Prediction of the Rate of Penetration while Drilling Horizontal Carbonate Reservoirs Using the Self-Adaptive Artificial Neural Networks Technique

Abstract

Rate of penetration (ROP) is one of the most important drilling parameters for optimizing the cost of drilling hydrocarbon wells. In this study, a new empirical correlation based on an optimized artificial neural network (ANN) model was developed to predict ROP alongside horizontal drilling of carbonate reservoirs as a function of drilling parameters, such as rotation speed, torque, and weight-on-bit, combined with conventional well logs, including gamma-ray, deep resistivity, and formation bulk density. The ANN model was trained using 3000 data points collected from Well-A and optimized using the self-adaptive differential evolution (SaDE) algorithm. The optimized ANN model predicted ROP for the training dataset with an average absolute percentage error (AAPE) of 5.12% and a correlation coefficient (R) of 0.960. A new empirical correlation for ROP was developed based on the weights and biases of the optimized ANN model. The developed correlation was tested on another dataset collected from Well-A, where it predicted ROP with AAPE and R values of 5.80% and 0.951, respectively. The developed correlation was then validated using unseen data collected from Well-B, where it predicted ROP with an AAPE of 5.29% and a high R of 0.956. The ANN-based correlation outperformed all previous correlations of ROP estimation that were developed based on linear regression, including a recent model developed by Osgouei that predicted the ROP for the validation data with a high AAPE of 14.60% and a low R of 0.629.

1. Introduction

The total cost of drilling a hydrocarbon well is time-dependent [1]. Rig time, which is affected by many factors, such as rate of penetration (ROP), is considered the most critical parameter for determining the total cost of drilling. Optimizing ROP has a significant impact on reducing the total cost [2].

ROP is affected by several parameters, which can be categorized into controllable and uncontrollable parameters [3]. The controllable parameters include weight-on-bit (WOB), rotation speed (RPM), pumping rate (GPM), torque (T), and standpipe pressure (SPP) [4,5]. All abbreviations are listed in Appendix A. The uncontrollable parameters include bit size and drilling fluid type, density, and rheological properties. The uncontrollable parameters affect each other, which complicates the quantification of their effect on ROP [6].

Several models have been generated to predict ROP, bue the accuracy of these models varies due to variation in the drilling parameters considered in each model, which considerably limits their applicability [7,8]. The main issue is how the drilling parameters control and affect the ROP [9]. Therefore, understanding the drilling data behavior is considered to be a key factor in the generation of a good ROP prediction model.

ROP prediction models can be classified into two categories, i.e., traditional models using empirical correlations and data-driven models [10]. The traditional models are mathematical functions based on linear regression, while the data-driven models use artificial intelligence (AI) techniques to evaluate ROP as a function of the drilling parameters, formation characteristics, and/or drilling fluid properties.

1.1. Linear Regression-Based Correlations for Rate of Penetration Estimation

The first mathematical equation for ROP estimation (Equation (1)) was developed by Maurer for rolling cutter bits [11] and to predict the ROP based on the rock strength, WOB, RPM, and drill bit size.

$$ROP= \frac{k}{S^2}{\left(\frac{WOB}{d_b}-\frac{W_t}{d_b}\right)}^{2}RPM$$

(1)

where ROP is in (ft/hr), K is the proportionality constant, S represents the rock compressive strength, WOB is in (Klb_f), W_t is the threshold bit weight, which is very small compared to the WOB and could be considered equal to zero for simplification, and d_b is the drill bit diameter (in).

Bingham [12] developed another ROP model (Equation (2)) to include the effect of rock strength in the proportionality constant and replaced the second constant in Equation (1) with a varying exponent.

$$ROP= k{\left(\frac{WOB}{d_b}\right)}^{a_5}RPM$$

(2)

where K is the proportionality constant, including the effect of rock strength, and a₅ is the WOB exponent.

Neither Maurer’s [11] nor Bingham’s [12] correlations consider the effect of formation compaction, bit hydraulics, differential pressure, and bit wear on ROP change, which lowers the predictability accuracy of ROP these correlations.

Bourgoyne and Young [13] developed the correlation in Equation (3) for drilling optimization and ROP estimation based on a multiple regression analysis of detailed drilling data. In this model, the authors improved ROP predictability by considering the effect of formation strength, depth, compaction, the pressure differential in the bottom hole, bit diameter, WOB, RPM, bit wear, and bit hydraulics on the ROP.

$$\frac{dD}{dt}= e^{\left[a_1+{\sum }_{j=2}^8{a}_j{x}_j\right]}$$

(3)

where D is the true vertical depth of the well (ft), t is the time, the coefficients a₁–a₈ are related to the drilling parameters, x₁–x₈ denote dimensionless drilling parameters calculated from the actual collected drilling variables, a₁ represents the formation strength parameter, a₂x₂ and a₃x₃ account for the formation compaction, a₄x₄ models the effect of the differential pressure, a₅x₅ accounts the effect of the WOB and bit diameter, a₆x₆ considers the effect of the RPM, a₇x₇ models the effect of drill-bit tooth wear, and a₈x₈ accounts for the bit hydraulic jet impact. The variable x_j can be calculated using Equations (4)–(10):

$$x_2=10,000-D$$

(4)

$$x_3=D^{0.69}\left(g_p-9.0\right)$$

(5)

$$x_4=D\left(g_p-ECD\right)$$

(6)

$$x_5=ln\left[\frac{\raisebox{1ex}{$WOB$}\!\left/ \!\raisebox{-1ex}{$d_b$}\right.- {\left(\raisebox{1ex}{$WOB$}\!\left/ \!\raisebox{-1ex}{$d_b$}\right.\right)}_t}{4.0- {\left(\raisebox{1ex}{$WOB$}\!\left/ \!\raisebox{-1ex}{$d_b$}\right.\right)}_t}\right]$$

(7)

$$x_6=ln\left[\frac{RPM}{100}\right]$$

(8)

$$x_7= -h$$

(9)

$$x_8= \frac{\rho q}{350 \mu d_n}$$

(10)

where g_p denotes the pore pressure gradient of the well (lb/gal), ECD is the equivalent circulation density of the drilling mud at depth D (lb/gal), h is the fractional tooth height worn away, ρ is the density of the drilling fluid (lb/gal), q represents the flow rate (gal/min), μ is the drilling fluid viscosity (cP), and d_n represents the bit nozzle diameter (in).

Osgouei [6] used linear regression to modify Bourgoyne and Young’s model to enable prediction of the ROP for directional and inclined boreholes. This model can be used for holes drilled with either roller cone or polycrystalline diamond compact (PDC) bits. In their model, Osgouei [6] considered additional drilling parameters to account for inclination and redefined the same parameters in case a PDC bit was used instead of a roller cone bit. The generalized form of the Osgouei model is summarized in Equation (11).

$$ROP= e^{\left[a_1+{\sum }_{j=2}^{11}{a}_j{x}_j\right]}$$

(11)

where the functions a₂x₂–a₈x₈ are revised forms of the parameters considered by Bourgoyne and Young’s model and the functions a₉x₉–a₁₁x₁₁ account for the hole cleaning conditions, which considerably affect ROP and well drillability, especially in directional and inclined wells [14]. The variables x₃–x₇ are defined in the same way as in Equations (5)–(9), x₂ and x₈ can be calculated according to Equations (12) and (13), respectively, and x₉–x₁₁ can be calculated using Equations (14)–(19).

$$x_2=8800-D$$

(12)

$$x_8= ln\left[\frac{F_j}{1000}\right]$$

(13)

For roller cone bits,

$$x_9= ln\left[\frac{\raisebox{1ex}{$A_{bed}$}\!\left/ \!\raisebox{-1ex}{$A_{well}$}\right.}{0.2}\right]$$

(14)

$$x_{10}= \left[\frac{v_{actual}}{v_{critical}}\right]$$

(15)

$$x_{11}= ln\left[\frac{c_c}{100}\right]$$

(16)

For PDC bits,

$$x_9= ln\left[\frac{\raisebox{1ex}{$A_{bed}$}\!\left/ \!\raisebox{-1ex}{$A_{well}$}\right.}{0.35}\right]$$

(17)

$$x_{10}= \left[\frac{v_{actual}}{v_{critical}}\right]$$

(18)

$$x_{11}= ln\left[\frac{c_c}{25}\right]$$

(19)

where A_bed and A_well are the cross-sectional areas of the cutting’s bed and drilled hole (in), respectively, v_actual and v_critical are the actual and critical velocities of the cuttings, respectively, and c_c is the annular cutting’s concentration. As reported by Osgouei [6], ROP can be estimated with an error of ±25% compared to the actual ROP, which is considerably high.

1.2. Application of Artificial Intelligence for Rate of Penetration Estimation

AI techniques are used extensively in applications related to different engineering and scientific research areas [15,16,17,18,19,20], including in the petroleum industry where they can solve complicated problems such as prediction of drill bit wear from drilling parameters [21], real-time predictions of alterations in drilling fluid rheology [22,23], lithology identification [24], prediction of total organic carbon for the evaluation of unconventional resources [25,26,27,28,29], estimation of the oil recovery factor [30,31], estimation of pore and fracture pressures [32,33], evaluation of the static Young’s modulus [34,35,36], estimation of the reservoir porosity [37], evaluation of the bubble point pressure [38], and the prediction of formation tops [39].

The use of AI techniques for ROP prediction was suggested by Bilgesu et al. [40] to overcome the weakness of the empirical correlations and to improve the accuracy of ROP predictability. Bilgesu et al. [40] developed two artificial neural network (ANN) models for ROP estimation in nine different formations drilled in several vertical wells in the United States. The first model estimated ROP as a function of bit type and diameter, formation type, bit tooth and bearing wear, mud circulation, gross hours of drilling, footage, WOB, and RPM, while the second model excluded the bit tooth and bearing wear from the inputs. The authors reported that both models predicted ROP with very high accuracy.

Two ANN models were developed by Amar and Ibrahim [41] to estimate ROP as a function of the depth, WOB, RPM, tooth wear, Reynolds number function, ECD, and pore gradient. Both ANN-based models predicted ROP with very low average absolute percentage error (AAPE) compared with the available ROP correlations that were developed based on linear regression.

Coupling of the different AI models with linear regression for ROP optimization for horizontal wells was suggested by Mantha and Samuel [42]. In their models, Mantha and Samuel [42] used RPM, WOB, GPM, and gamma-ray (GR) logs as inputs to estimate ROP. All suggested models predicted the ROP with high accuracy compared to actual field-measured ROP. No empirical correlations were extracted from these suggested models, and it is still difficult for authors to extract them in order to test future data.

Elkatatny [43] was the first to develop an empirical correlation for ROP estimation for vertical wells based on the extracted weights and biases of the optimized ANN model. The developed empirical correlation evaluated ROP as a function of the RPM, WOB, T, GPM, and standpipe pressure (SPP) combined with drilling fluid properties, including mud density (MW) and plastic viscosity (PV). Elkatatny’s [43] correlation predicted ROP with an AAPE of only 4% compared to other ROP correlations, which predicted ROP using the same data considered by Elkatatny [43] with an AAPE of more than 10%.

Another model for ROP estimation based on support vector machines was developed by Ahmed et al. [44]. This model estimated ROP based on RPM, WOB, T, GPM, and SPP and drilling fluid properties of the MW, PV, funnel viscosity, and solid concentrations. This model also estimated ROP accurately, with an AAPE of only 2.83%.

This study aimed to develop a new empirical correlation for ROP estimation in horizontal wells and carbonate formations as a function of RPM, T, and WOB, combined with conventional well log data including GR, deep resistivity (DR), and formation bulk density (RHOB). The correlations developed in this study were based on the weights and biases of the trained ANN model, which was optimized using the self-adaptive differential evolution (SaDE) algorithm.

2. Methods

Artificial neural networks (ANN) are a computing system designed to mimic the way biological systems such as human or animal brains are behaving [45,46]. ANNs were developed to help with classification, identification, estimation, and decision-making in various situations using a machine program. In this study, the simplest ANN form, multi-layered perceptron, which consists of a single input layer, single or multiple learning layers, and one output layer, was used. Firstly, the ANN systems were trained using labeled data (supervised learning) to perform the needed tasks [47], then the trained ANN model was used to predict the objective variables.

Several parameters control the performance of an ANN model, which comprise the size of the training dataset, the required training (learning) layers, the training neurons per learning layer, the training function, and the transfer function. Evaluating the performance of all combinations of these parameters is a time-consuming process.

Differential evolution (DE) is a robust optimization algorithm which has proved its effectiveness and accuracy in solving several numerical problems. The need to set initial DE control parameters, which are problem-dependent and therefore time-consuming, limits the application of DE. In 2005, the SaDE algorithm was developed by Omran et al. [48], which accelerated the optimization process since it did not need parameter tuning.

The SaDE algorithm was used in this study to select the optimum ANN design parameters required to predict ROP. An empirical equation for ROP calculation was also developed based on the weights and biases associated with the training and output layers and neurons of the optimized ANN model.

2.1. Data Preparation

In this study, the ANN model was trained using the drilling parameters RPM, T, and WOB, combined with conventional well log data including GR, DR, and RHOB as inputs to estimate the ROP as the output. All abbreviations are listed in Appendix A. The data collected in this study were from two different wells and all data used to train the ANN model (i.e., the inputs) were real-time data that extracted by the driller in the real-time base, meaning that the predicted ROP based on these variables was a real-time ROP.

To ensure a highly accurate ROP prediction using the ANN model, the collected data was preprocessed before being introduced into the ANN model to remove outliers. This step is necessary for processing any type of data with any AI technique [49]. The data were studied statistically to remove outliers and the standard deviation was considered to be the main parameter to judge the validity of the data, where all values within the range of ±3.0 standard deviations were considered valid; all others were considered outliers and removed from the data. After data preprocessing, 3531 of the data points collected from Well-A and 3600 of Well-B data points were considered valid for model training and testing.

2.2. Training the ANN Model

The dataset collected from Well-A (3531 data points) was used to build the ANN model, with 85% of this data randomly selected to train the ANN model and 15% used for testing the trained model. These percentages were selected based on the optimization process, as discussed later in this section.

The training dataset was first analyzed statistically to determine the ranges for the different parameters used to develop the ANN model; the results of this analysis are summarized in

Table 1. The statistical parameters for the training dataset (85% of the data collected from Well-A).

Statistical Parameters	RPM (rpm)	T (kft.lbf)	WOB (klbf)	GR (API)	DR (Ω.m)	RHOB (g/cm3)	ROP (ft/hr)
Minimum	59.8	3.06	5.55	8.80	0.64	2.13	59.8
Maximum	132	7.81	24.3	69.5	965	3.02	132
Range	72.2	4.75	18.8	60.7	963.9	0.89	72.2
Standard Deviation	17.5	0.75	2.64	8.5	212.4	0.16	17.5
Sample Variance	306	0.57	6.99	72.9	45114	0.02	306
Kurtosis	−1.14	0.16	1.55	1.96	6.09	−0.87	−1.14
Skewness	−0.49	−0.44	−0.94	1.27	2.49	−0.10	−0.49

. As indicated in Table 1, RPM ranged from 59.8 to 132, T was between 3.06 and 7.81 kft.lb_f, WOB was from 5.55 to 24.3 klb_f, GR was between 8.80 and 69.5 API, RHOB ranged from 2.13 to 3.02 g/cm³, and ROP was from 6.53 to 53.2 ft/hr. The ranges of the training data, as summarized in Table 1 were very important and needed to be considered when predicting ROP. To ensure high accuracy during ROP predictions, the new input dataset used for the ROP estimation had the same range as the training data.

Figure 1 compares the relative importance of the input parameters used to train the ANN model. The inputs used in this study were selected based on their relative importance to the ROP. As shown in Figure 1, the T and WOB had good correlation coefficients with the ROP, i.e., 0.67 and 0.61, respectively. RPM and DR had acceptable correlations with the ROP. Although GR and RHOB had low correlation coefficients of 0.07 and 0.13, respectively, with the ROP, they were considered as inputs in this study because they both directly affected the ROP. Changes in GR and RHOB gave indications about the change in the lithology, which led to a change in ROP. The use of GR to improve ROP predictability for horizontal wells was suggested previously by Mantha and Samuel [42].

After preprocessing the training data, the SaDE algorithm was applied to optimize the ANN model to select a combination of ANN design parameters to predict ROP with the minimum AAPE and highest correlation coefficient (R) and coefficient of determination (R²).

Different design parameters of the ANN model were studied during the optimization process, including the percentage of the training and testing data, training function, transfer function, number of hidden layers, and the number of neurons associated with every hidden layer. The effect of using 5% to 95% of the data collected from Well-A for training was evaluated during model optimization. The predictability of different training functions, such as Levenberg–Marquardt backpropagation, BFGS (Broyden–Fletcher–Goldfarb–Shanno) Quasi-Newton backpropagation (trainbfg), conjugate gradient, Bayesian regularization backpropagation, gradient descent with adaptive learning rate backpropagation, and gradient descent with momentum backpropagation, was evaluated. The effects on training accuracy of transfer functions log-sigmoid (logsig), tan-sigmoid, and pure-line were also evaluated. The use of 1–3 hidden layers, with 5–30 neurons per layer, was also studied.

The optimization process was conducted using SaDE in MATLAB^®. Based on the results of the optimization process, the use of the trainbfg function to train the ANN model and the logsig transfer function with a single hidden layer with 26 neurons, as listed in

Table 2. Combination of the optimum ANN design parameters for ROP prediction.

Parameter	Value
Training function	trainbfg
Transfer function	logsig
Number of hidden layers	1
Number of neurons	26

, optimized the ANN model for ROP prediction. The structure of the suggested ANN model for ROP prediction is shown in Figure 2, which shows that the ANN model had an input layer with 6 neurons (one neuron for every input variable), a single layer with 26 neurons, and one output layer with a single output neuron (i.e., ROP).

An empirical correlation based on the weights and biases of the optimized ANN model with the design parameters, as listed in Table 2, was then developed. In this step, the optimized ANN model was converted from a black box to a white box for easy use and to program the developed correlation for future use.

2.3. Testing and Validation of the Developed ANN-Based Correlation

After developing the empirical correlation, the remaining 531 data points of Well-A, which represented 15% of the Well-A data, were used to test the developed ANN-based correlation, which was then validated using the 3600 unseen data points collected from Well-B. Before testing or validating the developed correlation, the ranges of the testing and validation datasets were investigated and confirmed to fall within the range of the data used to train the ANN model, which is summarized in Table 1. This step was very important to ensure that the ROP was accurately predicted. The predictability of the developed ANN-based correlation for ROP for the validation data was compared with the predictability of four commonly used empirical correlations for ROP estimation, namely, Maurer’s [11], Bingham’s [12], Bourgoyne and Young’s [13], and Osgouei’s [6] correlations, as presented earlier in Equations (1)–(3), and (11).

2.4. Evaluation Criteria

Four metrics were considered in this study to evaluate the predictive power of the developed ANN model and ANN-based empirical correlation for ROP estimation, namely, AAPE, R², R, and a visualization check of the match between the actual and the predicted ROP. These metrics were considered sufficient to evaluate the predictability of the ROP for training, testing, and validation datasets.

3. Results and Discussion

3.1. Training the ANN model

The ANN model developed in this study was trained using 3000 datasets, including RPM, T, WOB, GR, DR, and RHOB as inputs to predict the ROP as an output. The training data was randomly selected from Well-A data and represented 85% of the total data available from Well-A. The SaDE algorithm was used to optimize the ANN model to predict the ROP; Table 2 summarizes the optimum ANN parameters for ROP estimation.

The training data and their corresponding actual and ANN predicted ROP values are summarized in Figure 3 as a function of well depth. The results in Figure 3 show that the ANN model predicted the ROP for the training dataset of Well-A with a low AAPE of 5.12% and a considerably high R of 0.960, confirming a good match between the actual and predicted ROP curves, indicating high accuracy of the developed ANN model in estimating ROP.

Figure 4 shows a cross-plot of the actual ROP versus the predicted ROP for the training dataset (85% of Well-A data), indicating high accuracy of the ANN model in estimating the ROP with an R² of 0.921.

3.2. Developing the ANN-Based Correlation

The proposed ANN-based empirical model was given by Equations (20)–(27).

$$RO{P}_n= {\displaystyle \sum }_{i=1}^N{w}_{2_i}\frac{1}{1+e^{-\left(w_{1_{i,1}}RP{M}_n+w_{1_{i,2}}{T}_n+w_{1_{i,3}}WO{B}_n+w_{1_{i,4}}G{R}_n+w_{1_{i,5}}D{R}_n+w_{1_{i,6}}RHO{B}_n+b_{1_i}\right)}}+b_2$$

(20)

where ROP_n is the normalized ROP, N represents the number of neurons in first hidden layer (26 neurons), i denotes the index of each neuron in the hidden layer, as shown in

Table 3. Weights and biases for the first hidden layer of the ANN-based ROP model.

i	w1i,1	w1i,2	w1i,3	w1i,4	w1i,5	w1i,6	b1i	w2i
1	0.472	−1.869	3.127	−1.312	0.232	−0.621	6.431	0.190
2	−0.778	−3.454	1.887	6.725	−2.769	−6.941	−5.566	−0.299
3	3.037	−4.214	−2.902	−5.927	1.752	−0.969	−4.733	0.396
4	3.279	2.073	−4.944	2.953	−0.896	6.679	−5.680	−0.654
5	−1.167	0.034	1.637	−4.192	−0.988	−3.395	2.298	−1.056
6	−5.614	0.971	−4.185	−2.940	−0.165	2.019	3.522	0.418
7	4.712	1.857	2.161	3.803	−1.979	−0.936	−3.041	0.859
8	−0.795	0.183	0.660	−4.797	−13.842	1.330	−14.488	−0.945
9	3.732	−4.144	−3.459	−1.063	3.321	−3.952	−1.752	−0.451
10	0.502	−10.524	5.801	1.869	−0.852	−6.240	−3.857	0.294
11	−0.317	−5.318	6.544	3.210	1.360	0.461	1.090	−0.326
12	4.212	0.773	0.592	1.958	1.470	2.034	−0.118	−0.388
13	2.910	0.433	3.394	0.904	−1.510	0.177	−0.756	0.195
14	−3.365	−6.367	−1.462	3.218	−5.920	4.702	−1.646	0.416
15	−2.423	3.326	0.483	−3.764	−3.841	−4.881	0.170	−0.324
16	2.190	5.445	0.414	−2.546	0.930	−2.493	−1.045	0.704
17	−1.443	2.033	8.548	4.160	3.550	9.202	−0.189	0.176
18	1.699	−0.890	−0.025	−0.947	−2.317	3.959	−0.119	−0.379
19	−2.209	3.899	−4.020	−7.417	1.532	1.539	−2.022	0.738
20	−1.240	2.891	−3.411	−8.762	1.554	0.939	−3.210	−0.762
21	3.398	−3.788	−4.602	1.518	−1.239	0.043	−0.422	−0.048
22	1.904	−1.289	6.963	−1.661	0.961	−3.726	−4.547	0.378
23	−3.985	1.949	−2.266	1.458	0.686	2.434	−4.530	−1.069
24	−0.462	−1.879	0.058	3.565	1.033	−2.105	−4.423	0.795
25	−0.804	0.043	0.667	−3.192	−10.184	1.230	−11.419	1.249
26	−5.700	−2.821	−1.045	1.765	2.478	−3.574	−6.170	−0.580

, w_1i is the weight associated with input and hidden layers for each input parameter, b_1i denotes the biases associated with the input and training layers, w_2i is the weight associated with the hidden and output layers, RPM_n, T_n, WOB_n, GR_n, DR_n, and RHOB_n represent the normalized RPM, T, WOB, GR, DR, and RHOB, respectively, and b₂ is the bias associated with the hidden and output layers, which was 0.248 in this case. All weights and biases needed for Equation (20) are listed in Table 3.

The normalized input parameters in Equation (20) were calculated using Equations (21)–(26).

$$RP{M}_n=0.0277\left(RPM-59.76\right)-1$$

(21)

$$T_n=0.421\left(TORQUE-3.0633\right)-1$$

(22)

$$WO{B}_n=0.1066\left(WOB-5.5520\right)-1$$

(23)

$$G{R}_n=0.0329\left(GR-8.8038\right)-1$$

(24)

$$D{R}_n=0.0021\left(DR-0.6449\right)-1$$

(25)

$$RHO{B}_n=2.2589\left(RHOB-2.1345\right)-1$$

(26)

The value of ROP_n, as calculated by Equation (20), was the normalized ROP, which was converted to the real ROP using Equation (27).

$$ROP= \frac{RO{P}_n+1}{0.0429}+6.5345$$

(27)

3.3. Testing the Developed ANN-Based Correlation

The ANN-based model developed in this study, as summarized in Equations (20)–(27), was then tested using the remaining unseen 15% of the data of Well-A. Figure 5 compares the actual ROP with the ROP values estimated using the ANN-based empirical equations (Equations (20)–(27)) for the Well-A test dataset. As indicated in Figure 5, the ANN-based empirical equations were able to predict the ROP with high accuracy, as indicated by the low AAPE of 5.80% and the high R of 0.951. Figure 5 shows high accuracy. Also shows high accuracy of the ANN-based correlation, where there was a good match between the actual and predicted ROP curves.

Figure 6 compares the actual ROP with the predicted value using the ANN-based empirical correlation. As indicated in Figure 6, the ANN-based equation predicted the ROP for the test dataset (15% of the data of Well-A) with high accuracy in terms of the R² (R² = 0.905).

3.4. Validation of the Developed ANN-Based Empirical Correlation

The accuracy of the ANN-based empirical correlation predictions using Equations (20)–(27) was validated using another set of unseen data collected from Well-B. Figure 7 shows the input variables of RPM, T, WOB, GR, DR, and RHOB used to predict the ROP in Well-B, and also compares the ROP values predicted using the ANN-based correlation (Equations (20)–(27)) with the actual ROP. As indicated in Figure 7, the AAPE and R of the predicted ROP were 5.29% and 0.956, respectively. The low AAPE and high R of the predicted ROP indicated the high accuracy of the developed ANN-based correlation. A visual check of the actual and ANN-based ROP values in Figure 7 also indicated a good match between the two curves (i.e., the actual ROP and the ANN-ROP), thereby confirming high accuracy of the ANN-based correlation.

Figure 8 compares the actual ROP with the predicted value using the ANN-based empirical correlations, as summarized in Equations (20)–(27) for the validation dataset (3600 data points collected from Well-B). As indicated in Figure 8, the ANN-based equation predicted the ROP for the validation dataset with a relatively high R² of 0.914, indicating high accuracy of the developed ANN-based correlation.

3.5. Comparison of the Predictability of the Developed ANN-Based Empirical Correlation with Available Correlations

In this section of the study, the accuracy of four currently available empirical equations for ROP estimation, namely, Maurer’s, Bingham’s, Bourgoyne and Young’s, and Osgouei’s correlations, was compared to the accuracy of the developed ANN-based empirical correlation in Equations (20)–(27). As shown in Figure 9, Maurer’s correlation was the least accurate correlation in terms of ROP estimation according to the validation data, with the highest AAPE of 19.81% and a very low R of 0.625. Bourgoyne and Young’s correlation predicted the ROP with an AAPE of 14.82% and an R of 0.622, while the ROP predicted by Bingham’s correlation had AAPE and R values of 14.62% and 0.634, respectively. Osgouei’s correlation was the best empirical correlation of the available correlations in terms of predicting the ROP, with the lowest AAPE of 14.60% and an R of 0.629.

The developed empirical correlation based on ANN, which was summarized in Equations (20)–(27), outperformed all available correlations in predicting ROP for the Well-B validation data, as indicated by the very low AAPE and very high R values of 5.29% and 0.956, respectively, as shown in Figure 9. Visual checking of the predicted ROP curves with the different correlations and the actual ROP shown in Figure 9 indicated an excellent match between the actual and ANN-based empirical correlation compared to all of the available correlations.

Figure 10 shows cross-plots comparing the actual ROP with the ROP values (3600 data points of Well-B) calculated using the different empirical correlations. As indicated in this figure, none of the available correlations accurately predicted the ROP, as indicated by the low R² values between the actual and predicted ROPs. Maurer’s and Bourgoyne and Young’s correlations predicted the ROP with R² values of only 0.390 and 0.387, respectively. Bingham’s and Osgouei’s correlations predicted the ROP with slightly higher R² values of 0.402 and 0.396, respectively. The ANN-based correlation predicted the ROP with a very high R² value of 0.914.

As shown in Figure 10a–d, the cross-plots of the actual and predicted ROP values using the previously developed correlations were highly scattered compared to the cross-plot for the ROP values predicted using the ANN (Figure 10e).

4. Conclusions

A new correlation to estimate ROP while drilling horizontal wells in carbonation formations was developed based on the extracted weights and biases of the trained and optimized ANN model. The ANN model was optimized using a self-adaptive differential evolution algorithm. The developed correlation estimated ROP as a function of the drilling parameters of the RPM, torque (T), and WOB, combined with conventional GR, DR, and RHOB well log data. The following observations were made:

• The optimized ANN model predicted the ROP for the training dataset (3000 data points) with an AAPE of 5.12% and a correlation coefficient (R) of 0.960;
• The developed correlation predicted the ROP for the testing dataset (531 data points) with AAPE and R values of 5.80% and 0.951, respectively;
• The developed ROP correlation outperformed a recently developed empirical correlation for estimating ROP in directional wells (the Osgouei model), which predicted the ROP for the validation data with a high AAPE and a low R of 14.60% and 0.629, respectively;
• The developed correlation predicted ROP for the validation dataset of Well-B (3600 data points) with an AAPE of only 5.26% and a high R of 0.956.