Numerical and experimental investigation of convective drying in unsaturated porous media with bound water

The heat and mass transfer in an unsaturated wet cylindrical porous bed packed with quartz particles was investigated theoretically for relatively low convective drying rates. Local thermodynamic equilibrium was assumed in the mathematical model describing the multi-phase flow in the unsaturated porous media using the energy and mass conservation equations to describe the heat and mass transfer during the drying. The drying model included convection and capillary transport of the free water, diffusion of bound water, and convection and diffusion of the gas. The numerical results indicated that the drying process could be divided into three periods, the temperature rise period, the constant drying rate period and the decreasing drying rate period. The numerical results agreed well with the experimental data verifying that the mathematical model can evaluate the drying performance of porous media for low drying rates. The effects of drying conditions such as the ambient temperature, the relative humidity, and the velocity of the drying air, on the drying process were evaluated by numerical solution.     

Effective equations for two-phase flow in porous media: the effect of trapping on the microscale

In this article, we consider a two-phase flow model in a heterogeneous porous column. The medium consists of many homogeneous layers that are perpendicular to the flow direction and have a periodic structure resulting in a one-dimensional flow. Trapping may occur at the interface between a coarse and a fine layer. Assuming that capillary effects caused by the surface tension are in balance with the viscous effects, we apply the homogenization approach to derive an effective (upscaled) model. Numerical experiments show a good agreement between the effective solution and the averaged solution taking into account the detailed microstructure.     

Ensemble phase averaged equations for multiphase flows in porous media. Part 2: A general theory

Most models for multiphase flows in a porous medium are based on a straightforward extension of Darcy’s law, in which each fluid phase is driven by its own pressure gradient. The pressure difference between the phases is thought to be an effect of surface tension and is called capillary pressure. Independent of Darcy’s law, for liquid imbibition processes in a porous material, diffusion models are sometime used. In this paper, an ensemble phase averaging technique for continuous multiphase flows is applied to derive averaged equations and to examine the validity of the commonly used models. Closure for the averaged equations is quite complicated for general multiphase flows in a porous material. For flows with a small ratio of the characteristic length of the phase interfaces to the macroscopic length, the closure relations can be simplified significantly by an approximation with a second order error in this length ratio. This approximation reveals the information of the length scale separation obscured during an averaging process and leads to an equation system similar to Darcy’s law, but with additional terms. Based on interactions on phase interfaces, relations among closure quantities are studied.     

Ensemble phase averaged equations for multiphase flows in porous media. Part 1: The bundle-of-tubes model

A bundle-of-tubes construct is used as a model system to study ensemble averaged equations for multiphase flow in a porous material. Momentum equations for the fluid phases obtained from the method are similar to Darcy’s law, but with additional terms. We study properties of the additional terms, and the conditions under which the averaged equations can be approximated by the diffusion model or the extended Darcy’s law as often used in models for multiphase flows in porous media. Although the bundle-of-tubes model is perhaps the simplest model for a porous material, the ensemble averaged equation technique developed in this paper assumes the very same form in more general treatments described in Part 2 of the present work (Zhang, D.Z., 2009. Ensemble Phase Averaged Equations for Multiphase Flows in Porous Media, Part 2: A General Theory. Int. J. Multiphase Flow 35, 640–649). Any model equation system intended for the more general cases must be understood and tested first using simple models. The concept of ensemble phase averaging is dissected here in physical terms, without involved mathematics through its application to the idealized bundle-of-tubes model for multiphase flow in porous media.     

Fractal porous media IV: Three-dimensional stokes flow through random media and regular fractals

The three-dimensional Stokes flow of a Newtonian fluid through random and/or fractal media is numerically determined. The permeability of these media is derived. Results relative to these structures are presented and discussed. The validity of the Carman equation and of a simple scaling argument is questioned.     

Viscosity enhancement in non-Newtonian flow of dilute aqueous polymer solutions through crystallographic and random porous media

We report results on the flow of dilute aqueous solutions of hydrolysed poly(acrylamide) (HPAA) through beds of spheres packed in simple cubic and body-centred cubic crystallographic arrays. Pressure drop measurements made across the arrays as a function of the flow rate have been used to estimate the specific viscosities of the HPAA solutions as a function of the superficial strain rate. It is found that greater non-Newtonian increases in the specific viscosity occur in the body-centred cubic array, which is thought to be due to the presence of trailing stagnation points, which are not present in the simple cubic array. Experiments have been performed using HPAA solutions in the presence of mono- and divalent cations at various concentrations and, for validity, have been compared with results obtained from a traditional randomly packed porous medium. In addition, a study of mechanical degradation of the polymer in flow through the crystallographic arrays has been carried out and reveals a greater rate of degradation in the body-centred array and also a significant increase in degradation with salt concentration.     

Fractal porous media III: Transversal Stokes flow through random and Sierpinski carpets

The transversal Stokes flow of a Newtonian fluid through random and Sierpinski carpets is numerically calculated and the transversal permeability derived. In random carpets derived from site percolation, the average macroscopic permeability varies as (- _c)^3/2, close to the critical porosity _c. This exponent is found to be slightly different from the conductivity exponent. Results for Sierpinski carpets are presented up to the fourth generation. The Carman equation is not verified in these two model porous media.     

Heat and brine transport in porous media: the Oberbeck-Boussinesq approximation revisited

This paper discusses the Oberbeck-Boussinesq approximation for heat and solute transport in porous media. In this commonly used approximation all density variations are neglected except for the gravity term in Darcy’s law. However, in the limit of vanishing density differences this gravity term disappears as well. The main purpose of this paper is to give the correct limits in which the gravity term is retained, while other density effects can be neglected. We show that for isothermal brine transport, fluid volume changes can be neglected when a condition is fulfilled for a dimensionless number, which is independent of the density difference and specific discharge. For heat transfer an additional condition is required. One-dimensional examples of simultaneous heat and brine transport are given for which similarity solutions are constructed. These examples are included to elucidate the volume effects and the corresponding induced specific discharge variations. Finally, a two-dimensional example illustrates the relative effects of volume changes and gravity.     

Numerical simulation and homogenization of two-phase flow in heterogeneous porous media

A mathematically rigorous method of homogenization is presented and used to analyze the equivalent behavior of transient flow of two incompressible fluids through heterogeneous media. Asymptotic expansions and H-convergence lead to the definition of a global or effective model of an equivalent homogeneous reservoir. Numerical computations to obtain the homogenized coefficients of the entire reservoir have been carried out via a finite element method. Numerical experiments involving the simulation of incompressible two-phase flow have been performed for each heterogeneous medium and for the homogenized medium as well as for other averaging methods. The results of the simulations are compared in terms of the transient saturation contours, production curves, and pressure distributions. Results obtained from the simulations with the homogenization method presented show good agreement with the heterogeneous simulations.     

Solutions of xanthan gum/guar gum mixtures: shear rheology,porous media flow,and solids transport in annular flow

Mixtures of xanthan and guar gum in aqueous solution were studied in two flow situations: simple shear and porous media. In addition, solids transport in vertical annular flow of sand suspensions was explored. The zero shear rate viscosity of the solutions displayed a pronounced synergy: the viscosity of the mixture is higher than that of the polymer solutions in a wide range of relative concentrations of the two polymers, in agreement with previous literature. However, at relatively high shear rates, the viscosity approaches the value of the more viscous xanthan gum solutions at mass fractions of xanthan gum between 0.1 and 0.15, and the degree of synergy substantially decreases. Stress relaxation experiments in simple shear indicate that the polymer mixtures exhibit a well-defined yield stress after relaxation that is absent in solutions of pure polymers. In porous media flow experiments, a synergistic behavior mimicking the shear flow results was obtained for the polymer mixtures at low shear rates. However, at a critical shear rate, the apparent viscosity in porous media flows exceeds the shear viscosity due to the elongational nature of flow in the pores. The solids transport capacity in annular flows is well-represented by trends in shear viscosity and stress relaxation behavior. However, the lack of viscosity synergy at high shear rates limits the applicability of the mixtures as a way to improve solids suspension capacity in annular flows.     

Two-phase flow in heterogeneous porous media III: Laboratory experiments for flow parallel to a stratified system

Two-phase flow in stratified porous media is a problem of central importance in the study of oil recovery processes. In general, these flows are parallel to the stratifications, and it is this type of flow that we have investigated experimentally and theoretically in this study. The experiments were performed with a two-layer model of a stratified porous medium. The individual strata were composed of Aerolith-10, an artificial: sintered porous medium, and Berea sandstone, a natural porous medium reputed to be relatively homogeneous. Waterflooding experiments were performed in which the saturation field was measured by gamma-ray absorption. Data were obtained at 150 points distributed evenly over a flow domain of 0.1 × 0.6 m. The slabs of Aerolith-10 and Berea sandstone were of equal thickness, i.e. 5 centimeters thick. An intensive experimental study was carried out in order to accurately characterize the individual strata; however, this effort was hampered by both local heterogeneities and large-scale heterogeneities.The theoretical analysis of the waterflooding experiments was based on the method of large-scale averaging and the large-scale closure problem. The latter provides a precise method of discussing the crossflow phenomena, and it illustrates exactly how the crossflow influences the theoretical prediction of the large-scale permeability tensor. The theoretical analysis was restricted to the quasi-static theory of Quintard and Whitaker (1988), however, the dynamic effects described in Part I (Quintard and Whitaker 1990a) are discussed in terms of their influence on the crossflow.Roman Letters A interfacial area between the -region and the -region contained within V, m² - a vector that maps onto , m - b vector that maps onto , m - b vector that maps onto , m - B second order tensor that maps onto , m² - C second order tensor that maps onto , m² - E energy of the gamma emitter, keV - f fractional flow of the -phase - g gravitational vector, m/s² - h characteristic length of the large-scale averaging volume, m - H height of the stratified porous medium , m - i unit base vector in the x-direction - K local volume-averaged single-phase permeability, m² - K - {K}, large-scale spatial deviation permeability - { K} large-scale volume-averaged single-phase permeability, m² - K ^* large-scale single-phase permeability, m² - K ^** equivalent large-scale single-phase permeability, m² - K local volume-averaged -phase permeability in the -region, m² - K local volume-averaged -phase permeability in the -region, m² - K - {K } , large-scale spatial deviation for the -phase permeability, m² - K ^* large-scale permeability for the -phase, m² - l thickness of the porous medium, m - l characteristic length for the -region, m - l characteristic length for the -region, m - L length of the experimental porous medium, m - characteristic length for large-scale averaged quantities, m - n outward unit normal vector for the -region - n outward unit normal vector for the -region - n unit normal vector pointing from the -region toward the -region (n = - n ) - N number of photons - p pressure in the -phase, N/m² - p ₀ reference pressure in the -phase, N/m² - local volume-averaged intrinsic phase average pressure in the -phase, N/m² - large-scale volume-averaged pressure of the -phase, N/m² - large-scale intrinsic phase average pressure in the capillary region of the -phase, N/m² - - , large-scale spatial deviation for the -phase pressure, N/m² - p_c , capillary pressure, N/m² - p _c capillary pressure in the -region, N/m² - p capillary pressure in the -region, N/m² - {p _c } ^c large-scale capillary pressure, N/m² - q -phase velocity at the entrance of the porous medium, m/s - q -phase velocity at the entrance of the porous medium, m/s - S_wi irreducible water saturation - S /, local volume-averaged saturation for the -phase - S _i initial saturation for the -phase - S _r residual saturation for the -phase - S ^* { }^*/}^*, large-scale average saturation for the -phase - S saturation for the -phase in the -region - S saturation for the -phase in the -region - t time, s - v -phase velocity vector, m/s - v local volume-averaged phase average velocity for the -phase, m/s - {v } large-scale averaged velocity for the -phase, m/s - v local volume-averaged phase average velocity for the -phase in the -region, m/s - v local volume-averaged phase average velocity for the -phase in the -region, m/s - v -{v } , large-scale spatial deviation for the -phase velocity, m/s - v -{v } , large-scale spatial deviation for the -phase velocity in the -region, m/s - v -{v } , large-scale spatial deviation for the -phase velocity in the -region, m/s - V large-scale averaging volume, m³ - y position vector relative to the centroid of the large-scale averaging volume, m - {y}^c large-scale average of y over the capillary region, m Greek Letters local porosity - local porosity in the -region - local porosity in the -region - local volume fraction for the -phase - local volume fraction for the -phase in the -region - local volume fraction for the -phase in the -region - {}^* { }^*+{ }^*, large-scale spatial average volume fraction - { }^* large-scale spatial average volume fraction for the -phase - mass density of the -phase, kg/m³ - mass density of the -phase, kg/m³ - viscosity of the -phase, N s/m² - viscosity of the -phase, Ns/m² - V /V , volume fraction of the -region ( + =1) - V /V , volume fraction of the -region ( + =1) - attenuation coefficient to gamma-rays, m^-1 - -      

Fluid flow in fractured and fracturing porous media: A unified view

Fluid flow in fractures that pre-exist or propagate in a porous medium can have a major influence on the deformation and flow characteristics. With the aim of carrying out large-scale calculations at reasonable computing costs, a sub-grid scale model has been developed. While this model was originally embedded in extended finite element methods, thereby exploiting some special properties of the enrichment functions, we will herein show that, using proper micro–macro relations, in particular for the mass balance, sub-grid scale models can be coupled to a range of discretisation methods at the macroscopic scale, from standard interface elements to isogeometric finite element analysis.     

渗流气体滑脱现象与渗透率变化的关系

气体在致密多孔介质中低速渗流时存在着因气体分子碰撞岩壁而引起的滑脱现象，它由介质的孔隙结构和气体分子的平均自由程共同决定。该现象使气测渗透率大于孔隙介质的绝对渗透率。介质中气体的低速渗流为黏滞流与滑脱流组成，各自所占比例与气体分子按自由程的分布有关。理论计算得到了低速气体渗流的气测渗透率Kg与绝对渗透率K0比值的关系式，实验结果与理论分析吻合。     

Gas cluster growth by solute diffusion in porous media. Experiments and automaton simulation on pore network

The study presented in this paper deals with the liquid–gas phase change by pressure decline of supersaturated CO₂ solutions in 2D porous media. The growth of the gas phase is studied experimentally and numerically as a function of supersaturation, wettability and gravity. Experiments are performed on a transparent etched network (micromodel) and simulations with a specific numerical automaton.In the experiments, the nucleation process, i.e. the occurrence of the gas bubbles, as well as the growth of these bubbles are visualised and analysed by means of a micro video camera and an image processing apparatus. The observations confirm the heterogeneous nature of nucleation and the disordered growth pattern of the gas phase. The analysis of the growth rate of a single gas cluster shows that this phenomenon is different from the compact growth of an isolated single bubble in the bulk. As previously predicted, the bubble growth by mass transfer and volume expansion in porous media is characterised by a pattern of the invasion percolation type under normal laboratory conditions.Numerical simulations of the growth pattern and the growth rate of a single gas cluster are performed with a numerical automaton. Based on a pore network modelling technique and a set of hypotheses derived from the observations, this automaton is first validated by comparing the numerical results with the experiments. Then, the automaton is used to conduct a sensitivity study. In particular, the influences of the Jakob number, pressure decline rate, Bond number, wettability and characteristics of the microstructure are investigated.     

高温高压气体在低含水率介质中迁移的数值模拟

针对气液两相非等温渗流模型高度非线性的特点,发展了适宜的数值离散方法。根据相态转换准则和控制方程的性质,采用最低饱和度法简化算法。空间离散方面,使用有限体积法;时间离散方面,设计了一套包含合理求解顺序的Picard迭代法,解决了方程组强耦合的问题。利用上述数值方法对高温高压气体的迁移行为进行数值模拟,证明了气体在低含水率介质和等效孔隙度的干燥介质内的运动基本一致,并分析了空腔内的气液相态转变过程。在此基础上,研究了多孔介质孔隙度和渗透率对气体压强演化和示踪气体迁移的影响。研究表明,孔隙度越小（相同渗透率）、渗透率越高（相同孔隙度）,示踪气体的迁移距离越远,并给出了估算不同孔隙度和渗透率下迁移距离的半经验公式。     

Analysis of hydrodynamic and thermal dispersion in porous media by means of a local approach

A pore scale analysis is implemented in this numerical study to investigate the behavior of microscopic inertia and thermal dispersion in a porous medium with a periodic structure. The macroscopic characteristics of the transport phenomena are evaluated with an averaging technique of the controlling variables at a pore scale level in an elementary cell of the porous structure. The Darcy–Forchheimer model describes the fluid motion through the porous medium while the continuity and Navier–Stokes equations are applied within the unit cell. An average energy equation is employed for the thermal part of the porous medium. The macroscopic pressure loss is computed in order to evaluate the dominant microscopic inertial effects. Local fluctuations of velocity and temperature at the pore scale are instrumental in the quantification of the thermal dispersion through the total effective thermal diffusivity. The numerical results demonstrate that microscopic inertia contributes significantly to the magnitude of the macroscopic pressure loss, in some instances with as much as 70%. Depending on the nature of the porous medium, the thermal dispersion may have a marked bearing on the heat transfer, particularly in the streamwise direction for a highly conducting fluid and certain values of the Peclet number.     

Numerical study of steady/unsteady flow and heat transfer in porous media using a characteristics-based matrix-free implicit FV method on unstructured grids

In this study, a high-resolution characteristic-based finite-volume (FV) method on unstructured grids [Int. J. Numer. Method Eng. 50 (2001) 11; Int. J. Heat Fluid Flow 21 (2000) 432] is extended by a matrix-free implicit dual-time stepping scheme for the numerical simulation of steady and unsteady flow and heat transfer with porous media. The method has been used to study the characteristics of a complex problem: flow and heat transfer in a channel with multiple discrete porous blocks, which was originally proposed by Huang and Vafai [J. Thermophys. Heat Transfer 8 (3) (1994) 563]. In addition, flow and heat transfer in a channel partially or fully filled with porous layers and containing solid protruding blocks with constant heat flux on its lower surface are also investigated in details. Hydrodynamic and heat transfer results are reported for both steady and transient flow cases. In particular, the effects of Darcy and Reynolds numbers on heat transfer augmentation and pressure loss are studied. An in-depth discussion of the formation and variation of recirculation is presented and the existence of optimum porous insert is demonstrated. At high Reynolds numbers the flow in the porous channel exhibits a cyclic characteristics although unlike the non-porous channel flow, the cyclic vortex development is only restricted to a small area behind the last solid block, while temperature changes more slowly and does not exhibit cyclic variations over a long period of time. It is shown that for all the cases studied altering some parametric values can have significant and interesting effects on both flow pattern as well as heat transfer characteristics.     

Stochastic and upscaled analytical modeling of fines migration in porous media induced by low-salinity water injection

Fines migration induced by injection of low-salinity water(LSW) into porous media can lead to severe pore plugging and consequent permeability reduction. The deepbed filtration(DBF) theory is used to model the aforementioned phenomenon, which allows us to predict the effluent concentration history and the distribution profile of entrapped particles. However, the previous models fail to consider the movement of the waterflood front. In this study, we derive a stochastic model for fines migration ...     

Numerical simulation of compressible gas flow and heat transfer in a microchannel surrounded by solid media

In this study, a numerical model is developed to investigate the coupled compressible gas flow and heat transfer in a microchannel surrounded by solid media. To accommodate the varying flow cross-section, the compressible gas flow model is established in a non-orthogonal curvilinear coordinate system. An iterative numerical procedure is employed to solve the coupled heat transfer and gas flow equations. The computer code for the compressible gas flow is first validated against two test problems, and then extended by including the heat conduction in the solid media. The effect of the inlet Mach number on the Nusselt number is examined. It is found that the pressure difference from the pyrolysis front to the heated surface is induced essentially by the gas addition from the channel wall, instead from the pyrolysis front. The necessity of accounting for the gas compressibility is clearly demonstrated when severe heating is applied. The pressure distribution obtained along the channel axial direction is useful for further structural analysis of composite materials.     

Freezing and thawing porous media: experimental study with a dielectric capacitive method

A capacitive sensor-based apparatus has been used to study the ice/water phase change in consolidated porous media subjected to freezing and thawing. This technique relies on the dielectric properties of water, ice, air, and the mineral substrate in the radio-frequency range. It gives directly the freezing and thawing temperature depressions and indirectly provides an estimation of pore size distribution through the Gibbs–Thomson relation. It also holds good promise for evaluating the amount of liquid water in frozen porous media by combining drying and freezing tests. To cite this article: T. Fen-Chong, A. Fabbri, C. R. Mecanique 333 (2005).