Percolation model of two-phase flow through porous media

A physical model of the process of two-phase flow of immiscible fluids through a porous medium is developed and used to make an analytical calculation of the dependence of the relative phase permeabilities on the saturation of the medium by one of the phases. The theory is compared qualitatively with experiment for a model capillary radius frequency function and quantitatively with numerical calculations made on a computer. In both cases good agreement is obtained. The pressure dependences of the phase permeabilities are analyzed. The question of residual saturation with the wetting fluid after completion of the displacement process is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 88–95, January–February, 1987.     

Vertical two-phase flow in porous media

New concepts are introduced to describe single-component two-phase flow under gravity. The phases can flow simultaneously in opposite directions (counterflow), but information travels either up or down, depending on the sign of the wavespeedC. Wavespeed, saturation and other quantities are defined on a two-sheeted surface over the mass-energy flow plane, the sheets overlapping in the counterflow region. A saturation shock is represented as an instantaneous displacement along a line of constant volume fluxJ _Q in the flow plane. Most shocks are of the wetting type, that is, they leave the environment more saturated after their passage. When flow is horizontal all shocks are wetting, but it is a feature of vertical two-phase flow that for sufficiently small mass and energy flows there also exist drying shocks associated with lower final saturations.     

Steady-state two-phase flow through planar and nonplanar model porous media

Transport in Porous Media - A comparative experimental study of ‘steady-state’ two-phase flow in two types of model porous media is made to determine the effects of nonplanarity on the...     

Determining equations of two-phase flows through anisotropic porous media

A new interpretation of the concept of relative phase permeability is given. Relative phase permeabilities are represented in the form of fourth-rank tensors. It is shown that in the case of anisotropic porous media functions depending not only on the saturation but also on the anisotropy parameters represented in the form of ratios of the principal values of the absolute permeability coefficient tensor correspond to the classical representation of the relative phase permeabilities. For a two-phase flow in anisotropic porous media with orthotropic and transversely-isotropic symmetry a generalized two-term Darcy’s law is analyzed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 87–94, March–April, 1998. The work was carried out with support from the Russian Foundation for Fundamental Research (project No. 96-01-00623).     

Interfacial drag of two-phase flow in porous media

In this paper, the influence of the interfacial drag on the pressure loss of combined liquid and vapour flow through particulate porous media is investigated. Motivation for this is the coolability of fragmented corium which may be expected during a severe accident in a nuclear power plant. Cooling water is evaporated due to the particles decay heat. To reach coolability, the outflowing steam has to be replaced by inflowing water.     

Mathematical model of two-phase fluid nonlinear flow in low-permeability porous media with applications

IntroductionItisasuccessfulexampleinadevelopmentstoryofscienceandtechnologyformechanicsoffluidsinporousmediatocombinewithengineeringtechnology .Fieldsinfluencedbythemechanicsinvolveddevelopmentofoil_gasandgroundwaterresources,controlonseawaterintrusionandsubsidenceandgeologichazards,geotechnicalengineeringandbioengineering ,andairlineindustry[1～ 7].Aproblemonnonlinearflowinlow_permeabilityporousmediaisbutonlyabasiconeindifferentkindsofengineeringfields,butalsooneoffrontlineresearchfieldsofmod…     

Investigation of two-phase flow in inclined macroinhomogeneous porous media

The results of experiments carried out in order to determine the principal characteristics of the process of displacement of one (nonwetting) fluid from inclined macroinhomogeneous porous media by another (wetting) fluid are presented. Irrespective of whether flow in inclined stratified nonhomogeneous formations or in zonally nonhomogeneous media (with a corresponding well distribution) is investigated, the term oblique stratification is used for describing these processes.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 125–131, November–December, 1992.     

Comparison theorems for problems of nonlinear anisotropic flow through porous media

Flow law constraints that make it possible to establish comparison theorems (analogs of the theorems of [1, 2]) for nonlinear flows in an anisotropic inhomogeneous medium are formulated. In the theorems obtained the changes in the values of the pressure head and, moreover, the flow rate, filter velocity and pressure head gradients for such perturbations of the problem as the depression of individual surfaces, changes in the given boundary values of the head, etc., are established. The strict monotonicity of the relation between the flow rate and the pressure head difference in a region of the enlarged stream tube type and the possibility of an increase in flow rate with increase in flow resistance are demonstrated. The question of the correspondence between the constraints introduced and certain common models of porous media is discussed. Kazan'. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 45–51, September–October, 1988.     

A resistance model for flow through porous media

A new model for resistance of flow through granular porous media is developed based on the average hydraulic radius model and the contracting–expanding channel model. This model is expressed as a function of tortuosity, porosity, ratio of pore diameter to throat diameter, diameter of particles, and fluid properties. The two empirical constants, 150 and 1.75, in the Ergun equation are replaced by two expressions, which are explicitly related to the pore geometry. Every parameter in the proposed model has clear physical meaning. The proposed model is shown to be more fundamental and reasonable than the Ergum equation. The model predictions are in good agreement with the existing experimental data.     

Modeling of two-phase flow in porous media with heat generation

The main purpose of this work is investigation of coolability of a boiling debris bed. The main governing equations are derived using volume averaging technique. From this technique some specific interfacial areas between phases are appeared and proper relations for modeling these areas are proposed. Using these specific areas, a modification for the Tung/Dhir model in the annular flow regime is proposed. The proposed modification is validated and the agreements with experimental data are good. Finally, governing equations and relations are implemented in the THERMOUS program to model two-phase flow in the debris bed in the axisymmetric cylindrical coordinate. Two typical configurations including flat and mounted beds are considered and the main physical phenomena during boiling of water in the debris bed are studied. Comparing the results with the one-dimensional analysis shows higher specific power of the bed.     

Nonisothermal immiscible compressible thermodynamically consistent two-phase flow in porous media

In this paper, we introduce a new model of the nonisothermal immiscible compressible thermodynamically consistent two-phase flow in a porous domain Ω. This model includes the term describing the skeleton and interphase boundary energies. In the framework of the model, we derive the equation for the entropy function in the whole Ω and then obtain the estimate of the maximal entropy of the system.     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

Discretization limits of lattice-Boltzmann methods for studying immiscible two-phase flow in porous media

Digital images of porous media often include features approaching the image resolution length scale. The behavior of numerical methods at low resolution is therefore important even for well-resolved systems. We study the behavior of the Shan-Chen (SC) and Rothman-Keller (RK) multicomponent lattice-Boltzmann models in situations where the fluid-fluid interfacial radius of curvature and/or the feature size of the medium approaches the discrete unit size of the computational grid. Various simple, small-scale test geometries are considered, and a drainage test is also performed in a Bentheimer sandstone sample. We find that both RK and SC models show very high ultimate limits: in ideal conditions the models can simulate static fluid configuration with acceptable accuracy in tubes as small as three lattice units across for RK model (six lattice units for SC model) and with an interfacial radius of curvature of two lattice units for RK and SC models. However, the stability of the models is affected when operating in these extreme discrete limits: in certain circumstances the models exhibit behaviors ranging from loss of accuracy to numerical instability. We discuss the circumstances where these behaviors occur and the ramifications for larger-scale fluid displacement simulations in porous media, along with strategies to mitigate the most severe effects. Overall we find that the RK model, with modern enhancements, exhibits fewer instabilities and is more suitable for systems of low fluid-fluid miscibility. The shortcomings of the SC model seem to arise predominantly from the high, strongly pressure-dependent miscibility of the two fluid components.     

Thermal stretching in two-phase porous media: Physical basis for Maxwell model

An alternate yet general form of the classical effective thermal conductivity model (Maxwell model) for two-phase porous materials is presented, serving an explicit thermo-physical basis. It is demonstrated that the reduced effective thermal conductivity of the porous media due to non-conducting pore inclusions is caused by the mechanism of thermal stretching, which is a combination of reduced effective heat flow area and elongated heat transfer distance (thermal tortuosity).     

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.     

Nonlinear waves and localized structures in two-phase flow through porous media

Three problems of two-phase flow through porous media are considered. In the first two flows in the region of near-critical saturations are investigated. Since under these conditions the active saturation and hence the phase permeability of one of the phases are small, it is important to take into account the delay in phase redistribution — such types of flow as nonlinear waves and localized structures become important. In the third problem it is assumed that the capillary jump is insignificant as compared with the phase pressures. It is shown that in this case localized structures may also occur.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 71–76, January–February, 1988.     

A new formulation of immiscible compressible two-phase flow in porous media

A new formulation is proposed to describe immiscible compressible two-phase flow in porous media. The main feature of this formulation is the introduction of a global pressure. The resulting equations are written in a fractional flow formulation and lead to a coupled system which consists of a nonlinear parabolic (the global pressure equation) and a nonlinear diffusion–convection one (the saturation equation) which can be efficiently solved numerically. To cite this article: B. Amaziane, M. Jurak, C. R. Mecanique 336 (2008).     

Diffusion in anisotropic porous media

An experimental system was constructed in order to measure the two distinct components of the effective diffusivity tensor in transversely isotropic, unconsolidated porous media. Measurements were made for porous media consisting of glass spheres, mica particles, and disks made from mylar sheets. Both the particle geometry and the void fraction of the porous media were determined experimentally, and theoretical calculations for the two components of the effective diffusivity tensor were carried out. The comparison between theory and experiment clearly indicates that the void fraction and particle geometry are insufficient to characterize the process of diffusion in anisotropic porous media. Roman Letters A interfacial area between - and -phases for the macroscopic system, m² - A _e area of entrances and exits of the -phase for the macroscopic system, m² - A interfacial area contained within the averaging volume, m² - a characteristic length of a particle, m - b average thickness of a particle, m - c _A concentration of species A, moles/m³ - c _o reference concentration of species A, moles/m³ - c _A intrinsic phase average concentration of species A, moles/m³ - c _a c _A–c _A, spatial deviation concentration of species A, moles/m³ - C c _A/c ₀, dimensionless concentration of species A - binary molecular diffusion coefficient, m²/s - D _eff effective diffusivity tensor, m²/s - D _xx component of the effective diffusivity tensor associated with diffusion parallel to the bedding plane, m²/s - D _yy component of the effective diffusivity tensor associated with diffusion perpendicular to the bedding plane, m²/s - D _eff effective diffusivity for isotropic systems, m₂/s - f vector field that maps c _A on to c _a , m - h depth of the mixing chamber, m     

A finite-strain model for anisotropic viscoplastic porous media: I – Theory

In this work, we propose an approximate homogenization-based constitutive model for estimating the effective response and associated microstructure evolution in viscoplastic (including ideally-plastic) porous media subjected to finite-strain loading conditions. The proposed model is based on the “second-order” nonlinear homogenization method, and is constructed in such a way as to reproduce exactly the behavior of a “composite-sphere assemblage” in the limit of hydrostatic loading and isotropic microstructure. However, the model is designed to hold for completely general three-dimensional loading conditions, leading to deformation-induced anisotropy, whose development in time is handled through evolution laws for the internal variables characterizing the instantaneous “ellipsoidal” state of the microstructure. In Part II of this study, results will be given for the instantaneous response and microstructure evolution in porous media for several representative loading conditions and microstructural configurations.