Homogeneity Equation for the Effective Elastic Properties of Heterogenous Rocks Developed From Machine Learning

• Published in: Journal of geophysical research. Machine learning and computation Vol. 2; no. 1
• Main Authors: Maalouf, E., Kareem, H., Chebli, H.
• Format: Journal Article
• Language: English
• Published: Wiley 01.03.2025
• Subjects: cracks; effective medium theory; elastic properties; homogeneity equation; neural networks; rock
• ISSN: 2993-5210; 2993-5210
• DOI: 10.1029/2024JH000483

This study develops an artificial neural network (ANN) model designed to estimate the effective elastic properties of heterogeneous rocks with inclusions. The ANN addresses the limitations of traditional effective medium models by avoiding theoretical assumptions, such as low inclusion percentages, and it incorporates the effects of inclusion length, spacing and orientation. This approach enables the accurate calculation of an equation that links dimensionless geometrical and physical features of rocks to their equivalent Young's modulus and Poisson's ratio. A two‐dimensional finite element model is employed to calculate, through dynamic loading, the effective Young's modulus and Poisson's ratio of an elastic matrix with properties comparable to sandstones and shales that has ellipsoidal inclusions representing cracks. While this study focuses on a single matrix and inclusion type, the results demonstrate that the method can be extended to any combination of matrix and inclusion elastic properties. The numerical model is validated against experimental and analytical solutions for simple geometries and then used to generate the training data, with inputs including crack length, crack density, aspect ratio, and distances between cracks. The ANN, though dependent on the numerical model for training and testing, operates much faster and can be applied to a broader range of scenarios. By training a single‐layer neural network, an accurate homogeneity equation is derived, relating effective elastic properties to dimensionless inclusion parameters. This approach to obtain a homogeneity equation has applications in rock mechanics, geophysics, and composite materials. Plain Language Summary In this work, an artificial neural network (ANN) model is established for estimating the effective elastic properties of heterogeneous rocks containing inclusions. The ANN improves on the shortcomings of the effective medium models by removing theoretical assumptions, such as low percentage of inclusions, and the model includes the influence of the inclusion length, spacing and orientation. This approach makes it possible to accurately determine an equation that correlates the geometric and physical properties of heterogenous rocks to their corresponding Young's Modulus and Poisson's Ratio. Numerical modeling is used to compute the effective Young's modulus and Poisson's ratio of an elastic matrix containing crack‐like ellipsoidal inclusions and train the machine learning model. It is shown that after a single‐layer neural network is trained, an accurate homogeneity equation is obtained for mapping effective elastic moduli onto dimensionless inclusion parameters of rocks with vertical and horizontal inclusions. While this work focuses on a single matrix and inclusion type, this approach can be applied to any combination of matrix and inclusion elastic properties. Key Points The neural network (NN) derived homogeneity equation estimates elastic properties of heterogeneous rocks with inclusions The NN bypasses theoretical assumptions of effective medium theories by considering crack length, spacing, density, and orientation Results focus on a sandstone‐like matrix and water‐filled cracks, showcasing the approach's applicability to other elastic properties