Prediction of Formation Compaction from LaboratoryCompressibility Data

I KOtWNKLIJKE SHELL EXPLORATIEEN

DIRK TEEUW PRODUKTIE LABORATORIUMRIJSWIJK, THE NETHERLANDS

ABSTRACT

This paper presents a new, simple and inexpensive

laboratory testing procedure for simulating realistic

reservoir compaction behavior both in cemented and

/riable rocks.

Core compaction is measured nowadays by

various types of equipment: oedometer, hydrostatic

and triaxial cells. Each method has its own

shortcomings in the reproduction of subsurface

displacement conditions; interpretation o{ tbe

experimental data is /urtber complicated by the

nonlinear compaction behavior o~ porous tricks.

A theoretical expression is derived wbicb

interrelates uniaxial and hydrostatic compaction,. . ,,. , J:-..: -- - / ~,~e ;m. qif,,

and wbzcb enables l~e preULLLLUU “j .,. .- .”-

reservoir compaction from hydrostatic cell

compaction data. This relation is veri{ied /or various

types of rock by comparative measurements in

oedometer and hydrostatic cells.

Based on these results, a new triaxial cell-type

compaction measurement /or friable rocks has been

developed. In addition, this method p?OUideS G=

independent evaluation of Poisson’s ratio.

INTRODUCTION

In order to predicr the compaction behavior ofreservoirs due to reduction in pore fluid pressure,

it is necessary to know the compaction or

compressibility characteristics of the reservoir

rock. This paper is concerned with the laboratorymeasurement of compaction properties of hard and

friable rock, allowing for the reservoir boundaryconditions.

In a theoretical analysis, Geertsmal has siiowii

that the cOtiipiULIUL1--...:- - ~f ~eSerVOirS depends on

reservoir geometry and that reservoirs which show

large lateral dimensions compared to their (vertical)thickness deform mainly in the vertical plane. For

Original manuscript received in Society of Petroleum Engineersoffice July 30, 1970. Revised manuscript received March 8, 1971.paper (SPE 2973) WaS presented at SPE 45th ‘nua~ ‘all

?vletallurgica$? and petroleumMeeting, held in Houston, oct. 4-7, 1970. c Copyr’ght 1971American Institute of Mining,Engineers, Inc.

lRefermces given at end of PaPer.

This paper will be printed in Transactions volume 251, which

will cover 1971.

SEPTEMBER, 1971 7-C=

such reservoirs disregarding lateral deformations,the actual compaction (z according to the linear

theory of elasticity amounts to

~z = cm A>. . . . . ...”””...

where cm = uniaxial compaction coefficient

Ap = reduction in pore fluid pressure.This uniaxial compaction is furthermore related

to the hydrostatic bulk deformation by the formula

()_A l+V (l_p)cb ,. .,... (2)cm-3 1-,

in which v = Poisson’s ratio of the rock

Cb = rock bulk compressibility

p = cma/c~, the ratio of rock matrix to

rock bulk compressibility.

For such reservoirs showing a measurablecompaction Cma << Cb, and the factor (1 –~) has a

negligible influence on Cm. Otherwise the value of

f3 can be calculated from experimental data on CbJ- 2

afiu “ma”For deteunining cm in the laboratory there are

two possible methods:1. direct measurement by using equipment which

simulates the reservoir boundary condition of zerolateral displacement. There are two ways of doingthis, the familiar oedometer cell test, and a modifiedtriaxial cell test — in which lateral deformation of

the sample is prevented by the appropriate

adjustment of the lateral stress. In addition, this

triaxial method enables an independent evaluationof v.

2. indirect measurement by measuring cb under

hydrostatic load and estimating v. in view of the‘ h-h~vi~r of potous rock,2 Eq. 2,noniiiiear i3iiiStiC “----- --- -

which is derived for linear elastic behavior, has tobe generalized.

This paper reports on an investigation into which

one of these different methods provides the mostrealistic compaction data for the evaluation of

in-situ reservoir compaction. For this purpose, the

experimental techniques are analyzed in detail,

and: as a result, a laboratory testing procedure whichenables the determination of realistic laboratory

compaction data, is recommended.

263

FIELD AND LABORATORYLOADWG CONDITIONS

The frame of a porous reservoir rock is frequently

loaded in the vertical plane by the weight of theoverburden; the horizontal frame

stress IS usually“ fluid

unknown. In the pores a certain. reservoirpressure exists.

If during production of reservoir

fluids the Pore ‘luid pressure ~ere~~re~~effective vertical stress, I.e.,

between overburden pressure and rese~voir fluidpressure, increases and the reservoir IS able to

compact.

Though the frame stress may be represented by acomplicated stress tensor, the internal stress due

to pore fluid pressure is a simple scalar quantity,

i.e., it has the same magnitude in every direction.The studies by Gassmannt3 ~ot’~ ~~~~~~mas

van der fiaap2 resulted In ofand

They demonstrated that theporoelasticity.

compaction behavior depends only on the effective

frame stress, i.e., the difference between external

and internal stresses. Therefore, to simulate

reservoir compaction in a laboratory experiment, It

only requires the application of the stress differenceinstead of the actual stresses.

Thus, experimentally.L: ___ 1,.-,-1 the SWrn.#eS

the Most attractive approacn IS LU W- . ..-

externallY, keeping the pore fluid pressure constant

and atmospheric. . .

Consequently the reservoir /oading cond!tlonsare simulated by (1) reproduction of the orlglnal

effective stress field at atmospheric pore pressure;(2) an increase of the effective external vertical

pressure equal to the pore pressure reduction in theactual reservoir;

and (3) the condition of zefo

lateral deformation, or its equivalent change meffective lateral stress.

TRANSLATION OF HYDROSTATIC INTOUNIAXIAL COMPACTION

In hydrostatic tests the loading conditions are

quite different from those prevailing in the reservoir.. . . r. -1> _...J~-~ +k- hvrlrostaticHence for apphcatlon to rlelu SLUU,==, . . . . . . . . . . . . .

compaction data need to be translated allowing for

these different conditions.For isotropic bodies of linear elasticity (Hooke’s

law) the translation is given by Eq. 2. However,for porous media, the strains generally do not

depend linearly on stress.2,6,7 In general, the

hydrostatic volumetric compaction, e, 1s

exponentially related to stress, ~g,the empirical

data being represented by the relatlons’2

n (3)e=a Ue~”” ”-o”” ““””””

where u and n are empirical constants for a given

rock. This means that the elastic modull of porousmedia are stress-dependent as well. For lsotroPlcrock under hydrostatic 10a~r~g t~~ee~~~~~~o~~

equal in all directions, as

and the rock will remain isotropic in the hydrostatictest. In uniaxial loading the vertical and lateral

stresses are different. As a result the elastic

Moduli also become different and the rock. behaves

mechanically anisotropicunder the applled load,

even though it was isotropic uncle-r the hydrostatic

load, and in the translation this difference must betaken into acCOunt.

In theoretical studies on mechanical beh~vlo.r O:Mindlin,8 DereSleWICZ

granular model packings,and Brandt 1° arrive at elastic parameters which

For consolidateddepend exponentially on stress” . .rocks similar empirical, exponential relatlons

are2 Introducing

given by Geertsma5 and van der Knaap.exponential elasticity into the general theory of

elasticity but assuming that POk SOfl’S ratlO, V, 1S

constant for all loading conditions,we arrive at the

. .relationship (Appendix A)

cz=+(~v)e’”””””””” . . (4)

which relates the uniaxial compaction, (Z,to the

hydrostatic compaction, e,at equal uniaxial and

hydrostatic pressures. This is actually a

generalization of Eq. 2, which also appears to hold

for nonlinear elastic behavior. (Strictly speaking,

this is only true for the condition of constant,

atmospheric POre Press,ure--.lf. ‘h! ‘ore, !~e~s~~~effects are neglected, [1 -~) 1s droPPea lronl ‘“=

equation, which is justifiable as long as ~~ ~> cma).

A similar expression for c~mpresslbllltyand

by Gassmann,3elasticity moduli, as given earner . .was derived for small stress and strain ,lncrements -

Over these increments the elastic coefficients were

considered constant. The present derivation showsthat the relation holds theoretically however large

the stress increments may be.

ANALYSIS OF MEASUR~G METHODS

OEIXWIETER TEST

In the oedometer (Fig. 1) the sample is loaded

vertical direction, and theuniaxiallY in the

corresponding change in height (volume) is

--- .I,rt=cl. The rigid cylinder wall prevents later+,,L&e...-. ---

deformation of the sample. In this test the reservoir

loading conditions are met reasonably well, providedsidewall friction is mi~lmized by the use of thin

samples, so that the uniaxial compaction data can

be applied directly to field studies.The accuracy of the experimental data is influenced

by the shape of the sample in two ways:1. The upper and lower surfaces of the .sarnple

must be flat and perpendicularto the cylindrical

surface, as any deviation will result in setting

effects with too large compaction over the low

pressure range of the test.2. The sample must fit precisely into the

oedometer. otherwise the requirement of zero lateral_——.deformation will not be met.

For clays and loose sands 11- 13 the method

works satisfactorily, with no difficulties being

shaping the samples. Forencountered over

consolidated rock it is very difficult to avoid a gap,

sOCIETY OF PETROLEUM ENGINEERS JOURNAL

264

be it small, between sample and cylinder wall. Ifthere is a gap on loading, the lateral St’es~~~become uncontrollable. In friable rock evengrain cementation can be destroyed:

res.ultmg in

disintegration of the rock. Due to this disturbanceof the sample, interpretation of the results In terms

of formation compaction becomes uncertain.So in spite of the realistic loading conditions,

the Oedometer test is not an attractive method of

taking measurements on friable rock; as for

well-consolidated rock the results depend on the

precise shape of the samPle.

TRIAXIAL TEST WITH ZEROLATERAL DEFORMATION

The triaxial test (Fig. 2) is used in the main todetermine the strength of porous media under

various conditions of stress. 14’5 The vert~cal andlateral stresses can be varied independently.

The sample is enclosed by an elastomer sleeve

which acts as a flexible, impermeable wal! aroundthe rock. Really this is a biaxial test In threedimensions, lateral behavior being assumed

isotropic.l-. I’n.. r ~eservoir loading conditions in theA“ ...LL.

triaxial test, anY lateral deformation of the sample

has to be counteracted by an appropriate lateral

stress. This can be achieved by simultaneouslymonitoring the change in height and pore volume ofthe sample during the experiment and makmg the

necessary adjustments to the lateral pressure

during measurement.The elastomer sleeve fits the sample tightly so

that there is no gap between sample and sleeve;

and as the accuracy of the measurement depends onthe ~hanin~ of the sample, the uPPer and 10wer_.. —=. ..O

surfaces have to be flat and perpendicular to the

I

II--IA

t

,,

FIG. 1 — UNIAXIAL COMPACTION cELL (a-~~.

SEPTEMBER, 1971

. .cylindrical surface. Any devlatlons, as in the

oedometer test, will result in setting effects with

too large compaction over the low pressure range.

In theory (Appendix B) the ratio of lateral to

vertical stresses is

‘H v I/n—=(~v) ‘ “””””””-”

(5)

‘z

where n is the exponent in the exponential

relationship of compaction vs pressure. In reverse,

it is possible to calculate u by establishing uH/uzfrom triaxial pressure data and n from triaxiai or

hydrostatic compaction data. In this way the ~iaxialtest Drovides an independent evaluation Of polsson’s.ratio of the rock.

Even though the triaxial test is laboriou: -aridtime-consuming, its unique experimental conditionsmake it extremely useful, e.g., as a check on the

other experimental methods.

HYDROSTATIC LOADING TEST

The hydrostatic loading test (Fig. 3) has been

widely used to measure the pore compressibility of. ----1 :4-.-J .--L 2,6,7 In this test, the sample isCO1l SUIIUCZLGU .~--.

enclosed by a thin jacket (elastomer or metal fed),

which, after being pressed around the core, fits thesample tightly. Thus no shaping problem ex:sts.

In

addition, the measurement is simple and rapid.The accuracy of the method is determined by the

validity of the translation and thus on the degree of

certainty in Poisson’s ratio. This is shown in Fig.

FIG. 2 — TRIAXIAL COMPACTION CELL.

26S

()1 l+V4, where the value of the translation factor~ ~v

is plotted vs v. Literature data show variation in vfrom 0.15to 0.4, seemingly without any relationship

to rock type. For the most frequent range of v, 0.25

to 0.35, t“he transition facto: ranges frQm. Q, 56 to

V.C,. This means that for most rocks using ann {7

averagevalueof v = 0.30 the error in translation

from hydrostatic into uniaxial compaction will be inthe order of t 10 percent.

If triaxial tests are made on the cores, Poisson’sratios may be determined from the triaxial data asindicated.

RESULTS OF COMPACTION EXPERIMENTS

ON VARIOUS POROUS MEDIA

In order to test the applicability of the variousmeasuring techniques, we made series of compaction

experiments in oedometer and hydrostatic cells on

different porous media. The triaxial cell was only. . .

used in measurements on field cores OrI W-~iC~

oedometer and h~-di~~tat~~ ~caFpcC~QflS had also

been determined.

---n..--.mm ANn UVnRf)sT’~TIc COMPACTIONSur.uum~lnm n.. - . . . . . . . . . .

OF CONSOLIDATED ROCK

For these measurements sandstone samples were

-Burette, 0.1 cm3/6 cm

TABLE 1 - POROSITY AND POISSON’S RATlO OFPOROUS TEST MEDIA

Porosity(percent) Poisson’s Ratio

Gildehaus sandstone 22 0.35Oberkirchen sandstone 19 0.26Kcyckeey de!~mite 18 0.27Kayakoey limestone 15 0.30

Packing of steel kteads 35 0.25

(400/.l)

taken from Gildehaus and Obernkirchen outcrops

(West Germany), which are known to be very

homogeneous and to have uniform properties.

Carbonate samples, a bioclastic limestone and avery-fine crystalline dolomite, were drilled fromcores taken in a Kayakoey well (Turkey). For bothcarbonate rocks vuggy porosity was less than 3percent.

The samples were loaded and unloaded in

successive cycles to check their reproducibility

and elastic behavior (maximum pressure 400 bat =

about 6,000 psi). Deformations oniy appeared to be

eiastic after the fii~~ ~~~ie. ‘~ ‘0 ‘riaxial

measurements were made, the Poisson’s ratio COUldnot be independently evaluated. For all rocks, we

., --- --J-...-.*8.. ~Qdobserved a constant ratio Detwccn UCUW.-...

~.t-ri on .s for pressures higher thannydro~t~ti~ ~Cmr-.-- ---

25 bar (about 350 psi). The magnitude of theseratios ~oints to acceptable Poisson’s ratios varying

from 0~25 to 0.35 (se~ Table I). This is also show;

o

in Fig. 5 for the sandstones and in Fig. 6 for the

carbonate rocks. In the latter figures, oedometer

#Manometer compaction data (dots) are compared to tiaia~iai

O-7U0kg/cm2 compaction (full line) computed from hydrostatic

data using Eq. 1 and the Poisson’s ratio of Table1. The sample compaction at 50 bar (about 700 psi)

1-——————r

mm

5mm

FIG. 3 — HYDROSTATIC COMPACTION CELL.

was taken as zero.

()

1 l+V.—3 l-v

1.0

0.8 -

0.6

0.4 — “— “—-—

0.2 ~

‘ob

0.1 0.2

v

()l+UFIG.4+ — Vs v.

l–v

SOCIETY OF PETROLEUM ENGINEERS JO URXAL266

OEDOMETER AND HYDROSTATIC COMPACTIONSOF AGGREGATES OF STEEL BEADS

The aggregates of steel beads were selected as

a model for elastic packings of loose particies, i.e.,aggregates with zero compressive strength. Thepackings were formed directly in the measuring

equipment according to the technique described byWygal.lb Their initial porosity amounted to 35

percent.In the first loading/unloading cycle (up to 700

bar = about 10,000 psi) deformations were partlyinelastic, porosities decreasing to 33.5 percent. Inthe following cycles the packings behaved almost

elastic, the permanent deformations being 10 percentor less of total deformations. NO triaxialmeasurements were made, so Poisson’s ratio of the

packings could not be evaluated.For these packings a reasonably constant ratio

~~~er%.e~was also between oedometer and

hydrostatic compaction for pressures above 25 bar(about 35o psi), which points to a poisson’s ‘atioof 0.25. In Fig. 7, for comparison, the oedometer~omPacCion~ (dQts) and uniaxial compaction (full

line) computed from hydrostatic data are plotted forthe second and third cycles. The sample compactionat 50 bar (about 700 psi) was taken as zero.

COMPARISON OF COMPACTION DATAMEASURED ON FIELD CORES

We have applied all three measuring methods to

samples of a sandstone formation, which was coredto investigate field compaction behavior. The coresshowed a rather large porosity range; low to medium

porosities were found in hard, well-cemented

sections, and higher porosities in the more friableones.

Uniaxial compaction have been measured by or. .computed from oedometer, hydrostatic and trlaxlal

loading experiments, and.L . . --..1.:-- ‘.,. mm=,-f;m”cne rC5ULLIIIg GU...y -- . . . . .

per ~C~t pre~~~re d~g~~ne is plotted in Fig. 8, as a

function of the in-situ porosity. For the translationof hydrostatic to uniaxial compaction for Porsson’s

ratio the average value of v = 0.30 was taken>

independent of porosity. Poisson’s ratios calculated

from the triaxial tests varied from 0.23 to 0.35, with

0.29 as the average.Fie. 8 shows that for the low-porosity values, as.

in the

c~ I3X10-3

t

2

11

hard, well-cemented rocks, all three methods

Cjddehoussandstone c~ I Obemkirchen sandstone

~om5!!y ~~ ~ porcmty 19 %

3xlo-’p

+&H--J, +4b+--J,FIG. 5 — COMPARISON OF UNIAXIAL COMPACTIONS

DERIVED FROM HYDROSTATIC DATA (CURVES) ANDOBTAINED IN OEDOMETER TESTS (DOTS) FOR

GILDEHAUS AND OBERNKIRCHEN SANDSTONES.

SEPTEMBER. 1971

f~ 1Koyaki3y dolomite q

I

Koyokby Imwstone

-s porosity 18’703xui’

p0r051ty15%3X!(!

I

FIG. 6 — COMPARISON OF UNIAXIAL COMPACTIONSDERIVED FROM HYDROSTATIC DATA (CURVES) ANDOBTAINED IN OEDOMETER TESTS (DOTS) FOR

KAYAKOEY CARBONATE ROCKS.

fz

8 XW-’

7 [Thd Ic.adq cyck

o

4 Y’5

1/o

&

3

F-

2

1

0loo200m Loo=m6m’m

w~ b5

FIG. 7 — COMPARISON OF UNIAXIAL COMPACTIONSDERIVED FROM HYDROSTATIC DATA (CURVES) ANDOBTAINED IN OEDOMETER TESTS (DOTS) FORARTIFICIAL PACKINGS OF STEEL BEADS (DIAMETER

400/L, INITIAL POROSITY 35 PERCENT).

-t~loxlo

In-situporosity,% Vb

FIG. 8 — COMPARISON OF UNIAXIAL COMPACTIONSOBTAINED FOR FRIABLE SANDSTONES BY DIFFER-

ENT MEASURING TECHNIQUES.

267

lead to about the same uniaxial compaction. For thehigher porosities in the friable rock, the triaxial

cell data agree well with those derived from

hydrostatic measurements, but in the oedometermeasurements much larger compaction are obtained.In addition, the samples remained intact inhydrostatic and triaxial measurements, whereas inthe oedometer experiments most of the samples weredestroyed. As the triaxial measurements prove thatthe friable rock can stand uniaxial loading without

destruction, ~he destructioii iii cxx?cmeter tests must

be due to an inadequate fit of the samples in thecell, resulting in clearly incorrect and too largecompaction values.

RECOMMENDED PROCEDURE FOR

LABORATORY COMPACTION MEASUREMENTSON FRIABLE AND WELL-CONSOLIDATED

ROCKS

In view of these findings we recommend the

following laboratory testing procedure:1. hydrostatic compaction measurements on suites

of samples systematically taken from cores of rhereservoir in question;

2. measurements of uniaxial compaction in rhetriaxial loading equipment on a limited amount ofsamples, selected on the basis of the results

obtained in the hydrostatic measurements. Inaddition, the triaxiaI measurements will enableindependent evaluation of Poisson’s ratio for the

,. ------- .rocK In ~-U CSLiU1l,

3. translation of hydrostatic into uniaxialcompaction with the aid of the simple relationshipgiven in Eq. 4, using the average Poisson’s ratio

established from the rriaxial measurements.

CONCLUSIONS

The uniaxial compaction of well-consolidated and

friable rocks can be measured by differenr methods.m, . .-:-1 ..= ● w;rh zero lateral deformation is thelne tria~i~. .-s. .. ..- _

most accurate, as it simulates the reservoir loadingconditions direcrly and has the most favorableexperimental conditions. In addition, triaxial tests

enable independent evaluation of the Poisson’sratio of rocks.

The hydrostatic compaction method provides asimple and rapid technique for routine compaction

measurements on well-consolidated and friable rock.A simple formula is given which allows the

translation of hydrostatic CO~~~CiiOtl data intouniaxial formation compaction.

For the translation of hydrostatic into uniaxialcompaction the Poisson’s ratio of the rock must beknown, For ~ ~;ven ~c~k these may be obtained from

rriaxial tests. For the rocks investigated thePoisson’s ratios varied between 0.25 and 0.35. The

use of an average value of 0.30 results for these---i.. :- . re]arive error of t 10 percent in predictedluL&> s,, - -------- .formation compaction.

Oedometer tesrs are not suitable for uniaxialcompaction measurements on friable cores. This is

due ro an inadequate fit of the samples in the

26S

oedometer cell, which may cause the destruction ofthe rock during the measurement. For well-consolidated rock uniaxial compaction obtained inoedometer tests agree well with those found in thetriaxial tests and those computed from hydrostaticcompaction data.

a=

b=

c1 =

C* =CL =“cm =

Cma =e=

E=

n=

.& =

q=

p=

(=

~z =

~.

(J.

‘e =

NOMENCLATURE

coefficient in expression for e

coefficient in expression for E. . .,- .

coefficient m tneoreticrd express~cn f~~

(&)/(d)

coefficient in theoretical expression for e

rock bulk compressibility

uniaxial compaction coefficient

rock matrix compressibility

hydrostatic compaction (relative change in

bulk volume)

Young’s modulus

exponent in expression for e

reduction in pore fluid pressure

exponent in expression for E

cma/cb, the ratio of rock matrix to rock

bulk compressibility

unidirectional strain

uni axial compaction (relative change inheight)

Poisson’s ratio

normal stress

effecrive normal stress

SUBSCRIPTS

H = horizontal direction

Z = vertical direction

x, y, z = directions in rectangular coordinate system

ACKNOWLEDGMENTS

The author wishes to thank ]. Geemsma, W. vander Knaap and L. J. M. Smits for stimulatingdiscussions; W. Moor and H. F. Lammerts forcarrying our the measurements; and the managementof SheH Research N. V., The Hague, The Netherlandsfor permission to publish this paper.

REFERENCES

1.

2.

3.

4.

.5.

Geertsma, J.: ( ~Problem~ of Rock Mechanics in

?etroieiiin %odw+ifim Engineering”, Ptoc., 1stw. . ..Internatl. Cong. Rock Mech., Lisbon (1966).

van der Knsap, W.: ttNonlinear Behavior Of Elastic

Porous Media”, Trarrs., AIME (1959) Vol. 216, 179-187.

(Cuber die Elastizit~t Por”6per i%dierl’~Gassman, F.: “’Vierteljahrschrift der Naturforschenden Gesellschaft,

Zurich (March, 1951) 1.

Biot, M. A.: , CGeneral ~eory Of Three-Dimensional

Ccn~e IidatiOn>>; ]. A~~L Pbys. (1941) Vol. 12, 155.. .

Geertsma, J.: “The Effect of Fluid Pressure Declineon Volumetric Changes of Porous Rocks”, Trans.,AIME (1957) Vol. 210, 331-430.

sOCIETY OF PETROLEUM ENGINEERS JOURNAL

6.

7.

8.

9,

10.

1L

12.

13.

14.

15.

16.

17.

18.

Hall, Howard N.: “Compressibility of ReservoirRocks”, Trans., AIME (1953) VO1. 198, 309-311.

Carpenter, C. B. and Spencer, G. B.: “Measurements~--.nl~t.=d=d Ofl-Bearingof Compressibility Of GW-------- ---

Sandstones”, RI 3S40, USBM (Oct., 1940).

Mindlin, R. D.: , ,Mechsni~s of Granular Media” !

PTOC., 2nd u. S. Natl. COng. APP1. Mech. P Ann Ar@,

Mich. (1954) 13.

Deresiewicz, H.: “Stress-Strain Relations for a SimpleModel of a Granular Medium”, J. Appl. Mecb. (1958)Vol. 25, 402.

Brandt, H.: ,(A study on the Speed of &.und in Porous

Granular Media”, J. Af@l. Mecb. (195S) Vol. 22, 479.

Bofset: H. G. and Reed, D. W.: “Experiment on

Compressibility of Sand”, ButL, AAPG (i!135j Vol.19, 1053.

Roberts, J. E. and de Souza, J. M.: “The Compressi-bility of Sands”, %oc., ASTM (1958) VOL 58, 1269.

van der Knaap, W. and van der Vlis, A. C.: “On the

Cause of Subsidence in Oil-Producing Areas”, Proc..7th World Pet. Cong., Mexico City (1967) Vol. 3, 85.—.. --A u ..~=i n 1.: The Measurement ofBishop, A. %. MM ,,e,...-., -. a..

Soil Properties in the Triaxial Test, 2nd ed., E. J.Arnold & Son, Ltd., London (1962).

WiIhelmi, Bernhard and Somerton, Wilbur H.: “Simul-taneous Measurement of Pore and Elastic Properties

of Rocks Under Triaxial Stress Conditions”, So c. Pet.Eng. J. (Sept., 1967) 283-294.

Wygal, R. J.: ,( Constmction Of Models that Simulate

Oil Reservoirs”, SOC. Pet. En& J. (Dec. ! 1963) 281-

286.

Holubec, 1.: “Elastic Behavior of Cohesionless Soil’\

]. Soil Mech. (i96Sj VGI. 94, 1215.

Timoshenko, S. and Goodier, J. N.: Theory O/

Elasticity, 2nd ed., McGraw-Hill Book Co., New York

(19.51).

APPENDIX A

RELATION BETWEEN COhf PACTIOFJS OF

ISOTROPIC ELASTIC POROUS MEDIA UNDERUNIAXIAL AND HYDROSTATIC LOADING

NONLINEAR ELASTICITY

As a rule, the elastic bulk deformations of poroussedimentary rock under hydrostatic pressure arefound to depend on stress ,2*6*7 the experimental

data being adequately represented 5*2 by

e= a o n, . . . . . . . . . . . . . (A-1)e

in which a and n are empirical constants for a given

rock and Ue the effective stress. Therefore theelastic bulk moduli are stress dependent as well,

which means that if the stresses in the horizontal

and vertical are different, there is also a differencein horizontal and vertical elastic moduli. Hence therock will behave mechanically anisotropic undertriaxial loading, even if it behaves isotropic underhydrostatic load. Such ani sotropic behavior can bedescribed using 4 instead of 2 elastic parameters,

e.g., Holubec17 distinguishes for cohesionless soildifferent vertical and lateral Young’s moduli andPoisson’s ratios, all dependent on stress.

Mindlin8 and Deresiewiczg have theoretically

SEPTEMBER, 1971

investigated the nonline= elastic behavior of amodel packing of spheres. For a cubic lattice

arrangement, the deformation of the packing is

related to the deformation of the single grains.

Using Hertz’s theorylg on the deformation Ofspheres and defining the elastic parameters as the

derivatives of strain to stress, they arrive at elasticmoduli which are exponentially related to stress bythe expression

. . . . . (A-2)

,. L. L- 6c:-:.=-P Ci ~S determined by them wnlcn cnc coe...b. b...elastic material constants of the separate grains.Thus they arrive at sets of stress-strain equationsin differential form.

Brandt1° investigated a more complicated modelaggregate of spheres. Using Hertz’s theory, hederived for the volumetric compaction under

hydrostatic preSSuie

2/3 (A-3)‘=c2ue’ ”””””””””””

in which the coefficient C2 is determined by the

elastic material constants of the constituting grains,the porosity of the packing and the number ofcontact points per grain. The coefficient C2 cannot

simply be compared to that obtained by integration. , ,: f ----of Eq. A-2 because or rne alfrercuc crrmpcsith. an_d

geometrical arrangement of the grains in the modelpackings. Nevertheless there is clearly similarityin Eqs. A-2 and A-3.

In sedimentary rock the grains are not perfectlyspherical nor are the grains stacked in a regular

geometric pattern. Hence it is impossible to express-~ ●L=. maher;altheir elastic properties in terms u~ .,,= ... . . . . . ..-

constants of the constituting grains. Nevertheless,

the empirical stress-dependent hydrostatic compac-tion (Eq. A-I) very much resembles Brandt’s “

theoretical expression (Eq. A-3), which suggests:hat, in analozv to Eq. A-2, the stress-dependent-.elastic parameters of porous rock may be representedby

~=$=@)% . . . . . . . .. (A-4)

in which b and q are constants for a given rock.

This expression combined with the assumption ofa constant Poisson’s ratio enables the descriptionof anisotropic mechanical behavior uader triaxialload with only two parameters instead of four, assuggested by Holubec. 17

GENERAL STRESS-STRAIN RELATIONSHIPSFOR ELASTIC POROUS BODIES

If we neglect for the time being pore-pressureeffects, the stress-strain relations from the general

theory of elasticity can be written in differentialform,

269

do do dodex=&%&-v-E#”””” (A-5)

x Y z

with analogous expressions for dey and dc=.Under in-situ loading conditions the vertical and

lateral normal stresses will be of different magnitude.We however assume that the lateral normal stresses

g% =Uy= UH will be equal. According to Eq. A-4,

the Young’s moduli in lateral and vertical directionswill be

‘H=b(%)q . . . . . . . . . . .. (A-6a)

‘z=b(oz)q ... . . . . . . ..(A-6bJ

These expressions introduced into Eq. A-5 after

SoKle rearrangement results in the stress-strain

relationships

d~.+ (uH) ‘qtMH- ; (Oz)-qduz

. . . . . . . . . . . . . . . . (A-7a)

dez = ; (uz) ‘qduz ‘q duH- ~ (uH)

. . . . . . . . . . . . . . . . (A-7b)

APPLICATION TO CONDITION OF UNIAXIALSTRESS WITH ZERO LATERAL DISPLACEMENT

Here the boundary condition of Eq. A-7 becomes

&f.f = O, which substituted into Eq. A-7a results in

do )- ~ r% -qdzH l-v [.~

(A-8)

H z“””

Eq. A-8 into Eq. A-7b gives

dcZ = (l+V) (1-2y)

b(l-v) (Uz)‘q dgz . (A-9)

Integration of Eq. A-g over the stress interval O to

crz finally gives

l-q

‘z = &’;) ‘;;!;) (Uz) . . (A-1o)

APPLICATION TOHYDROSTATIC STRESS CONDITIONS

In hydrostatic loading, the normal stresses equal

in all directions are Ue = CTH = Oz; so for theelastic properties: EH = Ez = bueq. For equal stress

increments &H = &z, Eqs. A-7a and A-7b provide

dc=dc =dc = 1-2V ~ -q*

H Zbee

. . . . . . . . . . . . . . . . (A-n)

After integration of Eq. A-11 over the stressinterval O to Ue, we obtain

1-2V l-q6— (A-12)

‘b(l-q)”e ”’’””””

For small strains the three-dimensional (relative)

change in bulk volume, e, may be approximated by

e = 3e, which substituted into Eq. A-12 gives

e.wuel-q . . . . . .( A-13)

RELATION BETWEEN HYDROSTATIC ANDUNIAXIAL COMPACTIONS

Comparing Eqs. A-10 and A-13 for equal pressuresUe in hydrostatic and ~z in uniaxial loadings, we

arrive at

= 1 l+V

‘z~f-v)e’ ”””” ”””” ”(A-14)

which relates the uniaxial compaction (2 to the

hydrostatic voiumetfi~ compaction e.

APPENDIX B

EVALUATION OF POISSON’S RATIOFROM TRIAXIAL TEST DATA

The formulas of Eq. A-1 and A-13 of Appendix Aboth express the total relative change in bulkvolume as a function of hydrostatic stress

ne=a~

e

- W22.).- Gel-q.e- 13(1-q)

It follows that n = 1 -q. Thus, by substitu% intoEq. A-8 – q = n -1, this equation becomes

(UH) ‘-1 doHn-1

= ~ (az) duz . (B-1)

Integration of Eq. B-1 over the stress intervals O to

OH and O to oz gives

(@n =+ (6z)n , . . . . . . (B-2)

and

‘H _ l/n(*V) ,“””””’”” (B-3)

‘z

which predicts for the triaxial test with zero lateraldeformation, the ratio of lateral to vertical stressesas a function of Poisson’s ratio of the rock andthe exponent n.

Alternatively Poisson’s ratio can be determined

from the experimental ratio of ‘JH/0.z and the

exponent n. The exponent n can be obtained fromhydrostatic tests as well as from triaxial tests.

270 SOCIETY OF PETROLEUM ENGINEERS JOURNAL

And as according to Eq. A-14, Cz is a linear function‘H v

of e, the relationship of (z VS az in the triaxial —=— . . ..- . . . . . . . (B-4)test shows the same exponent as that in the ‘z l-Vrelationship of e vs Oe in the hydrostatic test.

The theoretical value of the exponent n for Real vaIues of n for sandstones and carbonates

aggregates of perfect spheres amounts to n = 2/3. vary for given rocks; typical values are quoted in

For Iinear elastic media, n = 1, reducing Eq. B-3 to Refs. 2 and 5.

the well known equation ***

SEPTEMBER, 1971271