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.

中文翻译：

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用格林函数方法在弹性异质和非线性流体饱和介质中的应力演化

线性弹性的经典解析解被用作辅助解来解决流体饱和介质上的非线性和弹性异质问题。为简单起见，这里考虑均匀空间的二维开尔文解。材料非线性可能是由于不可逆变形或 4D 分析典型的非线性弹性响应造成的，正如这里所做的那样。一般过程依赖于离散搭配方法和定点迭代方法来构建位移场。通过将数值结果与嵌入无限空间的分层圆柱体的解析解进行比较，验证了该方法。该^ h- 收敛性以数字方式检查，说明用于离散边界的面数的强烈影响。基于位移范数的迭代过程的收敛对于接近均质的材料是准二次速率，并且随着弹性模量的对比度超过格林函数所考虑值的 15% 时下降到亚线性速率。然后将该方法应用于 2D 倾斜块区域，其中消耗性储层具有体积应变的弹性参数函数，以揭示 4D 效应。结果表明，速度变化对体积应变以及波传播方向的应变都很敏感。差异，包括由于场尺度结构响应引起的各向异性，