# Modified Richards’ Equation to Improve Estimates of Soil Moisture in Two-Layered Soils after Infiltration

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}(Figure 1). The hillslope had an average gradient of 34% (corresponding slope is 19 degrees). The dominant vegetation was primary perennial herbs and shrubs. According to the China Soil Database, the loamy soil is the main soil type in the study sites, with mean bulk density of 1.4576 g/m

^{3}. The loamy soil properties, including texture (content of sand, silt, and clay), bulk density, saturated water content, saturated hydraulic conductivity, and the fitting parameters to soil water retention curve are described in Table 1.

#### 2.2. Soil Moisture Measurement

^{3}/m

^{3}) ranging from 0 to 100% with accuracy of ±2%. Most infiltration models and equations assume that initial soil moisture is uniform among different depths, but in reality, initial water moisture is non-uniform in the natural soil profile. Monitoring instruments can record non-uniform initial water moisture during the infiltration process. The initial water moisture data recorded by the EM50 instruments were used to simulate the distribution of soil water during infiltration.

#### 2.3. Theory

#### 2.3.1. Governing Equations

_{0}.

#### 2.3.2. Modified Solution Equation with Parameters of Soil Properties for Upper Soil Layer

_{m}is water pressure head. Therefore, the solution of Equation (6) can be formulated as follows:

#### 2.3.3. Analytical Formulation with Slope for Deeper Soil Layer

_{0}and θn are initial and saturate soil water content. The sum of soil moisture is defined as follows [28]:

#### 2.4. Data Statistics and Analysis

## 3. Results

#### 3.1. Temporal Analysis of Soil Moisture in Vertically Layered Soils

#### 3.2. Comparison of Simulation Results of Modified and Original Richards’ Equation for Two Soil Layers

#### 3.3. Spatiotemporal Variability in Soil Moisture Estimates

## 4. Discussion

#### 4.1. Impact of Overland Flow on Simulations of Soil Moisture in Upper Soil Layer in Longer Rainy Periods

#### 4.2. Effect of Downslope Flow on Accuracy of Simulations in Different Slope Positions

## 5. Conclusions

- Although Richards’ equation is one of the most widely used infiltration equations in hydrological models, the original analytical solutions have a limited ability to estimate soil moisture traces during infiltration on a hillslope.
- The spatial–temporal variations in soil moisture are controlled by diverse environmental control factors, such as hillslope and soil properties. However, these factors are difficult to express using the original Richards’ equation because of the difficulty in summarizing the related parameters. As far as we know, this is the first attempt to express environmental factors using in situ layered parameters in the infiltration equation.
- The simulation results calculated by the modified Richards’ equation with layered parameters were better than those calculated by the original equation. However, the bias between the simulations using the modified Richards’ equation and the observed values was higher for the upper soil layers. The accuracy of simulations varied depending on the slope position and the length of the rainy period, because of the effects of lateral downslope flow and overland flow during infiltration.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Location of observation station and schematic diagram of monitoring instrument distribution.

**Figure 3.**Comparison between observed and simulated soil moisture by the modified Richards’ equation and original Richards’ equation (

**a**) at the depth of 10 cm and (

**b**) at the depth of 50 cm.

**Figure 4.**Comparison of simulated and observed soil moisture at various measurement points and soil depths.

**Figure 5.**Comparisons of the simulation results among different slope position. The depth is from 10 to 50 cm.

**Figure 6.**Estimated and observed soil moisture at different depths for two rainfall events with short intervals.

**Table 1.**Soil properties: texture (content of sand, silt, and clay), bulk density, saturated water content, saturated hydraulic conductivity, and the fitting parameters to soil water retention curve at this site.

Parameters | Texture | n | Sand | Silt | Clay | ρ | θn | D(θ) | α | Ks |
---|---|---|---|---|---|---|---|---|---|---|

(%) | (%) | (%) | (g/cm^{3}) | (cm^{3}/cm^{3}) | (/cm) | (m/d) | ||||

Values | loam | 0.5 | 49.44 | 45.04 | 5.52 | 1.4576 | 0.5 | 0.001 | 5 | 0.000132 |

**Table 2.**Temporal statistics (September 2016–August 2017) for mean soil moisture, the standard deviation, and the coefficient of variation in multiple soil layers.

Temporal Statistics | 0–10 cm | 10–20 cm | 20–30 cm | 30–40 cm | 40–50 cm |
---|---|---|---|---|---|

Mean (%) | 31.58% | 30.83% | 23.04% | 24.65% | 16.28% |

Standard Deviation (SD) | 0.037708 | 0.018525 | 0.006102 | 0.01508 | 0.015195 |

Coefficient of Variation (CV) | 11.94% | 6.01% | 9.33% | 6.12% | 2.65% |

**Table 3.**Simulation accuracy of modified equation and Richards’ Equation for top soil layer (10 cm).

Date | Root Mean Square Error (RMSE) | Bias | Relative Error | |||
---|---|---|---|---|---|---|

Modified Equation | Richards’ Equation | Modified Equation | Richards’ Equation | Modified Equation | Richards’ Equation | |

2016/9/26 | 0.0057 | 0.0083 | 0.0052 | −0.0053 | 2.41% | 2.19% |

2016/10/7 | 0.0108 | 0.0199 | 0.0090 | 0.0197 | 3.36% | 7.47% |

2016/11/21 | 0.0061 | 0.0050 | 0.0047 | −0.0032 | 1.84% | 1.18% |

2017/3/24 | 0.0010 | 0.0080 | 0.0010 | −0.0080 | 0.66% | 5.20% |

2017/4/13 | 0.0037 | 0.0066 | 0.0026 | −0.0061 | 1.68% | 3.75% |

2017/5/13 | 0.0272 | 0.0428 | 0.0030 | 0.02342 | 1.06% | 7.85% |

Average | 0.0091 | 0.0151 | 0.0042 | 0.0034 | 1.84% | 4.61% |

**Table 4.**Simulation accuracy of modified equation and Richards’ Equation for deeper soil layers (50 cm).

Date | Root Mean Square Error | Bias | Relative Error | |||
---|---|---|---|---|---|---|

Modified Equation | Richards’ Equation | Modified Equation | Richards’ Equation | Modified Equation | Richards’ Equation | |

2016/9/26 | 0.0100 | 0.0233 | 0.0024 | 0.0019 | 1.47% | 1.98% |

2016/10/7 | 0.0016 | 0.0150 | 0.0013 | −0.0052 | 0.57% | 1.84% |

2016/11/21 | 0.0026 | 0.0063 | 0.0024 | −0.0062 | 1.43% | 3.63% |

2017/3/24 | 0.0010 | 0.0080 | 0.0010 | −0.0080 | 0.66% | 5.20% |

2017/4/13 | 0.0037 | 0.0066 | 0.0026 | −0.0061 | 1.68% | 3.75% |

2017/5/13 | 0.0579 | 0.0569 | 0.0046 | −0.0008 | 3.79% | 1.57% |

Average | 0.0128 | 0.0194 | 0.0023 | −0.0041 | 1.60% | 2.99% |

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**MDPI and ACS Style**

Zhu, H.; Liu, T.; Xue, B.; A., Y.; Wang, G. Modified Richards’ Equation to Improve Estimates of Soil Moisture in Two-Layered Soils after Infiltration. *Water* **2018**, *10*, 1174.
https://doi.org/10.3390/w10091174

**AMA Style**

Zhu H, Liu T, Xue B, A. Y, Wang G. Modified Richards’ Equation to Improve Estimates of Soil Moisture in Two-Layered Soils after Infiltration. *Water*. 2018; 10(9):1174.
https://doi.org/10.3390/w10091174

**Chicago/Turabian Style**

Zhu, Honglin, Tingxi Liu, Baolin Xue, Yinglan A., and Guoqiang Wang. 2018. "Modified Richards’ Equation to Improve Estimates of Soil Moisture in Two-Layered Soils after Infiltration" *Water* 10, no. 9: 1174.
https://doi.org/10.3390/w10091174