Characteristics of capillary pressure curves

Journal of Petroleum Science and Engineering, 6 ( 1991 ) 249-261

249

Elsevier Science Publishers B.V., Amsterdam

Characteristics of capillary pressure curves Erle C. Donaldson a, Nina Ewallb and Baljit Singhc aDepartment of Petroleum and Geological Engineering, University of Oklahoma, Norman, OK 73019, USA b253 Shady Ridge Dr., Monroeville, PA, USA cUNOCAL, Santa Rosa, CA, USA (Received April 10, 1991; accepted after revision July 3, 1991 )

ABSTRACT Donaldson, E.C., EwaU, N. and Singh, B., 1991. Characteristics of capillary pressure curves. J. Pet. Sci. Eng., 6:249-261. The utility of capillary pressure curves can be extended to the evaluation of the fluid flow properties of rocks which provides another method for characterization of rocks. The capillary pressure hysteresis loop indicates wettability from infinitely water-wet to infinitely oil-wet by its position with respect to zero capillary pressure. The three-constant form of the hyperbola is a general expression for capillary pressure curves; hundreds of capillary pressure curves (from the literature and laboratory data) were represented by a hyperbolic function. The expression: ( l ) smooths laboratory data, (2) can be differentiated to obtain accurate calculation of the inlet saturation for centrifuge-derived data, (3) can be integrated for analysis of wettability and the energy required for fluid displacements, and (4) can be extrapolated to 100% single-phase saturation to obtain the correct threshold pressure.

Introduction

Many unsuccessful attempts have been made to develop a general analytic expression for capillary pressure curves because of the convenience that such an expression would offer for differentiation, integration, and smoothing of laboratory data that often contain random noise from experimental errors. Exponential curves tend to deviate in the center of the capillary pressure curve where it exhibits its maximum curvature. Polynomials deviate most at the points of highest pressure and lead to considerable error if they are used for determination of the areas under the curves. A 'spline-fit' with a polynomial can be used successfully in some cases, but it is awkward to use and becomes inaccurate if the points are far apart. The hyperbola, however, was found to fit the capillary pressure curves throughout the range of data. The value of the hyperbola as a curve-fitting

function for a wide variety of engineering application was discussed in 1972 by Hohmann and Lockard. Its application to capillary pressure curves was recognized and used by the principal author for more than 10 years. Proof of its general applicability to capillary pressure curves was established by Singh (1990) after examination of curves obtained from centrifuge data and curves reported in the literature (Leverett, 1942; Bruce and Welge, 1947; Purcell, 1949; Brown, 1951; Slobod et al., 1951; Killings et al., 1953; Hoffman, 1963; Donaldson et al., 1969; Bensten, 1974; Jennings et al., 1985; O'Meara, 1985; Anderson, 1986, 1987). Use of the hyperbola for capillary pressure curves allowed smoothing of experimental noise, differentiation, integration and extrapolation of the curves. Differentiation of the curves using an analytic expression which is an accurate representation of the data, enabled more accurate determination of the core inlet saturation for centrifuge data because both the

()920-4105/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

250

E.C. DONALDSON ET AL.

Nomenclature a

A,B,C

E~ g h L n

N

Pc P, PV Qrel

Q, r

r: rc ri

R, f

R' S

& Sl Swor U

v. v, O p (7

Centrifugal acceleration (cm/s 2) Constants of the three-constant equation of a hyperbola Displacement energy ( J / c m 3 o r B T U / b b l ) Gravitational acceleration (cm/s 2) Height of capillary rise (cm) Wettability index Core length Number of capillary tubes or number of data points Centrifuge speed (revolutions per minute) Capillary pressure (kPa or psi ) Capillary pressure at the inlet face of the core Capillary pressure at the outlet face of the core Threshold, or entry, capillary pressure Pore volume of the sample Relative volumetric flow in a core within a rage of radii Total flow rate of a group of capillaries Capillary radius (microns =/~m ) Average capillary radius Radius of a capillary (/tm) Radii of pores Radius of an interfacial curvature in a capillary (~m) Average pore entry radius Radius of centrifuge arm to the outer face of the core (cm) Saturation Average saturation Core inlet saturation, corresponding to Pc~ irreducible water saturation Water saturation corresponding to the residual oil saturation (So~= 1-Swor) Velocity (cm/s) Volume of an individual capillary Total volume of a group of capillaries Interfacial contact angle Density (cm3/g) Interfacial tension (milli N / m )

Hassler-Brunner ( 1945 ), and Rajah (1986) methods require differentiation. Subsequent use of the analytic expression for capillary pressure curves facilitated the determination of: ( 1 ) a pseudopore entry size distribution, (2) analysis of the single-phase fluid flow characteristics of the rocks, ( 3 ) estimation of the threshold (fluid entry) pressure, and (4) determination of wettability and displacement energy.

Air-displacing-water capillary pressure curves

The capillary pressure equation Capillary pressure curves (obtained by displacements of water by air and by mercury injection) are used principally to examine the pseudo pore-entry size distribution or rock samples (Ewall, 1985 ). Einstein and Muhsam ( 1923 ) determined the pore-entry radius of a clay filter by immersing the filter in ether and measuring the capillary pressure required to displace the ether with air. The pore radius was calculated using the capillary pressure equation (Eq. A4 ) which was developed for straight capillary tubes (Plateau, 1863; Adams, 1941 ). This equation (A4) may be rigorously applied only to straight capillary tubes; thus its application to porous media containing interconnected, tortuous, capillaries of variable shapes and sizes, may be viewed as qualitative only. An adequate method for the quantitative description of the porous systems of geologic materials has not been possible; hence, Eq. A4 remains in general use. Leverett (1941 ) applied capillary theory to sand and and rock samples.

Centrifuge rneth od Hassler and Brunner (1945) developed a centrifuge method for determination of capillary pressure for small cores by using a step increase of centrifuge speed (rpm) and allowing time for saturation equilibrium in the core at each step. The capillary pressure at the inlet face (end of the core closest to the center of rotation) of the core is calculated using Eq. A10. The average fluid saturation and the amount of fluid displaced is measured at each step from the initial conditions to the end of the run. Hassler and Brunner also presented an approximate method for correction of the av-

CHARACTERISTICSOF CAPILLARYPRESSURECURVES

erage saturation to obtain the inlet saturation corresponding to the inlet capillary pressure (Eq. A14). Rajan (1986) developed a general solution for Eq. A 14 that does not require the Hassler-Brunner assumption that the length of the core is negligible with respect to the arm of the centrifuge. Calculation of the inlet saturation using Eq. A 14, requires the inverse of the derivative of

o< 1 0

o~

0:s

0.~s

SATURATION (AVE. FROM CENTRIFUGE DATA

l+

Pc(data) = f(Sa)

2 51

the capillary pressure curve at each data point. If this derivative is obtained by graphical means, it introduces great inaccuracy to the final capillary pressure curve and to resulting analyses that use the curve, such as the poresize distribution. As mentioned above, Singh (1990) found that a least-squares solution of a three-constant hyperbolic function fit all types of capillary pressure curves and, therefore, could be used for differentiation and integration, which are both necessary for use of capillary pressure curves. The least-squares routine (program HYPER ) is presented briefly in the Appendix for convenience (Eqs. Al5, A16 and A l 7 ) . The capillary pressure at the inlet end of the core, Pc~ (Eq. A10) and the average saturation are obtained at each increment of centrifuge speed as listed in Table 1 and shown in Fig. I. A least-squares fit of the experimental data is obtained from program HYPER and then used to obtain the Hassler-Brunner calculated value of the inlet face saturation, S~, corresponding

..... Pc(HYPER) = f(Sa) ]

TABLE2 Fig. 1. The solid line is the capillary pressure versus the average core saturation from experimental data which contains random experimental errors. The dotted line is the smooth least-squares fit o f the data obtained from program HYPER; Pc~ = f ( S . ) . TABLE 1 Calculation of capillary pressure versus average saturation for Berea core (PV= 1.73, 0=0.17, K= 144, L=2.01 ). N (rpm)

V (displaced)

S~

Pc, (psi)

1300 1410 1550 1700 1840 2010 2200 2500 2740 3120 3810 4510 5690

0.30 0.40 0.50 0.60 0.70 0.75 0.80 0.90 1.00 1.05 1.10 1.20 1.25

0.827 0.769 0.711 0.653 0.595 0.567 0.538 0.480 0.422 0.393 0.364 0.306 0.278

4.135 4.865 5.879 7.071 8.284 9.885 11.843 15.293 18.370 23.818 35.518 49.769 79.219

Calculation of Si (Hassler-Brunner method)

S~

Pcl =f(S,)

dS,/dPc~

Pet dS,/dPct

St

1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.28

2.258 2.682 3.167 3.731 4.391 5.177 6.127 7.300 8.782 10.716 13.345 17.127 23.033 33.549 57.532 82.400

-0.036 -0.034 -0.031 -0.029 -0.027 -0.024 -0.022 -0.020 -0.017 -0.015 -0.013 -0.011 -0.008 -0.006 -0.004 -0.003

-0.081 -0.090 -0.099 -0.108 -0.117 -0.126 -0.135 -0.145 -0.154 -0.163 -0.172 -0.181 -0.190 -0.199 -0.208 -0.212

0.964 0.916 0.869 0.821 0.773 0.726 0.678 0.630 0.583 0.535 0.487 0.439 0.392 0.344 0.296 0.275

( 1 ) Program HYPER is used to obtain Pc1=f(Sa). ( 2 ) Pc~ is obtained for even increments of Sa, Column 2. (3) The H-B correction, using the inverse of the slope at each point, is obtained to make the correction which yields St corresponding to Pet: Sl =Sa+Pc~ dSJdPct. HYPER constants: A= -25.530, B= 17.612, C-4.506.

252

E.C. DONALDSON ET A L

90

25

8O 7o

ill

20

LU

n-60 ~15 ¢r n-

>-

n-

5 lo

< 20-

< O

it

[]

O

lO 0 0

0.25

0.5

0.25

0.75

--=-- Pc = f(Sa) --'~'-- Pc = f(Sl)

0.5

0.75

SATURATION (WETTING PHASE)

SATURATION (Water)

]

Fig. 2. Comparison of the capillary pressure as a function of the average saturation, S,, and the inlet saturation, S,, obtained from the Hassler-Brunner Eq. A 14.

Fig. 3. Extrapolation of t',~ =/(S~ ) to S~ = 1.0 gives the correct initial values of the capillary pressure curve and the correct threshold pressure ( / ; ) . The dotted line is occasionally drawn to follow an initial non-equilibrium data point.

to Pc~ (Table 2). Figure 2 compares Pc, as a function of Sa and Sl illustrating, graphically, the difference between the average and the inlet saturations corresponding to the inlet capillary pressure obtained directly from Eq. A 10. The correct capillary pressure curve is thus:

pressure. Since this is the case, one can use program HYPER to obtain the correct initial capillary pressure.

Pcl =f(S1 ).

Pore entry size distribution

Threshold pressure Extrapolation ofPc~ = f ( & ) to S~ = 1.0 yields the correct threshold (pore-entry) capillary pressure ( 1.97 psi for this core ). In many cases in the literature, capillary pressure curves are arbitrarily drawn as a curve from the first data point (less than S~ = 1.0) to zero, or approximately zero, because in some cases the first point obtained is slightly lower. This is shown as a dotted line in Fig. 3, compared to the correct capillary pressure curve. The first, and sometimes second, capillary pressure points are occasionally lower than the true values due to one or both of the following reasons: ( 1 ) incomplete initial saturation of the core which will cause the first point to be near zero, and (2) incomplete saturation equilibrium--the point was obtained before complete drainage, or injection of fluid, occurred for the applied

Ritter and Drake (1945) developed a method for determination of the pore-size distribution of porous materials using mercuryinjection capillary-pressure data. Burdine et al. (1950) applied the method to rock samples for analyses of the pore-entry size distribution ( Eq. A25 ) and calculation of permeability. When the derivative in Eq. A25 is evaluated graphically from experimental data, the experimental errors are exaggerated by the derivative leading to incorrect conclusions with respect to the pore-size distribution. The experimental data, however, can be smoothed by the least-squares fit of the data to a hyperbolic function and the resulting analytic expression can then be used to obtain more representative pore-size distributions for analyses of formation damage, changes of pore-size distributions by growth of bacteria in the po-

253

CHARACTERISTICS OF CAPILLARY PRESSURE CURVES

TABLE 3 Calculation of the pore-entry size distribution for the Berea core at even increments of S~, using the hyperbolic expression to obtain the derivatives Inlet saturation $1

Capillary pressure Pet

Pore entry radius r

Distribution, D ( ri)

1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.28

1.97 2.38 2.84 3.38 4.01 4.76 5.66 6.78 8.20 10.04 12.55 16.15 21.79 31.82 54.69 78.41

10.59 8.79 7.35 6.18 5.21 4.39 3.69 3.08 2.55 2.08 1.67 1.29 0.96 0.66 0.38 0.27

0.043 0.054 0.067 0.081 0.097 0.114 0.133 0.153 1.740 0.196 0.220 0.246 0.273 0.301 0.330 0.344

0.5

!°, 1 ~0.3 ~0.2 LU

¢,/) UJ

n- 0.1. ;, ~, ~ 8 1'o POREENTRYRADIUS,MICRONS

12

Fig. 4. Smooth pore-entry size distribution obtained using a hyperbolic function to represent the capillary pressure versus saturation data.

res, fluid flow characteristics, etc. (Donaldson, 1985). Table 3 lists the capillary pressure as a function of saturation [Pci =f(S1 ) ] obtained with program HYPER for a Berea core. The pore-entry size is calculated using Eq. A4 assuming that the cos O is equal to 1.0 at all saturations,

because air is non-wetting; the distribution is obtained from Eq. A25. A graphical presentation of the pores-size distribution as a function of pore-entry radius is shown in Fig. 4. The graph is a smooth curve showing that the core contains a relatively large number of small pores ( < 2/zm) compared to large pores in the sample. The distribution can be carried a step further to analyze the single-phase flow through the core with respect to the distribution of pore sizes. DeWitt ( 1951 ) presented a geological description of the Berea sandstone in Ohio.

Fluid flow properties The fluid flow velocity and residual saturations (irreducible water saturation and residual hydrocarbon saturation) are controlled by the pore-size distribution and wettability. Consequently, the distribution can be used to obtain an analysis of the fluid flow characteristics of a rock which presents another method for evaluation of reservoir rocks. Using a single phase, the term relative volumetricflow may be defined as the ratio of the "rate of fluid flow through an arbitrary interval (or range ) of pore sizes" to the "total flow rate of the core". The expression for the relative volumetric flow was derived using the theoretical argument for calculation of absolute permeability from the pore-size distribution which was presented by Burdine et al. (1950) and is shown in detail in the Appendix. The area under the pore-size distribution curve (Fig. 4) represents the pore volume (PV) of the porous medium: PVcal

=Area+ P V Siw

( 1)

An analytic expression of the size distribution as a function of pore radii (Fig. 4) was fit with a least-squares solution using program HYPER which was then used for integration. The range of pore sizes from 0 to 0.266/zm (Table 3) represents the range that contains the immobile, irreducible saturation whose volume is equal to the total pore volume mul-

254

E.C. DONALDSONET AL.

TABLE4

o,

Calculated relative flow with respect to the relative pore volume for Berea and Elgin sandstones Measured properties for Berea and Elgin, respectively: permeability--144, 3550 mD; porosity--0.170, 0.227; Siw-0.278, 0.096. Pore-size range

Relative pore volume

Relative flow

LL UJ

>

uJ n" E3 uJ

>o LIJ n"

Berea sandstone 0.00- 0.27 0.27- 3.71 3.71- 7.15 7.15-10.59

27.8 42.2 19.3 10.8

0.0 10.5 35.9 53.6

Elgin sandstone 0.00- 0.69 0.69- 5.50 5.50-17.62 17.62-23.4 23.40-26.5

9.6 18.3 42.2 19.4 10.4

0.0 0.9 27.5 39.9 31.5

tiplied by the irreducible saturation (Siw); this was selected as the first range of sizes with a flow rate equal to zero. The remaining groups of pore sizes (from 0.266/lm to the m a x i m u m pore size of 10.59 /~m) were arbitrarily divided into three groups, as shown in the first column of Table 4. The volume of each group of pore radii (second column of Table 4 ) was obtained by integration of the distribution curve (Fig. 4) between the limits of the radii for each group. The sum of the group pore volumes, for this case, was equal to the measured pore volume ( 1.73 cm3). The relative volume of each group of pore radii (in percent) to the total pore volume is listed in in column 2 of Table 4 and the relative flow for each group (Eq. A30) is listed in column 3. The histogram in Fig. 5 compares the relative pore volume to the relative flow rate (columns 3 and 4, Table 4). The pore size ranges from 0 to 0.266 ~tm, representing the irreducible saturation of 27.8% (of the total pore volume), does not conduct any of the fluid. The range of pore sizes from 0.27 to 3.7 # m represents 42.2% of the pore volume, but these po-

LU

_> LU n"

0.000 0.266 3,707

7,146 10.585

PORERADIUS(INCREMENTSOF 3.4 MICRONS) I~

Relativeporevol. ~

Relativeflow rate [

Fig. 5. Histogram showing the relationship between relative pore volume and relative flow rate for the Berea sandstone core. Approximately 28% of the pore volume contains immobile fluid, whereas the largest pores, consisting of only about 10% of the pore volume, conduct 54% of the fluid.

res only conducts 10.5% of the total fluid flowing. The pore size range of 3.7 to 7.1 /~m represents 19.3% of the pore volume and these pores conduct 35.9% of the fluid. Finally, the range of pore sizes from 7.1 to 10.6/zm only represents 10.8% of the pore volume, but it conducts 53.6% of the fluid. Thus, this analysis yields a direct comparison of the pore volumes of ranges of pore radii to the relative amount, or relative rate, of flow within the porous medium. This analysis can be used to examine formation damage due to plugging by precipitation, bacteria, particle movement, etc. The only requirement is obtaining centrifuge capillarypressure curves for air-displacing-brine for cores before and after damage. The analysis also provides a method for comparison of the fluid flow properties of different rocks. The Elgin sandstone (Cleveland, Oklahoma) has a very uniform particle-size distribution, resulting in properties that are very different from those of the Berea sandstone listed in the caption of Table 4. The pore-

2 55

CHARACTERISTICS OF CAPILLARY PRESSURE CURVES

size distribution versus pore radii for the Elgin sandstone was exactly modeled by a leastsquares fit of a power function (Y=A X B) rather than the hyperbola used for the Berea core.

The method selected for comparison of the flow properties was to base the Elgin bar chart on divisions of relative pore volumes which are equivalent to the divisions used for the Berea sandstone. Since the irreducible water saturations are different,and no flow occurs in pores that make up this volume, the first two divisions of the Elgin relative volume are equal to the first division of the Berea core. That is, 27.8% of the Berea's pore volume is occupied with non-flowing irreducible water whereas only 9.6% of the Elgin's pore volume is occupied by the irreducible water saturation. Nevertheless, the pores in the next 18.3% of the Elgin's relative pore volume are too small to conduct very much of the total flow (only 0.9%). The next group of pore sizes make up 42.2% of the pore volume but carry only 10.5%

ine

..............................................................................

C3

UJ =E LIJ nr

2

UJ ~>

5

uJ re

0

II

0.686 5.5 17.62 26.515 PORE RADIUS, MICRONS Relative pore vol. l

Relative flow rate ]

Fig. 6. Histogram of the relationship between relative pore volume and relative flow rate for the Elgin sandstone core. Only about 10% of the pore volume contains immobile fluid (compared to 28% for Berea sandstone), and the largest pores, consisting of about 10% of the pore volume, conduct 32% of the fluid flow (compared to 54% for the Berea core).

of the flowing fluid whereas the same relative volume of the Elgin conducts almost three times as much fluid (27.5%). The next group of pores in the Elgin sandstone, consisting of 19.3% of the pore volume, conduct almost equal amounts of fluid (35.9% for the Berea sandstone and 39.9% for the Elgin sandstone). The contrast between the final set of pores that make up 10.8% of the pore volume is much greater because 53.6% of the fluid flows through this volume in the Berea core, whereas only 31.5% is carried in this volume by the Elgin core. The comparisons are more obvious from the histograms of Figs. 5 and 6 where the Berea core shows a continual increase in flow capacity as the pore size increases, but the Elgin indicates that flow through corresponding relative volumes is rather constant by comparison.

Water-displacing-oil/oil-displacing-water capillary pressure curves Centrifugal displacements of water and oil are principally used to determine the USBM wettability index and the vertical saturation profile of thick reservoirs (Donaldson et al., 1969). Research experiments are sometimes conducted with refined oils, but measurements of the wettability index for petroleum reservoirs require use of the crude oil. Formation core samples should be used whenever possible and the measurements should be made at reservoir temperature if possible because the wettability is influenced by temperature, becoming more water-wet as the temperature is increased (Donaldson and Siddiqui, 1989 ). The USBM wettability index is based on the fact that the area under the positive oil-displacing-water capillary pressure curve (A1, Fig. 7 ) becomes larger, whereas the area under the negative water-displacing-oil curve (A2, Fig. 7) becomes smaller, if the wettability of the system changes to a more water-wet condition. The opposite takes place if the system becomes more oil-wet (Singh, 1990).

256

z.c'.

50

Berea Sandstone 4O Ed-1 = 5.51 J/ml Eo2

w n-7

M20 a. >n-

10

o 0 -10

-20

0

0.25 0.5 0.75 SATURATION (WATER)

Fig. 7. Wettability analysis ofa Berea sandstone core. The USBM wettability index is: lw=log (AI/A2). Positive numbers indicate water-wet conditions, whereas negative numbers oil-wet conditions. The areas under the capillary pressure curves represent the energy required for displacement of the fluid (Eo).

Complete, or infinite, water-wetting occurs when A2 vanishes (Iw=log ( A 1 / O ) = + o o ) ; similarly, an infinitely oil-wet surface may be attained in the laboratory (Iw=log (0/

Figure 7 compares the capillary pressure as a function of the average saturation and the inlet saturation which was obtained with the aid of program HYPER. The area under the positive curve (A 1 ) is greater than the area under the negative curve; hence, the wettability index ( + 0 . 3 1 9 ) indicates a water-wet system (the areas were obtained by integration of the expression obtained from a least-squares fit to a hyperbola). The areas (Eq. A31 ) represent the amount of energy required for displacement of the fluids and can be used to estimate the amount of energy that will be required to complete the waterflood of a reservoir (or any immiscible EOR displacement process). Figure 8 shows results obtained with a core which was cleaned with toluene and steam, and then treated with methoxysilane (General Electric Co. Dry Film-104) to make the surface oil-wet (Iw = - 0 . 2 5 4 ) . In becoming oilwet, the area under the positive curve (A1) diminished and the area under the negative curve (A2 ) increased.

25 ¸

A2 ) = - o o ) . Because the centrifuge is generally used for these experiments, it is necessary to correct the average saturation to the inlet saturation corresponding to the measured capillary pressure using either the Hassler-Brunner or Rajan method as discussed above for air-displacingwater capillary pressure curves. These corrections require the derivative of the capillary pressure curve. In addition, the determination of areas under the curves for wettability analyses requires the integral of the capillary pressure curves. Whenever these operations are conducted graphically, the inherent experimental errors in the data introduce considerable error and in some cases lead to incorrect conclusions based on the data. The data used for Fig. 7 was obtained from a Berea core that was cleaned with steam and saturated with brine and a 19 ° API crude oil.

D O N A L D S O N ET A L

- Treated with DF-104

Al•Berea

20

15

Ed-I = 1.05J/ml Ed-2 = 2.29J/ml Iw = -0.254

10

rr O.

+- +

0

+

~

...........

-10 +

-15-

-2ol

0

o~,s o15 -o.~5 SATURATION (WATER)

Fig. 8. Wettability analysis of the Berea sandstone core after treatment with methoxysilane to make its surface oil-wet. As the area A1 decreased and area A2 increased, the USBM wettability index became negative, indicating that the system was made more oil-wet by treatment with the silane solution (General Electric Co. Dry Film-104). Note also that the values of S~wand Swordecreased.

257

CHARACTERISTICS OF CAPILLARY PRESSURE CURVES

Conclusions A useful method for the characterization of the fluid flow properties of rocks, which is based on the pore-entry size distribution, has been developed. Methods for general core cleaning, restoring original wettability, and preserving the wettability of cores for long periods are presented. The hyperbola offers a general analytic expression for capillary pressure curves allowing accurate determinations of derivatives and integrals. A least-squares method for curve-fitting is presented. Extrapolation of the hyperbolic equation of a capillary pressure curve to 100% saturation yields the correct threshold pressure. The fluid displacement energy, calculated from the area under the capillary pressure curve, may be used to estimate the amount of energy required for a waterflood.

Appendix

Substituting ri for R in Eq. A2 yields the expression for capillary pressure in terms of the interfacial tension, contact angle and radius of the capillary tube: Pc = 2 e c o s O(1~re)

The equation for capillary pressure in a centrifugal force field developed by Hassler and Brunner (1945) and demonstrated for small cores by Slobod et al. ( 1951 ) is derived, beginning with the equation for capillary rise in a straight tube:

Pc = dpgh

a= u2/R ', and

(A1)

When the radii of curvature in two perpendicular directions are equal, as in a capillary tube, Eq. A 1 reduces to: Pc =2or(1/R)

(A2)

the special case of Plateau's equation (Eq. A2 ) may be used to derive the relationship for two phases in a capillary from the interfacial geometry of the wetting fluid in a capillary. When the radius of a spherical interface is larger than the radius of the capillary, and the two radii are related by the cosine of the contact angle as follows: cos O=rdri

(A6)

u=2R'N/60 (cm/s) Division by g yields the ratio of centrifugal acceleration to gravitational acceleration, and substituting into Eq. A6: (A7)

Pc =pgh=p(a/g)h

Capillary pressure as a function of interfacial tension and the radii of curvature (Plateau, 1983): Pc = ~ ( 1/g~ +_1/RE)

(A5)

The centrifugal acceleration is:

a/g= 47tER'N 2/(981 X 3600 ) Capillary pressure equations

(A4)

(A3)

= 1.1179X IO-SApNZhR ' (gf/cm 2)

(A8)

Integrating across the length of the core (from the inner, inlet face radius rl ) to the outer radius, r2:

P¢~ =Pc2 + 1.1179× lO- SAp [(r 2 - r ~ ) / 2 l (gf/cm 2)

(A9)

Capillary pressure and saturation gradients exist in the core under centrifugal force; but the only measured quantities are revolutions per minute (N) and the average saturation of the core (Sa). Hassler and Brunner assumed that the outer face of the core remains 100% saturated with the wetting phase at all centrifugal speeds; consequently, the capillary pressure at the outer face (Pcz) is zero. Melrose ( 1982 ) made an analysis of this condition and concluded that for a 100-mD core, the critical pressure exceeds the limits of capillary pres-

258

E.C. DONALDSONET AL.

sure attainable with the Beckman core analysis centrifuge and even for a 1000-mD sample, the critical pressure is approximately 552 kPa (80 psi). Therefore, the Hassler-Brunner boundary condition holds for all but the most unusual rock samples that would be tested with the Beckman core analysis centrifuge. This leads to the equation for centrifuge capillary pressure measurement adjusted to accommodate variable core lengths ): PcJ = 1.588)< IO-7)
(AI0)

A method for calculation of the inlet saturation (&) of the core can be obtained if the length of the core is considered negligible with respect to the radius of rotation (r~ = r2). Using the mathematical definition for the average saturation, Sa:

S~=I/L~SdL=I/(pgL)~Sd(pgL)

(All)

using the assumption that rl = r2, Eq. A 11 becomes synonymous to the inlet pressure of the

core, where Pcl =pgh, consequently:

S a = l / P c , f SdPc

(AI2)

When Eq. A 12 is differentiated and rearranged:

d(Sa Pcl ) = S dPc, and

(A13)

S, =Sc +Pc~ (dSa/dPc,)

(A14)

Program H Y P E R (least-squares solution of a hyperbola) The hyperbola can assume various shapes and although it is symmetrical, any part of the hyperbola can be used to represent a set of data which is either increasing or decreasing. The three-constant equation is: (A15)

Y= ( A + B X ) / ( 1 + CX)

When used for the capillary pressure equations: Y represents the capillary pressure and X the saturation. The following quantities are required: n, ZX, ~ Y, ~ X Y, Z X 2, Y~y2, E X y2, YX2yand EX~y2:

N U M ( 1 ) = (YX 2)X [ ( • X x Y ) ( Z X x y 2 ) _ ( 2 Y ) ( Z X 2 x y 2 ) ] + ( X X x Y) X [ (S.X) ( S.X 2 x y2 ) _ ( ~.Xx Y) ( S.X: x Y) ] +

(A16)

(ZX2× Y)× [(EY)(ZX2× Y)- (2X)(ZX×Y2)]

N U M ( 2 ) = nX [(YXZX Y ) ( E X x Y : ) - ( Z X x Y ) ( Z X 2 x y2)] + ( E X ) × [ ( E Y) ( X X 2 y 2 ) ( ~,Xx Y) ( E X × y2 ) ] + (YXX Y)X [ ( E X x y ) 2 - ( Y Y ) ( Y X 2 × Y ) ] N U M ( 3 ) = n x [(XX 2) ( ~ X x y 2 ) _ ( X X x Y ) ( y ~ X 2 x Y)] + (EX) X [ (Y~Y)(X2x Y ) - (EX) x ( X x y2) 1+ ( ~ X x Y) × [ (XX) ( 2 X × Y) - ( 5~Y) ( ~X2 ) ] nX [ (3~X2X y ) 2 ( 2 X 2 ) ( Z X 2 x y2) ] + DENOM = ( 2 X ) × [ ( £ X ) ( Y X 2 X y2 ) _ ( ~.X× Y) ( 2 X 2 × Y) l + ( YXx Y) × [ ( Z X x Y) ( )~X2 ) - ( 2 X ) (X 2 × Y) ] -

A = NUM( 1)/DENOM B = NUM(2)/DENOM C = NUM( 3 )/DENOM

-

(AI7)

Derivative: d Y / d X = ( B - A C ) / ( 1 + CX) 2

f Integral: J (A + B X ) / ( 1 + CX) d X = [BX/C+ ((AC-B)/C)

In( 1 + CX]~

259

CHARACTERISTICSOF CAPILLARYPRESSURECURVES

Procedurefor wettability analysis

Pore-size distribution The distribution function is defined as: D(ri) =dV/dr (A18) D(ri) d r = d V = P V d S

(A19)

Differentiating the capillary pressure equation (Eq. A4):

dr= (r2/2cr cos O) dPc

(A20)

Substituting Eqs. A4 and A20 into Eq. A19 and simplifying, yields the expression for the pore-size distribution:

D(ri)= (PcPV/r) dS/dPc (m 2)

(A25)

where: Pc is in pascals, PVis in m 3 and r (pore radius) is in m.

Relative volumetric flow Burdine et al. (1950) began with Poiseuille's equation (Eq. A26), which may be used to calculate the volumetric flow rate of a straight capillary tube of uniform size, assuming laminar, steady-state, incompressible flow. The total flow rate of a group of capillaries of various sizes may be determined from the sum of the flow rate in each tube (Eq. A27 ):

Q= (rtraAe) / ( 8 / ~ )

(A26)

Qt = (rtAe/8/zL) ~nir4a

(A27)

Defining the number of capillaries as the total volume of the tubes (Tt) divided by the volume of a single tube and substituting into Eq. A27, an expression for the total flow rate of the bundle of tubes is obtained in terms of the pore volume of the tubes and sum of the average pore radii (Eq. A29);

ni = Vt/ ( Xr2aL)

(A28)

Qt = (nzIP/8lzL) Y.~ (VirJrtL)

(A29)

The relative volumetric flow is obtained by dividing the flow rate for a range of pore sizes by the total flow rate for the group of capillaries summed over the entire pore-size distribution (Qt from I to t, Eq. A29): Qrel = ~ Im V i r a2/ ~

t1 V~ra2

(A30)

( 1 ) Oil-displacing-water (initial displacement): the core is first saturated 100% with brine which is then displaced with oil in the centrifuge to the irreducible water saturation (SiwL) at high speed. (2) Water-displacing-oil (negative capillary pressure curve): the core is covered with water and displaced to the water saturation equivalent to the water saturation equivalent to the residual oil saturation (Swor). HYPER is used to obtain the core inlet saturation using the Hassler-Brunner (1945) method. This yields Pcl =f(SL ), which is integrated from Siw to Swor to obtain the area under the capillary pressure curve (A2). The area under this curve represents the energy required to displace the oil from its initial saturation (Siw) to Swor. (3) Oil-displacing-water (positive capillary pressure curve). The core is covered with oil and the water is displaced from Swor to Siw2. Occasionally Siw~ is not equal to Siw2 due to a change of wettability of the core during the test. In addition, the first capillary pressure point may be low because of either incomplete initial saturation or insufficient time at the low centrifuge speed to allow equilibrium drainage of the core. HYPERis used to obtain Pcl =f(S~ ). Occasionally the Hassler-Brunner calculation yields inlet saturation greater than 1.0 for some values. When this happens, the only alternatives are to clean the core and begin again, or if only an estimate of wettability is required one may use the curve Pcl =f(Sa). Integration of the curve [Pc~=f(SL ) ] yields areaA1 which represents the energy required for displacement of the water by the oil. (4) The two areas are used to determine the USBM wettability index. In water-wet systems, the area under the negative (water-displacing-oil) curve is less than the area under the positive curve and approaches zero as the system becomes more water-wet, because water imbibes into the water-wet core, displacing oil, at zero capillary pressure. Thus, as the system

260

E.¢. DONALDSON ET AL.

becomes more water-wet, less energy is required for the displacement of oil by water. The opposite is true for oil-wet systems.

Iw = log (A 1/A2 )

(A32)

Core cleaning and storage Displacement energy and wettability The integral, or area, under the capillary pressure curve is expressed in pascals/cm 3 of fluid displaced when capillary pressure is in pascals and saturation is in percent pore volume (m 3 ); then (symbolically) for water displacing oil from S~wto Swo~: area= [Pc PV (S~w- Swor) ] = ( N / m 2) m 3 (Siw-Swo~) = NM = Joules

(A31 )

The USBM wettability index is defined a the c o m m o n logarithm of the area under the positive capillary pressure curve divided by the area under the negative curve. For water and oil, positive values indicate water-wet systems; a value near zero indicates neutral (or 50/50) wettability; and negative values indicate oil-wet conditions. Infinitely water-wet and oil-wet conditions can be attained in the laboratory.

A frequent problem encountered in using cores of all types for repeated analyses is adequate cleaning and storage of the cores to maintain the original wettability condition. Outcrop cores frequently contain humus materials that cannot be removed by the standard solvent cleaning methods, but these have a direct influence on the wetting behavior of the cores. Steam cleaning of cores has been used by the principal author for about fifteen years to: ( 1 ) remove humus materials from outcrop cores prior to use; (2) remove residual, highmolecular-weight asphaltenes and resins, that remain on cores contacted with crude oils and then cleaned by the standard toluene-alcohol extraction method; and (3) to return oilfield cores to original wettability (restored state ) for repeated analyses. The steam cleaning equipment is shown in Fig. 9: the cores are placed on a screen over boiling water and a condenser at the top returns water that aids in a leaching mechanism. The permeability of water-sensi-

Water in from tap Water out to drain ~. / /~' ..~ Solenoid valve electrically

.,,,,~_~.

Copper tubing condenser coil on inside of top

,~¢

_

tiedto the heaterswitch

,/I

-,

Door, hinged at top 1/8" rod screen

.....:;i~...,::.i......-::i....,::i.....cil,..,~::';~..~:L.

Knob on door

~:; ;--~:---:--'~--~i 1 / - / / ' 1 f 7:"-~:;~"::"" ( core (=):~"'"""=~/ I//-Y:::.::~"-.:!~-----'-----'191~, ,1 " " .

.

.

.

" " ~" ...... "

.

b

" " ~.~ / f | .

Water line to steam generator 1

.

.

.

--r\.4

~/2

(removab,e) " " Rock sample to be

-~

cleaned by steam

i)

woterlevel(regulated)

Heating element Stainless steel or aluminum sides and door insulated on outside

Fig. 9. Equipment used for steam clearring of outcrop and oilfield cores.

CHARACTERISTICSOF CAPILLARYPRESSURECURVES

tive cores does not change when the cores are cleaned by this method. Cores containing crude oil are first extracted with toluene, dried, and then subjected to the steam cleaning. To return the cores to the original wettability after steam cleaning, they are saturated first with the brine, the brine is displaced to Si,v with the crude oil, and the sample is heated in a sealed container at about 65 ° C for no less than 72 h. The best method found for preserving the wettability of oilfield cores for long periods (years) has been to immerse the core as quickly as possible in a glass bottle of crude oil from the reservoir and store it in a refrigerated room at about 5 ° C. References Adam, N.K., 1941. The Physics and Chemistry of Surfaces, 3rd ed. Oxford University Press, London, pp. 7-12. Anderson, W.G., 1986-1987. Wettability literature survey. J. Pet. Technol., 38 (Oct. '86): 1125-1144; 38 (Nov. '86): 1246-1262; 38 (Dec. '86): 1371-1278; 39 (Oct. '87): 1283-1300; 39 (Nov. '87): 1453-1468; and 39 (Dec '87): 1605-1622. Bensten, R.G. and Anli, J., 1974. Using parameter estimation techniques to convert centrifuge data into a capillary-pressure curves. SPE 1916, 49th Annu. Tech. Conf. Brown, H.W., 1951. Capillary pressure investigations. Trans. AIME, 192: 67-75. Bruce, W.A. and Welge, H.J., 1947. Restored-state method for determination of oil-in-place and connate water. API Prod. Div., Amarillo, Tex., May 22-23. Burdine, N.T., Gournay, L.S. and Reichertz, P.P., 1950. Pore size distribution of petroleum reservoir rocks. Trans. AIME, 198:195-204. DeWitt, Jr., W. 1951. Stratigraphy of the Berea sandstone and associated rocks in Northeastern Ohio and Northwestern Pennsylvania. Geol. Soc. Am. Bull., 62 (11 ): 1347-1370. Donaldson, E.C., 1985. Use of capillary pressure curves for analysis of production well formation damage. SPE 13809, Production Operations Symp., Oklahoma City, Okla., Mar 10-13:7 pp. Donaldson, E.C., Thomas, R.D. and Lorenz, P.B., 1969. Wettability determination and its effect on recovery efficiency. Soc. Pet. Eng. J., 9 ( 1 ): 13-20. Donaldson, E.C. and Siddiqui, T.K., 1989. Relationship

261 between the Archie saturation exponent and wettability. Soc. Pet. Eng. Formation Evaluation, (Sep.): 359362. Einstein, A. and Muhsam, H., 1923. Experimenteile Bestimmung der Kanalweite von Filtern. Dtsch. Med. Woch., 49: 1012-1013. Ewall, N.R., 1985. Relationship of pore size distribution to fluid flow. M.Sc. thesis, Pet. Eng., Univ. Oklahoma, Norman, Okla. Hassler, G.L. and Brunner, E., 1945. Measurement of capillary pressures in small core samples. Trans. AIME, 160: 114-123. Hoffman, R.N., 1963. A technique for the determination of capillary pressure curves using a constantly accelerated centrifuge. Soc. Pet. Eng. J., 3: 227-235. Jennings, J.W., McGregor, D.S. and Morse, R.A., 1985. Simultaneous determination of capillary pressure and relative permeability by automatic history matching. SPE 14418, 60th Annu. Tech. Conf. Kollings, C.R., Nielsen, R.F. and Calhoun, J.C., 1953. Capillary desaturation and imbibition in porous rocks. Pennsylvania State Univ., Mineral Industries Exp. Station, Bull., 62:55 pp. Leverett, M.C., 1941. Capillary behavior in porous solids. Trans. AIME, 142: 152-169. Melrose, J.C., 1982. Interpretation of mixed wettability states in reservoir rocks. SPE Tech. Pap. 10971, New Orleans, La., Sep. 26-29. O'Meara, Jr., D.J. and Crump., J.G., 1985. Measuring capillary pressure and relative permeability in a single centrifuge experiment. SPE 14419, 60th Annu. Tech. Conf. Plateau, J.A.F., 1863-1866. Statique exprrimentale et throrique des liquides soumia aux seules forces moleculaires. Smithsonian Inst. Annu. Rep., Ser. 1 ( 1863 ): 207-285; 2 (1864): 285-369; 5 (1865): 411-435; 6 (1866): 254-289. Purcell, W.R., 1949. Capillary pressures, their measurement using mercury and the calculation of permeability therefrom. Trans. AIME, 186: 39-48. Rajan, R.R., 1986. Theoretically correct analytic solution for calculating capillary pressure-saturation from centrifuge experiments SPWLA Trans., 27th Annu. Logging Symp. Houston, Tex., June 9-12. Pap. O, 18 pp. Ritter, H.L. and Drake, L.C., 1945. Pore size distribution in porous materials. Ind. Eng. Chem., Anal. Ed., 17 (12): 782-791. Singh, B., 1990. Capillary pressure phenomena of porous media. M.Sc. thesis, Pet. Eng., Univ. Oklahoma, Norman, Okla., 116 pp. Slobod, R.L., Chambers, A. and Prehn, Jr., W.L., 1951. Use of the centrifuge for determining connate water, residual oil and capillary pressure curves for small core samples. Trans. AIME, 192:127-134.