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

^{1}

^{2}

^{3}

^{*}

## 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

## References

- Richards, L.A. Capillary conduction of liquids in porous mediums. Physics
**1931**, 1, 318–333. [Google Scholar] [CrossRef] - Klute, A. Laboratory measurement of hydraulic conductivity of saturated soil. In Methods of Soil Analysis: Part 1 Physical and Mineralogical Properties, Including Statistics of Measurement and Sampling; John Wiley & Sons: Hoboken, NJ, USA, 1965; Volume 9, pp. 210–221. [Google Scholar]
- Hamilton, J.M.; Daniel, D.E.; Olson, R.E. Measurement of hydraulic conductivity of partially saturated soils. In Permeability and Groundwater Contaminant Transport; Zimmie, T.F., Riggs, C.O., Eds.; American Society for Testing and Materials: Baltimore, MD, USA, 1981; pp. 182–196. [Google Scholar]
- Nimmo, J.R.; Schmidt, K.M.; Perkins, K.S.; Stock, J.D. Rapid Measurement of Field-Saturated Hydraulic Conductivity for Areal Characterization. Vadose Zone J.
**2009**, 8, 142–149. [Google Scholar] [CrossRef] - Angulo-Jaramillo, R.; Vandervaere, J.P.; Roulier, S.; Thony, J.L.; Gaudet, J.P.; Vauclin, M. Field measurement of soil surface hydraulic properties by disc and ring infiltrometers: A review and recent developments. Soil Tillage Res.
**2000**, 55, 1–29. [Google Scholar] [CrossRef] - Reynolds, W.D.; Elrick, D.E.; Youngs, E.G.; Amoozegar, A.; Booltink, H.W.G.; Bouma, J. Saturated and field-saturated water flow parameters. In Methods of Soil Analysis. Part 4. Physical Methods; Dane, J.H., Topp, G.C., Eds.; Wiley SSSA: Madison, WI, USA, 2002; pp. 797–878. [Google Scholar]
- Bouwer, H. Rapid field measurement of air entry value and hydraulic conductivity of soil as significant parameters in flow system analysis. Water Resour. Res.
**1966**, 2, 729–738. [Google Scholar] [CrossRef] - Youngs, E.G. Estimating hydraulic conductivity values from ring infiltrometer measurements. J. Soil Sci.
**1987**, 38, 623–632. [Google Scholar] [CrossRef] - Bogena, H.R.; Huisman, J.A.; Oberdosrster, C.A.; Vereecken, H. Evaluation of a Low-Cost Soil Water Content Sensor for Wireless Network Application. J. Hydrol.
**2007**, 344, 32–42. [Google Scholar] [CrossRef] - Kizito, F.; Campbell, C.G.; Cobos, D.R.; Teare, B.L.; Carter, B.; Hopmans, J.W. Frequency, Electrical Conductivity and Temperature Analysis of a Low-Cost Capacitance Soil Moisture Sensor. J. Hydrol.
**2008**, 352, 367–378. [Google Scholar] [CrossRef] - Caputo, M.C.; De Carlo, L. Field measurement of hydraulic conductivity of rocks. In Hydraulic Conductivity—Issues, Determination and Applications; Elango, L., Ed.; Intech Open: London, UK, 2011; pp. 285–306. [Google Scholar] [CrossRef]
- Caputo, M.C.; De Carlo, L.; Masciopinto, C.; Nimmo, J.R. Measurement of field-saturated hydraulic conductivity on fractured rock outcrops near Altamura (Southern Italy) with an adjustable large ring infiltrometer. Environ. Earth Sci.
**2010**, 60, 583–590. [Google Scholar] [CrossRef] - Masciopinto, C.; Liso, S.I.; Caputo, M.C.; De Carlo, L. An Integrated Approach Based on Numerical Modelling and Geophysical Survey to Map Groundwater Salinity in Fractured Coastal Aquifers. Water
**2017**, 9, 875. [Google Scholar] [CrossRef] - Furman, A.; Ferré, T.P.A.; Warrick, A.W. Optimization of ERT Surveys for Monitoring Transient Hydrological Events Using Perturbation Sensitivity and Genetic Algorithms. Vadose Zone J.
**2004**, 3, 1230–1239. [Google Scholar] [CrossRef] - Perri, M.T.; De Vita, R.; Masciale, R.; Portoghese, I.; Chirico, G.B.; Cassiani, G. Time-lapse Mise-à-la-Masse measurements and modelling for tracer test monitoring in a shallow aquifer. J. Hydrol.
**2018**, 561, 461–477. [Google Scholar] [CrossRef] - Daily, W.; Ramirez, A.; LaBrecque, D.; Nitao, J. Electrical resistivity tomography of vadose water movement. Water Resour. Res.
**1992**, 28, 1429–1444. [Google Scholar] [CrossRef] - Garré, S.; Hyndman, D.; Mary, B.; Werban, U. Geophysics conquering new territories: The rise of “agrogeophysics”. Vadose Zone J.
**2021**, 20, e20115. [Google Scholar] [CrossRef] - De Carlo, L.; Battilani, A.; Solimando, D.; Caputo, M.C. Application of time-lapse ERT to determine the impact of using brackish wastewater for maize irrigation. J. Hydrol.
**2019**, 582, 124465. [Google Scholar] [CrossRef] - Camporese, M.; Cassiani, G.; Deiana, R.; Salandin, P. Assessment of local hydraulic properties from electrical resistivity tomography monitoring of a three-dimensional synthetic tracer test experiment. Water Resour. Res.
**2011**, 47, W12508. [Google Scholar] [CrossRef] - Binley, A.; Kemna, A. DC resistivity and induced polarization methods. In Hydrogeophysics; Rubin, Y., Hubbard, S.S., Eds.; Springer: Dordrecht, The Netherlands, 2005; Volume 50, pp. 129–156. [Google Scholar]
- De Carlo, L.; Berardi, M.; Vurro, M.; Caputo, M.C. Geophysical and hydrological data assimilation to monitor water content dynamics in the rocky unsaturated zone. Environ. Monit. Assess.
**2018**, 190, 310. [Google Scholar] [CrossRef] - Camporese, M.; Cassiani, G.; Deiana, R.; Salandin, P.; Binley, A. Coupled and uncoupled hydrogeophysical inversions using ensemble Kalman filter assimilation of ERT-monitored tracer test data. Water Resour. Res.
**2015**, 51, 3277–3291. [Google Scholar] [CrossRef] - Ferrè, T.P.A.; Bentley, L.; Binley, A.; Linde, N.; Kemna, A.; Singha, K.; Holliger, K.; Huisman, J.A.; Minsley, B. Critical steps for the continuing advancement of hydrogeophysics. Eos
**2009**, 90, 200. [Google Scholar] [CrossRef][Green Version] - Mboh, C.; Huisman, J.; Gaelen, N.; Rings, J.; Vereecken, H. Coupled hydrogeophysical inversion of electrical resistances and inflow measurements for topsoil hydraulic properties under constant head infiltration. Near Surf. Geophys.
**2012**, 10, 413–426. [Google Scholar] [CrossRef] - Rossi, M.; Manoli, G.; Pasetto, D.; Deiana, R.; Ferraris, S.; Strobbia, C.; Putti, M.; Cassiani, G. Coupled inverse modeling of a controlled irrigation experiment using multiple hydro-geophysical data. Adv. Water Resour.
**2015**, 82, 150–165. [Google Scholar] [CrossRef] - Camporese, M.; Paniconi, C.; Putti, M.; Salandin, P. Comparison of data assimilation techniques for a coupled model of surface and subsurface flow. Vadose Zone J.
**2009**, 8, 837–845. [Google Scholar] [CrossRef][Green Version] - Kitanidis, P.K. Prediction by the method of moments of transport in a heterogeneous formation. J. Hydrol.
**1988**, 102, 453–473. [Google Scholar] [CrossRef] - Govindaraju, R.S.; Das, B.S. Moment Analysis for Subsurface Hydrologic Applications; Springer: Dordrecht, The Netherlands, 2007; p. 296. [Google Scholar] [CrossRef]
- Ye, M.; Khaleel, R.; Yeh, T.-C.J. Stochastic analysis of moisture plume dynamics of a field injection experiment. Water Resour. Res.
**2005**, 41, W03013. [Google Scholar] [CrossRef][Green Version] - Farzamian, M.; Monteiro Santos, F.A.; Khalil, M.A. Estimation of unsaturated hydraulic parameters in sandstone using electrical resistivity tomography under a water injection test. J. Appl. Geophys.
**2015**, 121, 71–83. [Google Scholar] [CrossRef] - Singha, K.; Gorelick, S.M. Saline tracer visualized with three-dimensional electrical resistivity tomography: Field-scale spatial moment analysis. Water Resour. Res.
**2005**, 41, W05023. [Google Scholar] [CrossRef] - Caputo, M.C.; Nimmo, J. Quasi-steady centrifuge method of unsaturated hydraulic properties. Water Resour. Res.
**2005**, 41, W11504. [Google Scholar] [CrossRef][Green Version] - Turturro, A.C.; Caputo, M.C.; Perkins, K.S.; Nimmo, J.R. Does the Darcy-Buckingham Law Apply to Flow Through Unsaturated Porous Rock? Water
**2020**, 12, 2668. [Google Scholar] [CrossRef] - Turturro, A.C.; Caputo, M.C.; Gerke, H.H. Mercury Intrusion Porosimetry and Centrifuge Methods for Extended-Range Retention Curves of Soil and Porous Rock Samples. Vadose Zone J.
**2021**, 21, e20176. [Google Scholar] [CrossRef] - Caputo, M.C.; De Carlo, L.; Turturro, A.C. HYPROP FIT to Model Rock Water Retention Curves Estimated by Different Methods. Water
**2022**, 14, 3443. [Google Scholar] [CrossRef] - Andriani, G.F.; Pastore, N.; Giasi, C.I.; Parise, M. Hydraulic properties of unsaturated calcarenites by means of a new integrated approach. J. Hydrol.
**2021**, 602, 126730. [Google Scholar] [CrossRef] - Lappala, E.G.; Healy, R.W.; Weeks, E.P. Documentation of Computer Program VS2D to Solve the Equations of Fluid Flow in Variably Saturated Porous Media; Water Resources Investigations Report 83-4099; USGS: Denver, CO, USA, 1987; p. 193.
- Hsieh, P.A.; Wingle, W.; Healy, R.W. VS2DI—A Graphical Software Package for Simulating Fluid Flow and Solute or Energy Transport in Variably Saturated Porous Media; Water-Resources Investigations Report 99-4130; USGS: Lakewood, CO, USA, 2000; p. 20.
- Van Genuchten, M.T. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.
**1980**, 44, 892–898. [Google Scholar] [CrossRef][Green Version] - Rivett, M.O.; Wealthall, G.P.; Dearden, R.A.; McAlary, T.A. Review of unsaturated-zone transport and attenuation of volatile organic compound (VOC) plumes leached from shallow source zones. J. Contam. Hydrol.
**2011**, 123, 130–156. [Google Scholar] [CrossRef] [PubMed] - Troldborg, M.; Binning, P.; Nielsen, S.; Kjeldsen, P.; Christensen, A. Unsaturated zone leaching models for assessing risk to groundwater of contaminated sites. J. Contam. Hydrol.
**2009**, 105, 28–37. [Google Scholar] [CrossRef] - Jacques, D.; Šimůnek, J.; Mallants, D.; van Genuchten, M.T. Modelling coupled water flow, solute transport and geochemical reactions affecting heavy metal migration in a podzol soil. Geoderma
**2008**, 145, 449–461. [Google Scholar] [CrossRef] - Shentu, J.; Li, X.; Han, R.; Chen, Q.; Shen, D.; Qi, S. Effect of site hydrological conditions and soil aggregate sizes on the stabilization of heavy metals (Cu, Ni, Pb, Zn) by biochar. Sci. Total Environ.
**2022**, 802, 149949. [Google Scholar] [CrossRef] [PubMed] - Motz, E.C.; Cey, E.; Ryan, M.C. Vadose Zone Microbial Transport Below At-Grade Distribution of Wastewater Effluent. Water Air Soil Pollut.
**2012**, 223, 771–785. [Google Scholar] [CrossRef] - Sousa, M.R.; Jones, J.P.; Frind, E.O.; Rudolph, D.L. A simple method to assess unsaturated zone time lag in the travel time from ground surface to receptor. J. Contam. Hydrol.
**2013**, 144, 138–151. [Google Scholar] [CrossRef] - Jeong, J.; Park, E.; Han, W.S.; Kim, K.Y.; Oh, J.; Ha, K.; Yoon, H.; Yun, S.T. A method of estimating sequential average unsaturated zone travel times from precipitation and water table level time series data. J. Hydrol.
**2017**, 554, 570–581. [Google Scholar] [CrossRef] - Dahan, O. Vadose Zone Monitoring as a Key to Groundwater Protection. Front. Water
**2020**, 2, 599569. [Google Scholar] [CrossRef] - Day-Lewis, F.D.; Singha, K.; Binley, A.M. Applying petrophysical models to radar travel time and electrical resistivity tomograms: Resolution-dependent limitations. J. Geophys. Res.
**2005**, 110, B08206. [Google Scholar] [CrossRef][Green Version] - Cassiani, G.; Ursino, N.; Deiana, R.; Vignoli, G.; Boaga, J.; Rossi, M.; Perri, M.T.; Blaschek, M.; Duttmann, R.; Meyer, S.; et al. Noninvasive monitoring of soil static characteristics and dynamic states: A case study highlighting vegetation effects on agricultural land. Vadose Zone J.
**2012**, 11, vzj2011.0195. [Google Scholar] [CrossRef][Green Version] - Farzamian, M.; Autovino, D.; Basile, A.; De Mascellis, R.; Dragonetti, G.; Monteiro Santos, F.; Binley, A.M.; Coppola, A. Assessing the dynamics of soil salinity with time-lapse inversion of electromagnetic data guided by hydrological modelling. Hydrol. Earth Syst. Sci.
**2021**, 25, 1509–1527. [Google Scholar] [CrossRef] - Looms, M.C.; Jensen, K.H.; Binley, A.M.; Nielsen, L. Monitoring unsaturated flow and transport using cross-borehole geophysical methods. Vadose Zone J.
**2008**, 8, 227–237. [Google Scholar] [CrossRef][Green Version] - Dragonetti, G.; Farzamian, M.; Coppola, A.; Basile, A.; Monteiro Santos, F. In-situ estimation of soil hydraulic and hydrodispersive properties by inversion of Electromagnetic Induction measurements and soil hydrological modeling. Hydrol. Earth Syst. Sci.
**2022**, 26, 5119–5136. [Google Scholar] [CrossRef] - Binley, A.M.; Hubbard, S.S.; Huisman, J.A.; Revil, A.; Robinson, D.A.; Singha, K.; Slater, L.D. The emergence of hydrogeophysics for improved understanding of subsurface processes over multiple scales. Water Resour. Res.
**2015**, 51, 3837–3866. [Google Scholar] [CrossRef]

**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 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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