# Encoder–Decoder Convolutional Neural Networks for Flow Modeling in Unsaturated Porous Media: Forward and Inverse Approaches

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}score of over 91%, while the ED-CNN optimizer achieved an R

^{2}score of over 89%. The study highlights the potential of ED-CNNs as reliable and efficient tools for both forward and inverse modeling in the analysis of unsaturated flow.

## 1. Introduction

_{2}plume migration [41]. In addition to approximating state variables based on parameters characterizing the porous media and external forcing (as in forward simulations, see, e.g., [42]), ED-CNNs have also been used in the context of inverse modeling for direct estimation of input parameters from state variables without resorting to common data assimilation (e.g., Kalman filters) or calibration (e.g., Levenberg–Marquardt) algorithms. Studies using ED-CNN for parameter estimation of water flow and transport processes in porous media include, for instance, [28,43]. To the author’s knowledge, ED-CNNs have not been previously applied to forward or inverse modeling of water flow in unsaturated porous media.

## 2. Theoretical Framework and Methodology

#### 2.1. Encoder–Decoder Convolutional Neural Networks

#### 2.2. Image-to-Image Regression Modeling

^{D𝑖𝑛×𝐻×𝑊}→ ℝ

^{D𝑜𝑢𝑡×𝐻×𝑊}, where λ represents the relationship between the input and output images, and D

_{𝑖𝑛}and D

_{𝑜𝑢𝑡}are the dimensions of the input and output images, respectively. Advances in computing technology, such as the development of GPUs, have enabled the efficient computation of large matrices and images at the same time. This has contributed to deep CNN technology’s rapid growth and widespread adoption [36].

#### 2.3. Description of the Test Case

^{®}) to facilitate the visual observation of water movement. Assuming a 2D domain, the problem had impermeable top and bottom boundaries, and a variable pressure head was applied at the lateral boundaries by vertically moving two overflow outlets (Figure 1). The domain was initially saturated; overflow outlets, placed at the vertical sides of the experimental tank, were gradually lowered to drain water outside the tank, causing a decrease of the soil moisture from top to bottom through the experiment. This test case has been modeled both numerically and physically in the past (see [51]). A brief description of the numerical model and the experimental measurements is provided in the following two sub-sections.

#### 2.3.1. Numerical Simulations

^{−1}] is the specific storage, ${S}_{w}$[-] is the relative saturation and is defined as ${S}_{w}=\theta \times {{\theta}_{s}}^{-1}$, $H$[L] and $h$[L] are the piezometric and pressure heads, respectively. $H=h+z$, where $z$[L] is the elevation (positive in the upward direction). $k\left(h\right)$[LT

^{−1}] is the hydraulic conductivity and ${f}_{s}$[T

^{−1}] is the sink-source term. Water content and hydraulic conductivity are related to the pressure head using the Mualem–van Genuchten model [4,52]:

^{−1}] and $n$[-] are empirical parameters related to the mean and uniform pore size, respectively, ${k}_{s}$[LT

^{−1}] is saturated conductivity, and ${s}_{e}$ is effective saturation defined as ${S}_{e}=(\theta -{\theta}_{r})/({\theta}_{s}-{\theta}_{r})$.

^{3}·cm

^{−3}and 0.099 cm

^{3}·cm

^{−3}, respectively.

#### 2.3.2. Measurements from the Laboratory Experiment

#### 2.4. Model-Based Data Generation

#### 2.5. Model Training and Validation

^{2}score), the Root mean squared error (RMSE), and the percentage error, calculated as follows [55,56]:

## 3. Results and Discussion

#### 3.1. ED-CNN as Meta-Model

^{2}score variations with respect to the training sample size. This plot clearly shows that increasing the sample size from 100 to 300 improved the R

^{2}score from 0.77 to 0.91, whereas using 700 or more samples only slightly increased the model accuracy.

^{3}·cm

^{−3}to 0.02 cm

^{3}·cm

^{−3}. As demonstrated, ED-CNN predictions (Figure 5b) show significant similarity with the numerical model output soil moisture distributions (Figure 5a) with low rates of error (Figure 5c).

^{3}·cm

^{−3}, occurred at the last time step when the maximum drainage had been achieved. The spatial distribution of pixel-wise average RMSE for the 200 test cases in 4 time steps is depicted in Figure 6b. The figure shows that on average, the maximum error for each step occurred at the moving front of the saturation profile. To further assess the ED-CNN meta-model performance, the average RMSE histogram for the 200 test cases is presented in Figure 7. For the ED-CNN as a meta-model, the maximum value of the RMSE in a test sample was 0.079 cm

^{3}·cm

^{−3}, and 87% of the RMSEs were 0.023 cm

^{3}·cm

^{−3}. The distribution of the RMSE is close to a log-normal distribution skewed to the left.

#### 3.2. ED-CNN for Uncertainty Analysis

^{3}·cm

^{−3}and −0.014 to 0.0244 cm

^{3}·cm

^{−3}, respectively. The maximum error for both $\mu $ and $\sigma $ occurred at the moving front of the saturation profile.

#### 3.3. ED-CNN as an Optimizer

#### 3.3.1. Model Training and Validation Using Numerical Simulation Data

^{2}score reaches about 0.89 using 100 samples, whereas training the ED-CNN optimizer using more realizations does not significantly improve the accuracy. The RMSE histogram for ED-CNN as an optimizer is shown in Figure 10a–c. The maximum value of average RMSE for a test case for ${k}_{s}$ is 0.0074 cm·s

^{−1}, while 94% of RMSE values do not exceed 0.001 cm·s

^{−1}. Maximum RMSE for $\alpha $ is 0.02 cm

^{−1}but 82% of errors are less than 0.075 cm

^{−1}. The highest RMSE value for $n$ is equal to 1.29 and 95% of errors for test samples are less than 0.4. For all parameters, the RMSE distribution is skewed to the left and resembles a log-normal distribution. Moreover, to better compare real and predicted values for each parameter, a scatter plot was calculated and is visualized in Figure 10d–f. It is evident from the results that there is significant agreement between real values and ED-CNN predictions. The percentage errors between real values for all 200 test cases and ED-CNN predictions for ${k}_{s}$,$\alpha $, and $n$ are about 0.3%, 1.4%, and 1%, which indicates that the ED-CNN architecture meets the accuracy requirements and hydraulic parameters can be precisely estimated using this method.

#### 3.3.2. Comparison with Previous Studies

#### 3.3.3. Parameter Estimation Using Photographic Imaging Data

## 4. Conclusions

^{2}score of more than 91% for soil moisture predictions. For the ED-CNN optimizer, 100 epochs and 100 samples were required to train the model with an R

^{2}score of more than 89%. In soil moisture predictions, the maximum average RMSE is seen at the last time step, when maximum drainage has been achieved. The spatial distribution of pixel-wise average RMSE shows that the maximum error for each time step occurred at the moving front of the saturation profile. Future work may consider applying the developed methodology to (a) heterogeneous porous media, (b) a complete drainage–imbibition cycle, and (c) modeling solute transport in unsaturated porous media.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Schematic of proposed ED-CNN architecture (Conv: convolution layer, BN: batch normalization layer).

**Figure 4.**(

**a**) RMSE as a function of epochs and (

**b**) ${\mathrm{R}}^{2}$ score as a function of the training dataset size of the proposed ED-CNN meta-model for soil moisture predictions.

**Figure 5.**(

**a**) Numerical model-based, and (

**b**) ED-CNN-based soil moisture distribution in four time steps, in addition to (

**c**) the resulting spatial distribution of the absolute difference between numerical and ED-CNN results.

**Figure 8.**Spatial distribution of Monte Carlo- (

**a**–

**c**) mean and (

**d**–

**f**) standard deviation estimations based on the numerical model and ED-CNN, and their associated differences.

**Figure 9.**(

**a**) RMSE and (

**b**) ${\mathrm{R}}^{2}$ score of the proposed ED-CNN optimizer for soil moisture predictions.

**Figure 10.**(

**a**–

**c**) RMSE histograms of ${k}_{s}$, $\alpha $, and $n$, respectively, and (

**d**–

**f**) scatter plots of target vs. predicted values of ${k}_{s}$, $\alpha $, and $n$, respectively, estimated by the ED-CNN. Plots are based on 200 realizations.

**Figure 11.**Input image for the ED-CNN optimizer constructed from the photographic images of a sandbox flow tank in Belfort [51].

**Table 1.**Characteristics of the hypothetical probability distribution functions (PDFs) for the uncertain input parameters.

Parameter | Interval | Unit | |
---|---|---|---|

${k}_{s}$ | [0.001, 0.03] | [cm·s^{−1}] | |

$\alpha $ | Uniform | [0.001, 0.2] | [cm^{−1}] |

$n$ | [2, 8] | [-] |

Layer | Kernel Size | Resolution |
---|---|---|

Encoder | ||

Convolution + Batch normalization | 3 $\times $ 3 | 32 $\times $ 32 |

Downsampling1 | 2 $\times $ 2 | 16 $\times $ 16 |

Convolution + Batch normalization | 3 $\times $ 3 | 16 $\times $ 16 |

Downsampling2 | 2 $\times $ 2 | 8 $\times $ 8 |

Convolution + Batch normalization | 3 $\times $ 3 | 8 $\times $ 8 |

Decoder | ||

Convolution + Batch normalization | 3 $\times $ 3 | 8 $\times $ 8 |

Upsampling1 | 2 $\times $ 2 | 16 $\times $ 16 |

Convolution + Batch normalization | 3 $\times $ 3 | 16 $\times $ 16 |

Upsampling2 | 2 $\times $ 2 | 32 $\times $ 32 |

Convolution + Batch normalization | 3 $\times $ 3 | 32 $\times $ 32 |

Hyper-Parameter | Meta-Model | Optimizer |
---|---|---|

Optimizer | RMSprop | Adam |

Loss function | Mean squared error | Mean squared error |

Samples | 1000 | 1000 |

Test set | 200 | 200 |

Validation set | 300 | 300 |

Batch size | 12 | 24 |

Epochs | 150 | 300 |

Learning rate | 0.0001 | 0.0001 |

**Table 4.**Comparison of the results of previous studies on unsaturated flow parameter estimation with those of the current study.

Reference | Parameter Estimation Method | Estimated Hydraulic Parameters | Percentage Error (%) | ||
---|---|---|---|---|---|

${\mathit{k}}_{\mathit{s}}$ | $\mathit{\alpha}$ | $\mathit{n}$ | |||

[58] | EnKF | ${k}_{s}$$,\alpha $$,n$ | 3.2 | 3.6 | 1 |

[59] | EnKF | ${k}_{s}$ | 20 | - | - |

[60] | Analytical solution | ${k}_{s}$$,n$ | 10 | - | 10 |

[61] | GPPDE | ${k}_{s}$$,n$ | 3.3 | - | 8.93 |

[12] | EnKF | ${k}_{s}$$,\alpha $ | 1 | 1 | - |

[62] | Analytical solution | $\alpha $$,n$ | - | 3.2 to 28 | 0.4 to 8 |

Current optimizer study | ED-CNN | ${k}_{s}$$,\alpha $$,n$ | 0.3 | 1.4 | 1 |

**Table 5.**Comparison between ED-CNN estimation using experiment images and the numerical model with real values.

Parameter | Experiment Moisture Map as Input | Target Value |
---|---|---|

${k}_{s}$ | 0.35 | 0.37 |

$\alpha $ | 0.37 | 0.43 |

$n$ | 0.45 | 0.99 |

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## Share and Cite

**MDPI and ACS Style**

Hajizadeh Javaran, M.R.; Rajabi, M.M.; Kamali, N.; Fahs, M.; Belfort, B.
Encoder–Decoder Convolutional Neural Networks for Flow Modeling in Unsaturated Porous Media: Forward and Inverse Approaches. *Water* **2023**, *15*, 2890.
https://doi.org/10.3390/w15162890

**AMA Style**

Hajizadeh Javaran MR, Rajabi MM, Kamali N, Fahs M, Belfort B.
Encoder–Decoder Convolutional Neural Networks for Flow Modeling in Unsaturated Porous Media: Forward and Inverse Approaches. *Water*. 2023; 15(16):2890.
https://doi.org/10.3390/w15162890

**Chicago/Turabian Style**

Hajizadeh Javaran, Mohammad Reza, Mohammad Mahdi Rajabi, Nima Kamali, Marwan Fahs, and Benjamin Belfort.
2023. "Encoder–Decoder Convolutional Neural Networks for Flow Modeling in Unsaturated Porous Media: Forward and Inverse Approaches" *Water* 15, no. 16: 2890.
https://doi.org/10.3390/w15162890