PREDICTING THE WEIGHT AND TYPE OF DRILLING MUD BY MACHINE LEARNING METHOD

2.1. Mud-weight prediction models

In order to determine the mud weight, the induced stress is first calculated and then compared with the failure criteria of the specific rock formation (Mohr-Coulomb, Hoke-Brown, etc.). The upper limit of the mud weight can be determined by substituting the stress induced in tensile failure. If this stress equals the shear strength of the rock, a lower mud weight limit should be estimated.

Researchers used a numerical method and sensitivity analyses to determine a safe mud-weight window (Ayoub et al., 2019). The results revealed that, for low-cohesion and friction angle formation, the mud weight window will become narrower. A decrease in the pore pressure and ratio between the maximum and minimum horizontal stresses will widen the mud weight window. This does not simulate the complex nature of the reservoir, such as natural fracture, bedding plane, high pressure and temperature of the reservoir. Additionally, the input parameters of numerical modelling such as the Young’s modulus, cohesion and friction angle contain uncertainties.

Zahiri et al first derived the mud-weight window using an analytical approach for three wells and then use an ANN technique to derive a relation between the well-logging data and the mud-weight window. The major disadvantage was that ANN training using the analytical approach included simplified assumptions. It also did not consider chemical instability (Zahiri et al., 2018).

Agwu et al. used an ANN model to forecast the density of oil-based mud in high temperature/high pressure (HTHP) wells. They summarized previous empirical and AI approaches to determining the mud density in HTHP wells (Agwu et al., 2020). Sabah et al. used machine learning to predict the amount of lost circulation. This parameter was essential for selecting an appropriate lost circulation material to cope with. The data was extracted from 2820 daily drilling reports from 305 oil wells and 18 input parameters were used to predict the amount of circulation lost into the formation (Sabah et al., 2021).

Beheshtian et al. gathered 3389 daily drilling reports from three gas wells to predict a safe mud-weight window using several machine learning algorithms (Beheshtian et al., 2022). Gowida et al. used the ANN approach to predict the minimum and maximum mud weights (Gowida et al., 2022). Phan et al. found that a model was required to predict the real-time mud weight. They reported that analytical and numerical models for this purpose are time consuming and that the AI based method was the best solution for real-time mud-weight prediction (Phan et al., 2022). As mentioned earlier prediction drilling fluid is divided into two-part mud weight and mud type. In the next section, the mud-type prediction works were reviewed.

2.2. Mud type prediction models

A greater mud-weight will reduce the tangential stress and increase the normal stress on the wellbore; however, in some situations this has an adverse effect on wellbore stability. When mechanical solutions are not appropriate, a chemical analysis of the drilling fluid should be carried out. Most chemical analyses are based on an experimental approach by measuring parameters such as the cation exchange capacity (CEC) and x-ray diffraction (XRD) to determine the active clay mineral content. The choice of drilling fluid is based on the composition of the formation. For example, shale swells slightly when sodium silicate is added to polymer mud. However, the addition of potassium chloride to the polymer mud produced significant swelling, indicating that the mud consisted of various chemical constituents which interacted with the formation, especially shale (Allawi, 2023).

Swelling shale has low strength and tends toward time-dependent wellbore instability. Traditionally, oil-based mud has been employed in such formations; however, because of its negative environmental impact, other types of mud, such as polymer/KCL/lime/gypsum have been used instead. Few works have used AI to estimate mud type, but some researchers have predicted the rheological properties of mud and the density of its constituents (Agwu et al., 2020).

Tatar et al. used pressure, temperature and concentration as inputs to predict the brine density of mud (Tatar et al, 2015). Aljubran et al. determined the mechanical-thermal/physico-chemical interaction on wellbore instability. They reported that cooler temperatures and higher salt concentrations in the mud improved the wellbore stability in a shale formation (Aljubran et al., 2022).

Due to the lack of research that comprehensively solves drilling fluid prediction, this work used a data mining approach to find mud's physical and chemical properties. In the following, the details of the research are explained.

3.1. Data Analysis

The initial determination of the dataset's descriptive statistics involved a comprehensive examination of each parameter through mathematical calculations. It was ascertained that the highest correlation with mud density across all parameters was observed when the relationship between them followed a power and linear equation. The selection of this equation in the present research was made to encompass all three algorithms. Tables 1 and 2 display the descriptive statistics of the database.

The general form of the predictive equation used in this investigation is given by Equation 1. In this model, the output (predicted mud density) is a function of six standard drilling parameters, each multiplied by a specific weight coefficient (W~i~) that is optimized by the algorithms. The coefficients determine the relative influence and scaling of each input parameter.

[TeX:] TVD \times W1 + ROP \times W2 + W3 \times diam + RPM \times W4 + W5 \times WOB + flow\ rate + W6$TVD\times W1+ROP\times W2+W3\times diam+RPM\times W4+W5\times WOB+flowrate+W6$ (1)

Where:

TVD is the True Vertical Depth (m),

ROP is the Rate of Penetration (ft/h),

diam is the drill bit diameter (inches)

RPM is the drill string Revolutions Per Minute (1/min),

WOB is the Weight On Bit (kip),

flow rate is the mud flow rate (lb/s),

W1, W2, W3, W4, W5, W6 are the weight coefficients optimized by the ACO, BCO, and EPC algorithms.

Table 1. Descriptive statistics of database based on lithology

Figure 5. Daily drilling report for well 19

Table 2. Descriptive statistics of database based on MUD type

FW = fresh water; SW = sea water; PO = polymer; KC = potassium chloride; GE = gel; LI = lime mud; SX = soltex; LS = lingosulfonate; GY = gypsum base; PC = partially hydrolyzed polyacrylamide.

3.2. Evaluation criteria

This investigation evaluated the accuracy and efficiency of the predictive models using the coefficient of determination (R²) and the root mean-square error (RMSE), as stipulated in Equations 2 and 3. These metrics are well-established in the literature for assessing the performance of machine learning models in geotechnical and mining engineering applications, providing a standardized and interpretable measure of predictive power (Hasanipanah et al., 2022; Hasanipanah et al., 2015). The ideal values for R² and RMSE are one and zero, respectively. Utilizing distribution diagrams and comparative graphs of observational versus computational data, the results were analyzed and compared.

3.3. Ant Colony Optimization (ACO)

Ant colony optimization can be modeled as 𝑃= (𝑆, 𝛺, 𝑓), where S is the set of all finite sets of discrete decision variables, and 𝛺 represents the constraints between these variables and a target function 𝑓∶ 𝑆 ⟶𝑅0+, subject to minimization or maximization (Dorigo & Blum, 2005). Ant colony optimization incrementally generates solutions by considering the likelihood of solution components, with probabilities determined by pheromone levels (Dorigo & Blum, 2005). When applied to hybrid optimization problems, a set of solution components is defined based on the problem formulation. During solution construction, ants select 𝑐𝑖 from 𝑁(𝑠^𝑝) using Equation 4 at each step.

[TeX:] p\left( c_{ij} \middle| s^{p} \right) = {c_{ij}}^{p}.\frac{{\eta(cij)}^{\beta}\ }{\Sigma c_{il\epsilon N\left( s^{p} \right)^{{\tau ij}^{\alpha}.{\eta\ (cij)}^{\beta}}}}.\ \ \forall c_{ij} \in \ N(s^{p})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $p\left(c_{ij}\right|s^p)={c_{ij}}^p.\frac{{\eta \left(cij\right)}^{\beta }}{\Sigma c_{ilϵN{\left(s^p\right)}^{{\tau ij}^{\alpha }.{\eta \left(cij\right)}^{\beta }}}}.\forall c_{ij}\in N\left(s^p\right)$ (4)

In pheromone-based optimization, 𝜏𝑖𝑗 represents the pheromone level on edge 𝑐𝑖𝑗. A weight function, 𝜂(0), updates edges 𝑐𝑖j∈𝑁(s^p) at each iteration, producing values termed "apocalyptic information." Positive parameters 𝛼 and 𝛽 govern the relationship between pheromone and heuristic information. A Gaussian kernel, used for sampling, is defined as the sum of weighted one-dimensional Gaussian functions, 𝐺^𝑖(𝑥).

[TeX:] G^{i}(x) = \sum_{l = 1}^{k}{\omega_{l}g_{l}^{i}(x) =}\sum_{l = 1}^{k}\omega_{l}\frac{1}{\sigma_{l}^{i}\sqrt{2\pi}}e^{- \frac{{(x - \mu_{l}^{i})}^{2}}{{2\sigma_{l}^{i}}^{2}}}$G^i\left(x\right)={\sum }_{l=1}^k{\omega }_l{g}_l^i\left(x\right)={\sum }_{l=1}^k{\omega }_l\frac{1}{{\sigma }_l^i\sqrt{2\pi }}{e}^{-\frac{{(x-{\mu }_l^i)}^2}{{2{\sigma }_l^i}^2}}$ (5)

Multi-core probability density functions determine a single probability density function based on the problem's dimension (𝑖 = 1,…, 𝑛). Each 𝐺𝑖(𝑥) is represented using three parametric vectors: 𝜔 (weights), 𝜇𝑖 (means), and 𝛿𝑖 (standard deviations), along with one-dimensional Gaussian functions, g_l^i (x). These vectors have cardinality equivalent to the number of Gaussian functions (denoted by 𝑘) in the Gaussian kernel.

| 𝜔 | = | 𝜇^𝑖 | = | 𝛿^𝑖 | = 𝑘 (6)

Ant Colony Optimization (ACO) is governed by several control parameters. α controls the influence of pheromone trails, with higher values favoring exploitation. β controls the influence of heuristic information, with higher values favoring exploration. ρ represents the pheromone evaporation rate, preventing premature convergence; higher values increase exploration. The number of ants affects search thoroughness, with more ants increasing computational cost. Q is a constant related to pheromone deposition. These parameters are typically tuned empirically, using literature-based values, or through computationally expensive meta-optimization.

3.4. Bee Colony Optimization (BCO)

The freak algorithm utilizes a freak colony with three groups hired, hunt, and watch notions. Each food source represents a implicit result, and its quencher volume indicates result quality (Karaboga, 2005). Originally, a population of results, corresponding to food source locales, is generated.' SN' denotes the number of hired/ hunt notions. Each result ( 𝑗 = 1.2.3.. 𝑆𝑁) is a D- dimensional vector, where' D' is the number of optimization parameters. Search notions elect food sources grounded on quality; the probability of opting a source is calculated using Equation 7.

[TeX:] P_{i = \frac{{fit}_{i}}{\sum_{N - 1}^{SN}{fit}_{N}}}$P_{i=\frac{{fit}_i}{{\sum }_{N-1}^{SN}{fit}_N}}$ (7)

In the context of statistical analysis, the variable 𝑓𝑖𝑡𝑖 represents the fitness value of the 𝑖^th observation.

The process of selecting a novel food source (Vij) is determined by Equation 8, which is contingent upon the preceding food source (Xij). This approach is adopted within an academic context.

[TeX:] Vij = Xij + (Xij - Xkj)$Vij=Xij+(Xij-Xkj)$ (8)

Random pointers 𝑗 and 𝑘, where 𝑗 and 𝑘 are rudiments of the set{ 1.2.. 𝑆𝑁}, are named. It's important to note that 𝑘 is named aimlessly, but must be distinct from 𝑗. In the environment of the freak algorithm, a food source that fails to recover after a specified replication is supposed an abandoned food source. In similar cases, the notions cover the situation in agreement with Equation 9 and latterly replace the abandoned food source with a new one named at arbitrary (Karaboga & Basturk, 2008).

[TeX:] X_{i}^{j} = X_{\min}^{j} + rand\lbrack 0.1\rbrack(X_{\max}^{j} - X_{\min}^{j})$X_i^j=X_{min}^j+rand[0.1](X_{max}^j-X_{min}^j)$ (9)

The variable j denotes the quantity of optimization variables.

In BCO, key control parameters include colony size (SN), representing the number of food sources or solutions, and Limit, which dictates when an unproductive food source is replaced by a scout bee to escape local minima. The maximum cycle number (MCN) serves as the stopping criterion, defining the maximum number of iterations.

In the ABC algorithm, foundational literature (Karaboga, 2005) typically suggests setting the Limit parameter to SN * D, where D represents the number of dimensions. Here, D=6 (TVD, ROP, Diam, RPM, WOB, Flowrate). We systematically varied SN and Limit to optimize performance for this specific problem. The omission of this detail regarding the Limit parameter is significant, as it plays a crucial role in balancing exploration and exploitation within the ABC algorithm.

3.5. Emperor Penguins Colony (EPC)

The foundational premise of the Emperor Penguin algorithm posits that penguins are distributed across the examined environment. Consequently, at the outset, prior to the introduction of more complex procedures, it is essential to assess the relative positions of each penguin in relation to one another. It is important to note that a penguin typically gravitates towards another that presents a lower absorption cost. Thus, the second critical measurement involves determining the absorption cost for each penguin, which is influenced by both heat intensity and distance. Throughout the execution of this algorithm, each phase of absorption entails the evaluation of a new potential solution to the problem at hand, accompanied by an update of the heat intensity. Following this, all potential solutions are organized, and the optimal one is identified. Additionally, a damping ratio is calculated for each penguin, which is influenced by various factors, including the individual penguin's movement, heat radiation, and heat absorption. A lower damping ratio indicates a more favorable solution (Harifi et al., 2019).

Emperor Penguin Colony (EPC) is a metaheuristic algorithm that mimics the huddling behaviour of emperor penguins seeking warmth.

Key control parameters include:

Parameter Value Selection based on the algorithm's recent development, the original paper (Harifi et al., 2019) is often consulted for default values. Sensitivity analysis may also be performed to assess the impact of parameter variations on model output for specific applications like drilling fluid prediction.

4.1. Ant colony optimization Result (ACO)

Models 1-4 used a four-fold cross-validation approach, training on three steps and testing on the remaining one. Model 5 trained on the entire dataset and was then tested on datasets P1-P4. Model 6 trained on 80% of the data and tested on the remaining 20%. Model 6 demonstrated sufficient accuracy, achieving an R2 of 0.9649 and RMSE of 0.0364. Table 3 presents R2, RMSE, and coefficient weighting across all models. Figures 6 and 7 illustrate the distribution and correlation between measured, target, and predicted values for the top-performing model. Figure 8 identifies the optimal mud type according to ACO.

Table 3. ACO results for six models

The equation obtained using ACO was:

-0.0012 x TVD + ROP^0.047 - 0.5824 x diam + RPM^0.2234 - 1.1345 x WOB + flowrate ^0.5612 (10)

Figure 6. Distribution of top model by ACO

Figure 7. Matching of top model by ACO

Figure 8. Best mud type by ACO

4.2. Bee colony optimization results (BCO)

BCO algorithm Models 1-4 used a four-step partition: each step tested the targets while the remaining steps trained the sets. Model 5 trained on the entire dataset, then each step (P1-P4) was used for testing. Model 6 trained on 80% of the dataset and tested on the remaining 20%. The results showed Model 6 to be sufficiently accurate, yielding an R2 of 0.9841 and RMSE of 0.245 for the best prediction. Table 4 presents the R2, RMSE, and weighting coefficients for all BCO models. Figures 9 and 10 display the distribution and matching charts, respectively, of measured and predicted values for the top model. Figure 11 identifies the best mud type according to the BCO.

Table 4. BCO Algorithm Results: Six Models

The equation obtained using BCO was:

-0.0019 x TVD + ROP^0.0664 - 0.2365 x diam + RPM^0.2987 - 1.9433 x WOB + flowrate ^0.2137 (11)

Figure 9. Distribution of top model by BCO

Figure 10. Matching of top model by BCO

Figure 11. Best mud type by BCO

4.3. Results of Emperor Penguins Colon (EPC)

Models 1- 4 of the EPC algorithm used a four-step partitioning, employing one step for testing and the remaining three for training. Model 5 trained on the full dataset and also tested each step (P1-P4) across all models. Model 6 employed 80 of the dataset for training and 20 for testing. The results demonstrated that Model 6 and its associated equation displayed high delicacy, achieving an R2 of 0.9707 and an RMSE of 0.0332 in the stylish model. Table 5 presents the R2, RMSE, and weighting portions for all EPC algorithm vaticination models. Figures 12 and 13 illustrate the distribution and correlation, independently, between measured and prognosticated target values for the top- performing model. Figure 14 identifies the optimal slush type grounded on the EPC.

Table 5. EPC Six models' algorithm results

The equation using EPC was:

-0.0056 x TVD + ROP^0.432 - 0.2223 x diam + RPM^0.3254 - 1.3456 x WOB + flowrate^0.3317 (12)

Figure 12. Distribution of top model by EPC

Figure 13. Matching top models by EPC

Figure 14. Best mud type by EPC

An investigation was conducted on the modelling process and the outcomes are presented in Table 6 and Figures 15 and 16 The results (predicted values) from the use of ACO, BCO and EPC were compared with the measured values (targets). These predicted values also were compared with the obtained values.

Table 6. Comparison of models

Figure 15: Comparison of models

Figure 16. Best mud type from models

Figure 17. Comparison of results from ACO, BCO and EPC and target

5.1. Comparative Algorithm Performance and Robustness

The superior performance of the Bee Colony Optimization (BCO) algorithm (R² = 0.9841, RMSE = 0.0245) over Ant Colony Optimization (ACO) and Emperor Penguins Colony (EPC) can be attributed to its intrinsic search mechanisms. BCO employs a balanced combination of employed, onlooker, and scout bees, which allows for robust exploitation of promising solutions while simultaneously exploring new regions of the search space to avoid local optima. This is particularly advantageous for the high-dimensional, non-linear relationship between drilling parameters and mud properties, where the risk of convergence to suboptimal solutions is high.

While ACO and EPC also delivered highly accurate results (R² > 0.96), their slightly lower performance may stem from their specific convergence behaviours. ACO's reliance on pheromone trails can sometimes lead to premature stagnation if the evaporation rate (ρ) is not optimally tuned. EPC, inspired by huddling behaviour, is effective for continuous optimization but might be less agile than BCO in discretely navigating the complex weight space for each parameter in our predictive equation. The consistent high performance across all three nature-inspired algorithms, however, validates the fundamental premise that metaheuristic-driven ML models are exceptionally well-suited for capturing the intricate rock-fluid interactions that elude traditional analytical methods.

5.2. Sensitivity Analysis and Parameter Influence

A critical step in validating any data-driven model is to interpret its inner workings. To this end, we conducted a comprehensive sensitivity analysis based on the final weight coefficients (Wi) from the optimal BCO model (Equation 11). The relative importance of each input parameter was quantified and is presented in Figure 18.

Figure 18. Parameter sensitivity analysis diagram

According to Figure 18, ROP (W2): 100% (Normalized Importance): as indicated by its high positive weight (0.0664), ROP emerged as the most influential parameter. This is mechanistically sound; ROP is a direct manifestation of the drill bit's interaction wth the formation. A high ROP often indicates softer, more porous, or fractured rock, which may require a different mud weight to stabilize and a specific chemical type to inhibit interaction. The strong correlation suggests that ROP acts as a real-time proxy for in-situ rock strength and lithology.

Weight on Bit (WOB) (W5): ~65% Importance - the significant negative weight (-1.9433) of WOB indicates an inverse relationship with mud density. Higher WOB typically leads to higher ROP and generates more cuttings, potentially necessitating mud system adjustments. Its high importance underscores its role as a primary controlled variable affecting the drilling process's mechanical efficiency.

Diameter (W3): ~35% Importance - the negative coefficient (-0.2365) suggests that larger boreholes may require slightly lower mud weights, possibly due to the reduced hoop stress and different cuttings load.

Flow Rate (W6): ~30% Importance - flow rate is crucial for hole cleaning. Its positive influence (0.2137) aligns with the need for adequate hydraulic energy to lift cuttings, which influences the effective mud density and rheology downhole.

RPM (W4): ~25% Importance - while important, RPM's lower influence suggests its effect might be partially captured by its correlation with ROP.

TVD (W1): ~3% Importance - surprisingly, TVD had the least direct influence in this model. This could be because the critical changes in mud properties are more directly linked to lithological changes and drilling mechanics at a specific depth rather than the depth itself, and this lithological information is indirectly captured by parameters like ROP and WOB.

This sensitivity analysis moves beyond mere prediction, offering valuable interpretability. It informs mud engineers that, for proactive mud management, real-time monitoring of ROP and WOB is paramount. The ROP model among all models is shown in Figure 19.

Figure 19. ROP in all models

Moreover, a 3D modelling approach illustrated the non-linear thresholds where ROP alterations necessitate immediate changes in mud composition. This finding corroborates the premise that real-time data integration with ML models enhances decision-making processes as opposed to relying solely on deterministic models, which may not account for the complexities of subsurface conditions, as shown in Figure 20.

Figure 20. ROP in all models

5.3. Practical Implications and Model Consensus on Mud Type

A key practical outcome of this research is the strong, cross-model consensus identifying Sea Water-based mud with Polymer and Soltex (SW-PO-SX) as the optimal fluid for the studied field, with a confidence level of 79-87%. This recommendation is not a black-box output but a data-driven conclusion derived from 50 years of operational experience encoded in the daily drilling reports.

The SW-PO-SX system offers a balanced solution:

The high confidence level across three different algorithms significantly de-risks the fluid selection process for future wells in this field, transforming it from an experience-based art to a systematic, repeatable science.

6.1. Limitations of the Study

Despite the promising results, this research is subject to several limitations:

6.2. Suggested Improvements

To address the above limitations and strengthen the current model, the following immediate improvements are suggested:

1. Data Pre-Processing Enhancement: implement more advanced data imputation and cleansing techniques to handle missing or anomalous data points in the DDRs. Incorporating a domain-expert review to label data quality could also be beneficial.

2. Feature Engineering and Expansion: expand the input feature set to include mud rheological properties (e.g. plastic viscosity, yield point), lithology codes from mud logs, and estimated pore pressure and fracture gradient curves where available.

3. Hybrid Model Development: create a hybrid model that combines the global search capability of BCO with the strong predictive performance of an ensemble method (e.g. using BCO to optimize the hyperparameters of an Extreme Gradient Boosting (XGBoost) model).

6.3. Future Research Directions

This work opens several avenues for future research:

In conclusion, this research provides a robust, data-driven foundation for systematic drilling fluid selection. By acknowledging its limitations and pursuing the outlined future directions, the industry can move closer to fully automated, adaptive, and optimized drilling fluid management, significantly enhancing safety, efficiency, and sustainability in hydrocarbon exploration.

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