**Ahmed Abdulhamid Mahmoud, Salaheldin Elkatatny \* and Dhafer Al Shehri**

College of Petroleum Engineering and Geosciences, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia; g201205160@kfupm.edu.sa (A.A.M.); alshehrida@kfupm.edu.sa (D.A.S.)

**\*** Correspondence: elkatatny@kfupm.edu.sa; Tel.: +966-5-9466-3692

Received: 17 January 2020; Accepted: 28 February 2020; Published: 2 March 2020

**Abstract:** Prediction of the mechanical characteristics of the reservoir formations, such as static Young's modulus (Estatic), is very important for the evaluation of the wellbore stability and development of the earth geomechanical model. Estatic considerably varies with the change in the lithology. Therefore, a robust model for Estatic prediction is needed. In this study, the predictability of Estatic for sandstone formation using four machine learning models was evaluated. The design parameters of the machine learning models were optimized to improve their predictability. The machine learning models were trained to estimate Estatic based on bulk formation density, compressional transit time, and shear transit time. The machine learning models were trained and tested using 592 well log data points and their corresponding core-derived Estatic values collected from one sandstone formation in well-A and then validated on 38 data points collected from a sandstone formation in well-B. Among the machine learning models developed in this work, Mamdani fuzzy interference system was the highly accurate model to predict Estatic for the validation data with an average absolute percentage error of only 1.56% and R of 0.999. The developed static Young's modulus prediction models could help the new generation to characterize the formation rock with less cost and safe operation.

**Keywords:** static Young's modulus; sandstone formations; machine learning

### **1. Introduction**

Prediction of the mechanical characteristics of the reservoir formations, such as Young's modulus (E), is necessary for the evaluation of the wellbore stability, reservoir compaction, hydraulic fracturing, and formation control [1]. E is a mechanical parameter that gives an indication of the resistance of the rock samples when exposed to a uniaxial load [2]. On the other hand, static Young's modulus (Estatic) is a critical parameter needed to build the earth geomechanical model [3]. It is also used for fractures' designing and mapping [4,5]. While drilling hydrocarbon wells, Estatic is also needed with other mechanical and petrophysical properties to make a full description of the in-situ stresses to ensure wellbore stability [6].

Estatic varies significantly with the change in lithology [2,7]. Estatic for shale ranges from 0.69 to 6.89 GPa. For limestone, it is between 55.16 and 82.74 GPa, and for sandstone, it is between 13.79 and 68.95 GPa [7]. These ranges confirm the wide difference in Estatic from one formation type to another and the huge change within the same lithology. Therefore, it is necessary to estimate Estatic along the whole drilled hydrocarbon well.

Two methods for rock elastic parameters' estimation are currently available. These are the experimental laboratory method or the use of empirical correlations. The experimental laboratory method is based on conducting laboratory experiments on the rock samples using static or dynamic testing techniques. In the static technique, the sample is subjected to a uniaxial or triaxial load and the deformation of the sample is measured, while in the dynamic technique, shear and compressional

wave velocities along the tested sample are measured and then the sample's elastic parameters are calculated based on the shear wave (Vs) and compressional wave (Vp) velocity [8]. In the field, wireline logging tools are used to measure Vs and Vp. The dynamic Young's modulus (Edynamic) can then be evaluated based on Vs and Vp and using Equation (1).

$$\mathrm{E\_{dynamic}} = \frac{\rho \mathrm{V\_S}^2 \left(3 \mathrm{V\_P}^2 - 4 \mathrm{V\_S}^2\right)}{\mathrm{V\_P}^2 - \mathrm{V\_S}^2} \tag{1}$$

where ρ denotes the formation's bulk density in g/cm3, VS and VP are in km/s, and Edynamic is in GPa.

Several previous studies confirmed that the laboratory-measured Edynamic for the same rock sample is significantly greater than Estatic [9–11]. Edynamic could be 1.5 to 3 times greater than Estatic [12] and some recent studies reported that Edynamic could be ten times greater than Estatic [13,14]. The strain amplitude between the two experimentally testing methods is the main reason for this huge difference, which decreases as the rock strength increase [15].

The static elastic parameters are actually representative of the in-situ stress–strain conditions of the reservoir [16]. Accurate determination of the static elastic parameters requires conducting a time consuming and costly experimental tests on real core samples [17]. The common practice to decrease this high cost is to select core samples at specific intervals and conduct the experimental tests of these cores only. Then an empirical correlation between the laboratory-derived parameters and the conventional well log data will be developed based on the results of laboratory tests. The static moduli throughout the whole reservoir depths can then be predicted by calibrating the dynamic moduli using the developed correlations [4]. Because of the heterogeneity of the reservoir formations, the developed well log-based empirical equations are usually not generalized to all formation types. Therefore, different correlations need to be developed for every formation type to track the changes in the static parameters along the whole reservoir.

The correlation in Equation (2) was developed by Fei et al. [18] for the evaluation of Estatic for sandstone formations; this correlation evaluates Estatic as a function of Edynamic, which was developed based on 22 triaxial tests results.

$$\text{E}\_{\text{static}} = 0.564 \,\text{E}\_{\text{dynamic}} - 3.4941 \,\tag{2}$$

where Estatic and Edynamic are in GPa.

Mahmoud et al. [19] developed a set of equations to estimate Estatic for different types of formations. The main advantage of the correlations developed by Mahmoud et al. [19] is the ability to implement these correlations directly to evaluate Estatic without the need for Edynamic, these correlations are only a function of the bulk formation density (RHOB), compressional transit time (DTc), and shear transit time data (DTs).

Different recent studies confirmed the ability of machine learning techniques to accurately estimate rock mechanical properties. Abdulraheem et al. [20] optimized three machine learning models of the artificial neural networks (ANN), fuzzy logic model, and functional neural networks (FNN) for estimation of Estatic and the static Poison's ratio for the hydrocarbon reservoirs. The authors did not specify the reservoir rock formation type. The developed models confirmed their ability to estimate the reservoir rock mechanical properties.

In another study, Tariq et al. [21] developed three machine learning models of ANN, fuzzy logic, and support vector machine (SVM) to estimate Estatic for limestone formation. The ANN model overperformed the other machine learning models and the currently available empirical correlation for Estatic estimation.

Tariq et al. [22] developed empirical correlations for the estimation of the mechanical properties of Estatic, Poisson's ratio, and unconfined compressive strength based on the application of the artificial neural networks (ANN) and the use of the conventional well log data, the authors also did not specify the type of the formation they used in this study. The developed correlations improved their ability to accurately estimate the rock mechanical properties.

In 2017, Parapuram et al. [23] developed an ANN model to estimate the geomechanical properties of the upper Bakken shale based on well log data. The results of this study confirmed the ability of the ANN model to accurately estimate the rock mechanical properties.

Recently, in our previous study, Mahmoud et al. [24], we evaluated the use of the ANN in estimating Estatic for sandstone formations. Mahmoud et al. [24] reported that ANN is able to predict Estatic with very high accuracy, and it overperformed all available empirical equations currently in use.

Sustainable development can be defined as development that meets the needs of the present without compromising the ability of future generations. This study is aimed at evaluating the ability of four machine learning techniques namely ANN, SVM, FNN, and the Mamdani fuzzy interference system (M-FIS) in estimating Estatic for sandstone formations as a function of RHOB, DTs, and DTc. The new systems of static Young's modulus prediction are examples of the new development which will help the new generation to discover and extract the oil and gas at lower cost and with safer operation. The developed method depends on taking the reading from the well logging tools and applying the artificial neural network models to predict the static Young's modulus and provide a continuous profile of the elastic property through the whole reservoir. This will improve the time necessary for the decision on the required action based on given information.

#### **2. Theory of Machine Learning Techniques Considered in This Study**

The first machine learning technique used in this work was the ANN, which is a computing system that is designed to mimic the way the biological systems, such as the human or animal brains, behave. ANN is developed to identify, estimate, classify, or make a decision by using a machine program. ANN is available in different structures; the simplest ANN structure, which was used in this study, is called multi-layered perceptron (MLP) which consists of one input layer, one or several hidden (learning) layers, and one output layer, as shown in Figure 1 [25]. The ANN systems are trained originally using training data (supervised learning) to perform the needed tasks [26].

**Figure 1.** Artificial neural networks model with input layers of three inputs, one hidden layer, and an output layer.

M-FIS was the second machine learning technique used in this study, which combines the adaptive neuro-fuzzy inference system (ANFIS) and subtractive clustering, where ANFIS is a multilayer feed-forward adaptive network in which the incoming signal will be subjected to a particular function performed by each training node where every node has its own parameters pertaining (Figure 2). The hybrid learning procedure was performed in two steps; the first step was the forward pass in which

the functional signals representing the input data go forward and the least square formula was used to identify the parameters in the output layer (layer 5). The second step was the backward pass, in which the error rates propagate, in the opposite way, and the gradient method was implemented to update the parameters in the input layer (layer 1) [27].

**Figure 2.** Adaptive neuro-fuzzy inference system architecture of a model using two inputs layers and one output parameter.

The subtractive clustering is an unsupervised clustering algorithm that aims to examine the density of the available input data. Then it defines the point surrounded by the highest number of neighbors as the cluster's center. It then subtracts (removes) the other data points within a pre-specified fuzzy radius, and the subtractive clustering algorithm considers only the point defined as the cluster's center. This process is repeated to examine all input data points. Subtractive clustering generates the rules that approximate a function [28].

The third machine learning technique used in this study was the FNN model, compared to the ANN which uses the sigmoidal common model. The FNN model works with the generalized functional models. In FNN, the neuron's function is learned from the existing data, which means they are not constant. Therefore, the weights related to links are not needed because the neuron functions include the effect of weights [29]. FNN contains an input layer, an output layer and layers of computing units that are related to each other. In FNN, there are different arguments in neural functions instead of one argument, such as in ANN [30].

The fourth machine learning model considered in this study was the SVM, which is one of the most famous classifying algorithms developed by Vapnik [31] in the framework of statistical learning theory. It performs classification of the data optimally into two or more divisions by applying a multidimensional hyperplane; this hyperplane is set to classify the data based on the tuning parameters (design parameters) of the kernel, regularization parameter (C), gamma, and margin. In its nature, SVM is very similar to a neural network, where the use of SVM with sigmoid kernel function is almost identical to the use of the perceptron neural network, having two hidden layers. Although it was originally developed in the statistical learning theory, the SVM technique is applicable in regression and classification problems, and it is also suitable for solving non-linear problems [32].

#### **3. Applications of Machine Learning in Petroleum Engineering**

Machine learning techniques are used in several scientific and engineering fields since the early 1990s to solve complicated non-linear problems. Petroleum engineers and petroleum geologists use different machine learning techniques to solve problems related to petroleum industry, such as the characterization of the heterogeneous hydrocarbon reservoirs [33,34], evaluation of the reserve

of unconventional reservoirs [35–38], estimation of the rock mechanical parameters, such as the static Poisson's ratio in carbonate reservoirs [39] and the static Young's modulus for sandstone reservoirs [24,40], evaluation of the integrity of wellbore casing [41,42], optimization of drilling hydraulics [43], evaluation of pore pressure and fracture pressure [44,45], hydrocarbon recovery factor estimation [46,47], determination of the alteration in the drilling fluids rheology in real-time [48,49], optimization of rate of penetration [50,51], prediction of the formation tops [52], and others.

#### **4. Application to the Well Log Data**

The predictability of the machine learning models depends on the amount of training data points and the design parameters of every model. In this work, the machine learning model's design parameters and the selection of the optimum training data points were conducted based on the optimization process of all combinations of the design parameters, as will be discussed in the following sections.
