Nonlinear fluid flow laws for orthotropic porous media

Nonlinear fluid flow laws for orthotropic porous media are written in invariant tensor form. As usual in the theory of fluid flow through porous media [1, 2], the equations contain the flow velocity up to the second power. Expressions that determine the nonlinear resistances to fluid flow are presented and it is shown that, on going over from linear to nonlinear flow laws, the asymmetry effect may manifest itself, that is, the fluid flow characteristics may differ along the same straight line in the positive and negative directions. It is shown that, as compared with the linear fluid flow law for orthotropic media when for three symmetry groups a single flow law is sufficient, in nonlinear laws the anisotropy manifestations are much more variable and each symmetry group must be described by specific equations. A system of laboratory measurements for finding the nonlinear flow characteristics for orthotropic porous media is considered.     

低渗透多孔介质中的非线性渗流理论简

文中论述了低渗透性多孔介质中非线性渗流理论的几个问题,阐明了渗流流体的性质,指出了多孔介质对流体通过的选择性,提出了新的非线性渗流方程,用实验资料对其进行了验证,分析了该方程演变功能,表明它可以描述各种渗流规律.该方程的各项参数都可从实验中直接得到,应用方便,并且参数的物理意义明确.     

An overview on nonlinear porous flow in low permeability porous media

This paper gives an overview on nonlinear porous flow in low permeability porous media, reveals the microscopic mechanisms of flows, and clarifies properties of porous flow fluids. It shows that, deviating from Darcy's linear law, the porous flow characteristics obey a nonlinear law in a low-permeability porous medium, and the viscosity of the porous flow fluid and the permeability values of water and oil are not constants. Based on these characters, a new porous flow model, which can better describe low permeability reservoir, is established. This model can describe various patterns of porous flow, as Darcy's linear law does. All the parameters involved in the model, having definite physical meanings, can be obtained directly from the experiments.     

渗流力学研究的现状和发展趋势

渗流力学在能源、环境、水利、岩土、交通、生物等工程领域有广泛的应用，渗流力学经过约一个半世纪的发展，已经积累了相当多的成果。随着现代科学的发展以及生产实践需求的不断提高，渗流力学仍然有进一步完善和发展的广阔空间。本文简述了水利、环境、能源工程和生物学中的渗流力学问题，扼要概括了渗流力学理论研究的现状，并指出渗流力学在多孔介质描述、裂缝型介质渗流、多相多组分渗流、物理化学渗流、非线性渗流、非饱和渗流、微观渗流、渗流模拟等理论及相关方法和测试技术等方面的发展趋势。     

渗流力学的新发展

渗流力学是研究流体在多孔介质内运动规律的科学。自1856年法国工程师达西提出线性渗流定律以来,渗流力学一直在向前发展,但最近10多年来的发展更为迅速,其主要动向是:渗流力学应用范围比以前更广;渗流力学理论以较快的速度不断深化;渗流力学研究手段不断实现现代化。本文主要对几项新型渗流——非等温渗流,物理化学渗流,非牛顿流体渗流,生物流体渗流,细观渗流的发展作了综述。      

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.     

Upscaling Forchheimer law

We investigate the high velocity flow in heterogeneous porous media. The model is obtained by upscaling the flow at the heterogeneity scale where the Forchheimer law is assumed to be valid. We use the method of multiple scale expansions, which gives rigorously the macroscopic behaviour without any prerequisite on the form of the macroscopic equations. We show that Forchheimer law does not generally survive upscaling. The macroscopic flow law is strongly non-linear and anisotropic. A 2-point Padé approximation of the flow law in the form of a Forchheimer law is given. However, this approximation is generally poor. These results are illustrated in two particular cases: a layered composite porous media and a composite constituted by a square array of circular porous inclusions embedded in a porous matrix. We show that non-linearities are sources of anisotropy.     

From Single-Phase to Compositional Flow: Applicability of Mixed Finite Elements

In this paper we discuss the formulation of the governing equations that describe flow of fluids in porous media. Various types of fluid flow, ranging from single-phase flow to compositional flow, are considered. It is shown that all the differential equations governing these types of flow can be effectively rewritten in a fractional flow formulation; i.e., in terms of a global pressure and saturation (or saturations), and that mixed finite element methods can be accurately exploited to solve the pressure equation. Numerical results are presented to see the performance of the mixed methods for the flow equations in three space dimensions.     

Saturation–pressure relationships for two- and three-phase flow analogies for soft matter

Soft matter and porous media analogies are used to derive micro-scale informed pressure–saturation relationships for cell aggregates. These aggregates may consist of different cell types and an interstitial liquid. The extracellular matrix is the scaffold for these soft materials and is first taken as rigid. Extension to deformable material is also addressed.In tissue mechanics, micro-scale formulations are very often not sustainable from the computational point of view due to the complexity of the pore scale phase distributions and the large size of many problems of interest. Hence, models are formulated mainly at a larger scale, called the macro scale. We derive the relationships at that scale by exploiting theoretical results from two phase flow in porous media which incorporate information from the micro-scale. An example with an indirect validation of the obtained relationship is shown.     

Origin and quantification of coupling between relative permeabilities for two-phase flows in porous media

An extended formulation of Darcy's two-phase law is developed on the basis of Stokes' equations. It leads, through results borrowed from the thermodynamics of irreversible processes, to a matrix of relative permeabilities. Nondiagonal coefficients of this matrix are due to the viscous coupling exerted between fluid phases, while diagonal coefficients represent the contribution of both fluid phases to the total flow, as if they were alone. The coefficients of this matrix, contrary to standard relative permeabilities, do not depend on the boundary conditions imposed on two-phase flow in porous media, such as the flow rate. This formalism is validated by comparison with experimental results from tests of two-phase flow in a square cross-section capillary tube and in porous media. Coupling terms of the matrix are found to be nonnegligible compared to diagonal terms. Relationships between standard relative permeabilities and matrix coefficients are studied and lead to an experimental way to determine the new terms for two-phase flow in porous media.     

Free Flow at the Interface of Porous Surfaces: A Generalization of the Taylor Brush Configuration

A solution to the problem of shallow laminar water flow above a porous surface is essential when modeling phenomena such as erosion, resuspension, and mass transfer between the porous media and the flow above it. Previous studies proposed theoretical, experimental, and numerical insight with no single general solution to the problem. Many studies have used the Brinkman equation, while others showed that it does not represent the actual interface flow conditions. In this paper we show that the interface macroscopic velocity can be accurately modeled by introducing a modification to the Brinkman equation. A moving average approach was proved to be successful when choosing the correct representative elementary volume and comparing the macroscopic solution with the average microscopic flow. As the size of the representative elementary volume was found to be equal to the product of the square root of the permeability and an exponential function of the porosity, a general solution is now available for any brush configuration. Given the properties of the porous media (porosity and permeability), the flow height and its driving force, a complete macroscopic solution of the interface flow is obtained.     

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.     

考虑气体压缩性的多孔材料渗透率和惯性系数的测定

多孔介质材料的渗透率和惯性系数是决定多孔介质中流体流动特性的重要参数，一般需要通过实验进行测定，在测定渗透律和惯性系数量，选用气体作为工作介质可以为实验带来极大的方便，然而通常的实验都将气体看作不可压缩流体，直接根据Darcy-Forchheimer定律得到这两个参数，这种近似对实验条件如样品厚度、工作压力等提出了很多要求，本文提出了在考虑气体压缩性的情况下测定渗透率和惯性系数的方法，该方法可以大大降低实验时对样品厚度、工作压力等条件的要求。本文还根据该方法对多孔材料PVF进行了渗透率和惯性系数的测定，并对测量结果进行了验证。     

关于渗流中流线不封闭的特性和条件

本文对于流体在多孔介质中流动的特性进行理论研究和数值计算，提出两个关于渗流中流线不封闭的特性和条件，得到了在一般工程实际情况中的多孔介质区域内部不存在封闭流线的结论。本文以突变截面圆管中不可压缩渗流为算例，利用半人工瞬变方法进行数值计算，得到流体在充满多孔介质的突扩截面圆管和突缩截面圆管中流动时关于速度分布和压力分布的结果。由此表明，在突变截面附近的渗流区域中不存在回流和分离流，也不存在封闭的流线。渗流的这些流动特性不同于在无多孔介质的空间区域中的流动特性。     

Computational modelling of variably saturated flow in porous media with complex three‐dimensional geometries

A computational procedure is presented for solving complex variably saturated flows in porous media, that may easily be implemented into existing conventional finite‐volume‐based computational fluid dynamics codes, so that their functionality might be geared upon to readily enable the modelling of a complex suite of interacting fluid, thermal and chemical reaction process physics. This procedure has been integrated within a multi‐physics finite volume unstructured mesh framework, allowing arbitrarily complex three‐dimensional geometries to be modelled. The model is particularly targeted at ore heap‐leaching processes, which encounter complex flow problems, such as infiltration into dry soil, drainage, perched water tables and flow through heterogeneous materials, but is equally applicable to any process involving flow through porous media, such as in environmental recovery processes. The computational procedure is based on the mixed form of the classical Richards equation, employing an adaptive transformed mixed algorithm that is numerically robust and significantly reduces compute (or CPU) time. The computational procedure is accurate (compares well with other methods and analytical data), comprehensive (representing any kind of porous flow model), and is computationally efficient. As such, this procedure provides a suitable basis for the implementation of large‐scale industrial heap‐leach models. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical solution of the three‐dimensional fluid flow in a rotating heterogeneous porous channel

A numerical solution to the problem of the three‐dimensional fluid flow in a long rotating heterogeneous porous channel is presented. A co‐ordinate transformation technique is employed to obtain accurate solutions over a wide range of porous media Ekman number values and consequent boundary layer thicknesses. Comparisons with an approximate asymptotic solution (for large values of Ekman number) and with theoretical predictions on the validity of Taylor–Proudman theorem in porous media for small values of Ekman number show good qualitative agreement. An evaluation of the boundary layer thickness is presented and a power‐law correlation to Ekman number is shown to well‐represent the results for small values of Ekman number. The different three‐dimensional fluid flow regimes are presented graphically, demonstrating the distinct variation of the flow field over the wide range of Ekman numbers used. Copyright © 1999 John Wiley & Sons, Ltd.     

SPH-based numerical treatment of the interfacial interaction of flow with porous media

In this paper, the macroscopic equations of mass and momentum are developed and discretized based on the smoothed particle hydrodynamics (SPH) formulation for the interaction at an interface of flow with porous media. The theoretical background of flow through porous media is investigated to highlight the key constraints that should be satisfied, particularly at the interface between the porous media flow and the overlying free flow. The study aims to investigate the derivation of the porous flow equations, computation of the porosity, and treatment of the interfacial boundary layer. It addresses weak assumptions that are commonly adopted for interfacial flow simulation in particle-based methods. As support to the theoretical analysis, a two-dimensional weakly compressible SPH model is developed based on the proposed interfacial treatment. The equations in this model are written in terms of the intrinsic averages and in the Lagrangian form. The effect of particle volume change due to the spatial change of porosity is taken into account, and the extra stress terms in the momentum equation are approximated by using Ergun's equation and the subparticle scale model to represent the drag and turbulence effects, respectively. Four benchmark test cases covering a range of flow scenarios are simulated to examine the influence of the porous boundary on the internal, interface, and external flows. The capacity of the modified SPH model to predict velocity distributions and water surface behavior is fully examined with a focus on the flow conditions at the interfacial boundary between the overlying free flow and the underlying porous media.     

剧变截面圆管内渗流的数值计算方法

对于剧变截面圆管的渗流问题写出不可压缩渗流的基本方程组,对直接求解原始变量(速度和压力)的数值计算方法作出改进。先由非主流方向的运动方程计算压力,后由主流方向的运动方程计算主流方向的速度分量,再由连续性方程计算非主流方向的速度分量。这样可以避免在一般的求解原始变量方法中由连续性方程计算压力时出现的困难和麻烦。根据本方法和剧变截面圆管的特点,采用半交错不等距非正交贴体混合网格系。本文详细写出差分方程和迭代计算公式,对剧变截面圆管内的渗流算例进行数值计算。本方法的优点是简单和实用,在工程上具有较大的应用价值。     

The effective homogeneous behavior of heterogeneous porous media

The macroscopic equations that govern the processes of one- and two-phase flow through heterogeneous porous media are derived by using the method of multiple scales. The resulting equations are mathematically similar to the point equations, with the fundamental difference that the local permeabilities are replaced by effective parameters. The method allows the determination of these parameters from a knowledge of the geometrical structure of the medium and its heterogeneities. The technique is applied to determine the effective parameters for one- and two-phase flows through heterogeneous porous media made up of two homogeneous porous media.     

聚合物流体渗流机理研究

聚合物流体在多孔介质中渗流的研究是近年来有重大进展的领域。本文介绍从力学与物理方法进行渗流机理研究的思路、主要结果和当前活跃的研究课题。流体的非牛顿性对复杂边界条件下均匀流体力学效应的影响已得到了较好的定量处理;揭示了拉伸流粘弹特性对渗流影响的机理,其定量描述则尚有待努力。进而讨论了石油工程中十分重要的非均一流体渗流的新进展,包括大分子效应与粘性指进效应及其分形描述。对于上述物理效应的综合考虑将使聚合物渗流力学研究进入新的阶段。