Two level homogenization of flows in deforming double porosity media: biot-darcy-brinkman model

We present the two-level homogenization of the flow in a deformable double-porous structure described at two characteristic scales: the higher level porosity associated with the mesoscopic structure is constituted by channels in an elastic skeleton which is made of a microporous material. The macroscopic model is derived by the asymptotic analysis of the viscous flow in the heterogeneous structure characterized by two small parameters. The first level upscaling yields a Biot continuum model coupled with the Stokes flow. The second step of the homogenization leads to a macroscopic flow model which attains the form of the Darcy-Brinkman flow model coupled with the deformation of the poroelastic continuum involving the effective parameters given by the microscopic and the mesoscopic porosity features

Reactive Flows in Deformable, Complex Media

Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is variable, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, or biological systems. Such models include various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. Having this as the background theme, this workshop focused on novel techniques and ideas in the analysis, the numerical discretization and the upscaling of such problems, as well as on applications of major societal relevance today

Multilevel Schwarz Methods for Porous Media Problems

In this thesis, efficient overlapping multilevel Schwarz preconditioners are used to iteratively solve Hdiv-conforming finite element discretizations of models in poroelasticity, and an innovative two-scale multilevel Schwarz method is developed for the solution of pore-scale porous media models. The convergence of two-level Schwarz methods is rigorously proven for Biot’s consolidation model, as well as a Biot-Brinkman model by utilizing the conservation property of the discretization. The numerical performance of the proposed multiplicative and hybrid two-level Schwarz methods is tested in different problem settings by covering broad ranges of the parameter regimes, showing robust results in variations of the parameters in the system that are uniform in the mesh size. For extreme parameters a scaling of the system yields robustness of the iteration counts. Optimality of the relaxation factor of the hybrid method is investigated and the performance of the multilevel methods is shown to be nearly identical to the two-level case. The additional diffusion term in the Biot-Brinkman model yields a stabilization for high permeabilities. Additionally, a homogenizing two-scale multilevel Schwarz preconditioner is developed for the iterative solution of high-resolution computations of flow in porous media at the pore scale, i.e., a Stokes problem in a periodically perforated domain. Different homogenized operators known from the literature are used as coarse-scale operators within a multilevel Schwarz preconditioner applied to Hdiv-conforming discretizations of an extended model problem. A comparison in the numerical performance tests shows that an operator of Brinkman type with optimized effective tensor yields the best performance results in an axisymmetric configuration and a moderately anisotropic geometry of the obstacles, outperforming Darcy and Stokes as coarse-scale operators, as well as a standard multigrid method, that serves as a benchmark test

A study of the transient fluid flow around a semi-infinite crack

AbstractApplying the implicit finite difference approximation of the time derivative term, the diffusion equation governing fluid-flow around a crack in a fluid-infiltrated undeformable porous medium is transformed into a non-homogeneous modified Helmholtz’s equation. Then, Vekua’s theory regarding the solution of linear, second order, elliptical partial differential equations is employed for its solution and the corresponding Riemann function is found. Subsequently, the general solution of the Dirichlet initial-boundary value problem for a prescribed arbitrary distribution of pressure acting along a semi-infinite crack is found in the form of a Cauchy singular integral equation of the second kind. A numerical Gauss–Chebyshev quadrature scheme is proposed to solve this singular integral equation that is first applied to the steady-state problem and then to the transient problem. It is shown that the density of the Cauchy integral of the transient problem μˆ bears a simple similarity relationship with the steady-state problem μˆ0 of the form μˆ(x)≈(1-λ/0.4)μˆ0(x) for 0⩽x<∞,y=0, wherein λ=1/D·t, with D denoting the diffusivity coefficient and t the time. This solution is the first step towards the solution of transient fluid flow around multiple cracks and then of the coupled problem of a crack or cracks in deformable porous media and for the study of fluid-driven cracks in poroelastic media