Paper Sobre Experimentos de Presion Capilar

7/29/2019 Paper Sobre Experimentos de Presion Capilar

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T.P. 2640

EXPERIMENTS ON THE CAPILLARY P RO P ERTIESOF PORO(]S SOLIDS

JOHN C. CALHOUN, JR., MEMBER AIME, DEPT. OF PETROLEUM ENGINEERING, UNIV. OF OKLA.MAURICE LEWIS, JR., JUNIOR MEMBER AIME, and R. C. NEWMAN, STUDENT ASSOCIATE AIME

ABSTRACTA report is made of experimental

work performed on the capillary retention of water within porous solid systems. the displacement being accomplished with ai r and various organicliquids. A portion of the experimentswere designed to measure the loweringof vapor pressure of water within aporous solid with subsequent conversion of such vapor pressure data tocapillary pressure values. The form ofthe capillary pressure curve at highcapillary pressures has been elucidatedfrom this data. Certain theoretical approaches are presented to indicate thecorrelation between work done by previous investigators. The present work iscorrelated also with theory. Surfacearea values are calculated for the various core systems studied.

INTRODUCTIONLeverett' in 1941 presented a paperwhich gave the essential concepts of

capillary behavior in porOU!3 solids. Hispresentation included both theoreticalan d experimental aspects of the problem. He defined the term "capillarypressure" and applied it to an idealporous system. His work included thedefinition of a dimensionless quantitywhich was a function of fluid saturationand which could be correlated withporosity and permeability for clean,unconsolidated sands. His experimentalwork was performed by the drainage ofwater from packed columns of unconsolidated sands.

Subsequently, other authors havepresented experimental techniqueswhich permit the determination of capillary pressure data on small core samples. Data have been presented onporous systems other than unconsolidated sand and with the use of liquids

Manuscript received a t office of the BranchJanuary 24, 1949. Presented at AIME AnnualMeeting. Sa n Francisco, Calif., February 13-17.1949.1 References ar e given at end of paper.

July, 1949

other than water. Notable among theseare the works of Hassler, Brunner andDeahl; Bruce an d Welge: Amyx an dYuster,' and Purcell" Others' 1 haveindicated the applicability of such datato field problems without regard to thetheoretical aspects of the capillary pressure saturation function. Within the recent published literature there havealso been reported studies of the correlations between capillary pressure dataand other fundamental properties ofporous solids."'"

It was the intent of the present reported work to investigate the behaviornoted by water and various other liquids within porous systems in an attempt to amplify existing correlationsbetween capillary properties, surfaceproperties, and other fundamental characteristics of porous systems. The experimental work described here wasperformed as two different investigations bu t due to its nature it is reportedas separate phases of the same generaltopic.CAPILLARY PRESSURE AND

VAPOR PRESSURE LOWERINGI t has long been recognized that the

vapor pressure above the curved surface of a liquid is a function of thecurvature of the liquid surface. 1O Theca pillary pressure is also a functionof the curvature of the liquid surface.Both capillary pressure values an dvapor pressure lowering are, therefore,functions of the liquid saturation of aporous solid. A quantitative relationship between the vapor pressure lowering and the capillary pressure can besimply developed by considering a porous solid system containing water inequilibrium with its vapor.

The capillary pressure at any heightwithin the system is defined as

Pc = Pv - P, - - - - (1)where Pv is the pressure in the vaporphase and P, is the pressure in the

PETROLEUM TRANSACTIONS, AIME

liquid phase. Consider that both thewater and it s vapor are continuousphases within the system and that atany point the two phases are in equilibrium.

The gradients in the liquid and vaporcolumns within the porous system are:

dP , = plgdhdPv = pvgdh

By integrating these gradients from thepoint where h = 0 to the height h,values of P, an d Pv at the point harefound. Let the point where h = 0 bethat where Pc = _ At this point alsoPv is equal to Pvo, the equilibrium vaporpressure of the liquid above a flat surface.From the first of these integrationssince PI is independent of height:

P, = P1gh + PvoFrom the second, by substitutingp,. = ~ ~ , where M is the molecular

weight:In12. = ~ ghPvo RT

By combining the two last equationsto eliminate h, one obtains:P RT I PvI =--PI n-- _ PM Pvo vo

Substituting values for Pv and P, inequation (1) gives

PvIn ---P voP voor P - RTpv [ I ~ I 12.]_ P(> - M - pv n PyO YOwhich is approximately

I Pvon-Pv - - - - - - (2)This equation relates the vapor pres

sure above a curved surface with thecapillary pressure across the curvedsurface in terms of measurable quantities. The equivalent of this equationis given by Freundlich.'

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T.P.2640 EXPERIMENTS ON THE CAPILLARY PROPERTIES OF POROUS SOLIDS

There is much data in the literatureon vapor pressures within porous solidsat partial liquid saturations. Such dataar e often procured for adsorption measurements. To illustrate the conversionof data by equation (2), data on theadsorption of water vapor by silica gelwere taken from the literature." Thedata, given as pressure of adsorptionversus saturation, were converted tovalues of p, by equation (2). Thevapor pressure curves and corresponding capillary pressure curves are shownin Fig. 1. The hysteresis loop is similarto that shown by Leverett! on his capillary pressure curve3 for unconsolidatedsands.

FIG. 1 - CONVERSION OF VAPOR PRESSUREDATA TO CAPILLARY PRESSURE DATA. VAPOR

PRESSURE DATA FROM REFERENCE 11.

EXPERIMENTS ONVAPOR PRESSURE LOWERING

Equation (2) offered itself as a medium for evaluating capillary pressuresat low saturation values an d high values of the capillary pressure. With thisin mind an apparatus was constructedto measure the difference in vapor pressure above a flat surface of water andthat within a porous body at partialwater saturation. The apparatus usedis diagrammed in Fig. 2. Although itwas not sufficiently sensitive to detectsmall changes in vapor pressure, it sufficed to give accuracy over the rangeof saturation. The porous media employed were composed of consolidatedquartz. They were manufactured in thelaboratory by consolidating sand grainswith silica by deposition from colloidalsuspension an d by the hydrolysis oftetraethyl orthosilicate. The sand grainsbefore consolidation were in the sieve

190

FIG. 2 - DIAGRAMATIC SKETCH OF APPARATUS FOR MEASURING VAPOR PRESSURE

LOWERING.range, 140-170.

Data were taken on four separatecores. They were first saturated fullywith water and by the use of a conventional type capillary pressure cell

GRAD.PIPETTE

CORE

LEVELINGiWLB

FIG. 3 - DIAGRAMATIC SKETCH OF APPARATUS FOR MEASURING CAPILLARY PRESSURES

BY DISPLACEMENT.


(shown in Fig. 3) the capillary pressure curve for each core was determined in a routine fashion. The corewas then placed in the vapor pressureapparatus, maintained at a constanttemperature of 97F., where furtherdesaturation was accomplished by evacuating water vapor from the core sideof the system. After sufficient evacuationhad taken place, the system was closedan d allowed to come to equilibrium,after which the manometer read directly the amount of vapor pressure lowering. Saturations were determined by removing and weighing that portion of theassembly which contained the core. Itwas necessary after each weighing topurge the apparatus of the air whichentered when the assembly was unjointed for the weighing.

Both the routine capillary pressuredata and the vapor pressure loweringdata are given in Table 1. In Fig. 4are plotted curves which show the datataken by the customary method of obtaining capillary pressures for the coreslisted in Table 1 and for several othersimilar cores. The curves of Figs. 5,6 and 8 show the capillary pressuredata calculated from the vapor pressurecalculated from the vapor pressurepressure data shown in Fig. 4. Th elogarithm scale is used to permit showing the high values and at the sametime showing the lower plateau region.One experimental point on two of thesecurves was determined neither with theapparatus of Fig. 2 nor Fig. 3 but bymeans of an applied air pressure above00

: I ' I :- '--TTI---E-1----1-1--[-r I -i401--4-1++-4-'+-1--+-1-[ -'--+-r-I

I V-Ii: --r-I- Ir- -+--r- L --I, 1 I I I i1-----. -- -- f., i il I

30

1 I ~201-------\\\'\\-\+- T ~ - l - I I

c- I - - l ~ \ -;-il" I ,T !, I'!% ~ - - ~ 0 ~ 2 ~ - - ~ 0 4 ~ ~ 5 - . ~ M ' - ~ ~ 0 " - - ~ - r l t C

FiG. 4 - CAPILLARY PRESSURE CURVES WITHAIR DISPLACEMENT; 0 CORE NO.2; 'V CORENO.3 ; D CORE NO.5 ; CORE NO.6 ;o CORE NO.8; :,. CORE NO.9;

X CORE NO. 10.

July, 1949


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JOHN C. CALHOUN, JR., MAURICE LEWIS, JR. AND R. C. NEWMAN T.P. 2640

TABLE 1Capillary Pressure Data by Routine Displacement and by

Vapor Pressure .LoweringCore #2 Core #3

K ~ 2 1 0 0 rod. K ~ 8 5 5 md.. p ~ 0 . 3 2 2 . p ~ 0 . 2 9 1--

I Pc I PcSw Pyo/Py Cm Hg Sw ~ \ ' ( ) 2 ' ~ ~ C ~ 3 - = - _- - - - - - - - - - -0.830 5.43 0.712 5.380.603 6.19 0.486 5.930.578 6.34 0.276 8.480.459 7.05 0.192 17.460.394 8 26 0.142 36 40.361 8.85 0.0962 1.052 5,4300.315 9.93 0.0744 3.289 127,0000.248 17.6 0.0668 4.367 158,0000.197 22.1 0.05 60 6.369 198,0000.0667 1.304 28,350 0.0533 11.111 257,0000.0549 2 .016 75,000 0.0407 20.833 325,0000.0527 2.066 77,5000.0493 2.155 82,000a diaphragm on which the core wasplaced. These are the data at approximately 300 cm. Hg. on cores No.6 andNo.8.

Although no overlap of data betweenthe vapor pressure method and the

7,

6.

5"::;04,Za.U 3,"...J

0

0

0\

o , \0

0 1\"-b--.o 01 0 2 0 3 0 4 0.5 0.6 0.7 08 ogs.

FIG. 5 - C ,' PILLARY PRESSURE DATA CORENO. 20 DETERMINED BY DISPLACEMENT; D:ERMINED BY VAPOR PRESSURELOWERING.

IJ

IJ

0'

0

0 ,\\1.0 .......

,o 01 0, 04 ,). , 06 01 0 8 09FIG. 7 - CAPILLARY PRESSURE DATA CORENO. 60 DETERMINED BY DISPLACEMENT; DETERMINED BY VAPOR PRESSURELOWERING.

July, 1949

Core #6 Core #8K ~ 7 6 0 md . K ~ 3 0 0 0 md.

. p ~ 0 2 7 9 . p ~ 0 . 3 2 1Pc

~ 1 ~ Y o / p yPcSw Pvo/Py CmHg CmHg

0.811 50 5 0.856 5.190.530 6.02 0.691 5.890.343 7.51 0.558 6.860.246 12 60 0.464 8.990.181 25.3 0.346 13.00.146 45.0 0.252 23.60.144 305.0 0.201 38.80.103 1.101 10,250 0.121 305.00.079 3.344 129,000 0.0909 1.337 31,0000.069 4.796 168,000 0.0798 1. 782 61.9000.058 7.692 217,000 0.0676 2.941 11f-,0000.0536 9.709 243,000 0.0570 3.922 147,0000.0479 19.157 316,000 0.0496 5.917 191,0000.0382 28.903 360,000 0.0487 9.346 239,0000.0325 37.313 387,000 0.0375 10.526 252,0000.0319 24.096 340,000

routine displacement was procured, thedata give, in each case, continuouscurves from one hundred per cent saturation to zero saturation.

A significant point to this correlationis that there is, strictly speaking, no7/ )

1\\0

. \0 -o O! 0.2 0.3 0.4 0.5 06 0.7 0.8 0.9

FIG. 6 - CAPILLARY PRESSURE DATA CORENO . 30 DETERMINED BY DISPLACEMENT; DETERMINED BY VAPOR PRESSURELOWERING.

6.0

50

"::;vz73 0

0

10

1\\f-- If - - ;I

1\"- ~ r -?o 01 0 ... 03 0 4 OS 0.6 07 08 09

s.FIG. 8 - CAPILLARY PRESSURE DATA CORENO. 80 DETERMINED BY DISPLACEMENT; DETERMINED BY VAPOR PRESSURELOWERING.

PETROlEUM TRANSACTIONS, AIME

absolute irreducible minimum of saturation. All of the curves turn into thezero saturation axis at a finite positivecapillary pressure. Fig. 9 shows anexpanded plot of the capillary pressuresin the lower saturation region as calculated from the vapor pressure data.Within the range of average reservoirpressures, however, there is a saturation plateau, which from a practicalview point is equivalent to an irreducible minimum. However, this plateauis reached at capillary pressures somewhat above those normally taken as themaximum in routine laboratory work.From Fig. 4, for example, on all coresit appears that the irreducible minimum saturation would be approximately that saturation shown at the highestcapillary pressure, which is about 45cm. Hg. As shown by Figs. 5 to 8, however, all of these cores at a capillarypressure of 45 cm. Hg. are several percent away from the saturation plateau.

DISPLACEMENT WITHVARIOUS LIQUIDS

It is common to use the capillary displacement of water by air as a measureof the connate wate content of a givenreservoir material. Following Leverett'spresentation of capillary pressure inthe form

P ' = O ~ ~ J(Sw) (3)it has been suggested that, with twodifferent displacing fluids, the ratio ofcapillary pressures for a given value ofwater saturation would be the ratioof the 0 values. Work with systems otherthan air-water have b ~ e n reported byseveral authors. Hassler, et aI ' did notfind a constant ratio as expected. Pur-cell: whose work was done with mercury, gives good correlations betweencapillary pressure curves with waterand with mercury using a ratio ofocos e values rather than a ratio of 0values. Bruce and Welge' give a seriesof different irreducible water saturationsusing ai r and oil displacement.

In the present work several differentpure liquids were used to displace distilled water from three of the laboratoryprepared silica porous bodies. Liquidsused were toluene, benzene, cyclohexane, tetrade cane, and oleic acid. Airwas used as a displacing medium as

191


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TABLE 2Capillary Pressure Data Obtained by Displacement with Various Fluids

Core 1 - L K = 1.22 Darcys = 249Air Tetradecane Oleic Acid Toluene Benzene Hexane

Pc Pc Pc Pc Pc PcCmHg Sw CmHg Sw CmHg Sw CmHg Sw CmHg Sw CmHg Sw------ -- --- --- - - - - - - - - - - ------ - - - - -----1.6 98.01 0.5 98.80 0.2 99.08 1.7 88.1 0.8 99.19 1.43 98.892.7 96.46 1.4 93.38 0.3 96.63 2.1 76.0 1.3 96.60 1.74 90.113.4 91.22 2.1 84.4 0.7 91.73 2.4 57.7 1.4 91.68 2.14 77.204.2 79.0 2.4 71. 7 0.9 82.9 2.7 44.7 1.5 88.9 2.54 68.805.1 68.3 2.7 56.3 1.0 64.5 3. 0 33.3 1.6 83.7 3.40 47.306.2 41.6 3. 4 36.1 1.3 45.8 4.1 30.0 1.7 80.6 5.65 45.208.6 24.7 4.7 23.3 1.6 32.4 5.8 28.8 1.9 72.2 14.4 39.5015.4 22.9 8.1 21.4 2.0 26.0 9.8 28.0 2.0 70.4 14.7 40.2023.9 20.3 7.7 19.9 15.8 22.3 2.1 67.3 15.2 41.0016.0 22.6 2.2 65.4 15.7 42.102.7 62.0 16.3 43.002.9 58.0 16.6 43.603.4 55.0 16.5 43.307.3 54.3 16.7 43.9023.9 53.7 16.6 43.9016.8 44.20TABLE 3

0

0."3z7""..0

::: :::: ~ ~-- -- -

0.02 00 4

-"'r-.:: ~~ ~..." "' ~,"' ~ J\~ I \ '0.06 0.08 0.10

Sw

FIG. 9 - EXPANDED PLOT SHOWING CAPILLARY PRESSURES AT EXTREMELY HIGH

VALUES.

12f-- - - -+-- - tH\-- l -H\- -

Capillary Pressure Data Obtained by Displacement with Various Fluids 101---;--4-;Core 2 - L K = 1.83 Darcys = 251Air Tetradecane Oleic Acid Cyclohexane Toluene

Pc Pc Pc Pc PcCmHg Sw CmHg Sw CmHg !Sw CmHg Sw CmHg Sw-------------- - - ------ ---- - - - -----1.7 96.25 0.4 98.82 0.6 92.88 0.5 98.10 0.8 98.163.2 83.7 1.1 97.04 0.7 85.9 0.7 92.66 1.0 92.613.5 72.8 1.3 89.1 0.9 59.9 1.0 80.5 1.5 79.13.8 56.7 1.6 77.5 1.0 47.0 1.2 69.1 1.6 67.24.1 45.9 2.2 62.2 1.2 30.0 1.6 51.3 2.2 51.54.4 36.6 2.7 39.0 2.3 22.8 2.1 40.8 2.5 41.44.9 27.7 3.4 29.6 5.4 20.2 2.2 33.6 4.9 28.28.4 21.8 5.8 25.0 10.0 19.7 3.5 25.5 7.2 27.714.2 20.9 10.1 20.5 5.3 21.6 16.0 25.914.4 20.4 15.6 19.3 10.2 20.0 24.3 25.720.6 18.0 16.9 19.016.7 18.5TABLE 4

Capillary Pressure Data Obtained by Displacement with Various FluidsCore 3 - L K = 1.11 Darcys = 258Air Tetradecane

Pc PcCmHg Sw CmHg Sw- - - - - - - -0. 8 98.40 0.6 99.041.8 97.69 1.9 95.203.2 95.13 2.5 72.75 . 0 79.6 3. 8 51. 75.7 66.6 4.3 33.46.9 44.9 6.0 24.58.8 33.3 8.0 19.012.1 20.5 12.1 18.015.9 19.8 17.4 17.125.3 16.6well. Saturations were determined bythe volumetric measurement of theamount of displaced saturant in an apparatus similar to that shown in Fig. 3.The results of these tests are givengraphically in Figs. 10, 11 and 12.Tables 2, 3 and 4 list the data takenin chronological sequence. Not shownin these tables are several duplicateruns made with several of the displac.ing liquids.

The irreducible water saturationswhich were attained were approximately

192

I Oleic Acid HexanePc PcCmHg Sw CmHg Sw- - - - - - - - - - - - - - - - -0.7 91.60 0.1 99.080.75 77.4 2.2 94.201.1 61.5 3.0 83.41.3 48.8 3.5 70.82.2 35.2 3.8 56.72.9 26.6 4.2 41.34.9 20.7 4.4 35.07.3 18.7 6.9 27.214.5 25.814.7 26.515.2 27.516.1 28.917.0 30.8

equivalent on each core with the exceptions of benzene displacem"nt on core1.L, toluene displacement on cores lLand 2.L, and hexane displacement oncores IL and 3L. There is some basisfor believing that these apparently highvalues were due to faulty technique because there was on some of these particular tests an apparent regression ofthe desaturation process. It could alsobe true that the apparently high valuesof saturation found by displacementwith these fluids is a manifestation of a


5.

FIG. IO-CAPILLARY PRESSURE CURVES CORENO. IL; DISPLACEMENT OF WATER WITHVARIOUS FLUIDS. 0 AIR; tc, TETRADECANE;

2

0

8

4

2

0

x OLEIC ACID D HEXANE; V TOLUENE; BENZENE.

l

11 ~ h,'\1\ dd r--.t:-- t - -i \ r--: ;:


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JOHN C. CALHOUN, JR., MAURICE LEWIS, JR. AND R. C. NEWMAN T.P. 2640true characteristic of these core systems.

Interfacial tensions between waterand the various liquids were measured with the du Nouy tensiometer an dvalues were also taken from the International Critical Tables. This data isgiven in Table 5. It is apparent fromthese measurements that impuritiesmay have been present in some of theliquids which were employed.

21\\

\P, 1\ """1'-.6 \ ......,. ~ -1\ ~ ~ --4 \ -...... ~ :--- j~ - r--.r- --'" .. h- - ~~ t'ii~ t---- -- - '"""02 04 S.. 06 as 10

FIG. 12-CAPllLARY PRESSURE CURVES CORENO. 3-L. DISPLACEMENT OF WATER WITHVARIOUS flUIDS; 0 AIR; FIRST RUN;.p AIR,SECOND RUN; D HEXANE; l:::, TETRADECANE;x OLEIC ACID.

Fo r a given water saturation, valuesof capillary pressure with ai r displacing divided by the values for liquiddisplacing were always greater thanunity. These ratios are shown in Figs.13 and 14. The dotted lines on thegraphs show the ratios of the interfacial tension for ai r against water to

I I ~V IOR "L OL Ie A 10I ~ =:J_A- / ~ - - ; Ii .

. .A/ IIi - - _ i_ l

P ,........-' fCoR 2 '/

- - - - TE RAO ".2 ...-. ! - - ' -- 'I---- i-----' ~O 3' L f....:::' .. .- 1- - i--'V I-----' ORE 2' LIo 0.2 04 S>'t 0 ' 0&

FIG. 13 - RATIO OF EXPERIMENTAL CAPILLARY PRESSURES AND COMPARISON WITHINTERFACIAL TENSION RATIOS.

July, 1949

TABLE 5Interfacial Tensions(Dynes per centimeter)Temperature = 36C.

ConsecutiveReadings oWater- oWater- oWater-Air Ttradecane TolueneLiquid System

oWater- I oWaterBenzene Oleic Acid oWater- oWater-Hexane Cyclohexane- - ~ ~ ~ ~ ~ ~ ~ - ~ - - - - - ~ ~ - - - - ~ - ~ - ~ - ~ - - ~ - - I - - - - I ~ ' ~ - - ~ ' ~ ~l.23.4 .5.6.7 ..8.9 ..10.11.1213 ..14 ..15 . . . . . . . . . . .

70.270.0 32.5 26.331.7 26.231.7 25.931.4 25.731.4 25.7

InternationalCritical. . . . .Tables .. . 36.1(25C.)

that of the given liquid against water.In some instances, there is excellentagreement between the measured ratiosand theoretical ratios, in particular,hexane on core No. 3-L, benzene oncore No. 1-L and ,etradecane on coreNo. 1-L. In the .najority of instances,although the ratios at 100% saturationare near the expected theoretical valuesthe ratios bhow a decrease as desaturation proceeds. There is no readilyapparent explanation for this drop inratio.Theoretical Correlations:

In recent publications it has beensuggested, and the suggestions havebeen supported to a certain degree byexperimental data, that definite correlations can be obtained between variousfundamental measurable quantities ofporous bodies. Purcell' has offered the

! i ! !: ,..-;i I ~ 8[ . EN!:

Y . L ~ ! . /,.....- fORE IL-- -2 Yo IE 2 II !IT UEN

CO E (- l .-

FIG. 14 - RATIO OF EXPERIMENTAL CAPILLARY PRESSURES AND COMPARISON WITHINTERFACIAL TENSION RATIOS.

PETROlEUM TRANSACTIONS, AIME

24.6 8.5 45.7 31.524.6 8.5 45.6 28.525.4 8.4 45.8 24.323.7 7. 2 45.1 25.123.6 7. 8 46.5 23.145.7 21.5

(25C.) (20C.)35.00 I 15.59 51.1(25C.)

13.714.413.812.611.59. 89.28.89.4

following equation to compute the permeability from capillary pressure data:

K = F IJ' q, cos' (J dSw _ _ _ _ _ (4 )J1.02 Pc'oRose' has developed certain relativepermeability concepts on the basis ofthe Kozeny equation which he gives as :

q,K= A' t (5)

Rose furthermore states that from Leverett's J (Sw) function the followingIS true:

( 1 )'/2im J (Sw)= -Sw-+1.0 t (6)The J (Sw) function as given byLeverett is equation (3). To extend itsapplicability to all liquids it is nowwritten as:

Pc ~ K(S,,) = - - _- - - - - (7)o cos IJ 1>The value of the capillary pressure

as the saturation approaches unity isthe displacement pressure, PD. At thislimit, equation (7 ) is:

PD ~ KSw =-- - --) 1. 0 IJ cos IJ q,This can be rearranged to give anexpression for the permeability. Thus:K = [ J (Sw) 1.0]' q, 0'

Pn'cos' fi (8)

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Comparing equations (8) and (5)and considering equation (6) the following equality is found:A' = PD'

0' cos' eFor a straight bore capillary thisequality is known to be true. A is defined as the surface area per unit ofpore space which, for a capillary ofradius R and length L, would be:

A - 2 7r RL _ 2- 7r R'L - R

The displacement pressure is, in termsof R:2 0 cos ePD = R = Aocos e . (9)

which is the equation given above.Comparing equation (8) with equation (4) gives the following equality:

J1.0-.!:_ d S," =2 Po'o (10)An examination of the data presentedin this report leads to the conclusionthat:

1 _ 2 J l d ~ w ...... (11)PD' - Pc' owhich then produces the equalities thatfollow:

F = 4[J (Sw) LOr4or F=-t

and also

(12)

J1.01 " dSw- = 20 cos e --. . . (13)A' Po'oEquation (11) is given as a result of

observations made on the data given inTables 1, 2, 3 and 4. There is no readily apparent theoretical basis for therelationship. In examining the data ofTable 7 for ai r displacement only, thecapillary pressure curves were converted to plots of the reciprocal of Pc'versus Sw similar to that of Figure 15..These were then planimetered to obtainthe values of the integral and from therelationship of equation (11) values ofPD were calculated. Values of the integral and PD values are listed in Table 6.Comparison is also given in Table 6 between the calculated values of PD andthose read from the curves of Fig. 4 byextrapolation to a saturation of 1.0.

194

TABLE 6[1.0dSw Calc. Observed A ACore No. Displacing PD PD Cm'/Cm' (Carman Method)Fluid Pc ' CmHg CmHg Cm'/Cm')0Cm-'- - - - -2. . . . . . ..... Air .0186 5.18 5.20 960 840

3 ..... .. . Air .0223 4.72 5.15 872 11205 ... .. , . Air .0181 5.25 5.30 970 11256 ...... . . . . . . . Air .0228 4.68 4.60 866 11158. .. ... . .... Air .0172 5.40 4.70 998 7059 ...... ....... Air .0158 5.62 5.80 1,040 920

10 .. .... Air .0252 4.45 4.50 823 364

TABLE 7

Core No. DisplacingFluidto SwOPe 'Cm-2

Calc. Calc. APD o os IJ Cm'/Cm'CmHg dynes/cm- - - - - ------ - - - - - - - - - -I-L............... Air . . . . . . . . . " . . . .Tetradecane . . . . . . . .

O!eic Acid.Toluene . . . .2-L....... . . . . . Air ........ " .. "Tetradecane . . . . . . . .Oleic Acid . . . . . . . . . .Toluene . . . . . . . . . . . .Cyclohexane3-L." . . . . . . . . . . . . . . . . . . . . Air . . . . . . . . . . . . . . . .Tetradecane. . . . . .Oleic Acid . . . . . . . . . .Hexane . . . . . .

The data of Tables 2, 3 and 4 werealso replotted to give curves similar toFig. 15. Agreement with equation (11)was obtained for this data by noting thevalues of PD which balanced the equation. The values are given in Figs. 10,11 and 12. On these graphs the dashedlines are drawn to indicate the curvewhich satisfies equation (11). In Table

...--r - - r- ' , / 'I /: /1

IP:3 ViV ,,2 i / 1 Ii ,1, /l

/ -0.1 03 004 0.& 0.7 0& 09 1.0s.FIG. 15 - TYPICAL GRAPH FOR DETERMINA

J1.0dSwTlON OF Pe2oPETROLEUM TRANSACTIONS, AIME

.0278 4.25 785.109 2.14 36.2

.710 0.84 14.25.139 1.90 32.2

.056 2.99 554.228 1.48 35.6.890 0. 75 18.0.234 1.46 35.1.362 1.18 28.3

.023 4.62 854.090 2.35 36.6.560 0.945 14.7.056 2.99 46.5

7 are listed the integral values and thecalculated values of PD. The values ofo cos e in Table 7 were obtained fromthe calculated values of PD by referenceto the value of PD for ai r displacingwater, in which case the contact anglewas assumed to be zero.

It is of interest to note that the valuesof 0 cos e are in good agreement withthe previously tabulated value of Ii. Thefact that the agreement is close indiocates that in every instance the contactangle was probably zero or approxi.mately zero.

The internal surface area of the porous bodies was found by equation (9).Calculated values of PD and (j cos ewereused rather than the experimental values for this determination. The surfacearea values so found are given inTables 6 and 7.

It is possible also to determine theinternal surface area from the vaporpressure data which was taken. Thefollowing equation is given without deriviation as being applicable to theadsorption of a liquid within a porousmaterial 12

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JOHN C. CALHOUN, JR., MAURICE LEWIS, JR. AND R. C. NEWMAN T.P.2640

Pv =_ 1 _+ C - l ~ ( 1 4 )V(Pvo-Pv) VmC VmC Pvowhere V m= he volume of water adsorbed in a single layer on the entireporous surface.

V = he volume of water adsorbed atthe pressure, Pv

Pvo = he vapor pressure of waterabove a flat surface

C = a constant.Thus, by plotting the obtained vaporpressure data in the form of equation(14) the slope and intercept of theexpected linear plot will suffice to calculate Vm. A knowledge of the spacecovered by a single water molecule suffices then to calculate the surface areacovered. Figs. 16, 17 and 18 show theadsorption data for three cores asgiven in Table I plotted according toequation (14), using weights of wateradsorbed in place of volumes. Assumingthat one water molecule covers an areaof I1.S X 10-" cm', the total surface iscalculated in the three instances as 204,228, and 172 square meters per cubiccentimeter of void space. Values for thesurface area as given in Table 6 forthese cores are 872, 866, and 998 squarecentimeters per cubic centimeter ofvoid space. The latter values are morein agreement with that which would beexpected for uniformly packed spheresthan are the former.A third method of obtaining the internal surface area of these core speciments would be by use of the relationship:

J1.0- IA=-- - - Pc dSw5s-w-5s-A o - (IS)Leverett' has given the free surfaceenergy change per unit of pore volumebetween two different saturations of thesame core as:

JSw,L\F = - PcdSwSw,By choosing the limits of saturation aszero and one hundred per cent the surface energy change is :

L\F = Aas-w - A5s-Abecause the unit free surface energywhen the core is dry is 5S-A and theunit free surface energy when the coreis fully saturated is 5s-w. Combining thetwo expressions for L\F gives equation

July, 1949

,/---V-- , I--- i/V __ i -I-- -1 1f-------- 0 1---- ----' /I---} - - II~ I 1o 20 400

FIG. 16 - VAPOR PRESSURE DATA PLOTTEDAS WATER ADSORPTION FOR DETERMINATIONOF SURFACE AREA. CORE NO.3 .

I 1 /I / :2 I /i VI V.I I /0 yo 10 20 400 50p./W .CP,o-p,1


y !I:!> . / !

I / iV I i i/j , i i I, i,.I / i i// ' I 1 Ii I JIo 20 4000


( IS). This equation cannot be used tosolve for A because the interfacial tension values are not known. It could beused to obtain the difference in the interfacial tensions if A and the value ofthe integral were known. This has beendone for cores Nos. 3, 6 and 8, forwhich the capillary pressure function iscomplete from zero to full saturation. Achoice of the most logical A value wasnecessary for this calculations. Using thevalues obtained from the vapor pressure data yields values of (5S-A - 5s-w)of 214, 204, and 206 dynes per centimeter. Using the lower values of Awould have given negligible differences.

Bartell" has defined the above difference in interfacial tensions as theadhesion tension and for the system


silica-water he gives the value of(5S-A-5s-w) to be 7S.1 dynes per centimeter. It would appear from this thatthe A values as calculated by the adsorption method were too large.

Carman has evaluated the surfacearea of powders by a method basedupon permeability and porosity measurement, a discussion of which methodis given by Brunauer12. Using thismethod and the experimentally determined porosity and permeability valuesspecific surface areas were calculated.These are listed in Table 6. It will benoted that they do not agree numerically with A values calculated from PDbut the two are comparable in magnitude.It is apparent that two different sur

face area values are being measured.So far as flow properties and total voidspace are concerned the roughness ofthe internal surfaces will not be apparent. With the measurement of wettability and adsorption phenomena theintricacies of the surface will be apparent. The adsorption measurementshould give the higher values of Awhere there is roughness or etching ofthe internal surfaces. In the particularmeasurements reported here the porousbodies were prepared by consolidatingsand grains by depositing silica fromcolloidal suspension and by the hydrolysis of tefraethylorthosilicate. This deposition is in the form of fine flakyparticles, a condition which wouldenhance the amount of surface areawhich would be evident from a wettability or adsorption- measurement.SUMMARY AND CONCLUSIONS.

The experiments reported here leadone to the conclusion that for the practical purpose of obtaining connatewater values by capillary pressurecurves, caution should be used in choiceof a displacing liquid. The experimentsdemonstrate the desirability of determining such values at a maximum pressure which is a considerable distanceabove the capillary pressure plateau.From a strictly theoretical standpointthe experiments indicate that there isno "irreducible minimum" and thatzero water saturation is reached atsome finite value of capillary pressure.

The possible usefulness of vaporpressure measurements or adsorption

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data to give characteristics of porousreservoir materials has been outlined.Good agreement was obtained betweencapillary pressure data by displacementan d that calculated from the vaporpressure data.

Following the lead of previous authors, the capillary pressure data havebeen used to characterize the individualporous systems. It has been shown thatthe displacement pressure bears a defi-

. I h J Swte re atlOns Ip to ~ .Surface area values were calculatedfrom the displacement pressures, fromthe adsorption data an d from th e Carma n equation. It is ' apparent that difference quantities are being measuredby the different methods.

ACKNOWLEDGMENTSTh e experimental work which is re

ported in this paper was performed byMr. Newman an d Mr. Lewis while theywere graduate students in PetroleumEngineering at the University of Oklahoma. Further details of their work isembodied in the theses which theysubmitted in partial fulfillment of requirements for the Master's degree.During this period Mr. Newman heldthe Phillips Petroleum Company Fellowship in Petroleum Engineering andMr. Lewis held the Shell Oil CompanyFellowship in Petroleum Engineering.The authors wish to express their deepappreciation for this support, withoutwhich their work would have been seriously curtailed.

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NOMENCLATUREPc = Capillary pressurePn = Displacement pressurePv =, Vapor phase pressure at

equilibriumPvo = Equilibrum vapor pressure

above a fiat liquid surfacePI = Liquid phase pressure atequilibriumpv =Vapor phase densityPI = Liquid phase densityh = Height above a free water

surfaceM = Molecular weightR = Gas constantT = Temperature of porous solid

system.p = Porosity as a fractionK = PermeabilitySw = Water saturation, fractional15 = Interfacial tensionI5s-.< = Interfacial tension, silica airI5s-w = Interfacial tension, silica watero = Contact angleL'>F = Free surface energy change pe r

unit pore volumeA = Internal surface areaF = Lithology factor as defined by

Purcell= Tortuosity constant in Kozeny

equationREFERENCES

1. Leverett, M. c., "Capillary Behavio r in Porous Solids," Trans. AIME,142, (1941), 152-169.2. Hassler, G. L., Brunner, E., andDeahl, T. J. , "The Role of Capillarity in Oil Production," Trans.AIME, 155 an d 160. Reprint (1944-1945),153-172.

3. Bruce, W. A., and Welge, H. J.,"Restored State Method for Determination of Oil in Place and Connate Water," Oil and Gas Journal,156, No. 21 (July, 1947), 223-23S.


4. Amyx, J. W., and Yuster, S. T.,"Capillary Pressure in SecondaryRecovery," Producer's Monthly, 11,No.2 (December, 1946), 10-13.

5. Purcell, W. R., "Capillary Pressures - Their Measurement UsingMercury an d the Calculation ofPermeability Therefrom," Jnl. PetroTech. 1, No.2, (February, 1945).

6. Thornton, O. F., and Marshall,D. L., "Estimating InterstitialWater by the Capillary PressureMethod," Trans. AIME, 170, (1947).69-S0.

7. McCullough, J. J., Albaugh, F. W.,and Jones, P. H., "Determination ofthe Interstitial-Water Content ofOil and Gas Sand by LaboratoryTests of Core Samples," AP I Drilling and Production Practice,(1944), ISO-ISS.

S. Rose, Walter, Theoretical Generalizations Leading to the Evaluationof Relative Permeability," presentedat Petroleum Division, AIME,meeting at Dallas, Texas, October45-6, 1945.

9. Popovich, M. J., "A Study of th eRelationship Between Grain Sizean d Capillary Pressure Curves,"Producer's Monthly, 11, No. 12(October, 1947), 27-43.

10. Freundlich, H., Colloidal and Capillary Chemistry, E. P. Dutton an dCo., New York, 1926.

11. Anderson, J. S., Zeitschrift PhysicalChemistry 88, 191, 1914.12. Brunauer, S., The Adsorption of

Gases and Vapors, Volume I, Physical Adsorption, Princeton, 1945, p.303.

13. Bartell, F. E., an d Benner, F. C.,Adsorption at Solid-Liquid Interfaces. Fundamental Research onOccurrence an d Recovery of Petroleum, 1943, p. 94. * * *

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Documents

Paper Sobre Experimentos de Presion Capilar