Comprehensive analysis of stochastic processes in the mining and geological environment using the averaged operator split method

Introduction. This paper examines the application of methods for calculating complex geophysical processes occurring in structures characterized by a combination of regularity and entropic elements. These structures include both regions with distinct geological patterns and areas with irregularly distributed properties. The analysis is performed using the mathematical apparatus of partial differential equations, widely used in modern geophysics and the mining industry. Methodology and Research Methods. Methods for studying the splitting principle of the averaged operator as applied to nonlinear systems of equations in regular and randomly distributed structural elements are based on a modern range of approaches aimed at comprehensively studying the dynamics of complex systems subject to random disturbances and spatial heterogeneities of the environment. These approaches help investigate geodynamic processes in structures with changing geometry and physical and mechanical properties of rocks, reveal the mechanisms of stress - strain redistribution, and assess the stability of rock masses under uncertainty and variable external loading. Research Results and Discussion. This geophysical study examines the splitting of an averaged operator as applied to nonlinear systems of equations in regular and stochastically distributed framework structures of rocks. This method involves separating rapidly changing characteristics caused by microheterogeneity in the material structure from slowly evolving macroparameters reflecting global trends in the development of the geological massif. To achieve this, the initial differential equations are transformed by selecting appropriate scales that effectively account for these effects. The developed approach provides reliable estimates of the average statistical properties of the studied geoenvironments in the presence of a significant contrast between the sizes of porous inclusions and large structural elements of the rock. Conclusions. Averaging differential equations is an effective method for transforming the original complex system of equations into simplified models that preserve the fundamental physical and geological characteristics of the studied environments. This approach creates favorable conditions for successfully solving applied problems related to the mechanical state of rock masses, the properties of host rocks, optimizing production technologies, and developing advanced methods for studying rock masses. Further research areas include the development of specialized tools based on the proposed methodology. Such tools should include a set of methods and algorithms aimed at improving the efficiency of managing the mechanical state of rock masses, improving the characteristics of host rocks, optimizing mining processes, and creating new promising approaches to studying geological structures

Complex system · Nonlinear system · Partial differential equation · Stochastic process · Engineering and Agricultural Innovations · Geotechnical and Geomechanical Engineering · Heat Transfer and Mathematical Modeling