# Machine Learning Prediction of Nanoparticle Transport with Two-Phase Flow in Porous Media

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## Abstract

**:**

## 1. Introduction

## 2. Research Methodology

#### 2.1. Machine Learning Modeling

^{2}) correlation, mean absolute error (MAE), mean squared error (MSE), and root mean squared error (RMSE).

#### 2.2. Traditional Mathematical Modeling

## 3. Results and Discussion

_{w}, the relative permeability of oil K

_{rop}, and the relative permeability of water K

_{rwp}. Accordingly, we used the same datasets four times to predict the four different target variables: C, S

_{w}, K

_{rop}, and K

_{rwp}. The two phase artificial dataset composed of 38 features or independent variables which are time (t), space (x), (phi

_{0}) is the initial porosity, (phi) is the porosity of the medium, (rho

_{o}) is the oil density, (rho

_{w}) is the water density, (pc) is the capillary pressure, (k

_{0}) is the initial permeability, (K) is the absolute permeability, (k

_{ro}) is the oil relative permeability, (k

_{rw}) is the water relative permeability, (k

_{ro}

_{0}) is the initial oil relative permeability, (k

_{rw}

_{0}) is the initial water relative permeability, (k

_{rwp}) is the relative permeability of water when the surface of porous media is occupied nanoparticles, (k

_{rop}) is the relative permeability of oil when the surface of the porous media is occupied with nanoparticles, (theta

_{w}) is the ratio of the water relative permeability due to nanoparticles adhering, (theta

_{o}) is the ratio of oil relative permeability due to nanoparticle adhering, (g) is the gravitational acceleration, (S

_{w}) is the water saturation, (C) is the nanoparticles concentration in water, (C

_{s}

_{1}) is the volume of nanoparticles in water, (C

_{s}

_{2}) is the volume of nanoparticles entrapped in pore throats due to plugging, (D) is the diffusion- dispersion tensor, (lam

_{wp}) is the mobility ratios of water, (lam

_{op}) is the mobility ratio of oil, (lam

_{tp}) is the total mobility, (m) is the viscosity, (f

_{o}) is the flow fraction of oil, (S

_{or}) is the residual oil saturation, (S

_{iw}) is the irreducible water saturation, (a) and (b) are positive constant, (k

_{f}) is constant for fluid seepage allowed by the plugged pores, (d

_{np}) is the nanoparticles diameter, (gama

_{f}) is the coefficient of nanoparticles flow efficiency, (a

_{w}) and (a

_{o}) is surface area in contact with water and oil respectively.

_{rop}, phi, t, lam

_{wp}, and lam

_{op}are highly correlated with each other. On the other hand, C is correlated with C

_{s1}, C

_{s}

_{2}, t, lam

_{tp}, lam

_{wp}, lam

_{op}, and S

_{w}.

_{s}

_{1}, C

_{s}

_{2}, and t are the most important features that help in predicting the nanoparticles concentration. The score value calculated for each feature is varied from model to model. Figure 2 presents an example of the feature importance calculated for the DT model.

_{s}

_{1}, t, and C

_{s}

_{2}are the most important features in predicting the nanoparticles concentration C. A comparative analysis between different machine learning models is provided, and it was found that the ANN model has the lowest root mean square error value of 0.000216 and the highest ${R}^{2}$ value of 0.999999, as shown in Table 3.

_{rw}, k

_{ro}, lam

_{op}, lam

_{wp}, and lam

_{tp}. The score value calculated for each feature is varied from model to model. Figure 4 demonstrates the feature importance in predicting S

_{w}using DT, RF, and GBR models. It is shown from the figure that pc, k

_{rw}, k

_{ro}, k

_{rwp}, k

_{rop}, lam

_{wp}, lam

_{op}, lam

_{tp}, and m are the most important features in most models.

_{w}for all four ML models. The four models have been graphed twice: one time without scaling the datasets and the other time with scaling the dataset using the StandardScaler. It can be seen that predicting S

_{w}when the dataset is not standardized had very close results to predicted values when the dataset became standardized, and the values in the figure are correlated. Except in the ANN model, which requires that the dataset standardize prior prediction. The ANN model with the tanh activation function had the highest correlated values in the scatter plot.

^{2}value.

_{rop}. First, we checked the features’ importance for each model when the target variable was k

_{rop}, and we found out that each model selected different features. The most important feature found used in all models is lam

_{tp}; other common features used by RF and GBR are pc, k

_{ro}, and lam

_{op}. The score value calculated for each feature is varied from model to model. Figure 8 presents the feature importance of the DT, RF, and GBR models. Figure 8 demonstrates the feature importance in predicting k

_{rop}using DT, RF, and GBR models. It is shown from the figure that lam

_{tp}, lam

_{op}, pc, k

_{rw}, k

_{ro}, k

_{rwp}, k

_{rop}, lam

_{wp}, and S

_{w}

_{_Norm}are the most important features in most of the models.

_{rop}, especially ANN models with standardized dataset and activation function of sigmoid that had RMSE value of 0.000145 and R

^{2}value of 1.000000. Figure 9 presents the scatterplot between the actual and the predicted k

_{rop}. Table 7 presents the metric evaluation results of all models. Figure 9 demonstrates the actual values and predicted values of k

_{rop}for all four models. It is shown that the scatter plots of all four models have a positive correlation. The scatter plots of each of the four models tested have been graphed twice, one time without scaling the datasets and the other time with scaling the dataset using the StandardScaler. Both scatter plots of each model that predict the k

_{rop}had very close results when the dataset was standardized and not standardized in which the values in the figure are highly correlated, except in the ANN model, which requires that the dataset standardize prior prediction. The ANN model with the sigmoid activation function had the highest correlated values in the scatter plot.

_{rop}had high performance without tuning the hyperparameters. The 2D contour and the 3D surface of the hyperparameter tuning of the RF model are shown in Figure 10. In the RF technique, the model provided good performance. Moreover, running the GridSearchCV function, we found out the best parameters are max-features of 9, and n-estimators of 40 would give an accuracy score of 1. Moreover, the DT model has a good performance without tuning, and when we run the GridSearchCV function, it provides an accuracy score of 1, a max-depth of 24, and a max-features of 8 would lead to an accuracy score of 1. The GBR model provided good performance in general. Therefore, we did not tune the hyperparameters.

_{rop}when the medium surface is occupied with nanoparticles. Each 2D contour plot and 3D surface plot help to visualize the accuracy score of each two combinations of the selected hyperparameters in a color-coded manner. For the DT model, the two selected hyperparameters are max-depth and max-features. The yellow color code in both plots represents the highest accuracy score of 1 for both combinations when the max depth is 24, and the max features are 9. While for the RF model, the two selected hyperparameters are max-features and n-estimators. It is shown that max-features of 9 and n-estimators of 40 had the highest accuracy of a score of 1.

_{rwp}using DT, GBR, ANN, and RF algorithms. After evaluating all models, we found out that the RF model performed the best compared to other models. We checked the feature’s importance for each model when the target variable was k

_{rwp}. The important features in most of the models are k

_{rw}, k

_{ro}, K, t, C

_{s}

_{1}, C

_{s}

_{2}, S

_{w}, and pc. The score value calculated for each feature is varied from model to model. Figure 12 presents the feature importance of the DT, RF, and GBR models.

_{rwp}using DT, RF, and GBR models. It is shown from the figure that t, S

_{w}

_{_Norm}, C

_{s}

_{1_Norm}, C

_{s}

_{2_Norm}, pc, k

_{rw}, and k

_{ro}, are the most important features in most models.

_{rwp}. The lowest RMSE value is 0.000281, and the highest R

^{2}value is 0.999999. Table 9 presents the metric evaluation results of all models.

_{rwp}for all four models. The scatter plots of all four models have been presented twice; one time without scaling the datasets and the other time with scaling the dataset using the StandardScaler. The DT model shows a minor difference between the scaled dataset and the not scaled dataset. The GBR model, when the dataset was not scaled, had some mispredicted values compared to the scaled dataset. Moreover, the ANN model with a scaled dataset also had some mispredicted values. At the same time, the RF model of the not scaled dataset had the highest correlated values compared to the other models.

## 4. Conclusions

_{rwp}, while the ANN models performed better when the target variables were C, k

_{rop}, and S

_{w}. The features’ importance for each model has been determined, and it has been found that C

_{s}

_{1}, C

_{s}

_{2}, and t are the most important features that help predict the nanoparticle concentration. Moreover, the most commonly important features to predict water saturation are pc, k

_{rw}, k

_{ro}, lam

_{op}, lam

_{wp}, and lam

_{tp}. Moreover, the model parameters are tuned to improve the performance of machine learning models and the hyperparameters, including max features and the number of estimators. The GridSearchCV was used for RF hyperparameter tuning. It was found that the best parameters would give the highest accuracy. For predicting S

_{w}, the results of the non-scaled dataset were very close to the normalized dataset, while the opposite was true for the ANN model. The ANN model with ReLu, sigmoid, and tanh activation functions for the scaled dataset is all accurate with negligible differences.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Nanoparticle behaviors against pore volume. (

**a**) Nanoparticles concentration, (

**b**) Water saturation.

**Figure 3.**Scatter plots of ANN models with different activation functions (

**a**) “ReLu” activation function, (

**b**) “sigmoid” activation function, and (

**c**) “tanh” activation function, for target C. The dashed line represents a full coinciding between actual and predicted data.

**Figure 5.**Actual vs. predicted water saturation using machine learning techniques DT, RF, GBR, and ANN for S

_{w}. (

**a**) DT without scaling, (

**b**) DT with scaling, (

**c**) RF without scaling, (

**d**) RF with scaling, (

**e**) GBR without scaling, (

**g**) GBR with scaling, (

**f**) ANN without scaling with tanh, (

**h**) ANN scaling with tanh. The dashed line represents a full coinciding between actual and predicted data.

**Figure 6.**2D contour and 3D surface plots of hyperparameters tuning for (

**a**) DT, and (

**b**) RF for target S

_{w}.

**Figure 7.**Scatter plots of ANN models with different activation functions (

**a**) “ReLu” activation function, (

**b**) “sigmoid” activation function, and (

**c**) “tanh” activation function, for target S

_{w}. The dashed line represents a full coinciding between actual and predicted data.

**Figure 9.**Actual against predicted k

_{rop}using different machine learning techniques DT, RF, GBR, and ANN. (

**a**) DT without scaling, (

**b**) DT with scaling, (

**c**) RF without scaling, (

**d**) RF with scaling, (

**e**) GBR without scaling, (

**g**) GBR with scaling, (

**f**) ANN without scaling with tanh, (

**h**) ANN scaling with tanh. The dashed line represents a full coinciding between actual and predicted data.

**Figure 10.**2D contour and the 3D surface of hyperparameters tuning for (

**a**) DT, and (

**b**) RF for the target k

_{rop}.

**Figure 11.**ANN prediction with different activation functions (

**a**) “ReLu” activation function, (

**b**) “sigmoid” activation function, and (

**c**) “tanh” activation function, for the target k

_{rop}. The dashed line represents a full coinciding between actual and predicted data.

**Figure 13.**Actual vs. predicted k

_{rwp}using different machine learning techniques DT, RF, GBR, and ANN. (

**a**) DT without scaling, (

**b**) DT with scaling, (

**c**) RF without scaling, (

**d**) RF with scaling, (

**e**) GBR without scaling, (

**g**) GBR with scaling, (

**f**) ANN without scaling with tanh, (

**h**) ANN scaling with tanh. The dashed line represents a full coinciding between actual and predicted data.

**Figure 14.**2D contour and the 3D surface of hyperparameters tuning for (

**a**) DT, and (

**b**) RF for the target k

_{rwp}.

**Figure 15.**ANN predictions with different activation functions (

**a**) “ReLu” activation function, (

**b**) “sigmoid” activation function, and (

**c**) “tanh” activation function, for the target k

_{rwp}. The dashed line represents a full coinciding between actual and predicted data.

Metric | Formula |
---|---|

Mean absolute error (MAE) | $\mathrm{MAE}=\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}|actua{l}_{i}-predicte{d}_{i}|$ |

Mean squared error (MSE) | $\mathrm{MSE}=\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{(actua{l}_{i}-predicte{d}_{i})}^{2}$ |

Root mean squared error (RMSE) | $\mathrm{RMSE}=\sqrt{\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{(actua{l}_{i}-predicte{d}_{i})}^{2}}$ |

$\mathrm{R}\mathrm{squared}({R}^{2})$ | ${R}^{2}=1-\frac{{\mathrm{MSE}}_{\mathrm{model}}}{{\mathrm{MSE}}_{\mathrm{base}}}$ |

Variable | Count | Mean | Standard Deviation | Minimum Value | 25% | 50% | 75% | Maximum Value |
---|---|---|---|---|---|---|---|---|

t | 288,000 | 432,000 | 249,589.0149 | 0 | 216,000 | 432,000 | 648,000 | 864,000 |

x | 288,000 | 0.1 | 0.058025 | 0 | 0.05 | 0.1 | 0.15 | 0.2 |

phi_{0} | 288,000 | 3.00 × 10^{−1} | 5.55 × 10^{−17} | 3.00 × 10^{−1} | 3.00 × 10^{−1} | 3.00 × 10^{−1} | 3.00 × 10^{−1} | 3.00 × 10^{−1} |

phi | 288,000 | 0.99999 | 0.000037 | 0.999733 | 1 | 1 | 1 | 1 |

pc | 288,000 | 4221.978691 | 1598.255742 | −4765.365 | 4765.305428 | 4765.305428 | 4765.305428 | 4765.378608 |

k_{rop} | 288,000 | 9.01 × 10^{−1} | 2.88 × 10^{−1} | 6.86 × 10^{−9} | 1.00 × 10^{0} | 1.00 × 10^{0} | 1.00 × 10^{0} | 1 |

k_{rwp} | 288,000 | 9.13 × 10^{−1} | 2.55 × 10^{−1} | 1.55 × 10^{−8} | 1 | 1 | 1 | 1 |

S_{w} | 288,000 | 0.07101 | 0.185966 | 0.010101 | 0.010101 | 0.010101 | 0.010101 | 1 |

C_{s}_{1} | 288,000 | 0.392916 | 0.318226 | 0 | 0.08322 | 0.350227 | 0.669575 | 1 |

C_{s}_{2} | 288,000 | 0.392916 | 0.318226 | 0 | 0.08322 | 0.350227 | 0.669575 | 1 |

C | 288,000 | 0.778259 | 0.267433 | 0 | 0.682623 | 0.897228 | 0.966575 | 1 |

Metric | RMSE | MSE | MAE | ${\mathit{R}}^{2}$ |
---|---|---|---|---|

DT | 0.018360 | 0.000337 | 0.002237 | 0.995277 |

DT_sc * | 0.018620 | 0.000347 | 0.002308 | 0.995142 |

RF | 0.003747 | 0.000014 | 0.000232 | 0.999803 |

RF_sc * | 0.004788 | 0.000023 | 0.000251 | 0.999679 |

GBR | 0.027639 | 0.000764 | 0.008928 | 0.989297 |

GBR_sc * | 0.001669 | 0.000003 | 0.001128 | 0.999961 |

ANN_ReLu_sc * | 0.000216 | 0.000000 | 0.000202 | 0.999999 |

Metric | ANN (tanh) | ANN (sigmoid) | ANN (ReLu) |
---|---|---|---|

RMSE | 0.000242 | 0.000566 | 0.000216 |

MSE | 0.000000 | 0.000000 | 0.000000 |

MAE | 0.000213 | 0.000446 | 0.000202 |

${R}^{2}$ | 0.999999 | 0.999995 | 0.999999 |

Metric | RMSE | MSE | MAE | ${\mathit{R}}^{2}$ |
---|---|---|---|---|

DT | 0.000235 | 0.000000 | 0.000039 | 0.999998 |

DT_sc * | 0.000237 | 0.000000 | 0.000040 | 0.999998 |

RF | 0.000212 | 0.000000 | 0.000022 | 0.999999 |

RF_sc * | 0.000216 | 0.000000 | 0.000022 | 0.999999 |

GBR | 0.000610 | 0.000000 | 0.000156 | 0.999989 |

GBR_sc * | 0.000625 | 0.000000 | 0.000159 | 0.999989 |

ANN_tanh_sc * | 0.000125 | 0.000000 | 0.000065 | 1.000000 |

Metric | ANN (tanh) | ANN (sigmoid) | ANN (ReLu) |
---|---|---|---|

RMSE | 0.000125 | 0.000209 | 0.000310 |

MSE | 0.000000 | 0.000000 | 0.000000 |

MAE | 0.000065 | 0.000167 | 0.000083 |

${R}^{2}$ | 1.000000 | 0.999999 | 0.999997 |

Metric | RMSE | MSE | MAE | ${\mathit{R}}^{2}$ |
---|---|---|---|---|

DT | 0.000302 | 0.000000 | 0.000041 | 0.999999 |

DT_sc * | 0.000303 | 0.000000 | 0.000041 | 0.999999 |

RF | 0.000281 | 0.000000 | 0.000025 | 0.999999 |

RF_sc * | 0.000280 | 0.000000 | 0.000026 | 0.999999 |

GBR | 0.000442 | 0.000000 | 0.000088 | 0.999998 |

GBR_sc * | 0.000423 | 0.000000 | 0.000087 | 0.999998 |

ANN_tanh_sc * | 0.000182 | 0.000000 | 0.000047 | 1.000000 |

ANN_sigmoid_sc * | 0.000145 | 0.000000 | 0.000115 | 1.000000 |

ANN_ReLu_sc * | 0.000199 | 0.000000 | 0.000174 | 1.000000 |

**Table 8.**Performance evaluation of the ANN model with different activation functions for the target k

_{rop}.

Metric | ANN (tanh) | ANN (sigmoid) | ANN (ReLu) |
---|---|---|---|

RMSE | 0.000182 | 0.000145 | 0.000199 |

MSE | 0.000000 | 0.000000 | 0.000000 |

MAE | 0.000047 | 0.000115 | 0.000174 |

${R}^{2}$ | 1.000000 | 1.000000 | 1.000000 |

Metric | RMSE | MSE | MAE | ${\mathit{R}}^{2}$ |
---|---|---|---|---|

DT | 0.000857 | 0.000001 | 0.000130 | 0.999989 |

DT_sc * | 0.000877 | 0.000001 | 0.000130 | 0.999988 |

RF | 0.000281 | 0.000000 | 0.000025 | 0.999999 |

RF_sc * | 0.000581 | 0.000000 | 0.000053 | 0.999995 |

GBR | 0.007716 | 0.000060 | 0.001552 | 0.999099 |

GBR_sc * | 0.001261 | 0.000002 | 0.000270 | 0.999975 |

ANN_tanh_sc * | 0.026346 | 0.000694 | 0.003855 | 0.989259 |

ANN_sigmoid_sc * | 0.029402 | 0.000864 | 0.005519 | 0.986623 |

ANN_ReLu_sc * | 0.036650 | 0.001343 | 0.005107 | 0.979215 |

**Table 10.**Performance evaluation metrics of the ANN model with different activation functions for the target k

_{rwp}.

Metric | ANN (tanh) | ANN (sigmoid) | ANN (ReLu) |
---|---|---|---|

RMSE | 0.026346 | 0.029402 | 0.036650 |

MSE | 0.000694 | 0.000864 | 0.001343 |

MAE | 0.003855 | 0.005519 | 0.005107 |

${R}^{2}$ | 0.989259 | 0.986623 | 0.979215 |

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**MDPI and ACS Style**

El-Amin, M.F.; Alwated, B.; Hoteit, H.A. Machine Learning Prediction of Nanoparticle Transport with Two-Phase Flow in Porous Media. *Energies* **2023**, *16*, 678.
https://doi.org/10.3390/en16020678

**AMA Style**

El-Amin MF, Alwated B, Hoteit HA. Machine Learning Prediction of Nanoparticle Transport with Two-Phase Flow in Porous Media. *Energies*. 2023; 16(2):678.
https://doi.org/10.3390/en16020678

**Chicago/Turabian Style**

El-Amin, Mohamed F., Budoor Alwated, and Hussein A. Hoteit. 2023. "Machine Learning Prediction of Nanoparticle Transport with Two-Phase Flow in Porous Media" *Energies* 16, no. 2: 678.
https://doi.org/10.3390/en16020678