Fractals — An overview of potential applications to transport in porous media

We present an overview of the potential applicability of fractal concepts to various aspects of transport phenomena in heterogeneous porous media. Three examples of phenomena where a fractal approach should prove illuminating are presented. In the first example we consider pore level heterogeneities as typified by pore surface roughness. We suggest that roughness may be usefully modelled by fractal curves and surfaces and also cite experimental evidence for regarding pores as fractals. In the second example we consider a fractal network approach to modelling large-scale heterogeneities. The presence of features on all length scales in simple fractal models should capture the essential role played by the presence of heterogeneities on many scales in natural reservoirs. Studies of transport phenomena in such models may yield valuable insights into the problems of macroscopic dispersion. The final example concerns dispersion in multiphase flow. Here the fractal character is attributed to the distribution of the fluid phases rather than the porous medium itself. Again studies of transport phenomena in simple fractal models should help to clarify various problems associated with the corresponding phenomena in real reservoirs.     

LBM Simulation of Viscous Fingering Phenomenon in Immiscible Displacement of Two Fluids in Porous Media

This article presents the lattice Boltzmann simulation of viscous fingering phenomenon in immiscible displacement of two fluids in porous media. Such phenomenon generally takes place when a less viscous fluid is used to displace a more viscous fluid, and it can be found in many industrial fields. Dimensionless quantities, such as capillary number, Bond number and viscosity ratio between displaced fluid and displacing fluid are introduced to illustrate the effects of capillary force, viscous force, and gravity on the fluid behaviour. The surface wettability, which has an impact on the finger pattern, is also considered in the simulation. The numerical procedure is validated against the experiment about viscous fingering in a Hele-Shaw cell. The displacement efficiency is investigated using the parameter, areal sweep efficiency. The present simulation shows an additional evidence to demonstrate that the lattice Boltzmann method is a useful method for simulating some multiphase flow problems in porous media.     

Jots - a mathematical model of microscopic fluid flow in porous media

The model described in this paper is an approach to simulating flow through porous media on a microscopic scale. It is based on a variation of diffusion limited aggregation. The model is shown to match coreflood average saturation profiles and production histories as predicted by Darcy's equations while generating saturation distributions resembling viscous fingering. The model also is shown to simulate the limiting cases of infinite mobility ratio and zero flow rates as previously modeled by diffusion limited aggregation and percolation theory. With some simplifying assumptions, differential equations very similar to Darcy's equations are derived from the microscopic interpretation of fluid behavior in porous media used in this model.     

Viscous Fingering Instability in Porous Media: Effect of Anisotropic Velocity-Dependent Dispersion Tensor

The viscous fingering of miscible flow displacements in a homogeneous porous media is examined to determine the effects of an anisotropic dispersion tensor on the development of the instability. In particular, the role of velocity-dependent transverse and longitudinal dispersions is investigated through linear stability analysis and nonlinear simulations. It is found that an isotropic velocity-dependent dispersion tensor does not affect substantially the development of the instability and effectively has the same effect as molecular diffusion. On the other hand, an anisotropic velocity-dependent dispersion tensor results in different instability characteristics and more intricate finger structures. It is shown that anisotropic dispersion has profound effects on the development of the fingers and on the mechanisms of interactions between neighboring fingers. The development of the new finger structures is explained by examining the velocity field and characterized qualitatively through a spectral analysis of the average concentration and an analysis of the variations of the sweep efficiency and relative contact area.     

Two-dimensional adaptive Eulerian-Lagrangian method for mass transport with spatial velocity distribution

A Eulerian-Lagrangian scheme is used to solve the two-dimensional advection-dispersion equation. Concentration and its partial differential operator are decomposed into advection and dispersion terms. Thus, advection is formally decoupled from dispersion and solved by continuous forward particle tracking. Dispersion is handled by implicit finite elements on a fixed Eulerian grid. Translation of steep gradients of concentration in advection-dominated flow regimes, is done without numerical distortion. Continuous spatial distribution of velocities are evaluated by using Galerkin's approach in conjunction with Darcy's law based on hydraulic input data from each element. The method was implemented on coarse FE grid with linear shape functions, demonstrating no over/under shooting and practically no numerical dispersion. Simulations, covering a wide range of Peclet numbers, yield high agreement with analytic and practical results.     

A Fickian diffusion model for the spreading of liquid plumes infiltrating in heterogeneous media

Infiltration of water and non-aqueous phase liquids (NAPLs) in the vadose zone gives rise to complex two- and three-phase immiscible displacement processes. Physical and numerical experiments have shown that ever-present small-scale heterogeneities will cause a lateral broadening of the descending liquid plumes. This behavior of liquid plumes infiltrating in the vadose zone may be similar to the familiar transversal dispersion of solute plumes in single-phase flow. Noting this analogy we introduce a mathematical model for ‘phase dispersion’ in multiphase flow as a Fickian diffusion process. It is shown that the driving force for phase dispersion is the gradient of relative permeability, and that addition of a phase-dispersive term to the governing equations for multiphase flow is equivalent to an effective capillary pressure which is proportional to the logarithm of the relative permeability of the infiltrating liquid phase. The relationship between heterogeneity-induced phase dispersion and capillary and numerical dispersion effects is established. High-resolution numerical simulation experiments in heterogeneous media show that plume spreading tends to be diffusive, supporting the proposed convection-dispersion model. Finite difference discretization of the phase-dispersive flux is discussed, and an illustrative application to NAPL infiltration from a localized source is presented. It is found that a small amount of phase dispersion can completely alter the behavior of an infiltrating NAPL plume, and that neglect of phase-dispersive processes may lead to unrealistic predictions of NAPL behavior in the vadose zone.     

Vortex shedding from edges including viscous effects

This paper describes results obtained by using the inviscid Cloud-in-Cell vortex method to model the vortex sheet which is shed and rolls up from a single sharp edge. There is good agreement between these results and previous (Pullin 1978) computations of the development of the sheet in impulsively started incompressible inviscid flow. The Cloud-in-Cell method has been modified to include viscous diffusion calculated by finite differences on the mesh to give a mixed Eulerian-Lagrangian Navier-Stokes solver. This method has been shown to model the diffusing free vortex and the Stokes boundary layer quite accurately. It is used to compute impulsively started flow past sharp right-angled edges and edges with small rounding. The effect of viscous diffusion on the development of the shed vortex is discussed.

The method is also used to study the effect of rounding on the vortex shedding from a right-angled edge in oscillatory flow. This problem is particularly important in determining the roll damping and hence response of certain types of ship hull in waves. It is shown that the strength and effect of the shed vortices rapidly decrease as the ratio of the edge radius to the oscillation amplitude increases, and that at larger values of this ratio the mode of shedding changes from two vortices per cycle from one edge to a more complicated mode. The computed results are compared with flow visualisation using dye and neutrally buoyant particles in water flow around an oscillating edge.     

分形在油气田开发中的应用

分形用于测井资料分析，为油藏描述提供了新的工具，使得利用油藏数值模拟来模拟复杂结构的油气藏变得更加便利；分形油藏上的试井分析解决了一些以前难以解释的问题；用分形来描述裂缝和孔洞结构，提高了酸化压裂的设计水平；用分形理论来描述两相流，揭示了部分粘性指进的触发机理．总之，分形理论在油气田开发中的应用，促进了渗流力学基础理论的发展．     

多孔介质中的粘性指进与分形学

本文概述了多孔介质中粘性指进现象的研究成果,并对Ｈｅｌｅ－Ｓｈａｗ装置的模拟和三维多孔介质实体的实验进行了评价;介绍了分形学的基本概念以及近年来在指进研究中的应用．最后,对研究粘性指进与分形学的关系提出了一些建议．      

单组分多相系统驱替过程的格子Boltzmann模拟

结合格子Boltzmann方法中的Shan-Chen单组分多相模型,引入流体相间的内聚力和流体与 固体壁面间的黏附力,对二维孔隙网格中非浸润气相驱替完全浸润液相的过程进行模拟,流 体相间的交界面自然形成,整个驱替过程属于毛细指进. 随着毛细数的增加,黏性力的主导 作用增强,使得气相入侵的孔隙尺度减小,因此驱替形态随毛细数的不同有很大差别. 在微 重力的作用下,整个驱替过程受毛细力、重力和黏性力的共同作用,重力起到了稳定交界面 避免窜流的作用.     

Acoustical characterization of fluid-saturated porous media with local heterogeneities: Theory and application

A linear dynamic model of fully saturated porous media with local (either microscopic or mesoscopic) heterogeneities is developed within the context of Biot’s theory of poroelasticity. Viscoporoelastic behavior associated with local fluid flow is characterized by the notion of the dynamic compatibility condition on the interface between the solid and the fluid. Complex, frequency-dependent material parameters characterizing the viscoporoelasticity are derived. The complex properties can be obtained through determining the quasi-static poroelastic parameters, the properties of individual constituents, and the relaxation time of the dynamic compatibility condition on the interface. Relationships among various quasi-static poroelastic parameters are developed. It is shown that local fluid flow mechanism is significant only in the porous media with local heterogeneities. The relaxation time of the compatibility condition on the interface depends upon the details of local structure of porous media that control local fluid pressure diffusion. The new model is used to describe the velocity dispersion and attenuation in fully saturated porous media. The proposed model provides a theoretical framework to simulate the acoustical behavior of fully saturated porous media over a wide range of frequencies without making any explicit assumption about the structure of local heterogeneities.     

Miscible Displacements of Reactive and Anisotropic Dispersive Flows in Porous Media

The viscous fingering instability of miscible reactive–dispersive flows in a homogeneous porous media is investigated through nonlinear numerical simulations. In particular, the role of velocity-dependent transverse and longitudinal dispersions as well as the type and rate of auto-catalytic chemical reactions is analyzed. It is found that for a third-order auto-catalytic reaction, the higher the reaction rate, the more complex the finger structures. Furthermore, major differences between the flow development of third-order and second-order autocatalytic reactions are reported. In addition, the anisotropy and velocity dependence of the dispersion tensor are found to have a more profound effect on the fingering instability in the case of reactive flows than in the non-reactive ones. The qualitative characterization of the finger structures is explained by examining the flow velocity field and further quantified through an analysis of the average concentration and relative contact area.     

Large Scale Mixing for Immiscible Displacement in Heterogeneous Porous Media

We derive a large scale mixing parameter for a displacement process of one fluid by another immiscible one in a two-dimensional heterogeneous porous medium. The mixing of the displacing fluid saturation due to the heterogeneities of the permeabilities is captured by a dispersive flux term in the large scale homogeneous flow equation. By making use of the stochastic approach we develop a definition of the dispersion coefficient and apply a Eulerian perturbation theory to determine explicit results to second order in the fluctuations of the total velocity. We apply this method to a uniform flow configuration as well as to a radial one. The dispersion coefficient is found to depend on the mean total velocity and can therefore be time varying. The results are compared to numerical multi-realization calculations. We found that the use of single phase flow stochastics cannot capture all phenomena observed in the numerical simulations.     

Investigating the Temperature Dependency of Oil and Water Relative Permeabilities for Heavy Oil Systems

A look into the literature on the temperature dependency of oil and water relative permeabilities reveals contradictory reports. There are some publications reporting shifts in the water saturation range as well as variations in the relative permeability curves by temperature. On the other hand, some authors have blamed the experimental artifacts, viscous instabilities and fingering issues for these variations. We have performed core flooding experiments to further investigate this issue. Glass bead packs and sand packs were used as the porous media, and Athabasca bitumen with varying viscosities was displaced by hot water at differing temperatures. The unsteady-state method of relative permeability measurement was applied and the experimental data were history matched by a simulator that is tailor made to predict the relative permeabilities. The matches were obtained by varying the relative permeability correlation parameters. The results indicated that the initial water saturation has a direct relation with temperature, while residual oil saturation generally drops at higher temperatures. Although the water saturation range shifts, no direct and unique trend for either oil or water relative permeability is justified. The spread in relative permeabilities especially in the case of higher permeable cores suggests that viscous instabilities are present. As the same saturation shift happens by only changing the oil viscosity, the relative permeability variations with temperature can be attributed to oil to water viscosity ratio changes with temperature. Temperature dependency of relative permeabilities is more related to experimental artifacts, viscous fingering and viscosity changes than fundamental flow properties.     

Gravity Affected Lateral Dispersion and Diffusion in a Stationary Horizontal Porous Medium Shear Flow

Transport of dissolved species by a carrier fluid in a porous medium comprises advection and diffusion/dispersion processes. Hydrodynamic dispersion is commonly characterized by an empirical relationship, in which the dispersion mechanism is described by contributions of molecular diffusion and mechanical dispersion expressed as a function of the molecular Peclét number. Mathematically these two phenomena are modeled by a constant diffusion coefficient and by velocity dependent dispersion coefficients, respectively. Here, the commonly utilized Bear--Scheidegger dispersion model of linear proportionality between mechanical dispersion and velocity, and the more complicated Bear--Bachmat model derived on a streamtube array model porous medium and better describing observed dispersion coefficients in the moderate molecular Peclét number range, will be considered. Analyzing the mixing flow of two parallelly flowing confluent fluids with different concentrations of a dissolved species within the frames of boundary layer theory one has to deal with transverse mixing only. With the Boussinesq approximation being adopted approximate analytical solutions of the corresponding boundary layer system of equations show that there is no effect of density coupling on concentration distributions across the mixing layer in the pure molecular diffusion regime case. With the Peclét number of the oncoming flow growing beyond unity, density coupling has an increasing influence on the mixing zone. When the Peclét number grows further this influence is successively reduced until its disappearance in the pure mechanical dispersion regime.     

Characterization of the crossover from capillary invasion to viscous fingering to fracturing during drainage in a vertical 2D porous medium

We experimentally studied the displacement of a viscous wetting fluid (water) by an inviscid non-wetting fluid (air) injected at the bottom of a vertical Hele-Shaw cell filled with glass microbeads. In order to cover a wide parameter space, the permeability of the porous medium was varied by using different bead size ranges and diverse air flow rates were generated by means of a syringe pump. A LED light table was used to back illuminate the experimental cell, allowing a high speed camera to capture images of the drainage process at equal time intervals. The invasion occurred in intermittent bursts. Image processing of the bursts and fractal analysis showed successive transitions from capillary invasion to viscous fingering to fracturing during the same experiment, dependent on the medium permeability, the air injection flow rate, and the vertical position in the cell. The interplay between the capillary, viscous and gravity forces determines the nature of the invasion pattern and the transitions, from capillary invasion to viscous fingering with decreasing fluid pressure on one hand and from viscous fingering to fracturing with decreasing effective overburden pressure on the other hand.     

On the viscous dissipation modeling of thermal fluid flow in a porous medium

The problem of viscous dissipation and thermal dispersion in saturated porous medium is numerically investigated for the case of non-Darcy flow regime. The fluid is induced to flow upward by natural convection as a result of a semi-infinite vertical wall that is immersed in the porous medium and is kept at constant higher temperature. The boundary layer approximations were used to simplify the set of the governing, nonlinear partial differential equations, which were then non-dimensionalized and solved using the finite elements method. The results for the details of the governing parameters are presented and investigated. It is found that the irreversible process of transforming the kinetic energy of the moving fluid to heat energy via the viscosity of the moving fluid (i.e., viscous dissipation) resulted in insignificant generation of heat for the range of parameters considered in this study. On the other hand, thermal dispersion has shown to disperse heat energy normal to the wall more effectively compared with the normal diffusion mechanism.     

无黏流场中结构振动的气动非阻尼现象

研究跨音速飞行航天器的流-固耦合振动非阻尼现象的作用机理.研究表明,流场中的跨音速激波过程是此类非阻尼现象不可忽略的基本原因之一;由于流场对物体振动响应的滞后,形成了周期变化的分布气动力与物体运动之间有规律的相位差异,这种相位差可以引发局部的非阻尼的脉动力作用.这一机制与流场中的黏性过程如边界层和分离流动无关.单纯考虑黏性现象对非阻尼作用影响的认识是有缺陷的.     

Instability of sintering of porous bodies

Sintering models are discussed and used to analyze flow instabilities that may arise during preliminary compaction of powders. These instabilities can be at the origin of heterogeneities in the densification. The material is modeled as a viscoplastic thermal sensitive porous material. The modeling includes the limit case of a linear viscous material. The effects of sintering conditions (temperature and pressure in the case of pressure sintering) and the effects of material characteristics such as porosity, heat capacity, theoretical density, surface tension, particle size and creep parameters on stability of sintering are investigated. The heat release associated with the plastic flow is shown to sometimes have an important role. Stability criteria are derived and applied to the analysis of sintering and hot isostatic pressing, using various sintering models. These stability criteria can be used to optimize the densification process; one can control, for example, temperature so as to avoid any instability. Stability maps enabling an optimization of temperature–pressure regime in hot isostatic pressing are built for sample metal (nickel) powder.     

Acetone Droplet Behavior in Reacting and Non Reacting Turbulent Flow

Acetone droplet characteristics in reacting and non-reacting turbulent flow are predicted and compared to experimental data. Investigations are conducted to study the effects of surrounding environment properties on the velocities, dispersion, and evaporation of a relatively volatile spray fuel that featured a wide range of Stokes numbers. The simulations are performed in the framework of Reynolds Averaged Navier Stokes equations along with the Eulerian-Lagrangian approach in which 12 different classes of the dispersed phase. The phase transition is modeled by the Langmuir-Knudsen law that accounts for non equilibrium effects based on a consistent determination of the molar mass fraction on the droplet surfaces. For the droplet dispersion, the Markov sequence model is improved by adding a correction drift term to the fluid fluctuation velocity at the parcel position along the droplet trajectory. This correction term aimed at accounting for the non-homogeneity effects in the turbulent flow. The combustion is captured using the Bray-Moss-Libby model that is extended to account for the partially premixed spray combustion. The chemistry is described with the flamelet model using a recent detailed reaction mechanism that involves 84 species and 409 reactions for which the Lewis number is not set to the unity. Mean droplet velocities for reacting and non-reacting test cases are compared with experimental data. Good agreement is observed. The spray is interacting with the nozzle edge developing new classes and relatively dense region. Hence the RMS-velocities close to the nozzle exit plan demonstrate discrepancies. The droplets group combustion effect is found to be important in the modeling of the burning velocity which influences the flame propagation. Reasonable agreements between the numerical and the experimental results are also observed in the spray flux and temperature profiles.