# Nonlinear Autoregressive Neural Networks to Predict Hydraulic Fracturing Fluid Leakage into Shallow Groundwater

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}) between measured and predicted values is 0.923 and the mean squared error (MSE) is 4.2 × 10

^{−4}, and the values of R

^{2}= 0.944 and MSE = 2.4 × 10

^{−4}were obtained for the NAR-BR model. The results indicate the robustness and compatibility of NAR-LM and NAR-BR models in predicting fracturing fluid flow rate to shallow aquifers. This study shows that NAR neural networks can be useful and hold considerable potential for assessing the groundwater impacts of unconventional gas development.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Conceptual Model for Fracturing Fluid Migration along an Abandoned Well

^{3}of fracturing fluid [13] is injected into the domain during 2.5 days. Then, a 5-day shut-in period takes place. Afterwards, the production periods are modeled with outflow rates of 16 and 3.2 m

^{3}/day for an interrupted period of 2 and 13 years, respectively [13]. In the abandonment stage, the production is stopped. The fluid flow along the abandoned well and at the overburden aquifer interface is monitored. Note that we used data reported in Brownlow et al. [13], which are based on available industry data for the Eagle Ford Shale play in south Texas.

#### 2.2. Data Preparation and Analysis

#### 2.3. NAR Model

#### 2.4. Training Algorithms

#### 2.4.1. Levenberg–Marquardt

#### 2.4.2. Bayesian Regularization

#### 2.5. Network Architecture

#### 2.6. Performance Evaluation

^{2}) and mean squared errors (MSE). R

^{2}is a linear regression used to analyze the best fit between the measured values and model’s predicted values, given by:

## 3. Results and Discussion

^{2}for the training phase (Table 2) show that both models reached the best fitting performances in terms of evaluation criteria. For instance, R

^{2}values of higher than 0.99 indicated a strong correlation between the measured values and fitting values. According to Figure 5 and Table 2, we conclude that a sample of 446 data is adequate to train the NAR models.

^{2}values higher than 0.9 for the testing phase indicated the satisfactory performance of both developed models. For further comparison of evaluation indices, MSE of NAR-LM and NAR-BAR were 4.2 × 10

^{−4}and 2.4 × 10

^{−4}respectively, which proved the strong ability of the developed models in predicting the nonlinear behavior of fracturing fluid flow to the aquifer. The results suggest that NAR-BR model has a slightly better prediction performance compared with that of the NAR-LM model in terms of larger R

^{2}and smaller MSE.

^{2}value of the NAR-BR model is slightly higher.

^{−5}at epoch 9 for the NAR-LM model, while the NAR-BR model experiences the best performance (MSE = 1.3 × 10

^{−5}) for the training phase at epoch 16.

^{−3}m

^{3}/year) for the entire time period, indicating that the NAR-LM and NAR-BR models were able to efficiently predict fracturing fluid flow rate to the aquifer over the simulation period. Comparing the response lines and error plots of this figure further indicated the superior performance of the NAR-BR model.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The conceptual model used for simulations in Taherdangkoo et al. [17].

**Figure 2.**Fracturing fluid flow rate to the aquifer for the base-case model during the simulation time.

**Figure 6.**Performance of the NAR-LM model for the prediction of fracturing fluid flow rate to the aquifer: (

**a**) training and (

**b**) testing.

**Figure 7.**Performance of the NAR-BR model for the prediction of fracturing fluid flow rate to the aquifer: (

**a**) training and (

**b**) testing.

**Figure 8.**Convergence plots of the NAR models in terms of number of epochs: (

**a**) NAR-LM and (

**b**) NAR-BR.

**Figure 9.**Response of (

**a**) NAR-LM and (

**b**) NAR-BR models in predicting fracturing fluid flow rate to the shallow aquifer. The top panels show the modeled fluid flow rate presented in Figure 2. The bottom panels display the model’s performance error in the training, validation, and testing phases.

**Table 1.**Parameters used in base-case model and sensitivity analysis simulations [17].

Parameter | Unit | Base-Case Value | Min. | Max. | Source |
---|---|---|---|---|---|

Shale permeability | m^{2} | 1 × 10^{−19} | 1 × 10^{−21} | 1 × 10^{−18} | [12,13,34] |

Shale porosity | 0.01 | 0.01 | 0.05 | [7,34,35,36] | |

Overburden permeability | m^{2} | Depth-dependent | 1 × 10^{−17} | 1 × 10^{−15} | [7,37,38] |

Overburden thickness | m | 1600 | 900 | 2900 | [7,39,40] |

Salinity gradient | g/lm | 0.15 | 0.1 | 0.2 | [41,42] |

Fracturing fluid volume | m^{3} | 11,365 | 11,000 | 15,000 | [8,11,13] |

Abandoned well permeability | m^{2} | 1 × 10^{−12} | 1 × 10^{−17} | 1 × 10^{−12} | [43,44,45,46,47,48,49,50] |

Distance of fracture plane to well | m | 0 | 0 | 15 |

Statistical Parameter | NAR-LM | NAR-BR | |||
---|---|---|---|---|---|

Training | Validation | Testing | Training | Testing | |

R^{2} | 0.998 | 0.996 | 0.923 | 0.996 | 0.944 |

MSE | 1.07 × 10^{−5} | 1.2 × 10^{−5} | 4.2 × 10^{−4} | 1.3 × 10^{−5} | 2.4 × 10^{−4} |

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**MDPI and ACS Style**

Taherdangkoo, R.; Tatomir, A.; Taherdangkoo, M.; Qiu, P.; Sauter, M. Nonlinear Autoregressive Neural Networks to Predict Hydraulic Fracturing Fluid Leakage into Shallow Groundwater. *Water* **2020**, *12*, 841.
https://doi.org/10.3390/w12030841

**AMA Style**

Taherdangkoo R, Tatomir A, Taherdangkoo M, Qiu P, Sauter M. Nonlinear Autoregressive Neural Networks to Predict Hydraulic Fracturing Fluid Leakage into Shallow Groundwater. *Water*. 2020; 12(3):841.
https://doi.org/10.3390/w12030841

**Chicago/Turabian Style**

Taherdangkoo, Reza, Alexandru Tatomir, Mohammad Taherdangkoo, Pengxiang Qiu, and Martin Sauter. 2020. "Nonlinear Autoregressive Neural Networks to Predict Hydraulic Fracturing Fluid Leakage into Shallow Groundwater" *Water* 12, no. 3: 841.
https://doi.org/10.3390/w12030841