Homogeneous planar and two-dimensional mean-field antidynamo theorems with zero mean flow

In an electrically conducting fluid, two types of turbulence with a preferred direction are distinguished: planar turbulence, in which every velocity in the turbulent ensemble of flows has no component in the given direction; and two-dimensional turbulence, in which every velocity in the turbulent ensemble is invariant under translation in the preferred direction. Under the additional assumptions of two-scale and homogeneous turbulence with zero mean flow, the associated magnetohydrodynamic alpha- and beta-effects are derived in the second-order correlation approximation (SOCA) when the electrically conducting fluid occupies all space. Limitations of the SOCA are well known, but alpha- and beta-effects of a turbulent flow are useful in interpreting the dynamo effects of the turbulence. Two antidynamo theorems, which establish necessary conditions for dynamo action, are shown to follow from the special structures of these alpha- and beta-effects. The theorems, which are analogues of the laminar planar velocity and two-dimensional antidynamo theorems, apply to all turbulent ensembles with the prescribed alpha- and beta-effects, not just the planar and two-dimensional ensembles. The mean magnetic field is general in the planar theorem but only two-dimensional in the two-dimensional theorem. The two theorems relax the previous restriction to turbulence which is both two-dimensional and planar. The laminar theorems imply decay of the total magnetic field for any velocity of the associated turbulent ensemble. However, the mean-field theorems are not fully consistent with the laminar theorems because further conditions beyond those arising from the turbulence must be imposed on the beta-effect to establish decay of the mean magnetic field. In particular, negative turbulent magnetic diffusivities must be restricted. It is interesting that there is no inconsistency in the alpha-effects. The failure of the SOCA with the two-scale approximation to simply preserve the laminar antidynamo theorems at the beta-effect level is a further demonstration of the restricted validity of the theory and shows that negative diffusivity effects derived by approximation methods must be treated cautiously.     

A study on the Wilkins and Forchheimer equations used in coarse granular media flow

Complexity of the pore geometry and the random nature of flow velocity make it difficult to predict and represent post laminar flow through porous media. Present study experimentally investigates the applicability of Forchheimer and Wilkins equations for post laminar flow where Darcy’s law is invalid due to predominant inertial effect. It is observed that both porosity and media size have significant influence over the coefficients of the Forchheimer coefficients. To incorporate the effect of porosity and media size, behaviour of Forchheimer coefficients are investigated with hydraulic radius as characteristic length. An inversely proportional variation trend is found for all the present and earlier reported data. A new empirical relation between Forchheimer coefficients and hydraulic radius is obtained which can be universally applicable for all media size and porosity. Coefficients of the Wilkins equation are found to be non-deviating for different hydraulic radius in the present study and in the reported literature validating its applicability in predicting the non laminar flow through porous media. Further the Wilkins equation is modified after incorporating the correction factors for better applicability on the field.     

Permeability measurements of natural and experimental volcanic materials with a simple permeameter: Toward an understanding of magmatic degassing processes

Permeability measurement of quenched volcanic porous materials is an important approach to understand permeability development and degassing of vesicular silicic magmas. In this study, we developed a gas permeameter to measure permeability of natural samples and experimental products. The permeameter has broad measurement ranges of pressure difference (10¹–10⁵ Pa) and gas-flow rate (10^− 9–10^− 5 m³/s). These ranges enable us to measure viscous permeability in the range of 10^− 17–10^− 9 m² for 1 centimeter-scale samples, using the Forchheimer equation, which includes the inertial effect of gas flow permeating through the samples. In addition, we improved the procedure for performing permeability measurements of mm-sized products of decompression experiments. Although a previous study reported the first permeability data for vesicular silicic glass products of decompression experiments, we found an overestimation in their permeability data due to problems in sample preparation, especially for very low permeability samples. Our improved measurements give lower permeability values than those of Takeuchi et al. (2005)(Takeuchi, S., Nakashima, S., Tomiya, A., Shinohara, H., 2005. Experimental constraints on the low gas permeability of vesicular magma during decompression. Geophys. Res. Lett., 32, L10312 doi:10.1029/2005GL022491).     

Analytical approximations for flow in compressible,saturated, one-dimensional porous media

A nonlinear model for single-phase fluid flow in slightly compressible porous media is presented and solved approximately. The model assumes state equations for density, porosity, viscosity and permeability that are exponential functions of the fluid (either gas or liquid) pressure. The governing equation is transformed into a nonlinear diffusion equation. It is solved for a semi-infinite domain for either constant pressure or constant flux boundary conditions at the surface. The solutions obtained, although approximate, are extremely accurate as demonstrated by comparisons with numerical results. Predictions for the surface pressure resulting from a constant flux into a porous medium are compared with published experimental data.     

General statistical hydrodynamics in porous media

Summary The study of fluid flow in the ground is based upon the physics of flow through porous media. The author has recently proposed a theory (2) of such flow based upon the statistics of disordered phenomena which, however, was applicable to a special type of flow only. In the present paper, the earlier theory is developed into a general theory applicable to any type of microscopic flow equation. It is shown that the qualitative analogy which is observed between the equations of flow through porous media and the equations of flow through capillaries can be logically explained without the assumption of capillaric models. Thus, a theorem is proven stating that the flow through porous media is described by the superposition of two effects: firstly, one corresponding to the average flow through a set of small channels, and secondly, a dispersivity effect. Finally, the results are applied to a variety of flow equations such as laminar flow, turbulent flow, and molecular streaming, all of which may occur in groundwater flow.Published by permission of the Deputy Minister Department of Mines and Technical Surveys, Ottawa, Canada.     

THE THEORETICAL STUDY ON THE LAWS OF DRAG REDUCTION BY AERATION IN OPEN CHANNEL 1

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Coefficient of permeability determined by measurable parameters

Selection of a coefficient of permeability or the hydraulic conductivity in practical situations has required a fair amount of judgment and the expectation of accuracy can range over an order of magnitude. A determination of the coefficient based on measurable characteristics would greatly reduce the judgment aspect. The coefficient of permeability of a porous media can be derived from the following measurable parameters: the density and viscosity of the permeating fluid, the porosity of the media, the average hydraulic radius of the pores, and the gravitation constant. The hydraulic radius is calculated from the grain size distribution analysis of the porous media assuming spherical particles and a factor to account for the shape of the particles. The shape factor ranges from 1.0 to 1.4 in the extreme but, for common porous media, ranges only from 1.0 to about 1.1. All of the variables, except the shape factor, are measurable by standard test procedures. An important advantage of using this procedure for determining permeability is that the factors can be visualized as part of the physical flow process. The procedure assumes laminar flow and applies equally to liquid or gas fluids.     

Direct simulations of two-phase flow on micro-CT images of porous media and upscaling of pore-scale forces

Pore-scale forces have a significant effect on the macroscopic behaviour of multiphase flow through porous media. This paper studies the effect of these forces using a new volume-of-fluid based finite volume method developed for simulating two-phase flow directly on micro-CT images of porous media. An analytical analysis of the relationship between the pore-scale forces and the Darcy-scale pressure drops is presented. We use this analysis to propose unambiguous definitions of Darcy-scale viscous pressure drops as the rate of energy dissipation per unit flow rate of each phase, and then use them to obtain the relative permeability curves. We show that this definition is consistent with conventional laboratory/field measurements by comparing our predictions with experimental relative permeability. We present single and two-phase flow simulations for primary oil injection followed by water injection on a sandpack and a Berea sandstone. The two-phase flow simulations are presented at different capillary numbers which cover the transition from capillary fingering at low capillary numbers to a more viscous fingering displacement pattern at higher capillary numbers, and the effect of capillary number on the relative permeability curves is investigated. Overall, this paper presents a new finite volume-based methodology for the detailed analysis of two-phase flow directly on micro-CT images of porous media and upscaling of the results to the Darcy scale.     

Interface between turbulent flows above and within rough porous walls

This paper explores the concept of a macroscopic boundary between turbulent flows above and within rough permeable walls. The macroscopic boundary and the associated conditions for macroscopic flow variables have been thoroughly investigated for laminar, but not for turbulent flows. The literature on laminar flows follows two main conceptual models of the boundary: sharp boundary with step changes in macroscopic variables and gradual boundary with smooth changes of variables. The former approach is usually associated with the two-domain simulation models and the latter one with the single-domain models. This paper presents the derivation of the step conditions for velocity and shear stress at the macroscopic boundary between turbulent boundary layer and turbulent porous media flows. The physical meaning of the main terms in the shear stress condition is discussed in order to clarify the relationship between two-domain and single-domain simulation models.     

Seismic wave attenuation in fluid-saturated porous media

Contrary to the traditional view, seismic attenuation in Biot's theory of fluid-saturated porous media is due to viscous damping of local (not global) pore-fluid motion. Since substantial inhomogeneities in fluid permeability of porous geological materials are to be expected, the regions of highest local permeability contribute most to the wave energy dissipation while those of lowest permeability dominate the fluid flow rate if they are uniformly distributed. This dichotomy can explain some of the observed discrepancies between computed and measured attenuation of compressional and shear waves in porous earth. One unfortunate consequence of this result is the fact that measured seismic wave attenuation in fluid-filled geological materials cannot be used directly as a diagnostic of the global fluid-flow permeability.     

Radar determination of the spatial structure of hydraulic conductivity

Spatial variability of hydraulic conductivity exerts a predominant control on the flow of fluid through porous media. Heterogeneities influence advective pathways, hydrodynamic dispersion, and density-dependent dispersion; they are, therefore, a key concern for studies of ground water resource development, contaminant transport, and reservoir engineering. Ground-penetrating radar contributes to the remote, geophysical characterization of the macroscale variability of natural porous media. On a controlled excavation of a glacial-fluvial sand and gravel deposit in the Fanshawe Delta area (Ontario, Canada), the hydraulic conductivity field of a 45 x 3 m vertical exposure was characterized using constant-head permeameter measurements performed on undisturbed horizontal sediment cores. Ground-penetrating radar data were collected along the excavation face in the form of both reflection and common midpoint surveys. Comparison of geostatistical analyses of the permeameter measurements and the radar data suggests thatthe horizontal correlation structure of radar stack velocity can be used to directly infer the horizontal correlation structure of hydraulic conductivity. The averaging nature of the common midpoint survey is manifest in the vertical correlation structure of stack velocity, making it less useful. Radar reflection data do not exhibit a spatial structure similar to that of hydraulic conductivity possibly because reflections are a result of material property contrasts rather than the material properties themselves.     

Interpolation between Darcy–Weisbach and Darcy for laminar and turbulent flows

An equation describing flow in an open channel with obstacles is derived, following the conservation of momentum approach used by Bélanger and St. Venant. When the obstacles are all submerged the result yields the Darcy–Weisbach equation for turbulent flow in pipes and open channels. When the obstacles are only partially submerged the result leads to the governing equation in a porous medium. If the flow is turbulent the square of the velocity is proportional to the hydraulic gradient and if the flow is laminar, which is the usual case, the velocity is proportional to the hydraulic gradient. This last result is in agreement with Darcy’s law in porous media. Thus our equation interpolates between and reduces to, the two fundamental results of Darcy. In general our equation should prove useful in practice for open flow in a channel with both submerged and emerging obstacles.     

Propagation of seismic waves in patchy-saturated porous media: double-porosity representation

Propagation of harmonic plane waves is studied in a patchy-saturated porous medium. Patchy distribution of the two immiscible fluids is considered in a porous frame with uniform skeletal properties. A composition of two types of patches, connected through continuous paths, constitutes a double-porosity medium. Different compressibilities of pore-fluids in two porous phases facilitate the wave-induced fluid-flow in this composite material. Constitutive relations are considered with frequency-dependent complex elastic coefficients, which define the dissipative behaviour of porous aggregate due to the flow of viscous fluid in connected patches. Relevant equations of motion are solved to explain the propagation of three compressional waves and one shear wave in patchy-saturated porous solids. A numerical example is solved to illustrate dispersion in phase velocity and quality factor of attenuated waves in patchy-saturated porous materials. Role of fluid–solid inertial coupling in Darcy's law is emphasized to keep a check on the dispersion of wave velocities in the porous composite. Effects of patchy saturation on phase velocities and quality factors of attenuation are analysed using the double-porosity formulation as well as the reduced single-porosity equivalents.     

基于Kelvin-Voigt黏弹性骨架的含非饱和流体孔隙介质BISQ模型(英文)

本文综合考虑了在波传播过程中孔隙介质的三种重要力学机制——"Biot流动机制一squirt流动机制-固体骨架黏弹性机制",借鉴等效介质思想,将含水饱和度引入波动力学控制方程,并考虑了不同波频率下孔隙流体分布模式对其等效体积模量的影响,给出了能处理含粘滞性非饱和流体孔隙介质中波传播问题的黏弹性Biot/squirt(BISQ)模型。推导了时间-空间域的波动力学方程组,由一组平面谐波解假设,给出频率-波数域黏弹性BISQ模型的相速度和衰减系数表达式。基于数值算例分析了含水饱和度、渗透率与频率对纵波速度和衰减的影响,并结合致密砂岩和碳酸盐岩的实测数据,对非饱和情况下的储层纵波速度进行了外推,碳酸盐岩储层中纵波速度对含气饱和度的敏感性明显低于砂岩储层。     

储层岩石流动电位频散特性的数学模拟

利用储层岩石流动电位的频散特性评价复杂储层已经成为勘探地球物理领域关注的热点,但是目前还没有形成基于储层岩石储渗特性及电化学性质的具有普遍指导意义的理论方法和数学模型.本文利用微观毛管理论,通过随时间谐变条件下渗流场和电流场的耦合模型,建立了描述储层岩石流动电位频散特性的数学方法,定量分析了频率域储层岩石动态渗透率、动电耦合系数和流动电位耦合系数随储层岩石孔隙度、溶液浓度和阳离子交换量的变化规律.研究结果表明:储层岩石流动电位频散特性是储层流体惯性力与流体黏滞力相互作用的结果.储层岩石孔隙度越大,储层维持流体原有运动状态的能力越大,临界频率越小;储层岩石的溶液浓度和阳离子交换量对临界频率没有影响.储层岩石的孔隙度越大,流体流动能力越强,流动电位各耦合系数的数值越大;溶液浓度越小或阳离子交换量越大,孔隙固液界面的双电层作用越强,各耦合系数的数值越大.     

On pressure-work,viscous dissipation and the energy balance relation for geothermal reservoirs

This article examines the conditions under which the pressure-work and viscous dissipation terms should be retained in the energy balance relation for single (liquid water or vapor) and two-phase (liquid water and vapor) fluid flow through porous media. It is shown that if one wishes to retain the pressure-work term, then one must also keep the viscous dissipation term in the energy balance. Consideration of steady non-isothermal radial flow demonstrates that both pressure-work and viscous dissipation are liable to have negligibly small effects in single phase liquid water and in two-phase liquid-vapor systems. This conclusion is, however, not generally valid for pure vapor systems; in this case, pressure-work and viscous dissipation can produce significant variations in the computed reservoir response.     

A viscous boundary for transient analyses of saturated porous media

An absorbing boundary for saturated porous media is developed that can be used for transient analyses in the time domain. The elastic constitutive equations for the saturated porous media follow Bowen's formulation. The method consists of applying viscous tractions along the artificial boundary. The absorbing boundary behaviour is assumed linear and isotropic. Hadamard's conditions provide the speeds of the dilatational and shear waves that propagate in saturated porous media. Since these expressions are frequency independent, the intensities of the viscous tractions are evaluated in the time domain, and the two dilatational waves are accounted for. The viscous tractions are defined from the drained characteristics, assuming an infinite permeability, at variance with the traditional ‘undrained’ method based on undrained characteristics and a null permeability. Solid media and materials with low permeability are also retrieved as subcases. The results show that, at no additional cost, this ‘drained’ method is more accurate for all permeabilities than the ‘undrained’ method, which disregards the existence of the second dilatational wave. Copyright © 2003 John Wiley & Sons, Ltd.     

含黏滞流体各向异性孔隙介质中弹性波的频散和衰减

基于Biot理论,考虑液相的黏弹性变形和固液相接触面上的相对扭转,提出了含黏滞流体VTI孔隙介质模型.从理论上推导出,在该模型中除存在快P波、慢P波、SV波、SH波以外,还将存在两种新横波－慢SV波和慢SH波.数值模拟分析了6种弹性波的相速度、衰减、液固相振幅比随孔隙度、频率的变化规律以及快P波、快SV波的衰减随流体性质、渗透率、入射角的变化规律.结果表明慢SV波和慢SH波主要在液相中传播,高频高孔隙度时,速度较高;大角度入射时,快P波衰减表现出明显的各向异性,而快SV波的衰减则基本不变;储层纵向和横向渗透率存在差异时,快SV波衰减大的方向渗透率高.     

The Frequency (or Time) -Temperature Equivalence of Relaxation in Saturated Rocks

Laboratory data on dry and saturated rocks show that pore fluid has the most important effect on rock attenuation. It is known that viscous and inertial coupling between the frame of a porous rock and its pore fluid dissipates seismic energy by conversion to heat and hence cause attenuation. We show that attenuation peaks, in saturated rock have the same property as that of typical thermally activated relaxations. In the frequency domain, a plot of attenuation versus frequency shows an obvious systematic shift to higher frequencies with increasing temperatures. Similarly, the attenuation versus temperature curve moves to higher temperature with increasing frequencies. The attenuation peaks are somewhat broader than that for a Zener relaxation. A Cole-Cole distribution of relaxation times closely matches the attenuations. This behavior can be explained theoretically by local flow mechanisms.     

Effects of petrophysical parameters on attenuation and dispersion of seismic waves in the simplified poroelastic theory: Comparison between the Biot's theory and viscosity‐extended theory

The simplified macro‐equations of porous elastic media are presented based on Hickey's theory upon ignoring effects of thermomechanical coupling and fluctuations of porosity and density induced by passing waves. The macro‐equations with definite physical parameters predict two types of compressional waves (P wave) and two types of shear waves (S wave). The first types of P and S waves, similar to the fast P wave and S wave in Biot's theory, propagate with fast velocity and have relatively weak dispersion and attenuation, while the second types of waves behave as diffusive modes due to their distinct dispersion and strong attenuation. The second S wave resulting from the bulk and shear viscous loss within pore fluid is slower than the second P wave but with strong attenuation at lower frequencies. Based on the simplified porous elastic equations, the effects of petrophysical parameters (permeability, porosity, coupling density and fluid viscosity) on the velocity dispersion and attenuation of P and S waves are studied in brine‐saturated sandstone compared with the results of Biot's theory. The results show that the dispersion and attenuation of P waves in simplified theory are stronger than those of Biot's theory and appear at slightly lower frequencies because of the existence of bulk and shear viscous loss within pore fluid. The properties of the first S wave are almost consistent with the S wave in Biot's theory, while the second S wave not included in Biot's theory even dies off around its source due to its extremely strong attenuation. The permeability and porosity have an obvious impact on the velocity dispersion and attenuation of both P and S waves. Higher permeabilities make the peaks of attenuation shift towards lower frequencies. Higher porosities correspond to higher dispersion and attenuation. Moreover, the inertial coupling between fluid and solid induces weak velocity dispersion and attenuation of both P and S waves at higher frequencies, whereas the fluid viscosity dominates the dispersion and attenuation in a macroscopic porous medium. Besides, the heavy oil sand is used to investigate the influence of high viscous fluid on the dispersion and attenuation of both P and S waves. The dispersion and attenuation in heavy oil sand are stronger than those in brine‐saturated sandstone due to the considerable shear viscosity of heavy oil. Seismic properties are strongly influenced by the fluid viscosity; thus, viscosity should be included in fluid properties to explain solid–fluid combination behaviour properly.