SPE Members Abstract The main difficulties to overcome in dynamic displacement relative permeability measurements are capillary end effects and viscous fingering. The latter problem is particularly severe in three-phase systems which include gas. Both of these problems are greatly mitigated in the three liquid system presented here. Water, benzyl alcohol, and decane are the three immiscible liquids which play the role of water, oil, and gas in the conventional three-phase system. The interfacial tensions in this system are about a factor of ten smaller than the comparable tensions for the conventional system, so the capillary end effect is less. Furthermore, viscosity contrasts are diminished, thus lowering the chance of fingering. An extension of the Welge-JBN method to three phases is derived. Each displacement experiment follows a saturation trajectory across the ternary diagram. The method is used to calculate three-phase relative permeabilities along each trajectory. Finally, the data from all trajectories are combined to provide isoperms over a large portion of the ternary diagram. Also included are the two sets of two-phase relative permeabilities needed to apply the interpolative three-phase relative permeabilities needed to apply the interpolative three-phase relative permeability model of Stone. The measured isoperms indicate the model permeability model of Stone. The measured isoperms indicate the model overpredict the relative permeability to oil: oil flows at lower rates than predicted. Introduction Relative permeability is an important concept in the forecasting of field performance. The relative permeability to a given phase is a function of that phase's saturation and possibly a function of other phase saturations as well. The effective permeability is one of the major non-linear (in most cases) factors in the mass transport equations that are used in reservoir models, analytical or numerical. One can imagine determining relative permeabilities by solving the multiphase flow problem from first principles. However, since such an approach is presently untractable, relative permeabilities must necessarily be determined experimentally. The literature presents a vast number of papers relating to the determining of relative permeabilities to fluids in porous media. For a review of the subject see, for example, Honarpour et al. Although many papers discuss absolute permeabilities and two-phase relative permeabilities, few papers deal with three-phase relative permeabilities, few papers deal with three-phase relative permeabilities. Three-phase relative permeabilities are essential in permeabilities. Three-phase relative permeabilities are essential in forecasting the performance of saturated and retrograde condensate reservoirs, and of most tertiary recovery processes. Hence, the importance of understanding and predicting the simultaneous motion of three phases in porous media. There are two major methods involving continuous injection of fluids for determining relative permeabilities: the steady-state method and the unsteady-state method. In the steady-state method, a fixed mixture of fluids is injected into the core, until the same mixture is produced from the core. At this point, steady-state condition is produced from the core. At this point, steady-state condition is assumed and by applying Darcy's law for each individual phase, the relative permeability is determined straightforwardly. In the unsteady-state method, usually (but not necessarily) one fluid is injected. Recovery and pressure drop are measured as a function of pore volumes injected. The relative permeabilities as a function of pore volumes injected. The relative permeabilities as a function of saturation are determined using the combined Welge method and the Johnson Bossler and Naumann method (Welge/JBN). The advantage of the dynamic method over the steady-state method is that the number of the experiments required to map the saturation region is small. In the three-phase case, this advantage is even more significant because the dynamic displacement method yields three-phase relative permeabilities along a continuous saturation trajectory. The main limitation of the dynamic displacement method is that relative permeability data cannot be determined in regions for which there are permeability data cannot be determined in regions for which there are saturation shock fronts. Steady state studies of three-phase flow in porous media are not new in the petroleum literature. As early as 1941, Leverett and Lewis present a study of three-phase flow using the steady-state method. present a study of three-phase flow using the steady-state method. They conclude that the water relative permeability is solely a function of water saturation in an unconsolidated sand pack. However, they conclude that the gas relative permeability depends upon the relative amounts of water and oil, with the gas isoperms being convex in the direction of 100% gas saturation. The oil isoperms are more complex, depending on the amounts of water and gas in the sand pack. pack. P. 325

Two of the most important properties governing reservoir fluid flow are relative permeability and capillary pressure. Unfortunately, conventional techniques for measuring one are not without complications from the other. Worse yet, these complications become particularly inextricable for flow experiments on the short core samples typically available for analysis. The advantage of the present method is that it simultaneously measures both properties in an experiment where their effects have been largely decoupled. The present method entails using an automated centrifuge to measure fluid productions at several speeds and history-matching the production data with the aid of a numerical simulator. Although history-matching problems are, in general, not well posed, the present case is exceptional. In each speed step the asymptotic production is determined only by capillary pressure and the transient behavior is determined mainly by relative permeability. Consequently, the effects can be distinguished. A numerical simulator, tailor-made for centrifuge experiments, is presented along with typical results of history-matching. The error in a conventional capillary pressure is also estimated.

Centrifuge air/water capillary pressure measurements made at one atmosphere ambient pressure can have the liquid in tension (negative absolute pressure) and cavitation in the liquid saturated region should not occur with water-wet rocks having capillary entry pressure less than three atmospheres. Thermodynamics argue that water that is in hydrostatic equilibrium in the pore space that has been invaded by a gas that is also in hydrostatic equilibrium can be in chemical equilibrium with the water vapor in the gas. An experiment showed no evidence of cavitation in rocks with a capillary entry pressure of 50 psi. Introduction In part I we discussed the outflow boundary condition of zero capillary pressure. It was shown that some of the problems that have been attributed at the outflow boundary condition or cavitation could have been due to inadequate core cleaning or lack of equilibrium. For example, a core that was not adequately cleaned has a low capillary entry pressure and significant trapped water saturation in an air-brine centrifuge capillary pressure measurement. Conditions for the violation of the outflow boundary condition were defined. In Part 2 we discuss the question of cavitation of the liquid when the liquid is in tension. Cavitation is of concern with air-liquid capillary pressure measurements when the air pressure is one atmosphere and the capillary pressure exceeds one atmosphere. The liquid is then in tension (negative absolute pressure), and it is known that bulk liquid is not stable with respect to its vapor when it is in tension. If cavitation was to occur in the 100% liquid saturated region, the liquid production would occur before the capillary entry pressure was reached. This would result in an apparent capillary entry pressure that is less that the actual value. The question of cavitation in the liquid saturated region is addressed in the first section. If cavitation was to occur in the gas invaded region whenever the liquid was in tension, then the liquid could become discontinuous until the hydrostatic pressure in the continuous bodies of liquid is positive. The second section shows that liquid and gas that is in hydrostatic equilibrium can also be in thermodynamic equilibrium. The final section presents experimental results which show that a rock with a capillary entry pressure of 50 psi has the same capillary pressure curve regardless of whether the air pressure is one atmosphere or 300 psi, i.e., whether the liquid was in tension or not. The Appendix contains the derivations required to describe the nucleation of a bubble in a cavity. These results are used to describe cavitation in a liquid saturated region. The existence of water in tension in porous media is a common occurrence with the water vapor desorption method for determination of capillary pressure. Melrose has reviewed the literature on the negative pressures of liquids. Water can exist under greater negative pressures than is encountered in capillary pressure measurements made by the water vapor desorption, centrifuge, and porous plate methods and showed that the results were in agreement even though the water was under significant tension. It remains to prove that in porous media that liquid in tension exists as a stable state rather than a metastable state which is subject to random disturbances.

Summary The capillary pressure curve is estimated from centrifuge measurements withthe assumption that Pc=0 at the outflow end of the core. With the proper endpiece to support the core sample, this boundary condition is valid inpractically all circumstances. If the end piece is wetted by the producedliquid, however, experiments show that the capillary pressure curve is alteredsignificantly. Theoretical analysis demonstrates that film drainage candisplace the zero-capillary-pressure boundary condition to the bottom of theend piece or farther. Accordingly, we recommend that the displaced phase notwet the end piece; in the case of water, a teflon end piece is recommended overthe commonly used rubber one. When the end piece is not a problem, thecondition for 100% liquid saturation at the outflow face can be expressed as acritical Bond number, which is rarely exceeded. Saturation profiles for anexperiment just below the critical Bond number were in excellent agreement withpredictions. Pendant drops at the outflow face can result in a nonzerocapillary pressure at the outflow face, but this effect is insignificant. Goodagreement was obtained between mercury injection and centrifuge-measuredcapillary pressure curves. Attention is drawn to the importance of corejacketing, core cleaning, and equilibration times in providing reliablecentrifuge-measured capillary pressure curves. pressure curves. Purpose and Scope Purpose and Scope Although the centrifuge method for measuring capillarypressure has been used throughout the industry for more than 40 years, questions about its validity continue. The two most important questions concerncavitation and the outflow boundary condition of zero capillary pressure. Theboundary condition is investigated here; cavitation is discussed in Ref. 1. Before discussing the boundary condition, we raise several otherconsiderations. We begin with a brief description of the centrifuge method andcompare it with other methods for measuring capillary pressure curves. Then wepresent our modifications to the traditional pressure curves. Then we presentour modifications to the traditional centrifuge method. The next sectionexamines the topic of comparing capillary pressure curves, whether fromdifferent methods or from the centrifuge method applied to slightly differentcores. We then discuss important sources of error in centrifuge measurements. Having identified sources of error and methods of crosschecking results, weconsider the boundary condition. The first part of the boundary-conditionsection ignores the effects of the end piece supporting the core; the finalsections show under what circumstances the end piece is likely to beimportant. Centrifuge Method The centrifuge method entails increasing the centrifuge speed in steps andmeasuring at each step the amount of liquid produced from the core atequilibrium when all flow has ceased. An important assumption in analyzing thisexperiment is that Pc=0 at the outflow end. There are a variety of possible centrifuge capillary pressure experiments, depending on the drainage mode and fluids involved. Measurements usually aremade in the primary drainage mode where oil or air replace water in a core pluginitially 100% saturated with water. There are other possibilities however:e.g., secondary drainage from a core initially saturated at waterflood residualoil and imbibition into a core at irreducible water saturation. The conclusionsof the present study are relevant to all these experiments. For simplicity, thecore is assumed to be initially 100% saturated with the displaced phase. Thephase entering the core is called the invading phase. Centrifuge Measurement of Capillary Pressure Compared With Mercury, Porous-Plate, and Water-Vapor-Desorption Methods The two major alternatives to the centrifuge method are the porousplate andthe mercury-injection methods. Mercury injection has porousplate and themercury-injection methods. Mercury injection has the advantages that it isquick (results in a few hours) and can attain high pressures. Its disadvantagesare that wettability and clay effects are not modeled suitably. By comparison, the porous-plate method can use reservoir conditions and fluids relativelyeasily. However, weeks or months may be required to complete a measurement. Thecentrifuge method falls somewhere between these two. It is slower than mercurymethod (days vs. hours) but uses more representative fluids. It is much fasterthan the porous-plate method but does not provide reservoir conditions aseasily. The recently developed water-vapor-desorption method is not really analternative to the above methods but rather a means to extend them intohigh-capillary-pressure regimes. Several recent papers compare the equivalency of the various methods. Omoregie provided examples for which differences between mercury/air andcentrifuge curves result from clay and wettability effects. In addition, hedemonstrates how porous-plate measurements are affected by the time allottedfor equilibration. Porous-plate and centrifuge results differ when 4 days areallotted Porous-plate and centrifuge results differ when 4 days are allottedfor equilibration for each pressure in the porous-plate method but agree when10 days are allotted. In a comparison of the centrifuge, porous-plate, andwater-vapor-desorption methods, Melrose porous-plate, andwater-vapor-desorption methods, Melrose showed excellent agreement in a studyof matched Berea cores covering capillary pressures from 0.7 to 9.5 MPa. Thisagreement is particularly relevant to our cavitation investigation because thecentrifuge and water-vapor-desorption methods entail negative absolutepressures (i.e., tension) in the displaced phase, whereas the porous-platemethod has positive phase pressures. Ward and porous-plate method has positivephase pressures. Ward and Morrow obtained good agreement between the centrifugeand water-vapor-desorption methods when they used the latter method to extendcentrifuge curves of low-permeability sandstones to higher capillarypressures. Centrifuge Capillary Pressure Curves by Parameter Estimation The present method for calculating capillary pressure curves fromequilibrium data (measured or extrapolated) is a modified version of Hasslerand Brunner's traditional method. For the following derivation, we assume thatcavitation is not present and that Pc=0 at the outflow.

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