In-situ Determination Of Capillary Pressure And Permeability From Log

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IN-SITU DETERMINATION OF CAPILLARY PRESSURE, PORE THROAT SIZE AND DISTRIBUTION, AND PERMEABILITY FROM WIRELINE DATA by L. L. Raymer Schlumberger Well Services, Houston, Texas and P. M. Freeman Schlumberger Offshore Services, New Orleans, Louisiana

ABSTRACT

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The combination of wireline formation tester pressure measurements with conventional well log water saturation and porosity analysis permits the in-situ determination of a capillary pressure profile, a pore-size distribution profile, the mean effective pore radius, and the rock permeability. In this application of wireline data, multiple point measurements of fluid pressure recorded with the Repeat Formation Tester over the reservoir section define the fluid densities of the saturating fluids and the free water level and permit the calculation of the displacement pressure and the capillary pressure at any level of the reservoir. From these capillary pressure data, the water saturation versus depth profile obtained from the appropriate well log analysis can be transformed into a pore-size distribution profile and the mean effective pore radius can be defined. Variations in pore-size distribution over the vertical section of the reservoir can be recognized and quantified. Furthermore, permeability estimations can be made using the mean effective pore radius and the displacement pressure. INTRODUCTION Over the past decade, digital computers have been used more and more frequently for well log analysis. These computer-processed interpretations generally are presented as continuous plots of water saturation, porosity, lithology, and other petrophysical parameters versus well depth (either actual depth or true vertical depth). The water saturation versus depth plot, in addition to providing the obvious information concerning water and hydrocarbon saturations, can be transformed into a pore-size distribution plot. To do this, capillary pressure must be precisely known at each level throughout the reservoir. The multiple-point pressure measurement capability of the wireline Repeat Formation Tester can provide this capillary pressure data when the wellbore cuts both the hydrocarbon- and water-bearing columns of the reservoir. Multiple-point pressure measurements taken at various levels within the hydrocarbon and water columns define a pressure gradient within each column. From these pressure gradients, the densities of the saturating mobile fluids can be calculated and the free water level (hydrocarbon-water contact) can be located. Knowledge of saturating fluid densities and the free water level permits the water saturation versus depth plot from the computer-processed, or other, interpretation to be transformed into a capillary pressure versus water saturation plot.

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The capillary pressure versus water saturation can then, in turn, be transformed into a pore-size distribution plot provided some knowledge or assumption of interracial surface tension and contact angle is available. Fortunately, even when exact knowledge of surface tension and contact angle is not available, the terms can often be estimated to an acceptable degree. From the pore-size distribution plot, the mean effective pore-throat radius can be readily determined. Mean effective pore-throat

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radius provides a method to predict permeabilityy. Permeability can also be estimated using displacement pressure; displa=ent pressure is the capillary pressure observed in the reservoir at th~ depth at which water saturation is about 95 percent. The techniques to determine in situ these several petrophysical parameters are based upon well established petroleum engineering principles. However, the application of the principles to wireline measurements and well log computations is not so well recognized. We will not derive, nor attempt to prove the validity of, these basic petroleum engineering principles and equations in this paper. For that, the reader is referred to any petroleum engineering textbook. We will simply present the principles and equations, expand upon them where necessary, and demonstrate how they can be extended to wireline data. Determination

of Capillary Pressure, Pc, Profile

Determination of capillary pressure, Pc, makes use of the capillary pressure equation relating fluid pressures, fluid densities, and the elevation within a capillary system. That equation is:

pc=ph-pw=gh(pw-ph)

(1)

where Pc ph Pw h Pw ph J3

is is is is is is is

capillary pressure pressure within the hydrocarbon phase at a given elevation pressure within the water phase at a given elevation the elevation above the free water level (hydrocarbon-water contact) water density hydrocarbon density acceleration due to gravity

.-,.

Expressed in oilfield terms and rearranged, the relationship becomes:

Pc =

h(pw - ph) 2.3

(2)

where Pc h Pw Ph

is capillary pressure in psi is elevation above the free water level in ft is water density in g/cm3 is hydrocarbon density in g/cm3

Using this relationship, the true vertical depth scale of a computer-processed (or other) interpretation of water saturation versus depth computation can be resealed into capillary pressure. To do so simply requires that the fluid densities of the saturating hydrocarbon and water phases be accurately known and that the free water level be equally well established. The free water level is, by definition, the level at which capillary pressure is zero.

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Although hydrocarbon and water densities and the hydrocarbon-water contact can be approximated from a variety of well logging measurements (Pw from Rw determination, ‘h from VOLAN* processing, hydrocarbon-water contact from Rt, etc.), these approximations are not usually sufficiently accurate for the calculation of capillary pressure. The wireline Repeat Formation Tester is, however, an accurate source for this data. Multiple pressure measurements taken at several depths within the hydrocarbon and water columns of the reservoir can be used to generate a plot of pressure versus depth. These plotted pressure measurements define a pressure gradient over the hydrocarbonbearing interval of the reservoir and a pressure gradient over the water-bearing interval of the reservoir. From these pressure gradients, the fluid densities can be easily calculated. Expressed in oilfield units, the equation to do so is:

P=

AP/AD

(3)

0.4335 where P

AP/AD

is fluid density in g/cm3 is the pressure change, AP, over the appropriate depth interval, AD, in psi/ft

Iderdly, several pressure measurements should be made in both the hydrocarbon-bearing interval of the reservoir and the water-bearing interval — in fact, the more pressure measurements the better. In deviated wellbores, the depths at which the pressure measurements are made must be corrected to true vertical depth prior to determination of the pressure gradient and entry into Eq. 3, or Eq. 3 must include a correction term based on the cosine of the well deviation. The intersection of the two pressure gradient curves (the water-interval gradient and the hydrocarbon-interval gradient) locates the free water level. This is the level at which capillary pressure is zero. The free water level will always be lower in the reservoir than the hydrocarbon-water contact picked from well log analysis or indicated by production results. In some reservoirs, it maybe much deeper. Once fluid densities and the free water level have been accurately established, reservoir elevation can be resealed into capillary pressure using Eq. 1. This resealing and associated computations are illustrated on Fig. 1. Fig. 1 shows the computer-processed interpretation and wireline pressure measurement made in a Gulf Coast well. Seven Repeat Formation Tester pressure measurements were made within the reservoir — three in the water-bearing interval and four in the hydrocarbon-bearing interval. The measured pressures have been plotted versus the depths at which the measurements were made. The three lower measurements, taken in the water column, define a gradient with a slope of 0.467 psi/ft. Similarly, the four upper measurements, taken in the hydrocarbon column, define a gradient with a slope of 0.110 psi/ft. These gradients inserted into Eq. 3 yield the following fluid densities: Pw

=

AP/AD 0.4335

Ph

=

.

=

0.467 — 0.4335

1.08 g/cm3 for the water

.

=

0.110 — 0.4335

0.25 g/cm3 for the hydrocarbon

AP/AD 0.4335

*Mark of Schlumberger

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The low hydrocarbon density of 0.25 g/cm3 identifies the saturating hydrocarbon

as a gas.

The two pressure gradients intersect at 8020 ft. This is the free water level. Note that this is slightly lower than the gas-water contact which might be picked from the computer-processed interpretation water saturation curve or bulk volume analysis. With the fluid densities determined and the free water level located, reservoir elevations above the free water level can be resealed into capillary pressure values. For example, the capillary pressure at 8000 ft (20 ft above the free water level) is:

Pc =

h(~w - Ph)

=

20(1 .08-0.25)

2.3

2.3

= 7.2 psi

The capillary pressure at any elevation can be similarly calculated and the elevation depth can be resealed into capillary pressure. This has been done on Fig. 1. For clarity, the water saturation versus depth data from the computer-processed interpretation have been expanded in Fig. 2. The water saturation curve has been reversed to conform with the conventional petroleum engineering practice of presenting capillary pressure data. An elevation scale (depth above the free water level) and the calculated capillary pressure scale have been added. This reservoir is obviously of nonuniform pore structure and permeability. Note the variation in water saturations. If the sand were of uniform internal structure, the log-derived water saturation curve would follow one of the dashed curves sketched on the figure. This sand appears to be composed of rocks having at least three different pore geometries. Determination of Effective Pore Radius, rp Once the capillary pressure versus water saturation profile has been established, the effective pore radius can be predicted through capillary surface tension considerations. The relationship between capillary pressure and effective pore radius is:

Pc

=

2 y Cosfl (4)

‘P where Pc Y 8 rp

is is is is

capillary pressure interracial (surface) tension contact angle effective pore (capillary) radius

Evaluation of pore radius in terms of capillary pressure requires knowledge of both the interracial surface tension, y, and the contact angle, 6. Ideally, these should be determined in the laboratory. However, in the absence of measurements of interracial surface tension and contact angle, reasonable estimates usually can be made. Holcott observed that adhesion tension (the product of surface tension and cosine of contact angle, y cos 0) in many oil-bearing reservoirs was typically 30 dynes/cm at reservoir conditions. Hough, Rzasa, and Wood observed a similar consistency in adhesion tension of 35 dynes/cm in gas-bearing reservoirs.

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Using these values of adhesion tension as good approximations in the absence of measured values, Eq. 4 may be written as:

a

(5)

rP=p c

where rp is effective pore radius in Mm (microns) Pc is capillary pressure in psi a = 8.7 for oil-water at typical reservoir conditions = 10 for gas-water at typical reservoir conditions Effective pore radius can now be directly equated to capillary pressure. This has been done on Fig. 2. AS an example of the calculation, the following table relates pore radius, rp, to the capillary pressure, Pc, for the reservoir shown in Figs. 1 and 2 (a = 10 since this is a gas-bearing reservoir):

10

a

r p___

Pc

=

Pc

for Pc=O, rp=oo Pc= l,rp =10 Pc= 5,rp=2 Pc=lO, rp=l Pc = 20, rp = 0.5 The pore-radius scale determined by this method has been added to Fig. 2. The water saturation values now become a measure of the number of capillaries which are smaller or larger than a particular size. For example, in the high-permeability component of the sand (dashed line H), 50 percent of the capillaries are less than 7 wm in radius (50 percent are greater than 7 km); 30 percent are less than 4 Vm (70 percent are greater than 4 vm); 20 percent are less than 1 pm (80 percent are greater); and so on. Furthermore, the mean effective pore radius 7P is, by definition, that value around which the greatest change in water saturation occurs. For the high-permeability component of this rock, mean effective pore radius appears to be about 9 ~m. Of course, if adhesion tension is known from a laboratory measurement or from experience in the area, then Eq. 5 becomes

0.29 y COS 6 ‘P

=

Pc

,,..

The units of adhesion tension are in dynes/cm; the units of the other terms were defined in Eq. 5.

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.. Pore-Size

Distribution

The pore-throat radius versus water saturation plot can be used to construct a pore-size distribution plot. The technique is most easily understood by reference to Fig. 2. It has been noted that the reservoir appears to be composed of rocks exhibiting at least three different pore geometries. These three geometries are characterized by the dashed curves (labeled H, M, and L) superimposed upon Fig. 2. A pore-size distribution plot has been made for each by sorting the pore-throat frequency into 1-~m groups. For example, for dashed curve H, Fig. 2 indicates that 32 percent of the pore throats are less than 5 1/2 urn in radius and 28 percent of the pore throats are less than 4 1/2 Vm in radius. Therefore, in this rock, 4 percent of the pores have a radius of 5 pm f 1/2 pm. That number was plotted on Fig. 3. The remainder of the distribution curve was obtained in a similar manner, as were the distribution curves for the other two dashed curves (M and L). The log-derived water saturation versus depth computation identifies which pore-size distribution curve most characterizes the rock at a given level. For example, over the interval from 7991 to 8002 ft, the pore-size distribution H most characterizes pore geometry. Over the upper interval of the reservoir (above 7976 ft) and over the intervals from 7987 to 7990 ft and 8005 to 8007 ft, pore-size distribution is best characterized by distribution L. Other intervals are best characterized by the intermediate distribution M. It is obvious that distribution H has a coarser pore geometry (larger pores) than distributions L or M. In fact, the mean effective pore radius of the sand represented by distribution His about 9 vm. For the sand represented by distribution M, mean ef fective pore radius is 6.5 Mm; and for distribution L, it is only about 5 ~m. Generally speaking, the greater the mean effective pore radius, the greater the permeability and the coarser the sand. Thus, permeability over the interval from 7991 to 8002 ft can be expected to be greater than permeability elsewhere in the hydrocarbon-bearing portion of the reservoir. Calculation

of Permeability,

k, from Pore Radius

Several relationships are proposed in the literature to estimate permeability effective pore radius, Fp, is known. Most are derived from the Kozeny equation:

k=c~

SA

when the mean

(6)

where k $ c SA

is is is is

permeability porosity a shape factor a unit pore surface area

Porosity and surface area for a unit volume of rock can be approximated by:

(7)

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where

7P is n (c f

mean effective pore radius is number of capillaries per unit rock volume is effective pore length is the cubic dimension of the unit rock volume

Combining these equations, simplifying, rearranging, and converting to microns for pore radius and to millidarcies for permeability gives:

k = 250 C &p2

(8)

Geometrical considerations would suggest that c s 0.5; other experimentalists have recommended a value of 0.2 for c. Fortunately, some permeability versus mean effective pore radius data exist to better define c. Fig. 4 tabulates and plots that data. The numerical value recorded beside each data point is its porosity. This plot suggests a value of approximately 0.15 for c. The permeability equation (Eq. 8) therefore becomes

k = 37@~p2

(9)

where k is permeability in md (p is porosity (fractional) 7p is mean effective pore radius in pm Applied to the example of Figs. 1 and 2, this relationship yields a permeability of 920 md (o = 0.305, T = 9 Vm) for the rock represented by pore-size distribution H (in other words, the interval from 798 1 to 8002 ft). It should be noted that this relationship can be applied only when a transition zone is present and mean effective pore radius is, therefore, determinable. Also, the permeability so determined is the permeability of the rock at the transition zone depth; it may or may not be representative of the entire reservoir. In the reservoir of Fig. 1, it certainly is not. It is representative of only that portion of the reservoir over which the computed water saturation approaches the highpermeability dashed curve H. Calculation

of Permeability

from Displacement

Pressure, Pd

Still another technique to predict permeability utilizes the displacement pressure, Pd. Displacement pressure is that pressure which must be applied to a pore system before the nonwetting phase starts to displace the wetting phase. The technique is derived from work by Wyllie and Rose and Rose and Bruce. Wyllie and Rose proposed

(10)

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,,,

and Rose and Bruce developed

(11)

where T @ k ts Y Pd

tortuosity porosity permeability a shape factor interracial surface tension k displacement pressure

is is is is is

Combining the above relationships and, once again, converting to oilfield units gives:

k=

21y24)3

(12)

tsp2

d where k Pd Y ts $

is is is is is

permeability in md displacement pressure in pSi surface tension in dynes/cm a pore shape constant (usually about 2.25) porosity (fractional)

The displacement saturation.

pressure is the capillary pressure corresponding

to about

95 percent water

In low-permeability rocks, the displacement pressure, Pd, can be determined from wireline data with sufficient accuracy to calculate permeabilityy from the above relationship. In very highpermeability rocks, the displacement pressure probably cannot be obtained from well log measurements with sufficient accuracy to yield a reliable permeabilityy calculation. Nevertheless, as an exercise, the displacement pressure of Figs. 1 and 2 appears to be about 0.6 psi. This inserted into Eq. 12 would give a permeabilityy of 907 md. That is close to the 920 md obtained using mean effective pore radius; but the critical selection of the actual displacement pressure was probably influenced by the desired answer. SUMMARY

The process reported herein provides a method of combining pressure measurements made with a wireline formation tester tool with the continuous water saturation calculations from a well log analysis to determine, in situ, densities of saturating fluids, free water level, pore-throat radius, poresize distribution, and permeabilityy. The calculation process, based on established petroleum engineering equations, can be summarized as follows:

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1.

Determine Pw and ph from a plot of wireline pressure data versus depth over the hydrocarbonand water-bearing reservoir intervals:

APw/AD Pw =

0.4335 AP#AD

ph =

0.4335

2.

Free water level occurs where the APw/AD and Aph/AD

gradients intersect.

3.

Convert reservoir elevation (depth) above the free water level to capillary pressure:

Mpw-Ph)

Pc =

2.3

Pc = O at free water level 4.

Convert capillary pressure to pore radius:

a = 8.7 for oil a = 10 for gas 5.

Mean effective pore radius, Fp, is that pore radius at which greatest change in water saturation occurs.

6.

Displacement pressure, Pd, is the capillary pressure at Sw = % percent.

7.

Permeability can be predicted by

ccc k = 37 +Fpz

or k=9.4—

Y203 p2

d y= 30 for oil y = 35 for gas

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REFERENCES

1.

Pelissier-Combescure, J.; Pollock, D.; and Wittman, M.: “Application of Repeat Formation Tester Measurements in the Middle East, ” F+oc., SPE Middle East Technical Conference, Bahrain (March 1979).

2.

Calhoun, J. C.: Fundamentals Oklahoma (1953).

3.

Wyllie, R. J. and Rose, W. D.: “Some Theoretical Considerations Related to the Quantitative Evaluation of the Physical Characteristics of Reservoir Rock from Electrical Data,” Trans., AIME (1950) 189, 105-118.

4.

Rose, W. D. and Bruce, W. A.: “Evaluation Rock, ” Trans., AIME (1949) 186, 127.

5.

Slider, H. C.: Practical Petroleum Tulsa, Oklahoma (1976).

6.

Pittman, E. D.: “Influence of Porosity Type and Pore Geometry on Productive Capability of Sandstone Reservoirs,” AAPG Clastic Diagnosis School Manual, AAPG (1977).

7.

Pittman, E. D.: “Clay-Bearing Manual, AAPG (1977).

of Reservoir

Engineering,

Reservoir

Sandstone

Univ. of Oklahoma Press, Norman,

of Capillary Character in Petroleum Reservoir

Engineering

Methods,

Reservoirs, ” AAPG

Petroleum Publishing Co.,

Clastic Diagnosis

School .

8.

Hocott, C. R.: “Interracial Tension Between Water and Oil Under Reservoir Conditions,” Trans., AIME (1939) 132, 184.

9.

Hough, E. W.; Rzasa, M. J.; and Wood, B. B.: “Interracial Tensions at Reservoir Pressures and Temperatures, ” Trans., AIME (1951) 192, 57.

10.

Kozeny, J.: ‘‘Uber Kapillare Leitung des Wassers im Boden, “ Wien Akad. Wiss. Sitz. Berichte (1927) 136-2A, 271.

11.

Raymer, L. L,: “Elevation and Hydrocarbon Density Correction for Log-Derived Permeability Relationships, ” The Log Analyst (May-June 1981) 22, 3, 3-7.

12.

Katz, D. L.; Cornell, D.; Kobayshi, R.; Poettmann, F. H.; Vary, J. A.; Ellenbaas, J. R.; and Weinaug, C. F.: Handbook of Natural Gas Engineering, McGraw-Hill Book Company,” New York (1959). ABOUT THE AUTHORS

Lewis L. Raymer is manager of Marketing & Technique Services for Schlumberger Well Services in Houston, Texas. Since joining Schlumberger in 1955, he has served in various field, research, marketing, administrative, and managerial positions in the United States and South America. He is a graduate of Rice University. Lew has served the SPWLA as vice president-publications and editor of The Log Analyst. Philip M. Freeman graduated in 1962 from the University of London with a B.S. in physics. He joined Schlumberger the same year and has worked in Europe, Africa, U. S.A., South America, and the Far East in various field and management positions. He is presently marketing manager for Schlumberger Offshore Services in New Orleans. Phil is a member of SPWLA and the SPE of AIME.

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Figure I ACTUAL DEPTH- FT

Pc

‘P pm

psi 60

0.5

7960

L I I

20 -

M 50

15

7970

I

I

40

7960

I

I

30 1

7990

10 –

ccc 20

2

8000

I

5 –

\

10

8010

\

5

‘L

10 o– u (n ~N 1 ,,

~

-.

~ 0

40

WATER

-11-

80

Saturation

PORE THROAT

Figure 2

..-—-..— -

—._ 1

- %

DISTRIBUTION

- %

6020 )

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r

L

PORE THROAT

RADIUS - #m

Figure 3

\

029.5

0 15.3

19.0

1s.7 o

\

10 440

10 MEAN EFFECTIVE

1.0

c

PORE RADIUS - pm

..

Figure 4

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