# Research on Identifying Method of Freezing-Thawing Soil Hydraulic Properties

Zilong Wang, Qiuxiang Jiang, Qiang Fu, Tianxiao Li

► To cite this version:

Zilong Wang, Qiuxiang Jiang, Qiang Fu, Tianxiao Li. Research on Identifying Method of Freezing-Thawing Soil Hydraulic Properties. 6th Computer and Computing Technologies in Agriculture (CCTA), Oct 2012, Zhangjiajie, China. pp.359-366, 10.1007/978-3-642-36137-1_42. hal-01348252

HAL Id: hal-01348252

https://inria.hal.science/hal-01348252

Submitted on 22 Jul 2016

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Research on Identifying Method of Freezing-thawing
Soil Hydraulic Properties

Zilong Wang<sup>1,2</sup>, Qiuxiang Jiang<sup>1</sup>, Qiang Fu<sup>1</sup>, Tianxiao Li<sup>1</sup>

<sup>1</sup> College of Water Conservancy & Architecture, Northeast Agricultural University, P.R.
China 150030

<sup>2</sup> Postdoctoral Mobile Research Station of Agricultural and Forestry Economy Management,
Northeast Agricultural University, P.R. China 150030
wzl1216@163.com

**Abstract.** The identification of freezing-thawing soil hydraulic properties is the basis and key problem of studying soil water movement. Current identifying methods of the properties have many limitations, such as big error, high uncertainty and low operability. Thus, the identifying method of freezing-thawing soil hydraulic properties was researched in the research. The estimating parameters of soil freezing characteristic curve were calculated by classical statistical method. Targeting at the maximum absolute difference between measured and simulated value of unsaturated soil hydraulic properties, the particle swarm optimization algorithm was used to estimate the optimal values of soil water characteristic curve, unsaturated hydraulic conductivity and specific water capacity. The numerical simulation of freezing-thawing soil water movement proved that the measured and simulated value of soil unfrozen water content had high degree of agreement. The identifying method proposed in the research is effective and feasible.

**Keywords:** Freezing-thawing soil, Soil hydraulic properties, Identifying method, Particle swarm optimization

# 1 Introduction

Mathematical simulation is the method of researches on seasonal freezing-thawing soil water movement. The freezing-thawing soil hydraulic properties are the necessary data and the basis of quantification study precision whether in the analytic or numerical solutions. The hydraulic properties include soil freezing characteristic curve, soil water characteristic curve, unsaturated hydraulic conductivity (K), unsaturated diffusivity (D) and specific water capacity (C). For the later three properties can be related by the relational expression D=K/C, generally, two of them have to be identified.

The identifying method of freezing-thawing soil hydraulic properties includes direct manner and indirect manner [1]. The direct manner involves laboratory test and field experiment [2-3]. The indirect manner is the priority research area and can be found in many interrelated studies, such as pedo-transfer functions (PTFs), fractal,
soil morphology, numerical inversion and empirical formulae [4]. The current means have great error due to the complexity of freezing-thawing soil water movement [5-11]. After comprehensive analysis of current means, a new set of identifying methods of freezing-thawing soil hydraulic properties was proposed and its validity was verified in the study.

# 2 Basic theories

## 2.1 Soil freezing characteristic curve

Soil liquid water is not completely frozen when soil temperature is lower than 0°C and a certain amount of soil unfrozen water exists all along. The content of unfrozen water in frozen soil depends on earthiness, environmental condition and freezing-thawing history. Soil freezing characteristic curve reflects a dynamic balance relation between unfrozen water content and negative temperature in frozen soil and can be expressed as follows:

$$ \theta_u = AT_{abs}^{-B} \quad (1) $$

where $\theta_u$ is the soil unfrozen water content, $T_{abs}$ is the absolute value of negative temperature, A and B are the empirical coefficients related with earthiness.

Soil freezing characteristic curve is a normal straight line and can be obtained by regression analysis of experimental data.

## 2.2 van Genuchten-Mualem model

van Genuchten-Mualem model is an empirical model of calculating unsaturated soil hydraulic properties. It is widely used in soil water movement research for it can properly fit filed experimental data, although its mathematical form is complex [12-13]. The formulae of van Genuchten-Mualem model used in the study are as follows:

$$ |\psi| = \left\{ \left[ \left( \theta - \theta_r \right) / \left( \theta_s - \theta_r \right) \right]^{\frac{n}{1-n}} \right\}^{\frac{1}{n}} / a \quad (2) $$

$$ K = K_s \left\{ \left[ 1 - \left( a |\psi| \right)^{n-1} \left[ 1 + \left( a |\psi| \right)^n \right]^{\frac{1-n}{n}} \right] \right\}^{\frac{1}{n-1}} / \left[ 1 + \left( a |\psi| \right)^n \right]^{\frac{n-1}{2n}} \quad (3) $$

$$ C = (n-1)(\theta - \theta_r) \left\{ 1 - \left[ \left( \theta - \theta_r \right) / \left( \theta_s - \theta_r \right) \right]^{\frac{n}{n-1}} \right\} / |\psi| \quad (4) $$

where $\theta_s$ is the saturated soil water content, $\theta_r$ is the soil residual water content, $\psi$ is the soil matric potential, a and n are the empirical parameters.
The two empirical parameters in van Genuchten-Mualem model can be obtained by reversely solving the relation curve of the measured soil unfrozen water content data and the corresponding soil matric potential data. However, the curve is complicated and its parameter estimation is a nonlinear fitting problem, therefore the particle swarm optimization algorithm is performed in the study to solve the problem.

Particle swarm optimization (PSO) algorithm was proposed by Kennedy and Eberhart in 1995 [14-15]. It is a kind of self-adapting evolutionary computation technology and swarm intelligence algorithm on the basis of population search. Each solution of optimization problem is considered as a particle among the searching space in PSO algorithm. Each particle has its own position, speed and adaptive value (fitness) decided by a certain optimization problem. The best position that a particle has experienced during the flight is the best solution found by the particle (called the personal extremum, pbest). The best position that the whole group has experienced is the best solution found by the whole group (called the global extremum, gbest). Every particle constantly updates itself by using pbest and gbest and then creates a new population. At last, the whole population can comprehensively search the solution region and find the optimal values [16].

Let N be the particle population size, $x_i$ be the position of the i th particle ($i=1, 2, \dots, N$), $v_i$ be the speed and $f_i$ be the adaptive value of the particle. After the initial position and speed generate randomly, a particle updates itself by following the personal extremum pbest($t$) and the global extremum gbest($t$) in each iteration. At the moment of $t+1$, any particle $i$ can update its own position and speed by the following formulae:

$$v_i(t+1) = w v_i(t) + c_1 r_1(t) (pbest_i(t) - x_i(t)) + c_2 r_2(t) (gbest(t) - x_i(t)) \quad (5)$$

$$x_i(t+1) = x_i(t) + v_i(t+1) \qquad (6)$$

where $c_1$ and $c_2$ are the learning factors, $r_1$ and $r_2$ are the random numbers evenly distributing in the interval of (0, 1), w is the inertia weight.

Then the personal extremum of each particle and the global extremum of all particles can update by the following formulae:

$$pbest_i(t+1) = \begin{cases} x_i(t+1) & f_i(t+1) \ge f(pbest_i(t)) \\ pbest_i(t) & f_i(t+1) < f(pbest_i(t)) \end{cases} \qquad (7)$$

$$gbest(t+1) = x_{max}(t+1) \qquad (8)$$

where $f_i(t+1)$ is the adaptive value of the i th particle at the moment of $t+1$, $f(pbest_i(t))$ is the best adaptive value of the i th particle in its searching history, $x_{max}(t+1)$ is the position of the particle that corresponds with the biggest $f(pbest_i(t))$ among all the particles at the moment of $t+1$.

The two parameters ($a$ and $n$) are considered as the particle of PSO. The objective function of PSO can be expressed by the following formula:
$$ \min Q = \max(|Y_1 - y_1|, |Y_2 - y_2|, \dots, |Y_m - y_m|) \qquad (9) $$

where $Y_i$ and $y_i$ are the calculated and measured value respectively, m is the number of the measuring point in soil profile.

The algorithm will stop running when the difference between the former and later iteration value of the global optimal particle is less than error threshold. Thus, the convergence criterion of PSO can be expressed as follows:

$$ |a_{gbest}^p - a_{gbest}^{p-1}| \le e_1 \& |n_{gbest}^p - n_{gbest}^{p-1}| \le e_2 \qquad (10) $$

where $a_{gbest}$ and $n_{gbest}$ are the global optimal values of the model parameters (a and n) in each iteration, p and p-1 are the iteration number, $e_1$ and $e_2$ are the convergence thresholds of the model parameters (a and n).

# 3 Results and analysis

## 3.1 Fixing soil freezing characteristic curve

The soil profile in the study was divided into two layers, one extended from 0 to 100cm and the other one is under 100cm. The functional relations of soil unfrozen water content and negative temperature were built separately on the basis of the measured data from two different soil layers. The curve fitting situations were showed in Fig. 1.

<figure><img src="image_2.png" /></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Soil freezing characteristic curve</figcaption></figure>

The expressions of soil freezing characteristic curves of the two soil layers are as follows:

$$ \theta_u = 2E - 18(T + 273.16)^{6.935} \quad (0 \sim 100\text{cm}) \qquad (11) $$

$$ \theta_u = 6E - 32(T + 273.16)^{12.535} \quad (\text{Under } 100\text{cm}) \qquad (12) $$

where $\theta_u$ is the soil unfrozen water content, $T$ is the negative temperature.

## 3.2 Estimating unsaturated soil hydraulic properties

The PSO algorithm was programmed in Matlab 7.0. Set population size of particle swarm is 200; the maximum iteration time is 10000; the convergence thresholds as $1 \times 10^{-8}$ and $1 \times 10^{-4}$; the learning factors as 2 and the inertia weight changing from 0.9 to 0.1. After running the PSO algorithm program, the optimal solution of the objective function was found when the program ran to the convergence criterion and the model parameters of the two soil layers were listed in Table 1.

<table><caption>Table 1. Results of estimating model parameters</caption><thead><tr><th>Depth (cm)</th><th>θ<sub>s</sub></th><th>θ<sub>r</sub></th><th>a (cm<sup>-1</sup>)</th><th>n</th></tr></thead><tbody><tr><td>0~100</td><td>0.46735</td><td>0.02</td><td>0.000086</td><td>1.6939</td></tr><tr><td>Under 100</td><td>0.458</td><td>0.03</td><td>0.0684</td><td>1.1271</td></tr></tbody></table>

The soil water characteristic curves of the two layers were fixed by putting the parameters from Table 1 back in the van Genuchten-Mualem model, and showed in Fig. 2.
<figure><img src="image_4.png" /><figcaption>Fig. 2. Soil water characteristic curve</figcaption></figure>

The expressions of unsaturated hydraulic conductivity and specific water capacity in each soil layer were as follows:

$$ 
\left\{ 
\begin{aligned}
K &= \frac{K_s \left\{ \left[ 1 - (8.6 \times 10^{-5} |\psi|)^{0.6939} \left[ 1 + (8.6 \times 10^{-5} |\psi|)^{1.6939} \right]^{-0.4096} \right]^{2} \right\}}{\left[ 1 + (8.6 \times 10^{-5} |\psi|)^{1.6939} \right]^{0.2048}} && (0 \sim 100 \text{cm}) \\
C &= \frac{0.6939 (\theta - 0.02) \left[ 1 - \left[ (\theta - 0.02) / 0.44735 \right]^{2.4411} \right]}{|\psi|}
\end{aligned}
\right. 
(13) 
$$

$$ 
\left\{ 
\begin{aligned}
K &= \frac{K_s \left\{ \left[ 1 - (0.0684|\psi|)^{0.1271} \left[ 1 + (0.0684|\psi|)^{1.1271} \right]^{-0.1128} \right]^{2} \right\}}{\left[ 1 + (0.0684|\psi|)^{1.1271} \right]^{0.0564}} && (\text{Under } 100\text{cm}) \\
C &= \frac{0.1271 (\theta - 0.03) \left[ 1 - \left[ (\theta - 0.03) / 0.428 \right]^{8.8678} \right]}{|\psi|}
\end{aligned}
\right. 
(14) 
$$

### 3.3 Verifying validity of identifying method

In order to verify the validity of the identifying method of freezing-thawing soil hydraulic properties, the comparison between numerical simulated value and field
measured value is necessary. Thus, the numerical simulation of freezing-thawing soil water movement was conducted on a period in 2008.

The soil unfrozen water content distribution in December 19, 2008 was simulated and the freezing status was that the frozen depth had developed to 55cm. The comparison between simulated and measured values was showed in Fig. 3.

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Comparison between the measured and simulated soil unfrozen water content (2008-12-19)</figcaption></figure>

According to Fig. 3, the simulated and measured soil unfrozen water content in December 19, 2008 coincided with each other very well and had the same trend. The average relative error of the numerical simulation of soil unfrozen water content was 20.18%, therefore, the simulation was successful.

**4. Conclusions**

A new set of identifying methods of freezing-thawing soil hydraulic properties was proposed in the study. It includes two part, one part is fixing soil freezing characteristic curve by a regression and fitting analysis of measured data, the other part is using particle swarm optimization algorithm to estimate the empirical parameters of van Genuchten-Mualem model and then to obtain soil water characteristic curve, unsaturated hydraulic conductivity and specific water capacity. At last, a comparison and error analysis was performed on the simulated and measured data of freezing-thawing soil water movement for the purpose of verifying method validity. The results showed that the identifying method of freezing-thawing soil hydraulic properties proposed in the study had good effectiveness and suitability.
## Acknowledgements

The study was supported by the National Natural Science Foundation (51209039, 51279031), China Postdoctoral Science Foundation (2011M500632, 2012T50321), Heilongjiang Postdoctoral Fund (LBH-Z11231), Foundation of Heilongjiang Educational Committee (12521016), Doctoral Foundation of Northeast Agricultural University (2012RCB13) and Foundation of Harbin S&T Bureau (2010RFXXN043).

## References

1. Zhidong Lei, Shixiu Yang, Senchuan Xie.: Soil Water Dynamics. Tsinghua University Press, Beijing (1988)

2. Linfang Wang, Jun Fan, Quanjiu Wang.: Measurement of soil conductivity by automated disc tension infiltrometer. Transactions of the CSAE, 23(9), 72--75 (2007)

3. Bagarello V., Sgroi A.: Using the simplified falling head technique to detect temporal changes in field-saturated hydraulic conductivity at the surface of a sandy loam soil. Soil and Tillage Research, 94(2), 283--294 (2007)

4. Shaohui Xu, Jianli Liu.: Advances in approaches for determining unsaturated soil hydraulic properties. Advances in Water Science, 14(4), 494--501, (2003)

5. Hongsong Chen, Mingan Shao.: A review on indirect methods for estimating unsaturated soil hydraulic properties. Journal of Basic Science and Engineering, 10(2), 103--109 (2002)

6. Augeard B., Assouline S., Fonty A., et al.: Estimating hydraulic properties of rainfall-induced soil surface seals from infiltration experiments and X-ray bulk density measurements. Journal of Hydrology, 341(1/2), 12--26 (2007)

7. Minasny B., Hopmans J.W., Harter T., et al.: Neural networks prediction of soil hydraulic function for alluvial soils using multistep outflow data. Soil Science Society of America Journal, 68(2), 417--429 (2004)

8. Jinxia Yu, Huirong Guo, Qiong Fang.: Confirmation of the parameters of the unsaturated soil moisture movement: A case study of Loam in Zhumadian of Henan Province. Ground Water, 28(4), 25--27 (2006)

9. Yichang Wei, Zuoxin Liu, Lingling Kang.: Derivation and verification of soil hydrodynamic parameters in cinnamon soil. Journal of Hydraulic Engineering, (3), 81--86 (2004)

10. Mingan Shao, Quanjiu Wang, Robert H.: A simple infiltration method for estimating soil hydraulic properties of unsaturated soils I: Theoretical analysis. Acta Pedologica Sinica, 37(1), 1--8 (2000)

11. Mingan Shao, Quanjiu Wang, Robert H. A simple infiltration method for estimating soil hydraulic properties of unsaturated soils II: Experimental results. Acta Pedologica Sinica, 37(2), 217--224 (2000)

12. Mualem Y.: A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resources Research, 12, 513--522 (1976)

13. van Genuchten M.T.: A closed-form equation for predicting hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal, 44, 892--898 (1980)

14. Kennedy J., Eberhart R.C.: Particle Swarm Optimization. IEEE International Conference on Neural Networks, 1942--1948 (1995)

15. Eberhart R.C., Kennedy J.: A New Optimization Using Particle Swarm Theory. Proceedings of the sixth International Symposium on Micro Machine and Human Science, 39--43 (1995)

16. Zilong Wang, Qiang Fu, Qiuxiang Jiang. Soil nutrient management zones based on particle swarm optimization algorithm. Transactions of the CSAE, 24(10), 80--84 (2008)