Theory of dispersion of solutes in non-Newtonian flows through a circular tube

Summary In the present paperTaylor's analysis of the dispersion of a soluble matter in Newtonian flow through a circular tube is extended in the case of non-Newtonian flows of Eyring and Reiner-Philippoff model fluids. It has been shown here that the results for the Newtonian fluid can be deduced from the corresponding results of both the two types non-Newtonian flows. Few specific cases of both the two types of fluids have been studied. Aris modification ofTaylor's analysis is also applicable to non-Newtonian flows discussed here. The results may be useful in connection with the study of the dispersion of soluble salts in blood vessels. It may also be useful to physicians who wish to study molecular diffusion coefficients.     

Some exact solutions for the flow of fluid through tubes with uniformly porous walls

Summary This paper considers an incompressible fluid flowing through a straight, circular tube whose walls are uniformly porous. The flow is steady and one dimensional. The loss of fluid through the wall is proportional to the mean static pressure in the tube. Several formulations of the wall shear stress are considered; these formulations were motivated by the results from Hamel's radial flow problem, boundary layer flows/and boundary layer suction profiles. For each of these formulations exact solutions for the mean axial velocity and the mean static pressure of the fluid are obtained. Sample results are plotted on graphs. For the constant wall shear stress problem, the theoretical solutions compare favorably with some experimental results.Notations A, B, D, E constant parameters - a, b constant parameters - Ai(z), Bi(z) Airy functions - Ai, Bi derivatives of Airy functions - k constant of proportionality betweenV andp - L length of pores - p,p mean static pressure - p ₀ static pressure outside the tube - p ₀ value ofp atx=0 - Q constant exponent - R inside radius of the tube - T wall shear stress - T ₀ shear parameter - t wall thickness - U free stream velocity - ,u mean axial velocity - u ₀ value ofu atx=0 - V,V mean seepage velocity through the wall - v ₀ mean seepage velocity - x,x axial distance along the tube - z transformed axial distance - z ₀ value ofz atx=0 - mean outflow angle through the wall - cos - density of the fluid - wall shear stress - dynamic viscosity of the fluid - over-bar dimensional terms - no bar nondimensional terms The National Center for Atmospheric Research is sponsored by the National Science Foundation     

圆钢管混凝土构件弯矩-曲率滞回特性研究

首先确定了往复应力作用下组成圆钢管混凝土的钢材和核心混凝土的应力－应变关系模型、在此基础上,利用数值计算方法,对圆钢管混凝土构件弯矩－曲率滞回关系曲线进行了理论分析,其结果和试验结果吻合良好。最后,提出圆钢管混凝土构件弯矩－风率恢复力模型,给出模型中有关参数的计算公式,可为圆钢管混凝土体系弹塑性分析提供参考。     

基于孔隙网络模型的非牛顿流体驱替模式研究

本文提出了一种考虑了流体可压缩性与非牛顿流体剪切流变性的非稳态孔隙网络模型,用以研究非混相驱替中的毛管指进、黏性指进和两者之间的过渡区这三种不同的驱替模式.通过在孔隙网络模型中饱和非牛顿流体,研究不同注入毛管数和不同黏度比情况下的注水驱替模式.结果表明,非稳态孔隙网络模型可以模拟再现三种不同的流体驱替模式,并提出特征前缘流量用于区分三种驱替模式.毛管指进的入口压力的相对波动较大且手指的无序生长会抑制前缘的移动速度,因此特征前缘流量变化较小.黏性指进会抑制指进横向生长和向入口方向的流动,入口压力在突破时快速降低.在过渡区,驱替流体倾向于占据由更大半径孔喉组成的少量路径,并以更大的特征流量向出口处快速突破,导致更细的手指与较低的驱替效率.本文还对比了被驱替相为牛顿流体时的情况.结果表明,剪切流变性会减弱被驱替相黏性阻力的影响,从而使得过渡区的范围更宽.本研究有助于更好地研究复杂流体间黏性力与毛管力对驱替模式的影响,对于提高油气采收率和二氧化碳地质封存效率具有现实意义.

     

Estimates of effective permeability for non-Newtonian fluid flow in randomly heterogeneous porous media

An understanding of the interplay between non-Newtonian effects in porous media flow and field-scale domain heterogeneity is of great importance in several engineering and geological applications. Here we present a simplified approach to the derivation of an effective permeability for flow of a purely viscous power–law fluid with flow behavior index n in a randomly heterogeneous porous domain subject to a uniform pressure gradient. A standard form of the flow law generalizing the Darcy’s law to non-Newtonian fluids is adopted, with the permeability coefficient being the only source of randomness. The natural logarithm of the permeability is considered a spatially homogeneous and correlated Gaussian random field. Under the ergodic hypothesis, an effective permeability is first derived for two limit 1-D flow geometries: flow parallel to permeability variation (serial-type layers), and flow transverse to permeability variation (parallel-type layers). The effective permeability of a 2-D or 3-D isotropic domain is conjectured to be a power average of 1-D results, generalizing results valid for Newtonian fluids under the validity of Darcy’s law; the conjecture is validated comparing our results with previous literature findings. The conjecture is then extended, allowing the exponents of the power averaging to be functions of the flow behavior index. For Newtonian flow, novel expressions for the effective permeability reduce to those derived in the past. The effective permeability is shown to be a function of flow dimensionality, domain heterogeneity, and flow behavior index. The impact of heterogeneity is significant, especially for shear-thinning fluids with a low flow behavior index, which tend to exhibit channeling behavior.     

A numerical method for simulating non-Newtonian fluid flow and displacement in porous media

Flow and displacement of non-Newtonian fluids in porous media occurs in many subsurface systems, related to underground natural resource recovery and storage projects, as well as environmental remediation schemes. A thorough understanding of non-Newtonian fluid flow through porous media is of fundamental importance in these engineering applications. Considerable progress has been made in our understanding of single-phase porous flow behavior of non-Newtonian fluids through many quantitative and experimental studies over the past few decades. However, very little research can be found in the literature regarding multi-phase non-Newtonian fluid flow or numerical modeling approaches for such analyses.For non-Newtonian fluid flow through porous media, the governing equations become nonlinear, even under single-phase flow conditions, because effective viscosity for the non-Newtonian fluid is a highly nonlinear function of the shear rate, or the pore velocity. The solution for such problems can in general only be obtained by numerical methods.We have developed a three-dimensional, fully implicit, integral finite difference simulator for single- and multi-phase flow of non-Newtonian fluids in porous/fractured media. The methodology, architecture and numerical scheme of the model are based on a general multi-phase, multi-component fluid and heat flow simulator — TOUGH2. Several rheological models for power-law and Bingham non-Newtonian fluids have been incorporated into the model. In addition, the model predictions on single- and multi-phase flow of the power-law and Bingham fluids have been verified against the analytical solutions available for these problems, and in all the cases the numerical simulations are in good agreement with the analytical solutions. In this presentation, we will discuss the numerical scheme used in the treatment of non-Newtonian properties, and several benchmark problems for model verification.In an effort to demonstrate the three-dimensional modeling capability of the model, a three-dimensional, two-phase flow example is also presented to examine the model results using laboratory and simulation results existing for the three-dimensional problem with Newtonian fluid flow.     

Flow through anisotropic networks

This paper presents an analysis of Hagen-Poiseulle flow through plane random anisotropic networks of interconnected channels. Macroscopic permeability tensor of the network is expressed in terms of statistico-geometrical characteristics like the degree of anisotropy in channel orientations, average co-ordination number of the network and first two moments of channel length distribution. Analytical results are illustrated and verified using numerical analysis of flow in a simulated random network. The emphasis of the paper is on the effects of anisotropy on distributions of flow rates in channels. It is shown that, due to anisotropy the maximum flow rate generally occurs in channels that are not aligned along the direction of the macroscopic pressure gradient.     

Flow through anisotropic networks

This paper presents an analysis of Hagen-Poiseulle flow through plane random anisotropic networks of interconnected channels. Macroscopic permeability tensor of the network is expressed in terms of statistico-geometrical characteristics like the degree of anisotropy in channel orientations, average co-ordination number of the network and first two moments of channel length distribution. Analytical results are illustrated and verified using numerical analysis of flow in a simulated random network. The emphasis of the paper is on the effects of anisotropy on distributions of flow rates in channels. It is shown that, due to anisotropy the maximum flow rate generally occurs in channels that are not aligned along the direction of the macroscopic pressure gradient.     

Analytical modelling of viscous diapirism through a strongly non-Newtonian overburden subject to horizontal forces

We study the early stages of diapirism and analyse the gravitational and buckling instabilities of a buoyant viscous layer overlain by a layer of strongly non-Newtonian power-law rheology (when a power-law exponent tends to infinity). This situation models rocksalt under a layer of a perfectly plastic overburden. The growth rate of small perturbations on the interface between the two layers and the wavelength of the most unstable perturbations are found and compared with those of structures consisting of two Newtonian or two strongly non-Newtonian viscous layers. Effects due to the effective viscosity and thickness ratios between the two layers are assessed. Considering the effective viscosity of the overburden to be much greater than the viscosity of the buoyant salt layer, we obtain the following results. In the case of simple gravitational instability and no-slip boundary conditions, the instability pattern is similar to that in a strongly non-Newtonian power-law material. An increase in the thickness of the overburden decreases the dominant wavelength of the most unstable mode, while the dominant wavelength is lengthened in the case of Newtonian viscous layers. When the system of layers is subjected to either horizontal extension or shortening, and the upper boundary of the system is stress-free, the buckling instability overwhelms the gravitational instability, and the dynamic growth rate of the instability depends linearly on the effective viscosity ratio. We conclude that the introduction of strongly non-Newtonian power-law rheology into diapir overburdens greatly affects instability parameters such as growth rate and dominant wavelength of perturbations and hence, alters interdiapir spacings.     

Simulation of two-phase fluid flow through compactible reservoirs

Fluid-flow simulators used in the oil industry model the movement of fluids through a porous reservoir rock. These simulators either ignore coupling between the flow and concurring deformation of the solid rock frame or take it into account approximately, in the so-called loose or staggered-in-time mode. In contrast to existing simulators, the one we describe here fully couples two-phase (oil and water) flow to subsurface deformation and simultaneously accounts for all relevant physical phenomena. As such, our flow simulator inherently links time-dependent fluid pressures, saturations, permeabilities and flow velocities to stresses in the whole subsurface. These stresses relate to strains through the non-linear theory of elasticity, allowing us to model time-lapse changes in seismic velocities and anisotropy. The velocity variations manifest themselves in time shifts and reflection amplitudes that are conventionally measured from 4D seismic data. Changes in anisotropy produce time-dependent shear-wave splitting that can be used for monitoring the horizontal stresses.     

Flow of a double-diffusive stratified fluid in a differentially-rotating cylinder

Abstract

A study has been made of a basic state of axisymmetric flow, at large rotational Reynolds numbers, in a double-diffusive stratified fluid contained in a vertically-mounted, differentially-rotating cylindrical cavity. The aim is to describe the qualitative characteristics of the flow of a fluid, the density of which is stratified by two diffusive effects, i.e., temperature and salinity gradients. Attention is confined to situations in which the temperature and salinity gradients make opposing contributions to the overall density profile, the undisturbed stratification being gravitationally stable. Finite difference numerical solutions of the governing Navier-Stokes equations have been obtained using the Boussinesq approximation. The results are presented in a way that illustrates the explicit effects of double-diffusivity when the cavity aspect ratio, height/radius, is O(1). The principal non-dimensional parameters characterizing the flow field are identified. In the interior core, the primary dynamic balance is between the horizontal density gradient and the vertical shear of the prevailing azimuthal velocity. The effective stratification is seen to decrease as the double-diffusivity increases, even if the overall stratification parameter, St, is held constant. The solute field contains a very thin boundary layer structure at large Lewis numbers. The effective stratification increases with the Prandtl number. Results have been derived for extreme values of the cavity aspect ratio. For small cavity aspect ratios, the dominant dynamic ingredients are viscous diffusion and rotation. For large aspect ratios, the bulk of the flow field is determined by the rotating sidewall. In this case, the direct influence of the double-diffusivity is minor.     

Numerical studies of non-Newtonian mantle convection

Numerical model computations have been carried out to determine how the stress-dependence of non-Newtonian viscosity affects the flow structure of thermal convection. The viscosity laws have been chosen in accordance with present knowledge of upper mantle rheology, based on the diffusion and dislocation creep laws of olivine. The results show that there are important differences between the structures of Newtonian and non-Newtonian convection. While the Newtonian models are insufficient in some respects, the non-Newtonian solutions can explain the characteristics of the real mantle flow. However, this may require a faster plastic deformation than power law dislocation creep, at least in the high-stress regions of the mantle, e.g. at the active plate margins.     

Flow of viscous fluids through a porous deformable matrix

A self-contained account of mixture theory is presented as a framework for describing the flow of fluids, liquids and gases, through a porous deformable matrix, incorporating both mechanical and thermal effects. The theory comprises the conservation laws of mass, momentum and energy for each constituent and the mixture properties which describe the interactions between constituents. Mass transfer between constituents which arises during phase change and chemical reactions influences both conservation laws and mixture properties. An analysis of discontinuity conditions at a singular surface is presented, which would be needed, for example, to describe an advancing phase-change front. Details are presented for the flow of viscous fluids through a thermoelastic matrix undergoing infinitesimal deformation, a common model for underground reservoirs. The interactions of immiscible and miscible fluids are discussed. An essential ingredient is the relation between partial physical variables defined as mean values over mixture elements, and intrinsic variables defined with respect to the constituent elements.     

Ground subsidence due to uniform fluid extraction over a circular region within an aquifer

This paper examines the ground subsidence caused by the uniform extraction of fluids from a disc-shaped region located within a poroelastic halfspace. Biot’s theory of poroelasticity is used to examine the problem. The fluid extraction over the circular region at the interior of the poroelastic domain induces time-dependent axisymmetric surface settlements. The theoretical results are also used to examine the accuracy of a multi-physics computational scheme that can be used to examine more complex geological settings.     

Flow of a stratified fluid bounded by walls of finite thermal conductance

Abstract

A study is made of the behavior of a thermally stratified fluid in a container when the non-horizontal boundaries have finite thermal conductance. The theory of Rahm and Walin is briefly recounted. Numerical solutions to the Navier-Stokes equations for a Boussinesq fluid in a cylinder, adopting a Newtonian heat flux condition at the vertical sidewall, are presented. Results on the details of flow and temperature fields are given over ranges of the Rayleigh number Ra, the container aspect ratio H, and the sidewall conductance S. As S increases, the isotherms in the meridional plane are horizontal at small radii but they diverge at large radii. This creates temperature nonuniformilies in the horizontal direction, and convective motions result. The salient features of the interior temperature profiles are captured by the theoretical model. The velocity field is characterized by two oppositely-directed circulations. As Ra or S varies, the qualitative circulation patterns remain substantially unchanged, but the magnitudes of the convective flows differ by large amounts. The effects of the externally-imposed parameters on the flow and temperature structures are examined.     

Flow to an eccentric well in a leaky circular aquifer with varied lateral replenishment

Summary An analytical solution is obtained for the flow to an eccentric well in a leaky circular aquifer with lateral replenishment, both for steady and unsteady cases. The flows for external boundary conditions of constant head and zero flux, which were treated previously, follow in the limit from a more general boundary condition. Graphs are developed to show the influence of vertical leakage and lateral replenishment on the relationship between drawdown at the well and eccentricity.Other symbols are defined in the text as they occur.     

Flow at high Reynolds numbers through anisotropic porous media

An approximate expression is developed for the relationship between the hydraulic gradient (J), the specific discharge (q) and fluid and porous matrix properties in the case of saturated, steady and uniform (macroscopic) flow of a Newtonian liquid at high Reynolds numbers through a homogeneous anisotropic porous medium:

$\text{g}\text{J}\text{=(v}\text{w}{}^{\mathrm{(2)}}\text{+}\text{B}{}^{\mathrm{(4)}}\text{:}\text{qq}\text{/q+}\text{C}{}^{\mathrm{(3)}}\text{·}\text{q}\text{)·}\text{q}$

In this expression, the tensors w⁽²⁾, B⁽⁴⁾ and C⁽³⁾ denote properties of the solid matrix only. The tensors W⁽²⁾, and C⁽³⁾ are symmetrical; the tensor B⁽⁴⁾ is symmetrical only in the first and last pairs of indices. It seems that no mathematical expression with a finite number of parameters exists, which can serve as a universal exact expression for the sought relationship between J and q.     

利用统计地震模型从地震活动参数中获得流体信号

根据众所周知的库伦破裂准则,应力或孔隙压力的变化都会导致地震破裂。余震活动一般认为是地震应力变化引起的,它服从大森定律。而地震群则认为是通过地下流体侵入岩层所致。应力触发可以通过在地壳单独构建三维弹力应变模型分析,而地震引起的地下流体流动情况必须通过空隙压力变化和地震之间相互作用应力场的改变这样复杂的地震活动图像研究。我们看到ETAS模型是从复杂地震活动图像获得最初流体信号的一种适当工具。我们主要分析2000年发生在欧洲中部的波西米亚西北部Vogtland地区的大地震群。通过拟合随机的传染型余震序列模型(ETAS)我们发现应力触发在创建观测地震活动图像和解释观测事件破碎时间分布时起支配性作用。外力在直接触发引起地震活动中起到的作用是很有限的,这种外力被认为是由于流体入侵导致孔隙压力变化而产生的。然而时间上的反褶积表明显著的流体信号源于地震群。这些结果已被我们的仿真模型模拟研究证实,在这个研究中,流体侵入和应力在三维弹性半空间中沿断层面转移时触发地震。基于ETAS模型的反褶积过程能够揭示地下孔隙压力变化。