26 properties of reservoir rock

Chapter 26 Properties of Reservoir Rocks Daniel M. Bass Jr., C~hado School of Mmea*

Introduction This chapter deals with the fundamental properties of reservoir rocks. The properties discussed are (1) pomsi- 8-a measure of the void space in a rock: (2) pertneahi/it~-a measure of the fluid transmissivity of a rock; (3) fluid saturutim-a measure of the gross void space occupied by a fluid; (4) cupillaryprrssure rek- rim--a measure of the surface forces existing between the rock and the contained fluids; and (5) electrical cm- clucrivity offluid-saturated rocks-a measure of the conductivity of the rock and its contained fluids to electric current. These properties constitute a set of fundamental parameters by which the rock may be described quantitatively.

Typical core-analysis data are presented to illustrate the description of porous media by these fundamental properties.

Porosity Porosity is defined as the ratio of the void space in a rock to the bulk volume (BV) of that rock, multiplied by 100 to express in percent. Porosity may be classified accord- ing to the mode of origin as primary and secondary. An original porosity is developed during the deposition of the material, and later compaction and cementation reduce it to the primary porosity. Secondary porosity is that developed by some geologic process subsequent to deposition of the rock. Primary porosity is typified by the intergranular porosity of sandstones and the inter- crystalline and oolitic porosity of some limestones. Secondary porosity is typified by fracture development as found in some shales and limestones and the vugs or solution cavities commonly found in limestones. Rocks having primary porosity are more uniform in their

‘Thts author also Wrote the orlglnal chapter on this topic m the 1962 edition with coauthor James W. Amyx (deceased] D.M Bass Jr IS currently a petroleum COnSUltant

characteristics than rocks in which a large part of the porosity is induced. For direct quantitative measurement of porosity, reliance must be placed on formation samples obtained by coring.

Unit cells of two systematic packings of uniform spheres are shown in Fig. 26.1. The porosity for cubical packing (the least compact arrangement) is 47.6% and for rhombohedral packing (the most compact arrangement) is 25.96%. ’ Considering cubical packing, the porosity may be calculated as follows. The unit cell is a cube with sides equal to 2r where r is the radius of the sphere. Therefore, Vh=(2r)3=8r3, where V,] is the bulk volume. Since there are 8% spheres in the unit cell. the sand-grain volume, V,, , is given by

4ar’ v, = -

3

The porosity, $, is given by

where L’,, is PV. Therefore,

8r” -413irr3 4=

8r3 X100=(1--&00

=47.6%.

Of particular interest is the fact that the radii cancel and the porosity of uniform spheres is a function of packing only.

26-2 PETROLEUM ENGINEERING HANDBOOK

POROSITY: 47.6 % POROSITY= 25.96%

CuB~~~KXE;IDE RHOMBOHEDRAL OR CLOSE PACKED

Fig. 26.1-Unit cells and groups of uniform spheres for cubic and rhombohedral packing.

Tickell et al.’ has presented experimental data in- dicating that, for packings of Ottawa sand, porosity was a function of skewness of the grain-size distribution (see Fig. 26.2). Skewness is a statistical measure of the uniformity of distribution of a group of measurements. Other investigators have measured the effects of distribu-

In dealing with reservoir rocks (usually consolidated sediments), it is necessary to define total porosity and ef-

tion, grain size, and grain shape. In general, greater

fective porosity because cementing materials may seal off a part of the PV. Totalporosity is the ratio of the total

angularity tends to increase the porosity, while an in-

void space in the rock to the BV of the rock, while effective porosity is the ratio of the interconnected void space

crease in range of particle size tends to decrease

in the rock to the BV of the rock, each expressed in percent. From the reservoir-engineering standpoint, effec-

porosity.

tive porosity is the desired quantitative value because it represents the space that is occupied by mobile fluids. For intergranular materials, poorly to moderately well cemented, the total porosity is approximately equal to the effective porosity. For more highly cemented materials and for limestones, significant differences in total-porosity and effective-porosity values may occur.

-.08 -.04 0 +.04 +.08 SKEWNESS

Fig. 26.2-Variation of porosity with skewness of grain-size distribution.

Photographs of oilwell cores are presented in Fig. 26.3. 3 The pore configuration of the sandstones shown is complex, but the pores are distributed relatively uniformly. Complex pore configurations arise from the interaction of many factors in the geologic environment

Material having induced porosity, such as the carbonate rocks shown in Fig. 26.3, have even more com-

of the deposit. These factors include the packing and

plex pore configurations. In fact, two or more systems of pore openings may occur in such rocks. The basic rock

particle-size distribution of the framework fraction, the

material is usually finely crystalline and is called the “matrix.”

type of interstitial material, and the type and degree of

The matrix contains uniformly small pore openings that comprise one system of pores. One or

cementation. The influence of these various factors may

more systems of larger openings usually occur in carbonate rocks as a result of leaching, fracturing, or

be evaluated as statistical trends.

dolomitization of the primary rock material. Vugular pore openings are frequently as large as an ordinary lead pencil and usually are attributed to leaching of the rock subsequent to deposition. Fractures also may be quite large and contribute substantially to the volume of pore openings in the rock.

(b) (d)

Fig. 26.3-Oilwell cores. Consolidated sandstone: (a) wireline core, Lower Frio; (b) whole core, Seven Rivers. Vugular, solution cavities, and crystalline limestone and dolomite: (c) whole core, Devonian; (d) whole core, Hermosa.

PROPERTIES OF RESERVOIR ROCKS 26-3

Laboratory Measurement of Porosity

Numerous methods have been developed for the determination of the porosity of consolidated rocks having intergranular porosity. Most of the methods developed have been designed for small samples, roughly the size of a walnut. Since the pores of intergranular material are quite small, determining the porosity of such a sample involves measuring the volume of literally thousands of pores. The porosity of larger portions of the rock is represented statistically from the results obtained on numerous small samples.

In the laboratory measurement of porosity, it is necessary to determine only two of the three basic parameters (BV, PV, and grain volume). In general. all methods of BV determination are applicable to determining both total and effective porosity.

BV. Usual procedures use observation of the volume of fluid displaced by the sample. This procedure is par- ticularly desirable because the BV of specially shaped samples may be determined as rapidly as that of shaped samples.

The fluid displaced by a sample may be observed either volumetrically or gravimetrically. In either pm cedure, it is necessary to prevent fluid penetration into the pore space of the rock. This may be accomplished by (1) coating the rock with paraffin or a similar substance, (2) saturating the rock with the fluid into which it is to be immersed, or (3) using mercury, which by virtue of its surface tension and wetting characteristics does not tend to enter the small pore spaces of most intergranular materials.

Gravimetric determinations of BV may be accomplished by observing the sample’s weight loss when immersed in a fluid or the difference in weight of a pycnometer when filled with a fluid and when filled with fluid and the core sample. The details of gravimetric determinations of BV are best summarized by Example Problems I through 3.

Example Problem l-Coated Sample Immersed in Water. Given that the

weight of dry sample in air. A=20.0 g. weight of dry sample coated with paraffin, B=20.9 g

(density of paraffin=0.9 gicmj), and weight of coated sample immersed in water at 40°F.

C= 10.0 g (density of water= 1 .OO g/cm3). we can then calculate that

weight of paraffin=&A=20.9-20.0=0.9 g.

0.9 volume of paraffin = - = 1 cm3 ,

0.9

weight of water displaced=B-C=20.9- 10.0 = 10.9 g,

10.9 volume of water displaced = ~ = 10.9 cm3,

1 .o

volume of water displaced-volume of paraffin =10.9-1.0=9.9 cm3, and

BV ofrock, Vb,=9.9 cm3.

Example Problem 2-Water-Saturated Sample Im- mersed in Water. Given that

weight of saturated sample in air, D=22.5 g, and weight of saturated sample in water at 40”F,

E= 12.6 g. we can calculate that

weight of water displaced=D-E=22.5- 12.6 x9.9 g, and

9.9 volume of water displaced= - =9.9 cm3.

1.0 Therefore,

BV rock, If,,, =9.9 cm3.

Example Problem 3-Dry Sample Immersed in Mer- cury Pycnometer. Given A from Example Problem I and the following values,

weight of pycnometer filled with mercury at 20°C. F=350.0 g and

weight of pycnometer filled with mercury and sample at 20°C. G=235.9 g (density of mercury= 13.546 g/cm3).

we can calculate that weight of sample + weight of pycnometer filled with

mercury=A+F=20+350=370 g, weight of mercury displaced =A+F-G=370-

235.9= 134.1 g. and

134.1 volume of mercury displaced = - =9.9 cm3

13.546

Therefore, BV of rock=9.9 cm3. Determinations of BV volumetrically use a variety of

specially constructed pycnometers or volumeters. An electric pycnometer from which the BV may be read directly is shown in Fig. 26.4. The sample is immersed in the core chamber, causing a resulting rise in the level of the connecting U tube. The change in mercury level is measured by a micrometer screw connected to a low- voltage circuit. The electric circuit is closed as long as the measuring point is in contact with the mercury. The travel of the measuring point is calibrated in volume units such that the difference in the open-circuit readings with and without the sample in the core chamber represents the BV of the sample. Either dry or saturated samples may be used in the device.

Sand-Grain Volume (GV). The various porosity methods usually are distinguished by the means used to determine the GV or PV. Several of the oldest methods of porosity measurement arc based on the determination of GV.

The GV may be determined from the dry weight of the sample and the sand-grain density. For many purposes, results of sufficient accuracy may be obtained by using the density of quartz (2.65 g/cm3) as the sand-grain density.

For more rigorous determinations either the A.F. Melcher-Nutting4 or Russell’ method may be employed. The BV of a sample is determined: then this sample, or an adjacent sample, is reduced to grain size and the GV is determined. In the M&her-Nutting technique, all the measurements are determined


Type of sampling

Functions measured

Manner of measurement

Errors

TABLE 26.1~METHODS OF DETERMINING POROSITY

Effective Porosity Methods

Washburn-Bunting Porosimeter

One to several pieces per increment (usually one).

Solvent extraction and oven drying. Occa- sionally use retort samples.

PV and BV.

Reduction of pressure on a confined sample and measurement of air evolved. BV from mercury pycnometer.

Air from dirty mercury; possible leaks in system; incomplete evacuation caused by rapid operation or tight sample.

Stevens Porosimeter



Sand grain volume and unconnected PV and BV.

Difference in volume of air evolved from a constant-volume chamber when empty and when occupied by sample. BV by Russell tube.

Mercury does not become dirty. Possible leaks in system; incomplete evacuation caused by rapid operation or tight sample.

Kobe Porosimeter






Boyle’s Law Porosimeter






MICROMETER SCALE

ADJUSTING SCREW

OH INDICATOR

LIGHT

Fig. 26.4-Electric pycnometer

gravimetrically, using the principle of buoyancy (Exam- ple Problem 2). The Russell method uses a specially designed volumeter, and the BV and GV are determined volumetrically. The porosity determined is total porosity, $r. Thus,

Vb - vs +r=- x loo.

vb

From the data of Example Problem 2 and using a sand- grain density of 2.65 g/cm3,

vb =9.9 cm3,

20 v,=- =7.55 cm3,

2.65

and

9.9-1.55 +r= x 100=23.8%.

9.9

The Stevens6 porosimeter is a means of determining the “effective” GV. The porosimeter (Fig. 26.5) consists of a core chamber that may be sealed from atmospheric pressure and closed from the remaining parts of the porosimeter by a needle valve. The accurate volume of the core chamber is known. In operation, a core is placed in the core chamber; a vacuum is established in the


TABLE 26.1-METHODS OF DETERMINING POROSITY (continued)

Effective Porositv Methods Total Porosity Method

Core Laboratories, Core Laboratories, Dry Sample


Saturation* One to several pieces per increment (usually one).

Wet Sample Sand Density Several pieces per increment.

Several pieces for retort; one for mercury pump.

Solvent extraction and overy drying. Occa- sionally use retort samples.

PV and BV.

None. Solvent extraction and oven drying. Occa- sionally use retort samples.

Solvent extraction; then, in second step, crush sample to grain size.

Volumes of gas space, oil, and water. BV.

Sand-grain volume and unconnected PV, and BV.

BV of sample and volume of sand grains.

Weight of dry sample; weight of saturated sample in air: weight of saturated sample immersed in saturated fluid.

Weight of retort sample; volume of oil and water from retort sample; gas volume and BV of mercury-pump sample.

Difference in volume of air evolved from a constant-volume chamber when empty and when occupied by sample.

Weight of dry sample; weight of saturated sample immersed; weight and volume of sand grains.

Possible incomplete saturation.

Obtain excess water from shales. Loss 01 vapors through con- densers.

Possible leaks in system; incomplete evacuation caused by rapid operation or tight sample.

Possible loss of sand grains in crushing. Can be reproduced most accurately.

system by manipulating the mercury reservoir; the air in the core chamber is expanded repeatedly into the evacuated system and measured at atmospheric pressure in the graduated tube. The Stevens method is an adapta- tion of the Washburn-Bunting7 procedure, which will be discussed with the measurement of PV.

Example Problem 4-Determination of Grain Volume by Gas Expansion (Stevens Porosimeter). Given that

volume of core chamber, H= 15 cm3 and

total reading, 1=7.00 cm3 where

volume of air (first reading)=6.970, volume of air (second reading)=0.03, and volume of air (third reading)=O,

we can calculate effective grain volume=H-Z=8 cm3, bulk volume of sample (from pycnometer)= 10 cm3,

and effective porosity, 4, =[(lO-8)110] X 100=20%.

PV. All methods of measuring PV yield “effective” porosity. The methods are based on either the extraction of a fluid from the rock or the introduction of a fluid into the pore space of the rock.

The Washburn-Bunting 7 porosimeter measured the volume of air extracted from the pore space by creating a partial vacuum in the porosimeter by manipulation of the attached mercury reservoir. In the process, the core is exposed to contamination by mercury and, therefore, is

PULLEY.

MERCURY - RESERVOIR

--P GTOPCOCK

;RADUATED TUBE

-CORE :HAMBER

--CRANK

\ BRAKE

CONNECTING HOSE -

Fig. 26.5-Stevens porosimeter


Sample Number Type of Material

Approximate Gas Permeability

(md)

1 LImestone 1 2 Fretted glass 2 3 Sandstone 20 4 Sandstone 1,000

BZE Semiquartzltic sandstone 0.2 0.x Semlquartzitic sandstone 0.8 61-A Alundum 1,000 722 Alundum 3 1123 Chalk 16

1141-A Sandstone 45

TABLE 26.2-CHARACTERISTICS OF SAMPLES USED IN POROSITY-MEASUREMENT COMPARISONS

Porosity (O/O)

Average From Average From Value From

Average

17.47 28.40 14.00 30.29

3.95 3 94

28.47 16.47 32.67 19.46

Gas Saturation High Methods Methods Observation’

17.81 16.96 18.50 26.60 27.97 29.30 14.21 13.70 15 15 31.06 29.13 31.8

4.15 3.66 4.60 4.10 3.71 4.55

28.70 28.00 29.4 16.73 16.08 17.80 33 10 32.03 33.8 19.68 19.12 20.2

Value From Low

Observation”

16.72 27.56 13 50 26 8

3 50 3.48

27.8 16.00 31 7 100

not suitable for additional tests. The previously described Stevens method is a modification of the Washburn-Bunting procedure especially designed to prevent mercury contamination of the samples.

A number of other devices have been designed for measuring PV; these include the Kobe8 porosimeter, the Oilwell Research porosimeter, and the mercury-pump porosimeter. Kobe and Oilwell Research porosimeters are Boyle’s-law-type porosimeters designed for use with nitrogen or helium with negligible adsorption on rock surfaces at room temperature. The mercury-pump porosimeter is designed so that the BV may be obtained as well as the PV.

The saturation method of determining porosity consists of saturating a clean dry sample with a fluid of known density and determining the PV from the gain in weight of the sample. The sample usually is evacuated in a vacuum flask to which the saturation fluid may be ad- mitted by means of a separatory funnel. If care is exercised to achieve complete saturation, this procedure is believed to be one of the best available techniques for intergranular materials. Example Problem 5 illustrates the saturation technique of measuring PV.

Example Problem G-Effective Porosity by the Saturation Method. From the data of Example Prob- lems 1 and 2, we can calculate

weight of water in pore space=D-A=22.5-20 =2.5 g.

EXPANSION CHAMBER

KNOWN VOLUME COMPRESSED GAS SOURCE

CORE CHAMBER OF

KNOWN VOLUME

Fig. 26.6-Gas-expansion porosimeter for large cores.

2.5 g volume of water in pore space = ~

1 g/cm3 =2.5 cm3,

effective PV=2.5 cm’, and BV (Example Problem 2)=9.9 cm3.

Therefore,

2.5 ~,=~x100=25.3%.

Several methods of determining effective porosity are compared in Table 26.1.

Precision of Porosity Measurements. A group’ of ma- jor company laboratories conducted a series of tests to determine the precision of porosity measurements. The method used was either gas expansion or a saturation technique. Table 26.2 summarizes the results of these tests. Note that the gas-expansion method is consistently higher than the saturation method. This undoubtedly results from the fact that the errors inherent to each tend to be in opposite directions. In the case of the gas- expansion method, errors caused by gas adsorption would cause high values to be obtained while, for the saturation techniques, incomplete saturation of the sample would result in low values. The difference in the average values obtained by the two methods is about 0.8% porosity, which is approximately a 5% error for a 16% porosity sample. However, it is felt that all the methods commonly used to determine effective porosity yield results with the desired degree of accuracy if carefully performed.

Carbonate Rocks. Small samples, such as used in the routine techniques already discussed, yield values of porosity that do not include the effect of vugs, solution cavities, etc. The saturation methods of determining PV and BV are unsatisfactory because drainage will occur from the larger pore spaces. Therefore, it is necessary to use larger core samples and to determine the BV by measurement of the core dimensions or after coating the sample. The effective grain volume is obtained by using a large gas-expansion porosimeter of the type shown in Fig. 26.6. This porosimeter is based on Boyle’s law in


50

0 0 1000 2000 3000 4000 5000 6CCO

DEPTH OF BURIAL IN FEET

Fig. 26.7-Effect of natural compaction on porosity.

which high-pressure gas is equalized between two chambers. The porosity may be calculated from the measured pressures, the volume of either chamber, and the bulk-sample volume by the use of Boyle’s law.

Keltont” reported results of whole-core analysis. a method utilizing large sections of the full-diameter core. The following table of matrix vs. whole-core data summarizes a part of Kelton’s work. lo

Grollp

1 2 3 4 Matrix porosity, % bulk 1.98 1.58 2.56 7.92 Total porosity, % bulk 2.21 2.62 3.17 8.40

Matrix porosity is that determined from small samples; total porosity is that determined from the whole core. Whole-core analysis satisfactorily evaluates most carbonate rocks. However, no satisfactory technique is available for the analysis of extensively fractured materials because the samples cannot be put together in their natural state.

Compaction and Compressibility of Porous Rocks

The porosity of sedimentary rocks has been shown by Krumbein and Sloss ” to be a function of the degree of compaction of the rock. The compacting forces are a function of the maximum depth of burial of the rock. The effect of natural compaction on porosity is shown in Fig. 26.7. This effect is caused principally by the resulting packing arrangement after compaction; thus, sediments that have been buried deeply, even if subsequently uplifted. exhibit lower porosity values than sediments that have not been buried at great depth.

Geertsma12 states that three kinds of compressibility must be distinguished in rocks: (I) rock-matrix compressibility, (2) rock-bulk compressibility, and (3) pore compressibility. Rock-bulk compressibility is a combination of pore and rock matrix compressibility.

Rock-matrix compressibility is the fractional change in volume of the solid rock material (grains) with a unit change in pressure. Pore compressibility is the fractional change in PV of the rock with a unit change in pressure.

POROSITY-PER CENT

Fig. 26.8-Effective reservoir-rock compressibilities

The depletion of fluids from the pore space of a reservoir rock results in a change in the internal stress in the rock, thus causing the rock to be subjected to a different resultant stress. This change in stress results in changes in the GV, PV, and BV of the rock. Of principal interest to the reservoir engineer is the change in the PV of the rock. The change in rock-bulk compressibility may be of importance in areas where surface subsidence could cause appreciable property damage.

Hall I3 reported PV compressibility as a function of porosity. These data are summarized in Fig. 26.8. The effective rock compressibility in Fig. 26.8 results from the change in porosity caused by grain expansion and decrease in pore space because of compaction of the matrix.

Fatt I4 indicates that the pore compressibility is a function of pressure. Within the range of his data, he was unable to find a correlation with porosity.

725 Gradient (North

4 Gradient Sand 4 W.M.L(S.W.Tex.1

6 I I I I I I

35 30 25 20 15 IO 5 Formation Compressibility(Microsip1

1

Fig. 26.9-Depth vs. formation compressibility in abnormally pressured segment of an abnormally pressured reservoir.


/ I I

l l \eet

-I

l

l

l 0

J 35

Fig. 26.10-PV compressibility at 75% lithostatic pressure vs. initial sample porosity for limestones.

I I I I I I

0

Fig. 26.11-PV compressibility at 75% lithostatic pressure vs. initial sample porosity for consolidated sandstones.

100 I I I I I I

0 0 0

0 O 0” 08 0 0

0 O 0 0 00 0 0

0

00 00

@ OQ

0 0

0

B 0

\ 0 0 O 0 0 0 0 l

COR RELPlTlON \

0 0

H& LL�S -- 0 00 Oo o 0 0

@ >& . 0 00:

0 0 0 I

0 0

Fig. 26.12-PV compressibility at 75% lithostatic pressure vs. initial sample porosity for friable sandstones.

The additional significance of changes in porosity with the discovery of oil at deeper depths and new geological areas resulted in the need for a better understanding of the changes in porosity with the depletion of reservoir fluid pressure.

Hammerlindl I5 developed a correlation from measured field data that indicates the change in porosity compressibility with depth of burial of an unconsolidated sand (Fig. 26.9). Similar correlations have also been presented by others in the technical literature.

Considerable laboratory work has been performed recently in an attempt to understand better the effect of formation compaction on porosity. Newman I6 performed measurements on samples of limestone and consolidated, friable, and unconsolidated sandstones. He compared his results with those of Hall I3 and van der Knaap, ” as illustrated in Figs. 26.10 through 26.13. As noted in his data, an approximate correlation may exist between PV compressibility and porosity for limestones and consolidated sandstones. Little or no correlation exists between PV compressibility and porosity for friable and unconsolidated sandstones. By averaging the PV compressibility in 5% ranges of porosity, Newman at- tempted to correlate all four types of porous media. The results of this averaging technique are presented in Fig. 26.14.

The methods used to measure PV compressibility have come under discussion. PV compressibility can be measured in the laboratory by the hydrostatic (same pressure in all three directions) or the triaxial (different pressure in the z direction than in the x and y directions) techniques. The test samples also can be stress cycled

PROPERTIES OF RESERVOIR ROCKS

Fig. 26.13-PV compressibility at 75% lithostatic pressure vs. initial sample porosity for unconsolidated sandstones.

until the strain resulting from a supplied stress is constant, or the sample can be placed under stress and the strain measured for purposes of calculating PV compressibility.

Krug I8 and Graves I9 demonstrated that when a formation sample was stress cycled to a stable strain condition, the sample gave repeatable values of PV compressibility even when the sample was left in an un- stressed condition for 30 days or more. The data reported by Newman were for samples that were not stress cycled.

Lachance*’ compared PV compressibilities obtained by the hydraulic and triaxial methods. The reported results (Fig. 26.15) indicate a large difference in the magnitude of PV compressibilities obtained by the two methods. The triaxial data indicate that PV compressibility is essentially independent of the sample porosity. Newman’s I6 data were obtained by the hydrostatic method, whereas Krug’s I8 and Graves’ I9 data were obtained by the triaxial method.

In summary, rock compressibility is an important factor in reservoir evaluation. Oil reservoirs with high initial pressures and low fluid bubblepoint pressures are sensitive to the true value of PV compressibility. Gas reservoirs with initial reservoir pressures in excess of 6,000 psi are also sensitive to the value of PV compressibility. Newman, I6 Krug, I8 and Graves I9 all recommend that PV compressibility be measured on

Fig. 26.14-Class averages of PV compressibility vs. initial sample porosity.

Fig. 26.15-Calculated PV compressibility (porosity 2 8%).


-- . . --- Lf _-- -_- 6 3

II -- --- ] -_- ---I - _- --- --_ r/--i a t h,

.--------_

;

h-h,

Fig. 26.16-Schematic of Henry Darcy’s experiment on flow of water through sand.

samples from the reservoir in question when the PV compressibility may be significant in reservoir evaluation.

Permeability Introductory Theory

It is the purpose of this section to discuss the ability of the formation to conduct fluids. In the introduction to API Code 27,” it is stated that permeability is a property of the porous medium and a measure of the medium’s capacity to transmit fluids. The measurement of permeability, then, is a measure of the fluid conductivity of the particular material. By analogy with electrical con- ductors, the permeability represents the reciprocal of the resistance that the porous medium offers to fluid flow.

The following equations for flow of fluids in circular conduits are well known.

Poiseuille’s equation for viscous flow:

d2Ap v=- 321.LL ) . . . . . . . . . . . . . . . . ..I.......... (1)

Fanning’s equation for viscous and turbulent flow:

2dAp $=- fpL , . . . . . . (2)

where v = fluid velocity, cm/s, d = diameter of conductor, cm,

Ap = pressure loss over length L, dynes/cm2, L = length over which pressure loss is

measured, cm, p = fluid viscosity, Pa’s, p = fluid density, g/cm’, and f = friction factor, dimensionless.

A more convenient form of Poiseuille’s equation is

ar4Ap 4= , . . . . . . . . . 8clL (3)

where r is the radius of the conduit, cm, q is the volume rate of flow, cm”/s, and the other terms are as previously defined.

If a porous medium is conceived to be a bundle of capillary tubes, the flow rate q1 through the medium is the sum of the flow rates through the individual tubes. Thus,

qr=s i lljrj4, . . . .

P I 1

where n,, is the number of tubes of radius rj. If

(T/8) $ njr j4

j=l

(4)

is treated as a flow coefficient for the particular grouping of tubes the equation reduces to

qr=c$; . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...(5)

where

k

The pore structure of rocks does not permit simple classification of the flow channels. Therefore, empirical data are required in most cases.

In 1856, Darcy investigated the flow of water through sand filters for water purification. His experimental apparatus is shown schematically in Fig. 26.16.” Darcy interpreted his observations to yield results essentially as given in Eq. 7.

ht --h2 q=KA- . . . . . . . . . . . . . . . . . . . . . . L (7)

q represents the volume rate of flow of water downward through the cylindrical sandpack of cross-sectional area A and length L. h 1 and h2 are the heights above the standard datum of the water in manometers located at the in- put and output faces and represent the hydraulic head at


Points 1 and 2. K, a constant of proportionality. was found to be characteristic of the sandpack. Darcy’s investigations were confined to flow of water through sandpacks that were 100% saturated with water.

Later investigators found that Darcy’s law could be extended to fluids other than water and that the constant of proportionality K could be written as klp, where p is the viscosity of the fluid and k is a proportionality constant for the rock. The generalized form of Darcy’s law as presented in API Code 27 is presented in Eq. 8.

u,yz-; ($-gp;). . .

where s = distance in direction of flow, always

positive, us = volume flux across a unit area of the porous

medium in unit time along flow path S, z = vertical coordinate, considered positive

downward, cm, p = density of the fluid, g = acceleration of gravity,

- = pressure gradient along s at the point to ds which u refers, p = viscosity of the fluid, k = permeability of the medium, and

dt -= sin 8, where fl is the angle between s and ds the horizontal.

u, may further be defined as q/A where q is the volume rate of flow and A is the average cross-sectional area perpendicular to the lines of flow.

The portion of Eq. 8 in parentheses may be interpreted as the total pressure gradient minus the gradient caused by a head of fluid. Thus, if the system is in hydrostatic equilibrium, there is no flow and the quantity inside the parentheses will be zero. Eq. 8 may be rewritten as

kd u, =--@gz-p). . . . . . . . . .

fib

The quantity d(pgz-p)lds may be considered to be the negative gradient of a potential function b, where

a=p-pgz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(lO)

The potential function is defined such that flow will be from higher to lower values.

The dimensions of penmeabilit P

may be established from an analysis of Eq. 8 as k=L . In the cgs system of units, the unit of permeability would be cm*, a large unit for common usage; therefore, the petroleum industry adopted the darcy as the standard unit of permeability, which is defined as follows.

A porous medium has a permeability of one darcy when a single phase fluid of one centipoise viscosity that completely fills the voids of the medium, will flow through it under conditions of viscous flow at a rate of one cubic centimeter per second per square centimeter cross sectional area under a pressure or equivalent hydraulic gradient of one atmosphere per centimeter. ”

Fig. 26.17-Sand model for rectilinear flow of fluids.

Conditions of viscous flow mean that the rate of flow will be sufficiently low to be directly proportional to the potential gradient.

Darcy’s law holds only for conditions of viscous flow as defined. Further, for the permeability k to be a proportionality constant of the porous medium, the medium must be 100% saturated with the flowing fluid when the determination of permeability is made. In addition, the fluid and the porous medium must not react-i.e., by chemical reaction, adsorption, or absorption. If a reactive fluid flows through a porous medium, it alters the porous medium and, therefore, changes the permeability of the medium as flow continues.

Flow Systems of Simple Geometry

Horizontal Flow. Horizontal rectilinear steady-state flow is common to virtually all measurements of permeability. If a rock is 100% saturated with an incompressible fluid and is horizontal (Fig. 26.17), then dz/ds=O, dplds=dpi&, and Eq. 8 reduces to

4 -k dp ux=-=--3

A tth

which on integration becomes

kA(P I -P*) 9= cLL , .,.................... (11)

where k is the specific permeability. If a compressible fluid flows through a porous

medium, Darcy’s law, as expressed in Eq. 8, is still valid. However, for steady flow, the mass rate of flow rather than the volume rate of flow is constant through the system. Therefore, the integrated form of the equation differs. Considering steady rectilinear flow of compressible fluids, Eq. 8 becomes

b dp pux=---, .,..........,,........... (12)

ttb

or, for steady-state flow,

pux=p: =constant.


I FREE FLOW

II FLOW UNDER

HEAD h

III FLOW UNDER

HEAD h

Fig. 26.1 E-Sand model for vertical flow

The density-pressure relationship for isothermal conditions of a slightly compressible fluid may be expressed as

p=poe(‘p

and

ap=a”, . . . . . .(13) CP

where c is the fluid compressibility. Thus,

P 0 9 0 -b dp -k dp -z--c--

A P dx pc dx’

where q. is the volume rate of flow of a fluid of density PO.

On integration,

WP I -02) PO90 = . . . . . (14) cLcL

If terms of cp of second and higher order are neglected, the density can be expressed as

such that Eq. 14 reduces to

k4P I -P2) 40=

d .

The density-pressure relationship for isothermal conditions of an ideal gas may be expressed as

P P PPb -=- orp=-. . ,(15) Ph Pb Pb

Thus,

t”bqb b dp -=-_- A P dx’

where pb and qb are the density and volume rate of flow, respectively, at the base pressure, ph.

Substituting for p

Pbqb k dp -=--p-, . . . . . . . . . . . .

A cl&

which on integration yields

(16)

kA P,*-Pz~ q”=G . . . . . . . _. __ (17) Pb

Define p as (p, +p2)12, and yP as the volume rate of flow at p such that pqp =pbqh; then

k4PI -P2) 41, = uL .,.................... (18)

Therefore, flow rates of ideal gases may be computed from the equations for incompressible liquids so long as the volume rate of flow is defined at the algebraic mean pressure.

Vertical Flow. Fig. 26.18 shows three sandpacks in which linear flow occurs in the vertical direction.

First consider Case 1 (Fig. 26. IS)--when the pressure at the inlet and outlet are equal (free flow), such that only the gravitational forces are driving the fluids. Given

dz dp S=Z, -=l, and -=O, ds ds

the flow is then defined by

q&g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(19) P

Next consider Case 2-the case of downward flow when the driving head (difference in hydraulic head of inlet and outlet) is h (Fig. 26.18). We know that

dz, -ah z=l and z=-

L

Therefore, from Eq. 8,

. . . . . . . . . . .(20)


When the flow is upward and the driving head is h, Case 3 (Fig. 26.18), and z is defined as positive downward,

and

k u=+-

P

then

kApgh 4= . . .._.......................... (21)

Radial Flow. A radial-flow system, analogous to flow into a wellbore, is idealized in Fig. 26.19. If flow is considered to occur only in the horizontal plane under steady-state conditions, an equation of flow may be derived from Darcy’s law to be

2akMp, -P,, 1 4= ) . . . . (22) ~ ln rplr,,

where rI, is the radius at the external boundary at which pc (pressure at the external boundary) is measured, and r,v is the radius of the wellbore at which pa, (pressure at the wellbore) is measured. All other terms are as defined for linear flow.

Eq. 22 may be modified appropriately for the flow of compressible fluids. The details of modifying this equation are omitted because they are essentially the same as the ones used in the horizontal rectilinear flow systems.

After modification for variations in flowing volumes with changing pressures, Eq. 22 becomes for slightly compressible fluids

2xk&, -P,,,) h’ = . ,_..................... (23)

cp In r,,lr,,.

where w is the mass rate of flow, g/s, or

2mWpp -pw) 40 =

a~, In TJr,,.,

where q. is defined at the pressure p. where the density is po.

For ideal gases, Eq. 22 becomes

7rkh(p<,‘-~,,.~) 4/J = . (24) Wb ln r,,r,,

or

2dd-0, -p w) 911 = , . . . . cL ln r‘,,ra, (25)

where q,, is the volume rate of flow at the algebraic mean pressure (p, +p,,,)/2.

Fig. 26.19-Sand model for radial flow of fluids to central wellbore.

Conversion of Units in Darcy’s Law

It is convenient in many applications of Darcy’s law to introduce commonly used oilfield units. The following is a summary of the more common equations with the conversion factors to convert to oilfield terminology.

Linear-Flow Liquids (or Gases with Volume at Mean Pressure). Rate, BID, is given by

q=1.1271 kA(p I-p21 . . . . . . . LLL (26)

Rate, cu ft/D, is given by

q=6.3230 Wp I -~2)

. . . . . . . . . AL (27)

Radial-Flow Liquids (or Gases with Volume at Mean Pressure). Rate, liters per day, is given by

q=92.349x 103 Wp, -PM’)

. . . . p In r,/r,

(28)

Rate, cu ft/D, is

q-92.349x 103 kh(p, -p,v) cc ln r,/r,, . . (29)

Gases at Base-Pressure, p,,, and Average Flowing Temperature, Tf. Linear flow rate, cu ft/D. is given by

3.1615T,,kA(p, ’ -pz *) Yb = , . . . . . . . .

TfW&PlJ (30)


Fig. 26.20-Linear flow-parallel combination of beds.

and radial flow rate, cu ft/D, is given by*

19.88T~kh(p,2 -p,,,?) q/J= - .,............. (31)

Tfq.4,~p~ In rplr,,,

where k is in darcies; A is in sq ft; h is in ft; p , , p2, pp, p ,,.. and Ph are in psia; p is in cp; L is in ft; and re and T,, are in consistent units.

Since the previous equations describe the flow in the medium, appropriate volume factors must be introduced to account for changes in the fluids caused by any decrease in pressure and temperature from that of the medium to standard or stock-tank conditions. Permeability Conversion Factors. Following is a list of various unit conversions from darcy units to other systems of units.

kL!E- A(p/L) ’

1 darcy (d)= 1,000 millidarcies (md)

(cm3/s)cp =

cm ‘(atm/cm)

=9.869x 10 -’ (cm3/s) cp

,,* (dynIrm*)

=9.869x lop9 cm2

=1.062x IO-” sq ft

=7.324x lop5 (cu ft/sec)cp

sq ft(psi/ft)

(cu ft/sec)cp =9.697x10p4

cm’(cm water/cm)

=1.127 WD)cp

sq ft (psi/ft)

= 1.424 x 10 -’ (gal/min)cp

sq ft (ft water/ft)

Pe

Fig. 26.21~Radial flow-parallel combination of beds.

Flow Systems of Combinations of Beds

Consider the case where the flow system comprises layers of porous rock separated from each other by in- finitely thin, impermeable barriers as shown in Fig. 26.20. The average permeability k may be evaluated by Eq. 32.

(32)

C hj j=l

Fig. 26.21 shows that the same terms appear in the radial-flow network as in the linear system. The only difference in the two systems is the manner of expressing the length over which the pressure drop occurs. Because all these terms are the same in each of the parallel layers, an evaluation of the parallel radial system yields the same solution as obtained in the linear case.

Example Problem 6-Average Permeability of Beds in Parallel. What is the equivalent linear permeability of four parallel beds having equal widths and lengths under the following conditions?

Pay Horizontal Thickness Permeability

Bed (ft) (md) 1 20 100 2 15 200 3 10 300 4 5 400

i: kihj j& j=l .

ihi ’ j=I

k= (100x20)+(200x15)+(300x 10)+(400x5)

20+15+10+5

10,000 =--200 md.

50


AP, AP, AP, h

Fig. 26.22-Linear flow-series combination of beds.

Another possible combination for flow systems is to have the beds of different permeabilities to be arranged in series as shown in Fig. 26.22. In the case of linear flow, the average series permeability may be evaluated by Eq. 33.

L k= . . . . . . .(33)

LJ lkj J==I

The same reasoning can be used in the evaluation of the average permeability for the radial system (Fig. 26.23) so as to yield

kc In r,,/r,,.

’ In rjlr, -1

= h, j&l ‘.I

Example Problem 7-Average Permeability of Beds in Series. What is the equivalent permeability of four beds in series, having equal formation thicknesses under the following conditions?

Assume Bed 1 adjacent to the wellbore (1) for a linear system and (2) for a radial system if the radius of the penetrating wellbore is 6 in. and the radius of effective drainage is 2,000 ft.

Length Horizontal of Bed Permeability

_ @I Bed (md) 1 250 25 2 250 50 3 500 100 4 1,000 200

For a linear system,

k= ’

k Ljlkj j=l

Fig. 26.23-Radial flow-series combination of beds

Therefore,

j& 250+2.50+500+ 1,000

2.50 250 500 1,000

z+50+100+- 200

2,000 = 10+5+5+5 =80 md.

For a radial system,

k= log 2.000/0.5

log 25010.5 log 500/2so lop I .000/500 log 2.ooo11 .ooo + + +

25 SO loo 200

=30.4 md.

Permeability of Channels and Fractures

Only the matrix permeability has been discussed in the analysis to this point. In some sand and carbonate reservoirs, the formation frequently contains solution channels and natural or artificial fractures. These channels and fractures do not change the permeability of the matrix but do change the effective permeability of the flow network.

Circular Channel. Equating Darcy’s and Poiseuille’s equations for fluid flow in a tube, the permeability may be expressed as a function of the tube radius.

k=;, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(35)

where k and r are in consistent units. If r is in centimeters, then k in darcies is given by

r2 k=

8(9.869x 10 -9) I 12.50X 106r’,

where 9.869~ 1O-9 is a conversion factor from the previous list. Then, if r is in inches,

k= 12.50~ 106(2.54)*r2

=80~10~r*=20~10~d*,

where d is the diameter of the opening in inches.


(PI -P2YL (0)

~~~~~~

01 03 (p: -Op5:),2L

07 09

fb)

Fig. 26.24-Plol of experlmental results for calculation of permeability-(a) from k/F = qL/ VW, -P~)I; @I from ~~P=~~,P&L~P, 2 -P~*)I.

Therefore, the permeability of a circular opening 0.005 in. in radius is 2,000,OOO md.

Fracture. For flow throu 4

h slots of fine clearances and unit width, Buckingham’- reports that

12pL Ap=- h? ’

such that the permeability of the slot is given by

k=;. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(36)

When h is in centimeters and k in darcies. the permeability of the slot is given by

k= 12(9.869x 10-9)

=84.4x 10’h2,

and when h is in inches and k is in darcies, permeability is given by

k=54.4 x 106h2,

so that the permeability of a fracture 0.01 in. in thickness would be 5,440 darcies or 5,440,OOO md.

Physical Analogies to Darcy’s Law

Ohm’s law as commonly written is

where I = current, A,

E = voltage drop, V, and r = resistance of the circuit, Q,

but

L L r=p- or r= -,

A OA

where p = resistivity, Q-cm, u = llp=conductivity, L = length of flow path, cm, and A = cross-sectional area of conductor, cmZ.

Therefore,

I& PL

Comparing with Darcy’s law for a linear system,

,=!A%. P L

Note that

4 = 1,

k 1 and

I* P

4 E - =- L L’

The Fourier heat equation may be written as

d=k,aF,

where Q= rate of heat flow, Btulhr,

A = cross-sectional area, sq ft, AT = temperature drop, “F,

L = length of the conductor, ft, and kh = thermal conductivity, Btu/(ft-hr-“F).

A comparison with Darcy’s law indicates that

s=Q,

k -=kl,, and P

Ap AT -=- L L’


Electrical and heat models (based on these analogies) of rock and well systems frequently are used to solve fluid- flow problems involving complex geometry.

Measurement of Permeability

The permeability of a porous medium may be determined from samples extracted from the formation or by in-place testing. The procedures discussed in this section pertain to the permeability determinations on small samples of extracted media.

Two methods are used to evaluate the permeability of cores. The method most used on clean, fairly uniform formations uses small cylindrical samples called perm plugs that are approximately % in. in diameter and 1 in. in length. The second method uses full-diameter core samples in lengths of 1 to 1% ft. The fluids used with either method may be gas or any nonreactive liquid.

Perm-Plug Method. As core samples ordinarily contain residual oil, water, and gas, it is necessary that the sample be subjected to preparation before the determination of the permeability. The residual fluids normally are extracted by retorting or solvent extraction. The core is dried before permeability measurements are taken. Air commonly is used as the fluid in permeability measurements. The requirement that the permeability be determined for conditions of viscous flow is best satisfied by obtaining data at several flow rates and plot- ting results, as shown in Fig. 26.24, based on either Eq. 17 or 18. For conditions of viscous flow, the data should plot a straight line, passing through the origin. Tur- bulence is indicated by curvature of the plotted points. The slope of the straight-line portion of the curve is equal to k/p, from which the permeability may be computed. To obtain k in darcies, 4 must be in cm3/s, A in cm’, p 1 and p2 in atm. L in cm, and ~1 in cp.

A permeameter designed for the determination of the permeability of rocks with either gas or liquid is illustrated in Fig. 26.25. Data ordinarily are taken from this device at only one flow rate. To assure conditions of viscous flow, the lowest possible rate that can be measured accurately is used.

Example Problem S-Permeability Measurement. The following data were obtained during a routine penneability test. Compute the permeability of this core.

1. Flow rate= 1,000 cm3 of air at 1 atm absolute and 70°F in 500 seconds.

2. Pressure, downstream side of core= 1 atm absolute; flowing temperature, 70°F.

3. Viscosity of air at test temperature=0.02 cp. 4. Cross-sectional area of core=2.0 cm*. 5. Length of core=2 cm. 6. Pressure, upstream side of core= 1.45 atm

absolute.

PI VI fP?V2 =Pv

where 1 is upstream conditions and 2 is downstream conditions, and

P +Pz 1.45+1 P=- E-=1.225,

2 2

Fig. 26.25-Ruska universal permeameter.

and

1 x 1,000=1.225 V.

V=816 cm3,

_ v 816 -=1.63, 4=;=500

1.63x2x0.02 = x 1,000

2 x0.45

=72.5 md.

Assuming that the data indicated were obtained, but water was used as the flowing medium, compute the permeability of the core. The viscosity of water at test temperature was 1 .O cp.

v 1,000 4’-‘50040

t

and

.-

k=!!kx,,)oO= 2X1X2 ~ x 1,000=4,450 md.

A AP 2x0.45

Whole-Core Measurement. The core must be prepared in the same manner as perm plugs. The core is then mounted in special holding devices as shown in Fig. 26.26. The measurements required are the same as the perm plugs but the calculations are slightly different.


AIR PRESSURE -GASKET OPENING

END VIEW

TOP VIEW

SPL’T - TELOWMETER

GASKET1

Fig. 26.26-Clamp-type permeameter for large cores.

In the case of the clamp-type permeameter, the geometry of the flow paths is complex, and an appropriate shape factor must be applied to the data to compute the permeability of the sample. The shape factor is a function of the core diameter and the size of the gasket opening. The shape factor affects the quantity L/A in the previous equations.

Factors Affecting Permeability Measurements

In the techniques of permeability measurement previously discussed, certain precautions must be exercised to obtain accurate results. When gas is being used as the measuring fluid, corrections must be made for gas slippage. When liquid is the testing fluid, care must be taken that it does not react with the solids in the core sample. Also, corrections may be applied for the change in permeability because of the reduction in confining pressure on the sample.

Effect of Gas Slippage on Permeability Measurements

Klinkenberg I4 has reported variations in permeability determined by using gases as the flowing fluid from that determined by using nonreactive liquids. These variations were ascribed to slippage, a phenomenon well known with respect to gas flow in capillary tubes. The phenomenon of gas slippage occurs when the diameter of the capillary openings approaches the mean free path of the gas.

Fig. 26.27 is a plot of the permeability of a porous medium as determined at various mean pressures using hydrogen, nitrogen, and carbon dioxide as the flowing fluids. Note that for each gas a straight line is obtained for the observed permeability as a function of the reciprocal of the mean pressure of the test. The data obtained with the lowest-molecular-weight gas yield the straight line with the greatest slope, indicative of a greater slippage effect. All the lines when extrapolated to infinite mean pressure (l/j=O) intercept the permeability axis at a common point. This point is designated as the equivalent liquid permeability, kL. Klinkenberg and

b CARBON DIOXIDE I / I I I I 1

0 2 0.4 0.6 0.8 1.0 12 1.4

1

J

RECIPROCAL MEAN PRESSURE (ATM)-’

k!

Fig. 26.27-Permeability constant of core sample L to hydrogen, nitrogen, and COP at different pressures (permeability constant to iso-octane, 2.55 md).

others established that the permeability of a porous medium to a nonreactive homogeneous single-phase liquid was equal to the equivalent liquid permeability.

The linear relationship between the observed permeability and the reciprocal of mean pressure may be expressed as follows.

kg kL=- 1 +blp

=k, -ml, . . . . . . (37) P

where kL = permeability of the medium to a single

liquid phase completely filling the pores of the medium,

k, = permeability of the medium to a gas

completely filling the pores of the medium,

j = mean flowing pressure of the gas at which k, was observed,

b = constant for a given gas in a given medium, and

m = slope of the curve.

Reactive Fluids. While water commonly is considered to be nonreactive in the ordinary sense, the occurrence of swelling clays in many reservoir rock materials results in water’s being the most frequently occurring reactive liquid in connection with permeability determinations. Reactive liquids alter the internal geometry of the porous medium. This phenomenon does not vitiate Darcy’s law but rather results in a new porous medium, the permeability of which is determined by the new internal geometry.

While fresh water may cause the cementation material in a core to swell because of hydration, it is a reversible process. A highly saline water may be flowed through the core and return the permeability to its original value. The effect of water salinity on permeability is shown in Table 24.3. 25


TABLE 26.3-EFFECT OF WATER SALINITY ON PERMEABILITY OF NATURAL CORES (Grains/gal Chloride ion’)

Field Zone ~ - S 34

: 34 34 S 34 S 34 s 34 S 34 S 34 S 34 S 34 S 34 S 34 s 34 S 34 T 36 T 36 T T ;: T 36 T 36

k a

4,080 24,800 40,100 39,700 12,000 4.850

22,800 34.800 27,000 12,500 13,600 7,640

11,100 6,500 2,630 3,340 2,640 3,360 4,020 3,090

k 1,000 k 500 k 300

1,445 1,380 1,290 11,800 10,600 10,000 23,000 18.600 15,300 20,400 17,600 17,300

5,450 4,550 4,600 1,910 1,430 925

13,600 6,150 4,010 23,600 7,800 5,460 21,000 15,400 13,100

4,750 2,800 1,680 5,160 4,640 4,200 1,788 1,840 2,010 4,250 2,520 1,500 2,380 2,080 1,585 2,180 2,140 2,080 2,820 2,730 2,700 2,040 1,920 1,860 2,500 2,400 2,340 3,180 2,900 2,860 2,080 1,900 1,750

k 200

1,190 9,000

13,800 17,100 4,510

736 3,490 5,220

12,900 973

4,150 2,540

866 1,230 2,150 2,690 1,860 2,340 2,820 1,630

k 100 k w 885 17.2

7,400 147 8,200 270

14,300 1,680 3,280 167

326 5.0 1,970 19.5 3,860 9.9

10,900 1,030 157 2.4

2,790 197 2,020 119

180 6.2 794 4.1

2,010 1,960 2,490 2,460 1,860 1,550 2,280 2,060 2,650 2,460 1,490 1,040

‘Far example. ks means permeablhty lo air, hSoo means permeablhty 10 iresh water

means permeablllty 10 500.gram/gal chloride solul~on, and kw

Care must be taken that laboratory permeability values are corrected to liquid values obtained with water whose salinity corresponds to formation water.

Overburden Pressure. When the core is removed from the formation, all the confining forces are removed. The rock matrix is permitted to expand in all directions, partially changing the shapes of the fluid-flow paths inside the core.

Compaction of the core caused by overburden pressure may cause as much as a 25 % reduction in the permeabil i- ty of various formations, as observed in Fig. 26.28.2h Note that some formations are much more compressible than others; thus, more data are required to develop empirical correlations that will permit the correction of surface permeability for overburden pressures.

Factors in Evaluation of Permeability From Other Parameters

,Permeability. like porosity, is a variable that can be measured for each rock sample. To aid in understanding fluid flow in rocks and possibly to reduce the number of measurements required on rocks, correlations among porosity, permeability, surface area, pore size, and other variables have been made. The reasoning behind some of the correlations among porosity, permeability, and surface area is presented here to enable the reader to gain some understanding of the interrelation of the physical properties of rocks. Although these relations are not quantitative, they are indicative of the interdependence of rock characteristics.

Use of Capillary Tubes for Flow Network. The permeability of a tube derived from Darcy’s and Poiseuille’s equation is

k=f x

If a porous system is conceived to be a bundle of capillary tubes, then it can be shown that the permeability of the medium depends on the pore-size distribution and porosity. A flow network of tubes would be similar to layers of different permeability in parallel, such that the average permeability could be calculated by adapting Eq. 32 to read

m

C kjAj J=I

WI . . . . . . . . ..I.............. (38)

J=l

where kj is the permeability of one capillary tube and Aj is the area of flow represented by a bundle of tubes of permeability ki.

lCURVE SOURCE PERMEABILITY POROSITY -- nr rr.n.- . .^ --l?rruT

ti d 60 zt I t 3 ~WXIERN CALIFORNIA 335 25

:z 4 LOS ANGELES BASIN

2 CALIFORNIA I10 22

y 50 I I 0 1000

5 2000 3000 4000 5000

OVERBURDEN PRESSURE-PSIG

Fig. 26.28-Change in permeability with overburden pressure.


The quantities k, and A, can be defined in terms of the radius of capillary tubes.

A,=sn r ’ I I

and

7 k;+,

where n, is the number of tubes of radius r,.

5 A,=$A,, J=l

where 4 is the porosity of the flow network and A, is the total cross-sectional area of the flow network. By substitution, Eq. 38 reduced to

j&f

,,1

c n/r, 3 j= I 111 ( . . . . . . c ,l/r; * j=l

(39)

where li is the average permeability of the tube bundle. Note that the permeability of a bundle of tubes is a

function not only of the pore size but of the arrangement or porosity of the system.

Consider a system that comprises a bundle of capillary tubes of the same radii and length; k, the permeability, may be written from Eq. 39 as

7

k=$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(40)

The internal surface area per unit PV, A,,, may be defined in terms of the tube radius by

A,,=?. ,.,.....,.,,.,,_...,.........., .(41) r

Combining Eqs. 40 and 41 gives the permeability as a function of porosity and internal surface area. This function is

If l/K; is substituted for the constant l/z, the resulting expression is Kozeny’s equation wherein K; is defined as the Kozeny’s constant.

4 k=- K,A,” ~~~~..........,......___,,,,. (42)

Wyllie*’ derived the Kozeny equation from Poi- seuille’s law by using a specified flow network. The resulting permeability for this flow network is given by Eq. 43.

, . . . . . . . . . . . . . . . . . . . . . . (43)

where k = permeability of the porous medium,

F,Y = shape factor, L = length of the sample, and

L, = actual length of the flow path.

If

Lo 2

( > T =T= tortuosity of the porous medium

and K, =F,Y7=Kozeny constant, then Wyllie’s equation will reduce to the same form as Eq. 42.

Fluid Saturations In the previous sections of this chapter, the storage and the conduction capacity of a porous rock were discussed. To the engineer there is yet another important factor to be determined-i.e., the fluid content of the rock. In most oil-bearing formations, it is believed that the rock was completely saturated with water before the invasion of the rock by petroleum. The oil will not displace all the water from the pore space. Therefore, to determine the quantity of hydrocarbons accumulated in a porous rock formation, it is necessary to determine the fluid saturation (oil, water, and gas) of the rock.

There are two approaches to the problem of determining the fluid saturations within a reservoir rock. The direct approach is to measure. in the laboratory, the saturations of selected rock samples recovered from the parent formation. The indirect approach is to determine the fluid saturation by measuring some related physical property of the rock.

Factors Affecting Fluid Saturations of Cores The core samples delivered to the laboratory for fluid- saturation determinations are obtained from the ground by either rotary, sidewall, or cable-tool coring. In all cases, the fluid content of these samples has been altered by two processes. First, in the case of rotary drilling, the mud column exerts a greater pressure at the formation wellbore surface than the fluid in the formation. The dif- ferential pressure between the mud column and the formation fluids causes mud and mud filtrate to invade the formation, thus flushing the formation with mud and its fyltrate. The filtrate displaces some of the original fluids. This displacement process changes the original fluid contents of the in-place rock. Second. as the sample is brought to the surface, the confining pressure of the fluid column is constantly decreasing. The reduction of pressure permits the expansion of the entrapped water, ail, and gas. Gas. having the greater coefficient of expansion, expels oil and water from the core. Thus, the content of the core at the surface has been changed from that which existed in the formation. Because the invasion

PROPERTIES OF RESERVOIR ROCKS 26-2 1

of the filtrate precedes the core bit, it is not possible to use pressurized core barrels to obtain undisturbed samples.

Drill cuttings, chips, or cores from cable-tool drilling also have undergone definite physical changes. If little or no fluid is maintained in the wellbore, the formation adjacent to the well surface is depleted because of pressure reduction. As chips are formed in the well, they may or may not be invaded, depending on the fluids in the wellbore and the physical properties of the rock. In all probability, fluid will permeate this depleted sample, resulting in flushing. Thus, even cable-tool cores undergo the same two processes as was noted in the case of rotary coring, although in reverse order.

Sidewall cores from either rotary- or cable-tool-drilled holes are subjected to these same processes.

In an attempt to understand better the overall effect of the physical changes that occur in the core because of flushing and fluid expansion, Kennedy et al. 28 under- took a study to simulate rotary-coring techniques. The effects of both invasion and expansion because of pressure reduction were measured.

Schematics of the changes in saturation resulting from these two processes for oil- and water-based muds are shown in Fig. 26.29. For the water-based mud, the original displacing action of the water filtrate reduced the oil saturation by approximately 14%. The expansion to surface pressure displaced water and additional oil. The final water saturation was greater than the water saturation before coring. With oil-based mud, wherein the filtrate is oil, the displacing action did not alter the initial water saturations but did result in replacement of approximately 20% of the initial oil. On pressure depletion, a small fraction of the water was expelled, reducing the water saturation from 49.1 to 47.7 % The oil saturation was reduced by both processes from 50.9 to 26.7%. Thus, even when high water saturations are involved, up to approximately 50%, the water-saturation values obtained with oil-based muds may be considered to be representative of the initial water saturations in the reservoir. Hence, it is possible to obtain fairly representative values of in-place water saturations by using oil-based muds.

Attempts have been made to use tracers in the drilling fluid to determine the amount of water in the core that is caused by mud-filtrate invasion. The theory was that mud filtrate displaced only oil. Thus. upon recovering the core to the surface, the salt concentration of the core water could be determined. Thus, if the salt concentration in the reservoir water and the tracer concentration in the drilling fluid were known, the volume of filtrate and reservoir water in the core could be calculated. Because a large fraction of the initial reservoir water may have been displaced by the invading filtrate, the tracer method results in incorrect values for reservoir-water saturation.

To obtain realistic values of fluid saturation, it is necessary to choose the proper drilling fluid or to use indirect methods of saturation determination.

Determination of Fluid Saturations from Rock Samples

One of the most popular means of measuring fluid saturations of cores is the retort method. This method takes a small rock sample and heats the sample to

SATURATION AFTER MUD FLUSHING BEFORE PRE :SSURE REDUC TION

RESIDUAL SbTUkATlON SATURATION

WATER BASE MUD (a)

ORIGINAL SATURATION

T---l

SATURATION AFTER

RESIDUAL SATURATION

MUD FLUSHING BEFORE PRESSURE REDUCTION

OIL BASE MUD (b)

Fig. 26.29-Typical changes in saturations of cores flushed with water-based and oil-based muds.

vaporize the water and the oil, which is condensed and collected in a small receiving vessel. The retort method has several disadvantages as far as commercial work is concerned. The water of crystallization within the rock is driven off, causing the water-recovery values to be too high. The second error that occurs from retorting samples is that the oil, when heated to high temperatures, has a tendency to crack and coke. This change of hydrocarbon molecules tends to decrease the liquid volume. The fluid wetting characteristics of the sample surface may be altered during the retorting process as a result of the two previous factors. Before retorts can be used, calibration curves must be prepared for water and oils of various gravities to correct for losses and other errors. These curves can be obtained by run- ning “blank” runs (retorting known volumes of fluids of known properties). The retort is a rapid method for determining fluid saturations and, if the corrections are used, yields satisfactory results. It gives both water and oil volumes such that the oil and water saturations can be calculated from the following equations.

S&, “P

S, +,

P


Fig. 26.30—Laboratory layout for performlng routine core analysis

and

S,=l-S,-S,,

wheres, =s, =s, =v, =VP =

v, =

water saturation,oil saturation,gas saturation,

The otherextraction

water volume, cm3,pore volume, cm3, andoil volume, cm3.

method of determining fluid saturation is bywith a solvent. Extraction may be accom-

plished by a modified ASTM distillation method or acentrifuge method. In the standard distillation test, thecore is placed such that a vapor of either toluene, pen-tane, octane, or naphtha rises through the core. Thisprocess leaches out the oil and water in the core. Thewater and extracting fluid are condensed and collected ina graduated receiving tube. The water settles to the bot-tom of the receiving tube because of its greater density,and the extracting fluid refluxes over the core and intothe main heating vessel. The process is continued untilno more water is collected in the receiving tube. Thewater saturation may be determined directly by

S,,=v,;.VP

The oil saturation is an indirect determination. The oilsaturation as a fraction of PV is given by

so= w,, - w,, - w,VP-P* )

whereW‘II’ = weight of wet core, g,W td = weight of dry core, g,W, = weight of water, g,

VP = PV, cm3, andPO = density of oil, g/cm3.

The gas saturation is obtained in the same manner as theretort.

Another method of determining water saturation is touse a centrifuge. A solvent is injected into the centrifugejust off center. Because of centrifugal force, it is thrownto the outer radii and forced to pass through the coresample. The outflow fluid is trapped and the quantity ofwater in the core is determined. The use of the centrifugeprovides a very rapid method because of the high forcesthat can be applied. In both extraction methods, at thesame time that the water content is determined, the coreis cleaned in preparation for the other measurementssuch as porosity and permeability.

There is another procedure for saturation determina-tion that is used in conjunction with either of the extrac-tion methods. The core as received from the well isplaced in a modified mercury porosimeter in which theBV and gas volume are measured. The volume of wateris determined by one of the extraction methods. Thefluid saturations can be calculated from these data.

In connection with all procedures for determination offluid content, a value of PV must be established in orderthat fluid saturations may be expressed as percent of PV.Any of the porosity procedures previously described maybe used. Also, the BV and gas volume determined fromthe mercury porosimeter may be combined with the oiland water volumes obtained from the retort to calculatePV, porosity, and fluid saturations.

Porosity, permeability, and fluid-saturation determina-tions are the measurements commonly reported inroutine core analysis. A laboratory equipped for suchdeterminations is shown in Fig. 26.30.

Interstitial Water SaturationsEssentially, three methods are available to the reservoirengineer for the determination of interstitial water satura-tions. These methods are (1) determination from corescut with oil-based muds, (2) determination fromcapillary-pressure data, and (3) calculation from electric-log analysis (see Chap. 49).

Oil-Based Mud. The obtaining of water saturations byusing oil-based muds has been discussed. A correlationbetween water saturation and air permeability for coresobtained with oil-based muds is shown in Fig. 26.31.29


A general trend of increasing water saturation with decreasing permeability is indicated. It is accepted from field and experimental evidence that the water content determined from cores cut with oil-based mud reflects closely the water saturation as it exists in a reservoir, ex- cept in transition zones where some of the interstitial water is replaced by filtrate or displaced by gas expansion.

Fig. 26.32 shows permeability/interstitial-water relationships reported in the literature for a number of fields and areas. There is no general correlation applicable to all fields; however, an approximately linear correlation between interstitial water and the logarithm of permeability exists for each individual field. The general trend of the correlation is decreasing interstitial water with increasing permeability.

Capillary Pressure. Capillary pressure may be thought of as a force per unit area resulting from the interaction of surface forces and the geometry of the medium in which they exist. Capillary pressure for a capillary tube is defined in terms of the inter-facial tension between the fluids, a. the angle of contact of the interface of these two fluids and the tube. 0(., and the radius of the tube, VI.

This relationship is expressed in Eq. 44.

P,. = 20 cos 8,.

, ,,..,..................... (44) rt

where the angle 0,. is measured through the more dense fluid.

In a packing of spheres, the capillary pressure is expressed in terms of any two perpendicular radii of cur-

2 2 4 6 4 6 II t tic: t t ttn I 20 40 60 100 200 400

AIR PERMEABILITY, MILLIDARCYS

Fig. 26.31--Relation of air permeability to the water content of the South Coles Levee cores.

vature (these radii touch at only one point), r, and r7 _ and the interfacial tension of the fluids. This relationship is given in Eq. 4.5.

(45)

Comparing Eq. 4.5 with the equation for capillary pressure as determined by the capillary-tube method, it is found that the mean radius U is defined by

I 1 I 2 cos 0‘ _=-+-=-.--.-------' r rt r2 rf

rir2 rf y=-=-.---

rl +r2

2cos*,.. . . . . . . . . . . . . . . (46)

0 IO 20 30 40 50 60 70 80 90 100

LEGEND

I = HAWKINS 2: MAGNOLIA 3= WASHINGTON 4=ELK BASIN 5= RANGELY 6: CREOLE 7= SYNTHETIC ALUNDUM 8: LAKE ST. JOHN

C 9= LOUISIANA GULF COAST MIOCENE AGE-WELL A

IO=DITTO-WELLS BAND C

fil = NORTH BELRIDGE-CALIFORNIA

CORE ANALYSIS DATA

% INTERSTITIAL WATER

Fig. 26.32-Interstitial water vs. permeability relationships.


MEASURING APPARATUS

Fig. 26.33-Schematic of porous-diaphragm method of capillary pressure.

It is practically impossible to measure the values of rl and t-2 ; hence, they generally are referred to by the mean radius of curvature and empirically determined from other measurements on a porous medium.

The distribution of the liquid in a porous system depends on the wetting characteristics. It is necessary to determine which is the wetting fluid so as to ascertain which fluid occupies the small pore spaces. From packings of spheres, the wetting-phase distribution within a porous system has been described as either funicular or pendular in nature. In funicular distribution, the wetting phase is continuous, completely covering the surface of the solid. The pendular ring is a state of saturation in which the wetting phase is not continuous and the nonwetting phase is in contact with some of the solid surface. The wetting phase occupies the smaller interstices. As the wetting-phase saturation progresses from the funicular to the pendular-ring distribution, the volume of the wetting phase decreases and the mean radius of curvature or the values of rl and r2 tend to decrease in magnitude. Referring to Eq. 46, we see that if r I and r2 decrease in size, the magnitude of the capillary pressure would have to increase in value. Since r , and r2 can be related to the wetting-phase saturation, it is possible to express the capillary pressure as a function of fluid saturation when two immiscible phases are within the porous matrix.

Laboratory Measurements of Capillary Pressure

Essentially, five methods of measuring capillary pressure on small core samples are used. These methods are (1) the desatumtion or displacement process, through a porous diaphragm or membrane (restored-state method of Welge3’), (2) the mercury-injection method, (3) the centrifuge or centrifugal method, (4) the dynamic- capillary-pressure method, and (5) the evaporation method. *

Porous Diaphragm. The first of these, illustrated in Fig. 26.33, is the displacement or diaphragm method. The essential requirement of the diaphragm method is a

‘Since the method IS seldom used today, it WIII not be dwzussed The procedure con- slsts ot continuously monitoring the decrease m weight caused by evaporabon of a core sample lnitlally 100% saturated wlh a wetltng fluld. See Messner, E S : “Intetsbtial Water Determmation By An Evaporatmn Method.” Trans, AIME (1951) 192,269-74

permeable membrane of uniform pore-size distribution containing pores of such size that the selected displacing fluid will not penetrate the diaphragm when the pressures applied to the displacing phase are below some selected maximum pressure of investigation. Various materials including fritted glass, porcelain, and cellophane have been used successfully as diaphragms. Pressure applied to the assembly is increased by small increments. The core is allowed to approach a state of static equilibrium at each pressure level. The saturation of the core is calculated at each point defining the capillary-pressure curve. Any combination of fluids may be used: gas, oil, and/or water. Although most determinations of capillary pressure by the diaphragm method are drainage tests, by suitable modifications imbibition curves similar to Leverett’s may be obtained.

Mercury Injection. The mercury-capillary-pressure apparatus was developed to accelerate the determination of the capillary-pressure/saturation relationship. Mercury is normally a nonwetting fluid. The core sample is inserted in the mercury chamber and evacuated. Mercury is forced into the core under pressure. The volume of mercury injected at each pressure determines the nonwetting-phase saturation. This procedure is continued until the core sample is filled with mercury or the injection pressure reaches some predetermined value. Two important advantages are gained by this method: (1) the time for determination is reduced to a few minutes, and (2) the range of pressure investigation is increased because the limitation of the diaphragm’s properties is removed. Disadvantages are the difference in wetting properties and permanent loss of the core sample.

Centrifuge Method. A third method for determining capillary properties of reservoir rocks is the centrifuge method.3’ The high accelerations in the centrifuge increase the field of force on the fluids, in effect subjecting the core to an increased gravitational force. By rotating the sample at various constant speeds, a complete capillary-pressure curve may be obtained. The speed of rotation is converted into force units in the center of the core sample, and the fluid saturation is read visually by the operator. The advantage of the method is the increased speed of obtaining the data. A complete curve may be established in a few hours, while the diaphragm method requires days.

Dynamic Method. Brown3* reported the results of determining capillary-pressure/saturation curves by a dynamic method. Simultaneous steady-state flow of two fluids is established in the core. By the use of special wetted disks that permitted hydraulic pressure transmis- sion of only the selected fluid phase, the difference in the resulting measured pressures of the two fluids in the core is the capillary pressure. The saturation is varied by regulating the quantity of each fluid entering the core. Thus, it is possible to obtain a complete capillary- pressure curve.

Comparison of Methods of Measurement

Intuitively, it appears that the diaphragm method (restored state) is superior in that oil and water are used; therefore, actual wetting conditions are more nearly ap-

PROPERTIES OF RESERVOIR ROCKS

C#I = fractional porosity.

proached. Hence, the diaphragm method is used as the standard to which all other methods are compared. The mercury-injection test data must be corrected for wetting conditions before they can be compared with results from the restored-state method. If it is assumed that the mean curvature of an interface in rock is a unique function of fluid saturation, then the ratio of mercury to water capillary-pressure data is given by

+-+f+6.57. . . . . (47) ‘M’ 0,.

where urn is the surface tension of mercury and u, is the surface tension of water. Experimentation has shown the ratio to vary between 5.8 (for limestones) and 7.5 (for sandstones). Thus, no conversion factor can be defined that will apply to all rocks.

Good agreement of centrifuge data with those from the diaphragm method was reported by Slobod. 31 Unlike the mercury-injection method, there is no need of conversion factors to correct for wetting properties. The same fluids are used in the centrifugal and diaphragm methods.

Excellent correlation was obtained by Brown3’ between the diaphragm and dynamic methods. The dynamic data were obtained by simultaneous steady flow of oil and gas through the porous sample at a predetermined level of pressure difference between the fluids. Care was taken to maintain uniform saturations throughout the core as well as to conduct the test such that a close correspondence to drainage conditions existed.

If capillary-pressure data are to be used for determining fluid saturations, the values obtained should be comparable with those of other methods. Water distributions, as determined from electric logs, and capillary-pressure data are normally in good agreement. A comparison of these methods is shown in Fig. 26.34. 33 Shown also is the approximate position of the gas/oil contact as determined from other test data. In the gas-bearing portion of the formation there is no significant variation in water saturation with depth or method of determination. However, in a thin oil zone, such as that shown in Fig. 26.34. there is a significant variation in the water saturation with depth. Variations in water saturations with depth within an oil zone must be taken into account to determine accurately average reservoir interstitial-water saturations.

Water Saturation from Capillary-Pressure Data. In oilfield terms, the capillary pressure may be stated as

P,.=+(p, -p2), ....................... (48)

where h is in feet and p, and p2 are the densities of Fluids 1 and 2, respectively, in lbmicu ft at the conditions of the capillary pressure.

Converting Laboratory Data. To use laboratory capillary-pressure data, it is necessary to convert them to reservoir conditions. Laboratory data are obtained with a

DEPTH FEET BELOW SEA-LEVEL

Fig. 26.34-Comparison of water saturation from capillary pressure and electric log.

gas/water or an oil/water system that normally does not have the same physical properties as the reservoir water, oil, and gas.

Essentially two techniques, differing only in the initial assumptions, are available for correcting laboratory capillary-pressure data to reservoir conditions.

p,..R= uw><, cos 0,.,,

uwg cos 0,., p,., L

or

P,..R=~P& . . . . . . (49) (JL

where fJ wo = interfacial tension water/oil,

UWR = interfacial tension water/gas, Ocwo = water/oil contact angle, e,.,, = water/gas contact angle, subscript

R = reservoir conditions, and subscript

L = laboratory conditions.

Since the interfacial tensions enter as a ratio, pressure in any consistent units may be used together with the interfacial tension in dynes/cm.

Averaging Capillary-Pressure Data. Two methods have been proposed for correlating capillary-pressure data of similar geologic formations. The first correlating procedure is a dimensionless grouping of the physical properties of the rock and the saturating fluids. This function is called a J function34 and is expressed as

, ...................... .(SO)

where S,,, = water saturation, fraction of PV, P,. = capillary pressure, dyne/cm*,

u = interfacial tension, dyne/cm, k = permeability, cm*, and


90

81 z

72 I+ 2

63 3 g

54 a’ i: 3;

45 2 ,o

732 36 gs

2, 2 2 3- &

I8 z

3

20 30 40 50 60 70 80 90 IO8 WATER SATURATION,%

RESERVOIR FLUID DISTRIBUTION CURVES

Fig. 26.35~Series of capillary-pressure curves as a function of permeability.

Some authors alter the above expression by including the cos f3(. (where 8,. is the contact angle) as follows.

The J function originally was proposed as a means of converting all capillary-pressure data to a universal curve. There exist significant differences in correlation of the J function with water saturation from formation to formation such that no universal curve may be obtained, but the / function may be used to correlate the data from one formation.

The second method of evaluating capillary-pressure data is to analyze a number of representative samples and treat the data statistically to derive correlations that, together with the porosity and permeability distribution data, may be used to compute the interstitial-water saturations for a field. A first approximation for the correlation of capillary-pressure data is to plot water saturation against the logarithm of permeability for constant values of capillary pressure. A straight line may be fitted to the data for each value of capillary pressure, and average-capillary-pressure curves may be computed from permeability-distribution data for the field. The resulting straight-line equation takes the general form of

Fluid-distribution curves are reported for several values of permeability, ranging from 10 to 900 md in Fig. 26.35.” These data also may be considered to be capillary-pressure curves. The ordinate on the right reflects values of capillary pressure determined by displacing water with air in the laboratory. The ordinates on the left include the corresponding oil/water capillary pressure that would exist at reservoir conditions and the fluid distribution with height above the free-water surface.

The results of the statistical correlation previously discussed applied to the capillary-pressure data presented in Fig. 26.35 are shown in Fig. 26.36. The reader should note the linearity of the curves for each value of capillary pressure and the tendency of all capillary-pressure curves to converge at high permeability values. This behavior is what normally would be expected because of the larger capillaries associated with high permeabilities.

To convert capillary-pressure saturation data to height saturation, it is necessary only to rearrange the terms in Eq. 48 so as to solve for the height instead of the capillary pressure-i.e.,

P,. x 144 h,,=-, . . . . . . . . (52)

PW -PO

where hh = height above the free-water surface, ft, P II’ = density of water at reservoir conditions,

lbm/cu ft, PO = density of oil at reservoir conditions,

lbm/cu ft, and P,. = capillary pressure at some particular satura-

tion for reservoir conditions (it must be converted from laboratory data first, psi).

Example Problem 9-Calculation of Saturation Plane From Laboratory Capillary-Pressure Data. If

P<.,L = 18 psi for S,.=O.35, CJ 11’0 = 24 dynes. P M’ = 68 lbmlcu ft,

is “‘8 = 72 dynes, and PO = 53 lbm/cu ft,

then, from Eq. 49,

P(.,R = 18(24/72)= 18/3=6 psi,

and

Pc,R x 144 6 x 144 h= =-=58 ft.

P II’ -PO 68-53

Thus, a water saturation of 35 % exists at a height of 58 ft above the free-water surface.

To calculate the fluid saturation in the gas zone, it is necessary to consider all three phases: oil, water, and gas. If all three phases are continuous, it can be shown that

where PC., “‘R = capillary pressure at a given height above

the free-water surface determined by using water and gas,

PC, H%) = capillary pressure at a given height above the free-water surface, using water and oil, and

pC.O,q = capillary pressure at a given height above the free-oil surface, using oil and gas.


If the wetting phase becomes discontinuous, then the wetting-phase saturation takes on a minimum value, and, at all heights above the point of discontinuity, the wetting-phase saturation cannot be less than this minimum value. It is then possible to calculate the fluid saturations above the free-oil surface by the following relations.

1. S,. at h is calculated using oil and water as the continuous phases.

2. SL at h is calculated using oil and gas as the continuous phases and height denoted by the free-oil surface.

3. S,=l-SL andS,=SL-S,, where SL is the total liquid saturation, oil plus water, fraction.

Example Problem lo-Calculation of Water and Oil Saturation in Gas Zone From Capillary-Pressure Data. Let oil-zone thickness, h,, equal 70 ft and

u “‘R = 72 dynes, (JOf = 50 dynes, (J I,‘0 = 25 dynes,

P II = 68 lbm/cu ft

Pn = 7 Ibm/cu ft, and PO = 53 lbm/cu ft.

From Fig. 26.28 for a 900-md sample, let PI,,L = 54 psi by the method illustrated in

Example Problem 9, Pc,R = 18 psi,

h fi = height above free-water level= 120 ft, and S,,. = 16% at a height of 70 ft or greater (read

from curve).

Since the oil zone is only 70 ft thick, the height of 120 ft above the free-water surface must be at least 50 ft into the gas-saturated zone. The first step is to calculate the total fluid saturation, SL, using gas and oil as the continuous phases.

h,,,=h,,-h,=120-70=50 ft;

hj, Pc,.R = IJ~(PI, -P,h

where h&, is the height above free-oil level, ft.

P<,.R = 50/144x(53-7)=501144x46. P<,.R = 15.97,

and

P,.,[, = Pr-,R~=15.97~72!50=23 psi. 0 og

From Fig. 26.35, for a laboratory capillary pressure of 23 psi and a permeability of 900 md, the total wetting saturation, SL, equals 18 % Therefore.

S,,=SL-S,,,=18-16=2X

)- IOOC

700

500

400

300

__ u-l $ 200 Q: 8

3

2 IOC

z 2 7c s 2 50

% 40

3c

2c

I-

)-

I-

I-

IO- 0

-

- -

-

-

-

- -

-

-

- 20 40 60 80 WATER SATURATION (%)

Fig. 26.36-Correlation of water saturation with permeability for various capillary pressures.

and

It must be understood that the relationships used in Ex- ample Problem 10 for calculating the fluid saturations in the gas zone were based on continuity of all three phases. Since this is not normally the case, it might be expected that saturations somewhat different from the calculated values exist. Because the capillary pressure for a discontinuous phase could vary from pore to pore, it is impossible to ascertain the exact relationships that should exist. Hence, the preceding method of calculating fluid distributions is not exact but is usually as accurate as the data available for making the computation.

Electrical Conductivity of Fluid-Saturated Rocks Porous rocks comprise an aggregate of minerals, rock fragments, and void space. The solids, with the excep- tion of certain clay minerals, are nonconductors of elec- tricity. The electrical properties of a rock depend on the geometry of the voids and the fluids that fill the voids. The fluids of interest in petroleum reservoirs are oil, gas, and water. Oil and gas are nonconductors. Water is a


Fig. 26.37-Core-sample resistivity cell

conductor when it contains dissolved salts. Current is conducted in water by movement of ions and therefore may be termed electrolytic conduction. The resistivity of a material is the reciprocal of conductivity and commonly is used to define the ability of a material to conduct current. The resistivity of a material is defined by the following equation.

p=*A. ._.,.,.,,...,...,,.,,_,....,,.,, (53) L

where p = resistivity. I’ = resistance,

A = cross-sectional area of the conductor, and L = length of the conductor,

For electrolytes, p is commonly reported in Q-cm, and r is expressed in ohms, A in cm’, and L in cm. In the study of the resistivity of soils and rocks. it has been found that the resistivity may be expressed more conve- niently in n-m. To convert to 9-m from O-cm, divide the resistivity in Q-cm by 100. In oilfield practice. the resistivity in Q-m commonly is represented by the sym- bol R with an appropriate subscript to define the condition to which R applies.

Fundamental Concepts

The definition of electrical formation resistivity factor is perhaps the most fundamental concept in considering electrical properties of rocks. The formation resistivity factor as defined by Archie” is

F/$5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(54) R \I’

where R. is the resistivity of the rock when saturated with water having a rcsistivity of R,,

The second fundamental notion of electrical properties of porous rock is the resistivity index, IK, which is defined as

(55)

where R, is the true resistivity of the rock system at some particular value of water saturation and Ra is as previously defined.

Three idealized representations have been introduced in the literature from which the formation resistivity factor, FR, and the resistivity index , f~, have been related to the porosity, 4, and the rock tortuosity, 7.

From Wyllie’s2’ analysis the relations are

F& 4

and

IR =7(,--r-, SW

F,=d- 4

and

where T,, is the effective rock tortuosity at some water saturation.

Cornell and Katz3’ presented an analysis of a slightly different model. The relationships developed are as follows.

JR=&& se

Wyllie and Gardner 37 later presented an analysis based on a probability theory from which the following relationships were obtained.

FK=; 4’

and

I IR=--

s,,.’

From the analysis of the electrical properties of the foregoing idealized pore models, general relationships between electrical properties and other physical properties of the rock may be deduced. The formation resistivity factor has been shown to be some function of the porosity and the internal geometry of the rock system. In particular, the formation resistivity factor may be expressed in the following form.

FK=I@ -“I, (56)

where K is some function of the tortuosity and VI is a function of the number of reductions in pore-opening sizes or closed channels. It is suggested that K should be I or greater. The value of rn has been shown from theory to range from I to 2.


Both the formation resistivity factor, F,, and the resistivity index. IR, depend on ratios of path length or tortuosities. Therefore, to compute the formation resistivity factor or resistivity index from the relations derived from models, it is necessary to determine the electrical tortuosity. Since direct measurement of the path length is impossible, reliance has been placed primarily on empirical correlations based on laboratory measurements. Winsauer et al. 3X devised a method of determining tortuosity by transit time of ions flowing through a rock under a potential difference. The data obtained were correlated with the product FR+. The resulting correlation of formation resistivity factor, porosity, and tortuosity is given in Eq. 57.

1.67

=F,&

where L, is the actual length of flow path, and L is the length of the sample, or

T’.~“? =F&. . . . . . .(57)

The deviation from the theory is believed to be an indica- tion of the greater complexity of the actual pore system than that of the models on which the theory was based.

Archie suggested that the formation resistivity factor could be correlated with porosity and have the form

FR=c#-“‘, . . . ., ._.. . . .(58)

where 4 is the fractional porosity and m is the cementation factor. Archie further reported that the cementation factor probably ranged from 1.8 to 2.0 for consolidated sandstones and, for clean unconsolidated sands, was about 1.3.

Measurement of Electrical Resistivity of Rocks

Laboratory measurements of electrical properties of rocks have been made with a variety of devices. The measurements require a knowledge of the dimensions of the rock, the fluid saturation of the rock, the resistivity of the water contained in the rock, and a suitable resistivity cell in which to test the samples. A simple cell is shown in Fig. 26.37. 39 A sample cut to suitable size is placed in the cell and clamped between electrodes A. Current is then passed through the sample and the potential drop observed. The resistance of the sample is computed from Ohm’s law,

E r=-.

I

and R (the resistivity) is computed from

R,‘A L’

where A is the cross-sectional area of the sample and L is the length of the sample. The saturation conditions of the test may be established at known values before measurement or determined by an extraction procedure after measurement.

$ IO II: 0 k-

6

4 1 HUMBLE RELATION F= 5 ,t-j&tt-

-- 5 60 5 2 40

zi

5 20

F TIXIER S RELATION FOR UNCONSOLIDATED

2p-M CONSOLIDATED FORMATIONS 1,

2 4 6 8 IO 20 40 60 80 POROSITY, =/a

Fig. 26.38-Comparison of various formation resistivity factor correlations.

Empirical Correlation of Electrical Properties Archie, as previously mentioned, reported the results of correlating laboratory measurements of formation factor with porosity. He expressed his results in the form FRz4-m.

Winsauer et al. ” reported a similar relationship based on correlations of data from a large number of sandstone cores. This equation, commonly referred to as the Hum- ble relation, is

FR=0.62q!-2.‘5. . .(59)

In discussing the theory of the formation resistivity factor, it was stated that K should be greater than 1 and that m should be 2 or less. At this time, the discrepancy between theory and experiment must be attributed to the possible effect of conducting solids.

Improved correlations should result from considering other parameters, such as permeability, as variables in the relations.

A comparison of suggested relationships between porosity and the formation resistivity factor is shown in Fig. 26.38.


A COMPARABLE CLEAN SAND I

4 !,

1.0 IO 100

R,-WATER RESISTIVITY, OHM-METERS

Fig. 26.39-Effect of Interstitial clay on formation resistivity factors.

Effect of Conductive Solids

Investigations by Wyllie4’ Indicate that clays contribute substantially to the conductivity of a rock when the rock is saturated with a low-conductivity water. The effect of water resistivity on the formation resistivity factor for sands containing clay minerals is shown in Fig. 26.39. The formation resistivity factor for a comparable clean (clay-free) sand is a constant. The formation resistivity factor for the clayey sand increases with decreasing water resistivity and approaches a constant value at a

I -+ Rw

Fig. 26.40-Water-saturated rock conductivity plotted against water conductivity yields these measurements: (A) Suite 1 No. 40; (X) Suite 1 No. 21; (0) Suite 1 No. 4; (0) Suite 2 No. 13: (II!) Suite 6 No. 2.

water resistivity of about 0.1 Q-m. Wyllie proposed that the observed effect of clay minerals was similar to having two electric circuits in parallel-the conducting clay minerals and the water-filled pores. Thus,

Ro.\/z 1 1 1 FRO=- and -==++ FRR,,, \ . (60)

R, Roslr R,./ where

FRO = apparent formation resistivity factor, Rosh = resistivity of a shaly sand when 100%

saturated with water of resistivity R,,., R,., = resistivity caused by the clay minerals, R,,. = resistivity caused by the distributed water,

and FR = true formation resistivity factor of the rock

(i.e., the constant value of formation factor approached when the rock contains low-resistivity water).

The data presented in Fig. 26.40 represent graphically the confirmation of the relationship expressed in Eq. 60. The graphs were plotted by deWitte@ from data presented by Hill and Milburn.“3 The plots are linear and are of the general form

1 1 -=m-+b, . . . . . . . . . . . . . . . . . . . (61) Rosh R,i

where m is the slope of the line and h is the intercept. Comparing Eq. 60 with Eq. 61, note that m= ~/FR and b= l/R,.[. The curve labeled Suite 1 No. 40 indicates a clean sand because the line passed through the origin, thus having a zero intercept b= l/R,.,=O. Then l/Rosh =m(l/R,,)=(l/FRR,,), or Ro=F,R,,.. The remaining samples are from shaly sands, which have a finite conductivity of the clay minerals, as indicated by the intercepts of the lines. The linearity of the plots indicates that l/R,, is a constant independent of R,,. This phenomenon may be explained in terms of the ions absorbed on the clay. When the clay is hydrated, the absorbed ions form an ionic conducting path, which is closely bound to the clay. The number of absorbed ions is apparently little changed by the salt concentration of the interstitial water.

Eq. 60 may be rearranged to express the apparent formation resistivity factor in terms of R,., and FRR, .

K O”’ = R,. +R,.,/FR

and

FRY = &I

R,, +(R,.,IF,) ’

As R,,-+O,

RCl lim FRY =-

R,IIFR =FR.

Therefore, FRY approaches FR as a limit as R, becomes small. This behavior was observed in Fig. 26.39.


Hill and Milbum43 evaluated 450 samples from both sandstone and limestone formations. The formation resistivity factor was determined at a water resistivity of 0.01 R-m, a value at which the apparent formation resistivity factor, FRa, approaches the formation resistivity factor, FR. They designated the formation resistivity factor as FR,O,oI The data were fitted by the method of least squares to yield

F R.0,0,=1.4+-‘.78. . . . . ._ .(62)

This equation conforms to the theory previously discussed. They also fitted the data with K in Eq. 56 restricted to a value of 1. This yielded FR,P,o, =$ -’ 93, which corresponds closely to Archie’s onginal expression F=$ -‘,

In summary, Eqs. 58 (with m=2.0) and 59 have been used widely to represent the relation between formation resistivity factor and porosity. Both equations yield results satisfactory for most engineering purposes. However, we propose that Eq. 62 be considered as more valid because the data were taken to minimize the effect of clays. The selection of a particular relation should be based on independent observations on the formations or formations of interest in a given geologic province.

Resistivity of Partially Water-Saturated Rocks

A rock containing both water and hydrocarbon has a higher resistivity than when fully saturated with water. The resistivity of partially water-saturated rocks has been shown to be a function of the water saturation, S,,. . From theoretical developments, the following generalization may be drawn.

IR=K’S,,/‘, . . . (63)

where 1~ = R,/Ro, the resistivity index; K’ is some function of tortuosity; and n is the saturation exponent.

Archie compiled and correlated experimental data from various sources from which he suggested that the data could be represented by

IR =S,,. G. . . . (64)

Wyllie confirmed the suggested relationship for clean sands but found that the presence of clays [conductive solids) altered the relationship. A comparison of Ar- chic’s relationship and that for a core containing conductive solids is shown in Fig. 26.41. The change in the relationship depends on both the amount of clays and the water resistivity. Therefore. a general correlation for sands containing conductive solids is not available, although dewitte”’ has proposed a method of using Eq. 64 for evaluation of shaly sands.

Use of Electrical Parameters in Characterizing Porous Media

In the section on permeability, the Kozeny equation was developed as follows.

4 k=- F,A,‘7, . . . ..~~~~.... (65)

loo \

60 \

40 I \ I

z 20 2- WITHOUT CONDUCTIVE k ; IO m 56 ii

4

SOLIDS 2

I.01 .02 .04 08 .20 40 .80 WATER SATURATION

Fig. 26.41-Effect of conductive solids on the reslstivity-index- vs.-saturation relationship in Stevens sandstone core.

where k = permeability, C#I = porosity, fraction,

F, = shape factor, A,v = internal surface area/unit PV, and

r = Kozeny tortuosity.

7 has been shown to be a function of FR+i.e., 7= (FR#)-‘, where x ranges between 1 and 2. If the internal surface area is expressed in terms of the mean hydraulic radius, rH, by

A,& . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(66) rH

the general form of the relationship may be stated as

or

@H) 2 k

=F. (FR)X & . . . . . . . . . . . . . . . . . .._. (67) ~

Eq. 67 perhaps provides an improved basis for the correlation of characteristic physical data of porous media with the electrical formation resistivity factor.

The electrical properties of porous rocks, as discussed in the foregoing sections, form the basis of quantitative evaluation of electrical-logging records. In particular, the Humble relation (Eq. 59) has been used widely by service companies in estimating porosity from measurements made with “contact” resistivity devices such as the Microlog.‘” Eq. 64 forms the basis of inter- pretation of water saturations from deeper-penetration


resistivity devices such as the conventional resistivity curves of the standard electrical log or from the various “focused” electrical-resistivity devices.

Improvements in the statistical correlations of electrical properties of rocks will improve the results of such analyses.

Nomenclature A = cross-sectional area

A., = area of flow represented by a bundle of tubes of permeability k,

A, = internal surface area/unit PV h = constant for a given gas in a given medium c = fluid compressibility C = flow coefficient d = diameter E = voltage drop f = friction factor

F, = shape factor FR = formation resistivity factor

F~ri = apparent formation resistivity factor K = acceleration of gravity h = driving head

I? to = height above free-oil level IT,,~ = height above free-water level

/f, = thickness ofjth layer I = current

IH = resistivity index J = J function, Eq. 48 k = permeability of the medium

k,, = permeability to air k,y = permeability of the medium to a gas com-

pletely filling the pores of the medium kl, = thermal conductivity k, = permeability of one capillary tube kL = permeability of the medium to a single

liquid phase completely filling the pores of the medium

k,,. = permeability to fresh water km = permeability to 500-grain/gal chloride

solution K = constant of proportionality, Eq. 7

K’ = some function of the tortuosity. Eq. 63 K; = Kozeny’s constant

f. = length of flow path or length of the sample L,, = actual length of the flow path L.; = length ofjth layer UI = slope of the curve or cementation factor

n/ - ~ number of tubes of radius r, p = mean flowing pressure

A~YI = pressure loss over length L ~1~ = base pressure

Pe = pressure at the external boundary p,\. = pressure at the wellborc P,. = capillary pressure

Pc.,l, = capillary pressure. laboratory conditions P,.,,, = capillary pressure of mercury

pc~.i,,y = capillary pressure at height above free-oil surface, using oil and gas

P c .R = capillary pressure, reservoir conditions P,.,,. = capillary pressure of water

P c I,‘(, = capillary pressure at a given height above the free-water surface determined by using water and gas

P,,,,,, = capillary pressure at a given height above the free-water surface, using water and oil

q/, = volume rate of flow at the base pressure 40 = volume rate of flow of oil

q/J = volume rate of flow at the algebraic mean pressure (p, +p,, )/2

qr = total flow rate Q = rate of heat flow r = radius or resistance Y = mean radius

r, = radius at the external boundary TH = hydraulic radius r 1,’ = radius of the wellbore

R,., = resistivity caused by the clay minerals Rn = resistivity of the rock when saturated with

water having a resistivity of R,, Ro,I, = resistivity of a shaly sand when 100%

saturated with water of resistivity R,, s = distance in direction of flow, always

positive lTs = gas saturation SL = total wetting saturation S,, = oil saturation S,,. = water saturation T,f = average flowing temperature AT = temperature drop u., = volume flux across a unit area of the porous

medium in unit time along flow path s UY = volume flux across a unit area of the porous

medium in unit time along flow path x v = fluid velocity

Vh = BV V,, = oil volume VP = PV V, = sand-grain volume

V,,. = water volume w = mass rate of flow

WC.I, = weight of dry core W,.,,. = weight of wet core W,,. = weight of water

z = vertical coordinate, considered positive downward

0 = angle between s and the horizontal 19,. = angle of contact of the interface of two

tluids and the capillary tube Oc.,,,p = water/gas contact angle fI,.,,i, = water/oil contact angle

p = fluid viscosity p = fluid density or resistivity

~1, = fluid density at base pressure P 0 = oil density

u = IFT or conductivity ~I11 = surface tension of mercury


surface tension of water water/gas IFT water/oil IFT Kozeny tortuosity effective rock tortuosity fractional porosity effective porosity potential function

Key Equations in SI Metric Units

q=14.696x IO" k4p I -I+)

, . FLL

y=92.349x IO” MP, -P ,,.I c1 ]n r,lr,, ,

T/,kA(p, 2 -pz ‘) qb=23.1454x104 _

TfW<&P/, , .

4)) = I .4554 x 106 T/,vWp, * -P,, *)

ii.fzj~~pj, In rclr,, ’ ” ” ”

(26)

(28)

where q is in m”/d, k is in pm2, A is in m2, p is in kPa, p is in Pa’s, L is in m, I7 is in m, r is in m, and T is in “C.

References I.

2.

3 4:

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26 properties of reservoir rock

Documents

effec porosity

8r3 x100

isgivenby wherel

oneor cementation

isthe bulkvolume

haveevenmorecom ofthedeposit

lowerfriob wholecore

r 413irr3