SUMMARYIn this paper, a numerical convergence study of family of ux-continuous schemes is presented. The family of ux-continuous schemes is characterized in terms of quadrature parameterization, where the local position of continuity deÿnes the quadrature point and hence the family. A convergence study is carried out for the discretization in physical space and the e ect of a range of quadrature points on convergence is explored. Structured cell-centred and unstructured cell-vertex schemes are considered.Homogeneous and heterogeneous cases are tested, and convergence is established for a number of examples with discontinuous permeability tensor including a velocity ÿeld with singularity. Such cases frequently arise in subsurface ow modelling. A convergence comparison with CVFE is also presented.

SUMMARY This paper presents a fracture mapping (FM) approach combined with the extended finite element method (XFEM) to simulate coupled deformation and fluid flow in fractured porous media. Specifically, the method accurately represents the impact of discrete fractures on flow and deformation, although the individual fractures are not part of the finite element mesh. A key feature of FM‐XFEM is its ability to model discontinuities in the domain independently of the computational mesh. The proposed FM approach is a continuum‐based approach that is used to model the flow interaction between the porous matrix and existing fractures via a transfer function. Fracture geometry is defined using the level set method. Therefore, in contrast to the discrete fracture flow model, the fracture representation is not meshed along with the computational domain. Consequently, the method is able to determine the influence of fractures on fluid flow within a fractured domain without the complexity of meshing the fractures within the domain. The XFEM component of the scheme addresses the discontinuous displacement field within elements that are intersected by existing fractures. In XFEM, enrichment functions are added to the standard finite element approximation to adequately resolve discontinuous fields within the simulation domain. Numerical tests illustrate the ability of the method to adequately describe the displacement and fluid pressure fields within a fractured domain at significantly less computational expense than explicitly resolving the fracture within the finite element mesh. Copyright © 2013 John Wiley & Sons, Ltd.

We present a finite element scheme which combines the dual permeability method (DPM) and the extended finite element method (XFEM) to simulate coupled deformation and fluid flow in fractured porous media. The scheme incorporates spatial variability in fracture properties without requiring the fracture to be discretized or aligned with the computational mesh. DPM is used to describe the fluid flow interaction between the porous matrix and fractures, whilst XFEM is used to address the discontinuous displacement field within elements which intersect fractures. The method is strongly coupled and solves the stress and flow equations simultaneously for each time increment. DPM-XFEM uses the level set method (LSM) to define existing fractures and eliminates the need for their explicit discretization during simulation. The method performs well on coarse structured grids and does not require complex, difficult to generate meshes to conduct simulations. Comparison between the proposed method and the discrete fracture method (DFM) shows its ability to adequately determine the displacement and fluid pore pressure distribution within a fractured domain.

a b s t r a c tIn this paper, we propose and verify a numerical approach to simulate fluid flow in deformable porous media without requiring the discretization to conform to the geometry of the sealed fractures (possibly intersecting). This approach is based on a fully coupled hydro-mechanical analysis and an extended finite element method (XFEM) to represent discrete fractures. Convergence tests indicate that the proposed scheme is both consistent and stable. The contributions of this paper include: (1) a new junction enrichment to describe intersecting fractures in deformable porous media; (2) the treatment of sealed fractures. We employ the resulting discretization scheme to perform numerical experiments, to illustrate that the inclination angles of the fractures and the penetration ratio of the sealed fractures are two key parameters governing the flow within the fractured porous medium.