Holger Steeb

The size effect in foams and its theoretical and numerical investigation

Experimental data for foams lead to different values of the elastic moduli depending on the performed test, i.e. compression and tension tests give a different set of parameters than shear and bending tests. This may be explained by the size effect, which depends on the microstructure of the foams. Thus, in this paper, the behaviour of foams is investigated on the basis of both microscopic and macroscopic mechanical models. The microscopic approach is based on a lattice beam model. The solution of this model shows that the boundary–layer effect is strongly local but allows for the explanation of the size effect. Furthermore, the size effect can be included in the macroscopic continuum model by application of a Cosserat formulation. The extended continuum model allows for the independent fit of material parameters to different load cases, i.e. to compression and shear. The solution of the macroscopic Cosserat model permits a relation of the internal length–scale to the average cell size of the microstructure.

A hybrid-dimensional approach for an efficient numerical modeling of the hydro-mechanics of fractures

Characterization of subsurface fluid flow requires accounting for hydro-mechanical coupling between fluid-pressure variations and rock deformation. Particularly, flow of a compressible fluid along compliant hydraulic conduits, such as joints, fractures, or faults, is strongly affected by the associated deformation of the surrounding rock. We investigated and compared two alternative numerical modeling approaches that describe the transient fluid-pressure distribution along a single deformable fracture embedded in a rock matrix. First, we analyzed the coupled hydro-mechanical problem within the framework of Biots poroelastic equations. Second, in a hybrid-dimensional approach, deformation characteristics of the surrounding rock were combined with a one-dimensional approximation of the fluid-flow problem to account for the high aspect ratios of fractures and the associated numerical problems. A dimensional analysis of the governing equations reveals that the occurring physical phenomena strongly depend on the geometry of the hydraulic conduit and on the boundary conditions. For the analyzed geometries, hydro-mechanical coupling effects dominate and convection effects can be neglected. Numerical solutions for coupled hydro-mechanical phenomena were obtained and compared to field data to characterize the fractured rock in the vicinity of an injection borehole. Either approach captures convection, diffusion, and hydro-mechanical effects, yet the hybrid-dimensional approach is advantageous due to its applicability to problems involving high-aspect-ratio features. For such cases, the modeling of pumping tests by means of the hybrid-dimensional approach showed that the observed inverse-pressure responses are the result of the coupling between the fluid flow in the fracture and the rock deformation caused by fluid-pressure variations along the fracture. Storage capacity as a single parameter of a fracture is insufficient to address all aspects of the coupling.

Transition of effective hydraulic properties from low to high Reynolds number flow in porous media

We numerically analyze fluid flow through porous media up to a limiting Reynolds number of O(103). Due to inertial effects, such processes exhibit a gradual transition from laminar to turbulent flow for increasing magnitudes of Re. On the macroscopic scale, inertial transition implies nonlinearities in the relationship between the effective macroscopic pressure gradient and the filter velocity, typically accounted for in terms of the quadratic Forchheimer equation. However, various inertia-based extensions to the linear Darcy equation have been discussed in the literature; most prominently cubic polynomials in velocity. The numerical results presented in this contribution indicate that inertial transition, as observed in the apparent permeability, hydraulic tortuosity, and interfacial drag, is inherently of sigmoidal shape. Based on this observation, we derive a novel filtration law which is consistent with Darcys law at small Re, reproduces Forchheimers law at large Re, and exhibits higher-order leading terms in the weak inertia regime.

On attenuation of seismic waves associated with flow in fractures

Heterogeneity of porous media induces a number of fluid-flow mechanisms causing attenuation of seismic waves. Attenuation induced by squirt-type mechanisms has previously been analyzed for aspect ratios smaller or equal to 10 3 . Using a hybrid-dimensional modeling approach, particularly apt for large aspect ratio conduits, we numerically simulated deformation-induced fluid flow along two intersecting fractures to investigate the physics of attenuation related to the interaction of fracture-induced fluid flow and to leak-off. Attenuation related to fracture flow increases in magnitude with increasing geometrical aspect ratio of the fracture. The inherent time scales of both flow mechanisms do not influence each other, but the faster process is associated with stronger attenuation than the slower process. Models relying on simple diffusion equations have rather limited potential for approximation of pressure transients.

A low-cost X-ray-transparent experimental cell for synchrotron-based X-ray microtomography studies under geological reservoir conditions

A new modular X-ray-transparent experimental cell enables tomographic investigations of fluid rock interaction under natural reservoir conditions (confining pressure up to 20 MPa, pore fluid pressure up to 15 MPa, temperature ranging from 296 to 473 K). The portable cell can be used at synchrotron radiation sources that deliver a minimum X-ray flux density of 10(9) photons mm(-2) s(-1) in the energy range 30-100 keV to acquire tomographic datasets in less than 60 s. It has been successfully used in three experiments at the bending-magnet beamline 2BM at the Advanced Photon Source. The cell can be easily machined and assembled from off-the-shelf components at relatively low costs, and its modular design allows it to be adapted to a wide range of experiments and lower-energy X-ray sources.

About the transition frequency in Biot’s theory

Biots theory of wave propagation in porous media includes a characteristic frequency which is used to distinguish the low-frequency from the high-frequency range. Its determination is based on an investigation of fluid flow through different pore geometries on a smaller scale and a subsequent upscaling process. This idea is limited due to the assumptions made on the smaller scale. It can be enhanced for a general two-phase system by three properties: Inertia of the solid, elasticity of the solid, and frequency dependent corrections of the momentum exchange. They become important for highly porous media with liquids.

Waves in Residual-Saturated Porous Media

We present a three-phase model describing wave propagation phenomena in residual-saturated porous media. The model consists of a continuous non-wetting phase and a discontinuous wetting phase and is an extension of classical biphasic (Biot-type) models. The model includes resonance effects of single liquid bridges or liquid clusters with miscellaneous eigenfrequencies taking into account a visco-elastic restoring force (pinned oscillations and/or sliding motion of the contact line). For the quasi-static limit case, i.e., ω ↦0, the results of the model are identical with the phase velocity obtained with the well-known Gassmann–Wood limit.

Analysis of high-resolution X-ray computed tomography images of Bentheim sandstone under elevated confining pressures

A sample of Bentheim sandstone was characterized using high-resolution threedimensional X-ray microscopy at two different confining pressures of 1 MPa and 20 MPa. The two recordings can be directly compared with each other because the same sample volume was imaged in either case. After image processing, a porosity reduction from 21.92% to 21.76% can be deduced from the segmented data. With voxel-based numerical simulation techniques, we determined apparent hydraulic transport properties and effective elastic properties. These results were compared with laboratory measurements using reference samples. Laboratory and computed volumes, as well as hydraulic transport properties, agree fairly well. To achieve a reasonable agreement for the effective elastic properties, we define pressure-dependent grain contact zones in addition to mineral phases in the digital rock images. From that, we derive a specific digital rock physics template resulting in a very good agreement between laboratory data and simulations. The digital rock physics template aims to contribute to a more standardized approach of X-ray computed tomography data analysis as a tool to determine and predict elastic rock properties.

Characterization of Shape Memory Polymer Estane by Means of Dynamic Mechanical Thermal Analysis Technique

Commercially available shape memory polymer (SMP) Estane (designation: ETE75DT3 NAT022) is investigated by means of dynamic mechanical thermal analysis (DMTA) technique in torsion mode using the Modular Compact Rheometer MCR-301 (Anton Paar GmbH). Amplitude sweep tests have been run below and above the glass transition temperature to establish the linear viscoelastic range (LVR) in glassy and rubbery phase of this SMP for the correct physical interpretation of DMTA data. Temperature sweep tests were performed at various frequencies to study the influence of this parameter on values of the storage and loss moduli and the storage and loss compliances as well as the viscosities. These tests have been carried out in heating mode with different rates and at different strain amplitudes. The short- and long-term behavior of SMP Estane have been studied by frequency sweep tests performed at different temperatures and data have been transformed into time-domain properties by applying time-temperature superposition principles. All these DMTA data provide the experimental basis for the study of relaxation processes, property-structure relationships, and the shape memory effect in this little-known SMP.

Variational formulation of oscillating fluid clusters and oscillator-like classification. I. Theory

The present work develops the theoretical framework to describe oscillations of fluid clusters. The basic physical phenomena are presented and justified assumptions lead to the final set of equations for different types of oscillations (pinned/sliding). The special combination of a liquid cluster surrounded by a rigid solid matrix and a gas is investigated in more detail. Furthermore, a classification of oscillating fluid clusters is presented using a one-dimensional oscillator model. This classification includes three dynamic properties: mass, eigenfrequency, and damping whereas conceptual implementation and limitations for use in multiphase theories are clearly indicated. The frequency dependent flow profile leads to frequency dependence of the dynamic parameters. This is discussed and represented by dimensionless numbers.