**3.2 Determination of soil bulk density, porosity and air-filled porosity**

The thermo-TDR technique soil thermal property and water content data can be used to estimate soil structure changes [43, 48, 49]. The thermo-TDR technique can be applied to determine in situ ρb, *n* and *n*<sup>a</sup> based on three quantitative relationships of ρ<sup>b</sup> and θ with *C*, λ, e.g., de Vries, the Lu et al. and the Tian et al. models [45, 50, 51].

Thermo-TDR determinations of ρ<sup>b</sup> depend on the de Vries *C* model (hereafter *C*-based thermo-TDR method) and the Lu et al. or Tian et al. λ model (hereafter λbased thermo-TDR method) [45, 50, 51]. According to [45], soil *C* can be estimated as the weighted sum of volumetric heat capacities of soil solids, water and air. As the volumetric heat capacity of air is small compared to those for soil solids and water, soil *C* can be approximated as [9],

$$\mathbf{C} = \rho\_\mathbf{b}\mathbf{c}\_\mathbf{s} + \rho\_\mathbf{w}\mathbf{c}\_\mathbf{w}\boldsymbol{\theta} \tag{11}$$

From Eq. (11), ρ<sup>b</sup> is derived as,

$$\rho\_{\rm b} = \frac{C \text{-} \rho\_{\rm w} \text{c}\_{\rm w} \Theta}{\text{c}\_{\rm s}} \tag{12}$$

where cs is the specific heat of soil solids (kJ kg�<sup>1</sup> K�<sup>1</sup> ), ρ<sup>w</sup> is the density of water (1.0 g cm�<sup>3</sup> ), and cw is the specific heat of water (4.18 kJ kg�<sup>1</sup> K�<sup>1</sup> ) [9]. Once soil *C* and θ are determined from a thermo-TDR measurement, ρ<sup>b</sup> can be calculated with Eq. (12). It was pointed out that the *C*-based thermo-TDR method for determining ρ<sup>b</sup> was likely affected by changes in probe-to-probe spacing when inserting the sensor into soil [52, 53]. Liu et al. reduced such errors by increasing the rigidity of the sensor design, and obtained the continuous field ρ<sup>b</sup> for the tilled soil layers which changed over time with wetting and drying cycles [34, 48].

Because λ measurements using the heat pulse technique are not influenced by needle deflection, Lu et al. proposed the λ-based thermo-TDR method to determine in situ ρ<sup>b</sup> [54]. An empirical equation that related λ to ρb, θ, and soil texture was used [50],

$$
\lambda = \lambda\_{\text{dry}} + \exp\left(b - \theta^{-a}\right) \text{ } \theta > 0\tag{13}
$$

where *a* and *b* are shape factors that are estimated from ρ<sup>b</sup> and fractions of sand and clay,

$$\begin{cases} a = 0.67f\_{\rm cl} + 0.24\\ b = 1.97f\_{\rm sa} + 1.87\rho\_{\rm b} - 1.36f\_{\rm sa}\rho\_{\rm b} - 0.95 \end{cases} \tag{14}$$

where *f*sa and *f*cl are fractions of sand and clay, respectively, under the USDA soil textural classification system. The thermal conductivity of dry soils (λdry) relates linearly with *n* [46]. For mineral soils, setting soil particle density (ρs) as 2.65 g cm�<sup>3</sup> , λdry is calculated from [46],

$$
\lambda\_{\rm dry} = -0.56n + 0.51 = -0.56 \left( 1 - \frac{\rho\_{\rm b}}{\rho\_{\rm s}} \right) + 0.51 \tag{15}
$$

An iterative approach is used to numerically solve for ρ<sup>b</sup> because there is no explicit solution for ρ<sup>b</sup> from Eqs. (13)–(15). The nonlinear equation solver (*fsolve*) in MATLAB (Mathworks, Inc., Natick, MA) can be applied using an initial ρ<sup>b</sup> value of 1.0 g cm�<sup>3</sup> .

The empirical Lu et al. λ model introduced uncertainty in ρ<sup>b</sup> estimates, especially for coarse soils [50]. Thus, Tian et al. proposed a simplified version of the physically-based de Vries λ model to inversely estimate ρb, and they found that their λ model performed better than the *C* model and other empirical λ models [49]. When applying λ-based thermo-TDR methods on relatively dry soils, accurate θ inputs are required, because λ values of dry soils are insensitive to small θ changes. Therefore, Peng et al. used a combined approach to determine ρb: the *C*-based approach was used when θ was less than 0.10 m<sup>3</sup> m�<sup>3</sup> , and the λ-based approach was used at θ > 0.10 m<sup>3</sup> m�<sup>3</sup> [22].

Both *C*- and λ-based thermo-TDR methods rely on TDR determined θ values as inputs. Lu et al. introduced a heat pulse based approach to determine ρ<sup>b</sup> with only *C* and λ values [55]. This method relies on the de Vries *C* model and the Lu et al. λ model with known soil texture and cs as a priori, and calculates ρ<sup>b</sup> with an interactive procedure [45, 50]. The heat pulse based approach can be used when TDR θ is not readily available. Peng et al. [20] showed that on salt affected soils where the accuracy of TDR θ was greatly restricted, using the heat pulse based method provided more accurate determinations of θ and ρ<sup>b</sup> values than the thermo-TDR based method [8].

It is commonly recognized that a tilled soil layer undergoes great structural changes due to agricultural management and rainfall effects. The in situ measurements of ρ<sup>b</sup> in tilled soil layers using a thermo-TDR technique indicated that soil ρ<sup>b</sup> increased following tillage because rainfalls caused soil particles to settle and consolidate [48, 49]. **Figure 8** shows that soil ρ<sup>b</sup> increased and then leveled off, and the thermo-TDR method determined ρ<sup>b</sup> values mostly matched the core sample values.

With the thermo-TDR determined θ and ρb, soil *n* can be calculated with known soil particle density (ρ<sup>s</sup> = 2.65 g cm�<sup>3</sup> ),

$$m = 1 - \frac{\rho\_{\rm b}}{\rho\_{\rm s}} \tag{16}$$

Thus, the *n*<sup>a</sup> and degree of water saturation (*S*w) values can be calculated,

$$n\_{\mathbf{a}} = n - \theta \tag{17}$$

$$S\_{\mathbf{w}} = \frac{\theta}{n} \tag{18}$$

#### **Figure 8.**

*Dynamic thermo-TDR measured bulk density (ρb) values for two soil layers plotted along with independent ρ<sup>b</sup> values from soil core measurements. Both error bars and gray areas represent standard errors of the measurements (figure originally published in [49]).*

*Applications of Thermo-TDR Sensors for Soil Physical Measurements DOI: http://dx.doi.org/10.5772/intechopen.100285*

Fu et al. [56] showed that when applying the thermo-TDR technique in cropped soil, the influences of roots should be considered by using an extended mixing model based on Eq. (11),

$$\mathbf{C} = f\_{\mathbf{s}} \mathbf{C}\_{\mathbf{s}} + f\_{\mathbf{w}} \mathbf{C}\_{\mathbf{w}} + f\_{\mathbf{r}\mathbf{w}} \mathbf{C}\_{\mathbf{w}} + f\_{\mathbf{r}} \mathbf{C}\_{\mathbf{r}} \tag{19}$$

where *f*s, *f*w, *f*rw and *f*<sup>r</sup> are volume fractions of soil solids, soil water, root water and dry root, respectively; *C*<sup>s</sup> and *C*<sup>r</sup> are the volumetric heat capacity of soil solids and dry roots (assumed to be equal to the volumetric heat capacity of organic materials, 2.51 MJ m�<sup>3</sup> K�<sup>1</sup> at 20°C, [45]), respectively.

For a bulk soil sample with a volume of *V*, Eq. (19) can be rewritten as [57],

$$\mathbf{C} = \frac{m\_{\rm s}}{V}\mathbf{c}\_{\rm s} + \left(\frac{V\_{\rm w}}{V} + \frac{V\_{\rm rw}}{V}\right)\mathbf{C}\_{\rm w} + \frac{V\_{\rm r}}{V}\mathbf{C}\_{\rm r} \tag{20}$$

where *V*w, *V*rw and *V*<sup>r</sup> are the volumes of soil water, root water and dry roots, respectively, and *m*<sup>s</sup> is the dry mass of soil solids. By rearranging Eq. (27), the root zone ρ<sup>b</sup> can be derived as,

$$\rho\_{\rm b} = \frac{m\_{\rm s}}{V - V\_{\rm rw} - V\_{\rm r}} = \frac{C - \Theta\_{\rm total} C\_{\rm w} - \frac{V\_{\rm r}}{V} C\_{\rm r}}{c\_{\rm s} (1 - \Theta\_{\rm total})} \tag{21}$$

where

$$
\Theta\_{\text{total}} = \frac{V\_{\text{w}} + V\_{\text{rw}}}{V} \tag{22}
$$

where θtotal is defined as the sum of volumetric θ values of root and soil. Fu et al. report that when the maize root density is greater than 0.037 g cm�<sup>3</sup> , Eqs. (21, 22) should be used to estimate ρ<sup>b</sup> from thermo-TDR measured *C* and θ [57]. The soil profile root density distribution is needed to estimate ρ<sup>b</sup> in the root zone. **Figure 9** presents the results of thermo-TDR ρ<sup>b</sup> estimates and the actual ρ<sup>b</sup> in a maize root zone, and using the extended approach improves the accuracy of thermo-TDR ρ<sup>b</sup> estimates by accounting for the influence of roots during the maize growing season. Thus, it is important to consider the influence of roots when applying the thermo-TDR technique in crop fields.

#### **Figure 9.**

*Comparison of thermo-TDR soil bulk density (ρb) estimates from the original approach (Eq. (12)) and the extended approach (Eq. (21)). (figure originally published in [57]).*

## **3.3 Measuring soil ice contents during freezing and thawing**

Although in-situ determination of soil ice content during freezing and thawing is challenging, a thermo-TDR technique has been developed to measure soil liquid water and ice contents in partially frozen soils. Tian et al. report that thermo-TDR determined heat capacity and liquid water content in partially frozen soil can be used to determine soil ice content [4]. According to [45], the volumetric heat capacity of a partially frozen soil can be expressed as,

$$\mathbf{C} = f\_{\text{s}}\mathbf{C}\_{\text{s}} + \theta\_{\text{u}}\mathbf{C}\_{\text{u}} + f\_{\text{a}}\mathbf{C}\_{\text{a}} + \theta\_{\text{i}}\mathbf{C}\_{\text{i}}\tag{23}$$

where *f*s, θu, *f*<sup>a</sup> and θ<sup>i</sup> are the volume fractions of soil solids, unfrozen water, air and ice, respectively. *C*<sup>s</sup> (2.35 MJ m�<sup>3</sup> K�<sup>1</sup> ), *C*<sup>u</sup> (4.18 MJ m�<sup>3</sup> K�<sup>1</sup> ), *C*<sup>a</sup> (0.0012 MJ m�<sup>3</sup> K�<sup>1</sup> ), and *C*<sup>i</sup> (1.73 MJ m�<sup>3</sup> K�<sup>1</sup> ) are volumetric heat capacities of soil solids, unfrozen water, air and ice, respectively [58]. *C*<sup>a</sup> is very small compared to other soil constitutes which can be neglected. The term *f*<sup>s</sup> can be calculated from the ratio of ρ<sup>b</sup> and ρs.

Tian et al. reported that the heating strength of heat pulse measurements should be carefully controlled for measurements in partially frozen soil to minimize ice melting during the process [4]. Their results indicated that the heat pulse method failed to provide accurate thermal properties at soil temperatures between �5 and 0°C because of temperature field disturbances from latent heat of fusion. The optimized heating application strategy was found to be a 60-s heat duration (450 J m�<sup>1</sup> ) or a 90-s heat duration (450–900 J m�<sup>1</sup> ), and the *C*-based approach could only be applied at soil temperature ≤ -5°C. **Figure 10** shows the results of thermo-TDR determined ice contents on three soils with total water content (θt) of 0.15 m<sup>3</sup> m�<sup>3</sup> during freezing and thawing periods in a soil column experiment. Soil ice began to form when the temperature was below 0°C because of the supercooling effect. A large portion of latent heat was released during ice formation, which led to unstable thermo-TDR θ<sup>i</sup> values during this period. The measurement errors were within �0.05 m<sup>3</sup> <sup>m</sup>�<sup>3</sup> when soil temperatures were below -5°C [4].

Tian et al. reported that the *C*-based approach was prone to errors resulting from probe deflections due to ice expansion during freezing [5]. The λ-based approach using the simplified de Vries model was used to determine the ice content with inputs of λ, ρb, and TDR-θu, and it was also reported to perform well at temperatures of �1 and -2°C, thus extending the measurement range near 0°C. It was noted that both *C*-based and λ-based approaches required accurate ρ<sup>b</sup> information.

For soils experiencing seasonal or diurnal freezing and thawing cycles, Kojima et al. proposed an approach with TDR-θ determinations made before and after an imposed ice melting process caused by heating the soil surrounding the sensor [7]. The θ<sup>i</sup> value was equivalent to the difference between the two TDR-θ values, which represented the liquid water content and total water content in the soil. Their method only relied on the two TDR-θ values but required long measurement intervals and a relatively large heat input to melt the ice.

#### **3.4 Measuring heat, water, and water vapor fluxes in soil**

The thermo-TDR method is a useful tool that can be used in laboratory and field experiments to study transient in-situ properties and processes related to coupled heat and water transfer in soil. Heitman et al. used thermo-TDR sensors in a closed soil cell with imposed transient boundary conditions to obtain non-uniform temperature, water and thermal property distributions [3]. Thermo-TDR sensors were

#### **Figure 10.**

*Soil temperature dynamics, thermo-TDR measured ice contents (from Eq. (30)), and TDR evaluated ice contents (θt-θu) during freezing and thawing for soil samples with a water content of 0.15 m3 m<sup>3</sup> on sandy loam, silt loam and silty clay loam soils. Dashed lines indicate 0.05 m3 <sup>m</sup><sup>3</sup> error. (figure originally published in [4]).*

used to obtain soil thermal conductivity during wetting and drying processes on quartz sands for geothermal applications [59–62].

Significant improvements in both sensor configurations and theories have been made in fine-scale measurements of coupled water and heat transfer process in soil under field conditions, especially in near surface soils [63]. Based solely on the heat pulse function of the thermo-TDR sensor, the use of a series of such sensors aligned in a soil profile permitted the determination of soil heat fluxes, liquid water fluxes, and soil-water evaporation fluxes [64].

Based on Fourier's law, the one-dimensional heat flux density (*G*) can be calculated based on soil temperature gradient multiplied by soil thermal conductivity. A few studies have found that the reliability of heat flux density depends largely on the accuracy of λ determinations [65–67]. Soil temperature and λ could be measured simultaneously and in-situ with the heat pulse method. Besides, soil thermal conductivity models provided an alternative method to obtain λ. Ochsner et al. showed that the heat pulse probe worked well in obtaining λ and soil heat flux density under field conditions [66]. Peng et al. investigated λ model–based gradient methods to determine soil heat flux density [67]. Both heat pulse based and λ models based

#### **Figure 11.**

*The schematic view of thermo-TDR sensor measurement for state variables and parameters. (figure originally published in [22], replot in this context).*

gradient methods provided reliable near-surface heat flux with continuous and variable θ, ρb, λ and *T* measurements under field conditions that included soil disturbance or deformation [66, 67].

A heat pulse technique based on the sensible heat balance of near-surface soil layers was able to determine in situ soil water evaporation (*E*) rates [68, 69]. The sensible heat balance method determined soil water evaporation with time and depth [70–72]. Improvements in sensor configuration enabled the determination of soil temperature, heat fluxes and storage as well as latent heat at a mm-scale [73, 74]. Heat pulse measurements of soil water evaporation dynamics also made it possible to partition evapotranspiration under field conditions [75].

An analytical solution that related soil water flux density (*J*) to the maximum temperature difference at upstream and downstream sensing probes was developed [76]. Then a further simplified form was established using the ratio of downstream and upstream temperatures [77]. Studies demonstrated the accuracy of the heat pulse technique to determine soil water flux [76, 78–81]. Accurate measurements of soil water flux density are necessary to quantify infiltration, runoff, solute transport, and subsurface hydraulic processes.

### **4. Outlook**

**Figure 11** presents a flowchart of the uses and outcomes for the thermo-TDR method. Generally, the thermo-TDR determined state variables and physical parameters can be estimated with proper models and methods. The most promising aspect of the thermo-TDR technique is the capability to determine in situ bulk density, porosity, heat flux, water flux and vapor flux. These provide opportunities to study transient heat and water processes in field soils, including water evaporation, sensible and latent heat, and liquid water fluxes [64, 68, 69, 82].

#### **5. Conclusions**

This chapter includes descriptions of thermo-TDR sensors, methods for collecting and analyzing data, and reviews of current and potential thermo-TDR *Applications of Thermo-TDR Sensors for Soil Physical Measurements DOI: http://dx.doi.org/10.5772/intechopen.100285*

applications. The thermo-TDR sensor, which combines a heat pulse probe with a time domain reflectometry probe for soil thermal and electrical properties determinations, provides new opportunities for improved soil measurements on thermal properties, water content, bulk electrical conductivity, ice content, bulk density, air-filled porosity, heat flux, water flux, and vapor flux. The thermo-TDR technique has the potential to monitor in situ soil physical properties and processes for vadose zone soils.
