Factors Affecting Liquid-Liquid Relative Permeabilities of a Consolidated Porous Medium

1973 ◽  
Vol 13 (01) ◽  
pp. 39-47 ◽  
Author(s):  
E.J. Lefebvre du Prey
Keyword(s):  
Porous Medium ◽  
Stainless Steel ◽  
Relative Permeability ◽  
Viscosity Ratio ◽  
Sweep Efficiency ◽  
Factors Affecting ◽  
Continuity Equations ◽  
Relative Permeability Curves ◽  
Solid Liquid ◽  
Flow Laws

Abstract Many laboratory displacement tests have been performed to study factors affecting relative performed to study factors affecting relative permeability curves, residual saturations, and shape permeability curves, residual saturations, and shape of recovery curves. Three sintered artificial porous materials and pure fluid mixtures have been used for this systematic study. The factors were interfacial tension, the viscosity and the velocity of the fluids (in the dimensionless group / v), the wettability, and the viscosity ratio. The results can be used as guidelines for research on recovery processes. Introduction Waterflooding is by far the most common secondary recovery technique. Several processes (polymer injection, surfactant injection, hot water injection, etc.) attempt to improve sweep efficiency by affecting some of the factors involved in the displacement process. For orienting research on these processes, process. For orienting research on these processes, a good knowledge of how these parameters affect local displacement efficiency and over-all sweep efficiency in the reservoir is required. We present here an attempt to understand the effects of such parameters:at the microscopic level on the shape of relative permeability curves and the values of final saturations obtained by flooding, andat the macroscopic level, on the behavior of one-dimensional displacement. Effects of the morphology of the porous medium were not included in this study. We mainly examined the effects of fluid properties on fluid displacement in only three specific porous media. These three artificial sintered media were made of Teflon, stainless steel and alumina. For the following reasons they were very well suited to the systematic investigation undertaken:they are homogeneous and so the results are not subject to macroscopic heterogeneity effects;they are identical in the same series, thus permitting the results to be compared from one experiment to anothertheir constant and well-defined chemical composition makes it possible to perform wettability measurements outside the porous medium;they are consolidated like most of reservoir rocks;they have good mechanical properties and so can be washed successively without being altered andthe three media used correspond to three possible cases of wettability, i.e., Teflon is strongly oil-wet, alumina is strongly water wet, and stainless steel may have intermediate wettability depending on the fluids considered. Some of our results reported here (concerning experiments with Teflon at a viscosity ratio of one) were presented earlier. THEORY The elementary laws governing the distribution and flow of two phases in a porous material are quite well known:(1)viscous-flow laws in each phase (Navier and continuity equations),(2)phase (Navier and continuity equations),(3)solid-liquid boundary condition (zero velocity),(3)dynamic equilibrium laws of liquid-liquid interfaces (capillary law and continuity of velocities and viscosity stresses), and(4)solid-liquid-liquid contact line equilibrium laws with hysteresis and velocity dependency. Nevertheless the complexity of porous media, coupled with the difficulty of introducing the wettability law in a mathematical form, makes it impossible to go from flow properties on a microscopic pore scale to flow laws formulated on a macroscopic scale, i.e., relative permeability curves and capillary pressure curves. Use of dimensional analysis and reasoning with simple pore schemes are two ways of approaching this pore schemes are two ways of approaching this problem. problem. The parameters involved in the phenomenon under study are the following:fluid viscosities 1 and2,specific gravities P1 and P2,interfacial tension,pore dimension and morphology M,the wettability,system evolution prior to the moment of study K, andexternal conditions, i.e., a mean velocity v or a pressure gradient in the zone investigated. Three of these parameters, namely, M, and K, have complex meanings and cannot be specified by a single number. JPT P. 39

A Model for Two-Phase Flow in Consolidated Materials

1969 ◽  
Vol 9 (02) ◽  
pp. 221-231 ◽  
Author(s):  
R. Ehrlich ◽  
F.E. Crane
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Relative Permeability ◽  
Viscosity Ratio ◽  
Phase Flow ◽  
Unsteady State ◽  
Adequate Description ◽  
Two Phase ◽  
Phase Saturation ◽  
Relative Permeability Curves

Abstract A consolidated porous medium is mathematically modeled by networks of irregularly shaped interconnected pore channels. Mechanisms are described that form residual saturations during immiscible displacement both by entire pore channels being bypassed and by fluids being isolated by the movement of an interface within individual pore channels. This latter mechanism is shown to depend strongly on pore channel irregularity. Together, these mechanisms provide an explanation for the drainage-imbibition-hysteresis effect. The calculation of steady-state relative permeabilities, based on a pore-size distribution permeabilities, based on a pore-size distribution obtained from a Berea sandstone, is described. These relative permeability curves agree qualitatively with curves that are generally accepted to be typical for highly consolidated materials. In situations where interfacial effects predominate over viscous and gravitational effects, the following conclusions are reached.Relative permeability at a given saturation is everywhere independent of flow rate.Relative permeability is independent of viscosity ratio everywhere except at very low values of wetting phase relative permeability.Irreducible wetting-phase saturation following steady-state drainage decreases with increasing ratio of nonwetting- to wetting-phase viscosity.Irreducible wetting-phase saturation following unsteady-state drainage is lower than for steady-state drainage.Irreducible nonwetting-phase saturation following imbibition is independent of viscosity ratio, whether or not the imbibition is carried out under steady- or unsteady-state conditions. Experiments qualitatively verify the conclusions regarding unsteady-state residual wetting-phase saturation. Implications of these conclusions are discussed. Introduction Natural and artificial porous materials are generally composed of matrix substance brought together in a more or less random manner. This leads to the creation of a network of interconnected pore spaces of highly irregular shape. Since the pore spaces of highly irregular shape. Since the geometry of such a network is impossible to describe, we can never obtain a complete description of its flow behavior. We can, however, abstract those properties of the porous medium pertinent to the type of flow under consideration, and thus obtain an adequate description of that flow. Thus, the Kozeny-Carmen equation, by considering a porous medium as a bundle of noninterconnecting capillary tubes, provides an adequate description of single-phase provides an adequate description of single-phase flow. With the addition of a saturation-dependent tortuosity parameter in two-phase flow to account for flow path elongation, the Kozeny-Carmen equation has been used to predict relative permeabilities for the displacement of a wetting permeabilities for the displacement of a wetting liquid by a gas. It has long been recognized that relative permeability depends not only on saturation but permeability depends not only on saturation but also on saturation history as well. Naar and Henderson described a mathematical model in which differences between drainage and imbibition behavior are explained in terms of a bypassing mechanism by which oil is trapped during imbibition. Fatt proposed a model for a porous medium that consisted of regular networks of cylindrical tubes of randomly distributed radii. From this he calculated the drainage relative permeability curves. Moore and Slobod, Rose and Witherspoon, and Rose and Cleary each considered flow in a pore doublet (a parallel arrangement of a small and pore doublet (a parallel arrangement of a small and large diameter cylindrical capillary tube). They concluded that, because of the different rates of flow in each tube, trapping would occur in one of the tubes; the extent of which would depend upon viscosity ratio and capillary pressure. SPEJ p. 221

scholarly journals In-House Developed Kr Estimator: Implemented to Analyze the Sensitivity of Relative Permeability Data to Variations in Wetting Phase Saturation

2011 ◽  
Vol 29 (6) ◽  
pp. 817-825 ◽  
Author(s):  
Muhammad Khurram Zahoor
Keyword(s):  
Porous Medium ◽  
Pore Size Distribution ◽  
Pore Size ◽  
Relative Permeability ◽  
State Of The Art ◽  
Data Generation ◽  
Phase Saturation ◽  
Complex Models ◽  
Relative Permeability Curves ◽  
Very High

Reservoir surveillance always requires fast, unproblematic access and solution to different relative permeability models which have been developed from time to time. In addition, complex models sometimes require in-depth knowledge of mathematics for solution prior to use them for data generation. For this purpose, in-house software has been designed to generate rigorous relative permeability curves, with a provision to include users own relative permeability models, a part from built-in various relative permeability correlations. The developed software with state-of-the-art algorithms has been used to analyze the effect of variations in residual and maximum wetting phase saturation on relative permeability curves for a porous medium having very high non-uniformity in pore size distribution. To further increase the spectrum of the study, two relative permeability models, i.e., Pirson's correlation and Brooks and Corey model has been used and the obtained results show that the later model is more sensitive to such variations.

An Imbibition Model - Its Application to Flow Behavior and the Prediction of Oil Recovery

1961 ◽  
Vol 1 (02) ◽  
pp. 61-70 ◽  
Author(s):  
J. Naar ◽  
J.H. Henderson
Keyword(s):  
Porous Medium ◽  
Relative Permeability ◽  
Oil Recovery ◽  
Flow Behavior ◽  
Pore Geometry ◽  
Mathematical Treatment ◽  
Complete Wetting ◽  
Permeability Characteristics ◽  
Relative Permeability Curves ◽  
Wetting Fluid

Introduction The displacement of a wetting fluid from a porous medium by a non-wetting fluid (drainage) is now reasonably well understood. A complete explanation has yet to be found for the analogous case of a wetting fluid being spontaneously imbibed and the non-wetting phase displaced (imbibition). During the displacement of oil or gas by water in a water-wet sand, the porous medium ordinarily imbibes water. The amount of oil recovered, the cost of recovery and the production history seem then to be controlled mainly by pore geometry. The influence of pore geometry is reflected in drainage and imbibition capillary-pressure curves and relative permeability curves. Relative permeability curves for a particular consolidated sand show that at any given saturation the permeability to oil during imbibition is smaller than during drainage. Low imbibition permeabilities suggest that the non-wetting phase, oil or gas, is gradually trapped by the advancing water. This paper describes a mathematical image (model) of consolidated porous rock based on the concept of the trapping of the non-wetting phase during the imbibition process. The following items have been derived from the model.A direct relation between the relative permeability characteristics during imbibition and those observed during drainage.A theoretical limit for the fractional amount of oil or gas recoverable by imbibition.An expression for the resistivity index which can be used in connection with the formula for wetting-phase relative permeability to check the consistency of the model.The limits of flow performance for a given rock dictated by complete wetting by either oil or water.The factors controlling oil recovery by imbibition in the presence of free gas. The complexity of a porous medium is such that drastic simplifications must be introduced to obtain a model amenable to mathematical treatment. Many parameters have been introduced by others in "progressing" from the parallel-capillary model to the randomly interconnected capillary models independently proposed by Wyllie and Gardner and Marshall. To these a further complication must be added since an imbibition model must trap part of the non-wetting phase during imbibition of the wetting phase. Like so many of the previously introduced complications, this fluid-block was introduced to make the model performance fit the observed imbibition flow behavior.

The Flow of Foam Through Short Porous Media And Apparent Viscosity Measurements

1966 ◽  
Vol 6 (01) ◽  
pp. 17-25 ◽  
Author(s):  
S.S. Marsden ◽  
Suhail A. Khan
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Shear Rate ◽  
Relative Permeability ◽  
Oil Recovery ◽  
Apparent Viscosity ◽  
Viscosity Ratio ◽  
Surfactant Solution ◽  
Liquid Saturation ◽  
Liquid Displacement

Abstract Externally generated foam was injected continuously into short porous media. Both flow rate and pressure drop were measured. Liquid saturation was determined by electrical conductivity. Foam quality G, expressed as the ratio of gas volume to total volume, was varied from 0.70 to 0.96. As measured with a modified Fann VG Meter, apparent viscosity of this foam µa decreases with increasing shear rate but usually falls within the range of 50 to 500 cp. At a given shear rate, µa increases almost linearly with G. When measured with a Bendix Ultraviscoson, kinematic µa is independent of r but absolute µa increases with r from about 3 to 8 cp. The effective permeability-apparent viscosity ratio ke/µa decreases almost linearly with G for porous media of high permeability, but the rate of decrease becomes less for tighter ones. The relative permeability-apparent viscosity ratio kr/µa vs G data does not fall on a single line. The kr/µa ratio increases with liquid saturation in the porous medium and with surfactant concentration. Estimates of µa for foam in porous media vary from 30 to 100 cp. INTRODUCTION Although research on the development of a foam-drive, oil recovery process has been going on for almost a decade, most of the significant publications have appeared within the last several years. This illustrates well the rate at which interest in this process is accelerating. Bond and Holbrook1 were the first to describe the use of foam to improve oil recovery in their patent of 1958. They proposed that an aqueous foaming agent slug be injected into the formation and that this be followed by gas to produce a foam in situ. Fried2 studied the injection of foam into porous media which has already been subjected to conventional gas or water drives and found that gas could be used to drive a foam bank which would, in turn, displace additional oil in the form of an oil bank. He attributed the increased oil recovery to the high effective viscosity of foam flowing in porous media. His microscopic observations showed the importance of foam generation and regeneration within the porous medium. By injecting both air and aqueous surfactant solution, Bernard3 generated foams within the porous medium in which oil displacement was being studied. In a separate empirical test, he also measured the dynamic foaming characteristics of the same surfactants in water and/or oil. With some exceptions and for the seven surfactants studied, there seems to be a qualitative relationship between the efficiency of liquid displacement and the dynamic foaming test used. This relationship was not consistent enough to eliminate the necessity of actual foam flood tests in porous media for surfactant selection. In a study basic to gas storage in aquifers, Bennett4 described the displacement of brine by foam in consolidated porous media. Among other things, he stated that the ability of a surfactant solution to foam is more important than the stability of its foam. The presence of a foam bank between the displacing air and the displaced brine improved both breakthrough and ultimate recovery. In a continuation of this work Kolb5 attributed the great reduction in surfactant solution production rate as displacement by air progressed to a decrease in relative permeability to gas. These several effects reported by both Bennett and Kolb can all be attributed to the high apparent viscosity of foam which was obviously flowing in the porous media.

Numerical Investigation of the Effects of Core Heterogeneities on Waterflood Relative Permeabilities

1970 ◽  
Vol 10 (04) ◽  
pp. 381-392 ◽  
Author(s):  
John D. Huppler
Keyword(s):  
Numerical Simulation ◽  
Porous Medium ◽  
Pressure Drop ◽  
Difference Equation ◽  
Relative Permeability ◽  
Phase Difference ◽  
Two Dimensional ◽  
Relative Permeabilities ◽  
Two Phase ◽  
Relative Permeability Curves

Abstract Numerical simulation techniques were used to investigate the effects of common core heterogeneities upon apparent waterflood relative-permeability results. Effects of parallel and series stratification, distributed high and low permeability lenses, and vugs were considered. permeability lenses, and vugs were considered. Well distributed heterogeneities have little effect on waterflood results, but as the heterogeneities become channel-like, their influence on flooding behavior becomes pronounced. Waterflooding tests at different injection rates are suggested as the best means of assessing whether heterogeneities are important. Techniques for testing stratified or lensed cores are recommended. Introduction Since best results from waterflood tests performed on core plugs are obtained with homogeneous cores, plugs selected for testing are chosen for their plugs selected for testing are chosen for their apparent uniformity. We know, however, that uniform appearance can be misleading. For example, flushing concentrated hydrochloric acid through an apparently homogeneous core plug often produces "wormholes" in higher permeability regions. Also, we sometimes find that all core plugs from a region of interest have obvious heterogeneities, so any flooding tests must be run on nonhomogeneous core plugs. plugs. Nevertheless, relative permeabilities, as obtained routinely from core waterflood data, are calculated using the assumption that the core is a homogeneous porous medium. While it is obvious that porous medium. While it is obvious that heterogeneties mill affect these apparent relative permeabilities, there appear to be no experimental permeabilities, there appear to be no experimental results reported in the literature to indicate just how serious the problem is. Accordingly, a computer simulation study of core waterfloods was conducted to systematically examine the effects of different sizes and types of core heterogeneities on flood results. The study was performed by numerical simulation using two-dimensional, two-phase difference equation approximations to describe the immiscible water-oil displacement. For each simulation the permeability and porosity distribution of the heterogeneous core to be studied was specified; fluid flow characteristics of the system, including a single set of input relative-permeabilities curves, were stipulated The system was set in capillary pressure equilibrium at the reducible water saturation. Then a waterflood simulation was performed. From the resulting fluid production and pressure-drop data a set of production and pressure-drop data a set of relative-permeability curves was calculated using the standard computational procedure applicable to homogeneous cores. In this paper these calculated relative-permeability curves are denoted as "waterflood" curves to distinguish them from the specified input curves. The waterflood relative-permeability curves should closely match the input curves for homogeneous systems. Since the same set of input relative-permeability curves was used for all rock sections, deviations of the waterflood from the input relative-permeability curves gave an indication of the effects of heterogeneities. When the system was heterogeneous and there was good agreement between waterflood and input relative-permeability curves, then the heterogeneities did not strongly influence the flow behavior and the system responded homogeneously. MATHEMATICAL MODEL AND METHOD The waterflood simulations were carried out using two-dimensional, two-phase difference equation approximations to the incompressible-flow differential equations:* .....................(1) ....................(2) SPEJ P. 381

scholarly journals Mathematical modeling and experimental study on solid–liquid suspension separation in ultrasonic standing wave field

2021 ◽  
pp. 146134842110229
Author(s):  
Yajing Wang ◽  
Liqun Wu ◽  
Yaxing Wang ◽  
Yafei Fan
Keyword(s):  
Standing Wave ◽  
Sound Pressure ◽  
Acoustic Radiation ◽  
Sound Field ◽  
Radiation Force ◽  
High Energy ◽  
Factors Affecting ◽  
Liquid Suspension ◽  
Ultrasonic Standing Wave ◽  
Solid Liquid

A new method of removing waste chips is proposed by focusing on the key factors affecting the processing quality and efficiency of high energy beams. Firstly, a mathematical model has been established to provide the theoretical basis for the separation of solid–liquid suspension under ultrasonic standing wave. Secondly, the distribution of sound field with and without droplet has been simulated. Thirdly, the deformation and movement of droplets are simulated and tested. It is found that the sound pressure around the droplet is greater than the sound pressure in the droplet, which can promote the separation of droplets and provide theoretical support for the ultrasonic suspension separation of droplet; under the interaction of acoustic radiation force, surface tension, adhesion, and static pressure, the droplet is deformed so that the gas fluid around the droplet is concentrated in the center to achieve droplet separation, and the droplet just as a flat ball with a central sag is stably suspended in the acoustic wave node.

scholarly journals Biologics Formulation Factors Affecting Metal Leachables from Stainless Steel

2011 ◽  
Vol 12 (1) ◽  
pp. 411-421 ◽  
Author(s):  
Shuxia Zhou ◽  
Christian Schöneich ◽  
Satish K. Singh
Keyword(s):  
Stainless Steel ◽  
Factors Affecting ◽  
Formulation Factors
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