Multiphase Dispersion and Relative Permeability Experiments

1985 ◽  
Vol 25 (04) ◽  
pp. 524-534 ◽  
Author(s):  
M. Delshad ◽  
D.J. MacAllister ◽  
G.A. Pope ◽  
B.A. Rouse
Keyword(s):  
Relative Permeability ◽  
Single Phase ◽  
Phase Flow ◽  
Single Phase Flow ◽  
Relative Permeabilities ◽  
High Concentration ◽  
High Tension ◽  
Three Phase ◽  
Phase Saturation ◽  
Low Tension

Summary Experiments 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/microemulsion 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 data. 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. Introduction In this paper we report the initial results of a project1 to investigate the transport in porous media of several chemicals used in EOR. Specifically, we are studying the behavior of high-concentration, three-phase micellar formulations in beadpacks, sandpacks, and sandstone. The rheology, relative permeabilities, and dispersion coefficients have been the primary focus of this study to date. In this paper, we report on the last two parameters for a single polymer-free micellar formulation. These results are based on the theses of Delshad2 and MacAllister.3 The rheology of this and other EOR fluids is reported in Ref. 4. Oil recovery and history matching was done by Lin.5 A unique feature of this work was the way in which the relative permeabilities and dispersion experiments were combined into essentially the same experiment (see the section on procedures and materials). Since trapping has a profound effect on the efficiency of micellar/polymer flooding, another important feature is the measurement of microemulsion phase trapping at each relative permeability endpoint. These are believed to be the first direct measurement of this type. Literature Review No attempt will be made here to review the numerous high-tension relative permeability studies reported during the past several decades. Also, only a few of the classical single-phase flow dispersion studies will be mentioned. Low-tension data are much less extensive. Leverett,6 Mungan,7 du Prey,8 Talash,9 Bardon,10 Batycky,11 Klaus,12 and Amaefule and Handy13 are among the few who have reported results as a function of IFT. All of these results were for two-phase fluids. Furthermore, apparently only Talash, Klaus, and Amaefule and Handy used fluids containing sulfonates such as we are primarily concerned with, and then only at very low sulfonate concentrations. The general observations are that the relative permeability curves tend to increase and have less curvature as the IFT decreases or the capillary number increases. The residual saturations decrease simultaneously. Consistent with the capillary desaturation curves and theory reported by others,14–16 the nonwetting-phase saturation decreases first, then the residual wetting phase. It has been speculated for a long time that these curves will eventually become straight lines, but few if any of these experiments attained the ultralow IFT typical of optimal micellar fluids that would be necessary to test this idea.

An Improved Three-Phase Relative Permeability and Hysteresis Model for the Simulation of a Water-Alternating-Gas Injection

SPE Journal ◽  
2013 ◽  
Vol 18 (05) ◽  
pp. 841-850 ◽  
Author(s):  
H.. Shahverdi ◽  
M.. Sohrabi
Keyword(s):  
Relative Permeability ◽  
Oil Recovery ◽  
Gas Injection ◽  
Phase Flow ◽  
Relative Permeabilities ◽  
Water Alternating Gas ◽  
Three Phase ◽  
Three Phase Flow ◽  
Oil Water ◽  
Wag Injection

Summary Water-alternating-gas (WAG) injection in waterflooded reservoirs can increase oil recovery and extend the life of these reservoirs. Reliable reservoir simulations are needed to predict the performance of WAG injection before field implementation. This requires accurate sets of relative permeability (kr) and capillary pressure (Pc) functions for each fluid phase, in a three-phase-flow regime. The WAG process also involves another major complication, hysteresis, which is caused by flow reversal happening during WAG injection. Hysteresis is one of the most important phenomena manipulating the performance of WAG injection, and hence, it has to be carefully accounted for. In this study, we have benefited from the results of a series of coreflood experiments that we have been performing since 1997 as a part of the Characterization of Three-Phase Flow and WAG Injection JIP (joint industry project) at Heriot-Watt University. In particular, we focus on a WAG experiment carried out on a water-wet core to obtain three-phase relative permeability values for oil, water, and gas. The relative permeabilities exhibit significant and irreversible hysteresis for oil, water, and gas. The observed hysteresis, which is a result of the cyclic injection of water and gas during WAG injection, is not predicted by the existing hysteresis models. We present a new three-phase relative permeability model coupled with hysteresis effects for the modeling of the observed cycle-dependent relative permeabilities taking place during WAG injection. The approach has been successfully tested and verified with measured three-phase relative permeability values obtained from a WAG experiment. In line with our laboratory observations, the new model predicts the reduction of the gas relative permeability during consecutive water-and-gas-injection cycles as well as the increase in oil relative permeability happening in consecutive water-injection cycles.

Some Surprises in the Transport of Miscible Fluids in the Presence of a Second Immiscible Phase

1985 ◽  
Vol 25 (01) ◽  
pp. 101-112 ◽  
Author(s):  
Stanley C. Jones
Keyword(s):  
Steady State ◽  
Single Phase ◽  
High Water ◽  
Phase Flow ◽  
Single Phase Flow ◽  
Phase Mixing ◽  
Solvent Injection ◽  
The Core ◽  
Phase Saturation ◽  
Longitudinal Mixing

Jones, Stanley C., SPE, Marathon Oil Co. Abstract Displacements were conducted in Berea cores to gain insight into the mechanism of tertiary oil displacement and propagation by a micellar slug. Contrary to expectation, propagation by a micellar slug. Contrary to expectation, the first oil mobilized by micellar fluid was among the first oil (instead of the last oil) to be produced, giving the appearance of either viscous fingering or of unusually large dispersion. To eliminate the possibility of unfavorable mobility ratios caused by oil/water/surfactant interaction, we conducted several runs in which an injected hydrocarbon displaced another hydrocarbon, initially at residual saturation. In other experiments, water (the wetting phase) at irreducible saturation was displaced by a distinguishable injected aqueous phase. Injected hydrocarbon appeared in the produced fluids immediately after oil breakthrough, yielding behavior similar to the micellar-slug experiments. Even with a favorable viscosity ratio of less than 0.01, the apparent dispersion was huge. However, mixing zones in the wetting-phase displacements were quite normal and similar to those observed for single-phase flow. Nonwetting-phase fronts (injected hydrocarbon displacing resident hydrocarbon) are smeared much more than wetting-phase fronts because the entrance of hydrocarbon into smaller water-filled pore throats is delayed until the capillary entrance pressure is overcome by differences in the flowing oil and water pressure gradients. Oil might not be displaced from the smaller pores until long after oil breakthrough. Nonwetting-phase dispersion, which occurs in many EOR processes, can be expected to be one or two orders of magnitude greater than dispersion measured in single-phase-flow experiments. Entrance of the wetting phase, however, is not delayed; hence, wetting-phase Mixing zones are short. Introduction Experiments for this study were inspired by the question: How is residual oil, which has been mobilized by a micellar slug, transported? More specifically, does the first oil mobilized by a slug (near the injection end of a core) contact and mobilize oil downstream from it, which displaces more oil even farther downstream? If this were the case, the first oil to be produced would be the most-downstream oil (i.e., oil nearest the outlet). The last oil produced would be the first oil mobilized from the produced would be the first oil mobilized from the injection end of the core. This scheme is somewhat analogous to pushing a broom across a floor covered with a heavy layer of dust. The first dust encountered by the broom stays next to the broom. As the accumulated layer of dust in front of the broom becomes adequately compacted, it pushes dust ahead of it to from an ever-widening band or "dust bank" ahead of the broom. The dust farthest ahead of the broom is the first to be pushed into the dustpan, and the dust first encountered by the broom is the last to be pushed in. Or is this concept all wrong? Another model postulates that the oil first contacted by a micellar slug is mobilized and quickly travels away from the slug so that the downstream oil is contacted and mobilized by the slug, not by the first-mobilized oil. If this process were to proceed to its logical conclusion, the first-produced oil would proceed to its logical conclusion, the first-produced oil would be from the inlet end of the core, and the last-produced from the outlet end. Either of these two extremes would be modified by dispersion, which smears sharp fronts by mixing displaced and displacing fluids. Dispersion in porous media has been investigated extensively. Perkins and Johnston have reviewed several studies, mostly involving single-phase flow. The simultaneous injection of the water with light hydrocarbon solvents is a technique used to reduce solvent mobility and viscous fingering. Raimondi et al. performed steady-state experiments in which flowing performed steady-state experiments in which flowing water and oil were miscibly displaced by the simultaneous injection of water and solvent. They found that the longitudinal mixing coefficient for the hydrocarbon phase increased sharply with increasing water above the irreducible saturation. The displacement of the wetting phase was not greatly affected by the presence of the nonwetting phase. However, a large amount of oil that initially phase. However, a large amount of oil that initially seemed to be trapped by water was eventually recovered by continued solvent injection. Raimondi and Torcaso later found that some oil, particularly at high water-to-solvent injection ratios, was particularly at high water-to-solvent injection ratios, was trapped permanently, provided that injection rates, ratios, and pressure drops were unchanged in switching from water/oil to water/solvent injection. Fitzgerald and Nielsen also found that only part of the in-place crude was recovered by solvent injection. Moreover, solvent appeared in the effluent shortly after oil breakthrough. Oil recovery was further decreased when solvent and water were injected simultaneously. Thomas et al. reported slightly increased wetting-phase longitudinal mixing during simultaneous water/oil injection as the wetting-phase saturation decreased. Non-wetting-phase mixing increased substantially as the nonwetting-phase saturation decreased from 100%. SPEJ p. 101

Gas-Condensate Flow in Perforated Regions

SPE Journal ◽  
2007 ◽  
Vol 12 (01) ◽  
pp. 89-99 ◽  
Author(s):  
Mahmoud Jamiolahmady ◽  
Ali Danesh ◽  
Mehran Sohrabi ◽  
Rahim Ataei
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Finite Element Modeling ◽  
Relative Permeability ◽  
Single Phase ◽  
Phase Flow ◽  
Single Phase Flow ◽  
Fractional Flow ◽  
Fluid Properties ◽  
Perforated Completions

Summary The most crucial region with regard to affecting well productivity is the perforated region. Considerable effort has been directed to study this subject mathematically by many investigators, but they have been mainly focused on single-phase flow, while two-phase flow has received less attention. It has been demonstrated, first by Danesh et al. (1994) and subsequently by other researchers (Henderson et al. 1995; Blom et al. 1997; Ali et al. 1997), that the gas and condensate relative permeability (kr) can increase significantly by increasing the flow rate, contrary to the common understanding. This effect, known as positive coupling, complicates the flow of gas and condensate near the wellbore even further when it competes with the inertial forces at higher velocities typical of those around perforation tips. The flow of gas and condensate in the perforated region was studied in this work using a finite-element modeling approach. The model allows for changes in fluid properties and accounts for the positive coupling and negative inertial effects using a fractional-flow-based relative-permeability correlation. A sensitivity analysis on the impact of perforation characteristics such as density, phasing, length, and radius as well as that of fluid properties, rock characteristics, wellbore radius, fractional flow, and rate on well productivity was conducted, resulting in some valuable practical guidelines for optimum perforation design. Introduction The effect of perforation characteristics on the well flow efficiency has been studied by many investigators. Muskat presented the first analytical treatment of the problem (1943). In his analysis, perforations were represented by mathematical sinks distributed spirally around the wellbore but did not extend into the formation. Other early investigators used the finite-difference modeling technique to examine the productivity aspects of perforated completions (Harris 1966; Hong 1975). However, because of the limitations of the finite-difference method, these studies considered mostly unrealistic perforation geometries to avoid mathematical complexities. Later investigators applied the finite-element method, which models the geometry of the perforation with greater precision (Locke 1981; Tariq 1987). Tariq (1987) presented results of finite-element modeling of single-phase steady-state flow in perforated completions with and without the non-Darcy (inertial) effect for a linear core and a full 3D system. Although his results for single-phase flow are widely used, there are reports on lack of required accuracy at large perforation lengths and in the non-Darcy cases (Behie and Settari 1993; Jamiolahmady et al. 2006a).

Issue with Stone-II Three Phase Permeability Model, and A Novel Robust Fundamentals-Based Alternative to It

2021 ◽  
Author(s):  
Subodh Gupta
Keyword(s):  
Relative Permeability ◽  
Phase Flow ◽  
Residual Oil ◽  
Relative Permeabilities ◽  
Phase Permeability ◽  
Permeability Model ◽  
Oil Saturation ◽  
Three Phase ◽  
Three Phase Flow ◽  
Relative Permeability Curves

Abstract The objective of this paper is to present a fundamentals-based, consistent with observation, three-phase flow model that avoids the pitfalls of conventional models such as Stone-II or Baker's three-phase permeability models. While investigating the myth of residual oil saturation in SAGD with comparing model generated results against field data, Gupta et al. (2020) highlighted the difficulty in matching observed residual oil saturation in steamed reservoir with Stone-II and Baker's linear models. Though the use of Stone-II model is very popular for three-phase flow across the industry, one issue in the context of gravity drainage is how it appears to counter-intuitively limit the flow of oil when water is present near its irreducible saturation. The current work begins with describing the problem with existing combinatorial methods such as Stone-II, which in turn combine the water-oil, and gas-oil relative permeability curves to yield the oil relative permeability curve in presence of water and gas. Then starting with the fundamentals of laminar flow in capillaries and with successive analogical formulations, it develops expressions that directly yield the relative permeabilities for all three phases. In this it assumes a pore size distribution approximated by functions used earlier in the literature for deriving two-phase relative permeability curves. The outlined approach by-passes the need for having combinatorial functions such as prescribed by Stone or Baker. The model so developed is simple to use, and it avoids the unnatural phenomenon or discrepancy due to a mathematical artefact described in the context of Stone-II above. The model also explains why in the past some researchers have found relative permeability to be a function of temperature. The new model is also amenable to be determined experimentally, instead of being based on an assumed pore-size distribution. In that context it serves as a set of skeletal functions of known dependencies on various saturations, leaving constants to be determined experimentally. The novelty of the work is in development of a three-phase relative permeability model that is based on fundamentals of flow in fine channels and which explains the observed results in the context of flow in porous media better. The significance of the work includes, aside from predicting results more in line with expectations and an explanation of temperature dependent relative permeabilities of oil, a more reliable time dependent residual oleic-phase saturation in the context of gravity-based oil recovery methods.

Calculation of Imbibition Relative Permeability for Two- and Three-Phase Flow From Rock Properties

1968 ◽  
Vol 8 (02) ◽  
pp. 149-156 ◽  
Author(s):  
Carlon S. Land
Keyword(s):  
Pore Size Distribution ◽  
Pore Size ◽  
Size Distribution ◽  
Relative Permeability ◽  
Empirical Relation ◽  
Phase Flow ◽  
Three Phase ◽  
Phase Saturation ◽  
Phase Systems ◽  
Relative Permeability Curves

Abstract Relative permeability functions are developed for both two- and three-phase systems with the saturation changes in the imbibition direction. An empirical relation between residual nonwetting-phase saturation after water imbibition and initial nonwetting-phase saturations is found from published data. From this empirical relation, expressions are obtained for trapped and mobile nonwetting-phase saturations which are used in connection with established theory relating relative permeability to pore-size distribution. The resulting equations yield relative permeability as a function of saturation having characteristics believed to be representative of real systems. The relative permeability of water-wet rocks for both two- and three-phase systems, with the saturation change in the imbibition direction, may be obtained by this method after properly selecting two rock properties: the residual nonwetting-phase saturation after the complete imbibition cycle, and the capillary pressure curve. Introduction Relative permeability is a function of saturation history as well as of saturation. This fact was first pointed out for two-phase flow by Geffen et al. and by Osaba et al. Hysteresis in the relative permeability-saturation relation also has been reported for three-phase systems. Since saturations may change simultaneously in two directions in a three-phase system, four possible relationships arise between relative permeability and saturation for a water-wet system. The four saturation histories of this system were given by Snell as II, ID, DI and DD. I refer to the direction of saturation change (imbibition and drainage), with the first letter of the symbol indicating the direction of change of the water phase. As used in this paper, the second letter of the symbol refers to the direction of saturation change of the gas phase, i.e., D and I indicate an increase and decrease, respectively, in gas saturation. Only a few three-phase relative permeability curves have been published. Leverett and Lewis published three-phase curves for unconsolidated sand, and Snell reported results of several English authors for both drainage and imbibition three-phase relative permeability of unconsolidated sands. Three-phase relative permeability curves for a consolidated sand were published by Caudle et al. for increasing water and gas saturations (ID). Corey et al. reported drainage (DD) three-phase relative permeability for consolidated sands. Recently, Donaldson and Dean and Sarem calculated three- phase relative permeability curves from displacement data on consolidated sands, also for saturation changes in the drainage direction. The only published three - phase relative permeability curves for consolidated sands with saturation changes in the imbibition direction (II) are those of Naar and Wygal. These curves are based on at theoretical study of the model of Wyllie and Gardner as modified by Naar and Henderson. Interest in three-phase relative permeability has increased recently due to the introduction of new recovery methods and refinements in calculation procedures brought about by the use of large-scale digital computers. The scarcity of empirical relations for three-phase flow, and the experimental difficulty encountered in obtaining such data, have made the theoretical approach to this problem attractive. RELATIVE PERMEABILITY AS A FUNCTION OF PORE-SIZE DISTRIBUTION Purcell used pore sizes obtained from mercury-injection capillary pressure data to calculate the permeability of porous solids. Burdine extended the theory by developing a relative permeability-pore size distribution relation containing the correct tortuosity term. SPEJ P. 149ˆ

Three-Phase Relative Permeabilities from Displacement Experiments with Full Account for Capillary Pressure

1998 ◽  
Vol 1 (02) ◽  
pp. 92-98 ◽  
Author(s):  
H.M. Helset ◽  
J.E. Nordtvedt ◽  
S.M. Skjaeveland ◽  
G.A. Virnovsky
Keyword(s):  
Porous Media ◽  
Multiphase Flow ◽  
Capillary Pressure ◽  
Relative Permeability ◽  
Single Phase ◽  
Flow In Porous Media ◽  
Relative Permeabilities ◽  
The Core ◽  
Three Phase ◽  
Displacement Experiments

Abstract Relative permeabilities are important characteristics of multiphase flow in porous media. Displacement experiments for relative permeabilities are frequently interpreted by the JBN method neglecting capillary pressure. The experiments are therefore conducted at high flooding rates, which tend to be much higher than those experienced during reservoir exploitation. Another disadvantage is that the relative permeabilities only can be determined for the usually small saturation interval outside the shock. We present a method to interpret displacement experiments with the capillary pressure included, using in-situ measurements of saturations and phase pressures. The experiments can then be run at low flow rates, and relative permeabilities can be determined for all saturations. The method is demonstrated by using simulated input data. Finally, experimental scenarios for three-phase displacement experiments are analyzed using experimental three-phase relative permeability data. Introduction Relative permeabilities are important characteristics of multiphase flow in porous media. These quantities arise from a generalization of Darcy's law, originally defined for single phase flow. Relative permeabilities are used as input to simulation studies for predicting the performance of potential strategies for hydrocarbon reservoir exploitation. The relative permeabilities are usually determined from flow experiments performed on core samples. The most direct way to measure the relative permeabilities is by the steady-state method. Each experimental run gives only one point on the relative permeability curve (relative permeability vs. saturation). To make a reasonable determination of the whole curve, the experiment has to be repeated at different flow rate fractions. To cover the saturation plane in a three-phase system, a large number of experiments have to be performed. The method is therefore very time consuming. Relative permeabilities can also be calculated from a displacement experiment. Typically, the core is initially saturated with a single-phase fluid. This phase is then displaced by injecting the other phases into the core. For the two-phase case, Welge showed how to calculate the ratio of the relative permeabilities from a displacement experiment. Efros was the first to calculate individual relative permeabilities from displacement experiments. Later, Johnson et al. presented the calculation procedure in a more rigorous manner, and the method is often referred to as the JBN method. The analysis has also been extended to three phases. In this approach, relative permeabilities are calculated at the outlet end of the core; saturations vs. time at the outlet end is determined from the cumulative volumes produced and time derivatives of the cumulative volumes produced, and relative permeabilities vs. time are calculated from measurements of pressure drop over the core and the time derivative of the pressure drop. Although the JBN method is frequently used for relative permeability determination, it has several drawbacks. The method is based on the Buckley-Leverett theory of multiphase flow in porous media. The main assumption is the neglection of capillary pressure. In homogenous cores capillary effects are most important at the outlet end of the core and over the saturation shock front. To suppress capillary effects, the experiments are performed at a high flow rate. Usually, these rates are significantly higher than those experienced in the underground reservoirs during exploitation.

Three-Phase Relative Permeability Measurement Using a Nuclear Magnetic Resonance Technique for Estimating Fluid Saturation

1967 ◽  
Vol 7 (03) ◽  
pp. 235-242 ◽  
Author(s):  
D.N. Saraf ◽  
I. Fatt
Keyword(s):  
Nuclear Magnetic Resonance ◽  
Relative Permeability ◽  
Sand Sample ◽  
Phase Flow ◽  
Relative Permeabilities ◽  
The Core ◽  
Liquid Saturation ◽  
Three Phase ◽  
Three Phase Flow ◽  
Permeability Data

Abstract A method is described for measuring two- and three-phase relative permeabilities in sandstones or sand packs using a nuclear magnetic resonance (NMR) technique to determine fluid saturations Two- and three-phase relative permeabilities have been determined on Boise sandstone using the NMR technique of saturation measurement. Three- phase relative permeability to water was found to depend only on the water saturation, whereas three-phase permeability to oil depended on both the water and oil saturations. Relative permeability to gas in three-phase flow was found to depend only on the total liquid saturation. Introduction Three-phase relative permeabilities are extremely useful in calculating field performance for reservoirs being produced by simultaneous water and gas drives. Three-phase relative permeability data are also needed for analyzing solution gas-drive reservoirs which are partially depleted and are being produced by water drive. Some thermal recovery processes involve three-phase flow which require three-phase relative permeability data for predicting reservoir-behavior. Unfortunately three-phase relative permeability measurements have rarely been made. Also, because of the scarcity of three-phase data, it has not been possible to date to relate other measured rock characteristics to the relative permeabilities with a great certainty. Leverett and Lewis, Reid and Snells have reported three-phase relative permeability data on unconsolidated sands. Leverett and Lewis used ring electrodes spaced along the length of the sand sample to -measure the resistivity of the sample which was assumed to be monotonically related to brine saturation. Gas saturation was determined from pressure-volume measurements. Oil saturation was obtained by material balance on the cell containing the sand sample. This method is involved and time consuming. Another difficulty arises from the fact that the resistivity of the sand is a function not only of saturation of brine but also of the distribution and saturation history of the brine in the pore spaces. Reid used a gamma ray absorption technique for measuring liquid saturation. This method has the disadvantage that total liquid saturation rather than oil or brine saturation is all that can be measured and still another method is required to determine the saturations of individual components. Snells used a neutron bombardment method which also required a separate determination of the individual component saturations. Caudle et al. measured three-phase relative permeability on consolidated sandstones using vacuum distillation for determining fluid saturations. Distillation after each reading makes this technique very lengthy and time consuming. Corey et al and Naar and Wygal measured three-phase relative permeability on sandstones by the capillary- pressure method. Semipermeable diaphragm assemblies were used at each end of the core specimen to keep the water base in the core. Gravimetric methods were used to determine fluid saturations. Sarem recently repeated an unsteady-state method for measuring three-phase relative permeability on sandstones. This method is an extension of Weige's methods for measuring two-phase relative permeability. Although Sarem's method is simple and comparatively fast, the assumptions involved may oversimplify the problem. Sarem's assumption, that in all rocks relative permeability to each fluid will depend only on the saturation of that fluid, seems to be rather unrealistic. Neglecting capillary effects at the end of the core is also a weak assumption Donaldson and Deans measured three-phase relative permeability using a method similar to Sarem's. SPEJ P. 235ˆ

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