Computational Methods for Fracture in Porous Media: Isogeometric and Extended Finite Element Methods
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Computational Methods for Fracture in Porous Media: Isogeometric and Extended Finite Element Methods provides a self-contained presentation of new modeling techniques for simulating crack propagation in fluid-saturated porous materials. This book reviews the basic equations that govern fluid-saturated porous media. A multi-scale approach to modeling fluid transport in joins, cracks, and faults is described in such a way that the resulting formulation allows for a sub-grid representation of the crack and fluid flow in the crack. Interface elements are also analyzed with their extension to the hydromechanical case. The flexibility of Extended Finite Element Method for non-stationary cracks is also explored and their formulation for fracture in porous media described. This book introduces Isogeometric finite element methods and its basic features and properties. The rapidly evolving phase-field approach to fracture is also discussed.
The applications of this book’s content cover various fields of engineering, making it a valuable resource for researchers in soil, rock and biomechanics.
- Teaches both new and upcoming computational techniques for simulating fracture in (partially) fluid-saturated porous media
- Helps readers learn how to couple modern computational methods with non-linear fracture mechanics and flow in porous media
- Presents tactics on how to simulate fracture propagation in hydraulic fracturing
René de Borst
Dr. René de Borst is the Centenary Professor of Civil Engineering, University of Sheffield. Started his career at TNO and obtained his doctorate in 1986 at Delft University of Technology (with distinction). In 1988 was appointed Professor of Computational Mechanics at the Faculty of Civil Engineering of Delft University of Technology and in 1999 as Professor of Engineering Mechanics at the Faculty of Aerospace Engineering at the same university. In 2000 he was appointed as Distinguished Professor. In 2007 was appointed Dean of the Faculty of Mechanical Engineering and Distinguished Professor at Eindhoven University of Technology. Dr. De Borst has worked on several topics in engineering mechanics and in materials engineering, such as the mechanical properties of concrete, soils, composites and rubbers, in particular the development of mathematical and numerical models. His most significant work is on fracture mechanics, computational mechanics, and frictional materials. His research is of importance in civil engineering, structural engineering, and aerospace engineering.
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Computational Methods for Fracture in Porous Media - René de Borst
2017
Chapter 1
Introduction
Abstract
This chapter presents a general introduction into and a literature review of fluid-saturated deformable porous media, either fractured or fracturing, and the various discretization techniques that can be employed to solve the initial value problems that arise in transport in deformable, fracturing porous media.
Keywords
Fluid-saturated porous media; Fracture; Mass transport in cracks; Discretization techniques
1.1 Fracture in Porous Media
Fracture lies at the heart of many failures in natural and man-made materials. Fracture mechanics, as a scientific discipline in its own right, originated in the early 20th century with the pioneering work of Griffith (1921). Driven by some spectacular disasters in the shipbuilding and aerospace industries, and building on the seminal work of Irwin (1957), linear elastic fracture mechanics (LEFM) has become an important tool in the analysis of structural integrity.
Linear elastic fracture mechanics applies when the dissipative processes remain confined to a region in the vicinity of the crack tip that is small compared to the structural dimensions. When this condition is not met, e.g. when considering cracking in more heterogeneous materials like soils, rocks, concrete, ceramics, or many biomaterials, cohesive-zone models are to be preferred (Dugdale, 1960; Barenblatt, 1962). Cohesive-zone models remove the stress singularity that exists in linear elastic fracture mechanics. Fracture is then a natural outcome of the constitutive relations in the bulk and the interface, together with the balances of mass and momentum. Rice and Simons (1976) have provided compelling arguments in favor of the use of cohesive-zone models in fluid-saturated porous media by analyzing shear crack growth. Other arguments based on experimental evidence have been given in Valkó and Economides (1995).
The vast majority of the developments in fracture relate to solid materials. Occasionally, porous materials have been considered, but studies of crack initiation and propagation in porous materials, where the pores can be filled with fluids, are rather seldom found, at least until fairly recently. Indeed, the theory of fluid flow in deforming porous media has been practically confined to intact materials (Terzaghi, 1943; Biot, 1965; Coussy, 1995, 2010; Lewis and Schrefler, 1998; de Boer, 2000), and this holds a fortiori for numerical studies on fracture in porous media.
At the same time, fracture in heterogeneous, (partially) fluid-saturated porous media is a challenging, multi-scale problem with moving internal boundaries, characterized by a high degree of complexity and uncertainty. Moreover, fracture initiation and propagation in (partially) fluid-saturated porous materials occur frequently, indicating that there is a large practical relevance. The existence and propagation of cracks in porous materials can be undesirable, like those that form in human tissues, in rocks or salt domes is concerned. But cracking can also be a pivotal element in an industrial process, for example hydraulic fracturing in the oil and gas industry, Fig. 1.2. Another important application area is the rupture of geological faults, where the change in geometry of a fault can drastically affect pore pressures and local fluid flow as the faults can act as channels in which the fluid can flow freely (Rudnicki and Rice, 2006).
Figure 1.1 Fracture in the intervertebral disc, a fluid-saturated human tissue [Courtesy: J.M.R. Huyghe].
Figure 1.2 Simplified diagram of hydraulic fracturing on the horizontal part of a shale gas well [ http://bbc.com/news ].
The first approaches to analyze the propagation of fluid-saturated cracks were of an analytical nature (Perkins and Kern, 1961; Nordgren, 1972; Khristianovic and Zheltov, 1955; Geertsma and de Klerk, 1969). Idealized geometries of a single, fluid-filled crack were considered, the surrounding medium was taken as linear elastic, homogeneous and impervious, and an ad hoc leak-off term was introduced to account for the fluid loss into the surrounding medium (Carter, 1957). Linear elastic fracture mechanics was used to derive a crack propagation criterion. Invoking scaling laws, Detournay (2004) has put these works on a solid basis, and has identified that, depending on, inter alia, the values for the fracture toughness and the fluid viscosity, different propagation regimes can be distinguished. In case of viscosity-dominated propagation the classical square-root singularity at the crack tip no longer holds, and is replaced by a weaker singularity. Differentiation is made between four regimes: almost no leak-off vs. much leak-off, and viscosity vs. toughness dominated (Adachi et al., 2007).
1.2 The Representation of Cracks and Fluid Flow in Cracks
Ever since the first attempts to simulate fracture using the finite element method, there has been a debate on the most efficient and physically realistic method to model cracking. Essentially, there are two approaches: one can either represent cracks in a discrete manner, which dates back to Ngo and Scordelis (1967), or use a smeared or continuum approach (Rashid, 1968), see de Borst et al. (2004) for an overview and evolution of both approaches.
Because of their relative simplicity and ability to simulate complex crack patterns, at least in principle, smeared models have gained much popularity. However, this comes at a price. First, the introduction of decohesion renders continuum models ill-posed at a generic stage of crack propagation. In addition to this mathematical deficiency there is the physical argument that it is difficult, if possible at all, to translate the strains in the continuum model into discrete quantities like crack opening and crack sliding. Indeed, gradient-damage models (Peerlings et al., 1996; Frémond and Nedjar, 1996) and phase-field models (Francfort and Marigo, 1998) overcome the mathematical deficiency, but do not necessarily resolve the issue of quantifying discrete quantities like the crack opening. With the need to use cohesive fracture models which employ the crack opening and sliding as essential components in the constitutive relation in the crack, the difficulties to properly represent the crack opening only become a more pressing issue (Verhoosel and de Borst, 2013). The issue is also prominent when considering fluid transport in cracked porous media, as the possible difference between the fluid velocities inside and outside the cracks makes it difficult to quantify mass transport.
In the spirit of distributing discontinuities over a finite width, a model to capture fluid flow in a porous medium, which is intersected by multiple cracks, was proposed by Barenblatt et al. (1960). Fig. 1.3 shows the two different scales at which flow in fractured porous media is then considered: a microscopic scale at which we have interstitial pore fluid between grains, and a mesoscopic scale where fluid can flow almost freely in the cracks or faults. This idea was generalized to a deformable porous medium, which resulted in the double porosity model (Aifantis, 1980; Wilson and Aifantis, 1982; Khaled et al., 1984; Beskos and Aifantis, 1986; Bai et al., 1999), wherein Biot's theory for deformable porous media (Biot, 1941) was exploited. The double porosity model describes the effects of cracks on fluid flow and vice versa in a homogenized sense, but as in any distributed approach, the local interaction between crack propagation and fluid flow is not captured.
Figure 1.3 Two scales at which fluid can flow in a fractured porous medium: a microscopic scale with interstitial fluid between particles, and (nearly) free fluid within the fractures.
Returning to discrete crack models, it is noted that these have first been implemented by a simple nodal release technique (Ngo and Scordelis, 1967), and later, in a more elegant and versatile manner, using interface elements. Remeshing has been introduced to decouple the crack propagation path from the original mesh layout (Ingraffea and Saouma, 1985). Especially in three dimensions this can lead to complications and a considerable amount of remeshing. The extended finite element method (Belytschko and Black, 1999; Moës et al., 1999) has been proposed as an alternative, accommodating linear elastic fracture mechanics as well as cohesive fracture (Wells and Sluys, 2001; Moës and Belytschko, 2002; Remmers et al., 2003). It decouples the crack propagation path from the underlying discretization, and has been a main carrier of numerical approaches to fracture for more than a