# Estimating Unconfined Aquifer Diffusivity Using 1D Phase Spectral Analysis: A Case Study in the Middle Reach of the Hutuo River, North China Plain

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

^{3}to 4.9 × 10

^{4}m

^{2}/d, with a mean of 2.2 × 10

^{4}m

^{2}/d, which was consistent with the results obtained using power spectral analysis, pumping tests, and inverse numerical models. The phase spectral approach proposed in this paper can estimate the aquifer properties on a larger scale. If long time series of hydraulic heads are available, it can estimate hydrogeological parameters accurately and quickly. Considering the similarity of the linearized governing equations, it can also be applied to the river–aquifer system and the confined aquifer system.

## 1. Introduction

## 2. Mathematical Development

#### 2.1. Governing Equation

^{−1}] or [L], and $\mu $ is the specific yield [-]. When the hydraulic head, h, is considerably greater than the variation in hydraulic head, it can be approximately substituted by the saturated aquifer thickness, $\overline{h}$. Thus, Equation (1) can be modified as follows:

^{2}/T] is the unconfined aquifer diffusivity.

#### 2.2. Unit Response Function

_{0}(t) is the upgradient hydraulic head [L], ${\psi}_{\delta}$ is the unit response function of the unconfined aquifer [-], and * denotes the convolution integral.

#### 2.3. Phase Spectrum

_{0}(t), can be expressed as:

## 3. Field Case Study

#### 3.1. Study Area and Monitoring Data

_{0}(t), was considered acceptable.

#### 3.2. Results

^{2}/d, which was the comprehensive response to the long-term groundwater hydraulic head fluctuations.

^{2}/d, respectively, with a mean of 21,831 m

^{2}/d; those of monitoring well HTH03 were 5464, 3642, 29,975, and 49,854 m

^{2}/d, respectively, with a mean of 22,234 m

^{2}/d. In summary, the aquifer diffusivity estimated using the phase spectrum method ranged from 1.9 × 10

^{3}to 4.9 × 10

^{4}m

^{2}/d, with a mean of 2.2 × 10

^{4}m

^{2}/d. Moreover, as mentioned above, the mean aquifer thickness at the study site was approximately 50 m and the specific yield was 0.15–0.20. By using Equation (13), the hydraulic conductivity of the aquifer was estimated to be about 80 m/d.

#### 3.3. Validation

_{R}] is the power spectrum of an upgradient hydraulic head. Figure 4c,d presents the aquifer diffusivity power spectra calculated using Equation (14). Within a short period, the aquifer diffusivities obtained from the phase spectrum and power spectrum fluctuated in a similar pattern, with strong variation and uncertainty. Conversely, as the period increased, the power spectrum gradually stabilized at around 10

^{4}m

^{2}/d, with far less variation, especially for the significant periods of 924, 462, 308, and 231 days. For monitoring well HTH02, the corresponding aquifer diffusivities determined using the power spectrum method were 1291, 3746, 6183, and 30,574 m

^{2}/d, respectively, with a mean of 10,448 m

^{2}/d; those of monitoring well HTH03 were 3759, 9943, 12,601, and 82,215 m

^{2}/d, respectively, with a mean of 27,130 m

^{2}/d. These results are more or less in good agreement with those of the phase spectrum determined using Equation (12).

^{−3}to 3.0 × 10

^{−3}m/s) and specific yield from 0.18 to 0.21. Overall, previous studies reported hydraulic conductivities from 100 to 200 m/d and specific yields from 0.15 to 0.20. These results are basically consistent with those obtained using the phase spectrum method in the present study.

## 4. Discussion

## 5. Conclusions

^{3}to 4.9 × 10

^{4}m

^{2}/d, with a mean of 2.2 × 10

^{4}m

^{2}/d, which was consistent with the results obtained using power spectral analysis, transient pumping tests, and inverse numerical models.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Conceptual model of an unconfined aquifer system with assumptions of aquifer isotropy, i.e., no vertical flow, constant saturated aquifer thickness, and a unit transversal length. As the input signal of the system, the upgradient aquifer thickness or hydraulic head, h

_{0}(t), can be processed by the unconfined aquifer system to output the signal of the downgradient, h(x, t), at different locations.

**Figure 2.**(

**a**) Geographical location of the study site, highlighted by the red dash-line box. (

**b**) Google image of the study site, in which a hydrogeological profile (PM01) of three groundwater monitoring wells (HTH01, HTH02, and HTH03) was located perpendicular to the river.

**Figure 3.**(

**a**) Time series of the hydraulic head in monitoring wells HTH01–HTH03 along the PM01 profile. (

**b**) The linear trend was removed from the hydraulic head time series.

**Figure 4.**(

**a**,

**b**) Aquifer diffusivity phase spectra; (

**c**,

**d**) power spectra of monitoring wells HTH02 and HTH03. Dashed lines highlight significant periods.

**Figure 5.**Amplitude spectra of hydraulic head of monitoring wells HTH01–HTH03. Blue lines indicate significant periods. (

**a**) Amplitude spectra of hydraulic head of monitoring wells HTH01; (

**b**) Amplitude spectra of hydraulic head of monitoring wells HTH02; (

**c**) Amplitude spectra of hydraulic head of monitoring wells HTH03.

Well Names | Distance (m) | Well Depth (m) | Topographic Elevation (m) | Water Table Depth (m) | Aquifer Bottom Elevation (m) |
---|---|---|---|---|---|

HTH01 | 0 | 55 | 55 | 17.5–21.5 | −15 |

HTH02 | 380 | 56 | 54 | 16.0–21.5 | |

HTH03 | 690 | 55 | 54.5 | 16.0–21.5 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, B.; Wang, J.; Zhang, R.; Li, Y.; Kong, X.; Liu, Y.
Estimating Unconfined Aquifer Diffusivity Using 1D Phase Spectral Analysis: A Case Study in the Middle Reach of the Hutuo River, North China Plain. *Water* **2023**, *15*, 3012.
https://doi.org/10.3390/w15163012

**AMA Style**

Zhang B, Wang J, Zhang R, Li Y, Kong X, Liu Y.
Estimating Unconfined Aquifer Diffusivity Using 1D Phase Spectral Analysis: A Case Study in the Middle Reach of the Hutuo River, North China Plain. *Water*. 2023; 15(16):3012.
https://doi.org/10.3390/w15163012

**Chicago/Turabian Style**

Zhang, Baoyun, Junzhi Wang, Ruolin Zhang, Yasong Li, Xiangke Kong, and Yaci Liu.
2023. "Estimating Unconfined Aquifer Diffusivity Using 1D Phase Spectral Analysis: A Case Study in the Middle Reach of the Hutuo River, North China Plain" *Water* 15, no. 16: 3012.
https://doi.org/10.3390/w15163012