Steady-state Relative Permeability Research Articles

The difference between steady-state and unsteady-state relative permeabilities

In constructing simulations, it is commonly assumed that steady-state relative permeabilities can be used to describe displacements, which are unsteady-state flows. In order to test this assumption, a statistical structural model is used in the context of local volume averaging to examine the effects of interfacial tension, the interfacial viscosities, wetting and the viscosity ratio upon the capillary pressure and relative permeability functions in a displacement. The statistical structural model is a highly simplified idealization of the local pore structure, in which pores are randomly oriented in space, but no pore interconnections are recognized. While the statistical structural model is too simplified to give accurate descriptions of the relative permeabilities, it should be sufficient to determine whether significant qualitative differences between steady-state and unsteady-state relative permeabilities might be expected.

Relative Permeability Curves For Bitumen And Water In Oil Sand Systems

Abstract Relative permeability relationships for two- and three-phase flow are a vital part of numerical simulations, but are difficult to measure in the laboratory with any degree of certainty. When temperature is involved as an additional variable, the goal becomes more elusive. This paper presents steady state relative permeability measurements for Athabasca bitumen and water in resaturated sand packs. Injections of various bitumen/water mixtures were performed to cover the entire saturation range from irreducible water to residual oil. A total of thirteen experiments were performed, twelve vertically and one horizontally. Most measurements were conducted at 125 ºC. Additional measurements made at 175 ºC did not show any significant temperature dependence. A single set of bitumen-water relative permeability curves were obtained from the experimental data. The resulting curves exhibited an unconventional behaviour; the bitumen curve was found to be convex. This shape may be a reflection of the displacement process taking place. Furthermore, a comparison of experimentally obtained normalized relative permeability curves for Alberta heavy oils has shown that a wide variation existed among the reported data. Introduction In the last decade, computer-assisted simulation has become a major tool in evaluating and predicting reservoir performance. Description of reservoirs and understanding of oil displacement mechanisms due to primary, secondary or tertiary recovery processes are among the most important features of a good reservoir simulator. However, reliable input parameters are needed to carry out a satisfactory reservoir simulation and provide a predictive capability. These parameters have to be carefully measured in the laboratory. Caution should also be exercised in their interpretation and scale-up to reservoir conditions for use in reservoir simulation studies. Relative permeability is one such key parameter. Laboratory determined relative permeability functions are useful in analyzing immiscible displacement problems. The experimental techniques to measure relative permeability have been developed for light oil reservoirs. The measurements are performed either by the steady state (mixture njection) technique or the unsteady state (dynamic displacement) method. The latter has received more attention due to its faster turn-around time. However, both methods have inherent limitations. In dealing with heavy oils, the steady statemethod, although time-consuming, can provide the desired data. The purpose of this study was to determine the two-phase (oil-water) relative permeability-saturation behaviour for bitumen-containing unconsolidated water-wet sands. Steadystate measurements were conducted for the bitumen-hot waterclean unconsolidated sand system at 125 ºC. Some experiments were also conducted at 175 ºC. The shape of the resulting relative permeability curves was examined in terms of the displacement process taking place. The measured relative permeability data were also compared to the data available in the literature. Literature Review Relative Permeability Measurement The concept of relative permeability is an empirical extension of Darcy's Law from single to multiphase flow through porous media. Because the validity of this extension is uncertain, great care must be exercised when attempting to extend its area ofapplicability. Measurements of relative permeability have been performed since the mid-3Ds. Wycoff and Botset(1) described the relative permeability concept for oil, gas and water flowing simultaneously through a porous medium.

Influence of Interfacial Tension on Gas/Oil Relative Permeability in a Gas-Condensate System

SummaryTo understand better the effect of interfacial tensions (IFT's) on gas/oil relative permeabilities, with particular emphasis on those effective in condensate reservoirs, an experimental procedure was developed and used with the highly volatile methane/propane system. The objective was to measure steady-state relative permeabilities as functions of IFT. Thus the IFT was varied from 0.03 to 0.82 dynes/cm [0.03 to 0.82 mN/m], corresponding to pressures near the critical at a constant temperature of 70°F [21°C].Individual relative permeability curves obtained as a function of gas saturation approach the 45° [0.79-rad] diagonals for both gas and oil as the IFT is lowered. This supports the expectation that relative permeability curves for both gas and oil become straight lines as the IFT approaches zero. At the highest IFT for which experiments were performed (0.82 dynes/cm [0.82 mN/m]), the relative permeability curves approached those obtained for a nitrogen/kerosene flood for which the IFT is approximately 30 dynes/cm [30 mN/m].The most important conclusions derived from this work are that (1) the curvatures of the relative permeability curves diminish as the IFT is reduced, (2) the irreducible gas and liquid saturations approach zero as the IFT approaches zero, and (3) relative gas/oil permeabilities for gas-condensate reservoirs are altered from the normal relative permeabilities only at pressures, temperatures, and compositions close to the critical point.

Effect Of Temperature On Bitumen-Water End Point Relative Permeabilities And Saturations

Abstract This paper presents the results of an investigation on the effects of temperature on end point relative permeabilities for bitumen-containing systems. End point bitumen-water relative permeabilities and saturations were measured for constant rate floods through synthetic oil sand packs subjected to overburden pressure. A procedure was developed to obtain reproducible, clean, unconsolidated sand packs having porosities and absolute permeabilities of the same order of magnitude as rich oil sands. Bitumen and water displacement experiments were performed at constant temperatures and pressures representative of steamflood conditions. End point bitumen-water relative permeabilities and saturations were found to be independent of temperature in the range of 125 °C to 250 °C. The significance of this result can be important for reservoir engineers in predicting reservoir performance relative permeabilities for specific reservoir materials could be measured at some convenient temperature and be applied to the whole temperature range to which the reservoir is subjected to. Introduction The numerical ~imulation of thermal processes requires data on relative permeabilities and their dependence on temperature. However, reliable experimental data are not readily available. Thus, relative permeability is generally treated as an adjustable parameter for the history matching of past reservoir performance. Over the last thirty years, a number of laboratory studies have published conflicting results of temperature effects on two-phase relative permeabilities in porous media. Most experiments were performed on consolidated rocks, and only a few studies used reservoir cores and crude oils. Nakornthap and Evans(1) have summarized the temperature dependence of oil-water relative permeabilities reported in the literature. The general trend indicated that the oil and water relative permeability curves shifted to higher Water saturations. Using the unsteady State method, researchers working with heterogeneous porous media and oils of low to intermediate viscosity in the temperature range of up to about 150 °C reported that residual saturations and relative permeabilities were influenced by temperature(2–4). Other researchers working in the same temperature range with homogeneous sand cores and oils of intermediate viscosity reported the contrary(5,6). The only steady State relative permeabilities measured at elevated temperature (155 °C) were reported by Lo and Mungan(7). They found that temperature affected relative permeability curves in both water-and oil-wet systems similarly to the generally reported trends(1). They also indicated that, if the viscosity ratio remained constant, relative permeability curves were independent of temperature. The objective of this study was to determine the effect of temperature on relative permeabilities and saturations under controlled conditions using well-characterized systems. Twenty-five experiments were conducted on sand packs saturated with homogenized Athabasca bitumen and deionized water. The absolute permeabilities of the packs were measured at room and experimental temperatures. End point relative permeabilities and saturations were determined in the temperature range of 125 °C to 250 °C once a steady state condition with respect to the pressure drop across the pad bad been achieved. Experimental Apparatus and Procedures Apparatus The experimental apparatus was designed to measure two-phaserelative permeabilities under isothermal conditions up to 300 °C. A schematic diagram is shown in Figure 1.

A Dynamic Method For Measuring Relative Permeability

Abstract Currently available techniques for estimating effective permeabilities are valid only if the displacement is stable and stabilized. That is, only effective permeabilities which are applicable to steady-slate situations can be measured. As many field displacements are thought to be unstable there is a need to develop techniques capable of measuring unstable (unsteady-stale) effective permeabilities. This paper describes the progress made in developing one such method. The method uses the microwave attenuation technique to measure how saturation is distributed along the length of a core. This enables the estimation of the fraction of water flowing at any point along the core. Because a pressure profile is also available it is possible to relate the fraction flowing to the pressure gradient by means of the differential form of Darcy's law to obtain a dynamic estimate of the effective permeability at any given location (and saturation) along the length of the core. As data acquisition and storage is automated it is possible to obtain a complete set of relative permeability data in less than a day. Experimental results suggest that effective permeability is a unique function of saturation only if the displacement is steady-state, or if the displacement is stable and stabilized. If the displacement is unstabilized or if it is unstable, different effective permeability curves are obtained. Introduction One of the most important properties needed to make a waterflood prediction is the water-oil relative permeability characteristics of the reservoir rock. Such information may be obtained by either the steady-state or the external-drive techniques for measuring relative permeability. Steady-state methods require approximately one day to acquire a complete set of relative permeability data. External drive methods, on the other hand can acquire the same information in a few hours. However, it should be noted that both these methods have severe limitations when it comes to determining water-oil flow properties(1–4). Apart from being time-consuming, steady-state methods may not be applicable to systems where saturations change with time and show spatial gradients(1). Unfortunately, in most field situations the displacement is dynamic and thus the application of steady-state relative permeabilities to describe unsteady-state flow behaviour is questionable from a conceptual point of view. In this regard, it should be noted that many researchers have found disagreement between unsteady-state and steady-state relative permeabilities(3,5,6). Recently, Aleman-Gomez et al(7) concluded from their theoretical study of unsteady-state displacement that steady-state and unsteady-state relative permeabilities are similar only under certain limited conditions. They also commented that, if a discrepancy between steady-state and unsteady-state relative permeabilities was not observed, it was most probably because of the fact that the unsteady-state method was not capable of describing truly unsteady-state flow. In retrospect, the existence of this problem may have been deduced from the early work of Kimbler and Caudle(8) who concluded that the fluid distributions for steady-state flow differ significantly from those which pertain to unsteady-state flow. It is well known that the external drive method is based on the assumption that Buckley-Leverett(9) theory is valid.

Theoretical and Experimental Studies of Rate-Dependent Two-Phase Immiscible Flow

Summary Theoretical predictions are obtained with both Eulerian and Lagrangian methods for calculating saturation profiles in two-phase immiscible displacements in the presence of capillary pressure effects. The one-dimensional (1D) simulator WFLOOD is described with Lagrangian and Eulerian options for refined calculations of saturation distributions throughout all stages of a linear core flood. WFLOOD calculations are presented to demonstrate typical waterflood performance in a 1-m [3.3-ft] core with two different capillary-pressure functions at high and low flow rates. Steady-state and dynamic brine/tetradecane displacement experiments are described with Clashach sandstone cores that have radioactive ferrocene in the oil phase to measure saturation by a new nucleonic method. Some of the nonuniform character of these cores has been revealed by measurements. The PORES reservoir simulation model is used for a theoretical analysis of the experiments in which the nonuniform initial saturation distributions must be represented. It is shown that PORES, used in conjunction with measured steady-state relative permeabilities and measured static capillary pressure data, reproduces the time-dependent saturation profiles to within the accuracy of the measurements at high and low flow rates. The Lagrangian treatment in WFLOOD provides a theoretical benchmark that defines the levels of numerical dispersion present in the PORES Eulerian finite-difference calculations. A theoretical analysis is used to derive a constraint on the form of krw(dpc/dSw), as a function of Sw, which is valuable when capillary pressure functions Pc = Pc(Sw) are chosen for numerical modeling of immiscible displacements.

Multiphase Dispersion and Relative Permeability Experiments

AbstractExperiments in both Berea sandstone and sandpacks have been conducted to measure dispersion and steady-state relative permeabilities. Measurements have been made on both high-tension brine/oil and a low-tension, three-phase, brine/oil/surfactant/alcohol mixture. One interesting aspect of these experiments is the amount of microemulsion phase trapping. The endpoint microemulsion saturations for both the oil/microemulsion and brine/micro-emulsion phase pairs were high even at 10−3 dyne/cm [10−3 mN/m] interfacial tension (IFT). The dispersion was measured for each phase with radioactive and chemical tracers. The dispersivity was found to be a strong function of phase, phase saturation, porous medium, and IFT. Values of the dispersivity varied by two orders of magnitude over conditions investigated to date. Extremely early breakthrough of the tracer used in the oil phase (carbon 14) at high tension is especially remarkable. The brine tracer (tritium) curves were similar to that for 100% brine saturation except for a shift caused by material balance reasons. The classical solution to the convection-diffusion equation for single-phase flow has been generalized to multiphase flow and was used to aid in interpreting these data. This combination of relative permeability and dispersion in each phase of the experiment with a high-concentration, three-phase-microemulsion sulfonate formulation is believed to be new, and more directly applicable to commercial surfactant flooding than previously reported experimental results.

Effect of Capillary Number and Its Constituents on Two-Phase Relative Permeability Curves

Summary One primary goal of any enhanced recovery project is to maximize the ability of the fluids to flow through a porous medium (i.e., the reservoir). This paper discusses the effect of capillary number, a dimensionless group describing the ratio of viscous to capillary forces, on two-phase (oil-water) relative permeability curves. Specifically, a series of steady-state relative permeability measurements were carried out to determine whether the capillary number causes changes in the two-phase permeabilities or whether one of its constituents, such as flow velocity, fluid viscosity, or interfacial tension (IFT), is the controlling variable. For the core tests, run in fired Berea sandstone, a Soltrol 170™ oil/calcium chloride (CaCl2) brine/isopropyl alcohol (IPA)/glycerin system was used. Alcohol was the IFT reducer and glycerin was the wetting-phase viscosifier. The nonwetting-phase (oil) relative permeability showed little correlation with the capillary number. As IFT decreased below 5.50 dyne/cm [5.50 N/m], the oil permeability increased dramatically. Conversely, as the water viscosity increased, the oil demonstrated less ability to flow. For the wetting-phase (water) relative permeability, the opposite capillary number effect was shown. For both the tension decrease and the viscosity increase (i.e., a capillary number increase) the water permeability increased. However, the water increase was not as great as the increase in the oil curves with an IFT decrease. No velocity effects were noted within the range studied. Other properties relating to relative permeabilities were also investigated. Both the residual oil saturation (ROS) and the imbibition-drainage hysteresis were found to decrease with an increase in the capillary number. The irreducible water saturation was a function of IFT tension only. A relative permeability model was developed from the experimental data, based on fluid saturations, IFT, fluid viscosities, and the residual saturations, by using regression analysis. Both phases were modeled for both the imbibition and the drainage processes. These models demonstrated similar or better fits with experimental data of other water- and oil-wet systems, when compared with existing relative permeability models. The applicability of these regression models was tested with the aid of a two-phase reservoir simulator. Introduction As world oil reserves dwindle, the need to develop EOR techniques to maximize recovery is of great importance. Methods such as chemical flooding, miscible flooding, and thermal recovery involve altering the mobility and/or the IFT between the displacing the displaced fluids. Recovery efficiency was found to be dependent on the capillary number, defined asEquation 1 The viscous forces were defined as the fluid viscosity, flow velocity, and the flow path length. Capillary forces vary with the fluid IFT and the pore geometry of the medium.1 Taber defined the capillary number in terms of the pressure drop between two points, the flow length, and the IFT.2Equation 2 He concluded that as this ratio increased to a value of 5 psi/ft/dyne/cm [0.2 kPa/m/N/m] the ROS was reduced significantly. By decreasing the IFT by using surface-active agents, or by decreasing the path length by altering the field geometry, the capillary number could be increased. Others have shown similar results. Melrose and Brandner,3 for example, indicated that as the capillary number rose to a value of 10–4, the microscopic displacement efficiency, which accounts for the residual saturations to both oil and water, increased. The effects of the capillary number on the recovery of residual oil are given by Chatzis and Morrow4 and by other authors5 (Fig. 1). Few studies, however, have shown the effect of the capillary number on the two-phase flow between the residuals. The variables within this group have been researched, but their combined effect on relative permeabilities has been largely ignored. Several authors have noted that the viscosity ratio of oil and water alters the oil relative permeability but has little effect on that of water.6–8 Few or no changes by fluid flow velocity were observed, provided that no boundary effects were present during the core tests.9–11

A Model for Two-Phase Flow in Consolidated Materials

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

Fluid Distributions During Immiscible Displacements in Porous Media

Abstract For a wetting phase displacing a nonwetting phase from a porous medium the distribution of the residual fluid may depend on displacement conditions. Although this subject has been debated in the literature, only a few, experiments have been cited to support the various conclusions. Experimental results presented in this paper show that fluid distributions are dependent on imbibition procedures. Results agree qualitatively with predictions from the pore doublet model. if the rate of water imbibition is restricted, the nonwetting phase is trapped preferentially in the larger pores as expected. But if the rate of water imbibition is unrestricted, trapping occurs somewhat more in the smaller pores. These conclusions were deduced primarily from relative permeability measurements. Introduction Relative permeabilities are known to depend on the saturation history of the porous medium. For either continuously increasing or continuously decreasing wetting phase saturations, however, they have normally been assumed to be single-valued functions of saturation. Several investigators have compared relative permeabilities measured by different method and have reported acceptable agreement. Studies of simple models of the displacement process suggest that the distribution of residual fluids can be influenced by the displacement method. If this is so, relative permeabilities also should depend on the method of displacement. These predictions have not been supported by experimental data. The object of this paper is to investigate experimentally some of the predictions of these model studies. The particular question of importance is whether steady-state and unsteady-state methods should be expected to give the same values of relative permeability. Much of the reported data showing agreement between steady-state and unsteady-state methods have been obtained on unconsolidated sand. Johnson, Bossler and Neumann, however, compared steady-state and unsteady-state results for Weiler sandstone and found no significant difference. Levine made a detailed study of pressure and saturation distributions for a laboratory waterflood in a consolidated system but made no attempt to calculate performance from steady-state relative permeability data. Bail and Marsden in a somewhat similar set of experiments did attempt a comparison of observations with predictions but the results were inconclusive. Excluding gravity, two forces affect the distribution of wetting and nonwetting phases during an unsteady-state, immiscible displacement: viscous and capillary forces. If relative permeabilities are indeed the same when measured by steady- or unsteady-state methods, the fluid distributions at the same fluid saturations must be similar for both methods. Furthermore, the implication is that the relation between capillary and viscous forces is the same for both methods insofar as the effect on microscopic fluid distributions is concerned. Unsteady-state displacements are normally run at high rates to maximize the ratio of the viscous to capillary forces. The objective is to reduce the effect of capillary forces on macroscopic fluid distributions. For floods in which the phase which wets the porous medium displaces the nonwetting phase, capillary forces compete with viscous forces in determining fluid distributions. The residual nonwetting phase is discontinuous. Competitive aspects of viscous and capillary forces and the resulting effect on water-oil distribution in porous media have been illustrated in an elementary way using a pore doublet model. This model and predictions from it have been discussed at considerable length by Rose and Witherspoon, Rose and Cleary and by Moore and Slobod. Rose and Witherspoon show that so long as water invades the model at a rate equal to or greater than the free imbibition rate, the water will move through the larger capillary of a doublet first and trap oil in the smaller capillary. SPEJ P. 261ˆ

The Calculation of Waterflood Recovery From Steady-State Relative Permeability Data

Abstract The performance of laboratory water floods is compared with the flooding behavior calculated by the Buckley-Leverett techniques from steady-state relative permeability-saturation relations. Both the steady-state and unsteady-state experiments were conducted on the same sand-packed column. Relative permeabilities were calculated by the Welge technique from the unsteady-state waterflooding displacement experiments and compared with those measured by steady-state techniques. An evaluation of the various calculation procedures was made. It was demonstrated that steady-state relative permeabilities could be used to calculate waterflood performance. Introduction To calculate the waterflooding behavior of a given reservoir, relative permeability-saturation relations of the reservoir rack are required. These relative permeabilities are commonly measured on care samples by steady-state experiments such as the Penn State technique. The relative permeabilities obtained in steady-state experiments are calculated from oil and gas flow rates and pressure drops measured under conditions of equilibrium distribution of oil and water in the pore spaces with slow approach to equilibrium. On the other hand, in water floods the water saturation at a given point changes rapidly with time as the flood front passes. Extensive research has shown that the steady-state relative permeabilities to oil and water are independent of the rate of flow and the fluid viscosities. Research has also demonstrated that the waterflooding performance of homogeneous sands uniformly contacted by water is independent of rate when capillary pressure gradients, which are important in short laboratory columns, are made negligible. However, there is little experimental data in the literature comparing relative permeabilities calculated from steady-state flow experiments with those calculated from flooding experiments. In this paper, the production data of two water floods and the relative permeabilities calculated from a series of steady-state experiments on an unconsolidated sand column are presented along with the comparisons calculated by the Buckley-Leverett and by the Welge techniques. The purpose of this study was to determine if steady-state relative permeabilities could be used to predict the waterflooding performance. This study also permitted an evaluation of the Buckley-Leverett and Welge calculation procedures and allowed an insight into the nature of microscopic displacement of oil by water.