A study on relationships between measured crustal stresses and depth based on spherical shell model

Due to the exploitation of mineral resources in the deep underground space, it is more and more important to study the variation trend of deep in-situ stress field. In recent years the researches have updated many in-situ stress data. It is necessary to improve the existing estimation methods on this account. Based on the basic theory of elasticity and spherical shell model, this paper studied the effect of gravitational body force varying with depth and internal pressure, and obtained the theoretical formula of macro in-situ stress distribution; Employing the latest and widely distributed measured data, this paper fitted the crustal physical parameters in the theoretical formula, and obtained the fitting formula of vertical in-situ stress and average horizontal in-situ stress varying with depth. The results show that: the distribution of crustal stresses with depth is no longer linear on the macroscopic scale of the Earth; however, within the current depth range of human activities, basically the average in-situ stresses vary linearly with depth, which is closest to the results of the Dinnick hypothesis at γ=22 kN/m³ and λ=0.9; with the increase of depth, the average lateral pressure coefficient gradually approaches 1, which is closer to the measured data than the envelope proposed by Brown and Hoek; the high pressure in the crust will cause horizontal tensile stress, but the compressive stress generated by gravity is higher and therefore the vertical and horizontal in-situ stress in the whole crust are compressive stress. The research results provide a theoretical prediction model for the estimation of crustal stress distribution trend, especially in deep underground space.