Numerical Homogenization of the Absolute Permeability Using the Conformal-Nodal and Mixed-Hybrid Finite Element Method

The modeling of hydrocarbon reservoirs and of aquifer-aquitard systems can be separated into two activities: geological modeling and fluid flow modeling. The geological model focuses on the geometry and the dimensions of the subsurface layers and faults, and on its rock types. The fluid flow model focuses on quantities like pressure, flux and dissipation, which are related to each other by rock parameters like permeability, storage coefficient, porosity and capillary pressure. The absolute permeability, which is the relevant parameter for steady single-phase flow of a fluid with constant viscosity and density, is studied here. When trying to match the geological model with the fluid flow model, it generally turns out that the spatial scale of the fluid flow model is built from units that are at least a hundred times larger in volume than the units of the geological model. To counter this mismatch in scales, the fine-scale permeabilities of the geological data model have to be ‘upscaled’ to coarse-scale permeabilities that relate the spatially averaged pressure, flux and dissipation to each other. The upscaled permeabilities may be considered as ‘complicated averages,’ which are derived from the spatially averaged flow quantities in such a way that the continuity equation, Darcy’s law and the dissipation equation remain valid on the coarse scale. In this paper the theory of upscaling will be presented from a physical point of view aiming at understanding, rather than mathematical rigorousness. Under the simplifying assumption of spatial periodicity of the fine-scale permeability distributions, homogenization theory can be applied. However, even then the spatial distribution of the permeability is generally so intricate that exact solutions of the homogenized permeability cannot be found. Therefore, numerical approximation methods have to be applied. To be able to estimate the approximation error, two numerical methods have been developed: one based on the conventional nodal finite element method (CN-FEM) and the other based on the mixed-hybrid finite element method (MH-FEM). CN-FEM gives an upper bound for the sum of the diagonal components of the homogenized mobility matrix, while MH-FEM gives a lower bound. Three numerical examples are presented.