# Numerical Study of the Influence of Horizontal Spatial Distribution of Macropores on Water Infiltration

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Modeling Scheme

#### 2.2. COMSOL Simulation

^{−1}], H

_{p}is the pressure head [L] and $z$ is height, which is positive upwards [L]. As shown in Figure 1a, water flows in from the surface of the soil (pressure head equals 0) and out from the bottom of the soil (pressure head equals 0), and other boundaries are set as no-flow boundary conditions [58,59]. Water is driven entirely by gravity. Following Xin et al. [30], the macropore domain in the model is defined as the super-permeability zone [60,61]. The hydraulic conductivities of the macropore and matrix domain in the model are 1 m/s and 1.23 × 10

^{−5}m/s, respectively, and the porosities of the macropore and matrix domain in the model are 1 and 0.41, respectively [21].

#### 2.3. Statistical Parameters for Characterizing Macropore Position

#### 2.3.1. Spatial Dispersion, $\gamma $ (Position Relationship among Macropores)

^{2}] (as shown in Figure 2). Subsequently, the spatial dispersion, $\gamma $, is calculated as follows:

^{2}], and ${w}_{f}$ is the relative macroporosity at the soil surface (dimensionless).

#### 2.3.2. Spatial Deviation, ${\gamma}^{\mathrm{*}}$ (Position Relationship between the Macropores and Observation Area)

## 3. Results and Discussion

#### 3.1. The Effect of Macropore Number

#### 3.2. The Effect of Pore Size

#### 3.3. The Effect of Macropore Position

#### 3.4. Comparison between Number and Position Effects

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Generation Approach for Random Pore Sizes and Positions

## Appendix B. Spatial Dispersion (Weighted Mean Distance)

#### Appendix B.1. Case 1: When the Distance among Macropores Increases

**Figure A1.**Schematic diagram of case 1 at the soil surface, where the circles represent macropores and the red dots represent the total macropore centroid.

#### Appendix B.2. Case 2: When the Macroporosity Is Concentrated in One Macropore

**Figure A2.**(

**a**) is the relationship between ${x}_{3}$ and $\gamma $ for case 1; (

**b**) is the relationship between $\eta $ and $\gamma $ for case 2.

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**Figure 1.**(

**a**) is a schematic diagram of the model. (

**b**) is a schematic diagram of the simulation scheme.

**Figure 2.**The implication of spatial dispersion and spatial deviation, where circles represent macropores, red dots represent the total macropore centroid, and green dots represent the centroid of the observation area.

**Figure 3.**The relationship between four hydraulic evaluation indexes (the total infiltration fluxes (TIFs), the preferential infiltration fluxes (PIFs), the matrix infiltration fluxes (MIFs) and the proportion of preferential infiltration fluxes (PPIFs)) and the macropore number.

**Figure 4.**The relationship between pore size mean and four hydraulic evaluation indexes (which are TIFs (

**a**), PIFs (

**b**), MIFs (

**c**) and PPIFs (

**d**), respectively), and the relationship between pore size standard deviation and four hydraulic evaluation indexes (which are TIFs (

**e**), PIFs (

**f**), MIFs (

**g**) and PPIFs (

**h**), respectively) when $N$ = 9. The red lines are regression lines, and the red text is the expressions and correlation coefficients of these regression lines.

**Figure 5.**The relationship between macropore position and four hydraulic evaluation indexes (which are TIFs (

**a**), PIFs (

**b**), MIFs (

**c**) and PPIFs (

**d**), respectively) when $N$ = 9. The points are the simulated results, and the curves are the contour lines of the fitting surfaces whose correlation coefficients are 0.32113, 0.14672, 0.08565 and 0.12265, respectively.

**Figure 6.**The relationship between flow rate distribution at the bottom and macropore position at the surface when $N$ = 9, in which the black circles represent macropores at the surface. (

**a**–

**i**) show the cases of different spatial dispersion and spatial deviation. The spatial deviation decreases from top figures to bottom figures, and the spatial dispersion increases from left figures to right figures.

**Figure 7.**The competitive relationship between number and position in four hydraulic evaluation indexes (which are TIFs (

**a**), PIFs (

**b**), MIFs (

**c**) and PPIFs (

**d**), respectively). The dots are the simulated data, the meshes are the fitting surfaces, and the text is the correlation coefficients of the fitting surfaces. Red, blue, and green represent $N$ = 2, $N$ = 3, and $N$ = 4, respectively.

**Figure 8.**The competitive relationship between number and position in four hydraulic evaluation indexes (which are TIFs (

**a**), PIFs (

**b**), MIFs (

**c**) and PPIFs (

**d**), respectively). The dots are the simulated data, the meshes are the fitting surfaces, and the text is the correlation coefficients of the fitting surfaces. Red, blue, and green represent $N$ = 18, $N$ = 19, and $N$ = 20, respectively.

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## Share and Cite

**MDPI and ACS Style**

Zhang, R.; Huan, X.; Qian, J.; Xing, Y.
Numerical Study of the Influence of Horizontal Spatial Distribution of Macropores on Water Infiltration. *Water* **2023**, *15*, 3593.
https://doi.org/10.3390/w15203593

**AMA Style**

Zhang R, Huan X, Qian J, Xing Y.
Numerical Study of the Influence of Horizontal Spatial Distribution of Macropores on Water Infiltration. *Water*. 2023; 15(20):3593.
https://doi.org/10.3390/w15203593

**Chicago/Turabian Style**

Zhang, Ruigang, Xiaoxiang Huan, Jiazhong Qian, and Yueqing Xing.
2023. "Numerical Study of the Influence of Horizontal Spatial Distribution of Macropores on Water Infiltration" *Water* 15, no. 20: 3593.
https://doi.org/10.3390/w15203593