Finite volume method with muiltipoint flux and stress approximations using harmonic points for solving poroelasticity problems - ALBUQUERQUE, Pedro Victor Paixão (2023)
Modeling physical phenomena and how they interact with each another is at the core of Science and Engineering. In the present work, the phenomena of interest is the so called Poroelasticy, which is a field of science that studies the relationship between fluid flow and solid deformation within a porous media. This theory have several applications such as in Geotechnical and Petroleum Engineering, Hydrogeology and even in Medicine and Biology, to name a few. In the context of Petroleum Reservoir Engineering, until recently, the reservoir rocks mechanical response was neglected, to reduce simulations costs, since the main phenomena of interest was how the fluid flows inside the reservoir. The presence of a freely moving fluid in a porous rock modifies its mechanical response and, in return, this mechanical response influences the fluid flow inside the pore. The mathematical modeling of the aforementioned physical phenomena results in a set of partial differential equations which only have proper analytical solutions in simple, non-realistic cases. However, with the development of numerical and computational tools, approximate solutions can be obtained, thus allowing the understanding and prediction of the behavior of such physical phenomena. The mathematical model used in the present work is based on Biot’s theory of poroelasticity with the following assumptions for the solid phase: Quasi-static loading; Plane Strain; Infinitesimal Strain; Isotropic Linear Elasticity; Compressible Solid Matrix; and the following assumptions for the fluid phase: Single Phase Fluid; Slightly Compressible Fluid; Newtonian Fluid; Isotermic flow and; No gravitational effects. The set of Differential Equations were approximated via a unified finite volume framework, using a Multipoint Flux Approximation unsing Harmonic Points for both the fluid and solid equations, with a co-located variable arrangement and the Rhie-Chow interpolation, along with a Backwards Euler Scheme for temporal integration. The coupling between pressure and displacement was done via the fixed-strain split. The numerical modeling described in the present work is verified using benchmark problems found in the Poroelasticiy Literature. The results presented shows the numerical model is capable of producing robust and accurate approximated solutions, with both structured and unstructured meshes.