# Optimization of Aquifer Monitoring through Time-Lapse Electrical Resistivity Tomography Integrated with Machine-Learning and Predictive Algorithms

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## Featured Application

**In this work, Vector Autoregressive and Recurrent Neural Network algorithms are used to predict time-space evolution of a saline water plume in homogeneous and real aquifers.**

## Abstract

## 1. Introduction

_{2}geological storage and predicting the future behavior of soils in slopes [35,36,37]. However, only one example of RNN application to near surface geophysical data forecasting is reported in literature by Alali et al. [38], who used RNN to produce synthetic time-lapse seismic data.

## 2. Background Theory

#### 2.1. DC Electrical Method

_{AB}, Ampere) into the subsoil by using two current electrodes A and B and to measure the potential difference (ΔV

_{MN}, Volt) through two other electrodes M and N on the ground surface. The resistance of the material (R, Ω) through which the current flows can be obtained by:

_{a}, Ωm) and the configuration of electrodes, according to:

_{g}(m) is called geometric factor; it depends on electrodes geometry (array) and can be calculated by:

_{a}is called apparent resistivity and is defined as the ratio between the measured value of the parameter in question and its theoretical value in a homogeneous and isotropic medium of unit resistivity [39]. Therefore, in real cases, inverse methods must be applied to turn from the apparent resistivity to the true spatial distribution of this parameter.

#### 2.2. Predictive Methods

#### 2.2.1. Statistical Approaches

_{1}and y

_{2}, it is possible to forecast the values of these two variables at the future time t, using the data recorded for past n values. Furthermore, assuming the correlation between the two variables, the past value of both y

_{1}and y

_{2}will be used.

_{1,t}= c

_{1}+ ϕ

_{11,1}y

_{1,t−1}+ ϕ

_{12,1}y

_{2,t−1}+ e

_{1,t}

_{2,t}= c

_{2}+ ϕ

_{21,1}y

_{1,t−1}+ ϕ

_{22,1}y

_{2,t−1}+ e

_{2,t},

_{1,t}and e

_{2,t}are white noise processes that may be contemporaneously correlated. The symbols c

_{1}and c

_{2}are constants serving as the intercepts of the model.

_{ii}

_{,ℓ}represents the influence of the ℓ

_{th}lag of variable y

_{i}on itself. Instead, the coefficient ϕ

_{ij,ℓ}represents the influence of the ℓ

_{th}lag of variable y

_{j}on y

_{i}, where “lag” represents a fixed amount of passing time [47].

#### 2.2.2. Recurrent Neural Networks

_{ih}, W

_{hh}, and W

_{ho}that connect, the input to hidden layer, recurrent hidden layer to hidden layer itself, and hidden layer to output, respectively. The hidden layer captures information about what happened in the previous time step. It can be considered as the “memory” of the network.

_{t}) is computed that includes the differences between predicted and actual output (y

_{t}and y′

_{t}, respectively) at each iteration. At each time step, L

_{t}is given by cross-entropy, defined as follows:

_{ih}, W

_{hh}and W

_{ho}). The parameter α is called “learning rate”. It scales the gradient and thus controls the step size. When the gradient of the Loss function is calculated with respect to the weight matrixes, generally it is possible to incur in the well-known problem of the vanishing gradient. In fact, when the RNN is formed by many neural layers, there is the risk that the gradient tends to become smaller and smaller at each derivative step. Consequently, since the gradient tends to vanish over time, the network cannot learn properly about long-term time information kept in memory. This is a typical problem in all the types of deep neural networks formed by many hidden layers, and not exclusively in RNNs. In order to face that problem, a variant of RNN called “Long-Short Term Memory” (LSTM) algorithm can be used [49,50,51]. This allows for the network to include both short-term and long-term dependencies at a given time from many time steps before the current time and can solve the vanishing gradient problem effectively. A typical LSTM cell consists of three special “gates” called, respectively, the input gate, forget gate, and output gate. These allow for deciding what information must be included or excluded from the network memory and used for calculating the output. In other words, the effectiveness of the network is guaranteed by selecting the length of the sequence of historical data to be used as “memory information”. The LSTM network is designed to keep information in the memory only if required. The input gate is responsible for selecting the part of information that must be stored in memory and used for calculating the output. Instead, the forget gate is responsible for deciding what information is not relevant for calculating the output and should be excluded from the output computation. Finally, the output gate is set for selecting what information must be shown at each time step. It determines the value of the next hidden state that contains information on previous inputs, that is, the output gate determines what output (next Hidden State) to generate from the current Internal Cell State [52,53,54].

## 3. Datasets

#### 3.1. Real Scale Field Dataset

^{−5}ms

^{−1}[55].

#### 3.2. Small-Scale Laboratory Dataset

^{3}of homogeneous silica sand (95% SiO

_{2}) characterized by an average diameter equal to 0.09 mm (very fine sand), porosity of about 45–50%, and hydraulic conductivity in the order of 10

^{−5}ms

^{−1}[66]. An external hydraulic system was used to impose a constant hydraulic gradient during all the experiment while contamination occurs by continuously injecting 10 liters of salt water (NaCl solution with a concentration of 100 g/L) for 10 days. The source of contamination was at −0.15 m of depth (Figure 4).

## 4. Results

#### 4.1. Application of VAR and RNN Methods to Field Dataset

#### 4.2. Application of RNN to Laboratory Dataset

## 5. Discussion

#### 5.1. Field Experiment: Prediction of Saline Tracer Displacement

#### 5.2. Laboratory Experiment: Prediction of Contamination Evolution

_{b}, Ωm) and pore water electrical resistivity (ρ

_{w}, Ωm) according to the following equation:

_{e}will increase along with the increase of ions in water TDS is expressed in g/L and σ in mS/cm. Therefore, considering k

_{e}= 0.7 (saline water with σ from 1 till 45 mS/cm [74]), TDS (g/L) in the saturated zone can be obtained with the relation:

_{b}(Ωm) was obtained by CH-ERT monitoring, porosity was 0.45, m = 1.14, and n =1.5 [64,65], while saturation degree in the vadose zone was estimated by numerical models [66].

## 6. Summary of Results

- -
- In the field experiment, based on multiple monitor geoelectrical surveys in cross-hole configuration, multiple resistivity models changing over a period of 28 days of the tracer test were defined;
- -
- A variational statistical method was applied for predicting electrical resistivity variations;
- -
- Transport characteristics of the studied aquifer were evaluated through both measured and predicted electrical resistivity data;
- -
- This predictive approach can be applied for defining optimal aquifer management policy;
- -
- In the lab experiment, Recurrent Neural Networks to retrieve a Multivariate LSTM Forecast Dynamic Model of the tracer displacements over time were applied;
- -
- The predictions obtained through RNN are fully consistent with the evolution of the experimental system effectively observed, confirming the effectiveness of such a type of approach applied to predictive analysis of hydrogeological time-series;
- -
- The predictions are based on the history path of electric potentials recorded through multiple surveys including thousands of measurement points.

## 7. Conclusions

_{2}, and heat storage or extraction.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Pseudo section setting for ERT acquisition with Dipole Dipole (DD) array, (

**b**) cross borehole, and (

**c**) cross borehole with surface. A, B, M, and N are electrodes. D is borehole depth; S is borehole separation. D/S > 1.5. Adapted from [14].

**Figure 3.**(

**a**) Tracer test at Montalto Uffugo test site: (Formation A) shallow aquifer; (Formation B) shale; (Formation C) main aquifer, silty sands alternated to conglomerate and clay lenses; (Formation D) shale. B1–11 are boreholes. P1–2 are piezometers for CH-ERT. (

**b**) Pre-tracer test resistivity model and B11 stratigraphic log (from 41, modified); (

**c**) water electrical conductivity (normalized at 20 °C) monitored in P1 at the depth of 40 m; (

**d**) water electrical conductivity (normalized at 20 °C) logs measured 20 days after the salt injection (19 July), in boreholes B5 and P2. Adapted from [55].

**Figure 4.**Laboratory experimental set-up. A–F are the piezometers instrumented for CH-ERT. Distances are in cm. Modified from [65].

**Figure 5.**Subset of data (in terms of apparent resistivity) taken at several measurement points, in the set-up of Figure 4, over 20 time steps. Q is quadrupole.

**Figure 6.**Chart of four predictions (predicted future) vs. observations (“true” future) for a single quadrupole (Q1—field data). Remark: time step “0” corresponds to the first prediction time step. The training dataset consists of 10 time steps.

**Figure 7.**(

**a**,

**b**) Examples of RNN’s predictions (normalized electrical potentials vs. time); (

**c**) Loss function convergence trend (bottom-right panel). The training dataset consists of 13 time steps.

**Figure 8.**(

**a**) Some images of resistivity variation (d, %) after tracer injection. Black dots are electrodes, and P1, P2, and B5 are piezometers; (

**b**) electrical resistivity variation (d, %) curves along the piezometers P1, B5, and P2 from the CH-ERT dataset. Values at days 29 and 31 were estimated from CH-ERT predicted datasets.

**Figure 9.**(

**a**) True resistivity model obtained using ResIPy software of pre contamination condition; (

**b**) images of resistivity variation (d, %) after salt solution injection. Black dots are electrodes, C and D are piezometers.

**Figure 10.**In red is the 3D distribution of TDS > 10 g/L obtained from predicted CH-ERT. A–F are piezometers, V

_{c}is the contaminated volume. TDS values at day 24 were estimated from CH-ERT predicted datasets. The 3D distribution was obtained with Paraview software (version 5.6.0, Kitware, Sandia National Labs, Los Alamos nat. lab., ASC and ARL, https://www.paraview.org/, accessed on 13 June 2022).

**Table 1.**Description of the two datasets used in the present manuscript for testing Vector Autoregressive (VAR) and Recurrent Neural Network algorithms (RNN).

Electrical Resistivity Dataset | Number of Surveys in Training Data | Number of Surveys in Test Data |
---|---|---|

Field data | 2–20 CH-ERT | 5 CH-ERT |

(1836 data each) | (1836 data each) | |

Laboratory data | 2–20 CH-ERT | 5 CH-ERT |

(504 data each) | (504 data each) |

**Table 2.**Experimental measurements and corresponding predictions using Vector Autoregressive (VAR) statistical method.

Sequential ERT Surveys | Resistance at the Various Quadrupoles (Ω) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Q1 | Q2 | Q3 | Q4 | Q5 | Q6 | Q7 | Q8 | Q9 | Q10 | |

STEP 10 | 1.76 | 19.16 | 8.80 | 1.65 | 12.03 | 14.33 | 16.15 | 13.40 | 19.71 | 8.88 |

STEP 11 | 1.68 | 19.21 | 8.46 | 3.59 | 9.63 | 13.15 | 15.29 | 12.93 | 19.13 | 9.16 |

STEP 12 | 1.74 | 19.13 | 8.46 | 3.09 | 10.06 | 13.33 | 15.60 | 13.20 | 19.66 | 9.37 |

STEP 13 | 2.04 | 18.87 | 9.33 | 1.08 | 9.64 | 12.81 | 15.66 | 13.37 | 20.80 | 10.33 |

STEP 14 | 1.66 | 19.07 | 8.47 | 3.17 | 10.78 | 13.17 | 15.32 | 12.75 | 19.46 | 9.25 |

STEP 15 | 1.77 | 19.02 | 8.50 | 3.24 | 10-08 | 12.94 | 15.18 | 12.81 | 19.65 | 9.11 |

STEP 16 | 1.92 | 18.90 | 8.40 | 2.95 | 10.79 | 13.19 | 15.29 | 12.66 | 19.42 | 9.39 |

STEP 17 | 1.83 | 19.12 | 9.69 | 1.15 | 9.24 | 12.72 | 16.72 | 14.35 | 21.21 | 11.87 |

STEP 18 | 1.84 | 18.29 | 7.29 | 0.22 | 11.95 | 16.95 | 15.46 | 15.47 | 19.09 | 12.43 |

Sequential Predictions | Predicted resistance at the Various Quadrupoles (Ω) | |||||||||

PRED 1 | 1.85 | 18.80 | 8.47 | 1.38 | 10.95 | 14.07 | 15.80 | 14.12 | 20.05 | 11.10 |

PRED 2 | 1.86 | 18.77 | 8.47 | 1.25 | 11.04 | 14.12 | 15.84 | 14.19 | 20.09 | 11.24 |

PRED 3 | 1.87 | 18.75 | 8.47 | 1.12 | 11.12 | 14.18 | 15.88 | 14.27 | 20.13 | 11.37 |

PRED 4 | 1.88 | 18.72 | 8.47 | 0.99 | 11.21 | 14.24 | 15.92 | 14.35 | 20.17 | 11.50 |

PRED 5 | 1.89 | 18.69 | 8.47 | 0.88 | 11.29 | 14.30 | 15.96 | 14.43 | 20.22 | 11.63 |

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**MDPI and ACS Style**

Giampaolo, V.; Dell’Aversana, P.; Capozzoli, L.; De Martino, G.; Rizzo, E.
Optimization of Aquifer Monitoring through Time-Lapse Electrical Resistivity Tomography Integrated with Machine-Learning and Predictive Algorithms. *Appl. Sci.* **2022**, *12*, 9121.
https://doi.org/10.3390/app12189121

**AMA Style**

Giampaolo V, Dell’Aversana P, Capozzoli L, De Martino G, Rizzo E.
Optimization of Aquifer Monitoring through Time-Lapse Electrical Resistivity Tomography Integrated with Machine-Learning and Predictive Algorithms. *Applied Sciences*. 2022; 12(18):9121.
https://doi.org/10.3390/app12189121

**Chicago/Turabian Style**

Giampaolo, Valeria, Paolo Dell’Aversana, Luigi Capozzoli, Gregory De Martino, and Enzo Rizzo.
2022. "Optimization of Aquifer Monitoring through Time-Lapse Electrical Resistivity Tomography Integrated with Machine-Learning and Predictive Algorithms" *Applied Sciences* 12, no. 18: 9121.
https://doi.org/10.3390/app12189121