## Program – Abstracts of Invited Speakers AC.CES

#### René de Borst, Julien Vignollet and Stefan May: Fracture modelling in fluid-saturated porous media using extended and isogeometric finite element analysis

Since the pioneering work of Terzaghi and Biot the flow of fluids in deforming porous media has received considerable attention. Nevertheless, flow in fractured or fracturing porous media has received much less attention. Yet, the presence of damage, such as cracks, faults, and shear bands, can markedly change the physical behaviour. Indeed, the physics of the flow within such discontinuities can be very different from that of the interstitial fluid in the deforming bulk material.

Herein, we will develop a general numerical model for flow in progressively fracturing porous media. The theory includes flow inside stationary and propagating traction-free and cohesive cracks. The flow inside the evolving crack is assumed to be tangentially to the crack, so as to enable a two-scale approach. At the fine scale the flow in the cavity created by the (possibly cohesive) crack is modelled using a sub-grid scale model. Since the cross-sectional dimension of the cavity is small compared to its length, the flow equations can be averaged over the width of the cavity. The resulting equations provide the momentum and mass couplings to the standard equations for a porous medium, which hold on the macroscopic scale. Numerically, the two-scale model which ensues, imposes some requirements on the interpolation of the displacement and pressure fields near the discontinuity.

Advanced discretisation methods are needed to model a crack, which is essentially an internal free boundary. Exploiting the partition-of-unity property is one option. Another powerful method is isogeometric analysis. Therefore, we will first develop an isogeometric formulation for porous media. Initially, this will be done without taking cracks into account. Next, the extension will be made to include stationary and propagating cracks, such that it is possible to have fluid transport in the cracks. The extension can be done in two ways: either via lowering the order of the interpolation, or by using isogeometric interface elements. An issue that we will also address in this context is the proper integration of such elements, since standard Gauss integration tends to result in oscillatory traction profiles.

#### Bernardo Cockburn: An overview of the evolution of the HDG methods

We give a short overview of the development of the so-called hybridizable discontinuous Galerkin methods. In the framework of steady-state diffusion, we give the original motivation that prompted the introduction of these methods, provide their general definition, display three different ways of presenting them (which allow us of establish unexpected links with many other methods), and discuss their built-in stabilization mechanism. We then describe the successive stages of their development until the present day. We end by briefly describing their extensions to a variety of other problems of practical interest, and by describing ongoing work and some open problems.

#### Thomas J.R. Hughes: Isogeometric Analysis: Ten Years After

This October marks the tenth anniversary of the appearance of the first paper [1] describing my vision of how to address a major problem in Computer Aided Engineering (CAE). The motivation was as follows: Designs are encapsulated in Computer Aided Design (CAD) systems. Simulation is performed in Finite Element Analysis (FEA) programs. FEA requires the conversions of CAD descriptions to analysis-suitable formats form which finite element meshes can be developed. The conversion process involves many steps, is tedious and labor intensive, and is the major bottleneck in the engineering design-through-analysis process, accounting for more than 80% of overall analysis time, which remains an enormous impediment to the efficiency of the overall engineering product development cycle.

The approach taken in [1] was given the pithy name Isogeometric Analysis. Since its inception it has become a focus of research within both the fields of FEA and CAD and is rapidly becoming a mainstream analysis methodology and a new paradigm for geometric design [2]. The key concept utilized in the technical approach is the development of a new framework for FEA, based on rich geometric descriptions originating in CAD, resulting in a single geometric model that serves as a basis for both design and analysis.

In this talk I will make a few introductory remarks about Isogeometric Analysis, describe a few current areas of intense activity, and areas where problems remain open, representing opportunities for future research.

[1] T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs, Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement, Computer Methods in Applied Mechanics and Engineering, 194, (2005) 4135-4195.

[2] J.A. Cottrell, T.J.R. Hughes and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, Chichester, U.K., 2009.

#### Alfio Quarteroni: Reduced basis methods: principles, algorithms, applications

Projection-based reduced-order models (ROMs) provide efficient strategies to tackle parametrized partial differential equations, multi-query problems (control and inverse problems, for instance), and yield real-time simulations of complex problems.
Essential ingredients are the regularity and low dimensionality of solution manifolds, the use of Offline/Online computational stratagems,
the availability of a posteriori error estimates, and the use of low-dimensional approximation spaces.

In this talk I will recall the mathematical concepts behind Reduced Basis methods (a special family of ROM), provide both algebraic and geometrical interpretation, and address a few meaningful examples in simulation and control of haemodynamics problems

#### Klaus Regenauer-Lieb: Computational Reservoir Engineering and Unconventional Geomechanics

Unconventional Energy (Geothermal, Shale Gas, Shale Oil, Tight Gas, Coalbed Methane, Heavy Oil/Tar sands, Methane Hydrates) and mineral resources (deep in situ leaching) are trapped in a low porosity/permeability environment and are difficult to produce. Yet they define the most abundant resources available on our planet and are attractive targets for novel exploration, stimulation and production techniques. Recent setbacks in the Australian geothermal industry have revealed a regime of high tectonic stress, fluid overpressures, deeper and hotter resources where conventional empirical techniques fail. Similarly, unconventional gas production from central Australian shales at 3500-4000 m depth and ambient temperature conditions around 200°C have sharp decline curves and are difficult to render commercially viable. One important aspect is that conventional reservoir engineering approaches ignore the strong thermal-hydraulic-mechanical and chemical couplings (THMC) and therefore cannot adequately describe the time-dependent reservoir behavior. We propose that an unconventional geomechanics approach for reservoir engineering may resolve these issues. This unconventional geomechanics approach explicitly considers tight THMC couplings and associated creep processes.

In both the deep Australian geothermal and the unconventional shale gas reservoirs the role of diagenesis has for instance been found to be fundamental for the reservoir fluid flow. Diagenesis involves fluid release mineral reactions, where under mechanical deformation a nominally impermeable mineral can suddenly become permeable under critical conditions through localization instabilities. Such diagenetic reactions are thermally activated and in the shale gas example they are known to switch on suddenly in the diagenetic window between 100-200°C. In classical petroleum engineering approaches these interlayer water/gas release reactions are classically held responsible for cementation and are thought to significantly reduce porosity and permeability. However, we can show that for a critical competition of mechanical matrix deformation and capability of Darcy flow the contrary result is expected. Fluid channeling instabilities can form large interconnected fluid reservoirs in otherwise impermeable formations. We present analytical closed form solutions of these instabilities (periodic cnoidal waves) and discuss numerical solution techniques using unconventional geomechanics for unconventional reservoirs.

#### Jonathan Richard Shewchuk: Dynamic Meshing for Finite Element Simulations with Radically Changing Geometry

I describe algorithms and software for physically simulating materials that are substantially reshaped by fluid flow, plastic flow, fracture, and material merging. We combine Lagrangian finite element methods with algorithms for dynamic remeshing, which update a volume mesh undergoing radical deformations so that its tetrahedra do not deteriorate below a fixed quality threshold. Our dynamic mesher is conservative: it replaces as few tetrahedra as possible, and thereby limits the visual artifacts, artificial viscosity, and artificial plasticity that would be introduced if we repeatedly remeshed the domain from scratch. It also locally refines and coarsens a mesh, and even creates anisotropic tetrahedra, wherever a simulation requests it.

Our simulation method addresses a range of material behavior from purely elastic to highly plastic and from inviscid to viscous fluids, with particular advantages for materials in which these properties are mixed such as melting metals. I illustrate these capabilities with animations of fluid flow and deforming

elastoplastic objects.