A Numerical Approach for Arbitrary Cracks in a Fluid-Saturated Medium (original) (raw)
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An efficient numerical model for incompressible two-phase flow in fractured media
Advances in Water Resources, 2008
Various numerical methods have been used in the literature to simulate single and multiphase flow in fractured media. A promising approach is the use of the discrete-fracture model where the fracture entities in the permeable media are described explicitly in the computational grid. In this work, we present a critical review of the main conventional methods for multiphase flow in fractured media including the finite difference (FD), finite volume (FV), and finite element (FE) methods, that are coupled with the discrete-fracture model. All the conventional methods have inherent limitations in accuracy and applications. The FD method, for example, is restricted to horizontal and vertical fractures. The accuracy of the vertex-centered FV method depends on the size of the matrix gridcells next to the fractures; for an acceptable accuracy the matrix gridcells next to the fractures should be small. The FE method cannot describe properly the saturation discontinuity at the matrix-fracture interface. In this work, we introduce a new approach that is free from the limitations of the conventional methods. Our proposed approach is applicable in 2D and 3D unstructured griddings with low mesh orientation effect; it captures the saturation discontinuity from the contrast in capillary pressure between the rock matrix and fractures. The matrix-fracture and fracture-fracture fluxes are calculated based on powerful features of the mixed finite element (MFE) method which provides, in addition to the gridcell pressures, the pressures at the gridcell interfaces and can readily model the pressure discontinuities at impermeable faults in a simple way. To reduce the numerical dispersion, we use the discontinuous Galerkin (DG) method to approximate the saturation equation. We take advantage of a hybrid time scheme to alleviate the restrictions on the size of the time step in the fracture network. Several numerical examples in 2D and 3D demonstrate the robustness of the proposed model. Results show the significance of capillary pressure and orders of magnitude increase in computational speed compared to previous works.
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A continuum finite elements (FE) formulation involving discontinuity of the displacement field to simulate fracture mechanics problems for brittle or quasi-brittle solids is proposed. A homogeneous discontinuity is assumed in a cracked finite element, and a new simple stress-based implementation of the displacement discontinuity is introduced by an appropriate stress field correction to simulate, as usually done in an elastic-plastic classical FE formulation, the mechanical effects of the crack, i.e. the proposed formulation does not introduce any discontinuous displacement field by mean of special or modified shape functions. Both linear elastic and elastic-plastic behaviour of the non-cracked material can be considered. 2D fracture problems are solved by the proposed procedure, to predict the load-displacement behaviour as well as the crack patterns in brittle structures. The new proposed simple FE formulation for discontinuous problems is computationally economic, and maintains the internal continuity of the numerical model with well-known numerical benefits.
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An extension to a finite strain framework of a two-scale numerical model for propagating crack in porous material is proposed to model the fracture in intervertebral discs. In the model, a crack is described as a propagating cohesive zone by exploiting the partition-of-unity property of finite element shape functions. At the micro-scale, the flow in the cohesive crack is modelled as viscous fluid using Stokes' equations which are averaged over the cross section of the cavity. At the macro-scale, identities are derived to couple the local momentum and the mass balance to the governing equations for a saturated porous material. The resulting discrete equations are nonlinear due to the cohesive constitutive equations and the geometrically nonlinear kinematic relations. A Newton-Raphson iterative procedure is used to consistently linearise the derived system while a Crank-Nicholson scheme takes care of the time integration of the system. The derived model is used to analyse a quasi-static crack growth in confined compression under tensile loading.
Engineering Fracture Mechanics, 2018
The implicit formulation of the boundary element method is applied to bidimensional problems of material failure involving, sequentially, inelastic dissipation with softening in continuous media, bifurcation and transition between weak and strong discontinuities. The bifurcation condition is defined by the singularity of the localization tensor. Weak discontinuities are related to strain localization bands of finite width, which become increasingly narrow until to collapse in a surface with discontinuous displacement field, called strong discontinuity surface. To associate such steps to the fracture process in quasi-brittle materials, an isotropic damage constitutive model is used to represent the behaviour in all of them, considering the adaptations that come from the strong discontinuity analysis for the post-bifurcation steps. The crack propagation across the domain is done by an automatic cells generation algorithm and, in this context, the fracture process zone in the crack tip became totally represented.
2020
We show that for the simulation of crack propagation in quasi-brittle, two-dimensional solids, very good results can be obtained with an embedded strong discontinuity quadrilateral finite element that has incompatible modes. Even more importantly, we demonstrate that these results can be obtained without using a crack tracking algorithm. Therefore, the simulation of crack patterns with several cracks, including branching, becomes possible. The avoidance of a tracking algorithm is mainly enabled by the application of a novel, local (Gauss-point based), criterion for crack nucleation, which determines when the localisation line is embedded, and its position and orientation. We treat the crack evolution in terms of a thermodynamical framework, with softening variables describing internal dissipative mechanisms of material degradation. As presented by numerical examples, many elements in the mesh may develop a crack, but only some of them actually open and/or slide, dissipate fracture e...
Transport in Porous Media, 2014
In this paper, we present a general partition of unity-based cohesive zone model for fracture propagation and nucleation in saturated porous materials. We consider both two-dimensional isotropic and orthotropic media based on the general Biot theory. Fluid flow from the bulk formation into the fracture is accounted for. The fracture propagation is based on an average stress approach. This approach is adjusted to be directionally depended for orthotropic materials. The accuracy of the continuous part of the model is addressed by performing Mandel's problem for isotropic and orthotropic materials. The performance of the model is investigated with a propagating fracture in an orthotropic material and by considering fracture nucleation and propagation in an isotropic mixed-mode fracture problem. In the latter example we also investigated the influence of the bulk permeability on the numerical results.
Enriched finite elements for branching cracks in deformable porous media
Engineering Analysis with Boundary Elements, 2015
In 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.
Theory and numerics for finite deformation fracture modelling using strong discontinuities
International Journal for Numerical Methods in Engineering, 2006
A general finite element approach for the modelling of fracture is presented for the geometrically nonlinear case. The kinematical representation is based on a strong discontinuity formulation in line with the concept of partition of unity for finite elements. Thus, the deformation map is defined in terms of one continuous and one discontinuous portion, considered as mutually independent, giving rise to a weak formulation of the equilibrium consisting of two coupled equations. In addition, two different fracture criteria are considered. Firstly, a principle stress criterion in terms of the material Mandel stress in conjunction with a material cohesive zone law, relating the cohesive Mandel traction to a material displacement 'jump' associated with the direct discontinuity. Secondly, a criterion of Griffith type is formulated in terms of the material-crack-driving force (MCDF) with the crack propagation direction determined by the direction of the force, corresponding to the direction of maximum energy release. Apart from the material modelling, the numerical treatment and aspects of computational implementation of the proposed approach is also thoroughly discussed and the paper is concluded with a few numerical examples illustrating the capabilities of the proposed approach and the connection between the two fracture criteria.