The Effect of Wettability on Three-Phase Relative Permeability

We study three-phase flow in water-wet, oil-wet, and fractionally-wet sandpacks. We use CT scanning to measure directly the oil and water relative permeabilites for three-phase gravity drainage. In an analogue experiment, we measure pressure gradients in the gas phase to determine the gas relative permeability. Thus we find all three relative permeabilities as a function of saturation. We find that the gas relative permeability is approximately half as much in a oil-wet medium than in an water-wet medium at the same gas saturation. The water relative permeability in the water-wet medium and the oil relative permeability in the oil-wet medium are similar. In the water-wet medium the oil relative permeability scales as k _roS _o ⁴ for S _o>S _or, where S _or is the waterflood residual oil saturation. With octane as the oil phase, k _roS _o ² for S _o<S _or, while with decane as the oil phase, k _ro falls sharply for S _o<S _or. The water relative permeability in the oil-wet medium resembles the oil relative permeability in the water-wet medium for a non-spreading oil such as decane. These observations can be explained in terms of wetting, spreading, and the pore scale configurations of fluid.     

Analytical Solution for the Effect of Viscous Dissipation on Mixed Convection in Saturated Porous Media

A new analytical solution is introduced for the effect of viscous dissipation on mixed convection flow and heat transfer about an isothermal vertical wall embedded in Darcy and non-Darcy porous media with uniform free stream velocity. The effect of viscous dissipation on mixed convection in both regimes has been analyzed for both the aiding and opposing flows using Gebhart number, Ge _x =gx/c _p. The governing parameters are Re, Ra, Pe and Ge _x . The case of Re=0 corresponds to Darcy mixed convection region and Re/Pe is identified as the mixed convection governing parameter, Ra=0 leading to pure forced convection. A good agreement was found between the numerical and analytical solutions. It was found from the Nusselt number results that viscous dissipation lowers the heat transfer rate in both Darcy and Forchheimer flow regimes for aiding as well as opposing flows.     

Numerical Investigations of the Steady State Relative Permeability of a Simplified Porous Medium

The purpose of this paper is to investigate, by flow simulations in a uniform pore-space geometry, how the co and countercurrent steady state relative permeabilities depend on the following parameters: phase saturation, wettability, driving force and viscosity ratio. The main results are as follows: (i) with few exceptions, relative permeabilities are convex functions of saturation; (ii) the cocurrent relative permeabilities increase while the countercurrent ones decrease with the driving force; (iii) with one exception, phase 2 relative permeabilities decrease and phase 1 relative permeabilities increase with the viscosity ratio M=₁/₂; (iv) the countercurrent relative permeabilities are always less than the cocurrent ones, the difference being partly due to the opposing effect of the viscous coupling, and partly to different levels of capillary forces; (v) the pore-level saturation distribution, and hence the size of the viscous coupling, can be very different between the cocurrent and the countercurrent cases so that it is in general incorrect to estimate the full mobility tensor from cocurrent and countercurrent steady state experiments, as suggested by Bentsen and Manai (1993).(Now at AS Norske Shell, Norway.) e-mail:     

Influence of Porous Media and Airflow Rate on the Fate of NAPLs Under Air Sparging

Nonequilibrium air–water mass transfer experiments using a laboratoryscale singleair channel setup were conducted to investigate the influence of porous media and air velocity on the fate of nonaqueous phase liquids (NAPLs) under air sparging conditions. Benzene was used as a NAPL while silica sand 30/50 (dp₅₀=0.305mm, uniformity coefficient, UC=1.41) and silica sand 70/100 (dp₅₀=0.168mm, UC=1.64) were used as porous media. Air velocities ranged from 0 to 1.4cm/s. Mass transfer coefficients for the dissolution of NAPLs were estimated by numerical methods using a twodimensional dissolution–diffusion–volatilization model. The study showed that the presence of advective airflow in air channels controlled the spreading of the dissolved phase but the overall removal efficiency was independent of airflow rate. Removal efficiencies and dissolution rates of the NAPL were found to be strongly affected by the mean particle size of the porous media during air sparging. More than 50% reduction in the removal rate of benzene was found when silica sand 70/100 was used instead of silica sand 30/50. Mass transfer coefficients for the dissolution of benzene NAPL were estimated to be 0.0041cm/min for silica sand 70/100 and 0.227cm/min for silica sand 30/50. Increasing the air velocity from 0.6 to 1.4cm/s for silica sand 30/50 did not result in a higher removal rate. Quantitative estimation of the dissolution rates of benzene NAPL indicated that the dissolution rates (between 0.227 and 0.265cm/min) were similar in magnitude for the same porous media but different air flow rates. Based on the visualization study, air sparging may be used to control the spreading of the dissolved phase even when the glob of NAPL is several centimeters away from the air–water interface of the air channels.     

Similarity Solutions for Thawing Processes with a Heat Flux Condition at the Fixed Boundary

Similarity solutions for a mathematical model for thawing in a saturated semi-infinite porous medium is considered when change of phase induces a density jump and a heat flux condition of the type is imposed on the fixed face x=0. Different cases depending on physical parameters are analysed and the explicit solution is obtained if and only if an inequality for the thermal coefficient q ₀ is verified. An improvement for the existence of a similarity solution for the same free boundary problem with a constant temperature on the fixed face x=0 is also obtained. Sommario. Vengono considerate soluzioni di similarità per un modello matematico di disgelo di un mezzo poroso saturo semi-infinito allorquando il cambiamento di fase induce un salto di densità ed una condizione di flusso di calore del tipo viene imposta sulla faccia fissa x=0. Si analizzano differenti casi dipendenti da parametri fisici e la soluzione esplicita viene ottenuta se e solo se risulta verificata una diseguaglianzo per il coefficiente termico q ₀. Si ottiene altresi un miglioramento della condizione di esistenza di una soluzione di similarità per lo stesso problema al contorno libero con temperatura costante sulla faccia fissa x=0.     

Capillary Imbibition and Pore Characterisation in Cement Pastes

The kinetics of capillary imbibition in ordinary Portland cement pastes has been studied experimentally and theoretically. Nuclear magnetic resonance stray field imaging (STRAFI) has been used to record water concentration profiles for various ingress times. The profiles follow a t law and thus a master curve can be formed using the Boltzmann transformation. The distribution of pore sizes within the sample as measured by NMR cryoporometry shows a prominent peak at 100Å. A computer model of the pore structure was developed consisting of a lattice of interconnecting pores with a size distribution consistent with the cryoporometry results. The Hagen–Poiseuille law was used to describe the kinetics of the water in this pore structure. The best agreement between the computer simulations and the experimental master curve was obtained by using a narrower range of pore sizes than indicated by the cryoporometry results.     

Thermal Conductivity of Soils at Very Low Moisture Content and Moderate Temperatures

An analysis of soil thermal conductivity data shows that, at very low moisture content, this property first varies insignificantly and then begins to increase from a certain critical moisture content, whose value tends to depend on clay mass fraction. Two simple models evaluating the critical moisture content were developed; the first one is a fraction of the permanent wilting point; the second one is a simple linear function dependent on clay mass fraction. An insignificant variation of soil thermal conductivity is observed at 20°C, within a water pressure head ranging from 1×10³ to 1×10³m, while for higher temperatures (45–50°C) from 5000 to 100000m. Three extensions of the enhanced thermal conductivity model by Sundberg, namely SUN-1, SUN-2 and SUN-3, were proposed and tested. They produce an average root mean square error of 27%, 24% and 30%, respectively, with respect to experimental data. SUN-1and SUN-2 predictions could be further improved if better estimates of thermal conductivity at the dry state and the permanent wilting point were provided. SUN-3 is a simple model which does not require information about the PWP and critical moisture content. All SUN models have a potential for implementing the latent heat transfer component.     

Effects of NAPL Contaminants on the Permeability of a Soil‐Bentonite Slurry Wall Material

Soilbentonite slurry walls are designed to inhibit the subsurface movement of contaminants from hazardous waste sites. Although it is generally accepted that high concentrations of organic compounds will adversely affect soilbentonite slurry walls and clay liners, previous research investigating the effects of NAPLs on the conductivity of clay wall materials has been inconclusive. In this study the effects of various organics (benzene, aniline, trichloroethylene, ethylene dichloride, methylene chloride) on the effective conductivity of a typical soilbentonite slurry wall material were studied under two effective stress conditions, 200 and 52kPa. The hydraulic conductivity for the soilbentonite material permeated with water averaged 1.52×10^-8cms^-1. Compared to water, there was little change in conductivity when the sample was permeated with a solution containing a NAPL compound at its solubility limit, except for aniline. However, there was a one to two order of magnitude decrease in conductivity when the sample was permeated with a pure NAPL for all NAPLs tested. When the soilbentonite material was permeated with a water/NAPL/water/NAPL sequence, the conductivity decreased one to two orders of magnitude when a NAPL was introduced following water; however, when water was reintroduced after the NAPL, the conductivity increased to the initial hydraulic conductivity. The conductivity again decreased one to two orders of magnitude when the NAPL was reintroduced. This trend occurred for all NAPLs tested, and the fluid properties of the NAPL compounds alone did not account for the decrease in conductivity compared to water.     

Electrical charging of bodies as a consequence of charged-particle extraction by a hydrodynamic flow

We investigate the problem of electrical charging of bodies as a result of charged-particle extraction by a hydrodynamic flow. The analysis is performed in view of the application to the problem of motion electrification of aircraft caused by a stream of charged particles into the surrounding space. We formulate the appropriate system of nonstationary electrohydrodynamic equations. It is shown that in many applications the charging of electrically insulated bodies consists of two successive intermediate processes. The first process is the formation of charge Q on the body in time T₁ The second process consists of a change of the body potential (with a constant charge Q) as a consequence of the stream of charged particles into the outside space noted above. At the end of the second process (with duration T₂) the body potential is at . We also investigate the problem of charging a spherical source of neutral and charged particles. Using the analytical solution we find the quantities Q and and the characteristic times T₁ and T₂. It is shown that the time T₂ can exceed T₁ by several orders of magnitude. We formulate the problems of nonstationary electric fields during the extraction of several types of particles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 94–103, September–October, 1977.     

Characterization of Waterflood Saturation Profile Histories by the ‘Complete’ Capillary Number

A simple capillary model of free spontaneous imbibition and waterflood is introduced and used to interpret the complete capillary number CA for immiscible displacement in porous media. The complete capillary number is applied to characterize saturation profile histories obtained in sandpacks. Its value is equal to 1 in free spontaneous imbibition, and in a waterflood it is equal to the ratio of the water injection rate to the initial rate of free spontaneous imbibition. In situ saturation profiles in waterfloods in sandpacks at 1:1 viscosity ratio were determined by light transmission measurement. For CA 1 capillary forces controlled and the saturation rose practically uniformly along the entire length of the model from the initial connate water saturation to the final value, corresponding to residual oil saturation. For CA 1 viscous forces controlled and there was a step change in saturation from the connate water to the residual oil value, advancing as a piston. At intermediate values of CA the saturation profiles were S-shaped curves. The saturation profile histories in two packs of different lengths of the same sand were identical, within experimental error, at the same values of CA in the two packs, if plotted against normalized distance coordinates. The saturation profile histories measured in packs of the same length of two very different sands A and B were found to be similar for values of CA in the ratio (CA)_B/(CA)_A 5, for all values of CA.     

Analytical and Experimental Investigation of Squeeze-Film Dampers Executing Circular Orbits

We have referred here the first results obtained in the course of an experimental research intended to examine oil film pressure of a squeeze film damper (with inertialess lubricant) executing circular orbits. An analytical expression for pressure arising into the gap was obtained before, for the short bearing SFD, in the general case of offset circular orbits. The expression obtained has permitted, in a simple way, the examination of the influence of the offset on pressure. Two approximated expressions for pressure (the one for pressure as a function of time at a fixed measuring section, the other for pressure along the circumference of the bearing at a given time), corresponding to centered orbits, were compared with the exact ones. We observed that the limits within which the nondimensional offset can be considered negligible, change according to comparisons between circumferential pressure diagrams or between pressure diagrams as functions of time. In the second case they are much wider. The experimental results concerned offset orbits with nominal radius r=0.5C and offset <0.1. Good accordance between theoretical and experimental pressure trends has been found, as concerns the positive part of dynamic pressure. As concerns the negative part of dynamic pressure, tests have shown that cavitation takes place for a more or less wide length, depending on feeding groove pressure, other conditions being equal. Complete effective pressure diagram coinciding with the theoretical one has been found for particular feeding conditions. Oil film tensile stresses have been frequently detected with both complete and incomplete diagrams owing to cavitation. Maximum measured tensile stress was about 220kN/m². It was finally found that theoretical and experimental trends are coincident only if the SFD is not starved. Analytical expression for pressure in the long bearing SFD executing offset circular orbits is referred in the appendix.     

Well-Posedness for Hyperbolic Systems of Conservation Laws with Large BV Data

We study the Cauchy problem for a strictly hyperbolic n×n system of conservation laws in one space dimension assuming that the initial data has bounded but possibly large total variation. Under a linearized stability condition on the Riemann problems generated by the jumps in we prove existence and uniqueness of a (local in time) BV solution, depending continuously on the initial data in L¹_loc. The last section contains an application to the 3×3 system of gas dynamics.     

The effect of background particulates on the delayed transition of a heated 9:1 ellipsoid

An experimental study was done to quantify the effects of a variety of background particulates on the delayed laminar-turbulent transition of a thermally stabilized boundary layer in water. A Laser-Doppler Velocimeter system was used to measure the location of boundary layer transition on a 50 mm diameter, 9:1 fineness ratio ellipsoid. The ellipsoid had a 0.15 m RMS surface finish. Boundary layer transition locations were determined for length Reynolds numbers ranging from 3.0 × 10⁶ to 7.5 × 106. The ellipsoid was tested in three different heating conditions in water seeded with particles of four distinct size ranges. For each level of boundary layer heating, measurements of transition were made for clean water and subsequently, water seeded with 12.5 m, 38.9 m, 85.5 m and 123.2 m particles, alternately. The three surface heating conditions tested were no heating, T = 10°C and T = 15°C where T is the difference between the inlet model heating water temperature, T _i, and free stream water temperature, T . The effects of particle concentration were studied for 85.5 m and 123.2 m particulates.The results of the study can be summarized as follows. The 12.5 m and 38.9 m particles has no measurable effect on transition for any of the test conditions. However, transition was significantly affected by the 85.5 m and 123.2 m particles. Above a length Reynolds number of 4 × 10⁶ the boundary layer transition location moved forward on the body due to the effect of the 85.5 m particles for all heating conditions. The largest percentage changes in transition location from clean water, were observed for 85.5 m particles seeded water.Transition measurements made with varied concentrations of background particulates indicated that the effect of the 85.5 m particles on the transition of the model reached a plateau between 2.65 particulates/ml concentration and 4.2 particles/ml. Measurements made with 123.3 m particles at concentrations up to 0.3 part/ml indicated no similar plateau.     

Deformable-Drop Dynamics near an Airfoil Surface in Viscous Air

A locally monodisperse air-supercooled water drop flow past an airfoil is studied. The influence of the viscous boundary layer on the force exerted on the drops by the carrier gas and the effect of deformation of the drops on (i) their trajectories, (ii) the water mass flux density over the airfoil surface, and (iii) the wetted-surface length are considered. The effect of an airfoil electrostatic field on the dynamics of charged atmospheric drops is estimated. The role of the governing parameters, i.e. the airfoil angle of attack, the radius of the undisturbed spherical drop, and the drop charge is investigated.The possible origins of the discrepancy between the numerical and experimental results of foreign researchers are indicated.     

One-dimensional flows in electrogasdynamics

We consider electrogasdynamic (EGD) flow in the one-dimensional approximation. The conditions for transition through the speed of sound in EGD are formulated. The distribution of the basic parameters along the flow axis is presented. The concept of the efficiency of the EGD process is introduced, and its analytic expression is given.A comparison is made of experimental and theoretical data on the parameter distribution and efficiency in EGD processes.Notation p pressure - density - V velocity - E electric field intensity - dielectric constant - U potential - L characteristic length - k adiabatic exponent - M Mach number - i charge carrier index - 0 inlet - 1 exit     

Near-electrode layers at “Hot” electrodes

References [1, 2] present a definite approach to the problem of theoretically describing the near-electrode layers for a change of potential at hot electrodes located in a weakly-ionized gas stream. There it was assumed that the degree of gas ionization near the electrode and its temperature are such that the Debye length computed from parameter values near the surface of the electrode is smaller than the mean free path of the charged particles (d ). In this case, the problem of the near-electrode layer is separate from the problem of the distribution of quantities in the gas stream and its solution is used to formulate the boundary condition. Moreover, in solving the problem of the near-electrode layer, we can assume collision-free motion of the charged particles throughout this layer.In the indicated references the near-electrode layer problem was not solved. The quantities necessary for formulating the corresponding boundary conditions were assigned on the basis of physical considerations.In this article, some of the assumptions of [1, 2] are refined on the basis of the solution of model problems and a more detailed analysis of the effects in question. It is shown, in particular, that the theory is inadequate for describing near-electrode layers at electrodes made from a material that is a poor neutralizer for ions.On the basis of an analysis of solutions of model problems of the near-electrode layer for both ion-absorbing and ion-reflecting surfaces, an attempt is made to construct a semi-empirical theory of the near-electrode layer. For certain cases, experimental data are compared with calculated data and, in particular, the experimental constants needed to construct the theory are determined.Although the experimental and calculated data seem to agree, one should remember that the amount of data is still insufficient to make a final decision on the quality of the proposed theory. It is necessary to perform special carefully organized experiments to study near-electrode layers under different conditions. On the basis of an analysis of such experiments the experimental constants of the theory must be determined and then substituted in the calculation formulas. The results of such experiments may considerably modify our ideas about the phenomena taking place in near-electrode layers and may possibly require improvements in the theory.     

Water Impingement on a Vertical Wall due to Discontinuity Decay Above a Drop

The paper reports experimental data on the height of water impingement on a vertical wall during wave reflection due to discontinuity decay above an even bottom and a bottom drop. It is shown that for a bore type wave with a roller in its head, propagating over finitedepth water, the impingement height is proportional to the initial difference in freesurface level. In the case of a dry bottom in the tail water, this is also true for other types of waves formed during discontinuity decay.     

On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0     

Electrohydrodynamic flow in a two-dimensional channel with an axially disposed electrode-emitter

A study is made of the flow of an incompressible nonviscous unipolar charged liquid in a two-dimensional channel ¦ x ¦ <, ¦ y ¦ h with conducting walls and an axially disposed electrode-emitter (along y = 0). The charged particles have an arbitrary constant mobility. The charge distribution on the emitter is approximated as a unit step function. The problem is solved by linearizing the equations with respect to the electrohydrodynamic interaction. The behavior of the electrical parameters is determined, and the deformed profiles of velocity and pressure downstream of the zone in which the electrostatic forces are rotational in character are calculated. These profiles can be determined without having to solve the linearized partial differential hydrodynamic equations in the entire region occupied by the flow, although the profiles then depend on the distribution of the electrical parameters along the entire length of the channel.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 35–40, November–December, 1970.In conclusion the author wishes to thank A. B. Vatazhin for suggesting the problem and for his constant attention throughout the work.     

New concepts in relative permeabilities at high capillary numbers for surfactant flooding

Flooding oil reservoirs with surfactant solutions can increase the amount of oil that can be recovered. Macroscopic modelling of the process requires relative permeabilities to be functions of saturation and capillary number. With only limited experimental data, relative permeabilities have usually been assumed to be linear functions of saturation at high capillary numbers. The experimental data is reviewed, some of which suggest that this assumption is not necessarily correct. The basis for the assumption is therefore reviewed and it is concluded that the linear model corresponds to microscopically segregated flow in the porous medium. Based on new but equally plausible complementary assumptions about the flow pattern, a mixed flow model is derived. These models are then shown to be limiting cases of a droplet model which represents the mixing scale within the porous medium and gives a physical basis for interpolating between the models. The models are based on physical concepts of flow in a porous medium and so the approach described here represents a significant improvement in the understanding of high capillary number flow. This is shown by the fact that fewer parameters are needed to describe experimental data.Notation A total cross-sectional area assigned to capillary bundle - A ⁽ⁱ⁾ physical cross-sectional area of tube i - c ⁽ⁱ⁾ ordered configurational label for droplets in tube i - c configuration label for tube i (order not considered) - D defined by Equation (26) - E(...) expectation value with respect to the trinomial distribution - S _r ⁽⁾ fractional flow of phase - k absolute permeability - k _r relative permeability of phase - k _r ⁰ endpoint relative permeability of phase - L capillary tube length in bundle model - m ⁽ⁱ⁾ number of droplets of phase a occupying tube i - n exponent for phase a in Equation (2) - N number of droplets in bundle model - N _c capillary number - p pressure - p(c') probability of configuration c - Q ⁽ⁱ⁾ total volume flow rate in tube i - S saturation of phase - S flowing saturation of phase - S _r residual saturation of phase - S _r ⁽⁾ saturations when fractional flow of phase is 1 in the case of varying residual saturations for three-phase flow ( ) - t _c residence time for droplet configuration c - v ⁽ⁱ⁾ total fluid velocity in bundle tube i - , phase label - p pressure differential across capillary bundle - ⁽ⁱ⁾ tube conductivity defined by Equation (7) - viscosity of phase - interfacial tension - gradient operator - ... average over tube droplet configurations