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

Abstract

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.

Intelligent element: Coupling Green function approach and artificial intelligence to reduce discretization effort

“…The formulation of the intelligent element method is the combination of the method based on Green functions with the generalization capability of ANNs. The Green function method, presented by Peres et al., 65 consists of applying the principle of reciprocity to a pair of geomechanical systems, the first system is characterized by a classical problem that has an analytical solution (Green function) and the second is the problem to be solved. For the sake of illustration, without loss of generality, consider as the classical problem the case of a force applied in a linear elastic semi‐infinite medium (whose Green function was presented by Telles and Brebbia 66 and based on is shown in Appendix A) and as the desired problem a geomechanical reservoir being depleted, located in a stratified zone embedded in a semi‐infinite medium (Figure 2).…”

Non‐iterative Green's function approach for 2D unbounded heterogeneous fluid‐saturated media

“…Equation ( 1) is only applicable to homogeneous problems. In the case of heterogeneous problems, Peres et al 28 propose the inclusion of complementary tensor…”

“…The oil and gas recovery processes generate changes in the pressure of the fluids contained in the pores of the reservoir rock that cause changes in the displacement, strain, and stress fields of the geological formations. As demonstrated by Peres et al, 28 using the principle of virtual works it is possible to establish the following relationship between displacement, strain, and stress…”

“…In the original version of the GFA, the displacement field generated by a pore-pressure change is calculated using an iterative numerical scheme, which limits the computational performance of the method. In addition, the validation study presented by Peres et al 28 shows that the convergence rate of the iterative scheme decreases when the ratio among the mechanical properties of the rocks increases, reaching a point of no convergence when the ratio is greater than 100%. These limitations have prevented the application of the GFA in real problems.…”

“…The Green's function approach (GFA) proposed by Peres et al, [28][29][30] uses the classical Green's functions for point load as an auxiliary solution that together with the reciprocity theorem to capture the variation in the stress state of a geological formation subjected to processes of injection or extraction of fluids. This approach has no limitations in terms of geometry, number of layers, and heterogeneity of the geological profile and can be applied to materials with linear or non-linear behavior.…”