Macroscopic Dispersion

1964 ◽  
Vol 4 (03) ◽  
pp. 215-230 ◽  
Author(s):  
Joseph E. Warren ◽  
Francis F. Skiba
Keyword(s):  
Porous Medium ◽  
Monte Carlo ◽  
Molecular Diffusion ◽  
Monte Carlo Model ◽  
Laboratory Data ◽  
Miscible Displacement ◽  
Oil Field ◽  
Rock Properties ◽  
Porous System ◽  
Displacement Processes

Abstract An idealized miscible displacement process in a three-dimensional, heterogeneous porous medium has been studied via experimental computation based on a "Monte Carlo" model. The macroscopic dispersion which results solely from variations in the permeability of the porous system is related to the scale of the heterogeneity as well as the distribution function for the permeabilities. The interpretation of laboratory data is often subjective and may be misleading, since ambiguities are not always recognized. Furthermore, an experiment performed on a conventional oil-field core does not yield a valid measure of macroscopic dispersion for reservoir engineering purposes since the scale of heterogeneity that is significant in the laboratory is not significant in the reservoir. The analysis of field-performance data is discussed and it is shown that a unique characterization of the reservoir can not be obtained from this in formation alone. A possible method, based on a minimum of data, for estimating the spatial distribution of permeabilities in a reservoir is postulated. Introduction Many studies concerned with diffusion and dispersion in porous media have been undertaken because of the potential importance of miscible displacement processes for increasing the recovery of oil. The results from a large number of these studies have been published; in a recent paper Perkins and Johnston presented a comprehensive survey of the pertinent literature and excellent appraisal of the "state of the art".In this investigation, dispersion resulting from macroscopic variations in the properties of the porous medium is of primary interest. Dispersion, in this sense, is caused by fluctuations in the velocities of the individual fluid elements as they move through the porous system. This phenomenon is the result of inhomogeneities in the permeability and/or porosity, but it can be amplified by differences in the characteristics of the fluids involved. The objectives of this study are the following:to determine the qualitative manner in which macroscopic rock properties affect the observed dispersion coefficient;to evaluate the possible influence of macroscopic dispersion on laboratory experiments; andto appraise the significance of macroscopic dispersion with regard to reservoir performance. Investigations of this type are essential since many miscible displacement processes, while seemingly understood for a mechanistic point of view, do not behave in a predictable manner under field conditions. Although a plausible explanation is that the process is dominated by the environment, little serious effort has been devoted either to describing the reservoir itself or to determining the probable effect of the reservoir properties on the process. This study is intended to be exploratory in nature; however, it is hoped that it will provide additional insight into the physical problem and that it will suggest areas suitable for further investigation. EXPERIMENTAL COMPUTATION MODEL To limit the observed dispersion to that which results solely from variations in the properties of the porous medium, an idealized process in which one incompressible fluid displaces another identical fluid is simulated. This simulation is achieved by means of a modified Monte Carlo method - by tracking a number of mathematical particles (tracer) through the system according to the steady-state, single-phase velocity distribution. Then the system is characterized by the distribution of the residence times of the particles. Since any number of particles can be injected simultaneously (a true delta-function distribution) and only pressure conditions are imposed on the boundaries, the model is completely determinate. Furthermore molecular diffusion, microscopic dispersion, adsorption and/or reaction are automatically precluded from the model. SPEJ P. 215^

A Method for Predicting the Performance of Unstable Miscible Displacement in Heterogeneous Media

1963 ◽  
Vol 3 (02) ◽  
pp. 145-154 ◽  
Author(s):  
E.J. Koval
Keyword(s):  
Porous Medium ◽  
Heterogeneous Media ◽  
Research Effort ◽  
Heterogeneous Systems ◽  
Extended Period ◽  
Miscible Displacement ◽  
Longitudinal Dispersion ◽  
Miscible Displacements ◽  
Geometrical Effects ◽  
Displacement Processes

KOVAL, E.J., CALIFORNIA RESEARCH CORP., LA HABRA, CALIF. Abstract Practical miscible displacement processes will be characterized by fingering of the solvent into the oil. The fingering process is brought on by viscosity differences, and can be accentuated by channeling and longitudinal dispersion. The effects of these factors on the efficiency of unstable completely miscible displacements are combined in what is called the K-factor method. This method, analogous to the Buckley-Leverett method, predicts recovery and solvent cut as a function of pore volumes of solvent injected. Experimental data are included and show excellent agreement with theory for a wide variety of sandstone cores and viscosity ratios. Introduction Theoretical considerations, laboratory experiments, and pilot tests lead to the conclusion that miscible displacements in the field will be unstable. In an unstable miscible displacement, the solvent fingers through the oil. This fingering leads to early breakthrough of the solvent and an extended period during which both oil and solvent are produced.For such a system, there appear to be four principal factors which bring about or accentuate the effects of instability: longitudinal dispersion (including geometrical effects), channeling, viscosity differences and gravity differences. Other factors, such as diffusion and flooding rate, can also influence the effects of instability, but at the flooding rates considered in this report, circa 15 ft/D, they are unimportant. Longitudinal dispersion can be thought of as a spreading of the solvent front caused by the presence of microscopic inhomogeneities. Channeling of the solvent occurs when a porous medium has macroscopic inhomogeneities; i.e., gross permeability variations. Viscosity differences lead to fingering of the less viscous solvent. This difference in viscosity accelerates the growth of fingers along paths previously developed because of permeability variations. Gravity differences lead to overriding of the usually less dense solvent. Although gravity effects are generally small at a flooding rate of 15 ft/D, they would, nevertheless, unnecessarily complicate the interpretation of flooding experiments. For this reason, all the experiments reported herein were done with matched density fluids.Fingering and the resultant poor areal sweep were recognized early as the dominant influences on the efficiency and the economics of miscible displacement processes. Much research effort has been spent on ways to minimize fingering and increase areal sweep such as the use of graded viscosity slugs or water slugs. Some researchers attempted to work out ways to prevent fingering completely; i.e., to achieve a stable displacement through gravity or latitudinal dispersion stabilization. Others did not attempt to control fingering but obtained an economic process by merely recycling the solvent and sweeping pattern by pattern.During this period, all aspects of fingering came under close scrutiny. Some researchers reported on how fingering looks and how it is affected by viscosity ratio, geometry, and slug size. Peaceman and Rachford suggested a mathematical approach to the prediction of unstable miscible displacements in relatively homogeneous sand packs, but their work cannot be extended conveniently to heterogeneous systems. Hence for a heterogeneous system, no method is presently available for predicting solvent cut and recovery as functions of pore volumes of solvent injected.The purpose of this investigation was to attempt to fill in the gap in our knowledge concerning the prediction of performance of unstable miscible displacements. Necessarily, the system selected for study was a relatively simple one. The restrictions placed on the system were:The system was linear;The solvent was miscible in all proportions with the oil in place;The solvent was continuously injected into the porous medium;Gravitational effects were eliminated by matching densities andAll the flood rates were high and constant at 15 ft/D to avoid any small rate effect and to minimize any diffusion effects. To simplify and to indicate that both longitudinal dispersion and channeling arise from permeability variations, the effects which they cause or influence have been termed heterogeneity effects. SPEJ P. 145^

A New Method To Simulate the Effects of Viscous Fingering on Miscible Displacement Processes in Porous Media

1984 ◽  
Vol 24 (01) ◽  
pp. 56-64 ◽  
Author(s):  
Shapour Vossoughi ◽  
James E. Smith ◽  
Don W. Green ◽  
G. Paul Willhite
Keyword(s):  
Porous Media ◽  
Dispersion Equation ◽  
Viscosity Ratio ◽  
Laboratory Data ◽  
Viscous Fingering ◽  
Miscible Displacement ◽  
Concentration Profiles ◽  
Flow Function ◽  
Fractional Flow ◽  
Displacement Processes

Abstract Dispersion and viscous fingering are important parameters in miscible displacement. Effects of dispersion on concentration profiles in porous media can be simulated when the viscosity ratio is favorable. The capability to simulate viscous fingering is limited. This paper presents a new method to simulate effects of viscous fingering on miscible displacement processes in porous media. The method is based on the numerical solution of a general form of the convection-dispersion equation. In this equation the convection term is represented by a fractional flow function. The fractional flow function is derived from Darcy's law by using a concentration-dependent average viscosity and relative flow area to each fluid at any point in the bed. The method was extended to the description of a polymer flood by including retention and inaccessible PV. A Langmuir-type model for polymer retention in the rock was used. The resulting convection-dispersion equation for displacement by polymer was solved numerically by the use of a finite-element method with linear basis functions and Crank-Nicholson derivative approximation. History matches were performed on four sets of laboratory data to verify the model:an unfavorable viscosity ratio displacement,stable displacement of glycerol by polymer solution,unstable displacement of brine by a slug of polymer solution, anda favorable viscosity ratio displacement. In general, computed results from the model matched laboratory data closely. Good agreement of the model with experiments over a significant range of variables lends support to the analysis. Introduction Considerable effort has been directed to the study of dispersion phenomena in flow through porous media. Dispersion phenomena become important in EOR techniques, especially those involving the use of chemical slugs such as a micellar/polymer flood. Because the micellar solution is expensive, a carefully designed polymer buffer solution must be injected between the microemulsion and the drive water. This minimizes the effect of mixing and dispersion that otherwise would cause the micellar slug to lose its effectiveness. Aronofsky and Heller1 were among the first to use the diffusion or dispersion model to describe miscible displacement. This employs Fick's law of diffusion to describe the transport of mass within the zone containing both displacing and displaced fluids. The so-called convection-dispersion equation obtained by differential material balance has become generally accepted as the basis for analysis of miscible displacements. The dispersion equation has been solved numerically2–6 as well as analytically6,7 to obtain concentration profiles and dispersion coefficients. However, the prediction fails whenever viscous fingering occurs. Viscous fingering is the result of an unstable displacement of a more viscous fluid by a less viscous fluid. Finger-shaped intrusions of the displacing fluid into the displaced fluid have been observed and reported in the literature8–11 for miscible as well as immiscible displacements.

Monte Carlo Simulations of Phonon Backscattering in Low-Sized Nanowires

2009 ◽  
Author(s):  
Zan Wang ◽  
Yunfei Chen ◽  
Jiapeng Li ◽  
Ruijie Zhao ◽  
Yanyan Zhao
Keyword(s):  
Surface Roughness ◽  
Monte Carlo ◽  
Silicon Nanowires ◽  
Lattice Thermal Conductivity ◽  
Monte Carlo Model ◽  
Laboratory Data ◽  
Equivalent Diameter ◽  
Influence Coefficient ◽  
Roughness Effects ◽  
Electroless Etching

Monte Carlo model is used to study the roughness effects on lattice thermal conductivity of silicon nanowires. Based on the Matthiessen’s rule, a roughness influence coefficient is adopted to amend the relaxation time. Individual silicon nanowires growing by vapour-liquid-solid method are different from those making by aqueous electroless etching method in the surface roughness, which are both simulated over a temperature range of 15–315K with various equivalent diameters. The results are close to the laboratory data, proving the validity of the MC model and showing the effects of surface roughness on thermal transport decrease with the increasing of equivalent diameter.

Monte Carlo model for describing charge transfer in irradiated CCDs

1998 ◽  
Author(s):  
Dennis J. Gallagher ◽  
Raymond Demara ◽  
Gary Emerson ◽  
Wayne W. Frame ◽  
Alan W. Delamere
Keyword(s):  
Monte Carlo ◽  
Charge Transfer ◽  
Monte Carlo Model

Monte Carlo model of multi-site adsorption

1985 ◽  
Vol 8 (7) ◽  
pp. 364-365 ◽  
Author(s):  
J. Sedláček ◽  
L. Nondek
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Model
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