Machine Learning-Based Propped Fracture Conductivity Correlations of Several Shale Formations

Abstract

In hydraulic fracturing operations, small rounded particles called proppants are mixed and injected with fracture fluids into the targeted formation. The proppant particles hold the fracture open against formation closure stresses, providing a conduit for the reservoir fluid flow. The fracture’s capacity to transport fluids is called fracture conductivity and is the product of proppant permeability and fracture width. Prediction of the propped fracture conductivity is essential for fracture design optimization. Several theoretical and few empirical models have been developed in the literature to estimate fracture conductivity, but these models either suffer from complexity, making them impractical, or accuracy due to data limitations. In this research, and for the first time, a machine learning approach was used to generate simple and accurate propped fracture conductivity correlations in unconventional gas shale formations. Around 350 consistent data points were collected from experiments on several important shale formations, namely, Marcellus, Barnett, Fayetteville, and Eagle Ford. Several machine learning models were utilized in this research, such as artificial neural network (ANN), fuzzy logic, and functional network. The ANN model provided the highest accuracy in fracture conductivity estimation with R² of 0.89 and 0.93 for training and testing data sets, respectively. We observed that a higher accuracy could be achieved by creating a correlation specific for each shale formation individually. Easily obtained input parameters were used to predict the fracture conductivity, namely, fracture orientation, closure stress, proppant mesh size, proppant load, static Young’s modulus, static Poisson’s ratio, and brittleness index. Exploratory data analysis showed that the features above are important where the closure stress is the most significant.

Introduction

Hydraulic fracturing enables the exploitation of hydrocarbons from tight shale formations at commercial rates. A huge amount of slurry containing the proppant is placed in the fractures to keep them open and conductive under formation closure stresses. The ability of these fractures to deliver hydrocarbons is called propped fracture conductivity. It is described mathematically as the product of the proppant permeability and the width of the fracture.^1,2

Under controlled laboratory conditions, propped fracture conductivity is measured precisely using an American Petroleum Institute (API) conductivity cell. Extensive studies have been conducted on common shale formations (namely, Marcellus, Barnett, Fayetteville, and Eagle Ford) to understand the factors influencing propped fracture conductivity. Each shale formation is composed of different minerals with different amounts. For instance, Barnett is a clay-rich shale that may contain up to 50% clay minerals. In contrast, the Eagle Ford has a minor clay content and is made of more than 70% carbonate. Figure 1 shows a rough estimation of each shale’s mineralogical content.³⁻⁶

Figure 1.

Mineralogy of different shale formations.

Shale and proppant properties combined control the propped fracture conductivity along with the closure stress.^7,8 When the proppant load is low, the conductivity is mainly controlled by the shale properties. Contrary to the high proppant load, the proppant characteristics become more important in determining the fracture conductivity.^9,10

On the one hand, the shale-related properties have a major effect on propped fracture conductivity at low proppant concentrations. The fracture orientation effect relative to the bedding planes on fracture conductivity becomes more obvious when there is an intense anisotropy in the shale mechanical properties, that is, static Young’s modulus (E_static) and static Poisson’s ratio (PR).¹¹ Also, the fracture topography plays an important role, that is, the conductivity increases with the increase in the fracture surface roughness.¹² Knorr¹³ found that the increase in the fracture surface’s roughness decreases the pace at which the conductivity is declining. On the other hand, the increase in the proppant concentration yields a higher conductivity.¹⁴ Also, as the proppant size increases, the conductivity increases.¹⁵

Other factors could also influence the magnitude of fracture conductivity due to shale, fracture fluids, and proppant interactions. For instance, the short-run fracture conductivity loss could happen due to water damage, clay swelling, and proppant embedment.¹⁶⁻¹⁸ Also, the long-run fracture conductivity loss with production occurs due to fine migration, proppant crushing, and shale creeping.^10,19−21 Non-Darcy flow, due to high gas velocity, is known to reduce the effective proppant permeability.²² These interactions make the process of predicting propped fracture conductivity a challenging task.²³

There are several reported approaches in the literature to predict fracture conductivity. These models are either theoretical-based or empirical, driven from lab data. Zhang et al.²⁴ derived a theoretical model to predict shale fracture conductivity considering the damage caused by water. The model coefficients were obtained from experiments on the Barnett shale. Thus, it is only applicable to shales with similar properties and mineralogy. Jia et al.²⁵ provided a mathematical model to estimate nonspherical proppant conductivity, while Xu et al.²⁶ added the impact of fracture tortuosity. The complex discrete element method combined with computational fluid dynamics were used to estimate the propped fracture conductivity profile.²⁷ Many theoretical models have been proposed to predict propped fracture conductivity, but only a few empirical models were developed. In fact, generating sufficient experimental data that consider all factors impacting fracture conductivity is challenging. Kainer²⁸ employed multiple linear regression to generate empirical correlations based on experimental data for different shale types. He developed several models and determined the corresponding significant predictors for each type. The correlation coefficient obtained, however, is low.

The measurement of shale propped fracture conductivity in a laboratory setup is the most accurate approach but is a time-consuming and costly process. Predicting fracture conductivity from theoretical models is challenging as these models are complex and require many input parameters that might not be available. The previously developed empirical correlations are questionable and based on limited data. This suggests the need for a quick, cheap, and precise method to predict the propped fracture conductivity of different shale formations. The objective of this work is to develop such models using different machine learning (ML) approaches on vast experimental data.

Methodology

The experimental data for the prediction of propped fracture conductivity were taken from several published sources from the same research group.^8−10,12 We conducted an extensive literature review to collect the data using the same API conductivity cell and experimental procedures to ensure collection of a good number of consistent experimental data on several shales under a controlled environment. After data preprocessing, the total number of collected data points was 351. The relative importance of the features controlling fracture conductivity was examined through the correlation coefficient (CC) criterion. This study used Pearson, Spearman, and Kendall CC criteria to investigate the relationship between the features and fracture conductivity. Then the data were analyzed, and correlations were extracted using artificial neural network (ANN), fuzzy logic (FL), and functional network (FN) models.

Exploratory Data Analysis

Exploratory data analysis (EDA) is the process of investigating the relationships, patterns, and collinearity among the input parameters. The experimental data include investigations of fracture orientation, proppant mesh size, closure stress, proppant load, static E_static, PR, brittleness index (BI), and measured propped fracture conductivity. The fracture orientation is a categorical variable that can only be vertical or horizontal. Transforming the categorical variable into a dummy variable is a common practice to account for the different categories in the model. The presence of a categorical variable is represented by 1, while the absence is expressed by 0. Thus, the fracture orientation is dimensionless. The mesh size range was averaged and then converted into a particle diameter as shown in Table 1. The data were obtained from experiments on different shale formations such as Marcellus, Eagle Ford, Barnett, and Fayetteville. A complete statistical description of the data used for training is given in Table 2.

Table 1. Conversion of the Proppant Size from API to Inches.

mesh size (API)	mesh size (in)
30	0.0232
40	0.0165
50	0.0117
70	0.0083
100	0.0059

Table 2. Statistical Analysis of the Data Utilized for ML Modeling.

Pair Plots

A pair plot was used to observe both distributions of single variables and relationships between two variables. Pair plots were used to plot features when there were more than three dimensions. Figure 2 shows the pair plot of the overall propped fracture conductivity data. The pair plot was used to address the issue of redundant variables by figuring out the interrelationship between the selected features.

Figure 2.

Pair plot of the combined conductivity data set for different shale formations such as Marcellus, Eagle Ford, and combined Fayetteville and Barnett.

Relative Importance

ML models are data-driven, which means that the best way to feature selection is by picking the optimum input parameters. The optimum way is to find the linear CC between input parameters and the target parameter. The value of CC between the pair of two variables always lies between −1 and 1. A CC value close to “–1” shows a strong inverse relationship between the pair of variables, while a value close to “1” shows a strong direct relationship between the two variables. A CC value of zero shows that no relationship exists between the two variables.

In this study, Pearson, Spearman, and Kendall CC criteria were used to estimate the linear relationship between input parameters and the output parameter. The definitions of Pearson, Spearman, and Kendall CC criteria are given as eqs 123.

1

where x is the value of x-variable and y is the value of y-variable and k is the total number of samples. The Spearman correlation between two variables is equal to the Pearson correlation between the rank values of those two variables; while Pearson’s correlation assesses linear relationships, Spearman’s correlation assesses monotonic relationships (whether linear or not). If there are no repeated data values, a perfect Spearman correlation of +1 or −1 occurs when each of the variables is a perfect monotone function of the other.

2

where cov(x, y) is the covariance of the rank variables and γ_xγ_y are the standard deviations of the rank variables. A Kendall tau CC test is a nonparametric hypothesis test for statistical dependence between the pair of two variables. The Kendall tau CC is equivalent to the Spearman rank CC. While the Spearman rank CC is like the Pearson CC but computed from ranks, the Kendall tau correlation represents a probability.

3

where n_c is the number of concordant pairs, n_d is the value of the number of discordant pairs, and n is the total number of samples.

Figure 3 shows the Pearson, Spearman, and Kendall CC features with the natural log of propped fracture conductivity. The CC values of all input features based on Pearson criteria improved significantly by taking the natural logarithm of propped fracture conductivity. One might notice that the most significant parameter is the closure stress, which negatively correlates with the fracture conductivity. It is also confirmed that the proppant mesh size, proppant load, specimen E_static, and BI are equally important and contribute positively to fracture conductivity.

Figure 3.

Relative importance of the input parameters with the natural log of propped fracture conductivity.

Data Stratification

The data set was divided randomly into two parts: 70% of the data set was utilized for training the model, while the remaining 30% was kept for testing the model. Data were stratified randomly by Python. This splitting process ensured that the testing data fell within the range of the training data.

FN Model

The FN technique is the most advanced and simplest conventional neural network technique.²⁹ The FN is a supervised data learning technique mostly used for function approximation and regression purposes.³⁰ The FN algorithm proceeds with the learning from domain and data knowledge. A typical FN model is comprised of three units: an input storing unit, an output storing unit, and several layers of processing units. Unlike neural networks, the algorithm itself determines the network topology, structures, and unknown neural functions. In the conventional neural network model, the number of neurons is fixed and usually user-defined. The weights and biases (connections between the neurons) of the associated neurons are learned and estimated during the training phase.³¹ In contrast, in FN models, the number of neural functions is adaptive and varied during the structural learning and parametric learning phases.

FL Model

FL is a computing approach based on the degree of uncertainty instead of conventional sharp cutoffs. Contrary to the Boolean logic in which the variable truth values are either 0 or 1, there can be any real values between 0 and 1 in FL. Fuzzy sets are more realistic and mimic human reasoning in making decisions based on ambiguous or non-numeric data. FL is composed of four components, namely, rule base, fuzzification, inferential engine, and defuzzification. The rule base governs the decision making using a set of rules and conditions. Fuzzification turns the exact inputs known as crisp inputs into fuzzy sets, which are fed into the inferential engine. The latter evaluates the degree of matching between each rule and the fuzzy input to determine which rules should be fired. Finally, the defuzzification turns the fuzzy sets back into a crisp value.

ANN Model

ANN is a computing system that mimics an animal brain’s biological neural network. In this study, comprehensive numerical experimentation was performed to arrive at the optimum ANN model. The optimum selection of the hidden layer, the number of neurons in the hidden layer, optimum transfer function, optimum learning algorithm, and the best learning rate was selected based on the predicted values’ accuracy.

Results and Discussion

In this study, two scenarios were examined to train the ML models. The first scenario was to combine all the data sets of different formations and develop a single generalized model. The second scenario was to create an ML model for each formation separately.

The FN model was used on a combined data set to develop a generalized model. On the overall data set, FN predicted the propped fracture conductivity with R² of 0.853 on training and R² of 0.817 on testing. The root mean square error (RMSE) of the natural logarithm of conductivity was 0.051 on training and 0.104 on testing. The average absolute percentage error (AAPE) was relatively high, about 14.811% on training and 17.444% on testing. The training and testing cross plots are shown in Figure 4.

Figure 4.

Training and testing cross plots of the FN model on the overall data set.

A type of FL named adaptive network-based fuzzy inference system was employed to optimize the fuzzy inference system. The latter was created by generating a fuzzy inference system (genfis2) using subtractive clustering. In genfis2, the input and output membership functions were hardwired to Gaussian and linear, respectively. Also, the number of membership functions was two, while the cluster radius was set to 0.8. The trained FL models developed by the combined data set and each separate shale formation data set are summarized in Table 3.

Table 3. FL Model Accuracy Comparison.

	AAPE		RMSE		R2
formations	training	testing	training	testing	training	testing
overall	16.0861	11.6894	0.6418	0.6425	0.8341	0.8376
Marcellus	35.6522	31.2780	0.9418	0.8296	0.6882	0.7314
Eagle Ford	4.0536	3.8391	0.3118	0.3031	0.9153	0.9299
Barnett/Fayetteville	10.9987	8.7878	0.5296	0.4761	0.9180	0.9297

The trained model on the whole data set could predict propped fracture conductivity with R² of 0.83. The RMSE was 0.64, while the AAPE was 16.08% on training and 11.69% on testing. The training and testing cross plots are shown in Figure 5.

Figure 5.

Training and testing cross plots of the FL model on the overall data set.

On the overall data set, ANN was able to predict propped fracture conductivity with an R² of 0.895 on training and 0.892 on testing. The RMSE of the natural logarithm of conductivity was 0.030 on training and 0.057 on testing. The AAPE was quite high, about 11.860% on training and 7.869% on testing. The training and testing cross plots are shown in Figure 6. The hyperparameters of the proposed ANN model are listed in Table 4.

Figure 6.

Training and testing cross plots of the ANN model on the overall data set.

Table 4. Optimum Values for the Proposed ANN Model for Propped Fracture Conductivity on the Overall Data Set.

ANN parameters	optimum values
inputs	6
middle layer	1
neurons in the middle layer	10
training algorithm	Levenberg–Marquardt (LM)
learning rate, α	0.20
activation function of the middle layer	tangent sigmoidal
activation function of the outer layer	pure linear

ANN performed better than FN and FL. FN and FL models resulted in lower R², high AAPE, and high RMSE both during training and testing compared to the ANN model.

For the second scenario, the whole data set was segregated based on the formation type. An ANN model was trained separately for Marcellus, Eagle Ford, and combined Barnett and Fayetteville formations. The prediction performances of ANN models for each formation are shown on training and testing cross plots in Figures 789. Table 5 shows the comparison of the ANN model accuracy for the two scenarios.

Figure 7.

Training and testing cross plots of the ANN model on the Marcellus shale data set.

Figure 8.

Training and testing cross plots of the ANN model on the Eagle Ford shale data set.

Figure 9.

Training and testing cross plots of the ANN model on the combined Barnett shale and Fayetteville data set.

Table 5. ANN Model Accuracy Comparison.

	AAPE		RMSE		R2
formations	training	testing	training	testing	training	testing
overall	11.29	5.912	0.035	0.04	0.886	0.929
Marcellus	24.694	16.444	0.078	0.142	0.816	0.804
Eagle Ford	1.471	2.223	0.016	0.029	0.978	0.978
Barnett/Fayetteville	3.491	1.51	0.026	0.024	0.989	0.996

In this study, in addition to the trained ANN model, explicit empirical correlations to predict propped fracture conductivity of the shale formations were proposed. The normalized form of the proposed fracture conductivity is given by eq 4

4

where Y = w_1i,1 O_n + w_1i,2 Mesh_n + w_1i,3PL_n + w_1i,4σ_cn + w_1i,5E_staticn + w_1i,6υ_staticn + w_1i,6BI_n. The w₁, w₂, b₁, and b₂ are the weights and biases (see Table 6) of the propped fracture conductivity model. O_n, Mesh_n, PL_n, σ_cn, E_staticn, υ_n, and BI_n are the normalized values of fracture orientation, mesh size, proppant load, closure stress, E_static, PR, and BI. These values can be determined using eqs 5–11.

5

6

7

8

9

10

11

Table 6. Weights and Biases of the Optimized ANN Model on the Overall Data Set, Marcellus Shale Data Set, Eagle Ford Shale Data Set, and Combined Barnett and Fayetteville Shale Data Set.

The proposed equations to predict propped fracture conductivity W_kfn on the overall data set, Marcellus shale, Eagle Ford shale, and Barnett shale are given as eqs 12131415.

12

13

14

15

where W_kfn for each formation can be calculated using eq 4. The normalized parameters in eq 4 can be calculated using eqs 567891011. The minimum and maximum values of each input parameter are reported in Table 2.

Sensitivity Analysis

A sensitivity analysis was carried out to validate the proposed empirical correlation to predict the propped fracture conductivity on the overall data set. In the sensitivity analysis, one variable was changed and the other variables were kept constant at their average mean values, as listed in Table 2. Figure 10 shows the predicted propped fracture conductivity’s sensitivity results with the closure stress (σ_c) at different proppant loads such as 0.01, 0.025, 0.05, and 0.075 at horizontal and vertical fracture orientations. The σ_c was changed from its minimum value of 500 psi to a maximum value of 6000 psi. These minimum and maximum values were those on which the model was trained and are listed in Table 2. As evident from the sensitivity analysis results, the increase in the proppant load increases the propped fracture conductivity. The results obtained were very much aligned with the experimental observation.

Figure 10.

Sensitivity analysis of propped fracture conductivity with closure stress at different proppant loads for both horizontal and vertical fracture orientations.

Figure 11 shows the sensitivity results of the predicted propped fracture conductivity equation with the closure stress (σ_c) at different E_static values of shales such as 20, 25, and 30 GPa at horizontal and vertical fracture orientations. The closure stress was changed from its minimum value of 500 psi to a maximum value of 6000 psi. An increase in the E_static value of the shale increases the overall propped fracture conductivity. This proves that the developed empirical correlation is capable of predicting the propped fracture conductivity behavior if it is within the range on which the model is trained, as listed in Table 2.

Figure 11.

Sensitivity analysis of propped fracture conductivity with closure stress at different E_static values of the rock for both horizontal and vertical fracture orientations.

Similarly, the impact of the proppant load on the fracture conductivity at both vertical and horizontal directions is shown in Figure 12. The model could predict that increasing the proppant mesh size results in higher fracture conductivity. This behavior is well reported in the literature that identified that higher proppant permeability is obtained from large-sized proppant particles. This behavior can be reproduced by the model within the range of data in Table 2.

Figure 12.

Sensitivity analysis of propped fracture conductivity with closure stress at different mesh sizes for both horizontal and vertical fracture orientations.

Several BIs have been defined in the literature that helps to characterize unconventional resources. The most common ones are based on elastic properties, mineralogical composition, or strength parameters. The mathematical descriptions of the different BIs are expressed as eqs 161718. All these BIs are equivalent and yield high values for quartz-rich rocks.³²

16

17

18

In this work, the BI defined by elastic properties was used in developing different ML models. The other two methods could not be implemented as the mineralogical composition and strength parameters were not available for each sample. Figure 13 shows that the higher the rock elastic BI values, the higher the conductivity. Proppant impediment is lower in brittle formations and hence a wider fracture width is achieved.

Figure 13.

Sensitivity analysis of propped fracture conductivity with closure stress at different BIs for both horizontal and vertical fracture orientations.

Conclusions

This study provides a simple and accurate propped fracture conductivity correlation based on several ML methods, such as ANN, FL, and FN. More than 350 data points were collected from API conductivity experiments in four shale formations: Marcellus, Barnett, Fayetteville, and Eagle Ford. Based on the analysis of the collected data, the following can be concluded:

• 1-Pearson, Spearman, and Kendall CC criteria show that closure stress is the most important feature for predicting propped fracture conductivity. The proppant load, size, and formation elastic properties (i.e., E_static, PR, and BI) are also significant.
• 2-The ANN model provided the best result compared to FL and FN models. The R² of the general conductivity model was 0.89 and 0.93 for the training and testing data sets, respectively.
• 3-The formation-specific ANN models were more accurate than the general correlation except for Marcellus shale. Hence, it is advised to use the specific correlations for Barnett, Fayetteville, and Eagle Ford and the general correlation for Marcellus.
• 4-The provided correlations are easy to use and require simple input parameters such as fracture orientation, closure stress, proppant mesh size, proppant load, E_static, PR, and BI.
• 5-The provided correlations are data-driven and should be used only in the specified ranges.

Author Contributions

The manuscript was written through contributions of all authors.

The authors declare no competing financial interest.