# Analytical Solutions of Vertical Airflow in an Unconfined Aquifer with Rising or Falling Water Table

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Conceptual Model and Control Equations

#### 2.1. Conceptual Model and Assumptions

_{0}. The high-permeability aquifer III, the medium-permeability aquifer II and the lower part of high-permeability aquifer I are saturated with water. The uppermost aquifer I is capped by a low-permeability fine-sand layer. The vadose zone exists at the upper part of aquifer I and the fine-sand cap layer.

#### 2.2. Nonlinear Control Equation of Airflow

_{a}is the volumetric flux of air (L

^{−1}T

^{−1}), k is the intrinsic permeability of the porous medium (L

^{2}), k

_{ra}is relative permeability of porous medium to air (L

^{2}), μ

_{a}is dynamics viscosity of air (ML

^{−1}T

^{−1}), p

_{a}is the pressure of air (ML

^{−1}T

^{−2}), ρ

_{a}is the density of air (ML

^{−3}). Compared to the force caused by air pressure gradients, the force due to gravity is much smaller so it can be neglected under most conditions [36]. Then Equation (1) is linearized to

_{ma}is the mass flux of air (ML

^{−}

^{2}T

^{−1}), which is defined as

_{a}is the air gauge pressure relative to atmospheric pressure in the vadose zone expressed as a height of water column of a reference water density, which is defined as

_{w}is the density of liquid water (ML

^{−3}), ρ

_{w}gh

_{a}is relative air pressure in the vadose zone under the fine sand layer, D is the thickness of fine sand layer (L). When the relative pressure of air in the vadose zone is positive, the air in the sand column will be driven upward and the air mass of the vadose zone will be reduced. Conversely, if the vadose zone has a negative pressure, it will attract the air in the atmosphere to flow into the sand column, so that the air mass of the vadose zone will be increased. Assume that the air in the fine sand layer is evenly distributed, and the pore space is not filled with capillary water in the vadose zone. The total air mass in the vadose zone can be expressed as

^{2}), ϕ is the effective porosity of the coarse sand approximated as a constant.

_{a}p/p

_{a}, p

_{a}is the local atmospheric pressure, and ρ

_{a}is the air density under the local atmospheric pressure. The continuity equation of air flow is expressed as

_{a}(t). K

_{a}reflects the breathability of the fine sand layer.

#### 2.3. Feedback Equation for Groundwater Flow

^{3}T

^{−1}), K

_{1}, K

_{2}is the saturated hydraulic conductivity of unconfined aquifer I and II respectively (L

^{−1}T

^{−1}), h is the saturated water level of unconfined aquifer at any time. The pressure in the vadose zone is not equal to the atmospheric pressure; the head is not the height of the water table but (h + h

_{a}); h

_{u}is the head of the upper plate of the aquifer II; z

_{0}is the constant head. The rate of recharge or discharge per unit cross-sectional area is equal to the injecting or falling rate of surface of saturation multiplied by effective porosity of the coarse sand [2]:

## 3. Experimental Studies

_{s}) of the phreatic aquifers in the sand column is the same as the hydraulic conductivity of aquifer I (K

_{1}= K

_{s}) in the conceptual model (Figure 1). The aquifer III in the conceptual model represents the constant-head reservoir in the experiment, which is a boundary condition with a constant water head. Below we present experimental setup and results that show air pressure in the vadose zone induced by water table fluctuation.

#### 3.1. Experimental Setup

_{0}, and the height of the initial water level of the sand column is h

_{0}. The relative height between the two is reversed in the water injection experiment (z

_{0}> h

_{0}) and the drainage experiment (z

_{0}< h

_{0}). In order to capture the rapid changes of water level in the glass tube, a pressure sensor was put into the glass tube to record the water level automatically.

#### 3.2. Type Curves of Change in the Air Pressure Beneath the Low-Permeability Cap

_{2}O of water column during the experiment, the result of air pressure observation in the vadose zone is reliable.

#### 3.2.1. Results of the Drainage Experiments

_{0}− z

_{0}) (Figure 4), it is observed that greater value of (h

_{0}− z

_{0}) corresponds to greater NP and slower rate of water level drop, when thickness of fine sand layer is the same. This is because a certain volume of air was expanded in a short time in the vadose zone, but the atmospheric air could not enter in time via the fine sand layer. The greater amount of expanded air was, the greater the peak NP appeared.

#### 3.2.2. Results of the Injection Experiments

_{0}− z

_{0}was 30 cm, the maximum pressure was 10.4 cm of water column at 20 s. When the thickness of the fine sand was 5 cm and h

_{0}− z

_{0}was 30 cm, the peak air pressure reached about 17.6 cm of water column at ~ 30−35 s. This indicates that the thicker the low-permeability cap layer was, the higher the peak pressure in the vadose zone reached, and the later the peak value appeared. Corresponding to changes of the air pressure, the water level of the sand column increased gradually from the initial period, increased rapidly in the middle period, and reached stability in the late period.

_{0}− z

_{0}) but with the same thickness of the fine sand layer (Figure 5), the greater the value of (h

_{0}− z

_{0}) was, the greater the peak positive pressure was. Similar to the drainage experiments, this is because a certain volume of air was condensed in a short time in the vadose zone, but it could not escape to the atmosphere in time via the fine sand layer. The greater the amount of condensed air was, the greater the peak PP appeared.

#### 3.3. Limitation Remarks of the Experiments

## 4. Analytical Solutions

#### 4.1. Early Stage Analytical Solution

_{0}. The set of dynamic equations in this case can be simplified as

_{a}/h

_{a}

_{0}< 5%, which is approximated as zero. Then Equation (7) can be rewritten as

_{s}, C

_{a}, K

_{a}are constants. C

_{s}can be directly calculated based on the basic parameters of the experimental materials. From Equation (15) it can be seen that the early curve develops towards the maximum air pressure C

_{s}/C

_{a}, and the maximum pressure increases non-linearly with increasing thickness of fine sand layer thickness D.

#### 4.2. Middle Stage Analytical Solution

#### 4.3. Late Stage Analytical Solution

_{a}, the airflow equation in this case can be simplified as

_{c}is an integral constant, which has the same unit as the pressure. Since the air pressure also decays to a small value in the late-stage experiments, h

_{a}/(L − z

_{0}) can be approximated as zero, so the logarithmic approximation of the pressure is linear with time. From the ln-h

_{a}-t curve, K

_{a}/D can be derived.

#### 4.4. Estimating Parameters with Semi-Analytical Solutions

_{1}= K

_{2}= K

_{s}). The dynamic equations and the dynamic analysis features in early and late stages of the experiments show that the pressure in the vadose zone is controlled by the two parameters of K

_{s}/ϕ and K

_{a}/D. K

_{s}/ϕ is included in the C

_{s}parameter of Equation (16), which reflects the influence of the high-permeability layer (coarse sand) on the drainage and injection process. K

_{a}/D is included in the parameter C

_{a}of Equation (16), which reflects the influence of the low-permeability layer (fine sand) on the drainage or injection process. K

_{s}/ϕ = 0.299 cm/s can be calculated according to the properties of the experimental materials. C

_{s}= 9.89 cm/s can be calculated from Equation (17). In contrast, K

_{a}/D is an uncertain parameter.

_{a}/D was obtained from the results of the drainage experiments at 2 cm, 5 cm, and 7.5 cm thickness of the fine sand layer. The values of K

_{a}/D are 2.3 cm/s, 1.1 cm/s and 0.6 cm/s respectively (Figure 6a), which are generally inversely proportional to the thickness of fine sand layer. The corresponding K

_{a}is 4.5~5.5 cm

^{2}/s, a parameter reflecting the air permeability of the fine sand layer in the early stage of experiment. C

_{a}can be derived from Equation (16) as 0.28~0.36 s

^{−1}. The maximum pressure is mainly determined by C

_{s}/C

_{a}, and the maximum value tends to increase nonlinearly with the increase of the fine sand layer thickness D. When D→∞, the peak pressure will reach the limit value (h

_{0}− z

_{0}), which is the difference in the original water head.

_{a}/D are 12.9 cm/s, 9.5 cm/s, and 8.6 m/s respectively (Figure 6b), approximately inversely proportional to the thickness of the fine sand. The corresponding K

_{a}is 26~65 cm

^{2}/s, a parameter that reflects breathability of the fine sand layer in the late stage of drainage.

_{a}calculated from the early drainage data is one order of magnitude lower than that derived from the late drainage data, which indicates that the relative air permeability value of fine sand layer is not constant during the experiment. The reason for this is that in the fine sand there was residual moisture which weakened the breathability of the fine sand layer. In the late stage, the long-term drainage decreased the moisture content of the fine sand and increased the breathability.

#### 4.5. The Generic Behaviors of Airflow Revealed by the Analytical Solutions

_{s}/C

_{a}, and the maximum pressure non-linearly increases with the fine sand layer thickness D. In the middle stage, the maximum air pressure in the vadose zone is positively correlated with the rate of the water level change in the unconfined aquifer. In the late stage, the logarithmic approximation of the pressure is linear with time, and K

_{a}/D can be derived from the ln h

_{a}versus t curve.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Schematic diagram of setup for the injection experiment with constant head sources. Modified from Dong et al. [37].

**Figure 3.**Schematic diagram of setup for the drainage experiment with constant head sources. Modified from Dong et al. [37].

**Figure 4.**Changes of negative pressure (NP), cumulative outflow, and height of saturation surface with time in drainage experiments. (

**a**) Changes of relative air pressure with time in vadose zone (h

_{0}− z

_{0}= 40 cm). (

**b**) Changes of cumulative outflow with time (h

_{0}− z

_{0}= 40 cm). (

**c**) Changes of calculated height of saturation surface with time (h

_{0}− z

_{0}= 40 cm). (

**d**) Changes of relative air pressure with time in vadose zone (h

_{0}− z

_{0}= 30 cm). (

**e**) Changes of cumulative outflow with time (h

_{0}− z

_{0}= 30 cm). (

**f**) Changes of calculated height of saturation surface with time (h

_{0}− z

_{0}= 30 cm). In (

**c**,

**f**) the vertical lines subdivide the early, middle, and late stages.

**Figure 5.**Changes of the positive pressure (PP) with time in water injection experiments. (

**a**) z

_{0}− h

_{0}= 40 cm, (

**b**) z

_{0}− h

_{0}= 30 cm.

**Figure 6.**Comparison of observed air pressure and the approximate analytical solutions in the drainage experiments. (

**a**) curves derived from Equation (16) for the early period of the experiments. (

**b**) curves derived from Equation (25) for the late period of the experiments.

Experimental Groups | Initial Water Head Difference (∆H, cm) | Fine Sand Thickness (D, cm) | Observation Duration (hour) | |
---|---|---|---|---|

Drainage Experiments | 1 | 30 | 0.0 | 8.5 |

2 | 30 | 2.0 | 5 | |

3 | 30 | 5.0 | 7 | |

4 | 30 | 7.5 | 6 | |

5 | 40 | 2.0 | 9 | |

6 | 40 | 5.0 | 5 | |

7 | 40 | 7.5 | 7 | |

Injection Experiments | 1 | 30 | 2.0 | 3 |

2 | 30 | 5.0 | 3 | |

3 | 40 | 2.0 | 3 | |

4 | 40 | 5.0 | 3 |

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**MDPI and ACS Style**

An, R.; Dong, P.; Wang, J.-Z.; Zhang, Y.; Song, X.; Wan, L.; Wang, X.-S.
Analytical Solutions of Vertical Airflow in an Unconfined Aquifer with Rising or Falling Water Table. *Water* **2021**, *13*, 625.
https://doi.org/10.3390/w13050625

**AMA Style**

An R, Dong P, Wang J-Z, Zhang Y, Song X, Wan L, Wang X-S.
Analytical Solutions of Vertical Airflow in an Unconfined Aquifer with Rising or Falling Water Table. *Water*. 2021; 13(5):625.
https://doi.org/10.3390/w13050625

**Chicago/Turabian Style**

An, Ran, Pei Dong, Jun-Zhi Wang, Yifan Zhang, Xianfang Song, Li Wan, and Xu-Sheng Wang.
2021. "Analytical Solutions of Vertical Airflow in an Unconfined Aquifer with Rising or Falling Water Table" *Water* 13, no. 5: 625.
https://doi.org/10.3390/w13050625