A study of the applicability of the independent domain model of hysteresis to capillary pressure hysteresis in sandstone samples

A Study of the Applicability of the Independent Domain Model of Hysteresis to Capillary Pressure Hysteresis in Sandstone Samples FRANCIS S. Y. LAI, IAN F. MACDONALD, 1 FRANCIS A. L. DULLIEN, AND IOANNIS CHATZIS Department o f Chemical Engineering, University o f Waterloo, Waterloo, Ontario N2L 3G1, Canada Received November 20, 1980; accepted May 18, 1981 Imbibition, drainage, ascending scanning curve, and descending scanning curve data for a consolidated sandstone sample were obtained using a mercury porosimeter designed for convenient operation for these experiments. The sample was resin coated to minimize "small sample" surface effects. The data are used to assess the applicability of the independent domain model of hysteresis for consolidated materials. The model applies well for the permanent hysteresis loop data for the Clear Creek sandstone. Considering earlier studies as well, we conclude that the independent domain theory applies well for equilibrium behavior of both consolidated and unconsolidated materials. Lacking any rate-dependent concepts, the model, not surprisingly, does not apply for nonequilibrium unsteady flow data.

lary pressure, Pc = Pnw - Pw, between the nonwetting phase (II) (mercury in our experiments) and the wetting phase (I) (vacuum in our experiments). As the wetting phase is vacuum, the nonwetting phase pressure is numerically equal to the pressure difference, and, for convenience, we refer throughout the report simply to "pressure." At equilibrium the interfaces between the nonwetting and wetting phases are at positions within the porous medium that ensure mean radii of curvature corresponding to a capillary pressure, Pc, which is just equal to the applied pressure. An increase (decrease) in the applied pressure causes the interfaces to move so as to increase (decrease) the nonwetting phase saturation. If at some low pressure a domain is filled with a wetting fluid and, on increasing the pressure a change from state I to state II occurs at P1 (I ~ II), i.e., the domain is filled with the nonwetting phase, this corresponds to the so-called "drainage" process. Imbibition" into the domain occurs as the pressure is decreased to a value P2 (II ~ I).

INTRODUCTION

The idea of "domains" was introduced by Ewing (1) in his classical theory of magnetic hysteresis and considered further by N6el (2, 3). Everett et al. (4-6) and Enderby (7) in quite general treatments not restricted to magnetic hysteresis introduced the independent domain theory of hysteresis. Direct application to capillary condensation hysteresis in porous materials was made by Everett (8). According to this theory, hysteresis is due to the existence of a very large number of independent domains in the medium, at least some of which can exhibit metastability in pressure-saturation relations. Each independent domain is simply a collection of pores or capillaries all of which fill and empty as a group, as the saturation of the medium is increased or decreased, respectively. The saturation change is brought about by changing the pressure difference, i.e., the capil1 Author to whom correspondence should be sent. 362 0021-9797/81/120362-17502.00/0 Copyright© 1981by AcademicPress, Inc. All fightsof reproductionin any formreserved.

Journal of Colloidand Interface Science, Vol. 84, No. 2, December1981

INDEPENDENT DOMAIN MODEL: Pc HYSTERESIS Del

363

--DIRECTION OF PENETRATION~ ~DIRECTION OF WITHDRAWAL DeI (a)

~

De2

Dez

,

" II

~

D~

-~ - - ~ [z

rrl

lrz

(c) (b)

FIG. 1. Examples of independentand interdependentdomains: (a) independentdomainI; (b) independent domainII; (c) interdependentdomains I and II (9).

In general P1 and Pz are not equal, i.e., there is hysteresis, and the process is irreversible in the thermodynamic sense. If P2 is equal to P1, the process is reversible. An illustration of a possible explanation (9) of independent domains in terms of the pore structure, is given in Fig. la and b, where De~ are the diameters of the entry pores or pore necks and D~ are the pore or pore bulge diameters. At a given capillary pressure, corresponding to the minimum mean radius of curvature of the interface in a neck, all of domain I is penetrated by the nonwetting phase. At a higher value of the pressure all of domain II is penetrated at once. On decreasing the pressure, first all of II is emptied at a unique value of the capillary pressure, corresponding to the maximum mean radius of curvature of the interface in a bulge. Then, at another lower value of the pressure all of I is emptied, at once. Pores I and II are two independent domains. By contrast, in the case of interdependent domains (see Fig. lc) on increasing the capillary pressure, first only I1 is penetrated by a nonwetting phase. Domains 111, 12, and II2 are penetrated only at a higher capillary pressure. After the whole system has been filled, on decreasing the capillary pressure first only I12 empties. Domains 12, 111, and I1 empty only at a lower capillary pressure. Evidently, in this case domains I and II are not independent. The domains that would be observed, i.e., I1 and (111 + I2 + I12) on the one hand, and 112 and (12 + II1 + 11) on the other, intersect.

The total volume of mercury which enters the medium while the pressure is increased from zero to Prnax, and which is subsequently withdrawn from the medium when decreasing the pressure back to zero, may be divided (mentally) into a number of small elements. According to the independent domain theory, for a given medium, each element V~j is characterized by a pair of small pressure ranges, i.e., the pressure range of drainage, iPD ---> ~Pv + APD, and the pressure range of imbibition, JPI --> jPz - A P I . By virtue of the basic assumption of the independent domain theory, each volume element corresponds to a narrow range of domains of the medium, i.e., a large number of pores, endowed with definite physical identities and distributed over the medium, all of which fill and empty in the appropriate respective pressure ranges. All elements can be indicated according to their specifying pressure ranges in the (Pz, PD) plane of the rectangular coordinate system of Fig. 2. Drainage corresponds to the movement from left to right of a vertical plane parallel to the Pt axis, while the imbibition process is described by a vertical plane parallel to the P9 axis moving from top to bottom. The independent domain theory of hysteresis assumes that V~, suitably normalized, is a function of Pr and Po, and is characteristic of a particular medium. The calculation of the distribution of domain volumes necessitates experimental determination of either a set of so-called "descending" (or " r e w e t " for air-water systems) scanning loops associated with the Journal of Colloid and Interface Science, Vol, 84, No. 2, December 1981

364

LAI ET AL.

V

/

~

"

/~'----/jP~

///

o

is divided, and the better is the test of the theory. Based on the above theory, Poulovassilis (10), using glass beads known by the commercial name "Ballotini" as the medium, tried to predict " r e d r y " scanning loops from " r e w e t " scanning loops. Good agreement between theoretical predictions and experimental results was obtained. Topp and Miller (11) further tested the applicability of the theory by using two glass bead samples having different pore space geometries. The first sample, called "monodispersed," was composed of relatively uniform glass spheres about 180 tzm in diameter. The second sample, called "aggregated," was composed of much smaller glass beads, lightly sintered into aggregates roughly 200 tzm in diameter. They found that the theory fit the aggregated sample data somewhat better than the monodispersed sample data, and concluded that the theory is least satisfactory for media with narrow pore size distribution. Talsma (12) later applied the theory to two sands, one fine grained and the other coarse and angular grained. Results were moderately

/ B

iPo iPo'~ P~o,

.-

Po

FI~. 2. Perspective view of the distribution of domain volumesplotted above the PD, P~ plane.

secondary drainage curve or a set of so-called "ascending" (or " r e d r y " for air-water systems) scanning loops associated with the imbibition curve. If the independent domain theory is correct, both sets of data result in the same distribution of domain volumes. The greater the number of scanning loops in the set, the greater the number of volume elements, V~, into which the total volume

A - AVO METER M - MERCURY MANOMETER

- SAMPLE CELL RESIN CASING -SAMPLE

P~- 0 - I 0 0 p$1g P2" 0 - 2 0 0 psig

V 13

F - FLASK FOR TRAPPING Hg

1 ~ TO ATM

F -TO VACUUM

I

V7

P2

V 16

ATMOSPHERE V6 TO VACUUM SS TUBING 2 mm I.D. $LASS TUBE

VI FROM -'~ N2 CYLINDER

,VII ' VIO H9 RESERVOIR TO ATM

FIG.

'-=-

3. M e r c u r y p o r o s i m e t e r for study of capillary p r e s s u r e hysteresis loops.

Journal o f Colloid and Interface Science,

Vol. 84, No. 2, December 1981

INDEPENDENT

DOMAIN

good for the greater part of the hysteresis regions. Significant deviations were found in the high moisture content range in the finegrained sand. To our knowledge, the theory has not been tested for consolidated porous materials which are important in areas such as petroleum production. In the present research (13), a consolidated outcrop sandstone sample (Clear Creek) has been investigated. A modified mercury porosimeter has been used to obtain the hysteresis curves (14). EXPERIMENTAL

Mercury Porosimetry Experiments using mercury as the nonwetting fluid to penetrate an evacuated porous sample have been used extensively in the past to determine pore size distributions. The commercially obtainable porosimeters are designed primarily for the determination of the primary drainage curve (i.e., to measure mercury penetration into a sample initially containing no mercury). They usually consist of two separate systems: a vacuum chamber and a pressure chamber. During the experiment the test sample must be transferred from one chamber to the other to cover the whole range of data from subatmospheric to the maximum pressure. In the present work a mercury porosimeter of very simple design has been used in which it is not necessary to transfer the sample and which is very well suited for the study of the hysteresis loop, including the scanning curves.

The Modified Porosimeter This equipment was designed by Chatzis (14) with the aim of getting hysteresis scanning curves for porous samples. The apparatus, shown in Fig. 3, consists of a sample cell S, mercury manometer M, which can read out pressures in the range of 0 to 80 cm Hg, a glass tube G (i.d. = 2 mm), and a pressure gauge P (0-200 psi).

MODEL:

Pc H Y S T E R E S I S

365

The sandstone samples used were 3.8 cm in diameter and 1 cm in height. A liquid polyester resin was poured over the whole sample except for a small circular opening of about 5 mm diameter at one end. When the resin solidified, about an hour after adding the catalyst, it acted as an impermeable wall to mercury.

Experimental Procedure The experiment started with the sample evacuated to a pressure of about 8 × 10-3 cm Hg absolute (i.e., completely filled with wetting fluid). The primary drainage curve was obtained by injecting mercury, using nitrogen as the pressure source, and measuring the mercury content of the sample at several values of pressure as the pressure was increased slowly by small increments up to the maximum pressure used. In this and in all subsequent runs, the pressure was held constant for sufficient time at each reading value to ensure that equilibrium was reached. It usually took much longer for the interfaces to reach equilibrium at low pressures, especially at breakthrough; sometimes it took more than 30 rain between two readings. The pressure could be read with an accuracy of _+0.5 cm Hg when using the manometer, and of __-5 cm Hg when using the pressure gauge. The amount of mercury in the sample could be read with an accuracy corresponding to a wetting phase saturation (Sw) of _+0.2% (Sw is the percentage of the total pore volume occupied by the wetting phase). Only at this point are we ready to obtain data to test the independent domain theory. Since the theory only applies, even in principle, to reproducible permanent hysteresis loops, only the data bounded by the imbibition curve and the secondary drainage curve can be used to test the theory. (a) Ascending scanning curves. Starting at the maximum pressure, the pressure was decreased and data were obtained along the imbibition curve until the pressure was lowered to the value chosen as starting point Journal of Colloid and Interface Science, Vol, 84, No. 2, December 198l

366

LAI ET AL.

of the first scanning curve. The pressure was then increased and pressure-saturation data were obtained at intervals until the pressure was back to the maximum value. The pressure was then decreased again along the imbibition curve until the pressure was reduced to the (lower) starting point of the second scanning curve. The pressure was increased and pressure-saturation data were obtained at intervals until the pressure was back to the maximum value. The process was repeated until sufficient ascending scanning curves were obtained. The pressure was then lowered along the imbibition curve to zero pressure at a saturation corresponding to the residual nonwetting phase saturation. (b) Descending scanning curves. Starting at zero pressure [(P, Sw) = (0, 100-Snwr)], the pressure was slowly increased and data were taken along the secondary drainage curve until the pressure was raised to the value chosen as the starting point of the first scanning curve. (Snwr is the residual nonwetting phase saturation, i.e., that nonwetting phase fluid which cannot be removed by imbibition even when the pressure difference is lowered to zero.) The pressure was then decreased and pressure-saturation data were obtained at intervals until the pressure was back to zero. The pressure was then increased again along the secondary drainage curve until the pressure was raised to the (higher) starting point of the second scanning curve. The pressure was decreased and pressure-saturation data were obtained at intervals until the pressure was back to zero. T h e process was repeated until sufficient descending scanning curves were obtained. To facilitate the comparison between the experimental and the theoretically predicted scanning curves, scanning curves were started at approximately the same set of pressures in both the ascending and descending scanning processes. The Clear Creek sandstone sample had a porosity of 0.2085 and a permeability of 78 md. Journal of CoUoid and Interface Science, Vol. 84, No. 2, December 1981

The experiment was run twice; the experimental data obtained were found to be reproducible. The experimental scanning curves are presented in graphical form in Figs. 4 and 5. The volume distribution V~, was calculated from both the ascending and the descending scanning curves by the standard procedure used in the literature (10). To minimize the effect of experimental scatter, smooth curves were drawn through the data points of each scanning run. The Vii values were determined using these smoothed curves rather than the individual data points. The results of these calculations are shown in Tables I and II. Theoretical ascending scanning curves were calculated based on the experimental descending curves, and vice versa. The calculated scanning curves are compared with the experimental scanning curves in Figs. 6 and 7. Because it was not possible to start the experimental ascending and descending scanning curves at exactly the same sets of pressures, the intermediate pressures and saturations of the volume distribution tables are not exactly the same. Consequently, the predicted scanning curves correspond to slightly different starting conditions than the experimental curves. To eliminate this as a factor in judging the quality of the predictions, we interpolate between the actual experimental curves to obtain experimental scanning curves with the same starting conditions as those of the predicted scanning curves. In most cases, the shift from the nearest measured curve is very small. It is these interpolated curves which are compared with predictions in Figs. 6 and 7. The interpolation was done as follows. The scanning curves were replotted in dimensionless form as ( P - Pimb )/( PDr elmb) versus Sw, where P*mb and Por are the pressures from the imbibition and secondary drainage curves, respectively, at the saturation Sw corresponding to the pressure P on the scanning curve. The starting saturation S~, of a desired scanning curve was located -

-

INDEPENDENT DOMAIN MODEL:Pc HYSTERESIS 1000

I

i

I

367

I CLEAR

CREEK

I0o

izn "r

E (.~ S.

--

,~- SECONDARY DRAINAGE CURVE FROM DESCENDING SCANNING CURVES o - IMBIBITION CURVE FROM ASCENDING SCANNING CURVES A - IMBIBITION CURVE FROM DESCENDING SCANNING CURVES

I 0

•

SECONDARY DRAINAGE CURVE FROM ASCENDING SCANNING CURVES

×

SCANNING CURVE DATA POINT

a

PRIMARY DRAINAGE CURVE

•

IMBIBITION AND SECONDARY DRAINAGE CURVE FROM REPEAT RUN i i L_ I0 20 30 40

5O

60

I

90

I00

S w (%)

FIG.4. Primaryand secondarydrainagedata, primaryimbibitiondata, and ascendingscanningcurve data for Clear Creek sandstone.

on the plot and the difference between it and the starting saturation of the closest experimental curve ( S ~ , - Sw.) was determined and expressed as a fraction of the difference between the starting saturations of the two adjacent experimental curves (S~2,- Sw,). An interpolated curve was then drawn so that, for all values of (P -PImb)/(PDr- PImb), the quantity (Sw - S~I)/(S~ ~ - S~,) was constant and equal to ( S ~ , - S w ~ ) / ( S ~ - Sw,). This interpolated experimental curve was then replotted in dimensional form as P vs S~ on Figs. 6 and 7. DISCUSSION

Experimental Curves In principle, the initial slope of the primary drainage saturation curve should be infinite, with no nonwetting fluid (mercury)

penetrating the sample until a pressure equal to the capillary pressure of the largest pore necks is exceeded. In practice, the slope is finite in experimental tests. According to Wardlaw and Taylor (15), the finite slope is due to the invasion of the surface spaces, including entry via surface enlarged throats. This is a laboratory artifact resulting from the large surface-to-volume ratio of the small samples; in the field, the surface pores are a negligible portion of the total pore volume. In this work, the samples are penetrated by mercury from a small open area approximating a point to minimize the "small sample" effect (see Fig. 8a). Figure 8b shows a representative uncovered sample as commonly used in porosimetry studies. The effect of exposed area-to-volume ratio was investigated using two Berea sandstone samples of similar size and shape. Sample 1 was uncovered and Sample II was typical Journal of Colloid and Interface Science, Vol. 84, No. 2, December 1981

368

LAI ET AL. I000

I

I

I

1

CLEAR

CREEK

I00!

.-it. i= U

IO-

I

o

[

I

IO

20

I 30

I 40

50

s w I%!

FIG. 5. Secondarydrainage data, primaryimbibitiondata, and descendingscanningcurve data for Clear Creek sandstone. See Fig. 4 for symboldefinition. of the samples used in this work with 83% of its surface covered with resin. The initial slope of the pressure-saturation curve was much steeper for sample II than for sample I (Fig. 9). Excess mercury saturation in the uncovered sample persists until well after breakthrough indicating that the accessibility of pores remains higher in the boundary region than in the rest of the sample even after breakthrough. Eventually, however, the primary drainage curves of the covered and uncovered samples do merge into the same curve. Breakthrough occurs near the point where the primary drainage curve abruptly changes slope. We somewhat arbitrarily define the breakthrough pressure to be given by the Journal of Colloid and Interface Science, VoL 84, N o . 2, D e c e m b e r 1981

intercept of the tangent to the initial curve and the tangent to the nearly horizontal intermediate region curve. Breakthrough saturation is obtained from the intercept of the breakthrough pressure and the actual pressure-saturation curve. Breakthrough pressure for our material is about 78 cm Hg and is indicated by an arrow on Fig. 4. Chatzis (14) subsequently has modified the system by inserting a steel wire through the resin on the face opposite the hole (e.g., Fig. 8a) and using a current detector to measure breakthough. Although this is a less arbitrary, more accurate, preferable method of measuring breakthrough, a comparison indicates that our procedure gives quite accurate results for breakthrough pressure.

369

INDEPENDENT DOMAIN MODEL: Pc HYSTERESIS I000 -

I

I

I

I

-

A "T"

E

I00

u

10 0

)

I

I

I

10

20

50

40

50

S w (0/0)

FIG. 6. Predicted (from domain distribution diagram 1) and experimental ascending scanning curves for Clear Creek sandstone. ©, [] Experimental curve. Q, • Predicted points. - - -Predicted curve (fit to predicted points).

However, breakthrough Sw values from the procedure used are unreliable and usually high. Breakthrough pressure in secondary drain-

I000

I

age (about 66 cm Hg) is measured in a similar manner and also is indicated by an arrow on Fig. 4. From the experimental curves we can see that the breakthrough pressure in

I

I CLEAR

I

I

CREEK

-rE

lO0

eL.

I0 0

•

10

2O

30 Sw

40

50

(°/o)

Fro. 7. Predicted (from domain distribution diagram 2) and experimental descending scanning curves for Clear Creek sandstone. See Fig. 6 caption for symbol definitions. Journal of Colloid and Interface Science, VoL 84, No. 2, December 1981

370

LAI ET AL.

the primary drainage process is a little higher than that in the secondary drainage. A reasonable explanation is that the residual mercury following the primary imbibition process facilitates subsequent breakthroughs. Figure 10 shows schematically how this may occur in a small regular network. Breakthrough occurs at P2 in the primary drainage and at P3 (which is lower than P2) in the secondary drainage. This has also been shown in a much larger square network (50 × 50), using computer simulation, by Chatzis (14). His results are consistent with the above. In Fig. 10, entry pores or necks are represented by 1, 2, 3 and large pores or bulges are represented by 4, 5, 6 where the sizes D~ are in the order D 6 > D s > D 4 > D ~ > D2 > D1. Pi (where P6 < Ps < P, < P3 < Pz < P1) denote the applied pressures at which the capillary pressures corresponding to the Di are just equal to the applied pressure difference; i.e., they denote the pressures at which flow of nonwetting fluid in

~

.ff,'//J../J/7. 7/Jl,~--- I MPE R MEABL E I~'-';/~/'~-~" ~ /!-.~. (- ~ )' '.1' .~: :. ~ RESINWALL

t

MERCURY (a)

MERCURY (b) FIG. 8. Sandstone samples: (a) Resin-coated samples used in this study to minimize surface effects. (b) Uncoated sample typically used in porosimetry studies. Journal of Colloid and Interface Science,

Vol.84, No. 2, December1981

I000. 800~ 600~ 400 IL

~oo~

m

o

o

°

BEREA BE-I SAMPLEI (UNCOVERED) WT.= 2.o0671gm

~w~,o(o~OV~R~D,

IOOL~ o.. 4 0 f ~

i I L I / "f I 0 20 40 60 80 I00 120 VOLUME PERCENTSATURATION(WETTING PHASE) FIG. 9. Comparison of the effect of area exposed for penetration-to-volume ratio on capillary pressure-saturation curves for two Berea sandstone samples.

through necks 1, 2, 3 or out through bulges 4, 5, 6 can begin. Initially, the network is saturated with an incompressible wetting phase. Then the pressure is increased in a stepwise manner. No nonwetting phase enters the medium for pressures less than P3. The pores penetrated by the nonwetting phase at pressure Pz are denoted in Fig, 10a. The additional pores penetrated at pressure P2 also are denoted separately. Breakthrough occurs at P2 for the primary drainage process. Note that neck pore 2 in the lower middle of the diagram does not fill with nonwetting fluid because the bulges 4 and 5 at either end are already filled with nonwetting fluid and the wetting fluid in the neck pore is no longer continuously connected to the exit face and, therefore, has been trapped. Next, the applied pressure is decreased in a stepwise manner. When the pressure is lowered to P4, all of the nonwetting fluid outside the dashed lines on Fig. 10a has emptied from the medium. Therefore, the nonwet-

INDEPENDENT DOMAIN MODEL: Pc HYSTERESIS

WETTING PHASE

(

"3

¢

(b)

I

I NON-WETTING PHASE

(o)

(c)

FIG. 10. (a) Network showing that breakthrough occurs at a lower pressure in the secondary drainage process than in the primary drainage process. ~ denotes pores penetrated by nonwetting phase at P3 in primary drainage. ~ denotes pores subsequently penetrated at P2. (b) Flow path at breakthrough in primary drainage, P = P v (c) Flow path at breakthrough in secondary drainage, P = P~.

ting phase within the dashed lines has no path to the face of the medium and is trapped, i.e., it does not flow out when the pressure is lowered to P5 or, indeed, to zero. The pressure is again increased in a stepwise fashion for the secondary drainage process. At pressure P3, the nonwetting phase again penetrates pores of size D3 and D4, but now links up with the trapped nonwetting phase and breakthrough occurs through the top middle neck pore 3. The paths of flowing nonwetting phase at breakthrough for primary and secondary drainage are shown in Figs. 10b and c, respectively. For the Clear Creek sample, the secondary drainage and primary imbibition curves merge for pressures less than about 14 cm Hg. The bounds of the permanent hysteresis loop are (SwMi,, PMax) ---- (0%, 635 cm Hg) and (Swmax • 1 - STlwr, PMin) = (49.5%, 14 cm Fig). Nine ascending and nine descending

371

scanning curves were obtained giving a fairly fine mapping of the hysteresis region. At low saturations (and high pressures), the ascending scanning curves tended to merge with the primary imbibition curve and at high saturations (and low pressures), the descending scanning curves tended to merge with the secondary drainage curve. These approximately reversible behaviors suggest that the initial and final penetration by the nonwetting fluid are in regions of monotonically decreasing pore size primarily. At the low pressures, nonwetting fluid would be proceeding into the converging sections of the largest entry necks. The pressure at Which hysteresis first becomes evident in the descending scanning curves presumably would give an approximate measure of the throat size of the largest entry pore neck. This size should be larger than the secondary drainage breakthrough diameter which is the throat size of the smallest neck, not initially filled with residual nonwetting fluid, in the first continuous pathway of nonwetting fluid through the sample. At the highest pressures, the nonwetting fluid is probably penetrating the small nooks and crannies of the irregularly shaped pores. An alternate explanation would be that the smallest pores are uniform in diameter with necks and bulges essentially identical in size. Only eight ascending scanning curves are shown in Fig. 4 because the one commencing at 485 cm Hg coincides with the primary imbibition curve. Only seven descending scanning curves are shown in Fig. 5 because the two commencing at 45 cm Hg and 25 cm Hg are coincident with the secondary drainage curve. Therefore, the largest entry neck size is in the range 18.4-24.5/xm (corresponding to a scanning curve starting pressure in the range 60-45 cm Hg) and, as expected, is greater than the secondary drainage breakthrough diameter of about 17/xm. Throughout most of the hysteresis range, the scanning curves are similar in that they differ significantly in slope from the bounding curve at which they start and gradually Journal of Colloid and Interface Science, Vol. 84, No. 2, December 1981

372

LAI ET AL.

approach the bounding curve at which they terminate.

make up about 1/2 of the pore volume is clearly a major drawback. The representation of the entrapped nonwetting fluid in the distribution diagrams has been discussed by Poulovassilis (16). As shown in the distribution diagrams of the Clear Creek sample, a few of the independent domains (the so-called "negative domains") had negative contributions to volume which is physically unrealistic. If the data and curves fit to the data are sufficiently accurate that the negative domains are "real" for the process followed in determining the domain distribution functions given in Tables I and II, then the independent domain theory does not hold, at least not rigorously. The other possibility is that the negative domains result from the small experimental scatter combined with slight

Domain Theory The independent domain theory predicts that all pores will be filled with nonwetting fluid when the pressure reaches a sufficiently high value and that they will be emptied when the pressure is reduced to zero. That is to say, the independent domain theory of hysteresis does not consider those pore spaces which contain the so-called irreducible and residual saturations. From the experimental data on the sandstone sample, it can be seen that the volumes of the pores that contain the entrapped nonwetting phase contribute about 50% of the total sample volume. The fact that the domain theory is not applicable to pores that

TABLE I

Domain Distribution Diagram 1. V~j (vol%) for Clear Creek Sandstone from Experimental Descending Scanning Curves

49.5

0.I 0,6 0.8 9.7 16.2 6.2 6.4 3.5 3.0 3.0 ÷z~Sw(%) 49.4 48.8 48.0 38.3 22.1 15.9 9.5 6.0 3.0 O"-Sw(%) ÷

+

635 0.4

0.4 0.4

0.65

0.35

1.0 1.4

0.75

0.4

0.45

0.75

0.75

0.8

1.6 3.0

1.5

3,8 6.8

g

0.7

.8

2.55

.65

-0.2

1.0

0.65

0.2

0,35

3.95 10.75

2.1

E

1.2

8.05 8.8

1,5

5.8 -0.45

1.45

1

0

0

0

0

0

9.3 8.1

0.15! 2.95

4.6~

0.75

0

8.5 36.6

0.6

0.45

4.55

3.25! 2.65

0

0

0

0

0

0.2

0.7

o.4

p

0

0

0

~.8.1

25 0.I ~, 74 14 P (cm Hg) -+

I I .5

25

45

60

86

o 138

187

Drainage Journal of Colloid and Interface Science, Vol. 84, No. 2, December 1981

285

370

485

1.4 ¢9.5 635

373

INDEPENDENT DOMAIN MODEL: Pe HYSTERESIS TABLE II Domain Distribution Diagram 2. Vu (vol%) for Clear Creek Sandstone from Experimental Ascending Scanning Curves 0.I 0.7 1.6 9.4 14.6 6.6 7.0 4.0 2.7 2.8 ÷ASw(%) 49.5 49.4 48.7 47.1 37.7 23.1 16.5 9.5 5.5 2.8 O~,~w(%) + 635

u

0.3 485

.3 0.3

0.45

0.45

0.5

0.75

.9 1.2

0.55

1.8 3,0

0.9

1.35

0.8

].26

4.3 7.3

z

0.7

1,7

0.9

0,75

2.i

2.4

I.I

0.2

0.05

4.1 11.4

1.3

0

7.] 8.5

0.9

2.7

1.4

1.3

0.!

0

0

6.4 4.9

2.3

2.1

4.4

1.7

0.7

0

0

0

46

]1.2 36.1

I.I

-0.I

4.0

6.2

0.7

0

0

0

0

25

11.9 8.0

-0.4

0.] I I ÷ 14 .... P (cm Hg

-0.6

2.4

0

0

0

0

0

0

1,5 9.5

25

46

67

87

131

180

280

385

485

635

Drainage

uncertainties in curve fitting, especially where the slopes are steep. The presence of these few negative domains has little effect on the predicted scanning curves. It has been pointed out, by Dullien (9), that owing to the irregularity of pore structure it is unrealistic to expect to be able to find in real porous media definite volume elements each of which always is filled in a fixed pressure range and always is emptied in another fixed pressure range. It is more likely that, in general, to any two chosen pressure ranges of filling and emptying, respectively, there will correspond two different volumes. In other words, the domains are in general not independent. For this reason, we expect that the domain distributions obtained from the ascending and descending scanning curves will be different.

What we wish to determine is whether the differences are significant. Therefore, the interesting comparison to make is between the experimental ascending scanning curves and those predicted from the domain distribution obtained from the descending scanning curves and vice versa. If the curves are nearly the same, then the independent domain theory is approximately correct and pragmatically useful. Because it was not possible to start the ascending and descending scanning curves at exactly the same sets of pressures, some of the predicted scanning curves start at different positions on the boundary curves than do the closest experimental curves. However, making the plausible assumption that closely spaced scanning curves should be similar, we obtained interpolated experiJournal of CoUoid and Interface Science, Vol. 84, No. 2, December 1981

374

LAI ET AL.

mental scanning curves (as outlined earlier) which have the same starting points on the boundary curves as do the predicted curves. The predictions of the ascending scanning curves for the Clear Creek sample (Fig. 6) are remarkably good. For the most part, the shapes of the curves are correct. There is perhaps a slight trend from predicting too great a pressure change with saturation change at low pressures to predicting too small a pressure change with saturation change at high pressures. The predictions of the descending scanning curves for the Clear Creek sample (Fig. 7) are not as good as those for the ascending scanning curves, but are still quite good. The predicted change in pressure with saturation is too rapid at high pressures. The predicted scanning curves starting at 65 and 45 cm Hg actually lie outside the bounding secondary drainage curve, but otherwise the predicted curves are qualitatively correct in shape. The predictions can be improved by the following simple procedure. Since we know that the scanning curves must lie inside the boundary curves, whenever a predicted scanning curve lies partially outside the hysteresis loop, we simply replace it with one which is coincident with it within the loop, but which merges with the boundary curve at the point of intersection of the actual predicted curve. A quantitative comparison between the observed and predicted scanning curves is shown in Table III. With the exception of one predicted descending scanning curve lying outside and assumed to lie on the bounding cdrve, the correlation coefficients are very good. Furthermore, with the same exception, the slopes M of the relation between the observed (P0) and predicted (Pp) pressure values Po = M P P + C

[1]

are close to unity and the magnitudes of the intercepts C are small in comparison to the observed pressure values P0. In other words, the empirically established relation Journal of CoUoid and Interface Science, Vol. 84, No. 2, December 1981

T A B L E III Comparison between the Observed and Predicted Scanning Curves ~ Ascending scanningcurves Pressure range (cm Hg)

Correlation coefficient y

Slope M

Intercept C

25-78 44-- 145 60--200 86-340 135-415 180-440 285-560

0.982 0.9% 0.999 0.999 0.999 0.991 0.993

1.04 1.04 1.01 1.01 0.99 1.11 1.03

--6.58 --8.54 -- 1.30 1.17 7.55 7.36 10.08

Descending scanningcarves 45-- 14b 67-22 b 87-20 132-34 180-62 285-85 385-123 485-- 126

1.00 0.868 0.999 0.998 0.997 0.988 0.993 0.993

1.00 0.80 0.98 0.97 0.98 1.06 1.04 1.02

0.0 -- 1.07 0.704 3.54 4.48 3.67 4.68 13.49

a Linear regression is used to correlate the observed and predicted scanning data. M and C are the slope and intercept of the equation below. P0 = M P p + C , where P0 is the pressure observed, Pe is the pressure predicted. b The comparison is between the experimental curve and the secondary drainage curve, since all points predicted to lie outside the boundary curves are assumed to lie o n the nearest boundary curve.

[1] is significant and is a reasonable approximation of the relation P0 = PP.

[2]

Therefore, pragmatically, the independent domain theory applies quite well for the Clear Creek sandstone sample. One shortcoming, in principle, of the theory is that it allows only two possible states of a domain: pores are completely filled either with nonwetting or with wetting fluid. Pores are tacitly assumed to have the shape of ink bottles (Fig. l la) and all be of the same size; a pore like this will drain at PDn and imbibe at PDb- An alternate shape of

INDEPENDENT DOMAIN MODEL:Pc HYSTERESIS

375

the independent domain theory would not be applicable in many cases. We compared our results with those of (a) several other investigators for air-water beDn havior in soils and bead packs. We ranked the materials by our subjective judgment of how well the predicted scanning curves from (b) the independent domain theory fit the experiFro. 11. Pore shapes: (a) ink bottlepores, (b) variable mental scanning curves. In Table IV, the diameter pores. materials are listed in order from best agreement to worst agreement. When the qualitathe pore is shown in Fig. 1lb; this pore drains tive phrase describing the fit is the same, at Port, but imbibes wetting fluid gradually differences are small and the order of rankas the pressure is decreased before it fills ing has little significance. Other data included completely at P~b. A medium composed of in Table IV are the range of pore sizes reprepores like that in Fig. l lb effectively will sented by the pressure range of the hysterehave a broader distribution of bulge sizes sis loop; the ratio of the largest-to-smallest that a medium composed of pores like that pore size; the percentage of the loop presin Fig. l la. This would result in a steeper sure range corresponding to the middle 80% primary imbibition curve covering a wider of the hysteresis loop pore volume for both range of capillary pressures for the Fig. 1lb secondary drainage and primary imbibition, medium. This is not a unique explanation D(%), and I(%), (See Fig. 12); the ratio of of such a variation in primary imbibition these two percentages, D/l; and the ratio of curves, since a medium with a wide range the pressures of the endpoints of the middle of diameters o f ink bottle pores would also 80% of the pore volume for both secondary show similar behaviour in comparison to a drainage and primary imbibition. For qualitamedium with a narrow range of diameters tive purposes, we assume that the pressure of ink bottle pores, but it is a plausible one. range and ratio give a measure of the pore Another mechanism which will give a size range and ratio for both drainage and partly filled pore, and, therefore, a steeper imbibition, even though the assumption primary imbibition curve, is "filming" along usually is reliable and quantitative only for the pore wall by the wetting fluid. This phe- drainage. nomenon has been observed during the imD and I give a rough measure of the unibibition process by Chatzis (14) in his physi- formity of the pore neck and pore bulge size cal micromodels of pore networks, in which distributions. The smaller the value, the less the fluids used were oil and water. uniform the distribution, the flatter the presMaterials for which both boundary curves sure-saturation curve, and the closer the are similar and nearly flat in the transition material is to having a single pore size; the range should have pore structures which closer to 80% the more uniformly or evenly conform more closely with the assumptions the pore sizes distribute over the range of of the independent domain model than do pore sizes. The ratio D/I gives a measure of the pore structures of materials for which the similarity of the neck and bulge size disone or each boundary curve has a significant tributions. The closer to unity, the more slope and a wide pressure range. Since most similar in slope and range are the drainage real materials have a fairly broad distribu- and imbibition boundary curves. tion of irregularly shaped pores and have Although not shown, we also looked at boundary curves which are not flat and, in the same quantities for the middle 70% and general, not parallel, we would expect that middle 90%, and narrowest pressure range Journal of Colloid and Interface Science, Vol, 84, No. 2, December 198l

376

LAI ET AL. T A B L E IV G o o d n e s s of Fit and Various M e a s u r e s of the Pore Size Distribution for Several Porous Materials Mid 80% pore volume

Static equilibrium experiments P70-1 P62 Clear Creek P70-2 TA-Ad TA-Mo U n s t e a d y flow experiments TM-A T-C1L T-SiL TM-M T-SaL

(17) (10) (13) (17) (12) (12)

Very good Very good Good ~ very good Good Good Good

40--~ 300 60 --~ ~ 1.7 --~ 80 40 --* 300 15 --~ 180 45 --* ~

7.2 o0 46 7.6 12.5 o0

56 58 52 47 54 67

58 51 30 59 54 67

0.97 1.14 1.73 0.80 1.00 1.00

2.1 2.8 5.4 1.8 2.1 5.0

3.2 5.9 6.5 3.0 2.8 18

(11) (18) (18) (11) (19)

Fair ~ poor Fair --~ poor Poor Poor ~ very poor Poor ~ very poor

25 4.5 3.7 25 6

5.9 oo ~ 7.0 41

36 49 55 21 33

44 35 45 48 36

0.82 1.40 1.22 0.44 0.92

1.5 5.0 4.7 1.2 2.1

2.1 12 19 2.2 9.7

Journal of Colloid and Interface Science, Vol. 84, No. 2, December 1981

I (%)

D/I

Goodness of fit

80% of the hysteresis loop pore volume. The numbers change but the trends are essentially similar to those shown for the middle 80%. Our initial look at the results suggested that there was possibly a correlation between goodness of fit and the uniformity of the pore neck size distribution, i.e., with D, but this did not hold up. Further examination revealed that the studies could be divided into two groups. The experimental data for all materials down to and including TA-Mo are static, equilibrium data. The data of Topp et al. (11, 18, 19) for all the remaining materials are nonequilibrium data from unsteady flow experiments. The conclusion of Vachaud and Thony (20) that the independent domain theory does not apply also is based on nonequilibrium data. Furthermore, Topp et al. (21) found that data from unsteady flow experiments differed markedly from the data of static equilibrium experiments and the data of steady flow experiments which were in agreement with each other within experimental error. From our observation (13) that the time to reach equilibrium was appreciably longer for the drainage

D (%)

1 pressure ratio

Ref.

--* 150 ~ ~ ~ oo --~ 180 --> 250

Size ratio

D pressure ratio

Size range (~m)

results than for the imbibition results, we would expect that, in an unsteady flow experiment where the flow velocity is cycled Pmox PI

I

P3 P

Pz P4

Pmin

0

0,1Swmax

SWmat~ 0,9 clq x ~ Swmax

Sw PI - P2 D = Pmex_PmiXlO0

P3 - P4 I = ~ X p m orain x_

D PRESSURE RATIO = PI p'-'~

Z PRESSURE RATIO =

I00

P4

FIG. 12. Sketch to define the m e a s u r e s of uniformity of the pore size distribution.

INDEPENDENT

D O M A I N M O D E L : Pc H Y S T E R E S I S

at a constant rate, the drainage data will be further from equilibrium than will be the imbibition data. The failure of the independent domain theory in fitting these nonequilibrium data is not surprising; indeed, it would be surprising if it did apply. The data depend on the rate of change of the flow as well as on the pore structure. The independent domain theory has only pore structure parameters; it has no rate-dependent parameters. An interdependent domain theory which involves pore structure parameters only also would be expected to fail. In other words, we can not test the proposition that the domains are independent with the unsteady flow data, because we can not distinguish between assumptions concerning rate dependency and assumptions concerning independence as the source of poor performance. Turning our attention to the results from static equilibrium experiments, we see that the independent domain theory does quite well for all materials. Pragmatically, we can say that the independent domain theory is applicable for each of these materials. From Table IV, we can see that for these materials, the size range and uniformity of the pore structure varies substantially from one material to another and that there is no apparent trend in any of the parameters listed with goodness of fit. The parameter values for the unsteady flow experiments also vary with no apparent trend and the values overlap with those for the static equilibrium experiments. The values of D and I tend to be lower for the unsteady flow experiments, but this simply reflects the nonequilibrium nature of these results (21). The Clear Creek results indicate that the theory applies as well for consolidated media as for unconsolidated media. Clearly, the independent domain theory gives fairly accurate and acceptable predictions for the equilibrium behavior of most materials, contrary to our expectations based on t h e incompatibility of the actual pore structure and that assumed in develop-

377

ing the independent domain theory. The explanation possibly lies in the concept of conjugate pores proposed by Poulovassilis (22) and expanded on by Poulovassilis and Childs (23). Essentially, the idea is that for every pore space domain which fills with nonwetting fluid over a particular pressure interval and only partially empties over a second pressure interval resulting in a change in wetting phase saturation of -2xSw, there is another pore space domain which empties of nonwetting fluid over the second pressure interval above and only partially fills over the initial pressure interval resulting in a change in wetting phase saturation of + zXSw. Consequently, there would be no net change i n saturation and such an interdependent domain system would be indistinguishable from an independent domain system. Two interacting domains were considered merely for simplicity. In practise, there may be many interacting domains with a combined effect equivalent to an independent domain system. Another possible and we believe, more likely, explanation is suggested by some preliminary results by Chatzis (14) on the distribution of residual nonwetting phase following primary imbibition. Tentatively, we can speculate that much of the wetting phase which can be displaced in secondary drainage is in rather short pore volume segments (perhaps 1 or 2 pore bulges long) between residual nonwetting ganglia. The independent domain theory should apply better for such pore volume segments than for long interconnected pore volumes containing many pores of wide size distribution. This idea is being investigated further.

CONCLUSIONS

The independent domain theory applies well for Clear Creek sandstone. From the evidence examined here, we conclude that the model applies well for both consolidated and unconsolidated media provided that equilibrium data are used. On the other hand, Journal of Colloid and Interface Science, VoL 84, No. 2, December 1981

378

INDEPENDENT DOMAIN MODEL: Pe HYSTERESIS

the independent domain theory does not apply for nonequilibrium data which is not surprising as the model contains no ratedependent concepts. It should be noted that, even for equilibrium behavior, the independent domain theory can say nothing about the pores occupied by irreducible wetting phase or residual nonwetting phase, nor about the limits of the permanent hysteresis range. REFERENCES 1. Ewing, A., "Magnetic Induction in Iron and other Metals," Electrician Printing Company, London, 1891. 2. N6el, L., Th6orie des lois d'aimantation de Lord Rayleigh, 1, Cah. Phys. 12, 1 (1942). 3. N6el, L., Th6orie des lois d'aimantation de Lord Rayleigh, 2, Cah. Phys. 13, 18 (1943). 4. Everett, D. H., and Whitton, W. I., Trans. Faraday Soc. 48, 749 (1952). 5. Everett, D. H., and Smith, F. W., Trans. Faraday Soc. 50, 187 (1954). 6. Everett, D. H., Trans. Faraday Soc. 50, 1077 (1954); 51, 1551 (1955). 7. Enderby, J. A., Trans. Faraday Soc~ 51, 835 (1955); 52, 106 (1956).

Journal of CoUoidand Interface Science. Vol.84, No. 2, December1981

8. Everett, D. H., "The Solid-Gas Interface" (E. E. Flood, Ed.), Chap. 36, v. II pp. 1055-1110. Dekker, New York, 1967. 9. Dullien, F. A. L., "Porous Media-Fluid Transport and Pore Structure," pp. 18-20. Academic Press, New York, 1979. 10. Poulovassilis, A., Soil Sci. 93, 405 (1962). 11. Topp, G. C., and Miller, E. E., Soil Sci. Soc. Arner. Proc. 30, 156 (1966). 12. Talsma, T., Water Resour. Res. 6, 964 (1970). 13. Lai, F. S. Y., M.A.Sc. Thesis, University of Waterloo, 1979. 14. Chatzis, I., Ph.D. Thesis, University of Waterloo, 1980. 15. Wardlaw, N. C., and Taylor, R. P., Bull. Canad. Pet. Geol. 24, No. 2 (June 1976). 16. Poulovassilis, A., Soil Sci. 109, 154 (1970). 17. Poulovassilis, A., Soil Sci. 109, 5 (1970). 18. Topp, G. C., Water Resour. Res. 7, 914 (1971). 19. Topp, G. C., Soil Sci. Soc. Amer. Proc. 33, 645 (1969). 20. Vachaud, G., and Thony, J. L., Water Resour. Res. 7, 111 (1971). 21. Topp, G. C., Klute, A., and Peters, D. B., Soil Sci. So¢. Amer. Proc. 31, 312 (1967). 22. Poulovassilis, A., J. Soil Sci. 20, 52 (1969). 23. Poulovassilis, A., and Childs, E. E., Soil Sci. 112, 301 (1971).