Three-phase flow and gravity drainage in porous media

We present a theoretical and experimental treatment of three-phase flow in water-wet porous media from the molecular level upwards. Many three-phase systems in polluted soil and oil reservoirs have a positive initial spreading coefficient, which means that oil spontaneously spreads as a layer between water and gas. We compute the thickness and stability of this oil layer and show that appreciable recovery of oil by drainage only occurs when the oil layer occupies crevices or roughness in the pore space. We then analyze the distribution of oil, water and gas in vertical equilibrium for a spreading system, which is governed byα=γ _ow (ρ _o ?ρ _g )/γ _go (ρ _w ?ρ _o ), whereγ _ow andγ _go are the oil/water and gas/oil interfacial tensions respectively, andρ _g ,ρ _o andρ _w are the gas, oil and water densities respectively. Ifα>1, there is a height above the oil/water contact, beyond which connected oil only exists as a molecular film, with a negligible saturation. This height is independent of the structure of the porous medium. Whenα<1, large quantities of oil remain in the pore space and gravity drainage is not efficient. If the initial spreading coefficient is negative, oil can be trapped and the recovery is also poor. We performed gravity drainage experiments in sand columns and capillary tubes which confirmed our predictions.     

Effective elastic properties of porous materials: Homogenization schemes vs experimental data

In this paper, we focus on the prediction of elastic moduli of isotropic porous materials made of a solid matrix having a Poisson's ratio vm of 0.2. We derive simple analytical formulae for these effective moduli based on well-known Mean-Field Eshelby-based Homogenization schemes. For each scheme, we find that the normalized bulk, shear and Young's moduli are given by the same form depending only on the porosity p. The various predictions are then confronted with experimental results for the Young's modulus of expanded polystyrene (EPS) concrete. The latter can be seen as an idealized porous material since it is made of a bulk cement matrix, with Poisson's ratio 0.2, containing spherical mono dispersed EPS beads. The Differential method predictions are found to give a very good agreement with experimental results. Thus, we conclude that when vm=0.2, the normalized effective bulk, shear and Young's modulus of isotropic porous materials can be well predicted by the simple form (1 − p)² for a large range of porosity p ranging between 0 and 0.56.     

Mixing and Mass Transfer in Multicontact Miscible Displacements

In this work, we investigate the accuracy of some physical models that are frequently used to describe and interpret dispersive mixing and mass transfer in compositional reservoir simulation. We have designed a quaternary analog fluid system (alcohol?Cwater?Chydrocarbon) that mimics the phase behavior of CO₂-hydrocarbon mixtures at high pressure and temperature. A porous medium was designed using PolyTetraFlouroEthylene (PTFE) materials to ensure that the analog oil acts as the wetting phase, and the properties of the porous medium were characterized in terms of porosity, permeability and dispersivity. Relative permeability and interfacial tension (IFT) measurements were also performed to delineate interactions between the fluid system and the porous medium. The effluent concentrations from two-component first-contact miscible (FCM) displacement experiments exhibit a tailing behavior that is attributed to imperfect sweep of the porous medium: A feature that is not captured by normal dispersion models. To represent this behavior in displacement calculations, we use dual-porosity (DP) models including mass transfer between flowing and stagnant porosities. Two 4-component two-phase displacement experiments were performed at near-miscible and multicontact miscible (MCM) conditions and the effluent concentrations were interpreted by numerical calculations. We demonstrate that the accuracy of our displacement calculations relative to the experimental observations is sensitive to the selected models for dispersive mixing, mass transfer between flowing and stagnant porosities, and IFT scaling of relative permeability functions. We also demonstrate that numerical calculations substantially agree with the experimental observations for some physical models with limited need for model parameter adjustment. The combined experimental and modeling effort presented in this work identifies and explores the impact of a set of physical mechanisms (dispersion and mass transfer) that must be upscaled adequately for field-scale displacement calculations in DP systems.     

Effects of Material Symmetry on the Coefficients of Transport in Anisotropic Porous Media

The objective of this article is to highlight certain features of a number of coefficients that appear in models of phenomena of transport in anisotropic porous media, especially the coefficient of dispersion the second-rank tensor D _ij , and the dispersivity coefficient, the fourth-rank tensor a _ijkl , that appear in models of solute transport. Although we shall focus on the transport of mass of a dissolved chemical species in a fluid phase that occupies the void space, or part of it, the same discussion is also applicable to transport coefficients that appear in models that describe the advective mass flux of a fluid and the diffusive transport of other extensive quantities, like heat. The case of coupled processes, e.g. the simultaneous transport of heat and mass of a chemical species, are also considered. The entire discussion will be at the macroscopic level, at which a porous medium domain is visualized as a homogenized continuum.     

Evaporation From Confined Porous Media Due to Controlled IR Heating From Above

We report an experimental study of evaporation due to controlled infrared (IR) heating from above from an initially saturated confined porous medium consisting of nearly ‘mono-disperse’ particles which has been rarely used earlier. We have used three diagnostic tools simultaneously, evaporation rate measurements using a precision weighing balance, surface temperature measurements using IR imaging, and fluorescein dye mixed with water to visualize the drying front and the evaporation sites. IR images show that the first stage, so-called constant rate period (CRP), was maintained due to films of water reaching the top surface from the saturated region below. Gradually reducing evaporation rate in stage 1 is shown to be related to ‘shrinking evaporating patches’ on the top surface, clearly revealed as lower-temperature regions in the IR images. End of CRP coincides with disappearance of the low-temperature patches. We give end of CRP in terms of the average depth (L_cap) of the liquid level from the top surface at that time. L_cap and duration of CRP are strong functions of the porous medium bead size, transition to stage 2 happening earlier for coarser spheres. The obtained L_cap values deviated from the predictions of Lehmann et al. (Phys Rev E 77(5):056309, 2008) which we show is due to a small range of pore sizes in the current experiments. For both water and highly volatile n-pentane, we show that L_cap normalized by a length scale derived from gravity-surface tension force balance goes like Bo^0.20, for Bo varying from 2.0E ? 04 to 1.0E ? 01; Bo is the Bond number. Fluorescein dye imaging shows a different view of the evaporation stages. During CRP, highly concentrated deposits of the fluorescein dye particles, orange in colour, are seen in the top few bead layers. These orange deposits represent the sites on the beads surfaces where the evaporation has taken place. Even with external heating, evaporation from such a porous medium is limited to a finite depth from the evaporating end, similar to the observation by Lehmann et al. (2008) for isothermal evaporation in Hele-Shaw cell.     

A generalized approach to formulate the consistent tangent stiffness in plasticity with application to the GLD porous material model

It has been shown that the use of the consistent tangent moduli is crucial for preserving the quadratic convergence rate of the global Newton iterations in the solution of the incremental problem. In this paper, we present a general method to formulate the consistent tangent stiffness for plasticity. The robustness and efficiency of the proposed approach are examined by applying it to the isotropic material with J₂ flow plasticity and comparing the performance and the analysis results with the original implementation in the commercial finite element program ABAQUS. The proposed approach is then applied to an anisotropic porous plasticity model, the Gologanu–Leblond–Devaux model. Performance comparison between the consistent tangent stiffness and the conventional continuum tangent stiffness demonstrates significant improvement in convergence characteristics of the overall Newton iterations caused by using the consistent tangent matrix.     

Effects of anisotropy in permeability on the two-phase flow and heat transfer in a porous cavity

This paper reports on the results of a numerical study of convection flow and heat transfer in a rectangular porous cavity filled with a phase change material under steady state conditions. The two vertical walls of the cavity are subject respectively to temperatures below and above the melting point of the PCM while adiabatic conditions are imposed on the horizontal walls. The porous medium is characterized by an anisotropic permeability tensor with the principal axes arbitrarily oriented with respect to the gravity vector. The problem is governed by the aspect ratioA, the Rayleigh numberRa, the anisotropy ratioR and the orientation angle θ of the permeability tensor. Attention is focused on these two latter parameters in order to investigate the effects of the anisotropic permeability on the fluid flow and heat transfer of the liquid/solid phase change process. The method of solution is based on the control volume approach in conjunction with the Landau-transformation to map the irregular flow domain into a rectangular one. The results are obtained for the flow field, temperature distribution, interface position and heat transfer rate forA=2.5,Ra=40, 0≤θ≤π, 0.25≤R≤4. It was found that the equilibrium state of the solid/liquid phase change process may be strongly influenced by the anisotropy ratioR as well as by the orientation angle θ of the permeability tensor. First, for a given set of parametersA,Ra andR, there exists an optimum orientation θ_max for which the flow strength, the liquid volume and the heat transfer rate are maximum. There also exists an orientation θ_min=θ_max+π/2 for which these quantities are minimum. Second, when an anisotropic medium is oriented along the optimum direction θ_max, an increase of the permeability component along that direction will increase the flow and heat transfer rate in a same order while an increase of the other permeability component only has a negligible effect. For the parameter ranges considered in the present study, it was found that the optimum direction is lying between the gravity vector and the dominant flow direction.     

Extension of the bifurcation method for diffusion coefficients to porous medium transport

We present an extension of Bartlett's bifurcation method for the approximate computation of multicomponent diffusion coefficients in a gaseous mixture to diffusion in porous media. On behalf of the remark that the bifurcation coefficients F_i are merely proportional to the square root of the molar masses M_i, we state that Knudsen diffusion may also be represented through some bifurcation factor F_K. This approximation is tested in a variety of cases, displaying good results except for very light gas species.     

Homogénéisation d'un milieu élastique fortement hétérogèneHomogenization of a strongly heteregeneous elastic medium

We consider the homogenization of an elastostatic problem in a strongly heterogeneous periodic medium made of two connected components having comparable tensors of elastic moduli, separated by a third medium (soft layer), the thickness of which is of the same order ε than the basic periodicity cell, and such that its elastic moduli tensor becomes infinitely small following a rate ε^r, r>0. If r?2, we identify the homogenized problem. Otherwise, we have to assume moreover that there are no volume forces in the third medium. To cite this article: M. Mabrouk, A. Boughammoura, C. R. Mecanique 330 (2002) 543–548.     

Anisotropic deformation of thin films comprised of helical nanosprings

We investigate the mechanical anisotropy of thin films that consist of tantalum oxide (Ta₂O₅) helical nanosprings fabricated by dynamic oblique deposition. Not only the vertical but also the lateral stiffness of thin films is evaluated using specimens in which nanosprings are sandwiched between solid Ta₂O₅ layers. Lateral or vertical force is applied to the upper solid layer by a diamond tip built into an AFM. In particular, the lateral stiffness of a nanospring has never been reported before. Apparent shear and Young’s moduli, G′ and E′, of the thin films are 2–3 orders smaller than those of solid Ta₂O₅ film. Ratio E′/G′ of the two different nanosprings is 3.4 and 6.2, and about 2.5 for the solid film. The thin films show strong characteristic anisotropy that the solid one could hardly attain. The stiffness and its anisotropy strongly depend on nanospring shape.     

Energy balance and identification of hardening moduli in plastic deformation processes

The hardening moduli H_r and H_d of plastic deformation associated with the free energy and dissipation function in a representative material element are defined analytically and specified experimentally for three materials. Besides the stress–strain curve and work expended during the deformation process, variation of the hardening moduli with plastic deformation is also determined for austenitic steel, austenitic-ferritic steel and Fe–Si alloy.     

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.     

Radiation effect on Darcy free convection flow along an inclined surface placed in porous media

The paper investigates the effect of radiation on Darcy's buoyancy induced flow of an optically dense viscous incompressible fluid along a heated inclined flat surface maintained at uniform temperature placed in a saturated porous medium with Rosseland diffusion approximation employing the implicit finite difference method together with Keller box elimination technique. Both the streamwise and normal components of the buoyancy force are retained in the momentum equations. The numerical results show that as the buoyancy parameter, ξ, increases the local Nusselt number increases. The results for the locally nonsimilar solutions are compared with the locally similar solutions for small angle of inclination and approximate similar solutions along vertical surface. The effect of the conduction-radiation parameter, R _d , and the surface temperature excess ration, θ _w , on the local Nusselt number, the tangential velocity distribution and the temperature distribution are also shown graphically.     

Relative permeability and capillary pressure functions of porous media as related to the displacement growth pattern

Visualization experiments of the unsteady immiscible displacement of a fluid by another are performed on glass-etched pore networks of well-controlled morphology by varying the fluid system and flow conditions. The measured transient responses of the fluid saturation and pressure drop across the porous medium are introduced into numerical solvers of the macroscopic two-phase flow equations to estimate the non-wetting phase, k_rnw, and wetting phase, k_rw, relative permeability curves and capillary pressure, P_c, curve. The correlation of k_rnw, k_rw, and P_c with the displacement growth pattern is investigated. Except for the capillary number, wettability, and viscosity ratio, the immiscible displacement growth pattern in a porous medium may be governed by the shear-thinning rheology of the injected or displaced fluid, and the porous sample length as compared to the thickness of the frontal region. The imbibition k_rnw increases as the flow pattern changes from compact displacement to viscous fingering or from viscous to capillary fingering. The imbibition k_rw increases as the flow pattern changes from compact displacement or capillary fingering to viscous fingering. As the shear-thinning behaviour of the NWP strengthens and/or the contact angle decreases, then the flow pattern is gradually dominated by irregular interfacial configurations, and the imbibition k_rnw increases. The imbibition P_c is a decreasing function of the capillary number or increasing function of the injected phase viscosity in agreement with the linear thermodynamic theory.     

Rate form of the Eshelby and Hill tensors

Expressions are derived for the rates of change of the S and P tensors for transformed homogeneous inclusions in an anisotropic comparison medium undergoing prescribed changes of its elastic moduli. General results are obtained for ellipsoids and then reduced to yield explicit expressions in terms of the Stroh eigenvalues for cylindrical and disk-shaped inclusions in anisotropic solids and for spherical inclusions in isotropic solids. Applications are illustrated by solving the rate problem for an inhomogeneity in a large volume of a comparison medium, which is shown to be readily adaptable to standard averaging techniques for predictions of rates of change of overall moduli of composite materials experiencing evolution of phase moduli.     

Evaluation of molecular weight distribution from dynamic moduli

A method to evaluate molecular weight distribution (MWD) from dynamic moduli is presented here. It relies on the least-square fitting of the dynamic data to a model whose parameters depend on the MWD. In particular, the analytical solution for the relaxation modulus previously obtained from the double reptation model, with the Tuminello step relaxation function and the Generalized Exponential Function (GEX) describing the MWD (Nobile and Cocchini 2000), has been used. A Finite Element Approximation (FEA) has been applied to calculate dynamic moduli from the relaxation modulus as a function of MWD. The sensitiveness of the GEX-double reptation dynamic moduli on the model parameters has also been investigated and the results show that large changes of the M_w/M_n ratio weakly affect the dynamic moduli, while small changes of the M_z/M_w ratio significantly deform the dynamic moduli curves. The use of rheological data to obtain MWD, by the model used in this paper, will, therefore, be able to give rather well defined M_z/M_w ratios, while more uncertainty will be presented in the M_w/M_n results. The so-called GEX-rheological model for the dynamic moduli was applied to fit the experimental data of different polymers in order to obtain the best-fit parameters of the MWD of these polymers, without the need for the inversion of the double reptation integral equation. The stability of the results has been confirmed through the evaluation of the 90% confidence intervals for the first molecular weight averages. Finally, concerning the M_w and M_z values, the predictions obtained from the dynamic moduli measurements differ by less than 10% from those obtained from GPC measurements while, as expected, more uncertainty is present in the M_n predictions. Received: 6 February 2000 Accepted: 22 August 2000     

A computational study of some rheological influences on the “splashing experiment”

In various attempts to relate the behaviour of highly-elastic liquids in complex flows to their rheometrical behaviour, obvious candidates for study have been the variation of shear viscosity with shear rate, the two normal stress differences N₁ and N₂, especially N₁, the extensional viscosity, and the dynamic moduli G′ and G″. In this paper, we shall confine attention to ‘constant-viscosity’ Boger fluids, and, accordingly, we shall limit attention to N₁, η_E, G′ and G″.We shall concentrate on the “splashing” problem (particularly that which arises when a liquid drop falls onto the free surface of the same liquid). Modern numerical techniques are employed to provide the theoretical predictions. We show that high η_E can certainly reduce the height of the so-called Worthington jet, thus confirming earlier suggestions, but other rheometrical influences (steady and transient) can also have a role to play and the overall picture may not be as clear as it was once envisaged. We argue that this is due in the main to the fact that splashing is a manifestly unsteady flow. To confirm this proposition, we obtain numerical simulations for the linear Jeffreys model.     

Lie-group method of solution for steady two-dimensional boundary-layer stagnation-point flow towards a heated stretching sheet placed in a porous medium

The boundary-layer equations for two-dimensional steady flow of an incompressible, viscous fluid near a stagnation point at a heated stretching sheet placed in a porous medium are considered. We apply Lie-group method for determining symmetry reductions of partial differential equations. Lie-group method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system. The effect of the velocity parameter λ, which is the ratio of the external free stream velocity to the stretching surface velocity, permeability parameter of the porous medium k ₁, and Prandtl number Pr on the horizontal and transverse velocities, temperature profiles, surface heat flux and the wall shear stress, has been studied.     

The rheology of suspensions of porous zeolite particles in polymer solutions

We present data and predictive models for the shear rheology of suspended zeolite particles in polymer solutions. It was found experimentally that suspensions of zeolite particles in polymer solutions have relative viscosities that dramatically exceed the Krieger–Dougherty predictions for hard sphere suspensions. Our investigations show that the major origin of this discrepancy is due to the selective absorption of solvent molecules from the suspending polymer solution into zeolite pores. The effect raises both the polymer concentration in the suspending medium and the particle volume fraction in the suspension. Consequently, both the viscosity of the polymer solution and the particle contribution to the suspension viscosity are increased. We propose a predictive model for the viscosity of porous zeolite suspensions by incorporating a solvent absorption parameter, α, into the Krieger–Dougherty model. We experimentally determined the solvent absorption parameter by comparing viscosity data for suspensions of porous and nonporous MFI zeolite particles. Our results are in good agreement with the theoretical pore volume of MFI particles.     

Natural convection of nanofluid over vertical plate embedded in porous medium: prescribed surface heat flux

The aim of the present paper is to analyze the natural convection heat and mass transfer of nanofluids over a vertical plate embedded in a saturated Darcy porous medium subjected to surface heat and nanoparticle fluxes. To carry out the numerical solution, two steps are performed. The governing partial differential equations are firstly simplified into a set of highly coupled nonlinear ordinary differential equations by appropriate similarity variables, and then numerically solved by the finite difference method. The obtained similarity solution depends on four non-dimensional parameters, i.e., the Brownian motion parameter (N _b), the Buoyancy ratio (N _r), the thermophoresis parameter (N _t), and the Lewis number (Le). The variations of the reduced Nusselt number and the reduced Sherwood number with N _b and N _t for various values of Le and N _r are discussed in detail. Simulation results depict that the increase in N _b, N _t, or N _r decreases the reduced Nusselt number. An increase in the Lewis number increases both of the reduced Nusselt number and the Sherwood number. The results also reveal that the nanoparticle concentration boundary layer thickness is much thinner than those of the thermal and hydrodynamic boundary layers.