This paper was produced for the 2019 NAFEMS World Congress in Quebec Canada
This paper deals with the application of algorithms in the Finite Element Method (FEM) (Levin et al. 2016), (Levin et al. 2017), (Levin et al. 2015), (Smit et al. 1998), (Vdovichenko et al. 2016) for an estimation of effective elastoplastic properties (Mercier et al. 2012), (Castañeda and Willis 1999) of shales with kerogen inclusions.
Representative volumetric element (RVE) of a microstructure of a shale material with a kerogen undergo a series of numerical experiments. The kerogen is modeled as an elastoplastic inclusion in the matrix with continuous displacement under loading with normal stresses at the boundary. FEM discretization with RVEs allows to simulate arbitrary distribution of inclusions and their shapes versus simple geometries, which is only available in analytical solutions (Mauge and Kachanov 1994), (Talbot and Willis 2004), (Tsukrov and Kachanov 2000), (Tsukrov and Novak 2002). Another important feature of the developed numerical approach is that it accounts for the interaction between different inclusions, which is usually neglected in analytical models, although it can be a reason for underestimation of effective elastic parameters.
Linear effective elastic parameters have no strong relation to the value of external loading applied to the boundaries of RVE (Hashin 1962; Hashin and S. Shtrikman 1962; Hashin and Shtrikman 1962). The application of the model (Bagheri and Settari), (Myasnikov et al. 2016) in calculating the effective properties of the rock sample (periodicity cell) with elastic material behavior of the matrix with plastic inclusions will be discussed further. The presence of plastic zones near the boundaries of kerogen inclusions forming at certain levels of external loadings results in nonlinearity of effective properties of the RVE and, consequently, in their dependence on the value of the applied load.
In this work, by constructing a numerical model of the periodicity cell, we managed to overcome limitations of the analytical model (Hill 1963), (Kachanov et al. 1994) , (Mori and Tanaka 1973): a linear dependence between stresses and strains and stress concentrators of a simple shape. This generalization is made possible by FEA modelling of the boundary value problem at RVE and accounting for plastic deformations at the boundary of kerogen inclusions. Based on algorithms various characteristics of inclusions were implemented in the finite element model of CAE Fidesys (Vershinin et al. 2015),: density, shape, cohesion, friction angle, etc. The results of upscaling are presented for different values of external loadings, which demonstrate the dependence of their effective properties. The paper presents the results of numerical analysis of grid convergence, the impact of sizes of the periodicity cell and the size of inclusions on the results of upscaling. It also describes the coincidence of analytical and numerical solutions for a simplified elastic model and the difference between them in the case of a generalized elastoplastic model.