Finite element modeling of a multi-physics poro-elastic problem in multiscale media

Abstract We present an iterative algorithm for mathematical modeling of an elastic deformation process in a fluid-saturated fractured-porous medium. A three-dimensional multi-physics problem describes the coupled isothermal processes of the solid elastic deformation and slightly compressible fluid flow under external pressure. Mathematical models of these processes are connected via interface conditions for the pressure and density fields on the surface of a fractured-porous medium. For solving the multi-physics problem, a special multiscale procedure was developed. We use a heterogeneous multiscale finite element discretization on coarse polyhedral grids for the solid elastic deformation problem. Multiscale shape functions are constructed using special interface conditions for a hydrodynamic pressure on the surface of pores. We apply a discontinuous Galerkin method and a stabilized finite element discretization on fine tetrahedral grids for solving the hydrodynamics problem in fluid-saturated pores. In this case, we can realize an effective parallel procedure for solving the multi-physics problem. In each pore, hydrodynamics problems can be solved in parallel and independently. Verifications of the computational schemes are presented. We consider three-dimensional media with a different volume concentration of cracks and pores. Computational modeling results are presented. A time of solving the multi-physics problem using fine and coarse grids is shown.

[1]  Guido Kanschat,et al.  The local discontinuous Galerkin method for the Oseen equations , 2003, Math. Comput..

[2]  Thomas Rylander,et al.  Computational Electromagnetics , 2005, Electronics, Power Electronics, Optoelectronics, Microwaves, Electromagnetics, and Radar.

[3]  Jerzy Pamin,et al.  Thermo-mechanical XFEM-type modeling of laminated structure with thin inner layer , 2015 .

[4]  Tayfun E. Tezduyar,et al.  Finite element stabilization parameters computed from element matrices and vectors , 2000 .

[5]  Alessandro Russo,et al.  Bubble stabilization of finite element methods for the linearized incompressible Navier-Stokes equations , 1996 .

[6]  Jun Zhu,et al.  Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes , 2013, J. Comput. Phys..

[7]  Jaime Peraire,et al.  Discontinuous Galerkin Solution of the Navier-Stokes Equations on Deformable Domains , 2007 .

[8]  Xiangxiong Zhang,et al.  Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms , 2011, J. Comput. Phys..

[9]  Zhangxin Chen,et al.  On the heterogeneous multiscale method with various macroscopic solvers , 2009 .

[10]  Thomas J. R. Hughes,et al.  Stabilized finite element methods for steady advection-diffusion with production , 1994 .

[11]  Doghonay Arjmand,et al.  A finite element heterogeneous multiscale method with improved control over the modeling error , 2016 .

[12]  I. Zubko.  Computation of elastic moduli of graphene monolayer in nonsymmetric formulation using energy-based approach , 2016 .

[13]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[14]  Yalchin Efendiev,et al.  Generalized multiscale finite element methods for problems in perforated heterogeneous domains , 2015, 1501.03536.

[15]  Yalchin Efendiev,et al.  Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains , 2016, 1605.07645.

[16]  Brian T. Helenbrook,et al.  Application of p-Multigrid to Discontinuous Galerkin Formulations of the Poisson Equation , 2005 .

[17]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[18]  Chi-Wang Shu,et al.  Discontinuous Galerkin method for hyperbolic equations involving $$\delta $$-singularities: negative-order norm error estimates and applications , 2013, Numerische Mathematik.

[19]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[20]  E Weinan,et al.  The heterogeneous multiscale method* , 2012, Acta Numerica.

[21]  A. Younes,et al.  Influence of fluid flow and heat transfer on crack propagation in SOFC multi-layered like material with anisotropic porous layers , 2016 .

[22]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[23]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of , 1986 .

[24]  A. Zidane,et al.  A combination of Crouzeix-Raviart, Discontinuous Galerkin and MPFA methods for buoyancy-driven flows , 2014 .

[25]  J. P. Webb.  Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements , 1999 .

[26]  Yalchin Efendiev,et al.  A generalized multiscale finite element method for elastic wave propagation in fractured media , 2016 .

[27]  Andrea Crivellini,et al.  An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations , 2006, J. Comput. Phys..

[28]  HanYi Wang,et al.  Numerical modeling of non-planar hydraulic fracture propagation in brittle and ductile rocks using XFEM with cohesive zone method , 2015 .

[29]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[30]  Ivo Babuška,et al.  The p-Version of the Finite Element Method for Parabolic Equations. Part 1 , 1981 .

[31]  L. D. Marini,et al.  Two families of mixed finite elements for second order elliptic problems , 1985 .