Stress evolution in elastically heterogeneous and non‐linear fluid‐saturated media with a Green's function approach

Classical analytical solutions of linear elasticity are used as auxiliary solutions to solve non‐linear, and elastically heterogeneous problems on fluid‐saturated media. The 2D Kelvin's solution for a homogeneous space is considered here for simplicity sake. The material non‐linearity could be due to irreversible deformations or non‐linear elasticity response typical of 4D analysis as it is done here. The general procedure relies on a discrete collocation method and a fixed point iterative approach to construct the displacement field. The method is validated by comparing the numerical results with the analytical solution for a layered cylinder embedded in an infinite space. The h‐convergence is checked numerically illustrating the strong influence of the number of facets used to discretize the boundaries. The convergence of the iterative process based on the displacement norm is of a quasi‐quadratic rate for near homogeneous materials and declines to sub‐linear rates as the contrast in elasticity modulus exceeds 15% of the values considered for the Green's function. The method is then applied to a 2D tilted block region where the depleting reservoir has elasticity parameters function of the volumetric strain, to shed some light on the 4D effects. It is shown that the velocity changes are sensitive to the volumetric strain as well as to the strain in the wave propagating direction. Differences, including the anisotropy due to the structural response at the field scale, between the predictions based on this non‐linear isotropic elasticity and the classical R‐factor approach are finally discussed.