Enhancing sand production assessment through accurate determination of Young’s modulus and Poisson’s ratio

Abstract

Sand production is a critical concern in the petroleum industry, often leading to costly equipment damage, production downtime, and safety risks. Accurate prediction of sand-prone zones is essential for proactive sand management and optimized well design. Although several empirical and machine learning-based models exist to estimate static Young’s modulus (Es) and static Poisson’s ratio (νs)—key inputs for sand production prediction methods such as the Sand Production Index (B), shear modulus to bulk compressibility ratio (G/Cb), and Schlumberger Sand Index (S/I)—their performance varies widely, and their reliability has not been studied. This study addresses that gap by conducting a comparative evaluation of multiple estimation models for Es and ν_s from the literature, using a dataset of 100 samples with measured Es and ν_s values from existing models. The study investigates how different input models affect the accuracy of B, G/Cb, and S/I sand production predictions and rock type identification. Results demonstrate that while many models yield inconsistent outputs and often misclassify sanding zones, one of the evaluated models achieves near-perfect agreement with measured data (coefficient of determination = 0.9998, minimal root mean square error = 2.78E-17), leading to significantly more reliable sand production forecasts across all methods evaluated. By quantifying prediction inconsistencies and strengths among widely used models, this work provides critical insights into model selection for well-log interpretation. It highlights the risks of relying on poorly calibrated methods. The findings offer practical guidance for improving sand risk evaluation in the field.

Introduction

A substantial percentage of the world’s petroleum reservoirs are in weakly consolidated sandstone reservoirs. Thus, it is susceptible to sand production. The sand production can be caused by material degradation, drilling processes, start-up and shut-in operations, depletion of reservoir pressure, and high-pressure gradient¹. Sand production causes many issues in the petroleum industry, namely, a reduction in reservoir recovery². Methods for controlling sand production include mechanical, chemical, and rate control.

Sand production prediction is influential in the petroleum industry for managing and controlling sand problems in the early stages, before they become serious dilemmas³. Some methods were used to find the sand production. Li et al.⁴ used theoretical models to establish sand migration prediction depending on experimental data and its implications on the field of natural gas hydrate. Complete coverage of the screen with gravel can be critical to the long-term effectiveness of gravel packing to control the screen from erosion and sand production in the field, natural gas hydrate⁵.

The total drawdown (TD) is the difference between the reservoir pressure and the flowing bottom-hole pressure. When the reservoir pressure is static, no fluid is produced, and the pressure at the sand surface equals the static reservoir pressure. The reservoir can be in a dynamic state when the fluid is produced. The pressure at the perforations of the wellbore (sand face) must be lowered from the static reservoir to permit the fluid to flow toward the wellbore. The TD demonstrates how much pressure is relieved by the fluid flow from the reservoir toward the wellbore. The TD at which the sand production starts is known as the critical total drawdown (CTD)⁶. As deliberated, CTD can efficiently predict sand production. Some researchers, such as Kanj and Abousleiman⁷, Khamehchi et al.⁸, Alakbari et al.⁹, and Alakbari et al.^10,11 used data-driven approaches to find CTD.

Sand production prediction methods include content and concepts such as sand risk evaluation, critical sanding pressure drawdown predictions, sanding radius and size predictions, sand rate predictions, sanding cavity pattern predictions, etc. However, the scope of this study is to use standard methods for indicating sand production, such as the sand production index method (B), the shear modulus to bulk compressibility ratio (G/Cb), and the Schlumberger sand production index method (S/I)^12–14. These methods, B, Gg/Cb, and S/I, can be determined based on mechanical parameters, Young’s modulus (E), and Poisson’s ratio (ν). These methods, B, Gg/Cb, and S/I, can be determined using equations discussed in the methodology section.

The E can be found using some methods. Pigott et al.¹⁵ used amplitude variations with offset (AVO) inversion to obtain the E. The E can also be found by laboratory measurements using uniaxial compression experiments. Nonetheless, laboratory experiments have disadvantages, such as cost and time. The E, which is obtained directly from the experimental, is called the static E (Es); however, another E is the dynamic E (Ed), which is obtained from indirect acoustic measurements based on density compressional and shear wave velocities of the rocks using the following equation:

1

where: is the bulk formation density, g/cm³; is the shear wave velocity, km/s;

is the compressional wave velocity, km/s; is the dynamic Young’s modulus, GPa¹⁶.

Therefore, some studies, such as Heerden¹⁷, Christaras et al.¹⁸, Lacy¹⁹, Bradford et al.²⁰, Wang²¹, Wang and Nur²², Canady²³, and Fie et al.²⁴ used empirical correlations to estimate the E_s from experimental data. Mahmoud et al.²⁵ employed statistical methods to derive the Es equation from 300 datasets, utilizing bulk formation density (RHOB), shear time (DTs), and compressional time (DTc) as features. Brotons et al.²⁶ applied linear and nonlinear methods to estimate Es using 33 datasets, with Ed as input. Davarpanah et al.²⁷ obtained Es using non-linear logarithmic and power regression for specific rock types.

Machine learning methods are powerful mapping tools with an extraordinary capacity to approximate nonlinear multivariate functions²⁸. Alakbari et al.¹⁶ used the Gaussian process regression (GPR) approach to estimate Es from 1853 datasets collected from different locations. Mahmoud et al.²⁹ applied four machine learning methods—ANN (Artificial Neural Network), FNN (Feedforward Neural Network), SVM (Support Vector Machine), M-FIS (Mamdani Fuzzy Inference System)—to forecast Es of sandstone formations using conventional well logs, based on 592 data points. All models displayed outstanding precision, with correlation coefficients above 0.99. Kholy et al.³⁰ predicted Es of Caney shale by integrating ultrasonic lab measurements with well-log data and using Random Forest (RF) and Extreme Gradient Boosting (XGBoost) machine-learning models.

Poisson’s ratio (ν) can be obtained by using laboratory measurements, and this ν is known as the static ν (ν_s) from radial strain in the y-direction and the axial strain in the x-direction.

2

where is the radial strain in the y-direction, and is the axial strain in the x-direction.

The ν is determined using a dynamic method and known dynamic ν_dyn:

3

where is the compressional velocity, km/s; is the shear velocity, km/s; and is the dynamic Poisson’s ratio¹⁶.

Some studies, such as Kumar et al.³¹, Khandelwal et al.³², Ranjbar-Karami et al.³³, Brandås et al.³⁴, and Feng et al.³⁵ used empirical correlations and models to predict νs from experimental data. Gowida et al.³⁶ presented the ν_s correlation using the Artificial Neural Network (ANN) and based on 692 datasets. Some other studies used machine learning to predict the Es. Alakbari et al.³⁷ used a gated recurrent unit (GRU) deep learning approach to estimate the νs from 1691 datasets. They showed that their model is the best among the studied models³⁷.

In our previous studies, we mentioned the accuracy of the existing Es and ν_s models^16,37. The existing models to predict the Es are those of Mahmoud et al. ²⁵, Heerden ¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, Bradford et al.²⁰ correlations, and Alakbari et al.¹⁶ model. The existing models to predict the ν_s are Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², Brandas et al.³⁴ correlations, and Alakbari et al.³⁷ models. As we discussed_, sand production can be detected using B, G/Cb, and S/I methods, and these methods can be determined based on E and ν. The Es and ν_s instead of Ed and ν_d were used because they were from direct measurements. The measurements and predictions of the Es and ν_s effect on the B, G/Cb, S/I determinations to predict the sand production. The Es and ν_s can also be used to detect the rock types.

However, to our knowledge, no studies have investigated the effects of Es and νs determinations on the B, G/Cb, and S/I measurements for predicting sand production. Additionally, no research has been conducted to evaluate rock types based on Es and νs.

This study aims to review and compare sand-prediction methods using B, G/Cb, and S/I approaches, based on existing empirical correlations and models, as well as measured Es and ν_s. Additionally, rock types were identified using both the models and measured values of Es and νs. The study utilizes measured and predicted Es and ν_s to calculate the B, G/Cb, and S/I ratios, which are then used to forecast sand production. Furthermore, measured and predicted values of Es and ν_s were employed to classify rock types. The effectiveness of existing correlations and models in predicting Es and ν_s was compared against measured data to determine the B, G/Cb, and S/I parameters for predicting sand production and rock types based on these properties.

Methodology

Predictions of sand production based on static Young’s modulus (Es) and static Poisson’s ratio (ν_s)

In this study, first, the predicted Es values were determined from existing correlations and models, including those of Mahmoud et al.²⁵, Heerden¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, Bradford et al.²⁰ correlations, and Alakbari et al.¹⁶ model using 100 datasets from¹⁷ based on bulk formation density (RHOB), compressional time (DTc), and shear time (DTs). The predicted _s values were also determined from existing correlations and models, including Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², Brandas et al.³⁴ correlations, and Alakbari et al.³⁷ model applying 100 datasets from¹⁷ based on RHOB, DTc, and DTs. The measured datasets were collected from our previously published studies, Alakbari et al.¹⁶ and³⁷.

Sand production can be detected by using B (Eq. 4), G/Cb, (Eq. 5), and S/I (Eq. 8) methods based on the Es and ν_s.

The B can be determined using the following equation:

4

where:

B: Sand production index, psi.

Es: Elasticity modulus or static Young modulus, psi.

: Static Poisson’s ratio.

High reservoir sand will be produced when the B is less than 2×10⁶ psi^13,14.

The G/Cb can be found by applying the given equation:

5

where the G and Cb can be found by the following equations:

6

7

where:

G: Shear modulus, psi.

: Bulk compressibility, 1/psi.

Es: Elasticity modulus or static Young modulus, psi.

: Static Poisson’s ratio.

A threshold for sanding existed at = 0.8 . The ratio ( less than 0.7 indicates a high probability of sanding¹⁵.

8

9

where:

S/I: Schlumberger sand production index,

K: bulk modulus, MPa.

: bulk density, g/cc.

compression wave, s/ft.

shear wave, s/ft.

G: Shear modulus, psi.

The S/I of 1.24 indicates that the formation is likely to produce sand¹⁴.

In this research, we studied the effects of predicted Es values derived from existing correlations and models on the B, G/Cb, and S/I values. The predicted Es values, determined from existing correlations and models, include those of Mahmoud et al.²⁵, Heerden¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, Bradford et al.²⁰ correlations, and Alakbari et al.¹⁶ model. We also studied the effects of the predicted _s values that were determined from existing correlations and models on the B, G/Cb, and S/I values. The predicted _s values determined from existing correlations and models include those of Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², Brandas et al.³⁴ correlations and Alakbari et al.³⁷ model. In addition, the measured Es and _s were used as a benchmark or basic data to study their effects on B, G/Cb, and S/I values. The B, G/Cb, and S/I determinations are based on the Es and _s. Therefore, to study the effects of the Es values on the B, G/Cb, and S/I values, the _s values kept constant value with changing the Es values. To study the effects of the _s values on the B, G/Cb, and S/I values, the Es values kept constant values with changing the _s values. The different correlations and models were compared, and the optimal model was selected based on the values closest to the measured values, as the measured values are the most accurate for detecting sand production based on B, G/Cb, and S/I values.

Rock types based on static Young’s modulus and static Poisson’s ratio

Table 1 shows the sand consolidation as a function of Es. The rock types are based on _s display in Table 2.

Table 1.

Sand consolidation based on static Young’s modulus (Es)^38,39.

Sand’s type	Static Young’s modulus (psi)	Static Young’s modulus (GPa)
Zero-strength dry sand	50,000	0.34Szc
very weak damp sand	300,000	2.068
weakly cemented	500,000	3.447
weak more cemented	1000,000	6.894
a gray area	2,000,000	13.789
consolidated	3,500,000	24.131

Table 2.

Rock types are based on static Poisson’s ratio ( _s)⁴⁰.

Sand’s type	Static Poisson’s ratio
The soft rocks, namely clays	0.1—0.3
Medium rocks, such as sandstone	0.2—0.3
The hard rocks	0.3 – 0.4

As shown in Tables 1 and 2, the sand consolidation and rock types depend on the Es and _s values. Therefore, we studied the effects of the predicted Es values derived from existing correlations and models on sand consolidation and rock types. The predicted Es values derived from existing correlations and models include those of Mahmoud et al.²⁵, Heerden¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, Bradford et al.²⁰ correlations, and Alakbari et al.¹⁶ model. We also studied the effects of the predicted _s values that were determined from existing correlations and models on the sand consolidation and rock types. The predicted _s values that were determined from existing correlations and models include Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², Brandas et al.³⁴ correlations, and Alakbari et al.³⁷ model. In addition, the measured Es and _s were used as a benchmark basic data to study their effects on sand consolidation and on rock types. Finally, the correlations and models were compared, and the optimal model was selected based on the values closest to the measured values, as the measured values are the most accurate for detecting sand consolidation and rock types.

Statistical error analyses

Statistical error analyses (SEA) used in this study are the correlation coefficient (R), coefficient of determination (R²), and root mean square error (RMSE) to compare the models.

The correlation coefficient is an essential tool in SEA, as R² measures the strength and direction of the linear relationship between measured data and predicted or theoretical values. In SEA, an R² close to a positive one indicates that the model or measurement method precisely replicates the trend of the actual data. On the other hand, a value near zero shows weak agreement and potential error. A negative R² indicates that the model forecasts the opposite of the observed data, highlighting substantial methodological or analytical challenges. By examining how tightly data points cluster around a straight line, the R² offers insight into the precision of the data. The R² can be calculated by using the following equation:

10

where:

The coefficient of determination (R²) is typically used in SEA as it indicates how much of the variability in the observed data is explained by a predictive model. An R² value close to 1 indicates that most of the variation in the dependent variable is explained by the model, indicating that the errors between predicted and measured values are relatively small. On the other hand, a low R² suggests that the model does not account for much of the data’s behaviour, indicating higher unexplained error and potential issues with the model structure, missing variables, or measurement noise. By measuring the proportion of explained versus unexplained variance, R² delivers a strong indication of the model’s precision. The R-squared can be determined by using the following equation:

11

where:

The root mean square error (RMSE) is a significant metric in SEA because it provides a direct measure of the average magnitude of the prediction. By squaring the differences between predicted and observed values, RMSE highlights larger errors, making it highly sensitive to substantial deviations that may show model weaknesses or data quality issues. Taking the square root returns the error to the original units of the data, allowing for intuitive interpretation and comparison. A lower RMSE shows a model that closely matches observed data, whereas a higher value shows greater uncertainty, variability, or inaccuracies in predictions. Therefore, RMSE is crucial for evaluating the model’s accuracy. The RMSE is calculated from Eq. (12):

12

where is the deviation error that can be determined as follows:

13

i = 1, 2, 3, …, n.

Statistical error analyses were conducted for the B, G/Cb, and the S/I methods based on the Es values from the existing correlations while keeping the υ_s constant. Statistical error analyses were also conducted for the B, G/Cb, and S/I methods based on the υs values from the existing correlations while keeping the Es constant.

Results

Sand production prediction

As we discussed in the methodology, Sand production can be detected using the B, G/Cb, and S/I methods. These methods are determined based on Es and υ_s values. This study uses Es and υ_s obtained from existing correlations and models, including those of Mahmoud et al.²⁵, Heerden¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, Bradford et al.²⁰ correlations, Alakbari et al.¹⁶ model, Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², Brandas et al.³⁴ correlations, and Alakbari et al.³⁷ model to determine the B, G/Cb, and S/I methods to detect sand production.

In addition, the measured Es and _s were used as a benchmark or basic data to study their effects on B, G/Cb, and S/I values. The B, G/Cb, and S/I determinations are based on the Es and _s.; therefore, to study the effects of the Es values on the B, G/Cb, and S/I values, the _s values remained constant when changing the Es values. To study the effects of the _s values on the B, G/Cb, and S/I values, the Es values kept constant values with the _s values. In this study, 100 datasets from the same data in our existing studies were used¹⁷ and³⁵ were used to find the Es and υ_s. The Es can be determined using the bulk formation density (RHOB), compressional time (DTc), and shear time (DTs) by applying the existing models, including those of Mahmoud et al.²⁵, Heerden¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, Bradford et al.²⁰ correlations, and Alakbari et al.¹⁶ model. The υ_s can be determined using the bulk formation density (RHOB), compressional time (DTc), and shear time (DTs) by using the existing models, including Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², Brandas et al.³⁴ correlations, and Alakbari et al.³⁷ model.

Sand production index (B) method

The sand production index (B) can be calculated from Es and υ_s values. When the B is less than 2 , sand formation can produce sand. The B is determined using Eq. 4 based on changing Es values that were found from existing models, including Mahmoud et al.²⁵, Heerden¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, Bradford et al.²⁰ correlations, and Alakbari et al.¹⁶ model and keep the υ_s value constant to assess the effects of existing Es models on the B for detecting sand formation. In addition, B was determined using Eq. 4, based on changing the measured Es values while keeping the υ_s value constant, to assess the effects of the measured Es values on B for detecting sand formation. These values were used as basic data or a benchmark to detect sand production. All B values that were determined from the measured Es values and Es values from existing corrections and models, while keeping the υ_s value constant, were compared.

The B is also determined using Eq. 4, based on changes in υ_s values obtained from existing models, including Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², Brandas et al.³⁴ correlations, and Alakbari et al.³⁷ model and keeping the Es value constant to check existing υ_s models’ effects on the B for detecting sand formation. In addition, B was determined using Eq. 4, based on changing the measured υ_s values while keeping the Es value constant, to assess the effects of the measured υ_s values on B for detecting sand formation. These values were used as basic data or a benchmark to detect sand production. All B values that were determined from the measured υ_s values and υ_s values from existing corrections and models, while keeping the Es value constant, were compared.

Figure 1 shows the B values determined by varying the measured Es values and those from the existing Alakbari et al.¹⁶ (GPR) model while keeping the υ_s constant using Eq. 4. As shown in Fig. 1, samples from one to eight and 46 to 56 have sand production for the measurements, and Alakbari et al.¹⁶ (GPR) model values. However, other samples have no sanding for the measurements, and Alakbari et al.¹⁶ (GPR) model values. Alakbari et al.¹⁶ (GPR) model can accurately detect sand production using the B method based on the Es values that were determined using Alakbari et al.¹⁶ (GPR) model.

Fig. 1.

The sand production index (B) method is based on 100 datasets from Alakbari et al.¹⁶ model for the predicted Es values and the measured Es values from the same reference, Alakbari et al.¹⁶ while keeping the υ_s constant.

Figure 2 shows the B values determined by changing the measured Es values and Es values from the existing correlations, while keeping the υs constant, using Eq. 4. The existing Es correlations include those of Mahmoud et al.²⁵, Heerden¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, and Bradford et al.²⁰ correlations. As shown in Fig. 2, Mahmoud et al.²⁶ and Bradford et al.²⁰ correlations indicate that there is sanding for some samples that are not producing the sand for the measurements’ values. However, the previous correlations do not show any sanding in the sanding measurements. Therefore, some existing models fail to predict sand production when the B method is applied accurately. This study confirms that the Alakbari et al.¹⁶ (GPR) model surpasses some existing correlations to determine the Es for detecting sand production. On the other hand, the other existing correlations fail in detecting sand production based on their Es models.

Fig. 2.

The sand production index (B) method is based on 100 datasets for the measured Es values from Alakbari et al.¹⁶ and predicted Es values for other models used in the same reference, Alakbari et al.¹⁶ while keeping the υ_s constant.

Table 3 compares the performance of various predictive models using statistical metrics: correlation coefficient (R), coefficient of determination (R²), and root mean square error (RMSE). Among all models, the Gaussian Process Regression (GPR) model by Alakbari et al.¹⁶ stands out with near-perfect correlation (R = 0.9999), almost complete variance explanation (R² = 0.9998), and an extremely low RMSE (2.78E-17), indicating exceptional predictive accuracy. In stark contrast, the remaining models exhibit poor performance, with low or negative correlation values (R ranging from − 0.1741 to 0.2456), minimal explanatory power (R² mostly below 0.03), and significantly higher RMSE values, often exceeding 15. These results highlight the substantial superiority of the Alakbari et al.¹⁶ model in this context, while the others demonstrate limited reliability and predictive capability. The models used in this study to estimate Es and υ_s were not developed here but were adopted directly from previously published work by Alakbari et al.^16,37. These correlations were initially derived from extensive laboratory datasets and were independently calibrated and validated in the cited studies, where their high predictive accuracy was repeatedly demonstrated. In this study, we applied these established models solely to obtain Es and υ_s values, which were then used in Eqs. (4–9) to calculate the B, G/Cb, and S/I indices for sand production evaluation. No training or model fitting was performed, and the dataset used here was entirely independent of the datasets used in the original model development, eliminating the possibility of overfitting or data leakage. The consistently strong performance observed, therefore, reflects the inherent robustness and generalizability of the Alakbari et al.^16,37 models rather than methodological bias.

Table 3.

Statistical error analysis for the sand production index (B) method based on the Es values from the existing correlations while keeping the υs constant.

Model	R	R2	RMSE
Alakbari et al.16	0.9999	0.9998	2.78E − 17
Mahmoud et al.36	0.2456	0.0603	4.38E + 00
Heerden17	− 0.0944	0.0089	1.80E + 01
Davarpanah et al.27	− 0.1569	0.0246	1.88E + 01
Fie et al.24	− 0.1329	0.0177	7.64E + 00
Brotons et al	− 0.1329	0.0177	1.38E + 01
Lacy19	− 0.1329	0.0177	6.04E + 00
Canady23	− 0.1469	0.0216	1.81E + 01
Wang and Nur22	− 0.1329	0.0177	1.65E + 01
Wang21	− 0.1329	0.0177	5.59E + 00
Christaras et al.18	− 0.1329	0.0177	1.58E + 01
Bradford et al.20	− 0.1741	0.0303	2.84E + 01

Table 4 shows the number of sanding and non-sanding samples predicted by the B method when the E_s values from the existing correlations were used while keeping υ_s constant. The results indicate that only a limited number of models could detect sanding intervals, with both the measured data and the Alakbari et al.¹⁶ correlation identifying 19 sanding samples. Mahmoud et al.²⁵ and Bradford et al.²⁰ confirmed greater sensitivity, predicting 32 and 34 sanding samples, respectively. In contrast, all remaining models—comprising those by Heerden¹⁷, Davarpanah et al.²⁷, Fie et al.²⁴, Brotons et al., Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, and Christaras et al.¹⁸—classified all samples as non-sanding, failing to detect any sanding-prone intervals. The results prove significant variability in predictive capability among commonly used E_s correlations and emphasize the importance of selecting models that can reliably distinguish sanding from non-sanding conditions.

Table 4.

Number of sanding and non-sanding samples for the sand production index (B) method based on the Es values from the existing correlations while keeping the υs constant.

Model	Sanding samples	Non-sanding samples
Measured	19	81
Alakbari et al.16	19	81
Mahmoud et al.25	32	68
Heerden17	0	100
Davarpanah et al.27	0	100
Fie et al.24	0	100
Brotons et al	0	100
Lacy19	0	100
Canady23	0	100
Wang and Nur22	0	100
Wang21	0	100
Christaras et al.18	0	100
Bradford et al.20	34	66

Figure 3 shows the B values that were determined based on changing the measured _s values and _s values from the existing Alakbari et al.³⁷ (GRU) model while keeping the Es constant using Eq. 4. As shown in Fig. 3, all samples have no sanding for the measurements, and Alakbari et al.³⁷ (GRU) model values. Most of the Alakbari et al.³⁷ (GRU) model values match the measurement values. Therefore, Alakbari et al.³⁷ (GRU) model can adequately detect sand production using the B method based on _s.

Fig. 3.

The sand production index (B) method is based on 100 datasets from Alakbari et al.¹⁶ model for the predicted _s values and the υ_s measured values from the same reference, Alakbari et al.¹⁶ while keeping the Es constant.

Figure 4 shows the B values that were determined based on changing the measured _s values and _s values from the existing correlations while keeping the Es constant using Eq. 4. The existing correlations include Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, and Kumar et al.³¹ correlations. As shown in Fig. 4, Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, and Ranjbar-Karami et al.³³ correlations indicate sanding for most samples; however, the real values show no sanding. Kumar et al.³¹ correlation shows no sanding for most samples; however, the values are close to the sanding. Therefore, the existing correlations did not accurately indicate sand production.

Fig. 4.

The sand production index (B) method is based on 100 datasets for the measured υ_s values from Alakbari et al.¹⁶ and predicted υ_s values for Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, and Kumar et al.³¹ correlations used in the same reference as Alakbari et al.³⁷ while keeping the Es constant.

Figure 5 shows the B values that were determined based on changing the measured _s values and _s values from the existing Brandas et al.³⁴ and Khandelwal et al.³² correlations while keeping the Es constant using Eq. 4. As shown in Fig. 5, the Brandas et al.³⁴ correlation indicates that the sanding for all samples fails to show the proper sand production indications. Khandelwal et al.³² model shows no sanding for all samples, but most values are far from the measured values. Therefore, Brandas et al.³⁴ and Khandelwal et al.³² correlations could not prove the proper sand production indications.

Fig. 5.

The sand production index (B) method is based on 100 datasets for the measured υ_s values from Alakbari et al.³⁷ and predicted υ_s values for Brandas et al.³⁴ and Khandelwal et al.³² correlations used in the same reference as Alakbari et al.³⁷ while keeping the Es constant.

In summary, most of the existing correlations fail to indicate sand production using the B method based on Es and _s. However, Alakbari et al.¹⁶ (GPR) and Alakbari et al.³⁷ (GRU) models show the proper sand production indications and match the measured values for most samples using the B method Eq. 4.

Table 5 presents a performance comparison of various predictive models based on R, R², and RMSE. Among the models, Feng et al.³⁵ and Christaras et al.¹⁸ demonstrate strong performance, with high R values (0.8804 and 0.8704, respectively) and R² values above 0.75, indicating good predictive strength, though their RMSEs (~ 0.49) suggest moderate prediction error. Alakbari et al.³⁷ ‘s model also performs well, with slightly lower R and R², but achieves a significantly lower RMSE (0.1608), indicating better predictive accuracy. Gowida et al.³⁶ and Ranjbar-Karami et al.³³ report competitive R values above 0.86 but exhibit higher RMSEs (above 0.7), indicating lower precision. Kumar et al.³¹ and Khandelwal et al.³² perform poorly—Kumar’s model shows a negative R² and a high RMSE. In contrast, Khandelwal’s model exhibits a negligible R² and an extremely large RMSE (41.88), indicating a failure to capture the data’s trends. Brandås et al.³⁴ model shows weak correlation and limited predictive power, with an R² of just 0.08. Overall, the results highlight that while several models offer good correlation, their accuracy varies considerably, with Alakbari et al.³⁷ model striking the best balance between fit and precision.

Table 5.

Statistical error analysis for the sand production index (B) method based on the υs values from the existing correlations while keeping the Es constant.

Model	R	R2	RMSE
Alakbari et al.37	0.8315	0.6913	0.1608
Christaras et al.18	0.8704	0.7576	0.4977
Feng et al.35	0.8804	0.7751	0.4868
Kumar et al.31	− 0.5708	0.3259	0.3352
Gowida et al.36	0.8686	0.7545	0.7038
Ranjbar − Karami et al.33	0.8636	0.7457	0.8132
Khandelwal et al.32	− 0.0181	0.0003	41.8827
Brandås et al.34	0.2823	0.0797	2.4114

Table 6 shows the sanding and non-sanding numbers found when υ_s values from several published correlations were applied in the B-method calculations, with E_s held constant. The results show a wide disparity in the ability of these models to detect sanding-prone intervals. Both the measured υ_s values and the Alakbari et al.³⁷ correlation predicted no sanding occurrences, classifying all samples as non-sanding. Some models, however, showed moderate sensitivity, comprising Christaras et al.¹⁸ and Khandelwal et al.³², which identified 36 and 32 sanding samples, respectively. Other models produced far more extreme responses: Feng et al.³⁵ detected 40 sanding samples, whereas Gowida et al.³⁶ and Brandås et al.³⁴ classified nearly the entire dataset as sanding, and the Ranjbar-Karami et al.³³ model predicted sanding in all samples. These contrasting results underscore the strong influence of υ_s estimation methods on sand production assessment and highlight the need to choose correlations that deliver physically meaningful and geomechanically consistent predictions.

Table 6.

Number of sanding and non-sanding samples for the sand production index (B) method based on the υ_s values from the existing correlations while keeping the Es constant.

Model	Sanding samples	Non-sanding samples
Measured	0	100
Alakbari et al.37	0	100
Christaras et al.18	36	64
Feng et al.35	40	60
Kumar et al.31	0	100
Gowida et al.36	99	1
Ranjbar-Karami et al.33	100	0
Khandelwal et al.32	32	68
Brandås et al.34	100	0

Shear modulus to bulk compressibility ratio (G/Cb)

Sand production can be detected using the G/Cb method. When the G/Cb is less than 0.8 , it is indicated as sanding. The G/Cb is determined using Eq. 5 based on changing Es values that were found from existing models, including Mahmoud et al.²⁵, Heerden¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, Bradford et al.²⁰ correlations, and Alakbari et al.¹⁶ model and keep the υ_s value constant to assess the effects of existing Es models on the G/Cb for detecting sand formation. In addition, the G/Cb was determined using Eq. 5, in which the measured Es values were varied while keeping the υs value constant to assess the effects of the measured Es values on the G/Cb for detecting sand formation. These values were used as basic data or a benchmark to detect sand production. All G/Cb values that were determined from the measured Es values and Es values from existing corrections and models, while keeping the υs value constant, were compared.

The G/Cb is also determined using Eq. 5 based on changing υ_s values that were found from existing models, including Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², Brandas et al.³⁴ correlations, and Alakbari et al.³⁷ model, and keeping the Es value constant to check existing υ_s models’ effects on the G/Cb for detecting sand formation. In addition, the G/Cb was determined using Eq. 5 by changing the measured υ_s values while keeping the Es value constant to assess the effects of the measured υ_s values on the G/Cb for detecting sand formation. These values were used as basic data or a benchmark to detect sand production. All G/Cb values that were determined from the measured υ_s values and υ_s values from existing corrections and models, while keeping the Es value constant, were compared.

Figure 6 shows the G/Cb values determined by varying the measured Es values and those from the existing Alakbari et al.¹⁶ (GPR) model while keeping the υ_s constant using Eq. 5. As shown in Fig. 6, Alakbari et al.¹⁶ (GPR) The model indicates sanding for samples (1–8) and (44–57) and matches the measured values; furthermore, the Alakbari et al.¹⁶ (GPR) model has no sanding for the other samples and matches the measured values. Therefore, the Alakbari et al.¹⁶ (GPR) model indicates the proper sand production for all samples as the measurement values.

Fig. 6.

The shear modulus to bulk compressibility ratio (G/Cb) method is based on 100 datasets from Alakbari et al.¹⁶ model for the predicted Es values and the Es measured values from the same reference, Alakbari et al.¹⁶ while keeping the υ_s constant.

Figure 7 shows the G/Cb values determined by changing the measured Es values and Es values from the existing correlations, while keeping the υ_s constant, using Eq. 5. The existing Es correlations include those of Mahmoud et al.²⁵, Heerden¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, and Bradford et al.²⁰ correlations. As shown in Fig. 7, Mahmoud et al.²⁶ and Bradford et al.²⁰ models show the sanding for some samples that are not sanding for the measured values. However, other previous models do not show any sanding for the actual values. Therefore, all existing models in this figure fail to show the correct sand production using the G/Cb method based on Es (Fig. 7).

Fig. 7.

The shear modulus to bulk compressibility ratio (G/Cb) method is based on 100 datasets for the measured Es values from Alakbari et al.¹⁶ and predicted Es values for other models used in the same reference, Alakbari et al.¹⁶ while keeping the υ_s constant.

Table 7 presents a comparison of predictive model performance using three key statistical metrics: R, R², and RMSE. The Alakbari et al.¹⁶ model demonstrates outstanding predictive accuracy, with near-perfect R and R² values (0.9999 and 0.9998) and an extremely low RMSE (9.506E-06), indicating a highly reliable fit to the data. In contrast, all other models perform very poorly, with R values ranging from -0.2647 to 0.2075 and low R² values mostly below 0.07, suggesting weak or inverse correlations with the actual data. Additionally, their RMSE values are substantial (on the order of 10¹⁴), indicating substantial prediction errors and underscoring their lack of practical utility. Notably, the Mahmoud et al.²⁵ model shows a slightly positive R (0.2075) but still has an enormous RMSE (1.077E + 13). The overall comparison underscores the exceptional performance of the Alakbari et al.¹⁶ model compared to the other models, which fail to provide accurate or reliable predictions.

Table 7.

Statistical error analysis for the shear modulus to bulk compressibility ratio (G/Cb) method based on the Es values from the existing correlations while keeping the υs constant.

Model	R	R2	RMSE
Alakbari et al.16	0.9999	0.9998	9.506E − 06
Mahmoud et al.25	0.2075	0.0431	1.077E + 13
Heerden17	− 0.2262	0.0512	1.503E + 14
Davarpanah et al.27	− 0.2613	0.0683	1.562E + 14
Fie et al.24	− 0.2617	0.0685	2.878E + 13
Brotons et al	− 0.2607	0.0680	8.164E + 13
Lacy19	− 0.2594	0.0673	1.856E + 13
Canady23	− 0.2647	0.0700	1.402E + 14
Wang and Nur22	− 0.2644	0.0699	1.148E + 14
Wang21	− 0.2604	0.0678	1.591E + 13
Christaras et al.18	− 0.2605	0.0679	1.046E + 14
Bradford et al.20	− 0.1974	0.0390	4.226E + 14

Table 8 shows how variations in Es, based on earlier models, affect sanding detection when the G/Cb method is used with constant υs. The data and the model by Alakbari et al.¹⁶ produced identical results, identifying 22 sanding samples. Mahmoud et al.²⁵ predicted a higher sanding frequency with 34 samples, while Bradford et al.²⁰ showed the highest sensitivity, flagging 39 samples as sanding. Some models had limited detection capability—such as Fie et al.²⁴, Lacy¹⁹, and Wang²¹, which identified 6, 10, and 16 sanding samples, respectively. Conversely, several models, including those by Heerden¹⁷, Davarpanah et al.²⁷, Brotons et al., Canady²³, Wang and Nur²², and Christaras et al.¹⁸, did not predict any sanding intervals. The results demonstrate that the choice of Es correlation significantly impacts the G/Cb-based sanding assessment, with some models aligning closely with measured data, while others underpredict the sanding risk.

Table 8.

Number of sanding and non-sanding samples for the shear modulus to bulk compressibility ratio (G/Cb) method based on the Es values from the existing correlations while keeping the υs constant.

Model	Sanding samples	Non-sanding samples
Measured	22	78
Alakbari et al.16	22	78
Mahmoud et al.25	34	66
Heerden17	0	100
Davarpanah et al.27	0	100
Fie et al.24	6	94
Brotons et al	0	100
Lacy19	10	90
Canady23	0	100
Wang and Nur22	0	100
Wang21	16	84
Christaras et al.18	0	100
Bradford et al.20	39	61

Figure 8 shows the G/Cb values that were determined based on changing the measured _s values and _s values from the existing Alakbari et al.³⁷ (GRU) model while keeping the Es constant using Eq. 5. As shown in Fig. 8, Alakbari et al.³⁷ (GRU) model shows no sanding for most samples, the same as the measured values. Therefore, Alakbari et al.³⁷ (GRU) model confirms the proper sand production using the G/Cb method based on _s.

Fig. 8.

The shear modulus to bulk compressibility ratio (G/Cb) method is based on 100 datasets from Alakbari et al.³⁷ model for the predicted υ_s values and the υ_s measured values from the same reference, Alakbari et al.³⁷ while keeping the Es constant.

Figures 9 and 10 show the G/Cb values that were determined based on changing the measured _s values and _s values from the existing correlations while keeping the Es constant using Eq. 5. The existing correlations include Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², and Brandas et al.³⁴ correlations. As shown in Fig. 9, Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, and Ranjbar-Karami et al.³³ models indicate sanding for most samples that are not sanding for the measurements. Kumar et al.³¹ and Khandelwal et al.³² models show sanding for some samples, but the measured values do not. Brandas et al.³⁴ model indicates the sanding for all samples, Fig. 10. Therefore, the existing _s models in Figs. 9 and 10 fail to show the correct sand production using the G/Cb method.

Fig. 9.

The shear modulus to bulk compressibility ratio (G/Cb) method is based on 100 datasets for the measured υ_s values from Alakbari et al.³⁷ and predicted υ_s values for Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, and Khandelwal et al.³² correlations used in the same reference as Alakbari et al.³⁷ while keeping the Es constant.

Fig. 10.

The shear modulus to bulk compressibility ratio (G/Cb) method is based on 100 datasets for the measured υ_s values from Alakbari et al.³⁷ and predicted υ_s values for Brandas et al.³⁴ correlation used in the same reference as Alakbari et al.³⁷ while keeping the Es constant.

Table 9 compares the performance of several predictive models using R, R², and RMSE. The Alakbari et al.³⁷ model shows solid performance with an R of 0.8306 and a relatively low RMSE (~ 7.56E + 10), suggesting a good balance between correlation and prediction accuracy. However, Feng et al.³⁵ and Christaras et al.¹⁸ slightly outperform it in terms of R and R² (above 0.87 and 0.75, respectively), though their RMSE values are significantly higher (~ 2.3E + 11), indicating less precise predictions. Models by Gowida et al.³⁶ and Ranjbar-Karami et al.³³ also demonstrate strong correlation but suffer from even higher RMSEs, further reducing their reliability. In contrast, Kumar et al.³¹ and Khandelwal et al.³² exhibit poor performance, with negative R values and exceptionally high RMSEs, particularly for Khandelwal et al.³² ‘s model (1.92E + 13), indicating very low predictive accuracy. Brandås et al.³⁴ ‘s model offers moderate performance, with a low correlation (R = 0.32) and an RMSE in the order of 10¹². Overall, while several models exhibit strong correlation, the Alakbari et al.³⁷ model maintains a better balance of fit and predictive precision.

Table 9.

Statistical error analysis for the shear modulus to bulk compressibility ratio (G/Cb) method based on the υs values from the existing correlations while keeping the Es constant.

Model	R	R2	RMSE
Alakbari et al.37	0.83059	0.68989	75,629,215,983
Christaras et al.18	0.86893	0.75505	2.36463E + 11
Feng et al.35	0.87856	0.77186	2.31415E + 11
Kumar et al.31	− 0.57258	0.32785	1.57977E + 11
Gowida et al.36	0.86630	0.75048	3.37659E + 11
Ranjbar-Karami et al.33	0.86080	0.74098	3.93056E + 11
Khandelwal et al.32	− 0.01891	0.00036	1.92332E + 13
Brandås et al.34	0.32052	0.10273	1.01956E + 12

Table 10 presents the impact of υ_s correlations on sanding detection when using the G/Cb method under steady Es conditions. The measured υ_s values displayed a strong sanding response, with 87 samples below the threshold, whereas the Alakbari et al.³⁷ model yielded a slightly lower count of 74 sanding samples. Some correlations—like those proposed by Christaras et al.¹⁸, Feng et al.³⁵, Kumar et al.³¹, Gowida et al.³⁶, Ranjbar-Karami et al.³³, and Brandås et al.³⁴—classified the whole dataset as sanding, representing very high sensitivity in these models. Conversely, the correlation by Khandelwal et al.³² delivered a more moderate distribution, identifying 54 sanding samples. The results showed that υ_s-derived mechanical properties can substantially influence G/Cb predictions, highlighting the significance of indicating correlations that reflect realistic geomechanical behaviour to avoid overestimating sanding risk.

Table 10.

Number of sanding and non-sanding samples for the shear modulus to bulk compressibility ratio (G/Cb) method based on the υs values from the existing correlations while keeping the Es constant.

Model	Sanding samples	Non-sanding samples
Measured	87	13
Alakbari et al.37	74	26
Christaras et al.18	100	0
Feng et al.35	100	0
Kumar et al.31	100	0
Gowida et al.36	100	0
Ranjbar-Karami et al.33	100	0
Khandelwal et al.32	54	46
Brandås et al.34	100	0

Schlumberger sand production index (S/I) method

The Schlumberger sand production index (S/I) method can detect sand formation. When the S/I is less than 1.24 , it is indicated that there is sand in the formation. The S/I is determined using Eq. 8, based on changes in Es values derived from existing models, including those of Mahmoud et al.²⁵, Heerden¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, Bradford et al.²⁰ correlations, and Alakbari et al.¹⁶ model and keep the υ_s value constant to assess the effects of existing Es models on the S/I for detecting sand formation. In addition, the S/I was determined using Eq. 8 by varying the measured E_s values while keeping the υs value constant to assess the effects of the measured E_s values on the S/I for detecting sand formation. These values were used as basic data or a benchmark to detect sand production. All S/I values that were determined from the measured Es values and Es values from existing corrections and models, while keeping the υs value constant, were compared.

The S/I is also determined using Eq. 8, based on changes in υ_s values derived from existing models, including Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², Brandas et al.³⁴ correlations, and Alakbari et al.³⁷ model and keeping the Es value constant to check existing υ_s models’ effects on the S/I for detecting sand formation. In addition, the S/I was determined using Eq. 8 by varying the measured υ_s values while keeping the Es value constant to assess the effects of υ_s on the S/I for detecting sand formation. These values were used as basic data or a benchmark to detect sand production. All S/I values that were determined from the measured υ_s values and υ_s values from existing corrections and models, while keeping the Es value constant, were compared.

Figure 11 shows the S/I values determined by varying the measured Es values and those from the existing Alakbari et al.¹⁶ (GPR) model while keeping the υ_s constant using Eq. 8. As shown in Fig. 11, Alakbari et al.¹⁶ (GPR) model indicates the sanding for the same samples of the measured values. In addition, the Alakbari et al.¹⁶ (GPR) model shows no sanding for the same samples of the measured values. Therefore, the Alakbari et al.¹⁶ (GPR) model successfully shows the correct sand production using the S/I method based on Es.

Fig. 11.

The Schlumberger sand production index (S/I) method is based on 100 datasets from Alakbari et al.¹⁶ model for the predicted Es values and the measured Es values from the same reference, Alakbari et al.¹⁶ while keeping the υ_s constant.

Figure 12 shows the S/I values determined by varying the measured Es values and Es values from existing correlations, while keeping the υs constant, using Eq. 8. The existing Es correlations include those of Mahmoud et al.²⁵, Heerden¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, and Bradford et al.²⁰ correlations. As shown in Fig. 12, Mahmoud et al.²⁶ and Bradford et al.²⁰ models indicate the sanding for some samples that are not sanding. The other previous models do not show any sanding for the sanding samples. Thus, all existing models in this figure fail to predict sand production correctly using the S/I method based on Es, Fig. 12.

Fig. 12.

The Schlumberger sand production index (S/I) method is based on 100 datasets for the measured Es values from Alakbari et al.¹⁶ and predicted Es values for other models used in the same reference, Alakbari et al.¹⁶ while keeping the υ_s constant.

Table 11 compares the predictive performance of various models across three key metrics: R, R², and RMSE. The Alakbari et al.¹⁶ model once again demonstrates exceptional predictive capability, with an almost perfect correlation (R = 0.9999), near-total variance explanation (R² = 0.9998), and an extremely low RMSE (2.559E-17), indicating virtually flawless accuracy. Mahmoud et al.²⁵ also performs well, with a strong R of 0.9467 and R² of 0.8962, though with a higher RMSE of 4.81. The remaining models show a significant drop in performance, with R values mostly between 0.18 and 0.45, indicating weak to moderate correlation with the target variable. Lacy¹⁹, Wang²¹, and Brotons et al.²⁶ exhibit slightly better results among this group, with moderate R² values and RMSEs ranging from 13 to 26. However, Bradford et al.²⁰ performs poorly, with a negative R value and the highest RMSE (61.91), indicating minimal predictive reliability. Overall, the Alakbari et al.¹⁶ model clearly outperforms all others, with Mahmoud et al.²⁵ offering a respectable but distant second.

Table 11.

Statistical error analysis for the Schlumberger sand production index (S/I) method based on the Es values from the existing correlations while keeping the υs constant.

Model	R	R2	RMSE
Alakbari et al.16	0.9999	0.9998	2.559E-17
Mahmoud et al.25	0.9467	0.8962	4.810E + 00
Heerden17	0.2702	0.0730	3.562E + 01
Davarpanah et al.27	0.1859	0.0346	3.632E + 01
Fie et al.24	0.4031	0.1625	1.522E + 01
Brotons et al	0.4203	0.1766	2.561E + 01
Lacy19	0.4444	0.1975	1.335E + 01
Canady23	0.2808	0.0789	3.446E + 01
Wang and Nur22	0.3531	0.1247	3.096E + 01
Wang21	0.4275	0.1828	1.332E + 01
Christaras et al.18	0.4247	0.1804	2.986E + 01
Bradford et al.20	− 0.1403	0.0197	6.191E + 01

Table 12 presents the sanding and non-sanding classifications derived from the S/I method when E_s values from several published correlations were applied, with υ_s constant. The measured data and the Alakbari et al.¹⁶ correlation identified a few sanding intervals, with eight samples highlighted in each case. Mahmoud et al.²⁵ and Bradford et al.²⁰ confirmed somewhat higher sensitivity, predicting 17 and 16 sanding samples, respectively. On the other hand, most of the other models—comprising those by Heerden¹⁷, Davarpanah et al.²⁷, Fie et al.²⁴, Brotons et al., Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, and Christaras et al.¹⁸—classified all samples as non-sanding. The typical lack of sanding predictions across these models indicates that many Es correlations may not have sufficient resolution to trigger S/I threshold responses, emphasising the variability in model behaviour and the significance of selecting correlations capable of detecting subtle geomechanical changes relevant to sanding onset.

Table 12.

Number of sanding and non-sanding samples for the Schlumberger sand production index (S/I) method based on the Es values from the existing correlations while keeping the υs constant.

Model	Sanding samples	Non-sanding samples
Measured	8	92
Alakbari et al.16	8	92
Mahmoud et al.25	17	83
Heerden17	0	100
Davarpanah et al.27	0	100
Fie et al.24	0	100
Brotons et al	0	100
Lacy19	0	100
Canady23	0	100
Wang and Nur22	0	100
Wang21	0	100
Christaras et al.18	0	100
Bradford et al.20	16	84

Figure 13 shows the S/I values that were determined based on changing the measured _s values and _s values from the existing Alakbari et al.³⁷ (GRU) model while keeping the Es constant using Eq. 8. As shown in Fig. 13. Alakbari et al.³⁷ (GRU) model shows the same values of the measurements for most samples. Consequently, Alakbari et al.³⁷ (GRU) model exhibits the correct sand production using the S/I method based on _s.

Fig. 13.

The Schlumberger sand production index (S/I) method is based on 100 datasets from Alakbari et al.³⁷ model for the predicted _s values and the measured _s values from the same reference, Alakbari et al.³⁷ while keeping the Es constant.

Figure 14 shows the S/I values that were determined based on changing the measured _s values and _s values from the existing correlations while keeping the Es constant using Eq. 8. The existing correlations include Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², and Brandas et al.³⁴ correlations. As shown in Fig. 14, the existing models indicate no sanding for most samples. However, the existing models do not match the measured values using the S/I method based on _s, Fig. 14.

Fig. 14.

The Schlumberger sand production index (S/I) method is based on 100 datasets for the measured υ_s values from Alakbari et al.³⁷ and predicted υ_s values for Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², and Brandas et al.³⁴ correlations used in the same reference as Alakbari et al.³⁷ while keeping the Es constant.

In conclusion, Alakbari et al.³⁷ (GRU) model indicates the correct sand production for most samples using different methods, such as B, G/Cb, and S/I methods based on Es and _s. However, the other existing models fail to show the proper sand production for most samples.

Table 13 provides a comparative analysis of various predictive models based on their R, R², and RMSE. All models exhibit exceptionally high R and R² values, indicating strong correlations and excellent model fits. The Alakbari et al.³⁷ model stands out with an R of 0.9999, R² of 0.9998, and the lowest RMSE (0.02274), reflecting both high accuracy and minimal prediction error. Models by Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, and Ranjbar-Karami et al.³³ also achieve perfect R and R² scores (0.9999 and 0.9998, respectively). Still, their RMSE values are significantly higher, ranging from 0.10940 to 0.27501, suggesting reduced predictive precision despite the strong correlation. Kumar et al.³¹ ‘s model performs well, with a slightly lower R² (0.9995) and a moderate RMSE (0.05548). In contrast, Khandelwal et al.³² and Brandås et al.³⁴ show the weakest performance in this high-performing group, with lower R and R² values and notably higher RMSEs, particularly for Brandås et al.³⁴, whose RMSE exceeds 1.2. Overall, while many models demonstrate excellent fit, the Alakbari et al.³⁷ model provides the best balance between correlation and prediction accuracy.

Table 13.

Statistical error analysis for the Schlumberger sand production index (S/I) method based on the υs values from the existing correlations while keeping the Es constant.

Model	R	R2	RMSE
Alakbari et al.37	0.9999	0.9998	0.02274
Christaras et al.18	0.9999	0.9998	0.11151
Feng et al.35	0.9999	0.9998	0.10940
Kumar et al.31	0.9998	0.9995	0.05548
Gowida et al.36	0.9999	0.9998	0.20306
Ranjbar-Karami et al.33	0.9999	0.9998	0.27501
Khandelwal et al.32	0.9909	0.9819	0.21414
Brandås et al.34	0.9891	0.9783	1.22535

Table 14 displays the outcomes of the S/I method when υ_s values from some published correlations were used, while keeping Es constant. Most of the υ_s correlations showed nearly identical sanding responses, with Alakbari et al.³⁷, Christaras et al.¹⁸, Feng et al.³⁵, Kumar et al.³¹, Gowida et al.³⁶, Ranjbar-Karami et al.³³, and Khandelwal et al.³² each identifying only two sanding samples out of the full dataset. This uniformity indicates that these υ_s models have similar mechanical property trends when substituted into the S/I calculation. The correlation by Brandås et al.³⁴ displayed somewhat higher sensitivity, forecasting five sanding samples. The limited range of sanding detection across these correlations shows that υ_s variability has only a modest influence on S/I outcomes under constant Es conditions, reinforcing the relative stability of this index compared with other sanding prediction approaches.

Table 14.

Number of sanding and non-sanding samples for the Schlumberger sand production index (S/I) method based on the υs values from the existing correlations while keeping the Es constant.

Model	Sanding samples	Non-sanding samples
Alakbari et al.37	2	98
Christaras et al.18	2	98
Feng et al.35	2	98
Kumar et al.31	2	98
Gowida et al.36	2	98
Ranjbar-Karami et al.33	2	98
Khandelwal et al.32	2	98
Brandås et al.34	5	95

Detection of rock types

Rock types based on static Young’s modulus

Static Young’s modulus (Es) can be used to detect rock types, Table 1. This study applies the existing Es models, including those of Mahmoud et al.²⁵, Heerden¹⁷, Davapanah et al.²⁷, Fei et al.²⁴, Brotons et al.²⁶, Lacy¹⁹, Canady²³, Wang and Nur²², Wang²¹, Christaras et al.¹⁸, Bradford et al.²⁰ correlations, and Alakbari et al.¹⁶ model to detect rock types. The Es can be determined using the bulk formation density (RHOB), compressional time (DTc), and shear time (DTs) by applying existing models. In addition, measured Es were used to detect the rock types, and these points were used as basic data or benchmarks. In this study, 100 datasets from the same data as in our existing study¹⁶ were used to find the Es. As shown in Fig. 15, Alakbari et al.¹⁶ (GPR) model indicates that their values match the measured values for most samples. Therefore, Alakbari et al.¹⁶ (GPR) model can accurately detect rock types.

Fig. 15.

Rock types based on 100 datasets from Alakbari et al.¹⁶ model for the predicted Es values and the measured Es values from the same reference, Alakbari et al.¹⁶.

As shown in Fig. 16, all existing models show different rock types compared to most of the measured values. Thus, the existing models in the figure fail to detect rock types based on Es (Fig. 16).

Fig. 16.

Rock types based on 100 datasets for the measured Es values from Alakbari et al.¹⁶ and predicted Es values for other models used in the same reference, Alakbari et al.¹⁶.

In summary, all existing models in the figure fail to show the rock types based on Es, Fig. 16. However, the Alakbari et al.¹⁶ (GPR) model successfully indicates the correct rock types.

Rock types based on static Poisson’s ratio

Static Poisson’s ratio ( _s) can be used to identify the rock types, Table 2. Here, the _s values were determined from the existing models to identify the rock types. This study used the existing _s models, including Christaras et al.¹⁸, Feng et al.³⁵, Gowida et al.³⁶, Ranjbar-Karami et al.³³, Kumar et al.³¹, Khandelwal et al.³², Brandas et al.³⁴ correlations, and Alakbari et al.³⁷ model. The _s can be determined using the bulk formation density (RHOB), compressional time (DTc), and shear time (DTs), applying the existing models. Furthermore, measured _s values were used to detect the rock types, and these points were used as basic data or benchmarks. In this study, 100 datasets from the same data as in our existing study³⁵ were used.

As shown in Fig. 17, the Es values from Alakbari et al.³⁷ (GRU) model are close to the measured Es values for most samples. Consequently, the Alakbari et al.³⁷ (GRU) model can correctly identify the rock types. The other existing models show different rock types than those measured in most samples (Fig. 18). Therefore, the existing models in Fig. 18 fail to identify rock types based on upsilons. In conclusion, all existing _s models in the figure fail to detect the rock types for most samples, as shown in Fig. 18. However, Alakbari et al.³⁷ (GRU) model successfully identifies the rock types.

Fig. 17.

Rock types based on 100 datasets from Alakbari et al.³⁷ model for the predicted _s values and the measured _s values from the same reference, Alakbari et al.³⁷.

Fig. 18.

Rock types based on 100 datasets for the measured _s values from Alakbari et al.³⁷ and predicted _s values for other models used in the same reference, Alakbari et al.³⁷.

Comparative analysis of models

Table 15 shows a comparison of leading models used to predict Young’s modulus and Poisson’s ratio. In this study, the most frequently used models from the literature for estimating Young’s modulus and Poisson’s ratio were chosen to identify sand production techniques and classify rock types. Each model is defined by its main methodological approach, ranging from empirical correlations and regression methods to advanced artificial intelligence and hybrid fuzzy logic models. The strengths emphasize the unique benefits of each approach, including high predictive accuracy, ease of use in the field, and adaptability to different lithologies. On the other hand, the weaknesses reveal limitations, including the need for extensive calibration, reliance on specific datasets, and difficulties in applying the models to various geologic formations. This comparison provides a valuable reference for selecting suitable models for sand production risk assessment and the estimation of geomechanically properties in oil and gas reservoirs.

Table 15.

Comparative analysis of models.

Model	Key features	Strengths	Weaknesses
Mahmoud et al.25	Lithology-based static Young’s modulus correlations using log data and clustering	High accuracy due to clustering; useful for log-based modulus estimation	Limited generalizability across formations; requires clustering accuracy
Heerden17	Establishes general relationships between static and dynamic moduli of rocks	Widely accepted; provides base relationships for many models	May oversimplify complex rock behaviors; old dataset
Davapanah et al.27	Analyzes correlation between dynamic and static deformation moduli with statistical analysis	Robust analysis with large data sets; improved accuracy over simple models	Requires calibration; affected by rock heterogeneity
Fei et al.24	Empirical correlation models for static vs dynamic elastic parameters	Simple empirical models; easy to apply in field conditions	Empirical nature limits generalizability
Brotons et al.26	Compares static and dynamic modulus across various rock types for improved correlation	Comprehensive dataset and multi-method validation	Dependent on sample type; may not scale well to all lithologies
Lacy19	Dynamic rock mechanics testing aimed at fracture design optimization	Practical field application; optimized for fracture design	Limited to dynamic tests; static prediction indirect
Canady23	Method for full-range Young’s modulus correction using field data	Covers wide range of conditions; adaptable correction approach	Needs accurate calibration; assumes consistent conditions
Wang and Nur22	Distinguishes static and dynamic properties using core and log data	Well-documented differentiation of elastic properties	Core data dependent; may not fully reflect in-situ conditions
Wang21	Analyzes seismic and acoustic velocities; key insights into rock behavior	Foundational understanding of seismic and acoustic relationships	Broad insights; lacks specific predictive models
Christaras et al.18	Compares ultrasonic and mechanical resonance methods to static testing	Provides comparative reliability of indirect testing methods	Limited by method resolution; indirect estimations
Bradford et al.20	Elastoplastic modeling for assessing sand production risk in reservoirs	Real-world application; includes stress–strain behavior under production	Complex modeling setup; reservoir-specific calibration needed
Alakbari et al.16	GPR-based model for predicting static Young’s modulus in sandstones	Effective for heterogeneous sandstones; non-linear modeling capability	GPR models need large training datasets
Christaras et al.18	Comparison of moduli estimation methods for static elasticity	Validates accuracy of multiple measurement techniques	Measurement techniques may have resolution limitations
Feng et al.35	Piecewise linear empirical model for static Poisson’s ratio prediction	Accurate in predicting Poisson’s ratio from well logs	Piecewise fitting may oversimplify transitions
Gowida et al.36	Hybrid AI model to predict elastic behavior of sandstone rocks	Combines ML and physics; strong predictive power	Requires significant computation and training data
Ranjbar-Karami et al.33	Modified fuzzy inference system to estimate static rock properties	Incorporates expert knowledge; handles uncertainty well	Fuzzy rules need tuning; moderate interpretability
Kumar et al.31	Relates ultrasonic shear wave velocity to Poisson’s ratio	Effective for isotropic materials; validated correlations	Best suited to isotropic solids; less effective on complex rocks
Khandelwal et al.32	Correlates static properties with P-wave velocity for coal rocks	Simple and fast correlation for coal measures	Applicable to coal; limited beyond similar lithology
Brandas et al.34	Relates acoustic wave velocities to formation mechanical properties	Field-validated; applicable in multiple rock formations	Requires calibration with local conditions
Alakbari et al.37	GRU model for Poisson’s ratio prediction using deep learning	State-of-the-art AI; handles sequential data efficiently	Requires advanced training and data preprocessing

Table 16 presents a comparative summary of models developed to estimate static rock mechanical properties from data across a wide range of formations. While earlier studies, such as those by Heerden¹⁷, Davapanah et al.²⁷, and Fei et al.²⁴, focused on limited sample sizes (ranging from 22 to 55) and specific rock types or stress conditions, the models by Alakbari et al.^16,37 distinguish themselves through the use of significantly larger datasets (1,900 and 1,691 sandstone samples, respectively), enhancing model robustness and generalizability. Many traditional approaches rely on empirical correlations or laboratory measurements under controlled conditions, such as those by Wang and Nur²², Christaras et al.¹⁸, and Lacy¹⁹. In contrast, Alakbari et al.^16,37 adopted advanced machine learning techniques—specifically, Gaussian Process Regression (GPR) and a deep learning method (gated recurrent unit (GRU)) to model Young’s modulus and Poisson’s ratio using well-log data. The approaches of Alakbari et al.^16,37 not only improve prediction accuracy but also eliminate the need for core-based static testing. Moreover, while models such as those by Gowida et al.³⁶ and Ranjbar-Karami et al.³³ use AI-based techniques, they are generally limited in the type of information or sample size. Alakbari et al.’s^17,37 contributions stand out as the most comprehensive and scalable, especially for sandstone formations, offering practical applications in sand control, well log interpretation, and reservoir geomechanics.

Table 16.

Overview of models and data used for predicting rock mechanical properties.

Model	Number of samples	Types of formations	Key variables
Mahmoud et al.25	300	Sandstone, Limestone, Dolomite	Bulk density, compressional and shear wave transit times
Heerden17	55	Various rock types	Static and dynamic moduli under stress levels up to 40 MPa
Davapanah et al.27	40	Igneous, Sedimentary, Metamorphic	Static and dynamic modulus of elasticity, rigidity, and bulk modulus
Fei et al.24	22	Tight sandstones	Static and dynamic Young’s modulus and Poisson’s ratio under reservoir conditions
Brotons et al.26	33	Various rock types	Static and dynamic elastic modulus correlations
Lacy19	600	Various formations	Static and dynamic rock mechanical properties using ultrasonic, uniaxial, and triaxial tests
Canady23	Not specified	Various formations	Empirical method for full-range Young’s modulus correction
Wang and Nur22	Not specified	Reservoir rocks	Dynamic versus static elastic properties
Wang21	Not specified	Reservoir rocks	Seismic and acoustic velocities in reservoir rocks
Christaras et al.18	8	Various rock types	Comparison of ultrasonic velocity and mechanical resonance frequency methods
Bradford et al.20	Not specified	North Sea reservoir	Elastoplastic modeling for solids production risk assessment
Alakbari et al.16	1900	Sandstone rocks	Gaussian process regression-based model for static Young’s modulus
Feng et al.35	18	Various formations	Empirical method to predict static Poisson’s ratio via well logs
Gowida et al.36	Not specified	Sandstone rocks	Hybrid artificial intelligence model to predict elastic behavior
Ranjbar-Karami et al.33	33	Kangan and Dalan gas reservoirs	Modified fuzzy inference system for static rock elastic properties
Kumar et al.31	83	Isotropic solid materials	Correlation between ultrasonic shear wave velocity and Poisson’s ratio
Khandelwal et al.32	11	Coal measure rocks	Correlating static properties with P-wave velocity
Brandas et al.34	15	Formation mechanical properties	Relating acoustic wave velocities to mechanical properties
Alakbari et al.37	1691	sandstone formations	Bulk density, compressional, and shear wave transit times

Conclusion

Sand production can be effectively detected using the sand production index (B), shear modulus to bulk compressibility ratio (G/Cb), and Schlumberger sand production index (S/I) methods. These methods depend on accurate measurements of the static Young’s modulus (Es) and the static Poisson’s ratio (υs). Although various models in the literature estimate Es and υ_s using log-derived parameters such as bulk density (RHOB), compressional time (DTc), and shear time (DTs), their relative effectiveness in predicting sand production has not been systematically assessed until now.

This study used 100 datasets from our previous research^17,35 to evaluate how Es and υ_s values—derived from several established models—affect the accuracy of predicting sand production. By keeping one parameter constant while changing the other, we isolated the effect of Es and υs on the B, G/Cb, and S/I methods.

In this dataset, the sand production prediction B, G/Cb, and S/I values are based on the Es and υs from Alakbari et al.^16,37 models, which show the best agreement with the B, G/Cb, and S/I values based on the measured Es and υs. Therefore, Alakbari et al.^16,37 Es and υ_s models correctly predicted the sand production for most samples. In comparison, other previous models showed different B, G/Cb, and S/I values, indicating different sand production from the measured values. Alakbari et al.^16,37 models have Rs in the range from 0.83059 to 0.9999. The other previous models have Rs in the range from -0.0181 to 0.999.

The broader significance of these findings lies in their potential to enhance operational decision-making in reservoir development and sand control during well production. Inaccurate predictions of sand production can cause equipment damage, reduce productivity, and lead to costly interventions.

We recommend that industry practitioners select accurate Es and υ_s models for well-log analysis workflows, particularly during completion design and sand-management planning. Based on this dataset, Alakbari et al.^16,37 achieve high predictive accuracy while also improving the identification of sand-prone zones and rock types.

The limitations of the study

Despite offering valuable insights into the comparative performance of various Es and νs prediction models, this research is constrained by the size and diversity of its dataset. Therefore, the findings may not fully capture the variability in mechanical behaviour observed in heterogeneous formations, anisotropic reservoirs, or unconventional plays. Another limitation arises from the assumption that the sand production prediction methods (B, G/Cb, and S/I) respond uniformly to variations in Es and νs across all geological settings. The methods, primarily established for specific reservoir types, may lose accuracy when applied outside their intended domains. Consequently, while the findings strongly support the superiority of the Alakbari et al. models, further confirmation is needed to generalize these results to broader field applications.

Future research directions

Future studies should widen the dataset to encompass a broader range of lithologies, depositional environments, and stress regimes to advance the generalisability of model performance evaluations. Including core-scale laboratory measurements from diverse reservoirs would strengthen the basis for validating estimate methods for Es and νs.

Enhancing sand production prediction may also require adopting machine-learning and hybrid physics-based models that account for uncertainty, nonlinearity, and heterogeneity in reservoir rocks. Future research could investigate ensemble modelling, Bayesian inference, and deep learning architectures to strengthen the robustness of Es and νs predictions. Furthermore, integrating mechanical property estimation with real-time production data and downhole monitoring tools could facilitate dynamic sand risk forecasting.

Latin synonyms

B

Sand production index method

G/Cb

Shear modulus to bulk compressibility ratio

S/I

Schlumberger sand production index method

G

Shear modulus

Bulk compressibility

K

Bulk modulus

E_s

Static Young’s modulus or static elasticity modulus

E_d

Dynamic Young’s modulus

RHOB

Bulk formation density

DTs

Shear time

DTc

Compressional time

Compressional wave

Shear wave

R

Correlation coefficient

R²

Coefficient of determination

Greek synonyms

µsec

Microsecond

υ_s

Static Poisson’s ratio

Bulk density

ν_dyn

Dynamic Poisson’s ratio

Abbreviations

ANN

Artificial neural network

MPa

Megapascal (pressure unit)

GPa

Gigapascal (pressure unit)

GPR

Gaussian process regression

GRU

Gated recurrent unit

CTD

Critical total drawdown

COH

Cohesive strength

TVD

Total vertical depth

TT

Transit time

EOVS

Effective overburden vertical stress

MLR

Multiple linear regression

FL

Fuzzy logic

GRNN

Regression neural network

MLR

Multiple linear regression

GA-MLR

Genetic algorithm-evolved MLP

AVO

Amplitude variations with offset

RMSE

Root mean square error

FNN

Feedforward neural network

SVM

Support vector machine

M-FIS

Mamdani fuzzy inference system

RF

Random forest

XGBoost

Extreme gradient boosting

Author contributions

Fahd Saeed Alakbari ROLES: Methodology, Conceptualisation, Software, Writing—original draft. Syed Mohammad Mahmood ROLES: Validation, Writing—review & editing, Conceptualisation. Mahmoud M. Abdelnaby ROLES: Writing—review & editing, Conceptualisation, Supervision. Haithm Salah Hagar ROLES: Writing—review & editing. Funsho Afolabi ROLES: Writing—review & editing. Mysara Eissa Mohyaldinn ROLES: Writing—review & editing, Funding.

Funding

The authors sincerely acknowledge the financial support provided by Yayasan Universiti Teknologi PETRONAS (YUTP) under Cost Centres: YUTP FRG 015LC0-557 and YUTP PRG 015PBC-042 for supporting this study.

Data availability

The data that support the findings of this study are available from the corresponding author, upon request.

Declarations

Competing interests

The authors declare no competing interests.