An air porosimeter for the porosity of rocks

Int..L Rock Mech. M~n. StI. VoL 8, pp. 29-53. Pergamon Press 1971. Printed m Great Britain

AN AIR POROSIMETER FOR THE POROSITY OF ROCKSI" Y. V. RAMANA a n d B. VENKATANARAYANA National Geophysical Research Institute, Hyderabad, India

(Receired 4 April 1970) Abstract--A simple air porosimeter for the evaluation of porostty of rock samples that has been designed and constructed and which is based on the gas labs is fully described. Details of the a*r pores,meter, its cahbrat,on, accuracy of working, precautions to be taken, its advantages, the type of samples that it can handle are dzscussed. Particulars of standardization employing metal samples in which pore space has been art,ficially introduced by drilling, and repeatabdtty test results are also included. The results obtained by conventional saturation methods and theoret,cal empirical formulae are compared with those obtained due to the use of the air pores,meter. Results obtained from a study of some rock samples using the a,r pores,meter are presented, and the effect of porosity on other rock physical properties such as density, saturation, wave velocity are ind,cated graphically. The results of poros,ty, its dependency on surface area of the sample, are also reported. An express,on for the instrument constant is appended.

P V~ Vw V~ P* Pt pz Wt W2

= = = = = =

= = = Ps = Pt = Pz = p = g = ht = h2 = // = Po = Vo = I"t = 1:2 = K; K' = V* = a = x = K" = S =

NOTATION Percentage porosity Volume of pores or pore volume Volume occupied by grams or gram volume Bulk volume True or real porosity True specific gravity of rock in powdered form Apparent spe~.llic gravity of rock in bulk form Dry we,ght of sample Wet weight of surface dried rock sample Bulk denstty Pressure inside the pores,meter without sample Pressure ms,de the poroslmeter with sample Density of mercury Acceleration due to gravity Height of mercury m graduated tube without sample lleJght of mercury in graduated tube with sample Porosm~eter constant Atmospheric pressure lnmal volume of a~r in the poros~meter above the mercury after equalizat,on of levels Volume of air above mercury (without sample in chamber) corresponding to (ht) Volume of air above mercury (~lth sample in chamber) corresponding to (h2) Gas constants Volume of a~r excluding the locked up air in the graduated tube Area of cross sect,on of graduated tube Fle~ght between stopper and base up to U-curvature 1/~ Percentage sorption 1. INTRODUCTION

POROSITY is a n i m p o r t a n t p h y s i c a l p r o p e r t y o f r o c k t h a t is f r e q u e n t l y r e q u i r e d in the fields o f g e o l o g y , g e o p h y s i c s , p e t r o l e u m e n g i n e e r i n g a n d r o c k m e c h a n i c s . P o r o s i t y is o f t e c h n i c a l i n t e r e s t in t h e c e r a m i c a n d m e t a l l u r g i c a l i n d u s t r i e s . In e a r t h s c i e n c e s t h e r e is e v e r y n e e d t o "J"N.G.R.I. No. 70-197. ~ocx S l l - - c 29

30

Y. V. RAMANA AND B. VENKATANARAYANA

determine porosity in as large a number of rock samples as possible in order to get a representative value of porosity that could be used in interpreting the other physical properties of the rock. It is in. this context, that an attempt has been made to devise a porosimeter which is simple, quick, and easy to operate, and is effective to give repeatable results. Porosity is a rock property that affects the density, strength, sorption and in general rock elasticity. It contributes substantially in causing the variations in elastic wave velocities in rocks of a particular region; or in crustal rocks in terms of the depth; its effect is minimized or becomes negligible when any physical property is studied by subjecting the rock sample to an aU-sided pressure of about 1.5-2.0 kb [I]. The mode of formation of the rock and the subsequent geological history that it had undergone; the shape and orientation of the grains composing the rock giving rise to varied intergranular spacing; joints, fractures and microfractures caused naturally, or due to artificial man-made causes, such as, those due to the breaking of the rock, and its preparation by coring or cutting, etc. should account for the porosity observed in rock samples. Igneous rocks and metamorphic rocks generally have a very low degree of porosity as compared to the sedimentary rocks and this generalization is not without exceptions--for example vesicular basaltic rocks exhibit a porosity of 10--I5 per cent or even more while some compact limestones indicate porosity of hardly 3 per cent. The 'pore spaces' or 'voids" that cause the porosity in rock can conveniently be classified as of three types, viz. (a) External voids: (b) Open voids:

Those that may be present on tile surface of the prepared sample. Those that are inter-connected between themselves and linked to the external voids. Those that are neither interconnected between themselves, nor linked to the external voids, but are completely locked-up in the

(c) Closed voids:

rock mass.

Figure i is a representative sketch showing the three types of voids. Here, we are mainly concerned with the first two types of voids alone and the third factor does not come into the

.

.

.

.

--2

I

3 Fio. I. D~agram to show types of voids. !--Rock sample; 2--Open voids; 3minter-connected voids; 4--Locked voids.

AN AIR POROSIMETER FOR THE POROSITY OF ROCKS

31

p,cture; in other words we are concerned with the determination of the apparent porosity or effective porosity and not the true poroslt) for v, hose determination the locked pores also v, lll ha,,e to be considered. At this stage, it ts preferable to ha~e a picture of the bulk volume and grain volume, since these are the basic parameters required m the calculation of porosity of any specimen. By 'bulk volume" ts meant the total volume of the sample and this includes the volume of the grains and the pore spaces; while grain volume is the volume occupied by all the individual grains comprising the rock sample. All methods for the determination of poroslty involve some form or other to evolve the bulk and gram volumes and thereby help in estimating the pore volume. The air porosimeter presently being described wdl help in evaluating the grain volume of rock samples accurately and the bulk volume could be estimated by displacement m water or from the sample dimensions being taken into consideration. 2. PREVIOUS LITER~gTURE A good account of the early methods, which are essentially gray,metric or volumetric methods for the determination of the bulk or gram volume has been presented by Tic K[LL [2] The methods described therein include those for the determination of bulk volume, or gram volume, or both. Goodner, Russcl, Sutton, and Gealy are some of the early workers to have devised volumeters using mercury for the determination of bulk volume of test samplcs. Mclcher, Nutting and Brankstone devised methods for the evaluation of grain volume but their methods require the separation of grams that aggregate to compose the rock and are therefore dc',tructJble but glve a value of the true porosity. Washburn and Bunting, MacGee, Coberly and Stevens, and TICK! I.L et al. [3] describe method,, based on the gas laws for tile determmatlon of the grain volume of tile sample. Of these, Russcl's volumeter can be used for both bulk and gram volume determination. Tlckcll has discussed the merits and dements of each one of the above methods. The porosity of rocks is dependent on such factors like grain size, grain distribution, grain shape, gram orientatlon; density of the rock, amount of non-granular material or intergranular bmthng material, the degree of compaction due to pressure or metamorphism and the effect of them individually has been studied by FRASLRet al. [4]. HUGHLSand COOKE [5] describe a modified Boyle's law apparatus involving the evacuation of the air container for the determination of porosity of rock samples. BALAKRIStI'q'Aand SIIANKAR NARAYANA [6] employing a bigger sample chamber and an auxihary small dish in the bigger china dish kept at the bottom of the TIckell's instrument, determined the porosity of rock samples. SHANKAR NARAV^NA [7] studied the effect of surface area on poroslty of rock samples and found a decrease in porosity value with increasing surl:ace area. 3.

POROSITY EVALUATION

3 (a) Definition ofporosity The porosity of a rock (any sample) is the volume of the pores or air spaces or voids that it contains, expressed as a percentage of the total volume of the sample; i.e. Porosity --= P =

-

Volume of pores ) 100 Volume of rock q- volume of pores

(~:~-

(l)

32

Y.V. RAMANA AND B. VENKATANARAYANA

where Vs, V., lip denote the bulk volume, grain volume and pore volume of the rock respectively, and because Vb = (V. + V~). 3(b) Porosity by conrentional methods 3(bXi) Powder method. If P* is true porosity of rock, pt is the true specific gravity of the rock in the powdered form, and t'= is the apparent specific gravity of the rock in bulk form, then P* = \ -Pt- ~ t P2t/ I00. -

-

(2)

In this method the sample is destroyed due to powdering and cannot be recovered. 3(b)(ii) Saturation method. If Wt is the weight e r a dried piece of rock (dried for about 6 hr at 100°G) cooled and weighed and W2 is the weight of surface-dried rock sample after saturating it in water, then Wet weight-dry weight Porosity -----\Specim-~n vol-i-u~eemex d e ~ of water] × 1130 x loo.

(3)

The saturation method provides only the apparent porosity or effective porosity and not true porosity. 3(c) Davis' empirical relation In a study of sedimentary rocks (about 600 samples) D^vts [8] obtained an empirical relation for the estimation of porosity (P) from the bulk density (ps) and it is given by P = (2"~.6~44 P') 100

(4)

and the factor 2.654 denotes the average grain density of the mineral assemblage composing sandstones, shales, clays and silts. Thus, if the bulk density is known, it is possible to evaluate the porosity value using Davis' formula. 3(d) The air porosimeter The apparatus used here is essentially a gas (air) porosimeter involving a certain amount of suction and evacuation, and all gas porosimeters operate on the principle of the combination of Boyle's and Charles" laws. All air porosimeters give the value of effective porosity and not the true porosity. In the present work because the measurements are at room temperature, if Po is the atmospheric pressure, Pt and Pa are the gas pressures inside closed end of the porosimeter (see Fig. 2) without and with the sample and ht and h2 are the corresponding heights of the mercury column on the closed side in the graduated tube, then we have the basic relations from Boyle's Law: P, q- pg ht -----Po P, + pg ha = Po

(5) (6)

AN AIR POROSIMETER FOR THE POROSITY OF ROCKS

b - ~

33

1 .,'m

T 8 5cm

& T

I

+ i I~Cm

4 I

I.

5 5

7

90 -m

T l

24cr~

l--Funnel top 2--Pyrex stopper 3--Top half of B-40 joint 4--Bottom half of B-40 joint 3, 4---Sample chamber with ground joint 5--Steel clamp 6--Spring 7--Condenser 8--Graduated tube 9--Au~d~ary scale IO~Sample I I--Reference mark 12--Slot 13--Vertical wooden support 14--Mercury reservotr ! 5--Mercury 16---Movable wooden support for mercury reservoir 17--Plane m,rror 18--Thack-walled Polythene tubmg 19--R~g,d wooden base ~ Upper and lower hmlts of slot .¢

C---Caps S--Circular slots

A,r poros~meter F=o. 2.. Line drawing of air porosimetcr showing component parts.

34

Y. V. R A M A N A A N D B. V E N K A T A N A R A Y A N A

from which it follows that P t - - P2 = pg ( h , - - h t )

(7)

where t' is density of mercury, and g is acceleration due to gravity. Using equation (7) it can be shown (see Appendix) that V~ = fl (h, -- hL); or

V~ oc (h, -- hi) (8) where V o is grain volume and fl is an instrument constant. Thus by this method, the grain volume V~ of any sample could be determined from a knowledge of(h2 -- hi). Once this is known, the bulk volume Vbcould be easily determined either by taking the dimensions of the sample into consideration, or by displacement of the body in water or mercury. When Vg and Vb are known, the 'effective porosity" namely the porosity resulting due to the encompassing of the intercommunicating voids alone, could be evaluated using the relation that follows from the definition of porosity given below:

(9) where P is the percentage effective porosity, and ( V b - - Vo) denotes the 'pore volume" of the rock (Vp). 4. E X P E R I M E N T A L

4. I Description o f the apparatus

The air porosimeter is essentially a U-shaped glass instrument. Figure 2 shows a line drawing of the instrument showing the component parts while Fig. 3 is a photograph of the completed assemblage. The instrument as a whole is somewhat similar to the Boyle's law apparatus, while the left half alone is more similar to the Tickell's apparatus.The instrument is held in a vertical position by means of a rigid wooden stand 0 3 and 19) with metal clamps (5) to ensure sufficient strength to support the large weight of the mercury (15). The mercury reservoir (14) held on the right side is fitted with steel clamps to a movable wooden frame (16) that can be easily moved up and down and can be held at any intermediate position also with the aid of a metal bolt and nut arrangement. A small wooden projection of sufficient length at the back of the movable wooden frame running through a groove or slot (12) in the vertical wooden frame (13) enables the movement of the mercury reservoir smoothly in a vertical and linear fashion. Slots (5) in the base of the wooden stand holding the vertical frame provide easy movement of the transparent Polythene tubing 08) that holds the mercury linking the left and right halves of the instrument; and this arrests any possibilities of kink formation being developed in the connecting tube and keeps it taut. On the left half, at the top we have a ground stopcock (2) and a ground glass joint. The ground joint (B-40 Pyrex glass) forms the sample chamber (3 and 4) and it can take samples of up to l i in. diameter and 4 in. long. The funnel top (I) arrangement above the stopper is for easy transfer of mercury into the porosimeter. The condenser (7) below the ground joint is attached to a hollow thick-walled glass tube (8) with sutficiently wide bore as to avoid the corrections due to capillarity. The small tube linking the sample chamber to the condenser acts as the reference joint on which a reference mark (! I) is made--up to which point mercury has to be filled in the apparatus when the reservoir on the right is held at the topmost

N

1

:

=

= ~

[ IC, 3 I~ht+to~raph or" the a',+cmblcd air poroxt[lletct

RM f p

34]

FiG. 7. Phologr~ph to ~ho~,~. the dd'l'¢rcnt ~,ample~,

AN AIR POROSIMETER FOR THE POROSITY OF ROCKS

35

position and at the maximum level (A) of the groove. (A) and (B) are the limits of the slot at which equalization in levels and the heights in the graduated tube are to be noted respectively. The funnel top and the mercury reservoir are provided with suitable caps (C) to prevent dust and these require to be removed while working with the instrument. The plane mirror (17) and the auxiliary scale (9) arrangement held at the back of the graduated tube help in avoiding any errors due to parallax and facd~tate the observation of the correct reading of the meniscus height of the mercury column in the graduated tube. The use of a magnifying glass further helps in noting the correct meniscus height. The improvtzation of a right cylindrical spring (6) held in its horizontal positron at the bottom of the sample chamber of ] in. diameter and 1] m. long serves as a shock absorber while placing the core samples and thereby prevents any damage to the glass walls of the sample chamber. In fact, the spring has sufficient tension to absorb any shocks due to accidental shpping of the core while placing it in the chamber. 4.2 Operation o f the instrument I. After thorough cleaning and drying of the glass apparatus mercury ,s slowly transferred through the reservoir on the right or through the funnel on the left and while fill,ng the mercury the stopper is kept open and the reservoir is kept at the highest position (A) and care is taken to avoid the formation o f a i r bubbles. Mercury is filled up to a reference mark (I I ) made on the small tube connecting the sample chamber and the condenser and ~ ith this operation equahzation of mercury levels on the left- and right-hand sides of the apparatus is attained. Any locked-up air bubbles are allowed to escape by moving the mercury reservoir up and down to the extreme positions several times and later by adding any additional quantity of mercury, if required, to maintain the level up to the reference mark. For a ready check on the equalization of IcveN, a reference mark on tile reservoir sMc also is made. I!. The ground joints of the stopper and the sample chamber are smearcd with st, ffictent quantity of high-vacuum silicon grca~e and made air-tight by rubbing against the waiN, and the stopcock Is then kept in the 'closed" position. III. The mercury reservoir is brought down to the extreme Io~est position (B) and w~th tlu~ operation the mercury in the closed tube of the porosimetcr also moves down and stands at a particular height and this is becat, se the open end of the mercury rcscrvoir is exposed to the atmospheric pressure; and the reading corresponding to this level is noted from the graduatcd scale as (h0. The height (h,) so obtained is verified by repeating the operation at least thrce or four t~mes. IV. Later the cquahzation of levels at the top is again achieved by moving the reservoir back to its highest elevatzon. During this operation the stopper is kept m its 'open' positron. V. By operating the B-40 joint the sample to be studied is kept in the sample chamber and the joint is then closed and made air-tight. The stopcock also is later closed. VI. The mercury reservoir is again moved to its lowest extreme and the reading corresponding to the height of the mercury column m the capillary tube is read off the graduated scale for a second time and this gives the height (hz) obtained ~ith the sample in the sample chamber. By moving the mercury reservoir to the highest and lowest extremes, operation VI is repeated to verify the height (h,). VII. The difference in levels (hz -- h~) obtained from operations Ill and VI is a measure of the grain volume of the test sample, i.e.

(h~

-

h~)

oc

V,,

and this forms the basis for the working of the instrument.

36

Y. V. RAMANA AND B. VENKATANARAYANA

I&V

V IV

Vii

--

8Vl

FIG. 4. Operation diagram of air porosimctcr.

AN AIR POROSIMETER FOR THE POROSITY OF ROCKS

37

An operation dtagram depicting operations (I)-(VII) for quick and easy reference may be seen m Fig. 4. 4.3 Cahbration Before regular usage of the instrument for the evaluation of grain volume and hence porosity of rock samples, it requires to be calibrated. The instrument is calibrated using brass and aluminium disks of cylindrical shape. The right circular cylinders so used being metallic, are assumed to be free from any porosity, and therefore any rise in the mercury column in the closed side of the porosimeter due to the insertion of the calibrating sample, is due to the displacement of an equivalent volume of air. The calibration of the instrument can be attempted by using steel ball-bearings also. In actual pract=ce, the calibration of the instrument using right circular cylinders is repeated three times and later verified using ball-bearings also and the results so obtained were found to be easdy repeatable. The volumes of the metallic cyhndrical disks used for cahbration are first of all determined from sample geometry and further verified against displacement in water. The porosimeter readings on the graduated tube (h2 values) obtained against each one of the cahbrating samples are noted and these correspond to the respective grain volumes since the pore volume in them is neghgible. The data pertinent to the cahbration of the air porosimeter is given in Table I. In this table the results in the fifth column indicate the (hz -- h~) values for each metal sample. Using this data, it is possible to draw the calibration graph of volt=me against (h2 -- hi) and this is shown m Fig. 5. From Fig. 5 it will be possible to evaluate the grain volume corresponding to any (h2 -- h t) value obtained whde testing any rock sample or any other porous sample for the=r poros=ty.

E v

6 -

Cohbrolion Of porostmefer

c" E

I.,,,i e~

o 4

~e~e S ° j

-

=v E

~u c It,

jo

2L '^

°J° 15

3t2 Grain volume,

I

4b

1

~,O

i

75

cm 3

FIG. 5. Cahbratton graph for the evaluatton of gram volume From the cahbration data it is possible to evaluate the instrument constant (fl) and thts has been found to be 11.77. Therefore, it is possible to evaluate the grain volume by direct calculation using equation (8) from which it follows that V~ = /3 (h 2 -- ht), Alternatively, if the relation is taken as (h 2 -- ht) = K* Vwthen, the instrument constant K* = 1//3 can be shown to be 0-0850 and using this also the calculation of the grain volume is possible.

38

Y. V. R A M A N A A N D B. V E N K A T A N A R A Y A N A TAat.~ 1, CALtSS~AT~ONDATA

Sample No.

Core type

Volume Yr (crn 3)

l 2 3 4 5 6 7 8 9 10 11 12 13

Without sample Brass cores Brass cores Brass cores Brass cores Aiuminium Aluminium Brass Aluminium Aiuminium Aluminium Aluminium Aluminium

-7 12 16 20 2! 37 41 49 51 53 60 64

OF AIR I~ROSIMF['ER

Porosimeter reading (cm)

Dtfference in hetght of mercury colun~Jas (h, - hi) (cm)

35- 25 35" 82 36" 30 36-60 36- 90 36" 95 38-35 38- 75 39"45 39.60 39" 75 40.35 40- 70

n O" 57 t'05 1-35 1- 65 1- 70 3.10 3.50 4- 20 4- 35 4- 50 5- 10 5- 45

Instrument constant (statistical average): 0.0850 -~ K *

I

. . . . . .

3

't

~ .

IO

Flo. 6. Line sketch o f Tickall's ix)rosirr~tcr.

AN AIR POROSIMETER FOR THE POROSITY OF ROCKS

39

4.4 Compartson ~ ith Tickell's porosimeter For ready reference a sketch ofTickell°s poroslmeter (Fig. 6) is included and in the light of this the features of the present instrument are pointed out m this section. 1. A bigger sample chamber and a proportionate condenser arc used in the present instrument with their associated advantages enabling the study of bigger and therefore better representative rock samples for their porosity. 2. The placement of a spring in the sample chamber avoids any rupture of the glass walls due to even accidental slipping of the rock sample whde keeping it. 3. The provision of a plane glass mirror behind the capillary tube and the use of a magnifying glass avoid any errors due to parallax. 4. The stopcock used at the bottom end ofTickelrs porosimeter is not incorporated in the present apparatus and instead the bottom end is connected to a Polythene tube and linked to a reservoir held at the right side. The accidental slipping of this stopcock in TIckell's instrument meant loss of mercury or contamination of mercury. Also, the operation of two stopcocks unhke one in the present case is necessary in Tickell's apparatus. 5. The mercury reservoir of the present instrument is a substitute for the china dish held at the bottom of Tickell's instrument: or for the dish in the dish arrangement used by BALAKRISHNA and SIIANKAR NARAYANA [6]. 6. The rcmoxal of the stopcock described above (para. 4) and the incorporation of the reservoir described above (para. 5) have made the operation of the instrument much easier, and in fact, the operation is qtucker and much faster. This may be further elaborated as below. 6(i) The transfer of mercury and retilling of mercury after each measurement as rcquircd in the Tickcll's instrument Is avoided in the present arrangement. 60|) The associated air bubble formations, contamination of mercury with each transferring and its cleaning, possible accidcnt:tl spdhng and thcreforc the Io.~s of costly mcrcury are altogether avoldcd in the present arrangement and mercury remains tmtouched by hand. 6(lil) With propcr capping facilities on the mcrcury reservoir and the funnel end of the porosimeter, the present instrument i~ made dust-proof and does not need any attention and could be used at short notice as the mercury can be left in the unit (after tilhng once) unlike T~ckell's instrument. 60v) Errors due to air bubble formation are altogether avoided m the present working care being taken while initially setting tip, and their formation subsequently are ruled ont. 4.5 Adcantages of air poroshneter The advantages of the present porosimeter in relation to the Tickell's instrument have been discussed in the previous section. Tile other advantages are summarized below. I. Unlike the conventional method discussed in Section 3(b), in the present work the sample need not be allo~ed to come into contact ~lth any liquid and therefore avoid the sorptionai errors, or errors due to disintegration of grains, or losses due to chemical action. Further, even mercury, the porosimetcr liquid, does not come in contact with the sample and it does not affect the observations. 2. It is possible to study random-sized, unpolished rock samples for their porosity and therefore avoid errors due to polishing and sample preparation such as the filling of the external pores, smoothening of the surface pores, etc.

40

Y. V. RAMANA AND B. VENKATANARAYANA

3. The sample studted for its porosity is in no way affected in the test and therefore is nondestructive, and can be re-used any number of times for repeating, or subsequent verification of the porosity value. 4. With the aid of suitable modifications, being dtscussed in the next section, it will be possible to use the instrument for the porosity of even loose and weathered rocks, as also the porosity of clays, powders and sand. In summation, it may be observed that some of the advantages mentioned, are in general, the advantages in any form of an air porosimeter. 4.6 Sample preparation and precautions The majority of the rock samples studied in the present investigation are cylindrical cores of the respective rock types, the edges of which were lapped to be plane parallel. Only the samples of wood and limestone used in the surface area tests are of rectangular cross section and for these samples the opposite faces are made plane parallel by proper planing in the case of wood and by lapping all the six sides in the case of the limestone samples. Also, in preparing the rectangular bars much attention was bestowed to keep the bulk volume of the samples constant and this was realized by strictly retaining the size and sample geometry. Figure 7 is a photograph of the samples used in the present work and these include some of the aluminium samples in which pore space was artificially created. It is preferable to heat the rock samples and thereafter cool in a desiccator before they are used for the estimation of their porosity and this would ensure the elimination of the effect o f moisture, if any, in tile rock samples, thereby giving a better picture of the effective porosity of the rock. Also, whenever bulk volume of rock samples is required, it is desirable to determine, in particular the bulk volume of the rock samples due to displacement in mercury, instead of in water, in order to avoid any errors due to the absorption of water. 4.7 Temperature sensitivity In the present porosimeter, air and mercury are the two operating media on which the sensitivity of the instrument is dependent. Both are ideally suited for porosity measurements; air--because its nearly perfect expansibility and low viscosity easily penetrates the pores, interconnected pores and joints and fractures to give an accurate measurement of the porosity; while the advantages of mercury as a manometric liquid are too well known to be described here. However, since air is easily affected by external thermal changes the porosimeter is temperature-sensitive and therefore it is desirable to keep the temperature variations in the room where the porosimeter is being worked to at least within 5°C and this precaution is necessary to have repetitive and reliable results. But if temperature constancy cannot be maintained, temperature corrections will necessarily have to be applied but this problem is not too difficult and can easily be complied with. 4.8 Possiblefurther improvements The working of the porosimeter for further improvements by incorporating certain changes is dealt with in this section. 1. The meniscus level may be read more accurately by reducing the internal diameter of the graduated tube from ! cm to say about 5 ram and still avoid capillarity corrections, but this would lead to the lengthening of the tube and therefore the apparatus might become too long and hence unwieldy.

AN AIR POROSIMETER FOR THE POROSITY OF ROCKS

41

2. Alternatively, the chamber size and proportionately the condenser size could be mcreased; and an increase in chamber size would enable the handling of stdl bigger samples for porosity measurements; but an increase in bulb size would increase the cost of the unit as more mercury will necessarily have to be used. 3. With the aid of an auxiliary sintered glass d~sk placed at the bottom of the specimen chamber, it would be possible to measure the porosity of highly v,eathered rock samples, or loose materials having weak adhes,on between the grains. If porosity of sand or powders is required a container that may fit into the sample chamber may be used, but the calibration should be done with the container in place. 4. The capillary tube itself may be graduated instead of using the present scale arrangement and this is bound to facilitate easy and accurate reading of the level of the mercury meniscus. 5. It should be mentioned here, that the use of a plane mirror with a graduated scale etched on the mirror and the provision of an adjustable magnifying glass on the capillary tube, so as to slide alongside the graduated mirror would be an ideal arrangement to read the meniscus levels accurately. Alternatively a calibrated capdlary tube with a plane mirror at its back and a sliding magnifying lens could also be used to serve the same purpose of reading the mercury meniscus level correctly. Also, the use of a cathetometer may help in reading the meniscus level more accurately. Although these arrangements have not been provided, in the present working, it is proposed to incorporate some of these at an early date. 5. RESULTS AND DISCUSSION

5(a) Standardization with samples in which porosity is artificially intro&wed Before attempting to determine the porosity of rock samples, the working of the porosimeter with samples having a known porosity in them is inveshgated. Several aluminium ZO

i

--

Q

•

Oelermlned

o

Colculoled

o n

)

:\ i

~

~

r

35

Groin volume,

45

crn"

FroG. 8(a). Plot to show the porosMty in slandard samples of alumm~um.

42

Y. V. R A M A N A A N D B. V E N K A T A N A R A Y A N A

0 3568 cm dta hcte

t

b I_

,

3 176 crn

[-

Top a n d bottom view

0 3~6B cm dla ~-.~ -..._ _~.. _

~

/'/

/

I '

E ~)' e'- 1 O,

l t

07G cm J!I

I

Sechon on A B

! It;. 8(b). Sketch to show pore spaces in standard saraplcs.

TABLE 2. COMPARISON OF CALCULATED AND MEASURED POROSITY

No. of holes

Difference in height of Hg column (cm)

Grain vol. (cm ~)

Bulk vol. (cm 3)

0 1 2 3 4 5 6 9 12

3.40 3" 35 3"28 3"26 3" 20 3"17 3"15 3"00 2"87

40"2 39"6 38"8 38"5 37" 8 37"5 37"2 35"5 34"0

40 40 40 40 40 40 40 40 40

Porosity Determined Calculated (%) (%)

0 1"09 3"00 3"68 5" 49 6"40 7-00 11"25 15-00

0 I" 27 2"55 3"80 4- 68 6-35 7"54 11"41 15"25

Hole diameter: 0-3568 cm; sample length: 5.078 cm; sample diam~er: 3 -176 cm

AN AIR POROSIMETER FOR THE POROSITY OF ROCKS

43

samples of cylindrical shape and free from any pot holes were chosen and machined so as to have the same bulk volume in each one of the samples by strictly keeping the diameter and height of the samples to within one thou. Thereafter in each one of them, holes of varying numbers (0, 1, 2 ........ 12) are drilled end-to-end in a lengthwise direction using the same drill bit, and by so doing a known pore space is introduced in each one of the standard samples [see Fig. 8(a)]. The introduced porosity is estimated from the known hole size and the sample height and the number of holes, and the same is estimated using the air porosimeter described here. The data pertinent to the standardtzatton may be seen in Table 2, while a plot of estimated grain volume in the art~ficlal samples vs. porosity may be seen in Fig. 8. In Fig. 8 the results of both the calculated and experimentally determined values are plotted to present a picture of the deviation between the two. The results of Table 2 and Fig. 8 clearly indicate the good working of the present porosimeter and the deviation between estimated and measured values is less than ! per cent. 5(b) Repeatabihty tests and porosity measurements in rocks A number of rock samples from different locations in Andhra Pradesh, South India were collected and prepared for the present work. The majority of the rock samples are of sedimentary origin belonging to the Gondwana and Cuddapah formations. Besides these, some vestcular trap rocks and sills are also chosen. In all, results on eight sandstones, six of Gondwana, t~ o of Cuddapah; six shales of Cuddapah, two red and four grey coloured ; two sill rocks of the amygdaloidal variety; and some four vesicular trap rocks (basalts) from the Deccan plateau are presented in Table 3. Table 3 includes results from each type of area where the rock samples were collected. The results of samples A-I-A-6; B-7, B-8; C-9, C-10; D-I I-D-14; E-15, E-16; F-17, f:-19; and G-21, G-22 are those due to more than one core sample being taken out of a s,ngle large chunk of the collected rock from the indavidual locations. A perusal of the results wdl clearly reveal the reproducibility of the porosity values of A-I-A-6, B-7, B-8 etc. The variation in porosIty values from one core sample to another taken out of the same rock is of the order of I-2 per cent as is revealed by these results. Even the I-2 per cent variation in the porosity of the same rock from different samples may be due to slight differences in the rock introduced naturally or arhficially in its preparation, and need not necessardy be due to experimental errors, and m fact such small variations are reflected by the other parameters hke density and wave velocity also. Thus, these results are a further indication of the good working of the porosimeter and its suitability for the study of rock porosities. Results on a glass sample that has been tested for its failure due to the development o f microfractures under uniaxial load are also included in Table 3. The results on the glass sample clearly indxcate that, in what has been a sample with no porosity in its initml state, a porosity of as much as 5 per cent is introduced in it because of microfracturtzation. The results h~ted in Table 3 present the data on four rock types. Using this data the porosity against bulk density are plotted and this is shown in F~g. 9 and the results indicate a linear relationship between the two parameters, and a decrease in porosity with increasing bulk density may be observed. 5(c) Porosity by conrentional method and effect of saturation In Sectton 3(b)(ii) the conventional method of determining porosity has already been described. The results on ten rock samples obtained by the conventional saturation method are presented m Table 4. Results include the dry wezght and saturated weights. The two

-m 2.66 2.67 2-65 2-62

Cuddapah (Pulivendla) Cuddapah (Pulivendla) Cuddapah(Tadapat~) Cuddagmh(Tadapatn) Cuddapah(TadaPatH) Cuddapah (TadapatH) Cuddapah (Pulivendla) 2.86 Cuddapah (Pulivendla) 2.84 Deccan trap (Vikarabad) 2.55 Deccan trap (Vikarabad) 2"49 Gondwana (JanampeO Gondwana (Janampet)

Red shale Red shale

Grey shale Grey shale Grey shale Grey shale

Sill Sill

Vesicular trap Vesicular trap

Red shaly sandstone Red shaly sandstone

Fractured g l a s s (irregularshape)

D-If D-12 D-13 D-14

E-15 E-16

F-17 F-19

G-21 G-22

H-23 .

.

Cuddapah(Dhone)

C-9 C-10

13-8

.

1-98 1"97

2.44

2"46

Cuddapah(Dhone)

Loose & weathered sandstone & weathered sandstone

2.27 2.30 2"25 2.26 2.30 2.28

13-7

(Pedavegi) (Pedavegi) (Pedavegi) (Pedavegi) (Pedavegi) (Pedavegi)

4

Gondwana Gondwana Gondwana Gondwana Gondwana Gondwana

2

( V,)

Wave velocity

.

.

2"75 2-71

2" 75 2"74

2"94 2-95

2"77 2.78 2-80 2.85

2"75

2"75

2- 72 2" 72 2"73 2" 76 2- 73 2.75

5

.

2-40 2'72

4'85 4" 99

6"37 6"29

4-52 4.49 4-51 4.32

5.24 5.30

3.57 4-06 3-55 3"79 4"09 4"01

6

(g/cm 3) (g/cm 3) (km/sec)

Density Bulk Gram

3.162 3"166

2"526 2" 542

2"544 2"544

2"570 2"562 2" 576 2"562

2" 562 2"568

2"542

2"562

2"532 2"530 2"512 2"522 2"538 2"522

7

(cm)

Diameter

3.662 3'530

1-250 5" 902

6"462 5"000

2"464 2"290 1"900 1 "910

2- 200 1"844

2"362

3"420

6"456 4"834 4"182 3"738 4"700 4"112

8

(cm)

Height

POROSITY I~I R O C K SA.t,fPIJ~ BY AIR PORO61METER

Red sandstone Red sandstone Red sandstone Red sandstone Red sandstone Red sandstone

3

Rock type

3.

A-I A-2 A-3 A-4 A-5 A-6

Sample No.

Formation {location in Andhra Pradesh. S. India)

TABLE

16"02

29.00 28"00

36"50 29" 00

33"00 26-00

14"42 12"58 9- 76 9-41

9" 50 11-00

ll-00

17"00

32.00 24"00 21-00 18"00 24"00 21"00

9

I "30

1"55 1"50

2"72 2" 15

2"65 2"05

1-10 0"92 0- 75 0"70

0" 57 0"70

0"75

1"15

2"15 !"62 1"40 1"23 !"66 i"37

10

Difference in height Bulk of Hg vol. columns (cm ~) (cm)

15-20

18" 30 17"70

32"30 25" 50

31 "20 24"10

12"90 10"80 8" 70 8"20

6- 60 8'20

8-70

13"50

25"40 19-10 16"50 14"40 19-50 16"25

11

{cm ~)

Grain VO[.

5" II

36" 89 36" 78

10"95 12'06

5"76 7"30

I0" 54 11-76 10"20 12"80

28'80 25"50

20"90

20" 58

20"62 20"00 21 "42 20"00 22"61 19"75

12

{%)

Porosity

> Z >

> Z

< l'n

>

Z>

.<

AN AIR POROSIMETER FOR THE POROSITY OF ROCKS

45

oB

:2

ZS--

aE

o

o

I

- -

°°

,

Z 0O

50

.

.

.

.

~

x

Z 50 Bulk density,

i

3 O0

3 50

g / c m "5

F,G. 9. Relation I~t~een porosity and saturation.

saturatton weights presented are those obtained without evacuating the sample and wtth evacuation and the corresponthng two porosity values are also listed. It is evident from these results that a better value of porosity is obtained by evacuating the samples and later saturating them. Sorphon results obtained using the relation S-- (W,--;Vt)lOO wt

(1o)

TABLE 4. POROSITY IN ROCK SAMPLES BY CONVL'NTIONALMETIIOD

Sample No.

Initial wt (dry) (g)

W t after 120 hr in water (g)

Porosity by saturation (o~)

A-I A-2 A-3 A-4 A-5 A-6 B-8 F-19 G-21 G-22

72.8104 55"5812 46"6542 41.6295 54.2822 46-8884 27.3893 73"3726 55"7401 53"5700

76"7290 57"9890 49"1370 43"9830 56.75q0 49-4430 28"0487 74"4462 62"7380 59"9520

12"06 9"51 11"20 12"47 I0 02 11"85 5"44 2"91 22"85 20"54

Rocm: 8/I--o

W t after saturat,on and evacuat,on Sorption (g) (~) 78"1154 59-3030 50"3030 45.0180 57.9930 50"5060 28.6489 76.1128 63.7080 61-0260

7-28 6"69 7"82 8-13 6-83 77"71 4-59 3-73 14"29 13"91

Porosity after evacuation (o~,) 16"55 15"41 17"60 18"39 15-74 17"56 11-22 9"30 28"25 27"44

46

Y. V. RAMANA AND B. VENKATANARAYANA

v, here S is the percentage sorption indicating degree of saturation WI is dry weight of sample I4/2 is saturated weight after evacuation and saturation are also included in Table 4 and the relation between porosity and sorption is indicated in Fig. 10 which is not strictly linear.

oe

20

~

x"

.<

8. =o

x Gondwonafree sonds~one • Gondwanored sandstone o Greysandstone [

1

!

I

Sorphon. % FiG. 10. Effect of dcnsity on porosity.

5(d) Porosity by different methods The results obtained from a study of 10--15 rock samples for their porosity employing (i) conventional saturation method without evacuation; (ii) conventional method with evacuation and saturation; (iii) using the present air porosimeter; and lastly (iv) as calculated by using Davis' empirical relation [equation (4)], described in Section 3(e), wherever it is applicable, are compared in Table S. It may be seen that the theoretical values are somewhat higher than the experimentally observed results. And amongst the experimental values the results due to saturation method without evacuation are the lowest and those obtained due to evacuation and saturation are slightly better. The results obtained employing the present porosimeter are the best experimental values obtained and compare favourably with the theoretical values, although even they are lower by about 2-4 per cent ditk,ring from sample to sample. The low values encountered in saturation methods are possibly due to all the available pore spaces in the rock samples being incompletely filled with water; or due to the water being not retained in the pore spaces because of the larger nature of some o f the pores as may be seen in vesicular traps. Based on these results, it is not difficult to conchd¢ that the porosity values obtained by conventional methods cannot be relied upon completely and results obtained by air porosimeters give better representative values o f porosity that compare favourably with the theoretical calculated values of porosity. (Se) Surface area and porosity The fact whether the porosity of any sample depends on its surface area is of some interest in the study of rock porosity. For this, the research worker will necessarily have to prepare

AN AIR POROSIMETER FOR THE POROSITY OF ROCKS

47

TABLE 5. COMPARISON OF POROSITIES BY D|FFERE'qT METHODS Percentage porosity Conventional After e~acmethod, uation and saturation saturation by without conventional e,,acuation method

Sample No

By air porosimeter

A-I A-2 A-3 A-4 A-5 A-6 G-21 G-22 D-ll

20-62 20.00 21-42 20"00 22"61 23"00 36" 89 36" 78 10.54

D-12

11 "76

--

4'02

D-I 3

10"20

--

5' 57

D- I -t

12" 8 0

E- 15

5" 76

--

4- 70

L-I 6

7" 30

--

2' 02

1-19

12"06

•~

--_

12-06 9" 51 11 "20 12"47 10"02 11 "85 22" 85 20" 54 --

-

16"55 15"41 17-60 18"39 15"74 17"56 28" 25 27" 4.-t 3"92

I

20

23"22 21-40 24"43 23-82 21 "40 22"61 40" 75 41" 35 Formula not apphcablc Formula not applicable Formula not applicable Formula not applicable Formula not applicable |:ormula not applicable Formula not apphcablc

8" 27

-

2 91

3o

Calculated ",alue by D~,VlS' [8] formula

9' 30

o

2

\ \

•

0

0

I0--

3

.,,

I

IO

I

I

20

Surface

30

area,

I

40

I

50

cm 2

FIG. I I Relation between porosity and surface area o f hmestoncs (! and 2 from Shahabad, and 3, 4 and 5 from Huzurnagar) [7]. O, • --Limestones from Shahabad- A P. ~1, ×, ~ Limestones from Huzurnagar - A . P .

48

Y. V. R A M A N A

AND

B. V E N K A T A N A R A Y A N A

samples with the same bulk volume but of different surface areas in order to get any conclusive results. The results of SHANrOtrt NAIa.~YA,~A[7] obtained from a study of limestones between porosity and surface area keeping bulk volume constant (indicated in Fig. 11) are of some interest in the present context. His results reveal an exponentially decreasing porosity value with increasing surface area and the decay factor is more pronounced for samples (1), (2) and (4) as compared to his other samples (3) and (53, which also show the same trend but to a lesser extent. SHANKAZN~AYANA [7] concludes that porosity approaches a constancy with increase in surface area in relation to the bulk volume. Some detailed studies keeping the above factors in view are attempted in the present investigation. Two sets of samples in wood (deal) and limestone keeping the bulk volume the same and with different surface areas are specially prepared and studied using the air porosimeter. The results of this investigation are presented in Table 6. Measurements in each

80

0

n

Q

C~

O

60

. >.

---

40

0

g.

o

Wood

*

Ltmeslone

20

O--Oo0

O0~O

Q

I

75

Surface

area.

cm z

FiG. 12. Porosiw vs surface area results o f present invcstigatton.

case are repeated at least three times to avoid any personal errors; and observations seeking the help of different workers also presented no differing values of the porosity, recorded in this table. The samples chosen present a contrast; the wooden samples exhibiting a fairly high degree of porosity in them--as much as 75 per cent--whereas the limestones exhibit porosity of the order of only 5 per cent. Thus, in the present study it has been possible to investigate the effect of surface area in samples having low as well as high porosity in them. The full particulars of this work are revealed in Table 6 and these include the dimensions, surface area, bulk volume, grain volume and percentage porosity. The disparity in the values of porosity of wood for the different samples is about 1 per cent, while it is slightly more in the ease of limestone samples. The results of porosity against surface area due to the present investigation are plotted in Fig. 12. The results of Table 6 and Fig. 12 do not show any particular dependence of porosity on surface area. The decrease in porosity with increasing surface area as indicated by SllANKARNARAYANA[7] is not indicated in these studies. On the

AN AIR POROSIMETER FOR THE POROSITY OF ROCKS

49

o t h e r h a n d Ftg. 12 shows that p o r o s i t y ts not d e p e n d e n t on the surface a r e a and m any case the decreasing exponential tendency ts not indicated. However, the d a t a when plotted on an e x p a n d e d scale, on the c o n t r a r y , shows a slight increase in p o r o s i t y value with mcrease in surface area a n d from a physical s t a n d p o i n t this ts u n d e r s t a n d a b l e , since the n u m b e r o f e x t r a n e o u s pores a n d the n u m b e r o f interconnected pores that m a y be linked to the ext r a n e o u s pores multiplies with an increase in surface area. TABLE 6. POROSITY IN SAMPLES WITH SAME BULK VOLUME BUT DIFFERING SURFACE ~REA

Dtfference m height of Bulk mercury Gram vol. columns vol. (cnP) (cm) (cm 3)

Length

Breadth

He,ght

(cm)

(cm)

(cm)

Surface area (cm 2)

Wood Wood Wood Wood Wood Wood Wood Wood

2'518 3"422 3"824 5"006 6"824 6"618 10"128 10- 168

2"512 2"524 2"532 2 "592 0"968 2"510 1"286 2"486

2-536 1-918 1"684 l "276 2"518 0 836 1.284 0"650

38"18 40"08 40"76 46 04 52"46 55"18 55"34 67"02

16'05 16"57 16"30 16"86 16"64 15"79 16"71 16"43

0"35 0"35 0"35 0"35 0"30 0"35 0"35 0" 35

4"1 4. I 4-1 4- 1 4"1 4'1 4"1 4" 1

75.07 75"25 74 84 75-68 75 30 74 03 75"46 75 04

Lnnestones l.tmcstone~ Limestones Lmlestones lame,itones Ltmestones Lmaestones

2-548 3-646 3"376 10"116 6"922 7.680 10"208

2"484 2"482 2"540 1"276 2"544 2"496 2"600

2-500 1-714 1.946 1-214 0-982 0.874 0 662

37"82 39" 10 40" 18 53-48 53-82 56"12 70"04

15"82 15"51 16-69 15"67 17"29 16"75 17"57

1"30 I "25 I "35 1"25 I "40 1"35 1"40

15-2 14-8 15"9 14"8 16"5 15.9 16"5

4"80 4-57 4 73 5 55 4"56 5"07 6 08

Sample

Porosity (%)

Average percentage porosity: (a) wood" 75-08; (b) hmc,,tone: 5"05 5(f) Effect of porostty on wave velocity In T a b l e 3 the restdts o f c o m p r e s s t o n a l wave velocities o b t a i n e d using the ultrasonic travel time technique measuring the time o f transit through the sample as described by BAt.AKRtS,NA and RAMANA [9] and e m p l o y i n g i M/cs b a r i u m titanate crystals are included so as to c o m p a r e the physical p r o p e r t i e s and study the effect o f porosity on wave velocity. It may be seen that there is a wide range in velocity values from as low as 2 . 4 0 k m p s in sdty s a n d s t o n e to 6.37 kmps in sd[ rocks. In Fig. 13 the relation between wave velocity and porosity, velocity and bulk density, velocity and grain density are shown graphically. It is clearly seen that p o r o s i t y and bulk density are n o n - h n e a r l y related to wave velocity. T h e velocity value is found to increase exponentially with increasing bulk denstty whereas it e x p o n e n t i a l l y decreases with increasing p o r o s i t y and the two are s o m e w h a t c o m p l e m e n t a r y to each o t h e r a n d these also a c c o u n t for the linear relation between p o r o s i t y and bulk density o b t a i n e d earlier (see Fig. 9). In Fig. 13, the d a t a o f grain density against wave velocity is also plotted which also indicates a n o n - h n e a r and exponential tendency. A l t h o u g h the rocks chosen are ofdlfferent rock types, the grain density values hover a b o u t 2.75 g/era 3 and range between 2-71 and 2-95 g/cm 3 and a c o r r e s p o n d i n g change in the v e l o o t y values m a y be observed. T h e plots o f velocity against bulk density a n d velocity against grain density clearly indtcate the effect o f p o r o s i t y on the rock velocity. T h e results clearly show high

50

Y. V. R A M A N A A N D B. V E N K A T A N A R A Y A N A

x

Bulk density

~

/

r

j

;

• Groin density

60

o

Porosity

2

x

E

X

.["

,

I~0

•

•' OO

r

!

50

3 DO

3.50

3~

45

Oen~lly,

[

5

i

15

q/crn

Z~ Porosity,

0

)

3

%

FIG. 13. Ell'cot of porosity and density on wave velocity.

velocity values in sill rocks and lower values in sedimentaries and vesicular trap rocks. All the measurements are under ordinary room conditions and the effect of grain density on wave velocity does not seem to be appreciable but may be pronounced under different conditions of high pressure or temperature. 6. CONCLUSIONS

On the basis of the results obtained in the present investigation, it follows that (1) The air porosimeter described here is a reliable, easy to operate instrument suitable for the measurement of porosity of rock samples. (2) The instrument gives repetitive results and errors in repeatability do not exceed about l per cent. (3) The porosity measurement by the air porosimeter is rapid and each measurement takes less than 5 rain, and therefore where statistical value of porosity is required, the method is useful and accurate. (4) In comparison with Tickell's instrument, the present air porosimeter often several advantages without sacrificing the accuracy while on the other hand improving it.

AN AIR POROSIMETER

FOR THE POROSITY OF ROCKS

51

(5) The porosity value obtained from the use of the air porosimeter is more reliable and representative than by the conventional saturatton method. (6) The instrument is suitable for the study of porosity of rocks, coals, wood, ceramics and with some modifications for clays, powders, etc. (7) Porosity in the rocks studied is found to be related hnearly w~th bulk density and nonlinearly with sorption and wave velocity. (8) Porosity is either independent of surface area or increases shghtly with increase m surface area. A c k n o w l e d g e m e n t s - - T h e a u t h o r s are sincerely thankful to Dr S. BALAKRISHNA, Head o f Rock M e c h a n , c s Davtsmn for has keen interest a n d e n c o u r a g e m e n t ; to Shri. T. NARAVAN GOV, D for useful d~scuss~ons a n d to S h n . M. V. M. SA'rVANARAVASA RAO, for verification o f certain m e a s u r e m e n t s . T h a n k s are to D r HAR[ NARAIN, Dtreetor, for his permasslon to p u b h s h thas work.

REFERENCES I. RAMANAY. W. Elashc behavlour o f s o m e [ n d m n rocks u n d e r confined pressure. Int. J. Rock Mech. Attn. Scl. 6, 191-201 (1969). 2. TteKEtL F. G. The Examinatmn o f Fragmental Rocks, O . U . P . , L o n d o n (1950). 3. TIeKELL F. G., MECtlEM O. E. a n d McCURDV R. C. Some studies on the porosity a n d permeabdlty o f rock~. Trans. Am. Inst. Attn. metall. Engrs 103,256 0933). 4. FRASER | | . J , T[CKELL F. G., MECtlLM O. E. a n d McCURDV R. C. S o m e studies on the porosity a n d permeabdlty o f rocks. Trans. A m Inst. M m . metall. Engrs 103, 250-260 0933). 5. Huc, Hrs D S a n d C(~}KE C. E. T h e effect of pressure on the r e d u c t m n of pore v o l u m e of c o n s o h d a t c d sandstones. Geoph)mcv 18, 298-309 (1953). 6. BALAKRISIINA S. and SllANKAR NARAYANA G. A simple m e t h o d for m e a s u r i n g porosRy in rocks. Proc. imhan Acad. Sci. AS I, (5) 265-269 (1960). 7. SHANKAR NARAYANA (J. Ela,,hc c o n s t a n t s and porosaty values of h m e s t o n e s J. seient, imL R e s , imha B20, (7) 355-3.56 (1961). 8. D,~vls D H. E',tmlatmg porosaty o f ~ d m l e n t a r y rocks from bulk density. J Geol. 62, 102-107 (1954) 9. ~ALAKRI%HNAS. a n d RAMANA Y. W. Comprc~.'qona[ wave velocRaes m s o m e agneous rocks J. huhan geophys Un. I, (I) 45-56 0964).

APPENDIX In m a n o m e t e r s we have tile relauon (',ee [ ig. 14) P -I'o or pressure tnsMde the clo~ed vessel (P) is gwen by

= pgh

(A 1)

/~ -- 1'o + pgh where Po Is the atmospherJc pressure •". P = Po + P g O " -- )'t)Thas as the prmc~ple in all open-tube m a n o m e t e r s . O n the other h a n d tf Yl > Y2 then v,e have the relauon Po -- P = Pg(Yi -- Y2) OF Po = P -{- Pg (Yt -- Y2) (A 2) Ltke~a,,e, in the aar porosameter, tf Po as the atmospheric pressure, and Pt and P2 are the gas pressures inside the porosameter without a n d wath the sample, then ,.~e ha,,e tl~e basac re[auons from Boyle's law Pt + pght = Po

(A.3)

Pz + pghz = Po.

(A 4)

PI -- P . = Pg (h~ -- hi).

(A.5)

Subtracting (A.3) from (A.4) we get We know from Boyle's law Po Vo = P t

VL = K ;

or PI ~ - -

K

Vx

52

Y. V. R A M A N A A N D B. V E N K A T A N A R A Y A N A

>: i >:, I!

t

I j

t¥.

_L Monometer

FIG. 14. Manometer. and Po(Vo

K" v~

=K';

- - V , ) = P2 I/2

orP= = = 4 =

K--eoV, v=

where VI and Vz are the volumes of air in the two operations made without and with the sample, and V. is the grain volume o f the sample and Yo is the volume of air originally present in the porosimctcr above the mercury after closing the stopcock.

.'.eoVo-eoV,=K" or

K--Po

V,=K'.

Substituting for P= and P= in equation (7) which is the same as (A.5) we have on the left-hand side P t - - p= =

K

K--PoV,

V=

V=

(A.6)

= K ( V . -- V,) + Po V, V=

V~ Vz Vl = V* 4 - a ( x - h , ) But V, = V* + a ( x - h j ) where (V*) is the volume o f air excluding the volume of air locked in the cross-section a of the graduated narrow bore, and x is the total height from the base to the top of the stopper. .'. V~ V2 = V *= + a V * ( x - - h i ) + a V ° ( x - h2) + a = ( x -

h , ) ( x - h=).

We may assume that

V= V== V "= and from V=-

Vi = a (x -- h=) - a (x -- h=) =

a (hi

-- h=).

Substituting these in (A.6) and equating to (A.5) we have pt

_ p = == K a ( h l

--

h=) + Po V, V= == p g ( h = VeZ

--

h=)

(A.7)

AN AIR POROSIMETER FOR THE POROSITY OF ROCKS

53

and on slmphfication we get )"w -

fV .2 pg + Ka)

l"t Po

(h2 - hi).

(A.8)

But

I/*z 9 g + Kal = 3 = a constant.

Vl Po

I

(A.9)

Then V, = 3 (h2 -- hD ~,hich is the same as equation (8)

(A.IO)

or

h2 - h,~ I V~ ! = ~ = K* constant and tlus is the same as relation (8) of Section 3(d) Since V*, p.g. K,a, Vt. Po are all fixed quantities m each operauon, B a m o u n t s to be an instrument constant and therefore V~ against (h2 - hD plot will result m a linear graph enabling the calibration of the instrument.