Transverse Dispersion Coefficients Research Articles

Displacement of oil by Solvent at High Water Saturation

Abstract Displacements of laboratory oils by propane in long, consolidated sandstone cores in the presence of high water saturations have shown that oil recoveries approaching 100 percent may be realized by continuous water-propane injection, even for oil saturations close to residual oil. However, it was often necessary to inject many pore volumes of solvent to attain this high a recovery. Initial oil saturations were established by injecting water and oil at a constant ratio into the porous medium containing residual oil to a waterflood until a steady state was obtained. Propane and water were then injected in the same fixed ratio to displace the oil. These and other experiments indicate that in the presence of a high water saturation only part of the presence of a high water saturation only part of the oil is flowable. Part resides in locations that are blocked by water, and the oil in these stagnant locations is not flowable. This nonflowable oil, it is believed, can be recovered by molecular diffusion into the flowing propane of a water-propane displacement. Values for the saturation of hydrocarbon that is contained in the stagnant locations and values for the ratio of the longitudinal hydrodynamic-dispersion coefficient to displacement velocity were determined at various water saturations in the test sandstones. The data suggest that rock wettability may influence the stagnant saturation and that stagnant oil saturations may not be as large in reservoir rocks as they are observed to be in laboratory sandstones. Mass transfer between the flowing solvent and hydrocarbon components in the stagnant saturation was expressed by a first-order rate expression. Rough values for the mass transfer coefficients for the propane-trimethylhexane hydrocarbon pair were estimated from experiments. Computations using these values for mass transfer coefficients indicate that experiments in laboratory-size cores may show much poorer displacement efficiency than that which might actually occur in the field. Introduction Injection of water with light hydrocarbon solvents is a technique that may be used to partially control solvent mobility. The higher water saturation forced by water injection reduces the permeability to solvent flow, and the mobility of the solvent region is reduced relative to that of the oil-bank region. However, it also might be expected that this higher water saturation influences the microscopic unit displacement of oil by solvent to some degree. For example, as discussed by Thomas et al., two possible effects of high water saturation on the displacement mechanism come to mind. First, a miscible displacement in the presence of water is operating on a different pore-size distribution than if no water were present. Pore-size distribution and the dp term (product of the microscopic inhomogeneity factor and "effective" particle diameter) may considerably influence the magnitudes of transverse and longitudinal dispersion coefficients. Secondly, in a multiphase system the wetting phase may trap single pores or even isolate large fingers or dendrites of the nonwetting phase. The nonwetting phase in these dead-end pores or dendrites would be phase in these dead-end pores or dendrites would be nonflowing and might either be completely isolated by the wetting phase or might communicate with the flowing nonwetting fluid by diffusion through nonwetting fluid-filled pores. Aspects of miscible displacement in the presence of water have been investigated by a number of researchers. Fitzgerald and Nielson observed that the simultaneous injection of gasoline and water into a Berea sandstone core in a 1:2 ratio recovered only 36 percent of the Bradford crude oil left in the core after waterflooding, and that only 55 to 75 percent recoveries were obtained for simultaneous water-solvent injection into the core when it contained crude oil at connate water saturation. Moreover, these authors reported recoveries of only 60 to 80 percent when solvent alone was injected into the core to displace residual oil to a waterflood. Raimondi et al. injected ethyl benzene (oil) and water simultaneously into a Berea sandstone core to establish flowing oil and water saturations and then injected heptane (solvent) and water simultaneously into the core to miscibly displace the ethyl benzene.

A Mathematical and Experimental Examination of Transverse Dispersion Coefficients

Abstract Transverse dispersion has received considerably less treatment in the literature than has longitudinal dispersion. Different methods for determining transverse dispersion coefficients have been used in different investigations, and the results obtained have not been consistent enough to permit accurate generalizations as to the effect of various physical parameters on the magnitude of these coefficients. A numerical solution to the differential equation describing transverse dispersion in the absence of longitudinal dispersion was obtained to enable one to calculate the dispersion coefficient from experimental results. The more general dispersion equation including longitudinal dispersion also was solved numerically to give quantitative limits of a dimensionless group within which the assumption of negligible longitudinal dispersion is justified. Possible experimental procedures were examined, and one utilizing a cylindrical packed column was chosen for the determination of transverse dispersion coefficients. Values of these coefficients were determined for a system of two miscible organic fluids of equal density and viscosity, for two sizes of packing material over a wide range of flow rates in the laminar regime. The dispersion coefficient was found to decrease, for a constant value of the product of packing size and interstitial velocity, as the size of the packing material particles increased. Introduction Longitudinal dispersion has received extensive treatment in the literature, and consequently is better understood than its orthogonal counterpart, transverse dispersion. Many mathematical models of dispersion processes assume that transverse dispersion is rapid enough to damp out any radial concentration gradients and therefore may be neglected. Laboratory and production results, however, indicate that this is a poor assumption. Various experimental procedures for determining transverse dispersion coefficients have been used in previous investigations, but the results have generally been expressed by similar correlations. The transverse dispersion coefficients obtained, however, have often varied considerably for given values of the correlation parameters. We feel that further experimental determinations of transverse dispersion coefficients will help alleviate some of the inconsistencies in these empirical correlations. Background In a porous medium, the "effective molecular diffusivity" De is less than the molecular diffusivity D measured in the absence of a porous medium, due to the tortuous path which a diffusing molecule must travel. Various authors have reported values of the ratio De/D in the range of 0.6 to 0.7. When there is fluid flow within the porous medium, mass transfer occurs by convective dispersion as well as by molecular diffusion. These are separate phenomena and can be treated as such on a microscopic scale. However, the mathematical complexity is such that only extremely simple geometries could be considered, and the results hardly would be applicable to the complex geometries existent in actual porous media.

Dispersion Coefficients for Gases Flowing in Consolidated Porous Media

Abstract The best currently available description of the longitudinal mixing properties of a porous medium is an equation of the formEquation 1 which relates the effective longitudinal dispersion coefficient Dl to the molecular diffusion coefficient D0, the electrical resistivity factor F, the porosity f and a Peclet number. If the parameters dps and m are determined for a porous medium of known porosity and electrical resistivity factor, then a dispersion coefficient may be estimated for a given flow rate and a given gas pair. A new method, featuring on-line gas analysis by thermal conductivity and on-line data reduction by analog computation, was developed and used to determine these mixing parameters for eight naturally occurring sandstones and two dolomite samples. The exponent m of the above equation was found to vary between 1.0 and 1.5. The characteristic length dp s in the above equation was found to vary between 0.25 and 1.9 cm, with an average value of 0.4 cm for sandstones. Measurements were made on two cores in which paraffin wax had been deposited by evaporation from a pentane solution. They indicated that the presence of an immobile phase such as connate water could increase the dispersion coefficients significantly. INTRODUCTION While the petroleum and chemical industries have studied the mixing of miscible liquids flowing in consolidated porous media and of miscible gases flowing in unconsolidated porous media, relatively little data have been presented to describe the mixing of gases flowing through consolidated porous media. Such data are of particular interest to the gas storage industry. For instance, the U.S. Bureau of Mines is storing large quantities of a rich helium-nitrogen gas in contact with a natural gas in a dolomite reservoir. Since the rich gas occupies only 15 percent of the total reservoir volume, it is essential that the extent of rich gas-natural gas mixing be predicted and understood as a function of rock properties, pressure and rate of movement. This investigation was concerned only with the determination of longitudinal dispersion coefficients. It is understood that a transverse dispersion coefficient, which characterizes mixing perpendicular to the direction of flow, may be an order of magnitude less than the coefficient characterizing mixing in the direction of bulk flow.5,19 It should also be recognized that the use of any dispersion coefficient is in itself a simplification. It is necessary to assume that mixing in a porous medium may be characterized by the equationEquation 2 for flow in a single direction. A number of authors1 have pursued the mixing problem, not in terms of the so-called "dispersion model" described by Eq. 1, but in terms of a "mixing cell model". This model supposes that a porous medium is constructed of a large number of small mixing chambers and that the concentration of the diffusing component within each mixing chamber is uniform. Fick's law (Eq. 1) assumes that there is no gross by-passing of one fluid by another, and that there are not stagnant pockets of gas in the system under consideration as discussed by Coats and Smith.8 These assumptions are not always valid for flow through porous media and it is important to recognize the limitations upon Eq. 1.

Dispersion Measurement in Open Channels

The true longitudinal and transverse dispersion coefficients can be computed with time-concentration data collected in a two-dimensional tracer plume, which originated from an instantaneous point source. This fact can be of significant help in reducing the physical effort and the cost of field measurements. An empirical formula has been developed from flume data which is superior to the deterministic formulas derived for dispersion coefficient prediction. The same type of formula may be useful in prototype streams. There is not sufficient data presently available to adequately determine the utility of the empirical formula for use on prototypes.

Design of Laboratory Models for Study of Miscible Displacement

Abstract Scaled laboratory-model studies provide a powerful method for evaluation of a proposed oil-recovery process. In recent years, models have been used extensively to evaluate processes in which solvents displace oil, both for general cases and for specific reservoir conditions. Since the performance of a miscible flood in a horizontal reservoir can be significantly affected by transverse mixing between solvent and oil, this displacement mechanism must be accurately simulated in the scaled model studies. Unfortunately, precise scaling of transverse dispersion coupled with the requirement of geometric similarity requires impractically large laboratory models and long times for experiments.If scaling requirements for miscible displacements could be relaxed while accurate simulation of essential displacement mechanisms is maintained, the utility of model studies would be greatly enhanced. The purpose of the work reported herein was to evaluate the relative importance of various mechanisms affecting miscible displacement and to ascertain whether the essential features of the displacement process can be simulated even though some scaling groups are not satisfied. These studies were performed with completely miscible systems in linear, horizontal models packed with unconsolidated media.From the experimental results, a set of relaxed scaling criteria was formulated which allows the requirements of geometric similarity and equality of the ratio of viscous to gravity forces to be omitted for specified conditions. The relaxed criteria are valid whether transverse mixing is by molecular diffusion or by convective dispersion.Correlations which permit prediction of vertical sweep efficiencies in linear, horizontal reservoirs were developed from the experimental data when transverse mixing is by molecular diffusion, These same correlations may be used when transverse mixing is by convective dispersion if an empirically defined, effective, transverse dispersion coefficient is used in the description of the mixing process. The effective transverse dispersion coefficient correlation essentially duplicates the dispersion coefficient correlation for equal-viscosity, equal-density fluid systems. Experimental values for the effective transverse dispersion coefficient can be measured readily. Introduction One of the most effective methods for evaluation of miscible-displacement oil-recovery processes is that of displacements in laboratory models scaled to simulate reservoir conditions. For these laboratory studies to be meaningful, however, the essential displacement mechanisms affecting reservoir performance must be accurately simulated.Since the performance of a miscible flood in horizontal reservoirs, or in dipping reservoirs at high rates, can be significantly affected by transverse mixing of solvent and oil, this mechanism must be considered in the design of laboratory experiments. Unfortunately, precise scaling of transverse dispersion coupled with the requirement of geometric similarity requires impracticality large laboratory models and long experiment times. This difficulty seriously limits the utility of laboratory model studies.Craig, et al, demonstrated that geometric similarity is not required when mixing is unimportant. Their experimental data indicate, for the cases studied, that the displacement is sufficiently characterized by scaling the ratio of viscous-to-gravitational forces. This work suggests that relaxation of the requirement of geometric similarity and, possibly, other criteria might also be permissible when mixing is important, provided suitable groups describing the mixing process are scaled.The purpose of the work reported here was to evaluate the relative importance of various mechanisms affecting miscible displacement and to ascertain whether the essential features of the displacement process can be simulated even though some scaling groups are not satisfied. SPEJ P. 28^