# Characterisation of the Hydrogeological Properties of the Ntane Sandstone Aquifer Using Co-Seismic and Post-Seismic Groundwater Level Responses to the Mw 6.5 Moiyabana Earthquake, Central Botswana

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}tidal constituent to estimate the temporal variations of the storativity, transmissivity, and permeability of the Ntane sandstone aquifer (the main aquifer system) prior to and after the earthquake event. The storativity and permeability computed for borehole MH2 showed a decrease in magnitude of 3.17432 × 10

^{−4}and 1.85 × 10

^{−13}m

^{2}respectively, indicating that strong ground shaking at borehole MH2 might have consolidated the aquifer material, thus resulting in decreased aquifer permeability. The aquifer coefficient of storativity decreased by 2.85 × 10

^{−4}while permeability was enhanced by 0.047 × 10

^{−13}m

^{2}at borehole Z12836. Enhanced permeability might have resulted from increased/enhanced fracturing of the aquifer, fracture clearing and dynamic shaking.

## 1. Introduction

#### Problem Statement

_{2}tidal constituent to estimate the aquifer’s storativity, transmissivity and permeability prior and following the earthquake.

## 2. Materials and Methods

#### 2.1. Study Location

^{2}wellfield situated to the southwest of Kudumatse village in-southeast Botswana. The borehole Z12836 (23.510° S and 26.699° E) located in this wellfield is approximately 167 km from the Moiyabana earthquake’s epicentre (Figure 1).

^{2}within the Central “Kalahari Game Reserve (CKGR)”. The wellfield belongs to the Ghaghoo Diamond Mine, formerly known as Gope Mine. The remotely located Ghaghoo Diamond mine is approximately 45 km west of the eastern border of the CKGR. It is within this wellfield that borehole MH2 (22.626° S and 24.762° E) is located approximately 40 km from the Moiyabana earthquake’s epicentre (Figure 1).

#### 2.1.1. Geology

#### 2.1.2. Hydrogeology and Aquifer Parameters

^{2}/day and S = 0.0001, and for borehole Z12836, K = 0.488 m/day T = 33.696 m

^{2}/day and S = 0.0002 [41,42,43]. These storativity values are indicative of confined aquifer conditions. The Kudumatse area is considered a high transmissivity zone, with T ranging from 500–3300 m

^{2}/day, and this is attributable to heavy fracturing and the low cementation of the sandstone [42].

#### 2.1.3. Borehole Construction

#### 2.2. Methodology

#### 2.2.1. Sampling

#### 2.2.2. Borehole Hydrograph Analysis

#### 2.2.3. Tidal Analysis

_{2}tidal component of the groundwater level for borehole MH2 is likely to have minimal impact on the M

_{2}component of the groundwater level [9,21]. Moreover, since borehole MH2 and Z12836 are far from the coastline, ocean tide corrections were also deemed not very necessary [21].

_{1,}M

_{2}, N

_{2,}K

_{1}and S

_{2}waves, Table 2 [2,6,53]. Tidal waves greater in amplitude of the five major tidal components, i.e., M

_{2}and O

_{1}, are usually preferred to evaluate the groundwater-level response to earth tides because they have a greater signal-to-noise ratio [6,9,49]. However, in this study, only the lunar semi-diurnal (M

_{2}) wave constituent was used to estimate the aquifer hydraulic properties. This is because it is more stable, exhibits a large amplitude and the smallest root-mean-square error, and is less affected by external influences from solar radiation/thermal effect and diurnal barometric pressure effects than the O

_{1}tidal component [2,9,10,11,21,46,49].

_{2}and the diurnal S

_{2}tidal components and to avoid spectral leakage [10,54,55]. Thus, the response of groundwater level to the volumetric strain associated with the M

_{2}tidal constituent was obtained every 31 days using the Baytap-G software. The pre-earthquake period was taken as 31 days before the earthquake and 31 days after the earthquake for the post-earthquake period, allowing for a comparison to be made, and also to be determine the impact of the Moiyabana earthquake on the hydraulic properties of the aquifer. In order to estimate earthquake-induced changes in the hydraulic properties of the Ntane sandstone aquifer system, the groundwater level response to earth tides and atmospheric pressure before and after the Moiyabana earthquake’s occurrence were used.

#### 2.3. Mechanisms for Classifying/Determining the Hydrogeological Properties

#### 2.3.1. Estimating the Aquifer Storage Coefficient

^{−9}, hence:

_{2}wave, $\mathrm{f}\left(\mathsf{\theta}\right)=0.5{\mathrm{c}\mathrm{o}\mathrm{s}}^{2}\left(\mathsf{\theta}\right)$ and $\mathrm{b}\approx 0.908$ [53,57].

#### 2.3.2. Estimation of the Aquifer Transmissivity Using Phase Shift

#### The Horizontal Flow Model

_{2}tidal constituent. The Hsieh solution can be approximated for a realistic well geometry and aquifer properties as

_{0}is the discharge of the aquifer at the well, $\mathrm{K}\mathrm{e}\mathrm{r}\left(\mathsf{\alpha}\mathsf{\omega}\right)$ and $\mathrm{K}\mathrm{e}\mathrm{i}\left(\mathsf{\alpha}\mathsf{\omega}\right)$ are the Kelvin functions of order zero and ${\mathrm{K}\mathrm{e}\mathrm{r}}_{1}\left(\alpha \omega \right)$ and ${\mathrm{K}\mathrm{e}\mathrm{i}}_{1}\left(\mathsf{\alpha}\mathsf{\omega}\right)$ are the Kelvin functions of order one. $\mathsf{\omega}$ is the tidal fluctuation frequency of the tidal wave constituent, taking the value of $\mathsf{\tau}$ to be 0.5175/d for the M

_{2}wave. ${\mathrm{r}}_{\mathrm{w}}$ is the inner radius of the well casing. T and S are the transmissivity and storativity of the aquifer, respectively.

_{2}tidal constituent is 12.421 h (44,715.6 secs); the transmissivity was calculated using the observed phase shift taking the well-aquifer parameter $\mathrm{S}{\mathrm{r}}_{\mathrm{w}}^{2}/{\mathrm{r}}_{\mathrm{c}}^{2}=1\times {10}^{-4}$ or negative phase shift as illustrated in [9].

#### Vertical Flow Model

#### 2.3.3. Estimation of the Aquifer’s Permeability

_{c}and r

_{w}and the approximated value of S [8]. Then the estimated transmissivity is used to estimate the permeability of the aquifer. The relationship between $\mathrm{T}$ and permeability, $\mathrm{k}$ can be obtained using Equation (16) as follows:

^{−3}Pa·s, 10

^{3}kg/m

^{3}and 9.81 m/s

^{2}, respectively [61].

## 3. Results

#### 3.1. Borehole Hydrograph Analysis

^{−3}J/m

^{3}causes sustained groundwater level changes in wells. Seismic energy density is the maximum seismic energy required in a unit volume of rock or sediment to do work [4]. Thus, the seismic energy density $\mathrm{e}$ (J/m

^{3}) is calculated according to the empirical relationship between earthquake magnitude $\left(\mathrm{M}\mathrm{w}\right)$ and earthquake epicentral distance $\mathrm{r}$ (km) [4] as follows:

^{3}and 0.0299 J/m

^{3}respectively (Table 3), which are within the range of seismic energy density capable of triggering sustained groundwater level changes induced by earthquakes; these values are also consistent with previous studies using global datasets [4,63]. Similarly, most sustained groundwater level changes are bounded by $\mathrm{e}\times $ 10

^{−3}J/m

^{3}[4].

#### 3.2. Spectra Analysis

_{2}tidal component is greater than the M

_{2}tidal component (Figure 5). Therefore, because of this tidal component’s low signal-to-noise ratio, the interpretation of the tidal response of the M

_{2}tidal component at borehole MH2 requires more caution [55].

_{1}, K

_{1}, N

_{2}, M

_{2}and S

_{2}tidal constituents and indicate excellent tidal responses (Figure 6). The spectrum of the groundwater level indicates that fluctuations in the groundwater level in the boreholes occur at the same frequency as fluctuations in the pressure head in the aquifer induced by the Earth tides [49]. The N

_{2}, component is small, and shows at a slightly lower frequency than the M

_{2}component (Figure 6). Groundwater level data associated with inland wells show the presence of the lunar harmonics O

_{1}and M

_{2}representing the effect of Earth tides, as the influence of ocean tides should be negligible and lunar harmonics do not have periodic oscillations in atmospheric pressure [57]. At a frequency of 1.9324 cpd corresponding to the M

_{2}tidal component, only Earth tide influences are present [48].

_{1}) constituent are shown by the barometric pressure spectra, which occur at the same harmonic frequency as the K

_{1}observed in the barometric spectra in Figure 7 [9,10,48]. The atmospheric harmonic S

_{1}can therefore interfere with the water level tidal reaction at this frequency [58]. Therefore, to estimate the hydrogeological properties, this study uses the stable M

_{2}constituent, which is present in the groundwater level spectrum of the two boreholes (MH2 and Z12836), as it is less affected by barometric pressure loading contamination [55].

#### 3.3. Tidal Analysis

_{2}tidal constituent’s volumetric strain. Hsieh et al. [9] indicated that the earthquake-induced changes in transmissivity and storage coefficient can cause phase shift and amplitude variations of the tidal constituents.

_{2}was very small, which might be an indication that the storativity of the aquifer did not change significantly except after the Moiyabana earthquake.

_{2}tidal constituent in boreholes MH2 were both significantly changed by the Moiyabana earthquake (Figure 8b). The phase shift change from 10.592° to −3.434° after the earthquake took place, as observed in borehole MH2 (Figure 8b), may be indicating a decrease in the permeability of the aquifer.

_{2}constituent increased, and that the post-earthquake phase shift changed from negative to positive and an increased in permeability was observed in the aquifer. In [8], characteristic M

_{2}wave co-seismic and post-seismic phase shifts observed in the wells in California suggested an enhancement of the aquifer’s permeability as a potential cause for the phase shifts. Therefore, permeability enhancement following an earthquake event may be related to propagation of seismic waves.

^{−4}after the earthquake. The transmissivity and permeability of borehole MH2 showed a decrease with a magnitude of 0.5609 × 10

^{−5}m/s and 0.185 m

^{2}/day, respectively. The storativity decreased with a magnitude of 2.85 × 10

^{−4}while the permeability was enhanced by 0.047 m

^{2}after the Moiyabana earthquake occurrence for borehole Z12836.

## 4. Discussion

^{−4}and 2 × 10

^{−4}for boreholes MH2 and Z12836, respectively, using the Cooper-Jacob analysis method. These values are within reasonable ranges of storativity for confined aquifers i.e., 0.5 × 10

^{−5}to 5 × 10

^{−3}[61,66]. Obtaining specific storage estimates within the same order of magnitude as the tidal analysis approach is satisfactory and the findings are consistent with the pumping test results.

^{2}/day and 8.44 m

^{2}/ day, respectively. This indicates that the transmissivity estimate from the tidal analysis method is within an acceptable range and is consistent with the constant rate test result, despite minor deviations. Although the scale of the region sampled by the tidal response is small, the pumping test and the tidal response yield a similar order of magnitude of permeability values. This might be an indication that within the volume investigated by tides, the wells are efficiently interconnected with the fracture network [10].

^{−4}m

^{2}/s and 5.899 × 10

^{−13}m

^{2}, respectively. The distinction in the obtained estimates of the transmissivity and permeability using the tidal analysis and pumping test could be that pumping test results are affected by the average properties inside the effective radius of the borehole. Therefore, presence of secondary fissures could result in higher hydraulic conductivity values estimates from pump test data analyses [67]. However, for a well-aquifer system that responds to tidal strain, the wells may not efficiently interconnect with the fracture network, which leads to lower transmissivity estimates. Thus, the estimated tidal analysis transmissivity values of borehole Z12836 are slightly lower than the pumping test values by an order of a magnitude.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**(

**a**) Lithological log and description of borehole MH2 located in the Gope wellfield, indicating the position of the Ntane sandstone aquifer, the lower confining mudstones and the volcanic cap [44]; (

**b**) Lithological log and descriptions of borehole Z12836 in the CIC Energy wellfield indicating the position of the Ntane sandstone aquifer, the lower confining mudstones and the volcanic cap [43].

**Figure 3.**Borehole MH2 hydrograph showing the groundwater level variation during the period of monitoring.

**Figure 4.**Borehole Z12826 hydrograph showing the groundwater level variation during the period of monitoring.

**Figure 5.**Amplitude spectra of harmonic frequencies, cpd (cycles per day) obtained from Fourier transformation of the groundwater level at borehole MH2 (Major tidal components are labelled on the periodogram).

**Figure 6.**Amplitude spectra of harmonic frequencies, cpd (cycles per day) obtained from Fourier transformation of the groundwater level at borehole Z12836 (Major tidal components are labelled on the periodogram).

**Figure 7.**Amplitude spectra of harmonic frequencies, cpd (cycles per day) obtained from Fourier transformation of the barometric pressure at borehole BH7435 (Major tidal components are labelled on the periodogram).

**Figure 8.**(

**a**) Tidal amplitude variation, and (

**b**) phase variation measured every 31 days at borehole MH2. Error bars indicate the root-mean-square error (RMSE) of the tidal analysis.

**Figure 9.**(

**a**) Tidal amplitude variation, and (

**b**) phase variation measured every 31 days at borehole Z12836. Error bars indicate the root-mean-square error (RMSE) of the tidal analysis.

Borehole ID | Borehole Depth (mbgl) | Depth to Base (mbgl) | Water Strike (mbgl) | Rest Water Level (mbgl) | |||
---|---|---|---|---|---|---|---|

Kalahari Beds | Stormberg Basalt | Ntane Sandstone Fm | Mosolotsane Fm | ||||

MH2 | 501 | 71.8 | 394 | 425 | >501 | 211; 388 | 99.362 |

Z12836 | 187 | 26 | 104 | 173 | >187 | 58; 104; 112 | 34.19 |

**Table 2.**Major harmonic components of the tidal potential (adapted with permission from Munk& MacDonald [53], 1960, Munk & MacDonald).

Tidal Component | Description | Period (Day) | Frequency, cpd (Cycles per Day) |
---|---|---|---|

O_{1} | Principal lunar | 1.0758 | 0.9295 |

K_{1} | Lunar–solar | 1.3721 | 1.0029 |

N_{2} | Lunar elliptic | 0.5275 | 1.8957 |

M_{2} | Principal lunar | 0.5175 | 1.9324 |

S_{2} | Principal solar | 0.5000 | 2.0000 |

Borehole ID | Epicentral Distance (km) | Seismic Energy Density (J/m^{3}) |
---|---|---|

MH2 | 40 | 2.277 |

Z12836 | 167 | 0.0299 |

**Table 4.**Summary of the M2 wave tidal and well-aquifer system hydraulic parameters for borehole. MH2 and Z12836 before and after the earthquake occurrence.

BH ID | Date | Tidal Amplitude (mm) | Phase Shift (°) | Specific Storage Ss (10^{−6} m^{−1}) | Storativity (Dimensionless) S (10^{−4}) | Transmissivity T (10^{−5} m^{2}/s) | Permeability k (10^{−13} m^{2}) |
---|---|---|---|---|---|---|---|

MH2 | Mar 17 | 1.592 | 10.592 | 20.172 | 6.25332 | 9.7766 | 3.215 |

Apr 17 | 3.209 | −3.437 | 9.933 | 3.079 | 9.2157 | 3.03 | |

Δ | 1.617 | −14.029 | −10.521 | −3.17432 | −0.5609 | −0.185 | |

Z12836 | Mar 17 | 2.46 | −2.933 | 12.883 | 8.89 | 8.0638 | 1.191 |

Apr 17 | 3.514 | 12.363 | 8.752 | 6.04 | 8.3799 | 1.238 | |

Δ | 1.054 | 15.259 | −4.131 | −2.85 | 0.3161 | 0.047 |

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## Share and Cite

**MDPI and ACS Style**

Marema, T.M.; Molwalefhe, L.; Shemang, E.M.
Characterisation of the Hydrogeological Properties of the Ntane Sandstone Aquifer Using Co-Seismic and Post-Seismic Groundwater Level Responses to the Mw 6.5 Moiyabana Earthquake, Central Botswana. *Water* **2023**, *15*, 1947.
https://doi.org/10.3390/w15101947

**AMA Style**

Marema TM, Molwalefhe L, Shemang EM.
Characterisation of the Hydrogeological Properties of the Ntane Sandstone Aquifer Using Co-Seismic and Post-Seismic Groundwater Level Responses to the Mw 6.5 Moiyabana Earthquake, Central Botswana. *Water*. 2023; 15(10):1947.
https://doi.org/10.3390/w15101947

**Chicago/Turabian Style**

Marema, Tshepang Mmamorena, Loago Molwalefhe, and Elisha M. Shemang.
2023. "Characterisation of the Hydrogeological Properties of the Ntane Sandstone Aquifer Using Co-Seismic and Post-Seismic Groundwater Level Responses to the Mw 6.5 Moiyabana Earthquake, Central Botswana" *Water* 15, no. 10: 1947.
https://doi.org/10.3390/w15101947