Computational homogenization and reduced order modeling of dif- fusion processes in fluid-saturated porous media

Book, 2016

Fluid-saturated porous media are well-known for their manifold attenuation mechanisms caused by wave-induced fluid flow between microscopic or mesoscopic heterogeneities. If elastic waves propagate through such a medium, the pore space is heterogeneously com- pacted resulting in local pressure gradients. These pressure gradients are equilibrated by a redistribution of the viscous pore fluid (water, gas, oil, etc.), which causes pore pressure diffusion. Hereby, part of the wave energy is lost, and the wave is attenuated. In this contribution, we concentrate on the upscaling behavior of fluid-saturated rocks featuring double porosity, patchy saturation or networks of fluid conduits. Based on Biot’s quasi- static equations of consolidation and an appropriate hybrid-dimensional description of the fracture networks, we establish a computational homogenization framework. To this end, we assume the pressure diffusion to occur within mesoscopic volume elements much smaller than the macroscopic wave length. Hence, diffusion takes place on a length scale much smaller than the observer scale and is considered, from the macroscopic viewpoint, as a local process. The heterogeneous poroelastic medium, therefore, is substituted by a homogeneous macroscopic Cauchy medium with apparent viscoelastic properties. The material properties of the substitute model are derived making use of a novel order reduc- tion technique allowing for a numerically efficient treatment of the scale-transition. The macroscopic internal variables representing the overall viscoelasticity are interpreted as parameters controlling the activity of mesoscopic pressure modes identified by the Proper Orthogonal Decomposition (POD) technique. We show that the resulting decoupled sys- tem of evolution equations is equivalent to that of a generalized Maxwell-Zener model. In the second part of this contribution we weaken the locality constraint and allow for macroscopically observable seepage of the pore fluid. Hence, the character of the ho- mogenization problem changes and the macroscopic substitute medium is considered as a poroviscoelastic material. We derive a consistent computational homogenization scheme and enrich the proposed order reduction concept by the required additional features. The contribution is completed by numerous numerical investigations including validation tests and extensive discussions of our findings.