Influence of Internal Structure and Medium Length on Transport and Deposition of Suspended Particles: A Laboratory Study

A laboratory study was undertaken on the transport and the deposition of suspended particles (silt of modal diametre 6 μm) in three columns of different length, filled with glass beads or gravel. Tracer tests were carried out at various flow velocities by short pulses of a mixture of suspended particles/dissolved tracer. The breakthrough curves were competently described with the analytical solution of a convection dispersion equation with a first-order deposition rate and the hydro-dispersive parameters were deduced. For the same experimental conditions, the results showed a difference in the behaviour of the suspended particles transport and deposition rates within the two porous media tested. The internal structure of both media governs the particle-grain collision frequency as well as the particles trapping. The scale effect was highlighted and affects the dispersivity, the size exclusion effect, the recovery rates and the deposition rates. Longitudinal dispersion increases with mean pore velocity and is described with a nonlinear relationship. The dispersivity increases with the column length. The size exclusion effect is more important in the short column. The recovery rate increases with flow velocity and decreases while increasing column length. The deposition rates increases until a critical flow velocity then decreases. This critical velocity is also sensitive to the scale effect, and increases with the column length.     

Calculating the Anisotropic Permeability of Porous Media Using the Lattice Boltzmann Method and X-ray Computed Tomography

A lattice Boltzmann (LB) method is developed in this article in a combination with X-ray computed tomography to simulate fluid flow at pore scale in order to calculate the anisotropic permeability of porous media. The binary 3D structures of porous materials were acquired by X-ray computed tomography at a resolution of a few microns, and the reconstructed 3D porous structures were then combined with the LB model to calculate their permeability tensor based on the simulated velocity field at pore scale. The flow is driven by pressure gradients imposed in different directions. Two porous media, one gas diffusion porous layer used in fuel cells industry and glass beads, were simulated. For both media, we investigated the relationship between their anisotropic permeability and porosity. The results indicate that the LB model is efficient to simulate pore-scale flow in porous media, and capable of giving a good estimate of the anisotropic permeability for both media. The calculated permeability is in good agreement with the measured date; the relationship between the permeability and porosity for the two media is well described by the Kozeny–Carman equation. For the gas diffusion layer, the simulated results showed that its permeability in one direction could be one order of magnitude higher than those in other two directions. The simulation was based on the single-relaxation time LB model, and we showed that by properly choosing the relaxation time, it could give similar results to those obtained using the multiple-relaxation time (MRT) LB method, but with only one third of the computational costs of MRTLB model.     

Multiresolution Wavelet Scale Up of Unstable Miscible Displacements in Flow Through Heterogeneous Porous Media

We describe scale up of geological models of field-scale porous media using a new method based on the wavelet transformations. The porous media of interest contain broadly-distributed and correlated permeabilities. Wavelet transformation of the permeability field of such porous media coarsens the geological model from smallest to largest length scales, drastically reduces the number of equations to be solved, preserves the important information on the permeability field at all the relevant length scales, and yields numerical results for any fluid flow property that are as accurate as those that are obtained with the highly detailed geological model of the same porous media. To test this method, we carry out extensive computer simulations of unstable miscible displacement processes and the associated viscous fingering phenomenon in highly heterogeneous porous media, both with the fine-scale geological model and the coarsened model. Excellent agreement is found between the results of the two sets of simulations.     

Dynamics of the Water–Oil Front for Two-Phase, Immiscible Flow in Heterogeneous Porous Media. 2 – Isotropic Media

We study the evolution of the water–oil front for two-phase, immiscible flow in heterogeneous porous media. Our analysis takes into account the viscous coupling between the pressure field and the saturation map. Although most of previously published stochastic homogenization approaches for upscaling two-phase flow in heterogeneous porous media neglect this viscous coupling, we show that it plays a crucial role in the dynamics of the front. In particular, when the mobility ratio is favorable, it induces a transverse flux that stabilizes the water–oil front, which follows a stationary behavior, at least in a statistical sense. Calculations are based on a double perturbation expansion of equations at first order: the local velocity fluctuation is defined as the sum of a viscous term related to perturbations of the saturation map, on one hand, plus the perturbation induced by the heterogeneity of the permeability field with a base-state saturation map, on the other hand. In this companion paper, we focus on flows in isotropic media. Our results predict the dynamics of the water–oil front for favorable mobility ratios. We show that the statistics of the front reach a stationary limit, as a function of the geostatistics of the permeability field and of the mobility ratio evaluated across the front. Results of numerical experiments and Monte-Carlo analysis confirm our predictions.     

Flow characteristics of anisotropic structures constructed with porous layers

Artificial structures, serving as the solid matrix of anisotropic porous media and satisfying the requirement needed for flow visualization, were constructed with the perforated Polypropylene plates in order to improve the understanding of transport phenomena occurring in anisotropic porous media. This paper reports the regressed correlations of the experimental pressure gradient and filtration velocity data of three anisotropic and one isotropic porous media measured along two mutually orthogonal directions, which correspond to the principal axes of the permeability tensor, for the filtration velocities ranging from 0.2 to 12 mm/s with water as the fluid. To reflect the observed data, the regression equation with two types of deviations was formulated, in which the pressure gradient is represented by the sum of the linear and nonlinear terms of the filtration velocity. The physical model developed for the linear term assumes the solid matrix as repeated circular orifices when the filtration velocity approaches zero. The exponent of the filtration velocity in the nonlinear term was determined to be that of the Forchheimer extension. Also, four models for the coefficient of the nonlinear term were examined and the results were compared. The distribution of the residuals (the differences between the observed and the correlated values) validated the suggested regression procedure and the resulting correlations.     

Dynamics of the Water–Oil Front for Two-Phase, Immiscible Flow in Heterogeneous Porous Media. 1 – Stratified Media

We study the evolution of the water–oil front for two-phase, immiscible flow in heterogeneous porous media. Our analysis takes into account the viscous coupling between the pressure field and the saturation map. Although most of previously published stochastic homogenization approaches for upscaling two-phase flow in heterogeneous porous media neglect this viscous coupling, we show that it plays a crucial role on the dynamics of the front. In particular, when the mobility ratio is favorable, the viscous coupling induces a transverse flux that stabilizes the water–oil front, which follows a stationary behavior, at least in a statistical sense. Calculations are based on a double perturbation expansion of equations at first order: the local velocity fluctuation is defined as the sum of a viscous term related to perturbations of the saturation map, on one hand, plus the perturbation induced by the heterogeneity of the permeability field with a base-state saturation map, on the other hand. In this first paper, we focus on flows in stratified reservoirs, with stratification parallel to the mean flow. Our results allow to predict the evolution of large Fourier mode of the front, and the emergence of a stationary front, for favorable mobility ratios. Numerical experiments confirm our predictions. Our approach is applied to downscaling. Extension of our theory to isotropic media is presented in the companion paper.     

Flux in Porous Media with Memory: Models and Experiments

The classic constitutive equation relating fluid flux to a gradient in potential (pressure head plus gravitational energy) through a porous medium was discovered by Darcy in the mid 1800s. This law states that the flux is proportional to the pressure gradient. However, the passage of the fluid through the porous matrix may cause a local variation of the permeability. For example, the flow may perturb the porous formation by causing particle migration resulting in pore clogging or chemically reacting with the medium to enlarge the pores or diminish the size of the pores. In order to adequately represent these phenomena, we modify the constitutive equations by introducing a memory formalism operating on both the pressure gradient–flux and the pressure–density variations. The memory formalism is then represented with fractional order derivatives. We perform a number of laboratory experiments in uniformly packed columns where a constant pressure is applied on the lower boundary. Both homogeneous and heterogeneous media of different characteristic particle size dimension were employed. The low value assumed by the memory parameters, and in particular by the fractional order, demonstrates that memory is largely influencing the experiments. The data and theory show how mechanical compaction can decrease permeability, and consequently flux.     

三维非均匀不稳定渗流方程的二维不均匀粗化算法

在无源汇条件下,根据流过某一个横截面的流体流量等于流过这一横截面内所有精细网格的流体流量之和这一特点提出了粗化网格等效渗透率的计算方法。在粗化区内,利用直接解法求解二维渗流方程,再用这些解合成粗化网格的三维合成解,并由合成解计算粗化网格的等效渗透率。根据精度的要求采用了不均匀网格粗化,在流体流速大的区域采用精细网格。利用所得等效渗透率计算了粗化网格的某三维非均匀不稳定渗流场的压降解,结果表明三维非均匀不稳定渗流方程的二维不均匀粗化解非常逼近采用精细网格的解,但计算的速度比采用精细网格提高了80倍。     

Dispersion Measurements in Highly Heterogeneous Laboratory Scale Porous Media

Laboratory experiments and numerical simulations investigate conservative contaminant transport in a heterogeneous porous medium. The laboratory experiments were performed in cylindrical columns 1m long and 3.5cm inside diameter filled with spherical glass beads. Concentration breakthrough curves are measured at a scale much finer than the size of the heterogeneity. Numerical simulations are based on a random walk in a known constant velocity field. The heterogeneity is a distinct, discontinuous change in the local permeability field. Fluid flow is miscible, flowing in a saturated porous medium. Previous work has shown this to be a very poorly understood phenomenon. The measurements reported here help to better understand how dispersion evolves through and past a heterogeneity.     

Wetting front instabilities: a three-dimensional experimental investigation

Wetting front instability or fingering experiments were conducted in three-dimensional infiltration columns, featuring stratified fine-over-coarse texture porous media, to investigate the influences of various soil and wetting phase properties on finger diameter and propagation velocity. The system parameter varied in this study included permeability, system flux rate, media size and gradation uniformity, initial moisture content, viscosity, density, and surface tension. The influence of each parameter is discussed and compared, where applicable, to the finding of previous studies. Finger diameter and velocity data were acquired using a neutron radioscopy based, real-time imaging system. Through the use of the imaging system, a very accurate and reliable experimental data set was produced for three-dimensional fingering events.     

Wavelet upscaling based on piecewise bilinear approximation of the permeability field

A method for upscaling of permeability in heterogeneous porous media is presented. The upscaled field takes the form K = e ^Y , where Y, in two dimensions, is a piecewise bilinear function. The method is tested on a number of random permeability fields, with different integral scale/correlation length and variance. The numerical results show that this method conserves much more of the heterogeneous fingering than classical block-based upscaling methods, e.g., geometric mean.     

Experimental investigation of solute transport in large,homogeneous and heterogeneous,saturated soil columns

Laboratory tracer experiments were conducted to investigate solute transport in 12.5-m long, horizontally placed soil columns during steady saturated water flow. Two columns having cross-sectional areas of 10×10cm² were used: a uniformly packed homogeneous sandy column and a heterogeneous column containing layered, mixed, and lenticular formations of various shapes and sizes. The heterogeneous soil column gradually changed, on average, from coarse-textured at one end to fine-textured at the other end. NaCl breakthrough curves (BTC's) in the columns were measured with electrical conductivity probes inserted at 50- or 100-cm intervals. Observed BTC's in the homogeneous sandy column were relatively smooth and sigmoidal (S-shaped), while those in the heterogeneous column were very irregular, nonsigmoidal, and exhibited extensive tailing. Effective average pore-water velocities (v _eff) and dispersion coefficients (D _eff) were estimated simultaneously by fitting an analytical solution of the convection-dispersion equation to the observed BTC's. Velocity variations in the heterogeneous medium were found to be much larger than those in the homogeneous sand. Values of the dispersivity,=D _eff/v _eff, for the homogeneous sandy column ranged from 0.1 to 5.0 cm, while those for the heterogeneous column were as high as 200cm. The dispersivity for transport in both columns increased with travel distance or travel time, thus exhibiting scale-dependency. The heterogeneous soil column also showed the effects of preferential flow, i.e., some locations in the column showed earlier solute breakthrough than several locations closer to the inlet boundary. Spatial fluctuations in the dispersivity could be explained qualitatively by the particular makeup of the heterogeneities in the column.     

Modeling Transverse Dispersion and Variable Density Flow in Porous Media

A two-dimensional numerical model is used to study the nonlinear behavior of density gradients on transverse dispersion. Numerical simulations are conducted using d ³ f, a computer code for simulation of density-dependent flow in porous media. Considering a density-stratified horizontal flow in a heterogeneous porous media, a series of simulations is carried out to examine the effect of the density gradient on macro-scale transverse dispersivity. Changing salt concentration significantly affects fluid properties. This physical behavior of the fluid involves a non-linearity in modeling the interaction between salt and fresh water. It is concluded that the large-scale transport properties for high density flow deviate significantly from the tracer case due to the spatial variation of permeability, described by statistical parameters, at the local-scale. Indeed, the presence of vertical flow velocities induced by permeability variations is responsible for the reduction of the mixing zone width in the steady state in the case of a high density gradient. Uncertainties in the model simulations are studied in terms of discretization errors, boundary conditions, and convergence of ensemble averaging. With respect to the results, the gravity number appears to be the controlling parameter for dispersive flux. In addition, the applicability and limitations of the nonlinear model of Hassanizadeh (1990) and Hassanizadeh and Leijnse (1995) (Adv Water Resour 18(4):203–215, 1995) in heterogeneous porous media are investigated. We found that the main cause of the nonlinear behavior of dispersion, which is the interaction between density contrast and vertical velocity, needs to be explicitly accounted for in a macro-scale model.     

膨胀性非牛顿流体多孔介质流动的模拟分析

在表征体元尺度采用格子Boltzmann方法分析膨胀性非牛顿流体在多孔介质中的流动,基于二阶矩模型在演化方程中引入表征介质阻力的作用力项,求解描述渗流模型的广义Navier-Stokes方程．采用局部法计算形变速率张量,通过循环迭代得到非牛顿粘度和松弛时间．对多孔介质的Poiseuille流动进行分析,通过比较发现结果与孔隙尺度的解析解十分吻合,并且收敛较快,表明方法合理有效．分析了渗透率和幂律指数对速度和压力降的影响,研究结果表明,膨胀性流体的多孔介质流动不符合达西规律,压力降的增加幅度小于渗透率的减小幅度．当无量纲渗透率Da小于10-5时,流道中的速度呈现均匀分布,并且速度分布随着幂律指数的减小趋于平滑．压力降随着幂律指数的增加而增加,Da越大幂律指数对压力降的影响越明显．     

Vortex structure of turbulence over permeable walls

In order to understand the effect of the wall permeability on the turbulent vortex structure near porous walls, based on PIV experimental data, a probability density analysis of fluctuating velocities, statistical quadrant and quadrant-hole analyses of the Reynolds shear stress are performed. The investigated flow fields are turbulent channel flows whose bottom walls are made of porous media. The porous media used are three kinds of foamed ceramics which have almost the same porosity (∼0.8) but different permeability. From the discussions on those analyses, a conceptual scenario of the development of the vortex structure near a permeable wall is proposed for a moderate permeability Reynolds number case. It explains the reason why the near-wall long streaky structure tends to vanish near a porous wall with increasing wall permeability.     

????????????????????????

针对多孔介质网状纤维的渗流和生物体内组织液流动等问题，研究多孔介质中圆柱 方形阵列的低$Re$流动，以及流动所导致的圆柱表面的剪切应力. 鉴于迄今为止的研究均限于 圆柱实体模型的情况，重点分析将圆柱实体模型推广到可渗透时所导致的流动特性和表 面剪切应力的变化. 可渗透圆柱及圆柱阵列之间的间隙物使用具有不同渗透率的多孔介质模 型，基于Brinkman方程，配以适当的边界条件，采用有限体积法进行数值求解. 计算结果表 明，圆柱的渗透特性改变了流场的流动特性，显著减小了流动在圆柱表面所产生的剪切应力， 改变了阵列的渗透性.     

A LGA model for fluid flow in heterogeneous porous media

A lattice gas automaton (LGA) model is proposed to simulate fluid flow in heterogeneous porous media. Permeability fields are created by distributing scatterers (solids, grains) within the fluid flow field. These scatterers act as obstacles to flow. The loss in momentum of the fluid is directly related to the permeability of the lattice gas model. It is shown that by varying the probability of occurrence of solid nodes, the permeability of the porous medium can be changed over several orders of magnitude. To simulate fluid flow in heterogeneous permeability fields, isotropic, anisotropic, random, and correlated permeability fields are generated. The lattice gas model developed here is then used to obtain the effective permeability as well as the local fluid flow field. The method presented here can be used to simulate fluid flow in arbitrarily complex heterogeneous porous media.     

Foam Mobility in Heterogeneous Porous Media

It is shown experimentally that in situ generation of foam is an effective method for achieving gas mobility control and diverting injected fluid to low permeability strata within heterogeneous porous media. The experimental system is composed of a 0.395 porosity, 5.35 µm² synthetic sandstone and a 0.244 porosity, 0.686 µm² natural sandstone. The cores are arranged in parallel and communicate through common injection and production conditions. Nitrogen is the gas phase and alpha-olefin sulfonate (AOS 1416) in brine is the foamer. Three types of experiments were conducted. First, gas alone was injected into the system after presaturation with the foamer solution. Second, gas and foamer solution were coinjected at an overall gas fraction of 90% into cores presaturated with surfactant. Each core accepted a portion of the injected gas and liquid according to the mobility within the core. Lastly, gas and foamer solution were coinjected into the individual, isolated porous media in order to establish baseline behavior. The results are striking. It is possible to achieve total diversion of gas injection to the low permeability medium in some cases. The results also confirm previous predictions that foamed gas can be more mobile in lower permeability porous media.     

Numerical Simulations of Flows in a Heterogeneous Porous Medium

Numerical simulations are employed to investigate the fluid flow and pressure loss in a heterogeneous block within a composite porous medium. The mean permeability of the heterogeneous block is seen to affect the overall effective flux significantly. The heterogeneous parameters of the permeability field, such as the correlation length and variance, affect it quite differently. Because of the channelling effects, the effective flux depends strongly on the realization of the permeability for larger correlation length. Under a specific permeability field, higher effective flux results from smaller variances. The influences of the inertial factor are found to be insignificant within the range of practical interests.