric water content of soils rises.

4.2 Thermal diffusivity

In order to further research therelationship between Volumetric water content and its heat transfer capability, based on the temperature curves in Figs. 3~5 and theunsteady heat conduction theory in semi-infinite soild,the thermal diffusivity of soil samples are back calculated.

The temperature t in test samples can be calculated via the eqution from the classic theory of heat transfer, which is presented as:

$$t(x,t)=(t_0-t_w)erf\left(\frac{x}{2\sqrt{a\tau }}\right)+t_w\text{ (1)}$$

Where τ is the time, in s;x is the depth of soil,in m;α is the thermal diffusivity, in m² /s;t is the initial temperature , in ℃; tw is the boundary temperature , in ℃; erf( ) is the Gaussian Error function, which can be expressed as:

$$\text{erf(}\eta \text{)=}\frac{2}{\pi }{\displaystyle \underset{0}{\overset{\eta }{\int }}\exp (-x^2)}\text{dx (2)}$$

The least square method is applied to calculate the value of the thermal diffusivity, and the calculation method of minimum error is given as:

$$\min \Delta T(\alpha )={\displaystyle \sum _{i=1}^d[T({\alpha }_i)}-T_i{]}^2/d\text{ (3)}$$

Where α is the thermal diffusivity, in m²/s; T(α_i) is the temperature prediction value of the set thermal diffusivity on the sequence i, in ℃; Ti is the monitored temperature value in tests on sequence i, in ℃; d is the sequence number; ΔT(α) is the error value between the predicted value and the monitored value.

According to the data consulted (Oscar et al., 2018; Krishnaiah & Singh, 2004; Mikailsoy & Shein, 2019; Arkhangelskaya & Lukyashchenko, 2018), the thermal diffusivity of soil is between 1.22 × 10^-7m² /sand 8.04 × 10^-7m² /s. So the initial value α₀ is set as 1.00×10^-7m² /s, and the step value is set as 1.00 ×10^-10m² /s. The calculated thermal diffusivities of three kinds of soils with different volumetric water content are shown in Table 3.

Table 3: The calculated thermal diffusivities of three kinds of soils with different volumetric water content

Soil	Volumetric water content,m3m-3	Thermal diffusivity,m2/s×10-7	Error value
Sandy soil	0.008	1.494	0.450
	0.012	1.603	0.422
	0.020	2.467	0.925
	0.054	2.728	0.370
	0.091	3.143	0.606
Silty clay	0.025	1.603	0.722
	0.052	1.991	0.741
	0.084	2.423	0.490
	0.123	2.658	0.491
	0.170	3.068	0.602
Clay	0.030	1.358	0.702
	0.060	1.521	0.900
	0.091	2.162	0.403
	0.150	2.706	0.773
	0.203	3.363	0.569

Based on the calculated thermal diffusivity in Table 3, the temperature field of soil samples in tests can be predicted. For example, the simulated temperature curves of sandy soil sample (volumetric water content, 0.008m³ m^-3), the silty clay sample (volumetric water content, 0.084m³ m^-3) and the clay sample (volumetric water content, 0.091m³ m^-3)are compared to the test monitored results as shown in Fig. 8, respectively.