Method-Dependent Values of 

Bulk, Grain, and Pore 

Volume as Related to 

Observed Porosity

GEOLOGICAL SURVEY BULLETIN 1203-D

Met hod-Dependent Values of 

Bulk, Grain, and Pore 

Volume as Related to 

Observed Porosity

By G. EDWARD MANGER

EXPERIMENTAL AND THEORETICAL GEOPHYSICS 

GEOLOGICAL SURVEY BULLETIN 1203-D

Includes formulated differences for total 

and apparent porosity

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1966
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

For sale by the Superintendent of Documents, U. S. Government Printing Office 

Washington, D.C. 20402 - Price 15 cents (paper cover)
CONTENTS

Page

Abstract__ _ ___--___--____-_--_--_________-____-__-_-_____-_-___- Dl

Introduction. _____________________________________________________ 1

Acknowledgments. __________________-__-_____________-_-___------- 2

General relations of bulk, pore, and grain volume to porosity__________ 2

Elements of porosity determination._____________________________ 2

Effect of bulk volume on porosity______________________________- 2

Relation of surface pores to method of bulk-volume measurement__ 3

Methods of bulk-volume measurement.__________________________ 3

Dimensioning.____________________________________________ 3

Water displacement______________________________________ 3

Mercury displacement________________________ ___________ 4

Pore volume and grain volume._________________________________ 5

Formulated porosity differences___________________________________ 6

Basic variables of porosity differences.___________________________ 6

Quantities of formulated porosity differences.___________________ 6

Relation between specimen grain volume and density and grainpowder volume and density________________________-___-_____- 7

Relation between apparent and total void (pore) volume. __ _ ______ 7

Relation between water-displacement and mercury-displacement bulk

volume___________________________________________________ 7

Relation of observed to formulated porosities.____________________ 8

Extension of certain definitions___________-_-___________-_-_-_-__ 10

Difference equations_________________________________________ 10

Significance of formulated porosity differences...____________________ 11

Changes in values of porosity difference with varying porosity_____ 11

Effect of using the same (unadjusted) pore volume for apparent porosity based on mercury-displacement and water-displacement bulk

volume_____________________________________________________ 11

Equality of total and apparent porosity._________________________ 13

Larger apparent-porosity difference than total-porosity difference. _ _ _ 13

Experimental illustration of the formulated porosity differences.________ 15

Excess of total over apparent porosity__________________________ 15

Larger difference in apparent porosity than in total-porosity values_ _ 17

The value of y. ___________,-_____-_____-___-____--_____-___- 17

Summary_ _ _____________________________________________________ 19

References__ _-________-______-_-___-__--_-----_______-_--__-_-_-_- 20

m
IV CONTENTS

ILLUSTRATION

FIGURE 1. Graph showing excess of total porosity of lapilli tuff over

apparent porosity, plotted against grain-powder density DIG

TABLES

Page 

TABLE 1. Porosity difference equations------------------------------ D12

2. Total and apparent porosity of lapilli tuff, subunit T, Paintbrush

Tuff, the Nevada Test Site, Nev. _____-________- 14

3. Porosity differences of lapilli tuff, subunit T, Paintbrush Tuff,

Nevada Test Site, Nev_______-__-___----- 15

4. Ratios of bulk volume based on dimensioning, water displacement, and mercury displacement.____-____-___-_-__--_--- 18
EXPERIMENTAL AND THEORETICAL GEOPHYSICS

METHOD-DEPENDENT VALUES OF BULK, GRAIN, AND 

PORE VOLUME AS RELATED TO OBSERVED POROSITY

By G. EDWAED MANGER

ABSTRACT

Observed values of porosity are derived from method-dependent values of 

bulk, grain, and pore volume. Formulated porosity differences show the relation 

of observed porosity to these method-dependent values. At very low porosity, 

differences in measured bulk volume, caused by method of measurement, result 

in differences in porosity values that are about equal to the excess of the greater 

over the lesser bulk volume, per unit bulk volume; thus at very low porosity, 

such porosity differences may be greater than the porosity. At very low porosity, 

apparent porosity, based on measured mercury-displacement bulk volume, can 

be forced to approximately equal apparent porosity, based on a greater measured 

water-displacement bulk volume, by not decreasing the volume of absorbed 

water (as a measure of pore volume) used for the mercury-displacement bulk 

volume according to the excess of the water-displacement over the mercury-displacement bulk volume. Omission of such a decrease, however, ignores the 

fact that pore volume is not determinable independently of bulk volume, 

and at higher porosity the omission results in greater excess of observed apparent 

over observed total porosity. With an increase in porosity there is a decrease 

in magnitude, as percent porosity, of the effect of method-dependent values 

of bulk volume. Experimental results show that an observed excess of total 

over apparent porosity for specimens of lapilli tuff from the Nevada Test Site 

is due largely to the fact that the density of the grain powder that is used for 

the total-porosity determination is not typical of the whole specimen. Of an 

observed excess of 0.96 percent total porosity, occluded pores constitute no more 

than 0.12 percent porosity and may be absent. Experimental results suggest 

that an excess of water-displacement over mercury-displacement bulk volume for 

some rocks may persist to low values of porosity.

INTRODUCTION

Observed total porosity and observed apparent (or effective) 

porosity of a porous medium depend upon the method used to measure 

bulk volume. An excess of total over apparent porosity is caused 

by occluded, or sealed-off, pores and by adsorption that may be 

occurring while grain-powder density is being determined to obtain 

total porosity. Additionally, certain experimental conditions, such

Dl
D2 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

as incomplete saturation of permeable pores during the test for 

apparent pore volume, may cause differences between observed total 

and apparent porosity.

Difference in values of total and apparent porosity, as a result of 

occluded pores, has been discussed in several reports, including those 

of Hartmann, Westmont, and Morgan (1926) and Fancher, Lewis, 

and Barnes (1933). Differences in observed porosity that result from 

method-dependent values of bulk volume have been considered in 

several reports, including those by Steinhoff and Mell (1924) and 

Hartmann (1926). Apparently, however, no study has been made of 

the relation of the combined effect of method-dependent determination of bulk, grain, and pore volumes to observed porosity. By 

means of a formulation of porosity differences, the present study 

considers the relation of observed porosity to a combination of such 

effects, including those of certain deficiencies of experimental procedure.

ACKNOWLEDGMENTS

The writer is indebted to Irwin Roman and Joel H. Swartz, geophysicists retired from the U.S. Geological Survey, who generously 

gave valuable suggestions and provided assistance in deriving the 

formulations in this report. Charles C. Hawley, of the Geological 

Survey, assisted in the experiments. The work on which this report 

is based was done by the Geological Survey on behalf of the Albuquerque Operations Office of the U.S. Atomic Energy Commission.

GENERAL RELATIONS OF BULK, PORE, AND GRAIN 

VOLUME TO POROSITY

ELEMENTS OF POROSITY DETERMINATION

Total porosity of a porous medium is a measure of its total void 

volume and is the excess of bulk volume over grain volume, per unit 

of bulk volume. Experimentally, total porosity is usually determined 

as the excess of grain density over bulk density, per unit of grain 

density. Apparent porosity, often called effective or net porosity, is 

a measure of the apparent void volume of a porous medium and is 

determined as the excess of bulk volume over grain volume and 

occluded-pore volume, per unit of bulk volume. The excess of bulk 

volume over grain volume and occluded-pore volume is determined 

by the fluid capacity of the permeable pores of the porous medium.

EFFECT OF BULK VOLUME ON POROSITY

Because pore volume is the excess of bulk volume over grain volume, 

values of pore volume vary with method-dependent values of bulk 

volume. If pore volume is considered to be constant, as is sometimes 

assumed, the effect of variation in values of bulk volume on values of
BULK, GRAIN, AND PORE VOLUME AS RELATED TO POROSITY D3

total and apparent porosity may be stated as follows: At constant 

grain volume (constant grain density), an increase in bulk volume 

(decrease in bulk density) results in an increase in total porosity; at 

constant pore volume (as, for example, volume of absorbed water), 

an increase in bulk volume results in a decrease in apparent porosity. 

At constant grain volume (constant grain density), a decrease in bulk 

volume (increase in bulk density) results hi a decrease hi total porosity; 

at constant pore volume (as, for example, volume of absorbed water), 

a decrease hi bulk volume results hi an increase hi apparent porosity.

RELATION OF SURFACE PORES TO METHOD OF BULK-VOLUME

MEASUREMENT

Variations hi bulk volume depend hi part upon the degree to which 

surface pores are included hi or excluded from the particular bulkvolume measurement. Surface pores, for lack of a better term, may 

be defined as those parts of pores which remain at the surface of a 

specimen after the grains have been sliced or abraded to shape the 

surface of the test specimen. The problem presented by surface pores 

is that, with a change hi method of bulk-volume determination, the 

same incremental volume of surface pores is added to or subtracted 

from both pore volume and bulk volume, resulting hi a change hi the 

ratio of pore volume to bulk volume, or correspondingly, hi a change 

hi porosity.

METHODS OF BULK-VOLUME MEASUREMENT 

DIMENSIONING

One can reason that the linear dimensioning of a geometrically 

shaped specimen should give the best value of bulk volume, provided that two conditions are met: (a) the specimen has a shape 

for which volume can be uniquely determined; (6) the shape is established by slicing cleanly through the rock without forming depressed surfaces. Such surfaces may be caused by plucking of 

grains or larger pieces of material from the specimen. Because in 

practice it is difficult to satisfy these conditions, the method of 

dimensioning is not generally used.

Dimensioned measurements of a specimen that is not exactly 

shaped can be expected to be too large, and hence the determined 

bulk volume would be too large. Likewise, dimensioning of a 

specimen from which surface grains have been plucked can be expected to show a bulk volume that is too large, because the surface 

reentrants that result from the plucking are included as bulk volume.

WATER DISPLACEMENT

In the water-displacement method a water-saturated specimen is 

freed of excess water by touching its surface with thoroughly moist
D4 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

absorptive paper; the specimen is then weighed in air. This weighing 

is immediately followed by immersing the specimen in water and 

weighing it while submerged. The difference hi weight of the saturated specimen in air and in water is the weight of the water displaced, 

from which the volume of the displaced water, and hence the specimen, 

is obtainable. Frequently saturation and displacement are obtained 

by using, instead of water, liquids with low surface tension that 

provide better wettability.

The water-displacement bulk volume may be too large or too small. 

Consider that a nonporous specimen is exactly shaped geometrically 

and the resulting surface is smooth but wettable. Presumably the 

true bulk volume would be obtainable by dimensioning. Consider 

now that the usual procedure is followed and an attempt is made to 

saturate the specimen before it is immersed in water to obtain its 

bulk volume. Because the surface of the specimen is considered to 

be wettable, the residual water film that adheres to the saturated 

specimen before immersion may be expected to result hi a bulk volume 

value that is too large.

This result might also be obtained for a very finely porous specimen 

or for a specimen whose surface appears macroscopically to be nonporous. On porous specimens, however, the surface water film tends 

to fill the hollows between the grains but to leave the apices of the 

grains free of water. As a result, the dimensioned volume of a geometrically shaped porous specimen usually exceeds the water-displacement bulk volume. Consequently, it may be expected that 

total porosity based on water-displacement bulk volume is usually 

less than total porosity based on a larger dimensioned bulk volume. 

For a constant pore volume, apparent (or effective) porosity based 

on water-displacement bulk volume usually exceeds apparent 

porosity based on dimensioned bulk volume.

MERCURY DISPLACEMENT

In the mercury-displacement determination of bulk volume, a 

specimen (generally of lower specific gravity than mercury) is forcibly 

submerged in a bottle full of mercury and the overflow mercury is 

considered to equal the bulk volume of the specimen. Because of 

its high specific gravity and fugacity, mercury tends to fill the surface 

pores of a specimen (Hartmann, 1926). Consequently, the volume 

of overflow mercury may indicate a minimum volume. The dimensioned bulk volume of a porous specimen usually exceeds the mercurydisplacement bulk volume.

Not all investigators have reported that the mercury-displacement 

bulk volume is the smallest. For some specimens of refractory 

materials Steinhoff and Mell (1924) reported that they obtained a
BULK, GRAIN, AND PORE VOLUME AS RELATED TO POROSITY D5

slightly greater mercury-displacement bulk volume than waterdisplacement bulk volume. Also, T. H. McCulloh (written commun., 

July 22, 1964) concluded that "the entrapment of air bubbles against 

the surface of a test specimen immersed in mercury leads invariably 

to a bulk volume determination that is erroneously large. The 

rougher the specimen surface, the higher the error." Accordingly, 

the mercury-displacement method of bulk-volume determination 

should be used only if such entrapment of air bubbles can be overcome.

A possible cause for a greater- mercury-displacement than waterdisplacement bulk volume may be related to the removal of the 

excess water from the surface of the test specimen before it is immersed 

in water. If, in the process of removal of the excess water, some 

interior water, together with all the water of the surface pores, is 

removed, the water-displacement bulk volume may be less than the 

mercury-displacement bulk volume.

Although bulk volume based on mercury displacement may represent the smallest bulk volume, it does not necessarily follow that it 

is too small. As was mentioned, if the surfaces of the porous specimen 

are macroscopically smooth, the water-displacement bulk volume 

may be too large, because of a water film which adheres to the saturated specimen before it is immersed. For such a specimen, the 

mercury-displacement bulk volume may be the best value.

PORE VOLUME AND GRAIN VOLUME

Neither the total nor the apparent pore volume of a rock specimen 

is determinable independently of the bulk volume. The total pore 

volume is the excess of the method-dependent bulk volume over 

grain volume. Similarly, although it may not be immediately 

obvious, apparent pore volume is the excess of the method-dependent 

bulk volume over grain volume and occluded-pore volume. Consider, 

for example, that the bulk volume of a specimen is obtained by 

dimensioning, but the apparent pore volume is measured by the 

exhaustion of pore air into the mercury vacuum of a WashburnBunting porosimeter. The value for the volume of pore air nevertheless is the excess of the value for the mercury-submerged bulk 

volume over the grain volume and occluded-pore volume.

Although the fraction of surface pores included in or excluded from 

pore volume depends upon the method of measuring bulk volume, 

method-dependent values of bulk volume involve more than the 

volume of surface pores. Observed total and apparent pore volumes 

may therefore differ from the most acceptable pore volumes by more 

than the volume of surface pores.

Grain volume can be precisely determined if adsorption of water 

by the grain powder during the pycnometric determination of its

793-623 &6 2
D6 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

volume is negligible and if the powder includes the whole specimen 

or is typical of the specimen. Nutting (1930) stated that adsorption 

of water by very finely powdered quartz grains during the pycnometric determination of the powder volume may cause an error of 1 

or 2 percent in the grain density, although data to substantiate this 

point were not presented.

FORMULATED POROSITY DIFFERENCES

BASIC VARIABLES OF POROSITY DIFFERENCES

To formulate differences among observed porosities, it is necessary 

for the experimentally determined bulk and grain densities of the 

total porosities and the experimentally determined pore and bulk 

volumes of the apparent (or effective) porosities to be referred to the 

same variables. This is done by expressing the experimental porosities 

in terms of two variables bulk volume and grain volume.

QUANTITIES OF FORMULATED POROSITY DIFFERENCES

In the development of the formulated porosity differences, bulk and 

grain volumes are related to ideal quantities and to quantities based 

on measurements of specimens. These quantities are given in the 

following list.

Symbol Quantity Method of determination

Ideal quantities and quantities independent of a specimen

. 

Di.

V,.. $ ;.

DnVolume of specimen____________________. 

Density of 

Volume of grains of specimen_ _____________. 

Density of grains of specimen____________.__. 

Volume of powder portion of specimen..________. 

Density of powder portion of specimen__.._______. 

Density of water (obtained from tables relating density to

temperature). 

Density of mercury (obtained from tables relating density

to temperature).

Weight of pycnometer, dry______________.... 

Weight of pycnometer filled with water.._______.

Quantities based on measurements on a bulk specimen

w. 

w,

v»

V.

V,

-»' 

D'*

Specimen weight, weight of dry specimen or bulk weight,

or weight of grains, all equal in value. 

Weight of water-saturated specimen..__ .. . ..

Weight of saturated specimen in water. _ __ _.

Bulk volume by water displacement___ ____.____.

Weight of mercury displaced by specimen____.-._. 

Bulk volume by mercury displacement-.-,----_ --. 

Total void volume, defined as excess of water-displacement 

bulk volume over ideal grain volume, or

Apparent void (pore) volume by water absorption... . 

Apparent void (pore) volume adjusted for mercury-displacement bulk volume. 

Bulk density by water displacement_______........

Bulk density by mercury displacement_ _.. 

Weighed in air.

Specimen evacuated, then 

saturated with water and 

weighed in air.

Weighed by suspended immersion.

Capture and weighing.

W'mlD..

W'/VW '. 

W'/V'm.
BULK, GRAIN, AND PORE VOLUME AS RELATED TO POROSITY D7

Symbol Quantity Method of determination

Quantities based on measurements on powdered portion of specimen

W" ............

W"............

W"em

W...........

V". ........... iy. ...........

Weighed in air. W",-Wt. Weighed in air. 

W.»- W,-( W",.- W".). 

W" WIDV. W'fV".

RELATION BETWEEN SPECIMEN GRAIN VOLUME AND DENSITY 

AND GRAIN- POWDER VOLUME AND DENSITY

If the powder is typical of the specimen, the weight of the grains 

in the specimen is greater than the weight of the powdered portion 

by a factor k, so that

W'=kW", where k^l (1) 

As grain volume is proportional to weight,

Vg=kVp (2) 

and

D,=Dg=W'/Vg (3)

If, during pycnometry, water is adsorbed by the grain powder, there 

is an apparent decrease in the measured volume of the powder. 

This decrease introduces the factor a, whereby

(4) 

and

D"=W"/V"=kW"/k(l-a)Vf=W'/(l-a)Vg='Dg/(l-a) (5)

RELATION BETWEEN APPARENT AND TOTAL VOID (PORE)

VOLUME

Occluded pores cause the apparent void, or pore, volume as determined by water absorption (V'w) to be less than the total void 

volume (F'0), or less than V'w Vg. A smaller void volume corresponds to a larger grain volumes by a factor b. Hence,

where 0^6 (6)

RELATION BETWEEN WATEB-DISPLACEMENT AND MERCURYDISPLACEMENT BULK VOLUME

In a recent study (Manger, 1965), bulk volume by water displacement was found usually to exceed bulk volume by mercury displacement, although for one specimen the mercury-displacement bulk
D8 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

volume was in excess. The bulk-volume relation is therefore expressed as

~T^ v w V 

m _ /7\ y (7)

or

(8)

Usually 7 is small, and may be negative, zero or positive.

If apparent pore volume is obtained as the excess of the waterdisplacement bulk volume over the volume of absorbed water, the 

value of apparent pore volume used for mercury-displacement bulk 

volume should be adjusted according to the excess or deficiency of 

water-displacement bulk volume with respect to mercury-displacement bulk volume. If this adjustment is made, then

V' vm = V'w- (V'w - V'n} = V'w-yV'm (9)

RELATION OF OBSERVED TO FORMULATED POROSITIES

Total porosity <l> tw based on water-displacement bulk volume usually 

is obtained by determination of grain and bulk density and is expressed as

Observed total porosity so obtained is related to formulated porosity 

based on bulk and grain volumes by means of volume and density 

relations in the lists of quantities and equation 5. The relation is 

is expressed as

W W

_D"-D' W _D' W D"_V' w-(l-a)Vg f . 

0i«- ft" ~ w ~ V' w ( }

D' w

Total porosity <$> tm based on mercury-displacement bulk volume 

likewise usually is obtained by the determination of grain and bulk 

density and is expressed as

D" D'm

Observed total porosity so obtained is related to formulated porosity 

based on bulk and grain volumes by means of volume and density
BULK, GRAIN, AND PORE VOLUME AS RELATED TO POROSITY D9

relations in the lists of quantities and equation 5. The relation is 

expressed as

W W 

, J[)"-D'm_Drm D"_V'm -(l-a)Vg -

D'm

Apparent porosity 4>ew based on water-displacement bulk volume 

is obtained by determination of apparent pore volume and bulk volume 

and is expressed as

*.*=&* (14) 

y w

Observed apparent porosity so obtained is related to formulated 

porosity based on bulk and grain volumes by means of volume relations 

in equation 6 and is expressed as

f

* w

Apparent porosity <f> em based on mercury-displacement bulk volume 

is obtained by determination of apparent pore volume and bulk volume 

and is expressed as

, ' vm

V'r (16)

Observed porosity so obtained is related to formulated porosity 

based on bulk and grain volume by means of volume relations of

pnnntvirms equations Q 9 nnrl and ft 6 nnrl and is is (vsrvrAssprl expressed naas

, _ vm _ 

4>em TTV " 

'\7> V vw  (V\r 1 w  VV

-rrr

* m

V'

If the adjustment by means of equation 9 for the value of pore 

volume used for mercury-displacement bulk volume is not made, the 

relation of observed to formulated porosity in equation 17 becomes
D10 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

This is equivalent to the equation

^=1^' (19)

" m

as equation 6 designates that F'W=F'«, (1+&) Vg.

EXTENSION OF CERTAIN DEFINITIONS

As previously mentioned (p. D7), adsorption of water by thegrain powder 

during pycnometry results in an apparent decrease in grain volume. 

The definition of factor a, where 0 = a<Cl, can be extended to include 

all experimental conditions that cause an apparent decrease in grain 

volume, or an apparent increase in grain density, and consequently 

an increase in the value of total porosity. Also, as was mentioned, 

the presence of occluded pores causes apparent pore volume to be less 

than total pore volume. The definition of the factor b where O^b, 

can be extended to include all experimental conditions that cause an 

apparent decrease in pore volume, or an apparent increase in grain 

volume, and consequently a decrease in the value of apparent porosity. 

Thus both factors a and b signify an increase in the value of total 

porosity over that of apparent porosity.

Conditions that cause an excess in the value of total porosity over 

apparent porosity and their relation to the factors a and b are listed 

as follows:

Condition Indicative factor 

Adsorption of water by the grain powder______________-_---_-_ ________ a

Grain-powder density greater than density of the whole specimen_________ a

Occluded pores present____----_____________________________ ______ 6

Incomplete saturation of the permeable pores as a measure of pore volume. _ _ _ _ 6

There is one known experimental condition other than measurement 

error that can result in a seeming excess of apparent over total porosity. 

This condition arises if the grain-powder density is less than the grain 

density of the specimen. In terms of equations 4 and 5, this is 

equivalent to an increase in the volume of grain powder, or kV" = 

(1+a) Vg and D"=Dg/(l-\-a'). To include this experimental condition in the preceding equations would require changing the limits 

of the factor a to show 0 ^ a. Because the development of porositydifference equations is facilitated by restricting the value of factor a 

to 0^a<Cl, the difference equations in the following section do not 

include the condition for deficient grain-powder density.

DIFFERENCE EQUATIONS

Difference equations are based on two total porosities and two 

apparent (or effective) porosities (0/w, <j> tm, (f>ew, and <£ «), as is shown in 

table 1. These difference equations are derived from equations 11,
BULK, GRAIN, AND PORE VOLUME AS RELATED TO POROSITY Dll

13, 15, and 17, where the basic quantities are bulk volume and grain 

volume. As was mentioned, the factor a of equation 4 takes into 

account the effect of adsorption of water during the pycnometric 

determination of grain-powder density and all other conditions that 

increase the value of total porosity. The factor b of equation 6 allows 

for the effect of occluded pores and all other conditions that decrease 

the value of apparent (or effective) porosity. The factor 7 indicates 

the relation between values of bulk volume as determined by water 

displacement and mercury displacement.

SIGNIFICANCE OF FORMULATED POROSITY 

DIFFERENCES

CHANGES IN VALUES OF POROSITY DIFFERENCE WITH VARYING

POROSITY

Grain volume, or Vg, in table 1 appears only in the numerator of 

the fractional terms that are equated with observed porosity differences, and Vg decreases as porosity increases. Thus it follows that 

as porosity increases, there is a decrease in the excess of observed 

porosity based on water-displacement bulk volume over observed 

porosity based on mercury-displacement bulk volume. As porosity 

decreases, there is an increase in the excess of observed porosity based 

on water-displacement bulk volume. In the virtual absence of 

porosity, where Vg^V'w, for no adsorption and no occluded pores 

(0=a=&), differences in porosity values, except for the <fou, $«» and 

^tm ^cm zero differences, are equal to about 7, where, as indicated by 

equation 7, 7=(F/W -

EFFECT OF USING THE SAME (UNADJUSTED) PORE VOLUME FOR 

APPARENT POROSITY BASED ON MERCURY-DISPLACEMENT 

AND WATER-DISPLACEMENT BULK VOLUME

At low porosity the value of apparent porosity based on mercurydisplacement bulk volume can be forced to approximately equal the 

value of apparent porosity based on water-displacement bulk volume 

by not applying the adjustment indicated in equation 9, that is, by 

not decreasing the volume of absorbed water (as a measure of pore 

volume) by the excess of water-displacement over mercury-displacement bulk volume. If this decrease is neglected, those porosity 

differences that contain the quantity 4>em are decreased by the amount 

7. For example, in table 1, for no adsorption and no occluded pores, 

if the decrease or adjustment is included, <l)tw <t>em='YVg/V'w. If the 

decrease is omitted, <j> iw 4>em==(yVg/V'w) 7. Because Fe «F'w at 

low porosities, (yVg/V'w) 7«0 and seemingly <f> tw <&>?»« 0.

That such equality of porosity is forced is readily shown. According 

to table 1, the differences in values of total porosity solely because of 

the bulk-volume effect that is, for no adsorption and no occluded
D12 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

SJ g

porosi of ap 

V' t; m rption

s T3 -3

sorption occluded

iv bs

 d 

Sg

 Sg

d»

f- O

-e- -e- -e- -e- -e- -eS _9 ^ S | 8

 £ £ £ £ £ £
BULK, GRAIN, AND PORE VOLUME AS RELATED TO POROSITY D13

pores is <j>tw <t>tm=yVg/V'w. Consequently at very low porosity, 

where Vg ~V'w, <£«w <£«TO ~ 7. For no adsorption and no occluded 

pores, however, total and apparent porosity based on mercurydisplacement bulk volume must be equal, or 0(TO =<£m. If <t> tm is 

substituted for <j)em in the equation 0*w 4>em ~Q in the previous paragraph where the adjustment is not made, it seems that <£ tw <£*w ~0. 

But as mentioned, table 1 shows that in the absence of adsorption 

and occluded pores, at very low porosity 0tw 0te «7-

Where the mentioned adjustment or decrease by means of equation 

9 is not made, apparent porosity based on mercury-displacement bulk 

volume, or <£m, usually exceeds all other porosity values, provided that 

the water-displacement bulk volume is greater than the mercurydisplacement bulk volume. This excess becomes greater as porosity 

increases. This is because omission of this decrease causes all differences that contain the quantity 0ew to be diminished by the amount 

7. As an example, consider the <£««, - 0«w difference of table 1 for no 

adsorption and no occluded pores. If adjustment for pore volume is 

not made, 0/w <£««= (yVg/V'to) ~ 7- As porosity increases, Vg and the 

Vg/V'w ratio decrease, and the 4> tw <i>em difference becomes more negative, or the excess of observed apparent porosity (<£m) over observed 

total porosity (4> tw) increases. Where adsorption is effective and 

occluded pores are present, it is possible that <£<«,- 0eTO >0, that is,

Vg Is^V. t 

w

This, however, requires large values for the factors a and b.

EQUALITY OF TOTAL AND APPARENT POROSITY

According to table 1, for no adsorption and no occluded pores,

4lw <t>ew = 0<m <l>em = 0. However, (j>tw (j) tm = yVg/V'w.

Thus equality of total and apparent porosity signifies only that 

occluded pores are absent, but not that the best value of porosity has 

been obtained. Obtaining the best value of porosity depends upon 

obtaining the best value of bulk volume.

LARGER APPARENT-POROSITY DIFFERENCE THAN 

TOTAL-POROSITY DIFFERENCE

Table 1 shows that where adsorption is effective and the value of 

water-displacement bulk volume is greater then the value of mercurydisplacement bulk volume, or Vw > F'TO , the total-porosity difference 

(f>tw (f>tm is y(l-a)VglV'w . Where occluded pores are present and 

Vw^>V'm, the apparent-porosity difference 4>ew <$>em is 7(1+6) Vg/V'w. 

The excess of apparent-porosity difference over total-porosity difference is therefore y(d-\-V)VsIV'w. As porosity increases, V§ decreases
TABLE 2. Total and apparent porosity of lapilli tuff, subunit T, Paintbr

Specimen

15......... __ ............... _ ..

2..................................

7... . .. -

13..................................

8..................................

1 _ .... _______ ... ____ ....

3..................................

4... ............ .................

10..................................

11.......... -. . .- .

Average without specimens 

10 and 11 _- .

Grainpowder 

density 

(g cm-») D"

2.232 

2.273 

2.279 

2.304 

2.314 

2.319 

2.323 

2.361 

2.442 

2.502

2.335

2.301

Pore 

volume 

(ml) by 

water 

absorption

1.802 

1.511 

1.570 

2.343 

2.401 

1.994 

1.509 

2.685 

2.803 

1.449

Bulk volume (ml) by 

displacement of 

Water 

F'«

4.394 

4.253 

4.895 

6.202 

5.912 

5.103 

4.070 

6.554 

6.756 

4.629

5.277 

5.173

Mercury

V'm

4.333

4.170 

4.826 

6.235 

5.819 

5.025 

4.011 

6.436 

6.615 

4.493

5.196 

5.107

Bulk density dry 

(g cmr8) based on 

bulk volume

Water D'»

1.342 

1.490 

1.554 

1.396 

1.360 

1.425 

1.448 

1.366 

1.360 

1.585

Mercury

D'm

1.361 

1.519 

1.576 

1.388 

1.382 

1.447 

1.470 

1.381 

1.389 

1.633

* For the apparent porosity based on mercury-displacement bulk volume, the pore

volume indicated by 

displacement over m

V' m).
BULK, GRAIN, AND PORE VOLUME AS RELATED TO POROSITY D15

and the value of the excess y(a-\-b)Vg/V'vl decreases. The fact that 

the total-porosity difference is smaller than the apparent-porosity 

difference does not mean that total porosities are better approximations to the true value of porosity, because obtaining the best value 

of porosity depends upon obtaining the best value of bulk volume.

EXPERIMENTAL. ILLUSTRATION OF THE FORMULATED 

POROSITY DIFFERENCES

EXCESS OF TOTAL OVER APPARENT POROSITT

Porosity data for lapilli tuff from the Nevada Test Site have previously been reported (Manger, 1965). These data are shown in 

table 2 with the addition of values of 7. Table 3 lists porosity differences derived from table 2 and includes values for <£*«, &.«,, or 

values for the excess of total over apparent porosity where both 

porosities are based on water-displacement bulk volume. An average 

excess of total porosity of 0.96 percent porosity according to table 1 

suggests that adsorption is effective or that occluded pores are 

present. Deficiencies and excesses of total porosity in table 3, however, are apparently related to grain-powder density, or D". Figure 

1 indicates a strong dependence of deficiencies and excesses of total 

porosity on grain-powder density.

TABLE 3. Porosity differences of lapilli tuff, subunit T, Paintbrush Tuff, 

Nevada Test Site, Nev.

Specimen

15

2 ___ ....

7

13 . 

9

1- .. . _ _-. - - .- - ... . ...

3

4 - - . - 

10- _ _. 

11 _ _ 

£>"!

(g em-3)

2.232

2.273

2.279

2.304

2.314

2.319

2.323

2,361

2.442

2.502

2.335

2.301

*,.-*..'

(percent)

-1.14

-1.08 -.26

1.63

.62

-.52

.59

1.60

2.82

5.35

.96

.18

*««-*«.,1 

(percent)

0.85

1.28

.96 -.35

.95

.95

.95

- 1.06

1.19

1.92

.98

.83

*«-*..' 

(percent)

0.83

1.29

Q7

-.33

.95

OJ.

.93

1.08

1.25

2.08

1.00

.83

1 Grain-powder density.

2 Total porosity based on water-displaceraent bulk density minus apparent porosity based on waterdisplacement bulk volume.

3 Total porosity based on water-displacement bulk density minus total porosity based on mercurydisplacement bulk density.

4 Apparent porosity based on water-displacement bulk volume minus apparent porosity based on mercury-displacement bulk volume.

As mentioned, there is only one known cause, other than measurement error, for a deficiency of total porosity with respect to apparent 

porosity, namely, that the density of the grain powder used for the 

grain-density determination is less than the grain density of the 

specimen. Because of this indicated causal relationship, the coefficient of determination r2, where r is the coefficient of correlation
D16 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

-2

2.20 2.30 2.40 2.50 

GRAIN-POWDER DENSITY, IN GRAMS PER CUBIC CENTIMETER

FIGURE 1. Excess of total porosity of lapilli tuff (based on grain-powder density) 

over apparent porosity, plotted against grain-powder density. Both total 

porosity and apparent porosity are based on bulk volume determined by 

water displacement. After Manger (1965).

(Ezekiel and Fox, 1959, p. 130), shows that of 0.96 percent average 

excess total porosity for 10 specimens in table 3, an excess total 

porosity of 0.84 percent is due to the fact that on the average the 

grain-powder densities are greater than the grain densities of the 

whole specimens. Occluded pores, if present, do not include more 

than 0.12 percent of rock volume or 0.12 percent porosity. If in 

figure I the two points showing extremely high excesses of total
BULK, GRAIN, AND PORE VOLUME AS RELATED TO POROSITY Dl7

porosity are omitted, the grain-powder densities of the remaining 

eight specimens are equally distributed about zero median excess 

total porosity. In this modification, the coefficient of determination 

shows that of 0.18 percent average excess total porosity, an excess 

total porosity of 0.10 percent is due to greater density of grain powders 

than of whole specimens. Thus, occluded pores or spuriously indicated occluded pores (because of adsorption of water by the grain 

powder or because of experimental error) do not constitute more 

than 0.12 percent porosity.

The small value of 0.12 percent excess porosity suggests that mean 

apparent porosity based on water-displacement bulk volume should 

approximate correct porosity. As noted in the section "Equality 

of total and apparent porosity," however, this suggestion is true 

only if water-displacement bulk volume yields the most acceptable 

value for bulk volume.

LARGER DIFFERENCE IN APPARENT-POROSITY THAN IN TOTALPOROSITY VALUES

In table 3 apparent porosity based alternatively on water-displacement and mercury-displacement bulk volume shows a mean 

difference of 1.00 percent porosity. Total porosity alternatively 

based on corresponding bulk densities shows a mean difference of 

0.98 percent porosity. The slightly greater apparent-porosity difference is to be expected according to the equations of table 1, whence 

the difference between apparent-porosity difference and total-porosity 

difference, for V'u^>Vfm) is 7(a+6) Vg/V'w. The value of this fraction 

is small because the value of 7 is small (in table 2 the average value 

of 7 is 0.016); a<l; Ffl<l, if a specimen shows any porosity; and, 

although the upper limit for b is not defined, it is very unlikely ever 

to be as great as Vg.

THE VALUE OF y

The effect of method on values of bulk volume for lapilli tuff is 

indicated by the value of 7. In table 2, 7 ranges from 0.005 to 

0.030 and averages 0.016. The value of 7 is expressed in terms of 

bulk-volume ratio in equation 8 as V'w/V'm =l+y, where V'u and 

V'm are respectively water-displacement and mercury-displacement 

bulk volume.

Bulk-volume ratios for the lapilli tuff from the Nevada Test Site 

in table 4 are compared with bulk-volume ratios of some other porous 

materials. For specimens of lapilli tuff having total porosity of 

about 38 percent, total-porosity difference increases from 0.83 percent 

porosity to 0.98 percent porosity as the ratio of water-displacement 

bulk volume to mercury-displacement bulk volume hi creases from 

1.013 to 1.016. For Steinhoff and MelTs (1924) cubes of refractory
D18 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

material, having porosity of about 24 percent, a total-porosity difference of 1.4 percent porosity corresponds to a ratio for dimensioned 

to mercury-displacement bulk volume of 1.019. These relations 

suggest that for the lapilli tuff, a ratio of dimensioned to mercurydisplacement bulk volume would increase the total-porosity difference 

of table 3 to about 0.98+2(0.98 0.83), or to 1.13 percent porosity. 

The corresponding increase in apparent-porosity difference would be 

to 0.83+2(1.00 0.83), or to 1.17 percent porosity. Because among 

observed values of bulk volume the dimensioned values usually are 

the greatest, maximum difference in porosity value for the lapilli 

tuff, as dependent upon the bulk-volume measurement, is probably 

not much more than 1 percent porosity.

TABLE 4. Ratios of bulk volume based on dimensioning, water displacement, and

mercury displacement

Author

Steinhoff and Mell 

(1924). 

Hartmann (1926) _ -

This study 

All specimens

Without specimens 10 and 11-.

Material

Porous refractory 

material 2 . 

Refractory brick «_

Lapilli tuff « __-

Total porosity (percent) based on 

bulk volume determined byDimensioning

25.0

Water displacement

25.5 

38.65 

38.19

Mercury 

displacement

23.6 

24.6 

37.67 

37.36

Porosity 

difference 

(percent)

1.4 

.9 

.98 

.83

Bulkvolume 

ratio i

» 1. 019 

31.012 

 1.016 

 1.013

1 Equals 1+y (see equation 8).

2 73 cubes, 2 cm along the edges.

8 Computed from respective total porosities.

4 10 specimens of chamotte.

6 10 specimens, rounded and multiplanert, 3 cc to 8 cc in volume.

«Derived from measured bulk volumes.

The data of table 4 suggest that with decrease in porosity there 

is a persistence of much of the method-dependent difference between 

water-displacement and mercury-displacement values of bulk volume. 

For eight specimens of lapilli tuff having total porosity of about 

38 percent, the bulk-volume ratio is 1.013 and 7=0.013. For 

Hartmann's (1926) specimens of refractory brick having total porosity 

of about 25 percent, the bulk-volume ratio is 1.012 and 7=0.012.

The texture of the surfaces of the refractory brick and of the 

lapilli tuff probably are different, but, with decreasing porosity, 

the volume of the previously mentioned surface pores likely decreases. The slight decrease in bulk-volume ratio that accompanies 

a large decrease in porosity from lapilli tuff to refractory brick suggests 

that many of the method-dependent differences between waterdisplacement and mercury-displacement bulk volumes persist to low 

porosities. According to table 1, in the vicinity of zero porosity, 

because of method-dependent differences of bulk-volume values, a
BULK, GRAIN, AND PORE VOLUME AS RELATED TO POROSITY D19

difference in porosity of about 7 is to be expected. For materials 

having the smooth surface texture of the specimens of lapilli tuff, 

the preceding relations suggest a difference of about 1 percent in 

porosity values in the vicinity of zero porosity because of differences 

in method-dependent values of bulk volume.

SUMMARY

Formulations of porosity differences show the effect of methoddependent values of bulk volume and the effect of adsorption, 

occluded pores, and other qualifying conditions on observed values 

of porosity. There is an increase in the magnitude of these effects, 

particularly of the method-dependent effect of bulk volume, as 

porosity decreases. Because of differences in bulk volumes measured 

by water displacement and by mercury displacement, observed 

values of porosity of lapilli tuff at about 38 percent porosity differ 

by an average of 1 percent porosity. Formulated differences of 

porosity and ratios of observed water-displacement to mercurydisplacement bulk volumes suggest that, in the vicinity of zero 

porosity, materials having the surface texture of the lapilli tuff 

would show a difference of 1 percent porosity between porosity based 

on water-displacement and mercury-displacement values of bulk 

volume. For a substance having very low porosity, forced equality 

of porosity can be obtained by using the same pore volume for 

differently determined values of bulk volume, but this method ignores 

the fact that pore volume is determinable only as the excess of bulk 

volume over grain volume.

There seems to be no best method of bulk-volume measurement 

that is applicable to all porous substances. Mercury displacement 

usually furnishes minimum values and dimensioning usually furnishes 

maximum values for bulk volume. Because of the macroscopically 

smooth surfaces of the test specimens of lapilli tuff, determination of 

bulk volume by mercury displacement seems to be the most acceptable 

method for these specimens. Were the specimens coarse grained, the 

most acceptable method probably would be either that of water displacement or dimensioning. Because of its fugacity and high density, 

mercury may be expected to fill coarse-surface pores. As a result, for 

coarse-grained specimens the volume of displaced mercury would 

likely represent too small a bulk volume.

An excess of total over apparent porosity can result from factors 

other than the presence of occluded, or sealed-off, pores. For the 

lapilli tuff a seeming excess of 1 percent porosity is due to an average 

grain-powder density that is greater than the grain density of the test 

specimens. Occluded pores, if any are present, do not exceed 0.12 

percent porosity.
D20 EXPERIMENTAL AND THEORETICAL GEOPHYSICS

Equality of total and apparent porosity where based on the same 

value of bulk volume indicates only that occluded pores are absent. 

It does not indicate that the best value of porosity has been obtained. 

Obtaining the best value of porosity depends upon obtaining the best 

value of bulk volume.

REFERENCES

Ezekiel, Mordecai, and Fox, K. A., 1959, Methods of correlation and regression 

analysis [3d ed.]: New York, John Wiley & Sons, Inc., 548 p.

Fancher, G. H., Lewis, J. A., and Barnes, K. B., 1933, Some physical characteristics of oil sands, in 3d Pennsylvania Mineral Industries Conf. Proc., Petroleum 

and Nat. Gas Sec.: Pennsylvania State Coll. Mineral Industries Expt. Sta. 

Bull. 12, p. 65-171.

Hartmann, F., 1926, Die Bestimmung des Raumgewichts und der Porositat 

feuerfester Steine nach dem Quecksilber-und dem Wasser Verdrangungsverfahren: Fachausschtisse des Vereins deutscher Eisenhtittenleute Ber. 82, 

9 p.

Hartmann, M. L., Westmont, 0. B., and Morgan, S. F., 1926, The determination 

of the bulk and pore volumes of refractory materials: Am. Ceramic Soc. Jour., 

v. 9, p. 298-310.

Manger, G. E., 1965, The best value of porosity of lapilli tuff from the Nevada 

Test Site: U.S. Geol. Survey Prof. Paper 525-B, p. B146-B150.

Nutting, P. G., 1930, Physical analysis of oil sands: Am. Assoc. Petroleum 

Geologists Bull., v. 14, p. 1337-1349.

Steinhoff, E., and Mell, Maria, 1924, Porositatsbestimmungen an feuerfesten 

Steinen: Fachausschiisse des Vereins deutscher Eisenhtittenleute, Ber. 44, 

6 p.

O