# Interplays between State and Flux Hydrological Variables across Vadose Zones: A Numerical Investigation

^{1}

^{2}

^{*}

## Abstract

**:**

_{a}) and groundwater recharge (GR)) hydrological variables across vadose zones is critical for understanding ecohydrological and land-surface processes. In this study, a one-dimensional process-based vadose zone model with generated soil hydraulic parameters was utilized to simulate soil moisture, ET

_{a}, and GR. Daily hydrometeorological data were obtained from different climate zones to drive the vadose zone model. On the basis of the field phenomenon of soil moisture temporal stability, reasonable soil moisture spatiotemporal structures were reproduced from the model. The modeling results further showed that the dependence of ET

_{a}and GR on soil hydraulic properties varied considerably with climatic conditions. In particular, the controls of soil hydraulic properties on ET

_{a}and GR greatly weakened at the site with an arid climate. In contrast, the distribution of mean relative difference (MRD) of soil moisture was still significantly correlated with soil hydraulic properties (most notably residual soil moisture content) under arid climatic conditions. As such, the correlations of MRD with ET

_{a}and GR differed across different climate regimes. In addition, the simulation results revealed that samples with average moisture conditions did not necessarily produce average values of ET

_{a}and GR (and vice versa), especially under wet climatic conditions. The loose connection between average state and flux hydrological variables across vadose zones is partly because of the high non-linearity of subsurface processes, which leads to the complex interactions of soil moisture, ET

_{a}, and GR with soil hydraulic properties. This study underscores the importance of using soil moisture information from multiple sites for inferring areal average values of ET

_{a}and GR, even with the knowledge of representative sites that can be used to monitor areal average moisture conditions.

## 1. Introduction

_{a}) and groundwater recharge (GR)) hydrological variables across vadose zones are of great importance for many research and application purposes, and especially for understanding ecohydrological and land-surface processes in Earth’s critical zones [1,2,3]. Due to the high non-linearity and extreme complexity of hydrological systems, those state and flux variables tend to display significant spatiotemporal variability. For example, a wealth of field and numerical studies has shown that there was considerable variability in soil moisture at different spatiotemporal scales, as a result of its complex interactions with surrounding environments [4,5]. Likewise, flux hydrological variables, such as ET

_{a}and GR, may also vary substantially across landscapes, largely because of the marked spatial variations in soil texture, vegetation, and climatic conditions [1,6,7,8,9]. As such, accurate estimation of those state and flux variables presents grand challenges to scientific communities [10,11,12,13]. Given the tight connections between state and flux variables [12,14,15], it is thus imperative to elucidate the interplays among those variables to improve their estimation across vadose zones.

_{a}and GR is extremely limited due to the high operational and maintenance costs of measurement equipment [3,25]. As an effective alternative to resolve this issue, soil moisture data have been used to infer ET

_{a}and GR, such as through inverse modeling [15,26,27]. Given the significant spatial variability in ET

_{a}and GR, it naturally leads to the question as to whether soil moisture data from representative locations identified by the TSA method can be used to infer areal average values of ET

_{a}and GR. To answer this question, it requires the knowledge of the relationships of soil moisture with ET

_{a}and GR. Although previous modeling studies have revealed that soil moisture, ET

_{a}, and GR were affected by similar factors (e.g., soil, vegetation, and climate), the dependence of those variables on environmental factors might vary considerably [9,28,29]. Therefore, further investigation into the interplays among soil moisture, ET

_{a}, and GR is still warranted.

_{a}and GR under different climatic conditions and (2) to examine whether soil samples with average moisture conditions can also produce average hydrological fluxes, such as ET

_{a}and GR, across vadose zones. To this end, a commonly used process-based vadose zone model with generated soil hydraulic parameters was utilized to simulate soil moisture, ET

_{a}, and GR, as this type of modeling approach has been widely adopted for understanding hydrological processes across vadose zones (e.g., Vereecken et al. [30]; Wang et al. [31]; Martinez et al. [28]; also see the review by Vereecken et al. [5]). Based on the simulation results, the distribution patterns of MRD, ET

_{a}, and GR were first discussed, and then the correlations of MRD with ET

_{a}and GR were examined to understand the interdependence of those variables on climatic conditions and soil hydraulic properties. Finally, with the aid of the TSA method, the feasibility of using representative locations of soil moisture for estimating areal average values of ET

_{a}and GR was evaluated.

## 2. Methods and Materials

#### 2.1. Model Setup and Data Inputs

_{a}, and GR, due to the accuracy of its numerical algorithm [33]. The Hydrus-1D model is a one-dimensional process-based vadose zone model, which utilizes the Richards equation to simulate soil water movement in vadose zones:

^{3}/L

^{3}) is volumetric soil moisture content, t (T) is time, x (L) is spatial coordinate, h (L) is pressure head, K (L/T) is hydraulic conductivity, and S (1/T) is a sink/source term (e.g., root water uptake). To be concise, only brief information on the model setup is given here, and detailed mathematical descriptions of the simulated processes can be found in Simunek et al. [32]. Specifically, the length of the simulated soil columns was set to be 3 m with a total of 301 spatial nodes evenly distributed at 1 cm intervals. Additional numerical experiments showed that further increases in the number of the spatial nodes did not improve the simulation results. At the surface, an atmospheric boundary condition was chosen for calculating actual soil evaporation (E

_{a}), which primarily depended on potential soil evaporation (E

_{p}) and a predefined pressure head at the surface [34]. Surface runoff was allowed to occur when precipitation (P) intensity was greater than the soil infiltration capacity or the surface soil layer became saturated. A unit hydraulic gradient condition was imposed at the lower boundary of the simulated soil columns. As such, GR is defined here as the amount of water that passes downward across the lower boundary [6,31].

_{p}), soil matric potential, and root density distribution. Note that root water uptake was assumed to be equal to actual transpiration (T

_{a}) in this study. Daily potential transpiration T

_{p}was obtained by partitioning daily potential evapotranspiration (ET

_{p}) into T

_{p}and E

_{p}according to Beer’s law [36]:

_{p}= ET

_{p}× e

^{-k×LAI}

_{p}= ET

_{p}− E

_{p}

_{a}and T

_{a}was set to be equal to ET

_{a}.

_{p}based on maximum and minimum air temperatures [42]. The summary of mean annual P ($\overline{P}$) and mean annual ET

_{p}($\overline{E{T}_{p}}$) during the study period is provided in Table 1 for the four sites. Based on the land cover of the study sites, forest was assumed at Hangzhou and Guiyang, while grass and shrub were assumed at Tianjin and Yinchuan, respectively. Physiological parameter values for different vegetation types, which were used for computing root water uptake, were obtained from literature [43,44,45]. Finally, to minimize the impact of initial conditions on the simulation results, all simulations were repeated for four times (i.e., a total of 20 years) until the simulated soil moisture profiles were in equilibrium with atmospheric forcings, and the simulation results from the last repetition were used in the analysis.

#### 2.2. Generation of Soil Hydraulic Parameters

_{S}× S

_{e}

^{l}× (1 − (1 −S

_{e}

^{1/m})

^{m})

^{2}

_{r}(L

^{3}/L

^{3}) and θ

_{s}(L

^{3}/L

^{3}) are residual and saturated soil moisture content, respectively, K

_{S}(L/T) is saturated hydraulic conductivity, and S

_{e}(-) is effective saturation degree that is defined as (θ − θ

_{r})/(θ

_{s}− θ

_{r}). The parameters α, n, and l are shape factors: α(1/L) is related to the inverse of air entry pressure, n (-) is a measure of pore size distributions with m = 1 − 1/n, and l (-) is a measure of pore tortuosity and connectivity.

_{a}, and GR [9,28,48,49]. Here, the method of Zhang and Schaap [50] was utilized to generate the VGM parameters for loamy sand, sandy loam, and silt loam. This method is built on the assumption of either normal (θ

_{r}and θ

_{s}) or lognormal (α, n, and K

_{s}) distributions of the VGM parameters. The validity of this assumption has been verified by Zhang and Schaap [50]. Based on a Monte Carlo procedure, new samples were drawn from the parameter (normal or lognormal) distributions with means and standard deviations given by Zhang and Schaap [50] for each soil texture class. In addition, following Wang et al. [31] and Zhang et al. [51], the parameter l was fixed to be 0.5 in all simulations. In this study, a total of 200 samples were generated for each soil texture class and the statistical summary of the generated VGM parameters is provided in Table 2. Note that soil profiles with uniform soil textures were used in this study. The main purpose of using uniform soil profiles is to promote general understanding, and thus reach general conclusions regarding the impact of soil hydraulic properties on subsurface hydrological processes. By comparison, the use of non-uniform soil profiles might lead to site-specific conclusions (as done in the majority of field-scale soil moisture studies), which, however, deserves future investigation based on numerical approaches.

#### 2.3. Temporal Stability Analysis of Soil Moisture

_{a}and GR. The TSA method is based on the metric of relative difference (RD) of soil moisture, which is defined as:

_{ij}is soil moisture content at location i and time j, and ${\overline{\theta}}_{j}$ is the spatial average of soil moisture content at time j. With a series of RD over time, MRD at location i is given as:

## 3. Results and Discussion

#### 3.1. Comparison of Hydroclimatic Conditions across the Study Sites

_{a}(defined as $\overline{E{T}_{a}}/\overline{P}$) and GR ($\overline{GR}/\overline{P}$) were used in the following analysis to ensure the comparability of the simulated patterns of ET

_{a}and GR at different sites. With regard to the interannual variability, there were also pronounced variations in annual P at all sites; however, the interannual variability in ET

_{p}was less significant (Table 1). It should be noted that climate seasonality is another important climatic factor that affects ET

_{a}and GR [40,54]. Therefore, to examine the climate seasonality at the study sites, mean monthly P and ET

_{p}during the five-year study period are plotted in Figure 1. It can be seen from Figure 1 that the climates at the sites exhibited similar seasonal patterns with highest P and ET

_{p}occurring during summer months. As such, it is reasonable to argue that climate seasonality did not exert a significant impact on the comparison of the distribution patterns of the simulated $\overline{E{T}_{a}}/\overline{P}$ and $\overline{GR}/\overline{P}$ across the sites in this study.

#### 3.2. Simulated Patterns of MRD

_{a}and GR on the basis of the TSA method, it is necessary to first examine whether simulated soil moisture data could mimic the field phenomenon of soil moisture temporal stability as shown in previous field studies. To this end, Figure 2 shows the ranked MRD with associated SDRD obtained for different soil textures at the study sites. As previously explained, only soil moisture data simulated at 25 cm were used to compute MRD and SDRD in Figure 2. In general, the simulated MRD patterns were similar to those derived from field observations at various spatiotemporal scales [20,21,22,23,24]. In particular, Figure 2 shows that certain samples displayed consistently wetter soil moisture conditions (i.e., with larger MRD values) than others. More importantly, the ranges of both MRD and SDRD were in line with previously reported values obtained at watershed and regional scales [24,53,55]. For example, the range of MRD for loamy sand was from −0.765 to 1.287 at Hangzhou, from −0.797 to 1.324 at Guiyang, from −0.819 to 1.237 at Tianjin, and from −0.912 to 1.166 at Yinchuan (Table 3). It should be emphasized that there were variations in the statistical characteristics of MRD and SDRD produced by different modeling studies [28,29], which could be largely attributed to the differences in the synthetic soil hydraulic parameter datasets and hydrometeorological conditions used for simulations.

_{S}. It, thus, corroborates our previous conjecture that the statistical characteristics of MRD and SDRD produced by synthetic modeling approaches are partly dependent on the variability in soil hydraulic parameters generated for simulations. In summary, the results from the TSA demonstrate the viability of using synthetic modeling approaches for reproducing reasonable spatiotemporal structures of soil moisture as observed in the fields.

#### 3.3. Distribution Patterns of $\overline{E{T}_{a}}/\overline{P}$ and $\overline{GR}/\overline{P}$

_{a}and GR [57]. Rainfall water infiltrated into soils needs to first satisfy the atmospheric demand for ET

_{a}before becoming GR. Thus, the opportunity for the infiltrated water to be evapotranspired is much higher under drier climatic conditions due to the higher atmospheric demands for evapotranspiration, resulting in the increasing trend in $\overline{E{T}_{a}}/\overline{P}$ with increasing climate dryness.

_{a}and GR processes. Due to the higher infiltration capacities (e.g., Table 2 shows that K

_{s}becomes gradually smaller from loamy sand to sandy loam and silt loam), coarser soils tend to produce lower $\overline{E{T}_{a}}/\overline{P}$ and higher $\overline{GR}/\overline{P}$ ratios [40,58]. Interestingly, at the site of Yinchuan with very dry climatic conditions, the contrast in the distribution patterns of $\overline{E{T}_{a}}/\overline{P}$ and $\overline{GR}/\overline{P}$ was less obvious among different soil textures. Regardless of soil textural differences, most of P becomes ET

_{a}in arid regions due to the limited water supplies, which greatly weakens the dependence of ET

_{a}and GR on soil hydraulic properties. It, thus, suggests that the controls of soil hydraulic properties on ET

_{a}and GR vary with climatic conditions.

#### 3.4. Relationships of MRD with $\overline{E{T}_{a}}/\overline{P}$ and $\overline{GR}/\overline{P}$

_{p}) of MRD with $\overline{E{T}_{a}}/\overline{P}$ and $\overline{GR}/\overline{P}$ are reported in Table 4. Overall, there existed positive correlations between MRD and $\overline{E{T}_{a}}/\overline{P}$, while negative ones between MRD and $\overline{GR}/\overline{P}$ (Table 4). The positive dependence of ET

_{a}on soil moisture levels has been widely observed in the field [12,59]. Meanwhile, finer textured soils tend to have larger water holding capacities and lower GR [31], which leads to higher soil moisture levels and MRD values for finer soils, and thus the negative correlation between MRD and GR.

_{r}). Physically speaking, soil moisture levels cannot reach below θ

_{r}under natural conditions. As such, regardless of climatic conditions, soil moisture levels might still vary considerably due to the variability in soil hydraulic properties (Figure 4). In contrast, as previously illustrated, the dependence of ET

_{a}on soil hydraulic properties greatly weakened under drier climatic conditions, leading to the smaller variability in $\overline{E{T}_{a}}/\overline{P}$ for different samples (e.g., at Yinchuan). To further demonstrate the varying dependence of MRD and $\overline{E{T}_{a}}/\overline{P}$ on soil hydraulic properties with climatic conditions, r

_{p}of MRD and $\overline{E{T}_{a}}/\overline{P}$ with different soil hydraulic parameters was computed for sandy loam, and the results are reported in Table 5. It is clear from Table 5 that the control of θ

_{r}on MRD greatly strengthened with increasing climate dryness, while the impacts of soil hydraulic properties on $\overline{E{T}_{a}}/\overline{P}$ along the gradient of increasing climate dryness was less obvious. Therefore, owing to the differences in the dependence of MRD and $\overline{E{T}_{a}}/\overline{P}$ on soil hydraulic properties, the correlation between MRD and $\overline{E{T}_{a}}/\overline{P}$ varied across different climate regimes. Moreover, Table 4 shows that the correlation between MRD and $\overline{E{T}_{a}}/\overline{P}$ was also dependent on soil texture. It appears that correlations between MRD and $\overline{E{T}_{a}}/\overline{P}$ were stronger for coarser soils (e.g., loamy sand) under the same climatic conditions, probably due to the tighter hydrological connection between vadose zone and land surface processes.

_{r}, but associated $\overline{E{T}_{a}}/\overline{P}$ exhibited a clear negative relationship with θ

_{r}. It is clear that samples with average moisture conditions generally do not correspond with samples with average $\overline{E{T}_{a}}/\overline{P}$ values, especially for the study sites with wet climatic conditions.

## 4. Conclusions

_{a}, and GR were affected by both climate and soil, their dependence on soil hydraulic properties varied considerably with climatic conditions. In particular, the controls of soil hydraulic properties on ET

_{a}and GR greatly weakened under arid climatic conditions due to the limited water supplies, whereas the distribution of MRD still largely depended on soil hydraulic properties under the same climatic conditions. As such, the correlations of MRD with ET

_{a}and GR varied with climatic conditions. Moreover, the modeling results showed that samples with average moisture conditions did not necessarily correspond with those with average flux conditions, which resulted from the complex interactions of soil moisture, ET

_{a}, and GR with soil hydraulic properties. Although the TSA method has been widely used for selecting representative locations for monitoring areal average moisture conditions, the results of this study underscore the importance of using soil moisture information from multiple sites to infer areal average values of ET

_{a}and GR, even with the knowledge of representative soil moisture monitoring sites.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Mean monthly precipitation (P) and potential evapotranspiration (ET

_{p}) during the study period at (

**a**) Hangzhou, (

**b**) Guiyang, (

**c**) Tianjin, and (

**d**) Yinchuan.

**Figure 2.**Ranked mean relative difference (MRD) of soil moisture with one standard deviation (vertical bars) for different soil textures at the study sites. Simulated soil moisture data at 25 cm were used to compute MRD.

**Figure 3.**Distributions of simulated mean annual ratios of actual evapotranspiration ($\overline{E{T}_{a}}$) and groundwater recharge ($\overline{GR}$) over precipitation ($\overline{P}$).

**Figure 4.**Relationships of mean relative difference (MRD) of soil moisture with mean annual ratio of actual evapotranspiration ($\overline{E{T}_{a}}$) over precipitation ($\overline{P}$).

**Figure 5.**Distributions of mean annual ratio of actual evapotranspiration ($\overline{E{T}_{a}}$) over precipitation ($\overline{P}$). Samples with MRD < ±0.05 are used here to plot the distributions of $\overline{E{T}_{a}}/\overline{P}$. $\overline{E{T}_{a}}/{\overline{P}}^{*}$ is the average value of $\overline{E{T}_{a}}/\overline{P}$ from all samples.

**Figure 6.**Relationships of MRD and $\overline{E{T}_{a}}/\overline{P}$ with different soil hydraulic parameters for sandy loam at Guiyang. Samples with MRD < ±0.05 are highlighted in red.

**Table 1.**Mean annual precipitation ($\overline{P}$), potential evapotranspiration ($\overline{E{T}_{p}}$), and aridity index ($\overline{E{T}_{p}}$/$\overline{P}$) at the study sites.

Location | Latitude | Longitude | $\overline{\mathit{P}}\text{}(\mathbf{cm}/\mathbf{year})$ | $\overline{\mathit{E}{\mathit{T}}_{\mathit{p}}}\text{}(\mathbf{cm}/\mathbf{year})\text{}$ | $\overline{\mathit{E}{\mathit{T}}_{\mathit{p}}}$$/\overline{\mathit{P}}$(-) |
---|---|---|---|---|---|

Hangzhou | 30.23°N | 120.17°E | 150.9 (127.4–172.9) * | 111.8 (108.0–115.1) | 0.74 (0.62–0.89) |

Guiyang | 26.58°N | 106.73°E | 103.8 (73.5–137.1) | 101.7 (95.9–104.6) | 0.98 (0.74–1.40) |

Tianjin | 39.10°N | 117.17°E | 54.3 (35.6–75.7) | 104.1 (99.9–107.2) | 1.92 (1.37–2.81) |

Yinchuan | 38.47°N | 106.20°E | 20.8 (16.6–29.2) | 108.6 (106.8–110.5) | 5.22 (3.74–6.43) |

**Table 2.**Average values of soil hydraulic parameters for the van Genuchten–Mualem model generated for different soil textural classes.

Soil Texture | θr (cm^{3} cm^{−3}) | θs (cm^{3} cm^{−3}) | α (1/cm) | n (-) | K_{s} (cm/day) |
---|---|---|---|---|---|

Loamy sand | 0.064 (0.035) * | 0.386 (0.063) | 0.036 (0.039) | 1.908 (0.580) | 342.5 (864.1) |

Sandy loam | 0.076 (0.047) | 0.381 (0.068) | 0.040 (0.075) | 1.673 (0.359) | 125.4 (334.1) |

Silt loam | 0.098 (0.057) | 0.432 (0.077) | 0.006 (0.007) | 1.790 (0.475) | 62.8 (180.9) |

**Table 3.**Statistical summary of mean relative difference (MRD) and standard deviation of relative difference (SDRD) of soil moisture. Simulated soil moisture data at 25 cm were used to compute MRD and SDRD.

Location | Loamy Sand | Sandy Loam | Silt Loam | ||||||
---|---|---|---|---|---|---|---|---|---|

Range of MRD | Range of SDRD | Mean SDRD | Range of MRD | Range of SDRD | Mean SDRD | Range of MRD | Range of SDRD | Mean SDRD | |

Hangzhou | −0.765–1.287 | 0.046–0.428 | 0.139 | −0.741–0.799 | 0.041–0.402 | 0.131 | −0.690–1.210 | 0.022–0.264 | 0.085 |

Guiyang | −0.797–1.324 | 0.037–0.486 | 0.158 | −0.789–0.873 | 0.049–0.430 | 0.159 | −0.732–1.230 | 0.027–0.370 | 0.112 |

Tianjin | −0.819–1.237 | 0.043–0.462 | 0.133 | −0.977–0.891 | 0.053–0.397 | 0.134 | −0.793–1.064 | 0.032–0.301 | 0.091 |

Yinchuan | −0.912–1.166 | 0.035–0.397 | 0.114 | −0.889–1.410 | 0.043–0.664 | 0.111 | −0.890–1.485 | 0.024–0.199 | 0.073 |

**Table 4.**Spearman’s rank correlation coefficients of mean relative difference (MRD) of soil moisture with mean annual ratios of actual evapotranspiration ($\overline{E{T}_{a}}$) and groundwater recharge ($\overline{GR}$) over precipitation ($\overline{P}$).

Location | Loamy Sand | Sandy Loam | Silt Loam | |||
---|---|---|---|---|---|---|

$\overline{\mathit{E}{\mathit{T}}_{\mathit{a}}}/\overline{\mathit{P}}$ | $\overline{\mathit{G}\mathit{R}}/\overline{\mathit{P}}$ | $\overline{\mathit{E}{\mathit{T}}_{\mathit{a}}}/\overline{\mathit{P}}$ | $\overline{\mathit{G}\mathit{R}}/\overline{\mathit{P}}$ | $\overline{\mathit{E}{\mathit{T}}_{\mathit{a}}}/\overline{\mathit{P}}$ | $\overline{\mathit{G}\mathit{R}}/\overline{\mathit{P}}$ | |

Hangzhou | 0.679 | −0.668 | 0.431 | −0.429 | 0.461 | −0.462 |

Guiyang | 0.630 | −0.623 | 0.288 | −0.279 | 0.378 | −0.393 |

Tianjin | 0.482 | −0.478 | 0.123 | −0.145 * | 0.188 ** | −0.258 |

Yinchuan | 0.255 | −0.297 | 0.227 ** | −0.241 | 0.089 | −0.158 * |

**Table 5.**Spearman’s rank correlation coefficients of soil hydraulic parameters for sandy loam with mean relative difference (MRD) of soil moisture and mean annual ratio of actual evapotranspiration ($\overline{E{T}_{a}}$) over precipitation ($\overline{P}$).

Soil Hydraulic Parameters | Hangzhou | Guiyang | Tianjin | Yinchuan | ||||
---|---|---|---|---|---|---|---|---|

MRD | $\overline{\mathit{E}{\mathit{T}}_{\mathit{a}}}/\overline{\mathit{P}}$ | MRD | $\overline{\mathit{E}{\mathit{T}}_{\mathit{a}}}/\overline{\mathit{P}}$ | MRD | $\overline{\mathit{E}{\mathit{T}}_{\mathit{a}}}/\overline{\mathit{P}}$ | MRD | $\overline{\mathit{E}{\mathit{T}}_{\mathit{a}}}/\overline{\mathit{P}}$ | |

θ_{r} | 0.411 | −0.372 | 0.573 | −0.369 | 0.734 | −0.356 | 0.920 | −0.018 |

θ_{s} | 0.497 | 0.481 | 0.398 | 0.422 | 0.219 ** | 0.351 | 0.026 | −0.095 |

α | −0.012 | −0.176 * | 0.026 | −0.307 | 0.038 | −0.443 | 0.061 | −0.493 |

n | −0.568 | −0.270 | −0.578 | −0.351 | −0.548 | −0.363 | −0.294 | −0.156 * |

K_{s} | −0.357 | −0.406 | −0.263 | −0.301 | −0.109 | −0.172 * | 0.008 | 0.286 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Z.; Wang, T.; Zhang, Y. Interplays between State and Flux Hydrological Variables across Vadose Zones: A Numerical Investigation. *Water* **2019**, *11*, 1295.
https://doi.org/10.3390/w11061295

**AMA Style**

Wang Z, Wang T, Zhang Y. Interplays between State and Flux Hydrological Variables across Vadose Zones: A Numerical Investigation. *Water*. 2019; 11(6):1295.
https://doi.org/10.3390/w11061295

**Chicago/Turabian Style**

Wang, Zhaoxin, Tiejun Wang, and Yonggen Zhang. 2019. "Interplays between State and Flux Hydrological Variables across Vadose Zones: A Numerical Investigation" *Water* 11, no. 6: 1295.
https://doi.org/10.3390/w11061295