Two-phase flow in heterogeneous porous media: The method of large-scale averaging

The analysis of two-phase flow in porous media begins with the Stokes equations and an appropriate set of boundary conditions. Local volume averaging can then be used to produce the well known extension of Darcy's law for two-phase flow. In addition, a method of closure exists that can be used to predict the individual permeability tensors for each phase. For a heterogeneous porous medium, the local volume average closure problem becomes exceedingly complex and an alternate theoretical resolution of the problem is necessary. This is provided by the method of large-scale averaging which is used to average the Darcy-scale equations over a region that is large compared to the length scale of the heterogeneities. In this paper we present the derivation of the large-scale averaged continuity and momentum equations, and we develop a method of closure that can be used to predict the large-scale permeability tensors and the large-scale capillary pressure. The closure problem is limited by the principle of local mechanical equilibrium. This means that the local fluid distribution is determined by capillary pressure-saturation relations and is not constrained by the solution of an evolutionary transport equation. Special attention is given to the fact that both fluids can be trapped in regions where the saturation is equal to the irreducible saturation, in addition to being trapped in regions where the saturation is greater than the irreducible saturation. Theoretical results are given for stratified porous media and a two-dimensional model for a heterogeneous porous medium.     

Impact of Non-uniform Properties on Governing Equations for Fluid Flows in Porous Media

The macroscopic governing equations of a compressible multicomponents flow with non-uniform viscosity and with mass withdrawal (due to heterogeneous reactions) in a porous medium are developed. The method of volume averaging was used to transform local (or microscopic) governing equations into averaged (or macroscopic) governing equations. The impacts of compressibility, non-uniform viscosity, and mass withdrawal on the form of the averaged equations and on the value of the macroscopic transport coefficients were investigated. The results showed that the averaged mass conservation equation is significantly affected by mass withdrawal when a specific criterion on the size of the domain is respected. The results also showed that the form of the averaged momentum equations is not affected by mass withdrawal, by compressibility effects or by non-uniform viscosity, provided that the Reynolds number at the pore level is small. Nonetheless, the velocity field is affected by the heterogeneous reaction via the averaged mass conservation equation, and also by viscosity variations due to the presence of the volume-averaged viscosity (which value changes with position) in the averaged momentum equations. A new closure variable definition was proposed to formulate the closure problem, which avoided the need to solve an integro-differential equation in the closure problem. This formulation was used to show that the permeability tensor only depends on the geometry of the porous medium. In other words, that tensor is independent on whether the fluid is compressible/incompressible, has uniform/non-uniform viscosities, and whether mass withdrawal due to heterogeneous reactions is present/absent.     

Combined free and forced convection in a porous medium between two vertical walls with viscous dissipation

Viscous dissipation effects in the problem of a fully-developed combined free and forced convection flow between two symmetrically and asymmetrically heated vertical parallel walls filled with a porous medium is analyzed. The equation of motion contains the modified Rayleigh number for a porous medium and the small-order viscous dissipation parameter. Particular attention is given to the solutions near the critical Rayleigh numbers at which infinite flow rates are predicted. Information concerning the multiplicity of solutions at critical Rayleigh numbers is also deduced from perturbation solutions of the governing equation.     

L 1 Convergence to the Barenblatt Solution for Compressible Euler Equations with Damping

We study the asymptotic behavior of compressible isentropic flow through a porous medium when the initial mass is finite. The model system is the compressible Euler equation with frictional damping. As t ?? ??, the density is conjectured to obey the well-known porous medium equation and the momentum is expected to be formulated by Darcy??s law. In this paper, we prove that any L ^?? weak entropy solution to the Cauchy problem of damped Euler equations with finite initial mass converges strongly in the natural L ¹ topology with decay rates to the Barenblatt profile of the porous medium equation. The density function tends to the Barenblatt solution of the porous medium equation while the momentum is described by Darcy??s law. The results are achieved through a comprehensive entropy analysis, capturing the dissipative character of the problem.     

Thermal nonequilibrium,non-darcian forced convection in a channel filled with a fluid saturated porous medium—A perturbation solution

This paper concentrates on the analysis of the thermal nonequilibrium effects during forced convection in a parallel-plate channel filled with a fluid saturated porous medium. The flow in a channel is described by the Brinkman-Forchheimer-extended Darcy equation and the thermal nonequilibrium effects are accounted for by utilizing the two energy equations model. Applying the perturbation technique, an analytical solution of the problem is obtained. It is established that the temperature difference between the fluid and solid phases for the steady fully developed flow is proportional to the ratio of the flow velocity to the mean velocity. This results in a local thermal equilibrium at the walls of the channel if the Brinkman term which allows for the no-slip boundary condition at the walls is included into the momentum equation.     

Mathematical modelling of flow through consolidated isotropic porous media

A new mathematical model is proposed for time-independent laminar flow through a rigid isotropic and consolidated porous medium of spatially varying porosity. The model is based upon volumetric averaging concepts. Explicit assumptions regarding the mean geometric properties of the porous microstructure lead to a relationship between tortuosity and porosity. Microscopic inertial effects are introduced through consideration of flow development within the pores. A momentum transport equation is derived in terms of the fluid properties, the porous medium porosity and a characteristic length of the microstructure. In the limiting cases of porosity unity and zero, the model yields the required Navier-Stokes equation for free flow and no flow in a solid, respectively.     

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.     

Progress in the extension of a second-moment closure for turbulent environmental flows

An advanced second-moment closure for the double-averaged turbulence equations of porous medium and vegetation flows is proposed. It treats three kinds of second moments which appear in the double-averaged momentum equation. They are the dispersive covariance, the volume averaged (total) Reynolds stress and the micro-scale Reynolds stress. The two-component-limit pressure–strain correlation model is applied to model the total Reynolds stress equation whilst a novel scale-similarity non-linear $k''–''\epsilon$ two-equation eddy viscosity model is employed for the micro-scale turbulence. For the dispersive covariance, an algebraic relation is applied. Model validation in several fully developed homogeneous porous medium flows, porous channel flows and aquatic vegetation canopy flows is performed with satisfactory agreement with the data.     

Radiation effects on combined convection over a vertical flat plate embedded in a porous medium of variable porosity

The present paper is concerned with the study of radiation effects on the combined (forced-free) convection flow of an optically dense viscous incompressible fluid over a vertical surface embedded in a fluid saturated porous medium of variable porosity with heat generation or absorption. The effects of radiation heat transfer from a porous wall on convection flow are very important in high temperature processes. The inclusion of radiation effects in the energy equation leads to a highly non-linear partial differential equations which are transformed to a system of ordinary differential equations using non-similarity transformation. These equations are then solved numerically using implicit finite-difference method subject to appropriate boundary and matching conditions. A parametric study of the physical parameters such as the particle diameter-based Reynolds number, the flow based Reynolds number, the Grashof number, the heat generation or absorption co-efficient and radiation parameter is conducted on temperature distribution. The effects of radiation and other physical parameters on the local skin friction and on local Nusselt number are shown graphically. It is interesting to observe that the momentum and thermal boundary layer thickness increases with the radiation and decrease with increase in the Prandtl number.     

Asymptotic Research of Nonlinear Wave Processes in Saturated Porous Media

The problem of nonlinear wave dynamics of a fluid-saturated porous medium is investigated. The mathematical model proposed is based on the classical Frenkel--Biot--Nikolaevskiy theory concerning elastic wave propagation and includes mass, momentum, energy conservation laws, as well as rheological and thermodynamic relations. The model describes nonlinear, dispersive, and dissipative medium. To solve the system of differential equations, an asymptotic modified two-scales method is developed and a Cauchy problem for initial equations system is transformed to a Cauchy problem for nonlinear generalized Korteweg--de Vries--Burgers equation for modulated quick wave amplitudes and an inhomogeneous set of equations for slow background motion. Stationary solutions of the derived evolutionary equation that have been constructed numerically reflect different regimes of elastic wave attenuation: diffusive, oscillating, and soliton-like.     

Averaged Momentum Equation for Flow Through a Nonhomogenenous Porous Structure

This paper addresses the derivation of the macroscopic momentum equation for flow through a nonhomogeneous porous matrix, with reference to dendritic structures characterized by evolving heterogeneities. A weighted averaging procedure, applied to the local Stokes' equations, shows that the heterogeneous form of the Darcy's law explicitly involves the porosity gradients. These extra terms have to be considered under particular conditions, depending on the rate of geometry variations. In these cases, the local closure problem becomes extremely complex and the full solution is still out of reach. Using a simplified two-phase system with continuous porosity variations, we numerically analyze the limits where the usual closure problem can be retained to estimate the permeability of the structure.     

The Forchheimer equation: A theoretical development

In this paper we illustrate how the method of volume averaging can be used to derive Darcy's law with the Forchheimer correction for homogeneous porous media. Beginning with the Navier-Stokes equations, we find the volume averaged momentum equation to be given by $$\langle v_\beta \rangle = - \frac{K}{{\mu _\beta }} \cdot (\nabla \langle p_\beta \rangle ^\beta - \rho _\beta g) - F\cdot \langle v_\beta \rangle .$$ The Darcy's law permeability tensor, K, and the Forchheimer correction tensor, F, are determined by closure problems that must be solved using a spatially periodic model of a porous medium. When the Reynolds number is small compared to one, the closure problem can be used to prove that F is a linear function of the velocity, and order of magnitude analysis suggests that this linear dependence may persist for a wide range of Reynolds numbers.     

微纳米结构壁面对流场及动压润滑的影响研究

固体边界具有的微纳米结构将影响流体在近壁面处的流动行为,进而由于尺度效应改变流体在整个微间隙的流动或润滑规律.将壁面可渗透微纳米结构等效为多孔介质薄膜,采用Brinkman方程来描述流体在近壁面边界渗透层内的流动,并将其与自由流动区域的不可压缩流体Navier-Stokes控制方程耦合,在界面处的连续边界条件下求解和分析了速度分布规律和压力变化规律.针对恒定法向承载力的油膜润滑条件,进一步讨论了静止表面或运动表面的微纳米结构对近壁面流动行为的影响;并揭示了考虑壁面微纳米结构的流体动压润滑的油膜厚度和摩擦系数的变化规律.论文结果为具有可渗透微结构表面的微间隙流动与润滑提供了理论参考.     

Flow through isotropic granular porous media

A generalization of the Navier-Stokes equation is developed to include laminar flow through a rigid isotropic granular porous medium of spatially varying permeability. The model is based on a theory of interspersed continua and the mean geometrical properties of an idealized granular porous microstructure. The derived momentum transport equations are applicable to granular porous media over the entire porosity range from zero through unity. No restriction with respect to flow velocity is imposed, except for the assumption of laminar flow within the pores. The results provide useful and versatile equations and substantiate many of the empirical equations currently in use. One of the major advantages of the generalized momentum equation is its adaptability to numerical simulation.     

Dependency of Tortuosity and Permeability of Porous Media on Directional Distribution of Pore Voids

Single-phase fluid flow in porous media is usually direction dependent owing to the tortuosity associated with the internal structures of materials that exhibit inherent anisotropy. This article presents an approach to determine the tortuosity and permeability of porous materials using a structural measure quantifying the anisotropic distribution of pore voids. The approach uses a volume averaging method through which the macroscopic tortuosity tensor is related to both the average porosity and the directional distribution of pore spaces. The permeability tensor is derived from the macroscopic momentum balance equation of fluid in a porous medium and expressed as a function of the tortuosity tensor and the internal structure of the material. The analytical results generally agree with experimental data in the literature.     

Computation of Jump Coefficients for Momentum Transfer Between a Porous Medium and a Fluid Using a Closed Generalized Transfer Equation

The momentum transfer between a homogeneous fluid and a porous medium in a system analogous to the one used by Beavers and Joseph (J Fluid Mech 30:197–207, 1967) is studied using volume averaging techniques. In this article, we present a closed generalized momentum transport equation (GTE) that is valid everywhere and is expressed in terms of position-dependent effective transport coefficients, which are computed from the solution of associated closure problems previously reported. A combination of the velocity profiles from the GTE in the definition of the excess terms that define the jump coefficients allows their computation using numerical techniques. The calculations are in concordance with those resulting from the work of Goyeau et al. (Int J Heat Mass Transf. 46:4071–4081, 2003), showing a strong dependence with the porosity. In addition, the effects of the roughness of the boundary on the computation of the position-dependent permeability tensor in the inter-region are also analyzed.     

On the theory of one-dimensional unsteady motions of an ideal compressible fluid

A transformation is constructed of the independent variables and the unknown functions for the momentum and continuity equations of which one-dimensional unsteady motions of a perfect gas, relative to which the governing system of equations is invariant.When this transformation is used, the governing equation of state of the gas is transformed into a new equation which contains arbitrary parameters. This may enable approximation of the complex equation of state of a given medium to be carried out by selection of the parameters (in particular, for gases with respect of the equilibrium reactions taking place therein), and the use of this transformation may make it possible to reduce the problem to one with a simpler equation of state, for which the corresponding problem is more easily solved.The transformations investigated do not have singularities and do not impose any significant limitations on the hydrodynamic quantitiesthey are applicable both for variable entropy and for flows with shock waves.     

L'équation de Navier–Stokes avec mémoire et le transfert de quantité de mouvement dans un milieu poreuxThe Navier–Stokes equation with memory,and momentum transfer in a porous medium

This paper is concerned with the momentum transfer in a porous medium. The equations for the continuous equivalent medium are written by the averaging method but the closure is obtained using an extended thermodynamics. The resulting model corresponds to the Navier–Stokes equation, in which the force exerted by the solid matrix on the fluid satisfies of first order differential equation. This Navier–Stokes equation model with memory generalises the Darcy model with an integro-differential term. To cite this article: O. Séro-Guillaume, D. Calogine, C. R. Mecanique 330 (2002) 383–389.