# Time-Lapse ERT, Moment Analysis, and Numerical Modeling for Estimating the Hydraulic Conductivity of Unsaturated Rock

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{fs}[4].

_{fs}[5,6,7,8], while several criticisms increase the difficulties in performing such tests on rocks [9,10,11] because of the gap effect between the fragile probes and the rigid medium that could affect the uncertainty of the measurements.

_{s}, of rock in unsaturated conditions. Quantitative information about the water mass and the movement of the center mass of the water plume were extracted from ERT-derived water content observations by performing the moment analysis technique. At the same time, water content distribution in the unsaturated zone during the infiltration test has been predicted through several simulation runs of the falling head test by constraining the water head and varying the K

_{s}value. The moment analysis was performed for each simulation and compared with the ERT-derived calculations, in order to provide an accurate estimation of K

_{s}. The goals of this paper are as follows: (1) to evaluate the ability of the geophysical tool to gain quantitative information of the dynamics occurring in the unsaturated rock at field scale, otherwise unpredictable with traditional observations; (2) to verify the reliability of stochastic techniques, such as moment analysis, in the monitoring of unsaturated flow; and (3) to reduce the uncertainty of such predictions by integrating geophysical datasets into stochastic and deterministic approaches.

## 2. Materials and Methods

_{s}parameter in the unsaturated zone. On one hand, the time-lapse ERT datasets collected during the infiltration test were used to image the resistivity variations of the subsurface that, in turn, were converted into water content through Archie’s parameters calibrated in the laboratory. On the other hand, the unsaturated flow of the infiltration dynamics was simulated to predict the water content distribution in the rocky subsurface.

_{s}. The conceptual scheme of the proposed approach is described in the flowchart shown in Figure 1.

#### 2.1. Properties of the Investigated Rock

#### 2.2. The Hydrogeophysical Test

_{0}) and ρ(t

_{1}) are the resistivities of the rock (Ω∙m) at times t

_{0}and t

_{1}, respectively, and S

_{w}(t

_{0}) and S

_{w}(t

_{1}) are the saturation degrees at times t

_{0}and t

_{1}, respectively. This approach allows the simplification of Archie’s equation, as the “n” saturation index is the only unknown parameter to be calibrated. Archie’s calibration was performed in a laboratory on calcarenite core samples. In the laboratory, starting from the saturation condition, the resistivity–water content curve was recorded every five minutes during the drying process.

#### 2.3. Forward Hydrological Modeling: Richards’ Equation

^{3}·m

^{−3}); θ

_{r}is the residual water content (m

^{3}·m

^{−3}); θ

_{s}is the saturated water content (m

^{3}·m

^{−3}); h is water potential (kPa); α is a scale parameter inversely proportional to mean pore diameter (cm

^{−1}); and n and m are the shape parameters of soil water characteristic, m = 1 − 1/n, 0 < m < 1. According to [35], van Genuchten’s parameters were set as reported in Table 2.

_{hh}/K

_{zz}(dimensionless), the specific storage S

_{s}(m

^{−1}), the effective porosity φ (m

^{3}·m

^{−3}), and the initial moisture θ content (m

^{3}·m

^{−3}).

_{hh}/K

_{zz}and S

_{s}were set equal to 1 and 1.6 × 10

^{−4}m

^{−1}, respectively, although small variations do not cause significant changes in the model output. Moreover, effective porosity φ was set equal to 0.45 on the basis of previous tests performed on core samples and the initial content θ to 0.22 m

^{3}·m

^{−3}, as estimated from the ERT-derived value from Archie’s conversion.

_{s}. Several simulations scenarios were run with K

_{s}ranging from 0.1 cm·min

^{−1}to 1 cm·min

^{−1}(Table 2), according to the expected values reported in the literature.

_{1}= 20 min, t

_{4}= 50 min, t

_{8}= 100 min, t

_{12}= 153 min, t

_{13}= 173 min, t

_{14}= 196 min, and t

_{15}= 256 min after the start of the first injection, as reported in Table 1.

#### 2.4. Moment Analysis

_{00}, is the changes in water mass within the domain respect to the background (Equation (5)) and represents the water storage along the reference section, expressed in m

^{3}m

^{−3}.

_{01}normalized by the mass M

_{00}, defines the vertical center of mass of the plume at a given time, z, expressed by Equation (6).

## 3. Results

#### 3.1. ERT-Derived Water Content Outputs

^{3}·m

^{−3}estimated from Archie’s conversion, by denoting an almost homogeneous initial condition of the subsurface, as expected. When the first injection starts, water infiltrates below the ring, deepening over time until the end of the first injection (Figure 3b–d). The water content observed in the background conditions can be attributed to copious precipitation some days before the test, leading to high values of water content in the upper portion of the subsurface soon after the starting of the first injection.

#### 3.2. Moment Analysis Derived from the ERT Dataset

_{8}. After the second injection, the added infiltrated mass water causes a rise in the mass center, as clearly observed in the shape of the curve, reaching a value of 0.35 m at time t

_{15}.

#### 3.3. Numerical Simulations

_{s}(Figure 6b), scenario E overestimates the K

_{s}(Figure 6d), and scenario C approximates the true distribution of soil moisture (Figure 6c).

_{s}, the moment analysis for each simulation scenario was calculated for all five scenarios.

_{s}value fits well the simulated one in the range 0.25 < K

_{s}< 0.35 cm∙min

^{−1}(Figure 7).

## 4. Discussion and Conclusions

_{s}values (0.25 < K

_{s}< 0.35 cm∙min

^{−1}), which are consistent with those values measured in the laboratory on samples of the same rock. Specifically, the authors of [36] found a K

_{s}value ranging from 0.26 cm∙min

^{−1}and 0.43 cm∙min

^{−1}in constant head conditions, and 0.31 and 0.47 cm∙min

^{−1}in falling head conditions. Moreover, the authors of [32,33] obtained a K

_{s}of about 0.48 and 0.3 cm∙min

^{−1}, respectively, by considering the maximum value of the experimentally measured hydraulic conductivity function.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Workflow diagram describing the integrated hydrogeophysical approach used in the present case study for estimating saturated hydraulic conductivity, K

_{s}.

**Figure 2.**Experimental set up of the infiltrometer test: (

**a**) technical drawing of the experimental set up in a plan view; (

**b**) picture of 3D electrodes’ configuration arrangement.

**Figure 3.**ERT-derived water content at different time points: (

**a**) before the starting of the first injection; (

**b**) 20 min; (

**c**) 50 min; (

**d**) 100 min after the first injection; (

**e**) 25 min; (

**f**) 45 min; (

**g**) 68 min; (

**h**) 128 min after the second injection.

**Figure 6.**Comparison of the ERT-derived water content (

**a**) and the simulated one for three different scenarios: (

**b**) scenario A; (

**c**) scenario C; and (

**d**) scenario E.

Infiltration Test | Time Point (hh:mm) | Hydraulic Head (cm) | ERT Observation |
---|---|---|---|

11:20 | t_{0} | ||

Start first injection | 11:43 | 3.1 | |

12:03 | 2.0 | t_{1} | |

12:33 | 1.0 | t_{4} | |

Stop infiltration measurements | 12:43 | 0.8 | |

13:23 | t_{8} | ||

Start second injection | 14:01 | 3.3 | |

14:16 | 2.5 | t_{12} | |

14:36 | 2.1 | t_{13} | |

14:59 | 1.6 | t_{14} | |

Stop infiltration measurements | 15:09 | 1.3 | |

15:59 | 0.2 | t_{15} |

Parameter | Scenario A | Scenario B | Scenario C | Scenario D | Scenario E |
---|---|---|---|---|---|

Saturated K_{hh} (cm·min^{−1}) | 0.1 | 0.25 | 0.35 | 0.75 | 1 |

K_{hh}/K_{zz} | 1 | ||||

Specific storage, S_{s} (m^{−1}) | 1.6 × 10^{−4} | ||||

Effective porosity, φ | 0.45 | ||||

θ, initial moisture content (m^{3}·m^{−3}) | 0.22 | ||||

θ_{r}, residual moisture content (m^{3}·m^{−3}) | 0.02547 | ||||

α (cm^{−1}) | 0.07721 | ||||

n | 1.7541 |

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**MDPI and ACS Style**

De Carlo, L.; Farzamian, M.; Turturro, A.C.; Caputo, M.C.
Time-Lapse ERT, Moment Analysis, and Numerical Modeling for Estimating the Hydraulic Conductivity of Unsaturated Rock. *Water* **2023**, *15*, 332.
https://doi.org/10.3390/w15020332

**AMA Style**

De Carlo L, Farzamian M, Turturro AC, Caputo MC.
Time-Lapse ERT, Moment Analysis, and Numerical Modeling for Estimating the Hydraulic Conductivity of Unsaturated Rock. *Water*. 2023; 15(2):332.
https://doi.org/10.3390/w15020332

**Chicago/Turabian Style**

De Carlo, Lorenzo, Mohammad Farzamian, Antonietta Celeste Turturro, and Maria Clementina Caputo.
2023. "Time-Lapse ERT, Moment Analysis, and Numerical Modeling for Estimating the Hydraulic Conductivity of Unsaturated Rock" *Water* 15, no. 2: 332.
https://doi.org/10.3390/w15020332