Combined use of finite volume and network modelling for Stokes flow and permeability tensor computation in porous media

Recent developments in imaging techniques such as X-ray computed micro-tomography make possible 3D imaging of porous media micro-geometry at unrivalled resolutions. After segmentation, the resulting binary images can be used to define the grid necessary to compute the local fluid flow through the pores or the averaged permeability tensor of the porous material. Efficient and accurate codes are available to perform those computations, but at extremely high computational costs (memory and CPU time) when large volumes of data, as presently accessible, are considered. An alternative to the direct computation of pressure and velocity fields (DCPV) in the 3D image is the use of network models (NM) where the complex pore structure is represented by an equivalent network of pore bodies (PB) connected via pore throats (PT). This method produces a system of linear equations of moderate size that can be easily solved. Then, applicability of NM is not limited by computational costs, but by difficulties in building, from a discretized 3D image, an equivalent network summarizing the relevant topological (connectivity of pore bodies) and geometrical (resistance associated to each pore throat) aspects of a porous medium. In this work different crucial steps of this process are analysed in depth: starting from a 3D binary image of a porous sample, a skeleton of the pore space is built. This skeleton is analysed to define a graph where nodes correspond to PB and branches to connections intersecting PT. Finally, the partitioning itself is performed to precisely delineate the PB associated to the nodes and the PT. Boundary conditions are considered with particular interest since the NM has to be used for flow modelling and permeability tensor estimation. The resistance associated to each PT is evaluated solving a local flow problem that is theoretically introduced. An original solution is proposed to handle the cases where more than two pores partly share a PT. Application to real examples are presented

Applying a deformable surface model to identify foam beads in a sample of polypropylene foam and follow their evolution in a dynamic crash loading experiment

The objective of this work is to study the deformation of polypropylene foam during a dynamic crash loading. The first difficulty consisted in conceiving an experimental setup that would allow to visualise intermediate steps in the deformation of the foam, which requires a non-destructive imaging technique. Fast external surface imaging is not sufficient for an accurate study of the deformation, therefore our attention focused on X-ray tomography. Because crash loading time is much smaller than tomogaphic acquisition time (a few milliseconds vs. almost an hour), several interrupted crashes are applied (a dynamic loading with a constraint on the strain), in between which a micro-tomogram is acquired. If we assume that the foam behaviour is not modified by the interruptedness of the dynamic loading (as in quasi-static loading), then we obtain a series of tomograms showing the evolution of the foam sample during dynamic compression. The second difficulty, that is presented here, is to use this information to quantify the foam deformation (and subsequently use this experimental data for predictive modelling) at the mesoscopic scale, i.e. that of the beads. Polypropylene foam is a multi-cellular material, each bead (around 2mm in diameter) is composed of micrometric cells, making the separation of the beads a difficult image analysis problem, as compared to separation of foam bubbles in a metallic foam, for instance. The solution we propose extracts a representative volume inside each bead, in order to calculate, at each stage of the compression, values such as grain density. For this purpose, a first processing, consisting of a sequence of simple signal processing and discrete morphological operators, determines approximate bead centres. With a simple nearest neighbours approach, the position for a given bead is identified for each stage of the experiment. From each centre a deformable surface algorithm is applied: a spherical mesh is placed inside the bead and expands until reaching the bead walls. This allows us to visualise the deformation and measure the average density of each bead during the compression. Unfortunately, due to the difficulty in precisely identifying bead walls (even manually), complete bead volumes and wall thickness is not yet determined: the deformable surfaces contain only the majority of the inside of the beads, and there still remains significant intersticial volume which has no physical significance (it represents a volume in which the bead wall is located, volume that is a function of the image processing). Future work to obtain more accurate data would focus on methods such as skeletonisation of the intersticial volume

Characterization by X‐ray <scp>μCT</scp> of the air‐filled porosity of an agricultural soil at different matric potentials

peer reviewedTo describe various important soil processes like the release of greenhouse gases or the proliferation of microorganisms, it is necessary to assess quantitatively how the geometry and in particular the connectivity of the air-filled pore space of a soil evolves as it is progressively dried. The availability of X-ray computed microtomography (μCT) images of soil samples now allows this information to be obtained directly, without having to rely on the interpretation of macroscopic measurements using capillary theory, as used to be the case. In this general context, we present different methods to describe quantitatively the configuration of the air-filled pore space in 3D μCT images of 20 separate samples of a loamy soil equilibrated at different matric potentials. Even though measures using μCT on such multi-scale materials strongly depends on image resolution, our results show that in general, soil samples most often behave as expected, e.g., connectivity increases with higher negative matric potential, while tortuosity decreases. However, simple correlations could not be found between the evolution of quantitative descriptors of the pore space at the different matric potentials and routinely measured macroscopic soil parameters. A statistical analysis of all soil samples concurrently confirmed this lack of correspondence

Algorithmic aspects of converting surface mesh data to volumetric images

In image analysis, some processes might imply a change or conversion in the structure of the data. The structure types will depend on the processing method and applications, and can consist of pixel data, point sets, finite elements, vector fields, implicit surfaces, graphs, basic shapes (spheres, cylinders, or cubes), etc. The work presented here discusses the problem of converting a triangulated surface mesh to a 3D image, a need that arises for example when using active surface-type segmentation methods of 3D images, shape-fitting, or combining data from laser surface scanning with 3D imaging. During the course of numerous projects, two main classes of mesh-to-image conversions have appeared: those identifying voxels (pixels in a 3D image) that intersect the mesh, or voxels that are contained in the mesh, supposing it defines a closed surface

Lien entre la microstructure des matériaux poreux et leur perméabilité : Mise en évidence des paramètres géométriques et topologiques influant sur les propriétés de transport par analyses d'images microtomographiques

he objective of this work is to develop 3D image analysis tools to study the micronic pore structure of porous materials, obtained by X-ray microtomography, and study the relation between microgeometry and macroscopic transport properties. From a binarised image of the pore space, a complete sequence of processing (artefact filtration, skeletonisation, watershed, etc. ) is proposed for positioning and delimiting the pores. A comparison with available methods is performed, and a methodology to qualify the robustness of these processes is presented. The decomposition is used, firstly for extracting geometric parameters of the porous microstructure and studying the relation with intrinsic permeability; secondly to produce a simplified pore network on which to perform numerical simulations.Ce travail a pour but de concevoir des outils d'analyse d'image 3D de matériaux poreux, obtenues par microtomographie à rayons X, afin de caractériser géométriquement la structure micronique des pores et de mettre en évidence le lien entre microgéométrie et propriétés de transport macroscopiques. Partant d'une image segmentée, une séquence complète de traitements (filtrage d'artefacts, squelettisation, LPE, etc.) est proposée pour positionner et délimiter les pores. Une comparaison aux techniques existantes est faite, et une méthodologie qualifiant la robustesse des procédures est présentée. Cette décomposition est utilisée, premièrement pour extraire des descripteurs géométriques de la microstructure porale qui sont examinés en rapport avec la perméabilité intrinsèque ; deuxièmement pour aider à la construction d'un réseau de pores permettant d'effectuer des simulations numériques

Using Avizo for better Minecraft constructions

Minecraft is an independent video game released in 2009 that has quickly become a phenomenon, counting over 23 million registered users, and regarded as one of the best games to play at work [1]. It is a multiplayer sandbox game focused on creativity and building. The open world in which the player evolves is a set of cubes of equal size and textured according to what they represent (dirt, stone), defined on a regular 3D grid. The player is free to dig or mine such blocks, storing these in his inventory, and replacing them one by one, where desired. This is the core principle in the construction process. Even though this open world is defined as cubic blocks on a regular grid, it provides a strong potential for creating elaborate 3D shapes. The difficulty lies in recreating a shape that one wants to model, which is where Avizo has been a valuable tool. In most situations, 3D models are defined as meshes, typically triangulated 2D meshes. If one such model is to be reproduced in Minecraft, either one has to be able to mentally visualize with astounding precision the positions in space where individual blocks should be placed, or possess a tool that firstly converts the model from a set of triangles to a set of cubes on a regular grid, and secondly allows to efficiently locate the positions in 3D space where each block should be placed. To solve the first point, a module I have developed and integrated in Avizo takes as input a triangulated mesh and a 3D image (a 3D uniform scalar field), the latter providing the 3D grid. For each triangle in the mesh, the module determines what unit cubes, or voxels it intersects in the 3D image, and set the voxel values accordingly. In order for this process to be algorithmically efficient, a small number of voxels should be tested, and the cube/triangle intersection test should be very quick. The first point is handled by beginning with the voxels that contain the triangle vertices and iteratively examining the 26-neighbourhood of the voxels found to intersect the triangle. The second point is handled by decomposing the intersection test using various projections of the voxel centre along the grid diagonals and length measurements using the infinity norm. The center image in figure 2 is the result of such a conversion from the mesh shown in the left image. In order to display a 3D image as a set of cubes, another module has been integrated in Avizo. Finally, concerning the second point, by effective usage of the Boundingbox and LocalAxis (for displaying gridlines and axis positions), and OrthoSlice modules, the Minecraft player can easily determine, slice by slice, where to place each block in the game. One such result is the right image of figure 2: the blocks have been placed one by one from the ground up, using the setup described above in Avizo. Although the block placing is tedious, the resulting satisfaction and praise by fellow players is well worth the effort

Measuring porosity inside a cross-section of a tube

This macro (source code at the end of the document) was written to measure the porosity inside an axial crosssection in a tube. The macro makes four assumptions: 1. The cross-section is perpendicular to the tube axis 2. The tube thickness is uniform 3. No artefacts except noise are present in the images (e.g. beam hardening) 4. The solid appears in light gray over a dark background From the set of cross-sections to analyze, the macro first thresholds the images, find the region inside the tube and computes the porosity in that region