Onset of Mobilization and the Fraction of Trapped Foam in Porous Media

Usually, foam in a porous medium flows through a small and spatially varying fraction of available pores, while the bulk of it remains trapped. The trapped foam is under a pressure gradient corresponding to the pressure gradient imposed by the flowing foam and continuous wetting liquid. The imposed pressure gradient and coalescence of the stationary foam lamellae periodically open flow channels in the trapped foam region. Foam lamellae in each of these channels flow briefly, but channels are eventually plugged by smaller bubbles entering into the trapped region. The result is a cycling of flow channels that open and close throughout the trapped foam, leading to intermittent pulsing of foam flow in that region.The dynamic behavior of foam trapped in porous media is modeled here with a pore network simulator. We predict the magnitude of the pressure drop leading to the onset of flow of foam lamellae in the region containing trapped foam. This mobilization pressure drop depends only on the number of lamellae in the flow path and on the geometry of the pores that make up this path.The principles learned in this study allow us to predict the fraction of foam that is trapped in a porous medium under given flow conditions. We present here the first analytic expression for the trapped foam fraction as a function of the pressure gradient, and of the mean and standard deviation of the pore size distribution. This expression provides a missing piece for the continuum foam flow models based on the moments of the volume-averaged population balance of foam bubbles.     

Pore Network Investigation of Trapped Gas and Foam Generation Mechanisms

Transport in Porous Media - The mobility of gas is greatly reduced when the injected gas is foamed. The reduction in gas mobility is attributed to the reduction in gas relative permeability and the...     

Modeling and Experimental Validation of Rheological Transition During Foam Flow in Porous Media

Transport in Porous Media - Flow of nitrogen foam stabilized by alpha olefin sulfonate (C14-16 AOS) was studied in a natural sandstone porous media using X-ray Computed Tomography. Foam was...     

Steady-State Source Flow in Heterogeneous Porous Media

The flow of fluids in heterogeneous porous media is modelled by regarding the hydraulic conductivity as a stationary random space function. The flow variables, the pressure head and velocity field are random functions as well and we are interested primarily in calculating their mean values. The latter had been intensively studied in the past for flows uniform in the average. It has been shown that the average Darcy's law, which relates the mean pressure head gradient to the mean velocity, is given by a local linear relationship. As a result, the mean head and velocity satisfy the local flow equations in a fictitious homogeneous medium of effective conductivity. However, recent analysis has shown that for nonuniform flows the effective Darcy's law is determined by a nonlocal relationship of a convolution type. Hence, the average flow equations for the mean head are expressed as a linear integro-differential operator. Due to the linearity of the problem, it is useful to derive the mean head distribution for a flow by a source of unit discharge. This distribution represents a fundamental solution of the average flow equations and is called the mean Green function G _d (x). The mean head G _d(x) is derived here at first order in the logconductivity variance for an arbitrary correlation function (x) and for any dimensionality d of the flow. It is obtained as a product of the solution G _d ⁽⁰⁾(x) for source flow in unbounded domain of the mean conductivity K _A and the correction _d (x) which depends on the medium heterogeneous structure. The correction _d is evaluated for a few cases of interest.Simple one-quadrature expressions of _d are derived for isotropic two- and three-dimensional media. The quadratures can be calculated analytically after specifying (x) and closed form expressions are derived for exponential and Gaussian correlations. The flow toward a source in a three-dimensional heterogeneous medium of axisymmetric anisotropy is studied in detail by deriving ₃ as function of the distance from the source x and of the azimuthal angle . Its dependence on x, on the particular (x) and on the anisotropy ratio is illustrated in the plane of isotropy (=0) and along the anisotropy axis ( = /2).The head factor k ^* is defined as a ratio of the head in the homogeneous medium to the mean head, k ^*=G _d ⁽⁰⁾/G _d= _d ^–1. It is shown that for isotropic conductivity and for any dimensionality of the flow the medium behaves as a one-dimensional and as an effective one close and far from the source, respectively, that is, lim _x0 k ^*(x) = K _H/K _A and lim _x k ^*(x) = K ^efu/K _A, where K _A and K _H are the arithmetic and harmonic conductivity means and K ^efu is the effective conductivity for uniform flow. For axisymmetric heterogeneity the far-distance limit depends on the direction. Thus, in the coordinate system of (x) principal directions the limit values of k ^* are obtained as . These values differ from the corresponding components of the effective conductivities tensor for uniform flow for = 0 and /2, respectively. The results of the study are applied to solving the problem of the dipole well flow. The dependence of the mean head drop between the injection and production chambers on the anisotropy of the conductivity and the distance between the chambers is analyzed.     

A Steady-State Upscaling Approach for Immiscible Two-Phase Flow

The paper presents a model for computing rate-dependent effective capillary pressure and relative permeabilities for two-phase flow, in 2 and 3 space-dimensions. The model is based on solving the equations for immiscible two-phase flow at steady-state, accounting for viscous and capillary forces, at a given external pressure drop. The computational performance of the steady-state model and its accuracy is evaluated through comparison with a commercial simulator ECLIPSE. The properties of the rate-dependent effective relative permeabilities are studied by way of computations using the developed steady-state model. Examples presented show the dependence of the effective relative permeabilities and capillary pressures, which incorporate the effects of fine scale wettability heterogeneity, on the external pressure drop, and thereby on the dimensionless macro-scale capillary number. The effective relative permeabilities converge towards the viscous limit functions as the capillary number tends to infinity. Special cases, when the effective relative permeabilities are rate-invariant, are also studied. The applicability of the steady-state upscaling algorithm in dynamic displacement situations is validated by comparing fine-gridded simulations in heterogeneous reservoirs against their homogenized counterparts. It is concluded that the steady-state upscaling method is able to accurately predict the dynamic behavior of a heterogeneous reservoir, including small scale heterogeneities in both the absolute permeability and the wettability.     

Effect of Confining Wall on Properties of Gas Flow Through Metal Foam: An Experimental Study

While the Darcy and Forchheimer relations are widely applied to determine the permeability and the form drag coefficient of open-cell metal foam, they both assume that the porous medium is infinite in all directions, i.e., large enough so that the effect of any confining walls is negligible. Many researchers, however, pay little or no attention to the size of metal foam samples in pressure-drop studies. The size of a foam sample perpendicular to the flow direction may be small enough such that wall effects are significant. This article experimentally investigates the wall effect on the permeability and form drag coefficient for two types of open-cell aluminum foam subjected to airflow entering the foam in the Forchheimer regime. The Forchheimer equation was recast in two different manners, which resulted in new non-dimensional numbers that correlated very well with the diameter of the foam samples measured in cells. The correlations are valid for a confining-tube-diameter-based Reynolds number ranging from approximately 13,000 to 105,000, and for diameters ranging from 12 to 36 cells and 24 to 60 cells for 10- and 20-pore per inch foam, respectively. The obtained correlations allow for determining pressure drop given only the velocity and the diameter of an aluminum foam sample.     

Direct Observation of Trapped Gas Bubbles by Capillarity in Sandy Porous Media

We investigated the mechanism of residual gas trapping at a microscopic level. We imaged trapped air bubbles in a Berea sandstone chip after spontaneous imbibition at atmospheric pressure. The pore structure and trapped bubbles were observed by microfocused X-ray computed tomography. Distributions of trapped bubbles in Berea and Tako sandstone were imaged in coreflooding at a capillary number of 1.0 × 10⁻⁶. Trapped bubbles are of two types, those occupying the center of the pore with a pore-scale size and others having a pore-network scale size. In low-porosity media such as sandstone, connected bubbles contribute greatly to trapped gas saturation. Effects of capillary number and injected water volume were investigated using a packed bed of glass beads 600μm in diameter, which had high porosity (38%). The trapped N₂ bubbles are stable against the water flow rate corresponding to a capillary number of 1.0 × 10⁻⁴.     

Mechanistic Model of Steady-State Two-Phase Flow in Porous Media Based on Ganglion Dynamics

Recent experimental work has shown that the pore-scale flow mechanism during steady-state two-phase flow in porous media is ganglion dynamics (GD) over a broad and practically significant range of the system parameters. This observation suggests that our conception and theoretical treatment of fractional flow in porous media need careful reconsideration. Here is proposed a mechanistic model of steady-state two-phase flow in those cases where the dominant flow regime is ganglion dynamics. The approach is based on the ganglion population balance equations in combination with a microflow network simulator. The fundamental information on the cooperative flow behavior of the two fluids at the scale of a few hundred pores is expressed through the system factors, which are functions of the system parameters and are calculated using the simulator. These system factors are utilized by the population balance equations to predict the macroscopic behavior of the process. The dependence of the conventional relative permeability coefficients not only on the wetting fluid saturation S_wbut also on the capillary number, Ca, the viscosity ratio the wettability (⁰ a, ⁰ r), the coalescence factor, Co, as well as the porous medium geometry and topology is explained and predicted on a mechanistic basis. Sample calculations have been performed for steady-state fully developed (SSFD) and steady-state nonfully developed (SSnonFD) flow conditions. The number distributions of the moving and the stranded ganglia, the mean ganglion size, the fraction of the nonwetting fluid in the form of mobile ganglia, the ratio of the conventional relative permeability coefficients and the fractional flows are studied as functions of the system parameters and are correlated with the flow phenomena at pore level and the system factors.     

Bifurcation of a Main Steady-State Viscous Fluid Flow in a Plane Divergent Channel

The evolution of steady-state viscous incompressible fluid flows in a plane divergent channel is investigated. For the classical formulation of the Jeffery-Hamel problem a complete solution is given as a function of the determining parameters. For a fixed angle of divergence the behavior of the main unimodal flow is determined as a function of the Reynolds number. Critical values at which the flow pattern bifurcates and the steady-state unimodal flow ceases to exist are found. The mechanism of bifurcation is established and its diagram is constructed. This mechanism and the diagram were not previously known in the scientific literature in connection with the investigation of the Jeffery-Hamel problem. The critical Reynolds number at which bifurcation occurs is given as a function of the channel divergence angle. The results may be of interest for hydromechanical, technological, and geophysical applications.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 25–36.Original Russian Text Copyright © 2005 by Akulenko and Kumakshev.     

Influence of Precursor Gas Flow Rate on Fluoropolymer Coating Growth Rate During Hot Wire Chemical Vapor Deposition

The formation of a fluoropolymer coating by chemical deposition has been studied experimentally. It has been found that increasing the flow rate of the precursor gas leads to a decrease in the growth rate of the coating. Deposition conditions were analyzed, and the gas-dynamic parameters of the process were estimated. The estimates are consistent with experimental data.     

Simulation of Entry-Region Flow in Open-Cell Metal Foam and Experimental Validation

Open-cell metal foam is distinguished from traditional porous media by its very high porosities (often greater than 90 %), and its web-like open structure and good permeability. As such, the foam is a very attractive core for many engineered systems, e.g., heat exchangers, filtration devices, catalysts, and reactors. The flow field inside the foam is rather complex due to flow reversal and vigorous mixing. This complexity is increased by the possible presence of an entry region. The entrance region in metal foam is usually underestimated and ignored, just like its counterpart in traditional porous media. In this paper, the actual entry length is determined by simulation and direct experiment on commercial open-cell aluminum foam. It is shown to be dependent on flow velocity and to reach a constant value for higher velocities. The complex and intrinsically random architecture of the foam is idealized using a unit geometrical model, in order to numerically investigate the flow field and pressure drop inside the foam. The Navier–Stokes equations are solved directly, and velocity and pressure fields are obtained for various approach velocities using a commercial numerical package. The entry length is ascertained from the behavior of the velocity field close to the entrance. Comparisons to experimental data were also carried out. The commercial foam that was used in the experiment had 10 ppi and porosity of 91.2 %. Air was forced to flow inside the foam using an open-loop wind tunnel. Good qualitative agreement between the modeling and experimental results are obtained. The agreement lends confidence to the modeling approach and the determined entry length.     

Surfactant Concentration and End Effects on Foam Flow in Porous Media

Foaming injected gas is a useful and promising technique for achieving mobility control in porous media. Typically, such foams are aqueous. In the presence of foam, gas and liquid flow behavior is determined by bubble size or foam texture. The thin-liquid films that separate foam into bubbles must be relatively stable for a foam to be finely textured and thereby be effective as a displacing or blocking agent. Film stability is a strong function of surfactant concentration and type. This work studies foam flow behavior at a variety of surfactant concentrations using experiments and a numerical model. Thus, the foam behavior examined spans from strong to weak.Specifically, a suite of foam displacements over a range of surfactant concentrations in a roughly 7m², one-dimensional sandpack are monitored using X-ray computed tomography (CT). Sequential pressure taps are employed to measure flow resistance. Nitrogen is the gas and an alpha olefin sulfonate (AOS 1416) in brine is the foamer. Surfactant concentrations studied vary from 0.005 to 1wt%. Because foam mobility depends strongly upon its texture, a bubble population balance model is both useful and necessary to describe the experimental results thoroughly and self consistently. Excellent agreement is found between experiment and theory.     

水平管气液两相间歇流含气率研究

对大直径水平管内气液二相流动进行了实验，实验结果表明大直径水平管与小直径水平管具有不同的气液二相流动特征。分析和提出了适用于大直径水平管内间歇流液弹含气率模型，其计算值与实验值非常接近。     

Effects of Gas Trapping on Foam Mobility in a Model Fracture

Transport in Porous Media - In enhanced oil recovery, foam can effectively mitigate conformance problems and maintain a stable displacement front, by trapping gas and reducing its relative...     

Effect of Compressible Foam Properties on Pressure Amplification During Shock Wave Impact

A comprehensive study is made of the influence of the physical properties of compressible open-cell foam blocks exposed to shock-wave loading, and particularly on the pressure distribution on the shock tube walls. Seven different foams are used, with three different shock Mach numbers, and three different slab lengths. Foam properties examined include permeability, density, stiffness, tortuosity and cell characteristics. The investigations concentrate on both side-wall and back-wall pressures, and the peak pressures achieved, as well as collapse velocities of the front face and the strength and nature of the reflected shock wave. The consequences of deviations from one-dimensionality are identified; primarily those due to wall friction and side-wall leakage. The results presented are the most comprehensive and wide ranging series conducted in a single facility and are thus a significant resource for comparison with theoretical and numerical studies. The different foams show significant differences in behavior, both in terms of peak pressure and duration, depending primarily on their density and permeability.This paper was based on work presented at the 2nd International Symposium on Interdisciplinary Shock Wave Research, Sendai, Japan on March 1–3, 2005.     

A Study on Elliptical Gas Flow in Tri-porosity Gas Reservoirs

Elliptical flow is common in the near vertical fracture area and in anisotropic reservoirs. However, the classical radial flow models cannot provide a complete analysis for elliptical flow. This article presents a new mathematic model for gas elliptical flow in tri-porosity gas reservoirs. The differential equation of the new model is written in Mathieu equation, so that the solution can also be expressed by Mathieu functions. The numerical solution of the corresponding Mathieu functions ce _2n (ξ, −q), Ke _2n (ξ, −q) and their derivatives are obtained to derive the dimensionless pseudo pressure drop in Laplace space. The sensitivities of tri-porosity systems, including the parameters related to anisotropies C _De^2S and ξ _w, the storativity ratios ω _f and ω _m, and the interporosity flow coefficients λ_vf and λ_mf, are studied using Laplace numerical inversion. The new solution includes not only the factors considered in classic solutions in previous articles, but also incorporates the effect of reservoir anisotropy.     

The Effect of Asphalt Precipitation on Flow Behavior and Production of a Fractured Carbonate Oil Reservoir During Gas Injection

The first field data, collected over an 11 year period, are presented which indicate the possible effect of asphalt precipitation on the permeability and injectivity index of a fractured carbonate oil reservoir. The asphalt aggregates were formed during enhanced oil recovery by injection of a rich gas into the reservoir. The data indicate that, while at the initial stages of the operations the permeability and injectivity index decrease, at later times they appear to oscillate with the process time, with apparent oscillations' periods that depend on the heterogeneity of the reservoir. Two classes of plausible mechanisms that give rise to such oscillatory behavior are discussed. One relies on the changes in the structure of the reservoir's fractures, while the other one is based on asphalt precipitation in the reservoir. Computer simulations of flow and precipitation of asphalt aggregates in a pore network model of the reservoir are carried out. The results appear to support our proposition that asphalt formation and precipitation in the reservoir are the main mechanism for the observed behavior of the injectivity index. We also develop a stochastic continuum model that accurately predicts the time-dependence of the reservoir's permeability and injectivity index during the gas injection process.     

A New Stochastic Bubble Population Model for Foam Flow in Porous Media

Foam has been widely used as a mobility control agent for Improved and Enhanced Oil Recovery IOR/EOR, gas blocking, and acid diversion during matrix stimulation. The prediction of foam performance relies on macroscopic modeling. Traditionally, foam modeling approaches include fractional flow theories and population balance models. However, fractional foam models assume implicitly that foam is incompressible and do not account directly for the evolution of bubble population. The population balance models, instead, rely on the idea that foam mobility depends on bubble density and are more comprehensive. Yet, population balance models did not gain full acceptance thus far, because of their perceived complexity, with parameters that are hard to obtain experimentally. This article presents an improved foam model based on a simpler but realistic foam rheology and stochastic bubble generation ideas. Physical ideas in agreement with pictures emerging from recent foam studies using X-ray computed tomography form the basis for the new model.     

Correlation Between Foam Flow Structure in Porous Media and Surfactant Formulation Properties

Transport in Porous Media - The optimization of foam injection in porous media for enhanced oil recovery or soil remediation requires a large screening of surfactant formulations. Tests of foam...