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.

Mathematical Model of an Unstable Miscible Displacement

1963 ◽  
Vol 3 (02) ◽  
pp. 155-163 ◽  
Author(s):  
E.L. Dougherty
Keyword(s):  
Mathematical Model ◽  
Porous Media ◽  
Viscosity Ratio ◽  
Miscible Displacement ◽  
Factor H ◽  
Hyperbolic Partial Differential Equations ◽  
Mixing Process ◽  
Fractional Flow ◽  
One Dimensional ◽  
Pure Solvent

DOUGHERTY, E.L., CALIFORNIA RESEARCH CORP., LA HABRA, CALIF. JUNIOR MEMBER AIME Abstract A phenomenological theory for a one-dimensional unstable miscible displacement similar in type to the Buckley-Leverett model but including the effects of mixing is proposed An equation giving the fractional flow of pure solvent through an oil phase containing dissolved solvent is derived including the effects of gravity and nonhomogeneities. Numerical integration of the resulting pair of nonlinear hyperbolic partial differential equations gives volumes of dissolved and undissolved solvent as functions of space and time. Parameters in the model which characterize mixing due to dispersion were determined from experimental data; the parameters correlated with viscosity ratio. The results indicate that the rate of dispersive mixing is proportional to volumetric flow rate Incorporated into our equations were Koval's heterogeneity factor H, which characterizes mixing due to channeling. This allowed satisfactory predictions of displacement behavior observed in horizontal floods in nonhomogeneous cores; but in most cases the values of H which we used were considerably larger than those used by Koval. Introduction It has been shown that for favorable viscosity ratios, the diffusion equation with convection satisfactorily describes the behavior of a miscible displacement in a porous media. Numerous attempts to develop a satisfactory mathematical description for the case of unfavorable viscosity ratio have been reported, but they have met with only partial success. These attempts, which are reviewed by Koval, have taken two approaches:development of a flow model akin to the Buckley-Leverett approach for immiscible displacement neglecting the effects of mixing andsimultaneous application of Darcy's Law for flow and the diffusion equation with a convection term for mass transfer. We set out to construct a mathematical model of Type 1 including, though, the effects of mixing. Based upon a set of hypotheses, we derived a pair of non-linear hyperbolic partial differential equations which describe in a one-dimensional system the combined effects of flow and mixing. To work with these equations it was necessary to develop a fractional flow formula. The combined system was integrated numerically using the method of characteristics.In our equations the mixing process is characterized by four parameters. Three of these, labeled beta, p and p, account for dispersive type mixing. The fourth is the heterogeneity factor H, proposed by Koval to account for channeling due to nonhomogeneities in the porous media. Values of the parameters which would cause agreement between theory and experiment were determined by trial-and-error for miscible floods conducted in horizontal cores of both homogeneous and heterogeneous materials. Calculations were also performed for vertical floods in homogeneous cores.The purpose of this paper isto present the details of the mathematical analysis,to present the results of the calculations,to consider what light the results shed on the mixing process in an unstable miscible displacement, andto provide a firmer foundation for the correlation technique developed by Koval for predicting the behavior of unstable miscible floods. STATEMENT OF PROBLEM The problem is to construct a mathematical model which describes in one dimension the observed behavior in an unstable miscible displacement. The approach is phenomenological in that the equations are based on assumptions which violate certain physical precepts known to apply to the displacement phenomenon. However, the assumptions do allow us to account mathematically for the more essential phenomena. The work is of value if we can quantitatively predict experimental results which heretofore could not be predicted, even though the synthetic nature of the model belies complete explanation of the observations.We assume that the system is comprised of two contiguous flowing phases. One phase, which we call free solvent, has the properties of pure solvent. We call the fraction of the pore volume occupied by this phase the solvent saturation, designated s. SPEJ P. 155^

scholarly journals Stable and unstable miscible displacements in layered porous media

2019 ◽  
Vol 869 ◽  
pp. 468-499 ◽  
Author(s):  
Japinder S. Nijjer ◽  
Duncan R. Hewitt ◽  
Jerome A. Neufeld
Keyword(s):  
Porous Media ◽  
Numerical Simulations ◽  
Viscosity Ratio ◽  
Flow Direction ◽  
Relative Magnitude ◽  
Theoretical Modelling ◽  
Ambient Fluid ◽  
Miscible Displacements ◽  
Layered Porous Media ◽  
Displacement Processes

The effect of permeability heterogeneities and viscosity variations on miscible displacement processes in porous media is examined using high-resolution numerical simulations and reduced theoretical modelling. The planar injection of one fluid into a fluid-saturated, two-dimensional porous medium with a permeability that varies perpendicular to the flow direction is studied. Three cases are considered, in which the injected fluid is equally viscous, more viscous or less viscous than the ambient fluid. In general it is found that the flow in each case evolves through three regimes. At early times, the flow exhibits the concentration evolves diffusively, independent of both the permeability structure and the viscosity ratio. At intermediate times, the flow exhibits different dynamics including channelling and fingering, depending on whether the injected fluid is more or less viscous than the ambient fluid, and depending on the relative magnitude of the viscosity and permeability variations. Finally, at late times, the flow becomes independent of the viscosity ratio and dominated by shear-enhanced (Taylor) dispersion. For each of the regimes identified above, we develop reduced-order models for the evolution of the transversely averaged concentration and compare them to the full numerical simulations.

Experiments on Mixing During Miscible Displacement in Porous Media

1961 ◽  
Vol 1 (01) ◽  
pp. 1-8 ◽  
Author(s):  
William E. Brigham ◽  
Philip W. Reed ◽  
John N. Dew
Keyword(s):  
Porous Media ◽  
Breakthrough Curve ◽  
Glass Bead ◽  
Viscosity Ratio ◽  
Error Function ◽  
Breakthrough Curves ◽  
Miscible Displacement ◽  
Mixing Length ◽  
Theoretical Error ◽  
Function Relationship

Abstract The paper describes experiments on miscible displacement in various porous media and the results of these experiments. Both glass bead packs and natural cores were used. Bead diameters varied from 0.044 to 0.47 mm, and pack lengths varied from 83 to 678 cm. Natural cores used were Berea and Torpedo sandstone. By taking samples as small as 0.5 cc and using refractive index for analysis, the data on break through curves could be plotted to within ± 0.5 per cent. To plot the data correctly on error function paper, a parameter (Vp - v)/vV was used which allowed for the predicted growth of the front as it moved past the observer. The change in the amount of mixing (length of mixed zone) was studied by varying velocity, length of travel, bead size, viscosity ratio and pack diameter. When the displaced material was less viscous than the displacing material (favorable viscosity ratio), these changes were adequately predicted by theory. When natural cores were used, rather than glass beads, the amount of mixing was greatly increased - also qualitatively predicted by theory. In experiments with favorable viscosity ratios in which the ratio was varied from 0. 175 to 0. 998, it was found that the rate of mixing was changed by a factor of 5. 7. Thus, the rate of mixing is strongly affected by viscosity ratio, even when the theoretical error function relationship for mixing is valid. Experiments using fluids with viscosity ratios near 1.0 showed that the instability effects of even a slightly unfavorable viscosity ratio (1.002) caused disproportionately more elongated breakthrough curves than found with a favorable viscosity ratio (.998). When the viscosity ratio was as high as 5.71 these instability effects were much more pronounced, as evidenced by the shape of the breakthrough curve. The displacements at viscosity ratios above 1.0 no longer followed the theoretical error function curve.

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^

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