Upscaling of Hydraulic and Mechanical Parameters in Coupled Flow Deformation Simulation Problems

We investigate the effective elastic properties of heterogeneous rocks. This is done by using the finite element method (FEM) on systems with different types of heterogeneity. We look at configurations with known analytical solutions, namely, laminate composites and demonstrate that our numerical formulation produces a reasonable match for multiple combinations of elastic properties. We show that the stress component transversal to the composite is homogeneous, while the longitudinal component is piecewise constant in agreement with the theoretical assumptions. Next, we consider a system subdivided in N_x times N_y regions and investigate two types of heterogeneity: continuous and discrete. The former consists of systems with parameters sampled from a continuous probability density function (pdf), while the latter correspond to systems with a finite number material types. For the first case, we use a uniform pdf for the Young’s modulus and Poisson’s ratio and observed that the resulting distribution of effective moduli follow a strongly skewed distribution towards low values and, in addition, the distribution of local stresses is smeared out. In the case of a discrete distribution of properties, we consider a two-phase material with contrasting properties and distributed with a stiff background, and softer material distributed in isolated clusters. We observe that, in this case, the distribution of effective moduli follow a normal pdf while local stresses exhibit regions with extremely high values.