Prediction of the equivalent circulation density using machine learning algorithms based on real-time data

Abbreviations: AARE: absolute average relative error; AI: artificial intelligence; ANFIS: adaptive network-based fuzzy interference system; ANN: artificial neural networks; ARE: average relative error; ECD: equivalent circulation density; EM: equivalent mud weight; FIS: fuzzy inference system; GA: genetic algorithm; LLSVM: least square support vector machine; Mw: mud weight; PSO: particle swarm optimization; PV: plastic viscosity; Q: mud flow rate; R²: correlation coefficient; RD: relative deviation; RMSE: root-mean-squared error; ROP: rate of penetration; RPM: total downhole rotating speed in revolutions per minute; SD: standard deviation; SPP: standpipe pressure; TFA: total flow area; TVD: true vertical depth; WOB: weight on bit; YP: yield point

1.   Introduction

Nowadays, the process of exploring, drilling, producing, and managing reservoirs for gas and oil is a complex task due to the fact that traditional sources of these resources have already been identified and utilized for numerous years. Petroleum engineers are currently striving to adopt sophisticated technologies like machine learning to make informed decisions that minimize unproductive periods and expenses. Machine learning (ML) has been applied to various studies, primarily from 1993 by Hansen ^[1], where they used artificial neural networks (ANN) to successfully identify primary reflections from seismic data. In 2007, Kononov et al. ^[2] developed ANNs to compute travel times for a complete 3D volume model, and petrophysical and geomechanical properties of shale reservoirs. Fahd Syed et al. ^[3] presented a comprehensive review of artificial intelligence (AI) and ML applications in modeling such properties, as well as a systematic publication count that reflected the increasing interest in this subject.

ML has been utilized in drilling engineering for a long time. In 1990, Arehart ^[4] used ANNs to determine the grade (state of wear) of a drill bit while drilling. ANNs were applied by Moran ^[5] to predict the rate of penetration (ROP) in drilling operations, enabling a more accurate estimation of drilling time., In 2022 Sprunger et al.. ^[6] discussed the benefits of implementing machines in hydraulic fracturing operations in shale gas reservoirs. This study covered the design, interpretation, real-time prediction, and re-frac selection of hydraulic fracturing and concluded that machine learning is an accurate method for dealing with a large amount of fracturing data.

Additionally, ML has been previously applied in petroleum production engineering. Denney ^[7] utilized ANNs to pinpoint the potential restimulation locations for fractures based on the experience gained from the Red Oak oilfield. This paper ^[8] detailed the use of AI and ML for artificial lift selection and maintenance, equ ent malfunction detection. Additionally, it explained the workflow and effectiveness of each application.

There are many applications of ML in reservoir engineering. For example, in 1996, Yang and Kim ^[9] developed ANNs to find rock properties. Kohli & Arora ^[10] developed an ANN model to predict permeability from well logs. Fahd Syed et al. ^[11] discussed recent developments in the production performance estimation of shale gas in North America using machine learning, including key input parameters and methodology comparisons of different algorithms. In ^[12], the performance of shale gas reservoirs was evaluated, alongside understanding how artificial intelligence (AI) and machine learning (ML) were used for this purpose. The text also explained different ML approaches such as supervised and unsupervised methods, random forest, support vector machine, boosting techniques, clustering methods, and artificial network-based architecture. Moreover, Taha et al. ^{[13, 14, 15]} efficiently used ML algorithms to detect and remove noise from the shale gas production data without removing the main trends of the data profile. The results of these works helped improve the forecasting of the production, and can be applied to any other data.

Equivalent circulation density (ECD) is the total hydrostatic pressure of the mud in dynamic conditions. It displays the annular losses in a mud column ^[16]. ECD is one of the most important parameters for drilling operations monitoring, particularly for the little window between the formation and the fracture pressure ^[17]. Three crucial drilling fluid densities include the equivalent mud weight (EMW), ECD, and equivalent drilling fluid static density (ESD), with the first two being particularly significant. Estimating automated or real-time EMW during drilling allows for a quick evaluation for ECD, as well as the volume of the cuttings transported by the drilling fluid, which makes the drilling crew able to track and assess the effectiveness of hole cleaning in real-time. Hole cleaning in drilling operations must be optimized to increase the rate of penetration (ROP) and reduce hole issues. However, achieving optimal hole cleaning needs continuous and proper controlling and development, which are difficult to maintain. Poor hole cleaning can result in stuck pipes, reduced circulation, lower ROP, and increased torque and drag. The duration of the operations may be significantly impacted by these factors, which will ultimately raise the overall cost of drilling operations ^[18].

Drilling mud can detect the hydrostatic column pressure at any drilled depth. Mud density can either balance or unbalance the section of the wellbore. If the used drilling mud weight and ECD were high, overbalanced drilling occurs, which can significantly improve the control during operation ^[19]. On the other hand, several problems are associated with overbalanced drilling, such as a poor drilling rate, generating irregular differential pressure between both EMW and ECD with the formation, which elevates the potential for differential sticking, and increased mud chemical additions resulting in costs increments ^[19]. In addition, it can induce formation damage, particularly in the reservoir area. Hence, there are various advantages for calculating the desired mud weight and ECD overbalance throughout the drilling hole section. The best way to prevent these issues, besides achieving safe well drilling, maximizing drilling efficiency, and improving drilling performance, is understanding drilling fluid density, ECD, drilling fluid treatment, and chemical additives, in addition to continuously monitoring hole cleaning.

Marsh ^[20] described the idea and concept of drilling fluid properties, such as mechanical analysis, weight, consistency, characteristics of desired drilling fluid, treatment of drilling fluid, and chemical factors or additives. It has been found that the annular pressure losses, wellbore geometry, mud characteristics (density and viscosity), mud pumping rate, downhole pressure and temperature, and cuttings concentration have a significant effect on the ECD during the drilling operations ^{[21, 22]}. Ataga and Ogbonna ^[23] mentioned that the ECD and drilling pressure can be obtained through a mathematical model estimation, prediction utilizing artificial intelligence (AI) approaches, and by employing downhole sensors while drilling.

Erge et al. ^[24] and Rommetveit et al. ^[25] showed that although the downhole measurement is thought to be reliable and precise for ECD values, its application is uncommon due to the costly daily rate and operating restrictions, such as downhole pressure and temperature, that lead to tool failures.

There are several mathematical models for estimating the ECD in the literature, each with a different fluid type and set of input parameters. In the traditional models, numerous values and restrictions have been assumed for the downhole pressure, temperature, and mud types. The material balance calculation has been used to estimate ECD for the mud compositional study ^{[26, 27]}. Bybee ^[28] presented a mathematical equation for determining ECD with the mud static density and other mud-related characteristics. The model also takes the effect of solids concentration in the annulus into account.

The mathematical correlations that have been created are restricted to a few features and disregard many other input variables that affect the ECD. These variables include geometry, fluid rheological characteristics, drill string rotation, downhole pressure, temperature, cuttings dispersion, hole cleaning, swab, and the surge of drill pipe movements in the well; other factors have been neglected, but have an impact on the mud density ^{[29, 30]}. Ignoring these factors will have an impact on ECD prediction, resulting in incorrect evaluation of ECD, and causing well control issues while drilling ^[31].

The limitations of the ECD downhole device and the inadequate performance of mathematical correlations have prompted the emergence of a new approach for predicting equivalent circulating density (ECD) from drilling data in the petroleum industry. This approach involves the utilization of ANNs, which comprise the interconnected units or nodes, referred to as artificial neurons. These artificial neurons are designed to receive inputs and generate outputs based on their predetermined activation functions through a preconfigured network, which mimics the neurons found in the biological brain. ANNs have been recognized as one of the most successful artificial intelligence systems to date. The underlying principle of ANNs is rooted in their ability to emulate the learning capability of the nervous system, where the input data or signals received are utilized to generalize findings.

Baranthol et al. ^[32] performed field sampling measurements and validated their measurements using computer models. In ^[33], a correlation using AI and an ANN was created to estimate mud density as a function of the drilling fluid type, various pressures, and different temperatures. In ^[34], a study was conducted on the ECD under extreme heat and pressure to assess the ECD during drilling, in which a temperature profile was created using the CrankNicolson numerical discretizing scheme in the DD-Simulator, which is used to simulate the circulation of the wellbore. Ahmed et al. ^[30] created a model to predict how drill-string rotation will affect ECD. They discovered that the ECD would be more effective with a greater rotation and pump flow rates.

Elzenary et al. ^[35] used AI technologies, such as adaptive neuro-fuzzy inference systems (ANFIS) and ANNs, to find a relationship to predict the ECD while drilling and to compare the measured outcomes with these predictions. Ahmadi ^[36] used the ANFIS, enhanced particle swarm optimization ANFIS (PSO-ANFIS), and least square support vector machine (LLSVM) algorithms to calculate the ECD using only the mud's initial density, temperature, and pressure. Using the initial mud density, pressure, and temperature data, Ahmadi et al.. ^[37] investigated the prediction of ECD using PSO-ANN, fuzzy inference system (FIS), and a hybrid of genetic algorithm (GA) and FIS (GA-FIS). In terms of the R² and average absolute percentage error between the actual and projected values of ECD, the PSO-ANN model demonstrated a good level of prediction ability.

A model for ECD prediction utilizing the two AI techniques, ANN and ANFIS, was presented by Abdelgawad et al. ^[22]. While the ANFIS model was created using five membership functions with a gaussian membership function (gaussmf) as the input membership function and a linear type as the output membership function, the study provided an ECD-ANN model with one hidden layer with 20 neurons.

The radial basis function was used by Rahmati and Tatar ^[38] to create an ECD prediction model that demonstrated a strong prediction performance with an R² of 0.98 and an AAPE of 0.22. They used three different machine learning algorithms to predict the ECD while drilling horizontal sections using surface drilling data.

In Alkinani et al. ^[39], drilling parameters were used in addition to the hydraulics and mud properties, such as mud pumping rate, rheology of the mud (density, plastic viscosity, and yield point), total flow area (TFA) for the bit nozzles, revolutions per minute (RPM) for the drill pipe, and weight on bit (WOB), to predict the ECD using an ANN model that had just one hidden layer with 12 neurons.

Alsaihati et al. ^[40] developed a model using seven drilling parameters as inputs, where the SVM, RF, and FN were used to generate the model. The accuracy of the model that was created varied from an R² of 0.95 to 0.99 and the root mean square error (RMSE) ranged from 0.23 to 0.42. The author of this study did not create any machine learning-free ECD equations that could be employed. Gamal et al. ^[17] used only mechanical drilling parameters, such as mud pumping rate (GPM), ROP, drill string speed in RPM, stand-pipe pressure (SPP), WOB, and drilling torque, to predict the ECD using machine learning approaches such as ANNs and ANFISs, with an R² of 0.98 and an average absolute percentage error (AAPE) of 0.3% for the ANN, while the ANFIS recorded an R² of 0.96 and an AAPE of 0.7.

These outcomes demonstrated that ANN outperformed the other techniques. The literature makes it obvious that the AI models improved the ECD prediction; however, the models differ in terms of the input parameters, the data fed into the models, and the ECD prediction approaches. One of the issues with multiple studies in the literature is the need to utilizedownhole pressure and temperature as inputs to the prediction models. From an operational perspective, downhole sensors are needed to obtain these parameters with high accuracy for a better ECD prediction, and this will increase operational costs and time for the data collection.

The present study introduces a novel approach in which machine learning algorithms are primarily reliant on real-time drilling parameters from downhole sensor tools, including inclination angle, true vertical depth (TVD), annular temperature, mud flow rate (Q), mud weight (Mw), plastic viscosity (PV), yield point (YP), gel strength (0/10; referring to measurements taken after 10 seconds and 10 minutes, respectively), total flow area (TFA), annular pressure, standpipe pressure (SPP), rate of penetration (ROP), and total downhole revolutions per minute (RPM). To this end, a total of 4663 data points, obtained through measurement-while-drilling (MWD) from an offshore deviated well in Norway, were filtered and utilized to develop models for predicting ECD. The performance of these models was compared against each other, and the model demonstrating the highest accuracy underwent several statistical tests and was compared with other developed models ^[36].

2.   Data description, cleaning, and filtration

The study used actual drilling data that were collected by downhole sensors from a deviated section well during a drilling phase in Norway. The strategy that was followed while conducting this study is depicted in Figure 1. The input data selected based on experts' are shown in Figure 2, which represents inclination angle, annular temperature, annular pressure, TVD, Q, MW, PV, YP, gel (0/10), TFA, SPP, ROP, and RPM.

Key performance indicators (KPIs) have been determined by correlation analysis, as shown in Figure 3 and Table 1. To verify collinearity between the variables, scatterplots of each independent variable were created. According to the heat map, it is clear that some factors such as SSP, TFA, mud flow, and Mw have a variable direct correlation with the ECD, while some of the other parameters (PV, gel 10, inclination, annular pressure, gel 0, hole depth and, RPM) show an inverse correlation with the ECD; alternatively, annular temperature, YP, and ROP show a weak correlation with ECD.

Cleaning and filtering the data produces high-quality input parameters for the model. Therefore, a lot of effort has been done to clean the data, develop the AI model's training procedure, and maximize the correctness of the model results using the trained algorithm. If the accuracy is poor, then a process of retraining should be carried out in order to improve the accuracy and obtain the ideal model parameters for the ECD prediction's performance. Data cleaning is the process of locating and fixing problems with the data, such as data losses, corruptions, and errors, through employing statistical techniques like outlier detection and imputation. One of the cleaning processes that the model applied is the exclusion of the point if all the input parameters do not exist. Figures 4 and 5 show the preprocessed (filtration and removing) data for the inclination angle versus TVD depth before and after processing as an example of the cleaning process which has been done for the input parameters.

3.   Statistical analysis for parameters

The statistics features are crucial to the theoretical, practical, and long-term development of machine learning. These features can be used in problem framing, data comprehension, preparation, model evaluation, configuration, model selection, model presentation, and model predictions. Tables 2 and 3 show the statistical parameters of the whole data set. The first and second rows show the maximums and minimums of the data set, which shows the range of the selected data. The standard deviation is a measurement of dispersion, or how the data are distributed around the mean. The degree to which the data deviates from the normal distribution is measured by skewness. Any skewness is generally undesirable in models since it leads to a variance in estimation.

4.   Models development

Three models have been developed by using three machine learning algorithms. The first model used the passive aggressive regressor (par) algorithm. For the second model, the K neighbors regressor (knn) algorithm has been applied. The final model adopted the ANN with a Levenberg-Marquardt backpropagation algorithm.

4.1.   Passive aggressive regressor algorithm

This model was developed using a Python code. The total data points have been split into 80% for training and validation and 20% for the testing model. The selected sets of data were changed in an alternating pattern ten times, and the result of these folds is shown in Table 4.

Figure 6 shows the amount of loss in the value of residuals between the predicted ECD and the actual value of ECD. Notice that the residual values between the real and expected values by this model are very large. In addition, the accuracy of this model is very weak and unreliable, with a flawed value for the R², which is close to 25%, as shown in Figure 7.

4.2.   K Neighbors regressor algorithm

This model was also built using a Python script. The total data points have been divided into 80% for training and validation and 20% for testing. The selected sets of data were alternatively altered ten times, as shown in Table 5.

According to Figure 8, it is clear that the residual values between the predicted and actual ECD is centered around the values ranges from 0.05 to –0.05 (i.e., very small), and that this model can be trusted in the prediction for ECD.

Additionally, Figure 9 shows a high value of the R² for the values of ECD expected by this model.

4.3.   Artificial neural network (ANN)

This model was created using a MATLAB script. A total of 4, 663 points were obtained after data cleaning and filtering. The input data set is a function of the depth, which is a representation of the true vertical well depth. If either the input or output data for neural networks are very small, very large, or not normally distributed, data scaling should be performed ^{[41, 42]}. Therefore, the input parameters were normalized using Eq 1 to be between 0 and 1, while the output parameter (ECD) was transferred according to Eq 2, for a faster and more efficient training process.

where $i$ is the index number for each input parameter: inclination angle, annular temperature, annular pressure, TVD, Q, MW, PV, YP, gel (0/10), TFA, SPP, ROP, and RPM.

where, $y$ is the value of equivalent circulation density and $Yt$ is the transformed value of the equivalent circulation.

4.3.1.   Building the model

The total dataset was randomly partitioned into three subsets: 70% for training, 15% for validation, and 15% for testing purposes. The training data were utilized to develop the ANN model, with the target output being used to adjust the weights of each input. The calibration of these weights was performed using the feedforward backpropagation algorithm, which utilizes error backpropagation through the network to adjust and fine-tune the weights. It should be noted that the occurrence of overtraining (either overfitting or underfitting) in the model can diminish its ability to provide accurate estimations when applied to diverse conditions. Therefore, the training process is halted when the network's generalization reaches a certain threshold, and the verification set (data not employed to construct the network) is employed to assess the model's generalization capabilities. Subsequently, the accuracy of the ANN model is evaluated using the testing set. The architecture of the ANN model is illustrated in Figure 10.

A MATLAB script was created to go over 3, 840 possible combinations of different parameters (transfer function, training algorithm, and the number of neurons in the hidden layer(s)) as shown in Table 6 for one hidden layer network, and developed for two hidden layer networks to reach more than 230, 000 possible combinations with different parameters, referred to in Table 7.

The approach adopted in developing the model is depicted in Figure 11, which commences with a basic model comprising of a single hidden layer. The number of neurons is initially incremented, followed by the modification of the training algorithm, activation function of the hidden layer, and ultimately the activation function of the output layer. Subsequently, all possible combinations of the single hidden layer are explored, following which, the model is upgraded to two hidden layers, with modifications made to the number of neurons in the first hidden layer, the number of neurons in the second hidden layer, the training algorithm, activation function of the first hidden layer, the activation function of the second hidden layer, and the activation function of the output layer, in that order. Each combination is evaluated using the RMSE (Eq 3) and the R² (Eq 4). The combination yielding the lowest RMSE and highest R² was selected. The model and algorithm were then constructed based on this process, with the evaluation continuing until a satisfactory level of RMSE and R² was attained (i.e., R² overall > 0.9995 and RMSE < 0.0005). The program is designed to minimize the time required to construct and train the model.

where $N$ is the number of data points tested, ${E}^{Predicted}$ is the predicted equivalent circulation density, ${E}^{actual}$ is the corresponding to the actual ECD, and ${E}^{meanactual}$ is the average of the actual ECD.

4.3.2.   Results and discussion

The ANN model was created to predict the ECD according to Eq 6 with one hidden layer. The number of neurons in this hidden layer was optimized to be 34 according to Figures 12 and 13. The structure of this model is shown in Figure 14. The Levenberg-Marquardt backpropagation optimization was used as a training algorithm. TANSIG and LOGSIG were selected to be the hidden and output layers activation functions, respectively. This model displayedhigh accuracy and certainty when compared with the other two developed models.

where $Exp$ is the exponential function, $j$ is the number of neurons in the hidden layer, ${W}_{2j}$ is the weight of the resulting layer, $k$ is the number of input parameters, $W1$ is the weight of the hidden layer, $Xj$ is the standard input parameters (inclination angle, TVD, annular temperature, Q, MW, PV, YP, gel (0/10), TFA, annular pressure, SPP, ROP, and RPM), $b1$ is the bias of the hidden layer, and $b2$ is the bias of the resulting layer. The weights and bias values are illustrated in Appendix A.

The results showed a strong accuracy for the model in terms of ${\mathrm{R}}^{2}$, $RMS$, and loss function (Eq 5) for the training validation, testing, and overall, as shown in Table 8 and Figures 15–17.

Figure 18 represents the cross-plots for the ANN model results for the model training, validation, testing, and overall processes for estimating ECD values. The plots reveal that there is no unacceptable overestimation or underestimation of the anticipated value. Furthermore, they demonstrated the models' remarkable accuracy performance of predictions.

Figure 19 displays the agreement between the actual and predicted ECD values along the number of data points which have been used in the overall, training, validation, and testing of the ANN model.

4.3.3.   Statistical analysis for model results

To test the model, several statistical comparisons between the actual and anticipated data have been made. The absolute average relative error (AARE), standard deviation (SD), average relative error (ARE), relative deviation (RD), and mean square error (MSE) are used to evaluate the model based on the following Eqs (7–11). Table 9 and Figures 20–24 show the corresponding values of those statistical features.

4.3.4.   Comparison with other models

When compared to the other models, the developed ANN model has promising results. The comparison was based on the input parameters feeding the models, the number of data points used for creating these models, and the sources of this data. Additionally, this comparison takes into account the types of ML algorithms that were used to develop models. Furthermore, the results of several statistical tests were used to evaluate the accuracy of models. The developed model shows a high accuracy when compared to other models. This comparison is shown in Table 10.

5.   Conclusions

An ANN model was developed using MATLAB to predict the ECD based on a large dataset of data points obtained from downhole sensors during drilling operations. Preprocessing operations such as data cleaning, filtering, normalization, and transformation were carried out to prepare the data for training, validation, and testing of the model. Statistical tests were conducted to evaluate the performance of the developed model.

The statistical tests conducted include R², RMSE, ARE, AARE, SD, and MSE for the training, validation, testing, and overall datasets. The results of the tests indicate that the developed model does not suffer from neither underfitting nor overfitting. Moreover, the predicted ECD values from the developed model agrees well with the actual ECD values.

To further evaluate the performance of the developed model, the trend of the predicted data and actual data points along the number of the datasets was also analyzed. Additionally, the overall model performance was compared to other models based on input parameters, type of machine learning algorithm, number of data points, and results of statistical tests used to evaluate these models. The developed model showed a higher accuracy in predicting the ECD than other models considered.

In conclusion, the developed ANN model shows a promising solution for the optimization and automation of the drilling process by directly predicting the ECD. This can lead to minimizing hazards and reducing risks associated with drilling operations.

6.   Future work recomendation

Future research should investigate the performance of the developed model on datasets from different wells in various regions to confirm its capability to accurately predict ECD under different circumstances. In case of any deficiencies in the model's predictive ability, the model can be updated by including data from multiple wells located in different places to improve the training process and enhance its predictive capability.

Moreover, a parametric study can be conducted to examine the significance of each input element used to feed the model and to determine the effects of each input on the model's ability to predict the true value of ECD. This will help in understanding the relative importance of each input variable and may aid in improving the accuracy of the model predictions.

Overall, these future investigations will further enhance our understanding of the developed model and enable its application to a wider range of drilling operations, leading to an improved optimization and automation of the drilling process with reduced hazards and risks.

Conflict of interest

The authors declare that there is no conflict of interest in publishing this paper.

Appendix