51st U.S. Rock Mechanics/Geomechanics Symposium,
San Francisco, California, USA
2017. American Rock Mechanics Association
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ABSTRACT: 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 Nx times Ny 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.
Upscaling of flow parameters for reservoir flow simulation has been extensively studied [1, 2]. The reason for the interest lies on the fact that oil and gas reservoirs are usually very heterogeneous systems and, while computer power now allows the simulation of very large models , it is either impossible or impractical to represent details at all relevant scales.
Many methods have been suggested for the upscaling of flow parameters [4-7]. Most methods published to date, start with the assumption of a finite difference discretization and target the upscaling of transmissibilities. Alternative methods rely on the effective medium theory, which can only be applied to specific geometries .
Traditional approaches are based on analytical solutions, under the assumption of a specific connectivity and a given set of boundary conditions (local approximations). To remove these assumptions, global methods have been proposed  which involves the computation of a coarse representation that minimizes the global norm. A more practical approach, which in turn is more amenable for generalization, known as the global-local method has also been proposed .
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