15
AMERICAN INSTITUTE OF MINING AND METALLURGICAL ENGINEERS Technical Publication No. 1162 (CLASS G. P~TBOLEUY DIVIBION, NO. 102) DISCUSSION OF THIS PAPER IS INVITED. Discussion in writing (2 copies) ma be sent to the Secretary, American Inatitute of Mining and Metallurgjcal Engineers 29 Weet 39th gtreet, New York, N. Y. Unless special arran ement is made, disousaion of this paper dill close April 1. 1940. Any discurnon offered thereafter shoufd preferably be in the form of a new paper. Fundamental Phase Behavior of Hydrocarbons BY JOHN E. SHERBORNE,* JUNIOR MEMBER A.I.M.E. (Lon Angeles Meeting, October 1939) MUCH valuable scientific research has been performed in recent years on the subject of phase behavior of hydrocarbons. tl-ll Engineers employed in petroleum production are interesting themselves in this work as well as in methods of applying the fundamental data available to the solution of their various problems. Recently a number of papers have been published in which applica- tions of phase behavior have been made to specific cases pertaining to critical phen0mena.l2-'~ Little effort, however, has been made in the literature to show the relation between changes occurring in the critical region and the more common phase behavior, therefore it is believed that a presentation of the fundamentals of phase behavior with reference to hydrocarbons is timely. Study of phase behavior is not new. In the metallurgical field, knowledge of heterogeneous equilibria has advanced tremendously, particularly with reference to solid-solid and solid-liquid behavior. Much is known about vapor-liquid equilibria too, but few engineers are familiar with this subject.$ In a discussion of this sort, a definition of terms used is most impor- tant. Such terms as "pressure," "temperature " and "volume " need little definition other than mention of the units in which they are con- sidered. Pressure is expressed in pounds per square inch absolute. Temperature is usually expressed as degrees Fahrenheit or degrees Rankine. In considering thermodynamic and phase behavior, the use of the absolute. Rankine. scale is desirable. Volume is expressed as specific volume, such as cubic feet per pound. This will be recognized as the reciprocal of the specific weight. In considering systems composed of more than one component, it is sometimes desirable to consider molal Manuscript received at the office of the Institute Oct. 24, 1939. Petroleum Engineer, Union Oil Company of California, Compton, California. t Notable among the various investigators are Sage, Lacey and co-workers, Katz and Lindsly. Only a few selected references of these and other authors will be used. Numbers refer to references at end of paper. 1 Several books dealing with phase behavior are included in the bibliography. 16-20~44 -- Copyright, 1940, by the American Institute of Mining and Metallurgical Engineers, Inc. PmTnoLpn~ TECEXOLO~Y, February 1940. Printed in U. S. A .

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AMERICAN INSTITUTE OF MINING AND METALLURGICAL ENGINEERS

Technical Publication No. 1162 (CLASS G. P~TBOLEUY DIVIBION, NO. 102)

DISCUSSION OF THIS PAPER IS INVITED. Discussion in writing (2 copies) ma be sent to the Secretary, American Inatitute of Mining and Metallurgjcal Engineers 29 Weet 39th gtreet, New York, N. Y. Unless special arran ement is made, disousaion of this paper dill close April 1. 1940. Any discurnon offered thereafter shoufd preferably be in the form of a new paper.

Fundamental Phase Behavior of Hydrocarbons

BY JOHN E. SHERBORNE,* JUNIOR MEMBER A.I.M.E. (Lon Angeles Meeting, October 1939)

MUCH valuable scientific research has been performed in recent years on the subject of phase behavior of hydrocarbons. tl-ll Engineers employed in petroleum production are interesting themselves in this work as well as in methods of applying the fundamental data available to the solution of their various problems.

Recently a number of papers have been published in which applica- tions of phase behavior have been made to specific cases pertaining to critical phen0mena.l2-'~ Little effort, however, has been made in the literature to show the relation between changes occurring in the critical region and the more common phase behavior, therefore i t is believed that a presentation of the fundamentals of phase behavior with reference to hydrocarbons is timely.

Study of phase behavior is not new. In the metallurgical field, knowledge of heterogeneous equilibria has advanced tremendously, particularly with reference to solid-solid and solid-liquid behavior. Much is known about vapor-liquid equilibria too, but few engineers are familiar with this subject.$

In a discussion of this sort, a definition of terms used is most impor- tant. Such terms as "pressure," "temperature " and "volume " need little definition other than mention of the units in which they are con- sidered. Pressure is expressed in pounds per square inch absolute. Temperature is usually expressed as degrees Fahrenheit or degrees Rankine. In considering thermodynamic and phase behavior, the use of the absolute. Rankine. scale is desirable. Volume is expressed as specific volume, such as cubic feet per pound. This will be recognized as the reciprocal of the specific weight. In considering systems composed of more than one component, it is sometimes desirable to consider molal

Manuscript received at the office of the Institute Oct. 24, 1939. Petroleum Engineer, Union Oil Company of California, Compton, California.

t Notable among the various investigators are Sage, Lacey and co-workers, Katz and Lindsly. Only a few selected references of these and other authors will be used.

Numbers refer to references at end of paper. 1 Several books dealing with phase behavior are included in the bibliography. 16-20~44

-- Copyright, 1940, by the American Institute of Mining and Metallurgical Engineers, Inc. PmTnoLpn~ TECEXOLO~Y, February 1940. Printed in U. S. A.

2 FUNDAMENTAL PHASE BEHAVIOR OF HYDROCARBONS

or weight composition rather than volume. This will be considered more fully in discussing two-component systems.

System.-A system may be classified as a one, two, three, or multi- component system. From a strictly scientific point of view a component must be defined as a pure substance. Thus a system containing nothing but propane would be of one component, while one consisting of methane and propane would be of two components. I t follows that even the simplest of crude oils and natural gases are multi-component systems. However, it is sometime practical, under a wide range of conditions, to consider naturally occurring hydrocarbon mixtures as two-component systems in which natural gas and crude oil are the respective components.

Phase.-Under the proper conditions, any system may exist as one or. more phases. In the language of Willard Gibbs, a portion of matter homogeneous in the sense that its smallest mechanically isolable parts are indistinguishable from one another physically or chemically is a phase.

In Fig. 1 is shown a generalized pressure-temperature diagram for a one-component system. * In i t the lines represent the loci of equilibrium points. For example, any point on the line AB represents a condition of equilibrium between the solid and vapor phases for the particular pressure and temperature chosen. In like manner BD represents the solid-liquid equilibrium line while along the line BC liquid and vapor coexist. At no place does more than one phase exist except for the con- dition occurri~lg a t the boundary lines or their extensions, BF and BE, which represent the metastable states of supercooling and superheating, respectively. Thus, for the pressure P gas and liquid can exist together only a t the temperature T.

Phase Rule

Fig. 1 shows that as long as only one phase exists there are within the limits of the boundaries of that phase for a given pressure an infinite number of values for the temperature. If two phases coexist, there is only one va.lue of temperature for each value of pressure. Under such a condition, the system is said to have only one degree of freedom. In order for three phases to exist for a one-component system, the pressure and temperature are both fixed and the system is said to have no degrees of freedom. Physical behavior of this nature may be expressed as the phase rule, which may be given as:

P + F = 3 PI - - - - -- -

* Fig. 1 represents projections of curved surfaces onto a plane. The diagram has been generalized in order to show various points of interest, consequently it is not drawn to a scale applicable to any given system.

JOHN E. SHERBORNE

where P - number of phases present and F = number of degrees of freedom. This rule can be more generally expressed as:

P + F = C + 2

where C = number of components.

- TEMPERATURE - FIG. 1.-GENERALIZED PRESSURE-TEMPERATURE DIAGRAM FOR A ONI!I-COMPONENT

BYSTEM.

While the phase rule in itself is of little practical significance, it clearly demonstrates that with an increasing number of components the number of possible combinations rapidly increases.

In this figure, there are two other points, B and C, which are of interest. Point B, commonly referred to as the "triple point," represents the state in which gas, liquid and solid phases all exist simultaneously. Here, it will be noted, there are no degrees of freedom, the pressure and temperature being fixed. Point C represents the critical state of the system. It is the temperature a t which the properties of the liquid are identical with those of the gas. Since the pressure and temperature are fixed, here again no degrees of freedom exist and the state of the system is fixed. For a one-component system, the critical temperature is the highest temperature at which gas and liquid phases can both exist.

4 FUNDAMENTAL PHASE BEHAVIOR O F HYDROCARBONS

Since Fig. 1 shows only changes in pressure and temperature, an attempt has been made to show in Fig. 2 a series of surfaces representing the relation between changes taking place in the temperature, pressure and volume.*

Whether the system be in the solid, liquid or gaseous phase, a knowl- edge of the pressure, temperature and specific volumet is necessary and sufficient to completely establish the volumetric behavior of a one- component system. The relation may be expressed as

For obvious reasons the surfaces in Fig. 2 have been restricted to finite values of the three variables. Hence, for the limited range of

SPECIFIC VOLUME / FIG. 8 . -GENERAL~ZED PRESSURE-TEMPERATUlUbVOLUbfE DIAGRAM FOR A ONE-

COMPONENT BYSTICM.

pressure, temperature and volume covered by the surface ABGF only the solid phase exists, and in like manner, for the conditions represented by the surface BDHG, only solid and liquid phases coexist, and so on. The points GHI determine the triple-point line, all three points coinciding to form the point B in Fig. 1.

This diagram does not represent all known cases of one-component systems, water being one exception. However, it is applicable to most of the known pure substanoes.

t External fields of force, such as gravitational, electrical and magnetic and sur- face forces are assumed to be negligible. Unless otherwise noted, these assumptions will apply throughout this paper.

JOHN E. BHERBORNE 5

In general, only liquid, liquid-vapor and vapor phases need be investigated in considering hydrocarbon reservoir systems. Of particular interest is the region bounded by HCI, which represents the two-phase region, liquid plus vapor. Starting with the system a t point a, in which condition it is all liquid a t high pressure, holding the temperature con- stant a t t l and reducing the pressure from a to b results only in a change in the volume of the liquid until a t b evaporation starts. The volume of the system increases at constant pressure from b to d until a t d only an infinitesimal amount of liquid remains. Isothermal expansion of the gas takes place from d to e.

From the slope of the lines ab and de the compressibility of the liquid and gas respectively can be obtained. In the two-phase region the compressibility derivative becomes infinite. This derivative may be expressed as follows :

Compressibility (8) = - - (:;)* In a similar manner, from the slopes of the lines shown by mn, no, ob,

bd and dq, which represent isobaric changes in the volume of the system with temperature, the coefficients of expansion of the system for the conditions existing in the respective region chosen can be obtained. That is, from the slope of a line such as ob we obtain the expression:

Thermal expansion (a) = t$)p In Fig. 2 the critical point is given by C. At and above the critical

temperature t, it is not possible to distinguish between liquid and vapor. The fact that at the critical point the pressure, temperature and volume are fixed is shown.

The phase rule shows that for a two-component system it is possible for four phases to coexist. These may be any combination of liquid, solid or vapor, including four solid or four liquid phases. However, there have never yet been found more than two liquid phases coexisting in a two-component system. Although the production of oil from deeper zones of higher pressures and temperatures may disclose the coexistance of two or more liquid phases, this discussion will be restricted to the behavior of systems more common at the present time, those having only one liquid phase. Specifically, hydrocarbon systems as they occur in the reservoir will be considered to consist of varying mixtures of the component natural gas and the component crude oil.

6 FUNDAMENTAL PHASE BEHAVIOR OF HYDROCARBONB

Phase changes of a two-component system can best be shown by pressure, temperature and composition diagrams. While in some cases the use of molal composition is desirable, for practical purposes weight composition is most convenient. Using weight composition the crude oil or natural gas may be expressed as a percentage of the total weight of the system. This can be directly related to the gas-oil ratio providing the system is always assumed to be composed of two components. Unless it is stated otherwise, the weight composition will be used throughout the remainder of the paper.

FIG. 3.-GENERALIZED PRESSURE-TEMPERATURE COMPOSITION DIAGRAM FOR A TWO- COMPONENT SYSTEM.

In Fig. 3, based on a diagram by R o o ~ e b o o m , ~ ~ is shown a three- dimensional representation of the pressure, temperature, composition relations of a two-component system for which there is only one liquid phase.* The graph is plotted in rectangular coordinates in which the ordinate is pressure while the abscissae are composition and temperature. Point A represents 100 per cent of the pure substance A, natural gas, while point B represents 100 per cent of the pure substance B, crude oil. Point X represents a mixture of A and B of which the fraction XBIAB is component A, and the fraction XA/AB is component B. It should be

* Fig. 3 is a generalized diagram and is not drawn to scale for any specific binary system. The pressure and temperature of points A and B are shown a t some value above zero, and in order to avoid confusion resulting from too many lines, many of the details pertaining to the solid and associated phases in the region of low pressures have been omitted.

JOHN E. SHERBORNE 7

realized that the solid-liquid equilibria depicted in Fig. 3 are not descrip- tive of crude oil-natural gas systems.

A plane through the points CANIAODG represents the pressure- temperature diagram for the pure substance A ; likewise the points CBQHBPFE represent the pressure-temperature diagram for the pure substance B. These points, if projected onto one plane, give the diagram shown in Fig. 4. The similarity between Fig. 4 and Fig. 1 can be seer1 readily. Since for a given temperature the vapor pressure of the sub-

TEMPERATURE 4.-PROJECTION OF LIMITING COMPOSITION CURVES FOR TWO-COMPONENT

ONTO PRE8SUR.E-TEMPERATUBE PLANE. SYSTEM

stance A is greater than that for B, A is the more volatile and, conse- quently, represents gas in the gas-crude oil system.

Before considering pressure-temperature diagrams for mixtures of A and B, it is desirable to orient the phases on an isobaric graph of com- position and temperature taken a t pressure P . This is represented by the plane MNOPQK in Fig. 3 and is shown in Fig. 5. Choosing a composition represented by point a, Fig. 5, and following the changes of sbate that take place when the temperature is increased, one follows the line abcdejgp. The region ab represents a solid composed of a mixture of the two components in the proportion given by the composition a. At b, liquid of the composition y starts to appear until the solid in equi-

8 FUNDAMENTAL PHASE BEHAVIOR OF HYDROCARBONS

FIG. 5.-ISOBARIC COMPOSITION-TEM- r e t i vely. The hubble-point P E U T U B E SECTION SHOWING CONDITION OF TWO-COMPONENT SYSTEM SHOWN IN F I G . state is that in which an infinitesi-

librium with the liquid is pure, solid B. As the system changes from b to d the composition of the liquid follows the path yhd until at d all of the solid B disappears and liquid of the composition a exists from d to e. At point c, liquid of the composition given by h occurs in equilibrium with the solid B. The weight fraction of liquid present is given by the ratio ci:hi, while the proportion of the mixture that is solid is ch:hi. At

PRESSURE - PZ i P point e gas of composition shown M

Q by o starts to form and from e to g the composition of the vapor phase follows the path omg, while the liquid in equilibrium with it follows the path enq. At j an amount of gas of composition m , the weight fraction of which is expressed by the ratio jn lnm, is in equilibrium with liquid of the composition n, which exists in the

i proportion given by the ratio jmlmn. A trace of liquid of the composition q coexisting with gas

Y I of composition a exists a t g.

3 WHEN PRESSURE EQUALS P. ma1 amount of gas exists in equi- librium with the liquid phase, and in like manner, the dew-point state i s that in which a n injinitesimal amount of the liquid i s in equilibrium with the vapor phase. The temperature at point e is commonly known as the " bubble-point temperature " and is defined as follows :

The bubble-point temperature i s that temperature at which there is an injinitesimal amount of gas in equilibrium with the liquid phase, for a g i v ~ r ~ pressure and composition.

In like manner, the temperature at point g is known as the "dew-point temperature," and is defined as follows:

The dew-point temperature i s that temperature at which there i s an injinitesimal amount of liquid in equilibrium with the vapor phase, for a given pressure and composition.

It also is possible to speak of bubble-point pressure and bubble-point composition. Bubble-point composition may be expressed as bubble- point gas:oil ratio. The dew point may be treated in the same manner.

I I

SOLID I I

From g to p there is only gas. Two points, e and g, have

I special significance in petroleum 0 a P 1002

100% production, for they represent the A C O M P O S I T I O N B bubble ~ o i n t and the dew ~ o i n t .

JOHN E. SHERBORNE 9

In Fig. 6 is shown a temperature-molal composition diagram of the vapor-liquid region for the system methane-propane as it is at a pressure

200

100 F a" 0 ZJ t- 2 -100 W a = -200 ;-"

- 300

-400 0 20 40 60 80 100

METHANE PROPANE M O L A L COMPOSITION

FIG. 6.-TEMPERATURE-COMPOSITION DIAGRAM FOR TWO-COMPONENT SYSTEM, METHANE-PROPANE AT 600 LB. PER so. IN. PRESSWE.

Solid lines based on data from Sage, Lacey and Schaafsma (methane value from J. H. Perry, Handbook Chem. Eng., 619. New York, 1934. McCraw-Hill Rook Co.).

of 600 lb. per sq. in. absolute. Only the value for pure methanez2 and the values above 70°F.'0 are experi- mentally correct. The dotted lines are used solely to show the general nature of the vapor-liquid region.

A three-dimensional diagram of the liquid-vapor region for a two- component system is shown in Fig. 7. In it, the lines DLE, FMG, HNI, etc., represent bubble-point curves for the temperatures TO, TI 2 and Tz, respectively. A movement along any one of these lines would " represent an isothermal change in bubble-point pressure with compo- sition for the particular temper- ature chosen.

In like manner, the lines DQE, A FPG, HOI, etc., represent the dew- C O M P O S I T I O N

F I a . I 7.-GENERALIZED r ~ ~ ~ s s ~ ~ ~ - point curves for the respective TEMPERATURE-COMPOSITION DIAGRAM OF temperatures. TWO-COMPONENT BYBTEM IN RANGE OF

Fig. 8 shows an isothermal set- CONDITIONS UBUALLY FOUND I N PETRO- LEUM RESERVOIRS.

tion of composition versus pressure a t the temperature TI, Fig. 7. Here again the ordinate A F represents pure component A and BG represents pure substance B. Points F and G

10 FUNDAMENTAL PHASE BEHAVIOR O F IIYDROCARBONS

are the vapor pressures of the pure substances A and B, respectively, at the temperature for which the section is taken.

If a composition X is chosen and the pressure increased, the system starts as a gas and undergoes no change, except in volume, until a t the point P where an infinitesimal volume of liquid of composition d forms. This is the dew-point pressure. A further increase in pressure results in the formation of more liquid of a composition given by some point on the curve from d to M in equilibrium with gas given by some point

I I I I

A X B (GAS) COMPOSITION (OIL)

FIG. 8.-ISOTHERMAL PRESSURE-COMPOSITION DIAGRAM FOR HYPOTHETICAL CRUDE OIL-NATURAL GAS SYSTEhI AT TEMPERATURE TI.

on the curve between P and e . At b the weight fraction bc/ac of gas of composition a is in equilibrium with the weight fraction ablac of liquid of composition c. The total composition of the system is still given by X. By increasing the pressure still more! further quantities of the gas are liquefied, until a t M there is only an infinitesimal amount of gas in equilibrium with the liquid. This is the bubble-point pressure, for the temperature and composition in question.

Referring again to Fig. 7, the line DFCA represents the vapor-pressure curve for the component A and the line EGICB the vapor-pressure curve for the component B. If a mixture of A and B having a composition X is selected, the vapor-temperature relations will be expressed by the

2

S 4 0, 8 8 us

12: ,,, ?2* 5 $3 2 3E

a h-4

;

""3 2 B 0

w u

+ z; &

A

2 8 fa

il 9 '4 fa ilr

6 s i al l a; i F

12 FUNDAMENTAL PHASE BEHAVIOR OF HYDROCARBONS

plane LMNCXCROPQ instead of a line. Points CA and C B are the critical points of the pure components A and B, respectively. The critical point of the mixture is C x ; and for the example given it occurs a t a higher pressure and temperature than that of the more volatile com- ponent. The behavior shown in Fig. 7 is typical of that generally encountered in binary paraffin hydrocarbon systems.

Line CACRCB is the locus of points of maximum temperature for the two-phase region. Point CR, which is known as the cricondentherm,' or critical condensation temperature, is the highest temperature a t which liquid and vapor coexist for the mixture of composition X. As previously defined, for a one-component system the critical temperature was the highest temperature a t which the liquid and gas could occur together in equilibrium. I t was also defined as the temperature a t which the liquid and vapor phases were identical.

For a system composed of any number of components, the critical temperature i s a lways the temperature at which the liquid and vapor phases have identical properties. That it is not the maximum temperature at which the liquid and vapor phases can coexist for systems of two or more components is illustrated in Fig. 9, which is a pressure-temperature section for the system in Fig. 7 a t the composition X. The critical point, Cx, occurs a t a lower temperature than does the cricondentherm CR.

Because i t is not possible to show by means of Fig. 9 that the proper- ties of the liquid and gas are not identical a t the cricondentherm, a pressure-composition diagram, Fig. 10, for the two-component system methane-propaneg was constructed using two constant-temperaturc curves which were selected so that the points Cx and CR correspond to the composition X of Fig. 9. By the use of ratios, as employed in the dis- cussion of Figs. 3 and 5, i t can be shown that a t Cx the composition of the liquid is identical with that of the gas. This occurs a t 8g°F., while the cricondentherm CR occurs a t 130°F. At this latter point there is an infinitesimal amount of liquid of composition given by point a in equilib- rium with gas of composition X. Fig. 10 fails to show that the liquid and gas can coexist a t a pressure higher than that a t the critical point. This point, the maximum pressure a t which two phases can coexist, is shown by C, in Fig. 9. For a one-component system, the points C p , Cx and Cg coincide to form one point.

I t is convenient to assume that Fig. 9 represents a system of natural gas and crude oil, in considering what happens in production from n reservoir. In one case, illustrated by the line abc, there is liquid* (oil)

* Near the critical region it is difficult to distinguish between liquid and gas in the single-phase region. However, since a t the bubble point b there forms an infinitesimal

JOHN E. SHERBORNE 13

under a high pressure and at a temperature higher than surface tempera- ture but not higher than the critical temperature. No free gas exists. As the fluid rises in the well there is a decrease in pressure, from a to b, until at b gas starts to separate. This is the bubble point for the pressure and temperature shown. As fluid approaches the surface, the pressure continues to decrease and more and more gas comes out of solution until at the surface the point c is reached. While it is recognized that a temperature decrease occurs in the system between the reservoir and the surface, this change has been considered negligible for the purposes of this and the following illustrations.

In the second case, one in which the reservoir has a pressure similar to that of the first but a temperature above the critical temperature of the hydrocarbon system, only the gaseous phase* exists in the reservoir. This condition is shown by e. In coming to the surface the gas undergoes only a relatively small drop in pressure and no fluid appears. This is represented by some point z on the path of ej. If the pressure is decreased more, liquid starts to condense a t j and greater quantities of liquid form with further decrease in pressure, until at some point g a maximum amount of liquid condenses. Any further decrease in pressure results in a decrease in the amount of the liquid phase until the dew point is reached at h. At this point, the final drop of liquid vaporizes and only gas remains. This phenomenon, resulting in the formation of a liquid phase followed by its disappearance as a result of progressive increase or decrease in pressure, under these conditions of restraint, on a system of constant composition, has been called retrograde condensati~n.'~

I t is important to note that retrograde phenomena may occur as a result of change in pressure or temperature, but only when the composi- tion of the system remains constant. Changes in the reservoir condition that result from the selective withdrawal of either gas or oil cannot be classed as retrograde phenomena, although there is an apparent similarity.

Formation Volumes.-While the use of composition is convenient in illustrating changes that take place in multi-component systems, the effects that changes in pressure and temperahre have on the volume of constant or variable composition systems are of particular practical significance. The term "formation volume" as used is the ratio of the volume occupied by the hydrocarbons at subsurface equilibrium tempera- ture and pressure to a unit volume of oil as measured at 60°F. and a pressure of 14.73 Ib. per sq. in. abs. Liquid shrinkage is sometimes used instead of formation volume and may be defined as its reciprocal.

amount of the vapor phaee, and at the dew point j there forms an infinitesimal amount of the liquid phase, it is convenient to aesume that the single-phase fluid at the point a is liquid and that the phase at e i8 gaseoue.

See preceding footnote.

14 FUNDAMENTAL PHASE BEHAVIOR OF HYDROCARBONS

Closely allied with phase behavior in the treatment of petroleum- production problems is the effect upon the viscosity of the crude resulting from changes in pressure, temperature and composition. Since a number of papers have been published regarding t,his it will su5ce a t this time to point out that as much as eightfold changes in viscosity can occur throughout the ranges in pressure, temperature and compositior~ commonly occurring under producing conditions.

The applications of phase behavior to practical field problems are numerous, and have been discussed by many writer^.^^-^^ A review of these applications is beyond the scope of this paper, but it should be noted that in studies of reservoir conditions, in volumetric estimate of reserves, and in problems involving both homogeneous and hetero- geneous flow, a knowledge of phase behavior for the hydrocarbon system involved is requisit'e.

SELECTED BIBLIOGRAPHY

1. L. W. T. Cummings, F. W. Stones and M. A. Volante: Ind. and Eng. Chem. (1933) ae, 728.

2. P. Duhem: J d . Phys. Chem. (1897) 1, 273. 3. J. E. Gosline and C. R. Dodson: Amer. Petr. Inst. Drill. and Prod. Practice (1938)

423. 4. J. P. Kuenen: Ztsch. Phys. Chem. (1893) 11, 38. 5. Ibid. (1897) 24, 667. 6. B. E. Lindsly : Petr. Engr. (Feb., 1936) 7,34. 7. B. H. Sage and W. N. Lacey: Amer. Petr. Inst. Drill. and Prod. Practice (1935)

141. 8. Ibid. (1936) 158. 9. B. H. Sage, W. N. Lacey and J. G. Schaafsma: Ind. and Eng. Chem. (1934) 26,

214. 10. B. H. Sage, W. N. Lacey and J. G. Schaafsma: Amer. Petr. Inst. Prod. Bull. 212

(1933) 119. 11. H. S. Taylor, G. W. Wald, B. H. Sage and N. N. Lacey: Oil and Gas Jnl. (Aug. 10,

1939) 38,46. 12. E. 0. Bennett: Petr. Engr. (mid-year, 1939) 10, 50. 13. C. R. Horn: Oil Weekly (Sept. 11, 1939) 98, 27. 14. D. L. Katz: Amer. Prod. Inst. Drill. and Prod. Practice (1938) 435. 15. D. L. Katz and C. C. Singleterry: Trans. A.I.M.E. (1939) 132, 103. 16. F. V. L. Patten and C. I. Denny: Oil Weekly (Dec. 12, 1938) 92, 21. 17. A. Findley and A. N. Campbell: The Phase Rule and ita Application. New York,

1938. Longmans, Green and Co. 18. J. P. Kuenen: Verdampfung und Verflthsigung von Gomishen. Leipzig, 1906.

Barth. 19. A. C. D. Rivett: The Phase Rule. London, 1923. Oxford Univ. Press. 20. B. Roozeboom: Die Heterogenen Gleichgewichte vom Standpunkte der Phasen-

lehre, 11. Braunschweig. Viewig und Sohn.

JOHN E. SHERBORNE 15

21. H. S. Taylor: Treatise on Physical Chemistry. New York, 1935. D. Van Nostrand Co.

22. J. H. Perry: Chemical Engineering Handbook, 619. New York, 1934. McGraw- Hill Book Co.

23. C. E. Beecher and I. P. Parkhurst: Petr. Dev. and Tech. in 1926, A.I.M.E. (1927) G26, 51.

24. B. H. Sage and W. N. Lacey: Ind. and Eng. Chem. (1938) 30,829. 25. B. H. Sage and W. N. Lacey : Trans. A.I.M.E. (1938) 127,118. 26. B. H. Sage, B. N. Inman and W. N. Lacey: Ind. and Eng. Chem. (1937) 29,888. 27. B. H. Sage, W. R. Mendenhall and W. N. Lacey. Amer. Petr. Inst. Prod. Bull.

216 (1935) 45. 28. B. H. Sage, J. E. Sherborne and W. N. Lacey: Znd. and Eng. Chem. (1935) 27,

954. 29. B. H. Sage, J. E. Sherbome and W. N. Lacey: Amer. Petr. Inst. Prod. BUZZ. 216

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