# A comprehensive modeling in predicting the effect of various nanoparticles on filtration volume of water-based drilling fluids

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## Abstract

Filtration volume of drilling fluid is directly associated with the amount of formation damage in hydrocarbon reservoirs. Many different additives are added to the drilling fluid in order to minimize the filtration volume. Nanoparticles have been utilized recently to improve the filtration properties of drilling fluids. Up to now, no model has yet been presented to investigate the effect of nanoparticles on filtration properties of drilling fluids. The impact of various nanoparticles is investigated in this study. Artificial neural network is used as a powerful tool to develop a novel approach to predict the effect of various nanoparticles on filtration volume. Model evaluation is performed by calculating the statistical parameters. The obtained results by the model and the experimental results are in an excellent agreement with average absolute relative error of 2.6636%, correlation coefficient (*R*^{2}) of 0.9928, and mean square error of 0.4797 for overall data. The statistical results showed that the proposed model is able to predict the amount of filtration volume with high precision. Furthermore, the sensitivity analysis on the input parameters demonstrated that nanoparticle concentration has the highest effect on filtration volume and should be considered by researchers during process optimization.

## Keywords

Drilling fluid Filtration volume Nanoparticles ANN Modeling## Introduction

The first stage in the petroleum industry to access a hydrocarbon reservoir is the drilling operation. During drilling operation, drilling fluid is the most crucial and important components, which is usually called drilling mud. Among all types of drilling fluid (water-based, oil-based, and synthetic-based), water-based drilling fluids are the most widely used fluids in the drilling industry in comparison with the other types that is due to their higher cost, environmental issues, disposal problems, and health and safety issues (Mahmoud et al. 2016). Drilling fluids are responsible to carry out many different functions during the drilling process as controlling the formation hydrostatic pressure, preventing collapse of wellbore surface, transferring cuttings to the surface, suspending cuttings and additives, preventing formation damage by forming good mudcake on the wellbore surface, lubricating and cooling the bit and drilling string, and so on. Therefore, these fluids should be designed and manufactured well in such a specific characterization in order to handle the above functions. To modify or enhance a specific function or property, some additives are also added to the drilling fluid. The success of a drilling operation is closely related to the design and performance of the drilling fluid (Mahmoud et al. 2016). In order to minimize the encountered problems and to have low-cost drilling, the functions of drilling fluid must be optimized by improving the drilling fluid properties (Salih and Bilgesu 2017). Most of the problems encountered during the drilling operation such as stuck pipe, high torque and drag, surge and swab pressures, bit balling, shale swelling, and loss circulation are because of improper drilling fluid design (Bourgoyne et al. 1986) that are directly or indirectly related to hydraulic, rheological, and filtration properties of the drilling fluid (Salih and Bilgesu 2017). Therefore, one of the integral and essential tasks is to monitor and control the drilling fluid properties (Vryzas et al. 2018) that should be scheduled in the predefined drilling program. The effect of additives and also human must be added to the usual problems that are encountered during the drilling process that affect the drilling fluid properties.

Among the mentioned problems, one of the most crucial one is drilling fluid loss into the formation. Mitigating and controlling the fluid loss is essential to reduce both the formation damage and the cost of drilling fluid. So many fluid loss agents are added to the drilling fluid as fluid loss additives (Vryzas et al. 2016). The function of these additives is to make a filter cake on the wellbore surface in order to mitigate or prevent fluid loss into the formation, which is usually called mudcake (Barry et al. 2015). For this purpose, fluid loss agents must be optimized to have a homogeneous, thin, and low permeable mudcake. Whether the mudcake is thick or highly permeable, it would increase the probability of pipe sticking or the filtration volume of the drilling fluid, respectively.

By the arrival of nanotechnology, the special characteristics of nanoparticles have made them a potential additives in several fields of the sciences in order to mitigate, minimize, or solve the encountered problems, and moreover to improve the characteristic of a material or a process. Over the time, nanoparticles have also got involved in various fields of oil and gas industry. Recently, various nanoparticles are utilized by many researchers to improve the filtration properties of the water-based drilling fluids while maintaining the other drilling fluid properties optimal or to improve them.

The effect of silica (SiO_{2}) (Mahmoud et al. 2016, 2018; Salih and Bilgesu 2017; Parizad et al. 2018; Smith et al. 2018; Salih et al. 2016; Vryzas et al. 2015; Yusof and Hanafi 2015), ferric oxide (Fe_{2}O_{3}) (Mahmoud et al. 2016, 2017, 2018; Vryzas et al. 2015; Jung et al. 2011; Shakib et al. 2016), aluminum oxide (Al_{2}O_{3}) (Salih and Bilgesu 2017; Smith et al. 2018; Shakib et al. 2016), titanium oxide (TiO_{2}) (Salih and Bilgesu 2017; Shakib et al. 2016), copper oxide (CuO) (Shakib et al. 2016), stannic oxide (SnO_{2}) (Parizad and Shahbazi 2016), ferrosoferric oxide (Fe_{3}O_{4}) (Vryzas et al. 2016, 2018), and clay minerals (Shakib et al. 2016; Needaa et al. 2016) nanoparticles on filtration properties of water-based drilling fluid is investigated. The other types of nanoparticles as multi-walled carbon nanotube (MWCNT) (Ismail et al. 2014, 2016), iron oxide/clay hybrid (ICH) (Barry et al. 2015), aluminosilicate/clay hybrid (ASCH) (Barry et al. 2015), titanium oxide/polyacrylamide nanocomposite (Sadeghalvaad and Sabbaghi 2015), clay/silica nanocomposite (Cheraghian et al. 2018), nano-sized layered magnesium aluminum silicate (MAS) (Wang et al. 2018) are also utilized for this purpose. Also, the effects of nanoparticles along with the polymers, surfactants, and polymer–surfactant are dealt with in some other works (Fakoya and Shah 2018; Srivatsa and Ziaja 2011; Ahmad et al. 2017). The above investigations have showed that the addition of nanoparticles could improve the filtration properties of drilling fluid (Mahmoud et al. 2016, 2017, 2018; Salih and Bilgesu 2017; Vryzas et al. 2015, 2016, 2018; Barry et al. 2015; Parizad et al. 2018; Smith et al. 2018; Salih et al. 2016; Yusof and Hanafi 2015; Jung et al. 2011; Shakib et al. 2016; Parizad and Shahbazi 2016) and consequently could mitigate the formation damage.

In recent years, by the development of technology, artificial intelligence tools have widely applied in order to model nonlinear problems in various fields of science. Artificial neural network (ANN) is one of the most common tools that are utilized. Predicting the filtration properties of drilling fluids is also investigated in some recent researches in the absence of nanoparticles (Jeirani and Mohebbi 2006); however, no model is presented to see the effect of nanoparticles on filtration properties of drilling fluids yet.

In this study, the influence of various nanoparticles on filtration volume is considered by applying an ANN model as a novel method. For this purpose, a total of 1003 data points are gathered from the most recent researches found in the literature. Then, the ANN model is developed for accurate prediction of filtration volume of water-based drilling fluid as a function of effective parameters such as nanoparticle type, nanoparticle concentration, KCl salt concentration, temperature, pressure, round per minute (RPM), and time. The assessment of the proposed model is evaluated by statistical analyses. Finally, a sensitivity analysis is performed to determine the most sensitive parameters on the filtration volume.

## Preview of artificial neural network

*O*

_{i}is the output of

*i*th neuron,

*Y*

_{i}is the

*i*th input to the function,

*σ*is transfer function,

*N*is the number of neurons in each layer,

*w*

_{ji}is the weights that connects the

*i*th neuron to the other neurons of the next layer for

*j*th input parameter,

*P*

_{j}is the

*j*th input parameters, and

*b*

_{i}is the bias of the

*i*th neuron.

*i*th neuron and

*j*th input parameter,

*MC*is the momentum constant, \(dw_{{ij{\text{prev}}}}\) is the previous weight change for

*i*th neuron and

*j*th input parameter,

*LR*is the learning rate, and

*gw*

_{ji}is the weight gradient descent.

## Methodology

### Data collection

In this study, a data collection is used from the literature that relates the filtration volume of water-based drilling fluid to the effective input parameters. As said before, various nanoparticles are used by researchers in order to improve the drilling fluid properties. The data bank is gathered from those researches that the effect of nanoparticles on the filtration volume of water-based drilling fluid has been investigated. Some of the previous studies have also presented the effect of nanoparticles simultaneously with various polymers and surfactants. Such studies have not contributed in this research in order to model and investigate the effect of nanoparticles solely. Moreover, it is necessary to use dependable experimental results to construct a reliable network.

_{2}(Parizad et al. 2018), Fe

_{2}O

_{3}(Vryzas et al. 2015; Mahmoud et al. 2017), CuO (Shakib et al. 2016), Al

_{2}O

_{3}(Shakib et al. 2016), SnO2 (Parizad and Shahbazi 2016), and Clay (Shakib et al. 2016). The experimental data in these studies have included the impact of nanoparticle type, nanoparticle concentration (wt%), KCl salt concentration (wt%), temperature (°F), pressure (psi), RPM (round/min), and time (s) on filtration volume (ml) of water-based drilling fluid. The input parameter named nanoparticle type is considered as one of the input parameters to see the effect of different nanoparticles. A total number of 1003 data points are used for predicting model, which is described later in this paper. These data points are randomly divided into two categories that are named training data and testing data. In order to check the performance of the model in predicting the target, these two categories must be apart from each other and do not have any points in common. For this purpose, the training data consist of about 80% of the main data points that are 803 data points, and the remaining 20% of the main data points are used as testing data that are 200 data points. The statistical description of the data bank used in this study is given in Table 1.

Statistical description of the data bank used in this study

Parameter | Minimum | Maximum | Average | SD |
---|---|---|---|---|

Nanoparticle type | SiO | |||

Nanoparticle concentration (wt%) | 0 | 7.5 | 0.807527 | 1.281187 |

KCl salt concentration (wt%) | 0 | 6 | 1.857428 | 2.6723 |

Temperature (°F) | 77 | 300 | 166.193 | 77.85525 |

Pressure (psi) | 100 | 500 | 314.2572 | 140.8071 |

RPM (round/min) | 0 | 100 | 6.979063 | 25.47938 |

Time (s) | 0 | 1800 | 683.0076 | 608.018 |

Filtration volume (ml) | 1 | 65 | 8.955189 | 8.168526 |

### Model construction

*R*

^{2}. Trial and error process is done for several times to find the optimal network. The best network is found with 3 layers (2 hidden layers and output layer) that the number of neurons is 21, 9, and 1 in first, second, and third layer, respectively. The workflow of model construction, model training, and sensitivity analysis (which is described later) is represented in Fig. 2. Training parameters of the proposed model are given in Table 2.

Training parameters of the proposed model

Parameter | Description |
---|---|

No. of total data | 1003 |

No. of training data | 803 |

No. of testing data | 200 |

Network type | Feed-forward back-propagation |

Training function | TRAINLM |

Adaption learning function | LEARNGDM |

No. of total layers | 3 |

No. of hidden layers | 2 |

Transfer function (hidden layers) | TANSIG |

Transfer function (output layer) | PURELIN |

Max. epochs | 1000 |

### Error assessment

*R*

^{2}). As the criterion that is used in finding the optimal network, mean-squared error (MSE) is calculated as one of the most commonly used parameters. Also some other statistical parameters are calculated including root-mean-squared error (RMSE) that is the root of MSE; average relative error (ARE) that is the average value of RE; average absolute relative error (AARE) that is the absolute value of ARE; relative deviation (RE) that is the relative difference between actual and predicted values; and standard deviation (SD) between the actual and predicted data. Equations 5 to 11 express the definitions of the above parameters:

- 1.Correlation coefficient (
*R*^{2}):$$R^{2} = 1 - \frac{{\mathop \sum \nolimits_{i = 1}^{N} \left( {Y_{i}^{pr} - Y_{i}^{ac} } \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{N} \left( {\overline{{Y^{ac} }} - Y_{i}^{ac} } \right)^{2} }}$$(5) - 2.Mean-squared error (MSE):$$MSE = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left( {Y_{i}^{pr} - Y_{i}^{ac} } \right)^{2}$$(6)
- 3.Root-mean-squared error (RMSE):$$RMSE = \left[ {\frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left( {Y_{i}^{pr} - Y_{i}^{ac} } \right)^{2} } \right]^{0.5}$$(7)
- 4.Average relative error (ARE):$$ARE\left( \% \right) = \left( {\frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \frac{{Y_{i}^{pr} - Y_{i}^{ac} }}{{Y_{i}^{ac} }}} \right) \times 100$$(8)
- 5.Average absolute relative error (AARE):$$AARE\left( \% \right) = \left( {\frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left| {\frac{{Y_{i}^{pr} - Y_{i}^{ac} }}{{Y_{i}^{ac} }}} \right|} \right) \times 100$$(9)
- 6.Relative deviation (RD):$$RD\left( \% \right) = \frac{{Y_{i}^{pr} - Y_{i}^{ac} }}{{Y_{i}^{ac} }} \times 100$$(10)
- 7.Standard deviation (SD):$$SD = \left[ {\frac{1}{N - 1}\mathop \sum \limits_{i = 1}^{N} \left( {\frac{{Y_{i}^{pr} - Y_{i}^{ac} }}{{Y_{i}^{ac} }}} \right)^{2} } \right]^{0.5}$$(11)

In the above equations, *N* is the number of data; \(Y_{i}^{ac}\) is the actual data; \(Y_{i}^{pr}\) is the predicted data; and \(\overline{{Y^{ac} }}\) is the average of the actual data.

### Sensitivity analysis

The concepts of sensitivity and uncertainty analysis are one of the essential parts of modeling and statistical studies. Sensitivity analysis determines the contribution of each input parameter to the output parameters of a model or a data series. In other words, sensitivity analysis quantifies how the input parameters would influence the output parameters. Also, it is defined as the contribution of each input parameter to the uncertainty in model predictions (Hammonds et al. 1994).Uncertainty analysis determines the uncertainty in each input parameter in model predictions (Hammonds et al. 1994).

Sensitivity analysis methods could be categorized as two points of view. Based on the concept, sensitivity analysis methods are categorized as deterministic and statistic (Zhou 2014). Statistical methods dealt with sensitivity analysis after the uncertainty analysis stage, while deterministic methods dealt with sensitivity analysis first and then go through the uncertainty analysis (Cacuci et al. 2005). Based on the factor space of interest, sensitivity analysis methods are categorized as local and global (Yang 2011). Local methods investigate the model behavior by varying each input parameter at a time (Zhou 2014). Global methods investigate the model behavior by varying all parameters over their ranges simultaneously (Zhou 2014).

Among the various approaches for sensitivity analysis, Monte Carlo method is one of the most utilized methods in several fields of engineering. In this method, random samples of the input parameters are generated by various distribution sampling types over the ranges of parameters. Then, it repeatedly simulates the model several times that each time a different set of generated parameters of the distribution is utilized. The output of the model is the probability of output results (Bonate 2001). Monte Carlo method provides an effective approach to assess the influence of several interacting parameters that could exhibit a wide range of uncertainties (Santoso et al. 2019). The generated runs by this method may not be representative of the full-physics models, and it requires an excessive number of runs (Santoso et al. 2019). In order to reduce the number of runs and develop an efficient model, design of experiment (DoE) is developed (Santoso et al. 2019).

*P*

_{i}is the

*i*th input parameter,

*P*

_{i,j}is the

*j*th value of the

*i*th input parameter, \(\overline{{P_{i} }}\) is the average of

*i*th input, \(Y_{j}^{pr}\) is the

*j*th predicted filtration volume,

*N*is the number of overall data, and \(\overline{{Y^{pr} }}\) is the average of the predicted filtration volume.

The results of the sensitivity analysis are given in the Results and Discussion section.

## Results and discussion

*y*=

*x*line is plotted in this figure to evaluate the precision of the proposed model. The precision of the model is determined via the tight accumulation of data points around the

*y*=

*x*line. The amount of this precision is usually measured by correlation coefficient. The correlation coefficient is calculated by fitting the best line that passes through the data, which has the lowest amount of this coefficient between all the other lines that could pass from the data.

*R*

^{2}for training, testing, and overall data in this figure are 0.9938, 0.9904, and 0.9928, respectively. These values exhibit the efficiency of the employed ANN model to predict the filtration volume. The other statistical parameters are represented in Table 3. The table shows that the values of MSE, AARE, and SD for overall data are 0.4797, 2.6636, and 0.0585, respectively. It is clear from the table that the proposed model has the high performance and could estimate and predict the filtration volume precisely.

Statistical parameters of the proposed ANN model

Dataset | | MSE | RMSE | ARE (%) | AARE (%) | SD |
---|---|---|---|---|---|---|

Training data | 0.9938 | 0.3715 | 0.6095 | 0.0672 | 2.4713 | 0.0544 |

Testing data | 0.9904 | 0.9140 | 0.9560 | 0.4126 | 3.4356 | 0.0728 |

Overall data | 0.9928 | 0.4797 | 0.6926 | 0.1361 | 2.6636 | 0.0585 |

Statistical values of RD (%)

Dataset | Minimum | Maximum | Average |
---|---|---|---|

Training data | − 0.9401 | 0.2150 | − 6.7248e−04 |

Testing data | − 0.5444 | 0.3582 | − 0.0041 |

Overall data | − 0.9401 | 0.3582 | − 0.0014 |

## Conclusions

In this study, an ANN model is developed to predict the effect of nanoparticles on filtration volume of water-based drilling fluids. For this purpose, 1003 data points are gathered from the most recent researches. Influencing parameters including in this data bank are nanoparticle type, nanoparticle concentration, KCl salt concentration, temperature, pressure, RPM, and time. Nanoparticles that are engaged are SiO_{2}, Fe_{2}O_{3}, CuO, Al_{2}O_{3}, SnO_{2}, and clay. The superiority of the model confirms by evaluating the statistical analyses with AARE, *R*^{2}, and MSE values of 2.6636%, 0.9928, and 0.4797, respectively, for total data. Moreover, sensitivity analysis showed filtration volume is more sensitive to nanoparticle concentration and this parameter should be considered by researchers during process optimization. The proposed ANN model is capable and efficient and comprehensive in predicting the influence of various nanoparticles on filtration volume of water-based drilling fluids.

## Notes

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