Three dimensional simulation of fluid flow in X‐ray CT images of porous media

A numerical scheme is developed in order to simulate fluid flow in three dimensional (3‐D) microstructures. The governing equations for steady incompressible flow are solved using the semi‐implicit method for pressure‐linked equations (SIMPLE) finite difference scheme within a non‐staggered grid system that represents the 3‐D microstructure. This system allows solving the governing equations using only one computational cell. The numerical scheme is verified through simulating fluid flow in idealized 3‐D microstructures with known closed form solutions for permeability. The numerical factors affecting the solution in terms of convergence and accuracy are also discussed. These factors include the resolution of the analysed microstructure and the truncation criterion. Fluid flow in 2‐D X‐ray computed tomography (CT) images of real porous media microstructure is also simulated using this numerical model. These real microstructures include field cores of asphalt mixes, laboratory linear kneading compactor (LKC) specimens, and laboratory Superpave gyratory compactor (SGC) specimens. The numerical results for the permeability of the real microstructures are compared with the results from closed form solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical manifold method for dynamic nonlinear analysis of saturated porous media

It is well known that the Babuska–Brezzi stability criterion or the Zienkiewicz–Taylor patch test precludes the use of the finite elements with the same low order of interpolation for displacement and pore pressure in the nearly incompressible and undrained cases, unless some stabilization techniques are introduced for dynamic analysis of saturated porous medium where coupling occurs between the displacement of solid skeleton and pore pressure. The numerical manifold method (NMM), where the interpolation of displacement and pressure can be determined independently in an element for the solution of u–p formulation, is derived based on triangular mesh for the requirement of high accurate calculations from practical applications in the dynamic analysis of saturated porous materials. The matrices of equilibrium equations for the second‐order displacement and the first‐order pressure manifold method are given in detail for program coding. By close comparison with widely used finite element method, the NMM presents good stability for the coupling problems, particularly in the nearly incompressible and undrained cases. Numerical examples are given to illustrate the validity and stability of the manifold element developed. Copyright © 2006 John Wiley & Sons, Ltd.     

About Darcy's law in non-Galilean frame

This paper is aimed towards investigating the filtration law of an incompressible viscous Newtonian fluid through a rigid non-inertial porous medium (e.g. a porous medium placed in a centrifuge basket). The filtration law is obtained by upscaling the flow equations at the pore scale. The upscaling technique is the homogenization method of multiple scale expansions which rigorously gives the macroscopic behaviour and the effective properties without any prerequisite on the form of the macroscopic equations. The derived filtration law is similar to Darcy's law, but the tensor of permeability presents the following remarkable properties: it depends upon the angular velocity of the porous matrix, it verifies Hall–Onsager's relationship and it is a non-symmetric tensor. We thus deduce that, under rotation, an isotropic porous medium leads to a non-isotropic effective permeability. In this paper, we present the results of numerical simulations of the flow through rotating porous media. This allows us to highlight the deviations of the flow due to Coriolis effects at both the microscopic scale (i.e. the pore scale), and the macroscopic scale (i.e. the sample scale). The above results confirm that for an isotropic medium, phenomenological laws already proposed in the literature fails at reproducing three-dimensional Coriolis effects in all types of pores geometry. We show that Coriolis effects may lead to significant variations of the permeability measured during centrifuge tests when the inverse Ekman number Ek⁻¹ is 𝒪(1). These variations are estimated to be less than 5% if Ek⁻¹<0.2, which is the case of classical geotechnical centrifuge tests. We finally conclude by showing that available experimental data from tests carried out in centrifuges are not sufficient to determining the effective tensor of permeability of rotating porous media. Copyright © 2004 John Wiley & Sons, Ltd.     

A mathematical model of concentrate solids content for the wet drum magnetic separator

Low concentrate density from wet drum magnetic separators in dense medium circuits can cause operating difficulties due to inability to obtain the required circulating medium density and, indirectly, high medium solids losses. The literature is almost silent on the processes controlling concentrate density. However, the common name for the region through which concentrate is discharged—the squeeze pan gap—implies that some extrusion process is thought to be at work. There is no model of magnetics recovery in a wet drum magnetic separator, which includes as inputs all significant machine and operating variables.A series of trials, in both factorial experiments and in single variable experiments, was done using a purpose built rig which featured a small industrial scale (700 mm lip length, 900 mm diameter) wet drum magnetic separator. A substantial data set of 191 trials was generated in this work. The results of the factorial experiments were used to identify the variables having a significant effect on magnetics recovery.It is proposed, based both on the experimental observations of the present work and on observations reported in the literature, that the process controlling magnetic separator concentrate density is one of drainage. Such a process should be able to be defined by an initial moisture, a drainage rate and a drainage time, the latter being defined by the volumetric flowrate and the volume within the drainage zone. The magnetics can be characterised by an experimentally derived ultimate drainage moisture. A model based on these concepts and containing adjustable parameters was developed. This model was then fitted to a randomly chosen 80% of the data, and validated by application to the remaining 20%. The model is shown to be a good fit to data over concentrate solids content values from 40% solids to 80% solids and for both magnetite and ferrosilicon feeds.     

Wide-angle seismic velocities in heterogeneous crust

Seismic velocities measured by wide-angle surveys are commonly used to constrain material composition in the deep crust. Therefore, it is important to understand how these velocities are affected by the presence of multiscale heterogeneities. The effects may be characterised by the scale of the heterogeneity relative to the dominant seismic wavelength (λ); what is clear is that heterogeneities of all scales and strengths bias wide-angle velocities to some degree. Waveform modelling was used to investigate the apparent wide-angle P -wave velocities of different heterogeneous lower crusts. A constant composition (50 per cent felsic and 50 per cent ultramafic) was formed into a variety of 1- and 2-D heterogeneous arrangements and the resulting wide-angle seismic velocity was estimated. Elastic, 1-D models produced the largest velocity shift relative to the true average velocity of the medium (which is the velocity of an isotropic mixture of the two components). Thick (width > λ) horizontal layers, as a result of Fermat's Principle, provided the largest increase in velocity; thin (width ≪λ) vertical layers produced the largest decrease in velocity. Acoustic 2-D algorithms were shown to be inadequate for modelling the kinematics of waves in bodies with multiscale heterogeneities. Elastic, 2-D modelling found velocity shifts (both positive and negative) that were of a smaller magnitude than those produced by 1-D models. The key to the magnitude of the velocity shift appears to be the connectivity of the fast (and/or slow) components. Thus, the models with the highest apparent levels of connectivity between the fast phases, the 1-D layers, produced the highest-magnitude velocity shifts. To understand the relationship between measured seismic velocities and petrology in the deep crust it is clear that high-resolution structural information (which describes such connectivity) must be included in any modelling.     

The reflection of plane waves in a poroelastic half-space saturated with inviscid fluid

This paper discusses surface displacements, surface strain, rocking, and energy partitioning during reflection-of-plane waves in a fluid-saturated poroelastic half-space. The medium is modeled by Biot's theory, and is assumed to be saturated with inviscid fluid. A linear porosity-modulus relation based on experimental data on sandstones is used to determine the material parameters for Biot's model. Numerical results in terms of angle of incident waves and Poisson's ratio are illustrated for various porosities and degrees of solid frame stiffness. The results show that the amount of solid frame stiffness controls the response of a fluid-saturated porous system. A poroelastic medium with essentially dry-frame stiffness behaves like an elastic medium, and the influence of pore fluid increases as dry-frame stiffness is reduced. The effects of a second P-wave become noticeable in poroelastic media with low dry-frame stiffness.     

Détermination expérimentale des paramètres équivalents d'un milieu poreux hétérogène en écoulements uniforme ou radial

Tracer tests are carried out in a heterogeneous porous medium that has a 3D correlated random distribution of the permeabilities. The fitting of numerical models provides the values of equivalent permeability and macrodispersivity characterizing a 2D homogeneous horizontal medium. Different flow configurations are studied: uniform, radial and pump and treat (doublet). The fitted parameter sets are independent of the flow type, except for the doublet. They are greater than the values predicted by stochastic theories, due to the small number of correlation lengths explored by the tracer and the limited extension of the experimental set-up. To cite this article: C. Danquigny, P. Ackerer, C. R. Geoscience 337 (2005).     

多空隙组合地质单元渗流理论分析与试验

岩体通常是以孔隙岩石为基质并包含各种成因的裂隙和溶隙的复杂多空隙组合地质材料。为研究孔隙-裂隙-溶隙多空隙组合介质的渗流特性,在简要介绍多孔介质渗流Darcy定律、平板窄缝流Poiseuille定律和圆形管道流Darcy-Weibach理论的基础上,分析了几种孔隙-裂隙-溶隙组合情况的多空隙介质渗流特性,推导出相应组合的等效渗透系数KE,给出了描述多空隙组合介质KE的一般表达式,讨论了影响各种空隙组合KE的主要因素。以普通砖模拟岩石基质,通过砖身钻孔填充及砖间缝隙填充,试验模拟了岩溶岩体和裂隙岩体中溶孔和裂隙填充后的渗透情况。试验结果与所推求理论公式计算出的KE在同一量级且误差很小,很好地验证了所推导的多空隙组合介质渗透系数表达式的有效性。     

VTI介质多波速度与各向异性系数求取及应用

针对具有垂直对称轴的横向各向同性(VTI)介质,研究了多波速度与地层各向异性系数之间的关系。采用适应大偏移距以及强各向异性介质的双平方根方程,进行速度和各向异性系数分析,并将批量计算与交互解释相结合,实现了高精度各向异性介质多波速度与各向异性系数提取,并形成了一套利用多波地震资料求取地震波纵、横波速度及各向异性系数的方法和软件。所求取的参数可用于多波地震资料处理,提高多波资料的成像质量,也可为地层地质解释提供参考。实际资料的应用效果证明了该方法和软件的正确性及有效性。