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．     

Modelling of Power-law Fluid Flow Through Porous Media Using Smoothed Particle Hydrodynamics

The flow of non-Newtonian fluids through two-dimensional porous media is analyzed at the pore scale using the smoothed particle hydrodynamics (SPH) method. A fully explicit projection method is used to simulate incompressible flow. This study focuses on a shear-thinning power-law model (n < 1), though the method is sufficiently general to include other stress-shear rate relationships. The capabilities of the proposed method are demonstrated by analyzing a Poiseuille problem at low Reynolds numbers. Two test cases are also solved to evaluate validity of Darcy’s law for power-law fluids and to investigate the effect of anisotropy at the pore scale. Results show that the proposed algorithm can accurately simulate non-Newtonian fluid flows in porous media.     

Non-Newtonian flow in microporous structures under the electroviscous effect

Single phase non-Newtonian microporous flow combined with the electroviscous effect is investigated in the pore-scale under conditions of various rheological properties and electrokinetic parameters. The lattice Boltzmann method is employed to solve both the electric potential field and flow velocity field. The simulation of commonly used power-law non-Newtonian flow shows that the electroviscous effect on the flow depends on both the fluid rheological behavior and pore surface area ratio significantly. For the shear thinning fluid with power-law exponent n < 1, the fluid viscosity near the wall is smaller and the electrovicous effect plays a more important role compared to the Newtonian fluid and shear thickening fluid. The high pore surface area ratio in the porous structure increases the electroviscous force and the induced flow resistance becomes important even to the flow of Newtonian and shear thickening fluids.     

Transient spherical flow of non-newtonian power-law fluids in porous media

The transient spherical flow behavior of a slightly compressible non-Newtonian, power-law fluids in porous media is studied. A nonlinear partial differential equation of parabolic type is derived. The diffusivity equation for spherical flow is a special case of the new equation. We obtain analytical, asymptotic and approximate solutions by using the methods of Laplace transform and weighted mass conservation. The structures of asymptotic and approximate solutions are similar, which enriches the theory of one-dimensional flow of non-Newtonian fluids through porous media.     

The Falkner–Skan Wedge Flows of Power-Law Fluids Embedded in a Porous Medium

The non-Darcy flow characteristics of power-law non-Newtonian fluids past a wedge embedded in a porous medium have been studied. The governing equations are converted to a system of first-order ordinary differential equations by means of a local similarity transformation and have been solved numerically, for a number of parameter combinations of wedge angle parameter m, power-law index of the non-Newtonian fluids n, first-order resistance A and second-order resistance B, using a fourth-order Runge–Kutta integration scheme with the Newton–Raphson shooting method. Velocity and shear stress at the body surface are presented for a range of the above parameters. These results are also compared with the corresponding flow problems for a Newtonian fluid. Numerical results show that for the case of the constant wedge angle and material parameter A, the local skin friction coefficient is lower for a dilatant fluid as compared with the pseudo-plastic or Newtonian fluids.     

Displacement of a Newtonian fluid by a non-Newtonian fluid in a porous medium

This paper presents an analytical Buckley-Leverett-type solution for one-dimensibnal immiscible displacement of a Newtonian fluid by a non-Newtonian fluid in porous media. The non-Newtonian fluid viscosity is assumed to be a function of the flow potential gradient and the non-Newtonian phase saturation. To apply this method to field problems a practical procedure has been developed which is based on the analytical solution and is similar to the graphic technique of Welge. Our solution can be regarded as an extension of the Buckley-Leverett method to Non-Newtonian fluids. The analytical result reveals how the saturation profile and the displacement efficiency are controlled not only by the relative permeabilities, as in the Buckley-Leverett solution, but also by the inherent complexities of the non-Newtonian fluid. Two examples of the application of the solution are given. One application is the verification of a numerical model, which has been developed for simulation of flow of immiscible non-Newtonian and Newtonian fluids in porous media. Excellent agreement between the numerical and analytical results has been obtained using a power-law non-Newtonian fluid. Another application is to examine the effects of non-Newtonian behavior on immiscible displacement of a Newtonian fluid by a power-law non-Newtonian fluid.     

THE RESEARCH PROGRESS OF LATTICE BOLTZMANN METHOD IN NON-NEWTONIAN FLUID FLOW

格子玻尔兹曼方法（lattice Boltzmann method,LBM）能够直接计算局部剪切速率并可以达到二次精度,因此在非牛顿流动数值模拟中展现出一定优势。尽管已证实LBM 对于非牛顿流动的适用性,但是LBM 需要通过即时调节BGK（Bhatnagar-Gross-Krook）碰撞项中的松弛时间来实时反映黏度改变,当松弛时间接近1/2 时,迭代会出现数值不稳定现象。该文对LBM 在非牛顿流体研究中的进展进行了总结,介绍了增加数值稳定性的方法并对结果的精度进行了比较,在此基础上对LBM 在非牛顿研究中的进一步发展进行了展望。     

Mixed Convection in Non-Newtonian Fluids Along an Isothermal Vertical Cylinder in a Porous Medium

Mixed convection in power-law type non-Newtonian fluids along an isothermal vertical cylinder in porous media is studied. The problem is solved by means of a finite difference method for the case of uniform wall temperature. Results for the details of the velocity and temperature fields as well as the Nusselt number have been presented. The viscosity index ranged from 0.5–1.5.     

Mass transfer effects on the non-Newtonian fluids past a vertical plate embedded in a porous medium with non-uniform surface heat flux

 The present study is devoted to investigate the influences of mass transfer on buoyancy induced flow over vertical flat plate embedded in a non-Newtonian fluid saturated porous medium. The Ostwald–de Waele power-law model is used to characterize the non-Newtonian fluid behavior. Similarity solution for the transformed governing equations is obtained with prescribed variable surface heat flux. Numerical results for the details of the velocity, temperature and concentration profiles are shown on graphs. Excess surface temperature as well as concentration gradient at the wall associated with heat flux distributions, which are entered in tables, have been presented for different values of the power-law index n, buoyancy ration B and the exponent λ as well as Lewis number Le. Received on 26 April 2000     

Pressure-driven and electroosmotic non-Newtonian flows through microporous media via lattice Boltzmann method

The lattice Boltzmann method is developed to simulate the pressure-driven flow and electroosmotic flow of non-Newtonian fluids in porous media based on the representative elementary volume scale. The flow through porous media was simulated by including the porosity into the equilibrium distribution function and adding a non-Newtonian force term to the evolution equation. The non-Newtonian behavior is considered based on the Herschel–Bulkley model. The velocity results for pressure-driven non-Newtonian flow agree well with the analytical solutions. For the electroosmotic flow, the influences of porosity, solid particle diameter, power law exponent, yield stress and electric parameters are investigated. The results demonstrate that the present lattice Boltzmann model is capable of modeling non-Newtonian flow through porous media.     

Creeping flow through a model fibrous porous medium

Pressure losses and velocity distributions were measured for creeping flow through three model fibrous porous media. The three models consisted of square arrays of circular rods with solid volume fractions of 2.5, 5 and 10%. Measurements of flow resistances are in good agreement with theoretical predictions after wall effects are accounted for using Brinkman’s equation. Two-dimensional velocity vector maps were obtained in each array using particle image velocimetry. The velocity distributions are necessary for identifying non-Newtonian effects in flows with viscoelastic fluids.     

Self-similar solutions of non-Newtonian fluid boundary layer in MHD

The self-similar solutions of the boundary layer for a non-Newtonian fluid in MHD were considered in [1, 2] for a power-law velocity distribution along the outer edge of the layer and constant electrical conductivity through the entire flow. However, the MHD flows of many conducting media, which are solutions or molten metals, cannot be described by the MHD equations for non-Newtonian fluids.The self-similar solutions of the boundary layer for a non-Newtonian fluid without account for interaction with the electromagnetic field were studied in [3].In the following we present the self-similar solutions for the boundary layer of pseudoplastic and dilatant fluids with account for the interaction with an electromagnetic field for the case of a power-law velocity distribution along the outer edge of the layer, when the conductivity of the fluid is constant throughout the flow and the magnetic Reynolds number is small.Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 2, No. 6, pp. 77–82, 1967The author wishes to thank S. V. Fal'kovich for his interest in this study.     

幂律流体在多孔介质中径向渗流的分形模型

基于分形理论及毛细管模型,本文研究了非牛顿幂律流体在各向同性多孔介质中径向流动问题,推导了幂律流体径向有效渗透率的分形解析表达式．研究结果表明,幂律流体径向有效无量纲渗透率模型和Chang and Yortsos’s模型吻合很好;同时还得出幂律流体径向有效渗透率随孔隙率、幂指数的增加而增加,随迂曲度分形维数的增加而减少．     

An analysis for forced convection heat transfer from external surfaces to non-Newtonian fluids

A theoretical analysis has been proposed for the forced convection heat transfer from external surfaces immersed in non-Newtonian fluids of the power-law model. The integral treatment previously introduced for Newtonian fluids has been successfully extended to the non-Newtonian fluids over a flat plate and a wedge of an arbitrary included angle. The integral momentum and energy equations are transformed into a pair of characteristic equations, which can readily be solved for the velocity shape factor and the boundary layer thickness ratio, once the exponents in the expressions for the power-law model, free stream velocity and wall temperature variation are specified. It has been also found that an asymptotic expression derived under the assumption of large Prandtl number, is valid practically for all power-law fluids, and hence, can be used for a speedy, and yet accurate estimation of the local heat transfer to non-Newtonian fluids.     

Non-Newtonian effect on natural convection flow over cylinder of elliptic cross section

The non-Newtonian effect in the boundary layer flow over a horizontal elliptical cylinder is investigated numerically. A modified power-law viscosity model is used to correlate the non-Newtonian characteristics of the fluid flow. For natural convectionflows, the surface of the cylinder is maintained by the uniform surface temperature(UST)or the uniform heat flux(UHF) condition. The governing equations corresponding to theflow are first transformed into a dimensionless non-similar form using suitable transformations. The resulting equations are solved numerically by an efficient finite difference scheme. The numerical results are presented for the skin friction coefficient and the local Nusselt number with the eccentric angle for different values of the power-law index n. The local skin friction coefficient and the local Nusselt number are found to be higher and lower, respectively, for the shear thickening fluids(n > 1) than the other fluids(n≤1).The effects of different elliptical configurations on the average Nusselt number are also presented and discussed for both conditions of the surface temperature.     

Numerical Algorithms for Network Modeling of Yield Stress and other Non-Newtonian Fluids in Porous Media

Many applications involve the flow of non-Newtonian fluids in porous, subsurface media including polymer flooding in enhanced oil recovery, proppant suspension in hydraulic fracturing, and the recovery of heavy oils. Network modeling of these flows has become the popular pore-scale approach for understanding first-principles flow behavior, but strong nonlinearities have prevented larger-scale modeling and more time-dependent simulations. We investigate numerical approaches to solving these nonlinear problems and show that the method of fixed-point iteration may diverge for shear-thinning fluids unless sufficient relaxation is used. It is also found that the optimal relaxation factor is exactly equal to the shear-thinning index for power-law fluids. When the optimal relaxation factor is employed it slightly outperforms Newton??s method for power-law fluids. Newton-Raphson is a more efficient choice (than the commonly used fixed-point iteration) for solving the systems of equations associated with a yield stress. It is shown that iterative improvement of the guess values can improve convergence and speed of the solution. We also develop a new Newton algorithm (Variable Jacobian Method) for yield-stress flow which is orders of magnitude faster than either fixed-point iteration or the traditional Newton??s method. Recent publications have suggested that minimum-path search algorithms for determining the threshold pressure gradient (e.g., invasion percolation with memory) greatly underestimate the true threshold gradient when compared to numerical solution of the flow equations. We compare the two approaches and reach the conclusion that this is incorrect; the threshold gradient obtained numerically is exactly the same as that found through a search of the minimum path of throat mobilization pressure drops. This fact can be proven mathematically; mass conservation is only preserved if the true threshold gradient is equal to that found by search algorithms.     

Permeability mapping in porous media by magnetization prepared centric-scan SPRITE

The ability of porous media to transmit fluids is commonly referred to as permeability. The concept of permeability is central for hydrocarbon recovery from petroleum reservoirs and for studies of groundwater flow in aquifers. Spatially resolved measurements of permeability are of great significance for fluid dynamics studies. A convenient concept of local Darcy’s law is suggested for parallel flow systems. The product of porosity and mean velocity images in the plane across the average flow direction is directly proportional to permeability. Single Point Ramped Imaging with T ₁ Enhancement (SPRITE) permits reliable quantification of local fluid content and flow in porous media. It is particularly advantageous for reservoir rocks characterized by fast magnetic relaxation of a saturating fluid. Velocity encoding using the Cotts pulsed field gradient scheme improves the accuracy of measured flow parameters. The method is illustrated through measurements of 2D permeability maps in a capillary bundle, glass bead packs and composite sandstone samples.     

Non-steady MHD flow for a non-Newtonian fluid with variable conductivity

This paper deals with a similarity solution for the unsteady flow of a conducting non-Newtonian power-law in-compressible fluid, when a porous plate is moving uniformly in the presence of a transverse magnetic field, assuming that the electrical conductivity is a function of the velocity. The aim of this analysis is to determine the velocity and the effect of variation of the electrical conductivity on the solution. The basic equations have been solved by applying the perturbation method for small and large values of the magnetic interaction parameterN. The main features of the exact solution is that it represents shear flow.     

Analysis of a benchmark solution for non-Newtonian radial displacement in porous media

We present an analytical formulation useful to interpret the key phenomena involved in non-Newtonian displacement in porous media and an analysis of the results obtained by considering the uncertainty associated with relevant problem parameters. To derive a benchmark solution, we consider the radial dynamics of a moving stable interface in a porous domain saturated by two fluids, displacing and displaced, both non-Newtonian of shear-thinning power-law behavior, assuming the pressure and velocity to be continuous at the interface, and constant initial pressure. The flow law for both fluids is a modified Darcy’s law. Coupling the nonlinear flow law with the continuity equation, and taking into account compressibility effects, yields a set of nonlinear second-order partial differential equations. Considering two fluids with the same flow behavior index n allows transformation of the latter equations via a self-similar variable; further transformation of the equations incorporating the conditions at the interface shows for n<1 the existence of a compression front ahead of the moving interface. Solving the resulting set of nonlinear equations yields the positions of the moving interface and compression front, and the pressure distributions; the latter are derived in closed form for any value of n. A sensitivity analysis of the model responses is conducted both in a deterministic and a stochastic framework. In the latter case, Global Sensitivity Analysis (GSA) of the benchmark analytical model is adopted to study how the effects of uncertainty affecting selected parameters: (a) the fluids flow behavior index, (b) the relative total compressibility and mobility in the displaced and displacing fluid domains, and (c) the domain permeability and porosity, propagate to state variables. The relative influence of input parameters on model outputs is evaluated by means of associated Sobol indices, calculated via the Polynomial Chaos Expansion (PCE) technique. The goodness of the results obtained by the PCE is assessed by comparison against a traditional Monte Carlo (MC) approach.     

LBM simulation of fluid flow in fractured porous media with permeable matrix

To analyze and depict complicated fluid behaviors in fractured porous media with variably permeable matrix, an integrated discrete computational algorithm is proposed based on lattice Boltzmann method (LBM). This paper combines with the external force model and statistical material physics to effectively describe the feature changes while the fluid passes through the fractures within the permeable matrix. As an application example, a two dimensional rock sample is reconstructed using the digital image and characterized with different feature values at each LBM grid to distinguish pores, impermeable and permeable matrix by stating its local physical property. Compared with the conventional LBM, the results demonstrate the advantages of proposed algorithm in modeling fluid flow phenomenon in fractured porous media with variably permeable matrix.