Hydraulic fracturing extracts oil and gas from deep underground, with fracture conductivity being crucial for efficient production. Traditional lab techniques for measuring conductivity are costly and time-consuming. This paper explores using machine learn ing, specifically multivariate regression, to predict fracture conductivity based on experimental data like Poisson’s ratio and proppant size. Optimizing these models can enhance hydraulic fracturing efficiency in shale formations. Hydraulic fracturing, often referred to as fracking, is a technique used to extract oil and gas from deep underground formations. This process involves injecting a high -pressure fluid mixture into the rock layer, creating fractures through which hydrocarbo ns can flow more freely to the production well. A crucial aspect of hydraulic fracturing is fracture conductivity, which refers to the ability of the created fractures to allow the flow of hydrocarbons. High fracture conductivity is essential for maintaini ng efficient hydrocarbon production. It depends on various factors, including the proppant type and size, the closure stress, and the properties of the fracturing fluid. Optimizing these factors ensures that the fractures remain open and conductive over th e lifespan of the well ( Montgomery, 2013; Holditch, 2006; Sharma et al., 2014). Determining fracture conductivity in situ is a very challenging task and conducting lab me asurements using Hassler core technique on shale are expensive and time co nsuming . The Hassler core experiment is a laboratory technique that simulates in -situ conditions to measure the permeability and conductivity of fractures in core samples. This method involves placing a core sample within a Hassler sleeve, applying confining pressu re, and injecting fluid to simulate the hydraulic fracturing process. Researchers measure the resultant fracture conductivity, which is a key parameter influencing the efficiency of hydrocarbon extraction from shale formations. (Wu et al. , 2017 , Wu et al. , 201 9). Machine learning is b eing used to advance scientific computing in a variety of fields, such as fluid mech anics , (Raissi et al ., 2019 , Zhang et al. , 2020 , Wang e l al., 2017) solid mechanics , (Haghighat et al., 2020 , Zheng et al., 2022, Arora et al., 2022 ) materials science etc (Kim et al., 202 1). Machine learning techniques can also be applied to predict the fracture conductivity of new shale formations based on models built using data collected from experiments. Good experimental data is key to building a well -performing prediction model. In order to build a model to predict fracture conductivity, the following information about various shale samples needs to be recorded while conducting experiments: Poisson’s rat io, Young’s Modulus, Temperature, Closure pressure, Proppant particle size, and Sand concentration. The wider the range of the above -mentioned parameters, the better the model will be. There are various machine learning techniques that can be used to build the above model . In this paper we will discuss about multivariate reg ression. Multivariate Regr ession To build a robust multivariate regression model for predicting fracture conductivity, several mathematical steps and techniques should be adopted: 1. Data Representation and Feature Matrix Construction : Begin by organizing the collected data into a matrix Xwhere each row represents an individual shale sample, and each column represents a feature (e.g., Poisson’s ratio, Young’s Modulus, Temperature, Closure pressure, Proppant particle size, Sand concentration). Let y be the vector of observed fracture conductivity values. Mathematically, this can be represented as: where n is the number of samples and p is the number of features. 2. Model Formulation : The multivariate regression model assumes a linear relationship between the dependent variable y and the independent variables X. The model can be expressed as: 3. Parameter Estimation : The coefficients β can be estimated using the Ordinary Least Squares (OLS) method, which minimizes the sum of the squared residuals. The OLS estimator is given by: 4. Regularization (if necessary) : To prevent overfitting, especially with high -dimensional data, regularization techniques such as Ridge Regression (L2 regularization) or Lasso Regression (L1 regularization) can be applied. Ridge regression minimizes: and Lasso regression minimizes: where λ is the regularization parameter. 5. Model Validation and Performance Evaluation: After fitting the model, its performance should be evaluated using metrics such as the coefficient of determination (R²), mean squared error (MSE), and root mean squared error (RMSE). Cross -validation techniques, such as k-fold cross -validation, can provid e a more robust assessment of model performance. Mathematically: 6. Model Interpretation and Application : Once the model is validated, interpret the coefficients to understand the relationship between the features and fracture conductivity. Apply the model to predict fracture conductivity for new shale formations by inputting their corresponding feature valu es into the regression equation. By following these mathematical steps, a comprehensive and accurate multivariate regression model can be developed, providing valuable insights for predicting the fracture conductivity of shale formations.