JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. Bll, PAGES 25,149-25,159, NOVEMBER 10, 1996
Body waves in poroelastic media saturated
by two immiscible fluids
Kagan
Tuncay
•andM. YavuzCorapcioglu
Departmentof Civil Engineering,
TexasA&M University,CollegeStation
Abstract. A studyof bodywavesin elasticporousmediasaturatedby two immiscible
Newtonianfluidsis presented.We analyticallyshowtheexistence
of threecompressionalwaves
andonerotationalwavein an infiniteporousmedium.The firstandsecond
compressional
wavesare analogoustothefastandslowcompressionalwaves
in Biot'stheory.The third
compressionalwave
is associated
with thepressure
differencebetweenthefluid phasesand
dependent
on theslopeof capillarypressure-saturationrelation.
Effectof a second
fluid phase
on thefastandslowwavesis numericallyinvestigatedfor
Massilionsandstone
saturatedby air
andwaterphases.A peakin theattenuationofthefirstandsecondcompressionalwaves
is
observed
at highwatersaturations.Boththefirstandsecondcompressional
wavesexhibita
dropin thephasevelocityin thepresence
of air. The resultsarecomparedwith theexperimental dataavailablein theliterature.Althoughthephasevelocityof thefirstcompressional
and
rotationalwavesare well predictedby the theory,thereis a discrepancy
betweenthe experimental andtheoreticalvaluesof attenuationcoefficients.
The causesof discrepancy
are explained
basedon experimentalobservations
of otherresearchers.
Introduction
Biot'stheoryofwave propagationin linear,elastic,saturated porousmedia has been the basis of many velocity and
attenuationanalyses[Blot, 1956a,b].Biot demonstratedthe
presenceof two compressionaland one rotational wave in a
porousmedium saturatedby a singlefluid phase.The first
compressional
wave (also referredto as fast wave) is analogousto the compressionalwave in elasticmedia. The second
compressional
wave is slowerand highlyattenuated.Because
of its highlydissipativebehavior,thiswaveis very difficult to
observe.The secondcompressional
wavewasfirst observedby
Plozza[1980] (also seeBcroqnan [1980]). However, we must
note that neither of thesewavespropagateas a wave in the
fluid or in the matrix alone, both traveljointly in the matrix
and pore fluid. All waves are dispersedand attenuated.
Attenuation
arises because of the relative movement
of solid
andfluid phases.At low frequencies,attenuationis relatedto
the permeabilityof solidmatrix. Sincelaminar flow assumption does not hold for high frequencies,the theory needs
modifications[Biot, 1956b]. We refer to Corapciogluand
Tmzcay[1996]for an extensivereview of Biot's theory.
In additionto thethreetypesof bodywaves,othertypesof
wavesexistcloseto theinterfaces.Analogousto elasticmedia,
Dcr½sicv•cx[1962] and Jones[1961] showedthe existenceof
surfacewaves in saturatedporous media. They examined
surfacewavesby consideringthe couplingof the transverse
and the fast compressionalwaves. Later Tajuddin [1984]
presenteda study of Rayleigh waves consideringall three
types of body waves. Love waves which appear due to
stratificationof the Earth were investigatedby Deresisvdcz
[1961, 1964, 1965]and Chattopadhyayand Ds[1983].
In contrastto porousmedia saturatedby a singlefluid,
wavepropagationin porousmedia saturatedby multiphase
fluids received limited
attention
from
the researchers.
The
generaltrend is to extend Biot's formulation developedfor
saturatedmedium by replacingmodel parameterswith the
ones modified for the fluid-fluid or fluid-gas mixtures
[Doraszd•o, 1974; Mo•hizula; 1982; Murphy, 1984; PrMs st
a/., 1992].This approachresultsin two compressional
waves
and has been shown to be successful
in predictingthe first
compressionaland rotational wave velocitiesfor practical
purposes.
Brutsasrt[1964]whoextendedBiot'stheoryappears
to be the first one to predict three compressional
waves.The
third compressionalwave arisesdue to presenceof a second
fluid phasein the pores.BrutsasrtandLuthin [1964]provided
experimentaldata which agreeswith the resultsof Brutsasrt's
[1964]theory.The third compressional
wavewasalsopredicted by Garg and Nayœsh[1986] and Santosst al. [1990]. We
should note that in these studies,although same type of
governing equations were employed, the expressionsfor
materialconstantsdiffered.Miksis [1988]developeda model
for waveattenuationin partially saturatedrocksbasedon the
local three-phasephysicsin the pore space.Tuncay[1995]
derivedthe governingequationsand constitutiverelationsof
elasticporousmedia saturatedby two Newtonian fluids by
employingthe volume-averaging
technique.
In this study,startingfrom the governingequations,we
investigatethe types of waves and their characteristicsin
infiniteisotropicporousmediasaturatedby two fluids.We
showthat threecompressionalwavesand onerotational wave
•NowatIzmirInstitute
ofTechnology,
Cankaya,
Izmir,Turkey. exist.All wavesaredispersedandattenuated.First andsecond
Copyright1996by the AmericanGeophysicalUnion.
Paper number 96JB02297.
0148-0227/96/9 6JB-02297 $09.00
compressionalwavesare similar to thosein saturatedmedia.
We showthat the third compressional
waveoccursbecauseof
the existenceof capillarypressure.Becauseof the very low
velocityandhighattenuationofthethirdcompressional
wave,
25,149
25,150
TUNCAY
AND CORAPCIOGLU:
WAVE PROPAGATION
an experimentalobservationdoesnot appearto be possible.
Finally, we compare the resultsof model with experimental
data obtained in terms of phase velocities and attenuation
coefficientsof the first compressionaland rotational waves.
IN POROUS
MEDIA
•, 2A3IflK/l •l ( 1-a s)(A2
+K:);
•,3a3lf•f/l (1- •)( 1-a s)(A2
+K•)
(5)
•2! lal 2 ;
•22
A,=K,•(1- as)[K•s(1
- as)(K2
+A2/•)
Final Set of Governing Equations
(6)
+K2A2A,(1
-•)/•]
We startby introducingthe governingequationsfor elastic
porousmedia saturatedby two immiscibleNewtonian fluids,
for example,water and air. In the derivation,we assumethat
there is no massexchangebetweenthe phasesand the phases
are at rest. Furthermore, the solid phaseis assumedto be
isotropic,experiencing
smalldeformationsand providingall
shearresistanceof the porousmedium. Momentum transfer
terms are expressed in terms of intrinsic and relative
permeabilitiesassumingthe validity of Darcy's law. At very
low saturations(i.e., residualsaturations),the theorymay not
be valid sincenot all phasesof the medium are interconnected.
Furthermore, the theory employedin this study is limited
to low-frequencywave propagation. Let L, ] and •-be the
characteristiclengths of the macroscopicscale, averaging
volume, and pore scale,respectively.The requiredcondition
for the volume averagingis •' << ] << L. In this studywe
assumethat this requirementis satisfied,and furthermore,if
), is the wavelengthof the wave,we assume] < < X. Therefore
we limit the presentstudyto low-frequencywaves.In the case
of two pore fluids, the low-frequencylimit should be lower
than that of a single fluid case sincethe size of averaging
volumewould be larger to includethe presenceof two pore
fluids.In the caseof a singleporefluid, the sizeof the averagingvolumeisdeterminedby theporestructureonly.However,
when two fluids are presentin the pores, the orientation of
fluid phasesis an important factor on the size of averaging
a2•A3
I - K•K•$•(1-S•)(1-a•)[-K:s(1e•)+A2A
,I ;
(7)
•311•13
(8)
where
dP
all•l•$_ft•;
a2l •$1(1_•l)
dS,
A3=A ,(K•A2• +K•K• +K:A2(1- •))
(9)
(10)
+K•(1
-as)(K•(1
- •)+
In (4)-(10),I•r, I•., as,S,, andPeap
arethedrainedbulk
modulusof the solidmatrix, bulk modulusof phasei, volume
fractionofthesolidphase,saturationof thenonwettingphase,
andcapillary
pressure,
respectively.
Peap,
which
isequal
tothe
pressuredifferencebetweenthe nonwettingand wetting
phases,is assumedto be a function of the saturation of the
nonwetting
phase,S• only.Assuming
the validityof Darcy's
law, C• andC2 become
(1-as)2dUl
volume.
(11)
Kkr/
In addition to assumingan isotropicelasticporousmatrix
saturated by two immiscible Newtonian fluids, validity of
Darcy's Law, and low-frequencywave propagation, we
furthermoreassumedthe validity of the relationshipbetween
capillary pressureand a fluid saturation,and negligibilityof
whereK, gi, ka are the intrinsicpermeability,viscosity,and
crosspermeabilitiesknown as the Yuster effect.
relativepermeabilityof phasei, respectively.
Equations(1)
The governingequationsare obtainedin termsof displaceand (3) reduceto Blots [1956a]equationsfor a singlefluid
mentsof solid and fluid phasesas [ Tu•cay, 1995]
•l (1-••)2(1
-•)2la
2
(12)
phaseby settingSl andA2to zero.
(@s;•-=V((z,,+
7 )V'•s+
Zl:V'T+
al•V'uz) (1)CompressionalWaves
The formulationfor the compressional
wavesis obtained
by applyingthe divergenceoperatorto (1)-(3),
_
_
_
\02es ß 2
0•1
Ps;-•-=al
mY
•s+
•l2V2•
l+•13V2•2+
CI
('•-•')
•2 •
(13)
(3)
whereanglebracketsand overbarare thevolumeaverageand
intrinsicaverageoperators,respectively.Subscriptss, l, and
2 correspondto the solid phase, nonwetting phase, and
wettingphase,respectively.
In (1)-(3),Piandui arethedensity
anddisplacement
vectorof phasei, respectively.
Gtr is the
drained
shear
modulus
of thesolidmatrix.Theconstants
aij
are given by
z,,Aa-Ks[A, as(K•A:S•+K•K, +K•A2(1-•))
+Kfi•r,(1-a •)(K•( 1-•)+ •K: +A:)]
\o•eI
O•10• s
(14)
• o•s
(15)
Pl/'•-/'
=/221V2•s
+•22V2•
1+R23V2•2
--Ci
(-•--•-)
\•e2
2
2
P2;-•l•31V
•$+
•32
V•,+•3V2•2
--•(•--•)
whe• %=V.uj andall =all+4Gt/3. Thedflamtional
plane
ha•onic wavespropagatMgalongthez d•ction a• givenby
TUNCAY
AND CORAPCIOGLU:
WAVE PROPAGATION
IN POROUS MEDIA
25,151
where
Bjisthewaveamplitude,
1[isthewavenumber,
toisthe In general,for a givenfrequencyto, the polynomialin (19) has
frequency,and i is the imaginarynumber.In general,1[is a
complexnumber.For convenience
we rewrite(16) as
e/..•
-•. •t•z-•0
(17)
three complex roots and the wave number [, has six roots.
However,onlythreeof theserootsarephysicallypossible,that
is, the amplitudes of waves should decreaseso that the
imaginarypart of [ must be greaterthan zero. This implies
the existenceof three compressional
wavesin a poroelastic
where[i and[r aretheimaginaryandrealpartsof [, respec- medium saturated by two immiseible fluids. When we set
tively. The imaginarypart of the wavenumber[i is usually
A2=0, we find Z4=0 whichindicatesthat one of the comcalledtheattenuationcoefficient.The phasevelocityisdefined
asc=(o/[r Substitution
of (16)in (1)-(3)yields
pressionalwavesis associatedwith the pressuredifference
betweentwo fluid phases,that is, capillarypressure.
RotationalWaves
The formulation for the rotational wavesis obtainedby
applyingthe curl operatorto (1)-(3)
(18)
+i,•-q-q
qq
B•
0
( \a•ci,fool,
acl,]
q -qoq
G
0
(25)
-
whichimpliesthat for nonzerosolutionsthe determinantof
the coefficientmatrix mustbe equalto zero. This equationis
known as the dispersionequation in wave mechanics.The
determinantcan be expressedas
Z•Xa
+•2 +Z3X+
Z4.0
whereflj=Vx•j.•e substitution
ofhamonic
waves
asgiven
by Eq. (16) in Eqs. (24)-(26) yields
(19)
oo
where
X=toe/[
2.Thecomplex
coefficients
of(19)aregiven
by
•2
(27)
(20)
-i
+i,,,-G-G
qq
,
Z2w•
q
-q
C•
0
B•
0
-
The dispersionequationis the determinantof the coefficient
matrix
x•(zlx+z9.o
(21)
(i)2
+2a C, p +2a
and is in the form of
p
(28)
where
(29)
•s)2
(22)
Equation(28)showsthe existenceof a singlerotationalwave
in porousmedium saturatedby two immiseiblefluids.
Numerical
2 2 a,3(2a12az•-a,•a22)
Z4-'ai,(a22a•3-a23)-a,2•3+
(23)
Results
We solve(19) and (28) for Massilion sandstonesaturated
by air and water. The material propertiesof Massilion
25,152
Table
TUNCAY
1. Material
AND CORAPCIOGLU:
Parameters
of Massilion
WAVE PROPAGATION
Sandstone
Saturatedby Air and Water Phases
Parameter
Symbol
Bulkmodulus
ofsolid
matrix,
GPa
Bulkmodulus
ofsolidgrains,
GPa
Shear
modulus
ofsolid
matrix,
GPa
Kfr
Ks
Gfr
Density
of
solid
grains,
kgm
3
Intrinsic
permeability,
m
•
•s
Bulk
modulus
ofwater,
GPa
•2
Density
ofwater,
kg/m3
-
-
2000
-
1900
-
1.02
35.00
1.44
2650.00
0 77
997.00
HZ
Hz
15OO
-
,',
,- 14.00 -
1•oo-
12oo
-
1.10
11oo-
Bulkmodulus
ofair,MPa
K1
0.145
Viscosity
ofair,Pas
I•l
18x10'6
2000
8 18oo-
lxlO-3
P1
-
,•.1700
18oo-
2.25
Density
ofair,kg/m
3
500
........ •OOO
Hz
500
Hz
....
- -
v
is2
Viscosity
ofwater,
Pas
P1
Value
9xlO-13
Volume
fraction
ofsolidphase
2200
2100
IN POROUS MEDIA
o.o
1ooo
After Muz'phy[1982,1984]
o.'•
o.'•
Water
o.'•
o.'•
•.b
soturation
Figure 2. Effectof watersaturationon thephase
velocityof P1.
sandstone
takenfrom Murphy [1982, 1984]are listedin Table
1. V• G•nuchteds [1980] closedfern exp•ssions for the
capflla• p•ssu•-saturation •lations a• mployed to obtain
tion in the phasevelocityof P1 becauseof 1% air is almost
50%.We observea similartrendin the phasevelocityof the
second
compressional
wave,P2(Figure3).Thethirdcompressionalwave,whicharisesbecause
of the pressure
difference
between
thefluidphases,
hasthelowestphase
velocity
(Figure
4). Asexpected,
P3 vanishes
whenonlyonefluidphaseexists.
P3'smaximumvelocityin the frequencyrangeof interestis
lessthan 1 m/swhichmakesit verydifficultto observe,if not
where
P•pisthecapillary
pressure
(N/m2),
S2isthewater impossible.Figure 5 illustratesthe phasevelocityof the
saturation,
St2is the irreducible
watersaturation,
Sin2
is the rotational wave as a function of saturation. We note that the
upperlimit of water saturationin a two-phasemedium,m= ldrop in the phasevelocityis not observedin the rotational
I/n, and •xandn arematerialparameters.Parametersof (31) wave.The phasevelocityof S showslineardependence
on
arechosen
asa=0.025,n=10,andSt2=0.0.
We usetherelative watersaturation.
Thejumpsin P1 andP2 canbeexplained
by
permeabilitycurvesof W.ycœoffand Botset[1936]shownin
theconsiderable
changes
in compressibility
of porefluidswith
Figure 1.
a highlycompressible
phase,that is, air.
Figures2-5 illustrate the effect of water saturationon the
Figures6-9 illustratethefrequency
dependence
of phase
phasevelocities
at four differentfrequencies.
From thispoint velocities
of bodywaves.Thedropin thephasevelocityof P1
on we will representthe compressional
wavesby "P" and can alsobe seenin Figure 6. ComparingFigures6-8, we
rotationalwave by "S." Sincethereare threecompressional observe
thatthephasevelocities
of P2 andP3 approach
zero
waves,we will numberthem accordingto the magnitudeof whenthe frequencyapproacheszero. Thus we canconclude
their phasevelocity,P1 beingthe fastest.We observea sudden that P1 is a waveanalogous
to the compressional
wavein
drop in the phasevelocityof the first compressional
wave elastic
solids.
Figure8shows
thatthemaximum
phase
velocity
becauseof highcompressibility
of air (Figure2). The reduc- of P3 islessthan6 m/s.Therefore,
P1 andP2 correspond
to
dP=•dS•• (9).VanGenuchten
proposed
that
80-
1.O
P2
0.8
o.s
........
....
-
2000 Hz
-
..•3
0.4.-0.2
500
Hz
1000
Hz
15OO Hz
..............
7.:.:.:..-.:'.::.::..
..
-
ø'øo.
%
o.,
Wciter
o.'a
o.a
1.
Figure 1. Relativepermeabilitycurves[after Wyakoffand
Bot•t, 1936].
0
0,
o.'•
saturation
o.•
Water
o.'•
o.',
1.'o
saturation
Figaro3. Effectof watersaturation
onthephase
velocityof P2.
TU]qCAY
AND CORAPCIOGLU:
WAVE PROPAGATION
2300
IN POROUS MEDIA
25,153
-
P1
500
Hz
2100
-
........
........ •000
Hz
500
HZ
2000
S,=.1.0
S,=.0.8
....
HZ
---
S,--0.3
,•, 1900E
._
• 1700 -
(:3
•
1500 -
1 ,.300 -
o
0.12
o.o
o.4
Water
0.18
0.6
1,0
1 lOO
•'o
saturation
, ;•o
';'•'oo
Frequency
..................
1 oooo
1 ooooo
(Hz)
Figure 4. Effect of water saturationon the phase
velocityof P3.
Figure 6. Frequencydependence
of thephasevelocityof P1.
the fast and slow wavesin Biot's theory. Figure 9 demonstratesthe frequencydependenceof the rotational wave. We
observethat as the water saturationincreases,
the frequency
dependence
of phasevelocityof S becomesmore significant.
than that of water which explainswhy P2 is much more
attenuatedin air-saturatedsamplesthan in water-saturated
Figure 10 showsthat the attenuation coefficientof P1 starts
from zero and gradually increasesas the water saturation
increases.We observea suddendrop when the water saturation approachesunity. We shouldnote that viscosityof air is
much smallerthan that of water which explainswhy the first
compressionalwave is so little attenuatedin air-saturated
specimens.
In caseof a moreviscousfluid thanair, oneshould
expecta curvestartingfrom a nonzeroattenuationcoefficient.
we can conclude that P1 is a wave, whereas P2 and P3 are
ones[Nagy ct M., 1990;Nagy, 1993].The attenuation coeffi-
cientof P3isillustratedin Figure12.Ascanbeseenin Figure
Figures 10-13 show the effect of water saturation on the
12, the attenuationcoefficientbecomesvery largewhenthe
attenuationcoefficient,that is, the imaginarypart of wave watersaturationapproaches
zeroor unity,thatis,singlepore
number. We should note that the attenuation coefficient
fluid.Furthermore,
theattenuation
coefficient
ofP3isalways
calculatedis the part due to the contributionof the relative muchlargerthanthatof P1 andP2. Considering
lowvelocity
movementbetweensolid and fluid phases(equations(11)- andhighattenuationof P3, we believethat an experimental
of P3 is not possible.From the abovediscussion,
(12)). Inelasticenergylossesare not taken into consideration. observation
The
attenuation
coefficient
of P2 as a function
of water
saturationisillustratedin Figure 11.Again,we observea peak
associated
with a dissipativetype of behavior.However, it
shouldbe notedthat uponreflectionand transmissionof P1
at an interface,significantamountsof waveenergycouldbe
partitionedinto P3 and dissipatedinto the mediumunder
idealcircumstances.
This wouldalter the amplitudes
of the
observable P1 waves and would not be accounted for in solid
elasticityor evenin Biot'sporoelasticity
theory.
We alsoobserve
thatthepresence
of air reduces
thephase
velocity and increasesthe attenuation coefficientof P2 which
in theattenuationcoefficientaroundS2=0.8.We shouldnote makesan experimentalobservationof P2 alsodifficult. Since
experiment,it iswell knownthat
that the attenuation of the secondcompressionalwave is Plon•'s [1980]first successful
specimens
haveto be vacuum- impregnated
determinedby thekinematicviscosity,that is, theviscosity-to water-saturated
densityratio. The kinematic viscosityof air is much higher
to eliminate the remanent air saturation before the slow
compressional
waveP2appears.
Similarly,air-saturated
speci1 ooo
800
900
-
........
....
500
1000
Hz
Hz
1500
2000
Hz
Hz
-
700-
600
800-
........
S,--1.0
S.-0.8
....
S.-0.5
/
•
•'500
400
700
600
300
-
200
100
500
0.(
o.'•
o.'•
Water
o.'•
[
0.8
saturation
Figure 5. Effect of water saturationon the phase
velocity of S.
1.O
0
' , ........ •r,•r•---,-'•",
,'•,",,"'
I
I
•o
• oo
I
1 ooo
Frequency
I
• oooo
I
• ooooo
i
(Hz)
Figure 7. Frequency
dependence
of thephasevelocityof P2.
25,154
TUNCAY
AND CORAPCIOGLU:
WAVE PROPAGATION
IN POROUS MEDIA
0.04
P3
P1
//ø
........
$.i0.8
....
S mO.5
s:-o.3
::•,.<:
..... •, 0.03
/?
500
1000
1500
2000
........
....
- - -
Hz
Hz
Hz
Hz
/
d
!
• 0.02 -
//
!
!
/,
.c:2-
/
Q_
.•, /
• 0.01 -
...................
'
'
'
,,,,11
,
,
'
,,,,,I
• ooo
•requency
,
1 oooo
,
'
,,1•11
i
-0.2c....•,._-_-.F.::::.".:.:.-'..";
.................
----"'-'""-'--0.4
0.6
0.•l
1.0
I
,
• ooooo
I
Water
(Hz)
Figure 8. Frequencydependence
of the phasevelocityof P3.
Figure 10. Effect of water saturation on the attenuation
coefficient
menshaveto becompletelydriedto removethe residualwater
saturationbeforeP2 appears[Nagy ½tM., 1990].Thereforeas
notedearlier,it seemsto be highlyprobablethat any degreeof
partial saturation eliminates P2. Figure 13 illustrates the
attenuationcoefficientof the rotational wave. Although the
saturation
of P 1.
thesamples.
Sinceweassume
isotropyandhomogeneity
in the
calculations,someof the discrepancies
betweenour model
results and the experimental data can be associatedwith
Domenico's
conclusions.
Anothercommentwasmadeby ¾/n
½tal [1992]who experimentally
examinedthe effectof open
t•endissimilartothatofP1,wedonotobserve
a peakinthe
and dosed boundaries on the attenuation coefficient of P1.
attenuationcoefficient.ComparingFigures10and 13,we find
They found that the attenuationcoefficientdependson
saturationhistoryaswell asdegreeof saturationandboundary flow conditions.¾/n ½tM. [1992]observeda signif'leant
differencein the attenuationof P1 during drainageand
imbibition.Mucl•hy [1982]providedexperimentaldata on
low-frequencywave attenuationsand phase velocitiesin
Massilionsandstone
saturatedbyair andwater.The material
that attenuation
coefficients of P1 and S are in the same order
of magnitude.
ComparisonWith Experimental Data
Although there are severalexperimentalstudiesin the
literature on the phasevelocitiesand attenuationsof body
wavesin elasticporousmediasaturatedby two fluids,mostof
themarelimitedto high-frequency
wavepropagation.Among
the onesdealingwith high-frequencywave propagation,one
can note Domenico [1974, 1976], Gregory [1976], Elliot and
Wiley [1975]. Domenico [1974, 1976] who investigatedthe
effectsof water saturationon reflection,refraction,andphase
velocityof bodywaves,pointedout the difficultiesin conducting experimentswith multiphase fluids. Dom½•co [1974,
1976]demonstratedthat the drainageprocessusedin many
laboratorystudiesresultsin heterogeneous
gasdistributionin
900
properties of Massilion sandstoneare shown in Table 1.
Parameters
of (31)arechosen
as•=0.025,n=10,andSa=0.0.
We usetherelativepermeabilitycurvesof WyckotTa•dBotsct
[1936]shownin Figure 1.
Figure 14 showsvery favorablecomparisons
of model
calculatedand experimentalphasevelocitiesof P1 and S. As
predicted
byourmodel,experimental
datashowajumpin the
phasevelocity of P1 close to full water saturation. After a
significant
reduction,thephasevelocityincreases
asthewater
saturationdecreases.
The phasevelocityof S alsoincreases
as
the water saturation decreases.
I
-
//
Sw--1.0
,•,1200
ß'"
|
........
....
S.-0.3
....
m
850 -
500
Hz
1500 Hz
//
--- 2000Hz / "'
• •oo
/ /
\\
//, "'-'" ',,\\
.... ""
. .. ,,' /.'/'
m
I
•,
800-
•
..........
o
750
10000
Frequency
o.o
o.'•
(Hz)
Figure 9. Frequencydependence
of the phasevelocityof S.
o.g
Water
100000
o.'•
o.'•
1.o
saturation
Figure 11. Effect of water saturation on the attenuation
coefficient
of P2.
TUNCAY
100000
AND CORAPCIOGLU:
WAVE PROPAGATION
-
IN POROUS MEDIA
25,155
2200-
200080000
Experimental
-
18OO
Ii
....
Li,,,
'
60000-
500 Hz
Hz
1000
........
••
•,
15oo •z
--- 2000
.z
Numerical
-
Data
Results
..•
,,
.•.1600-
[li
•,,
-•,1400
I•rst compressional
wave (571--647
Hz)
•. 1200 -
40000-
•x
20000-
0
0.0
z
I
0/2
I
0.4
Water
0.6
0.'•
.•o
1000
800
-
600
-
400
1.'0
0.0
Rotational
: -
wave
-
i
o.'•
0.4
Water
saturation
. r --.
.
(365--3B5
- v
Hz)
- = :
,.....
i
o.'•
0.8
.
_ _
•.'o
saturation
Figure 12. Effect of water saturationon the attenuation Figure 14. Comparisonof predictedandexperimentalphase
coefficient
velocities of P1 and S.
of P3.
As mentionedpreviously,ourmodelconsiders
theattenua- of water saturation for different values of permeability.
tion due to the relative movementof solid and fluid phases. Althoughboth experimentaland theoreticalresultsexhibit a
Inelastic
losses or contact
relaxation
are not taken into
consideration.Thuswe canonly partially compareour results
for the attenuation coefficients. Y/n ½taZ [1992] observedan
intrinsicattenuationat moderatewater saturations,probably
due to the moist rock frame which tends to behave like a
peakin theattenuation,thematchisnot favorable.As seenin
Figure 16, attenuationof P1 at 600 Hz increasesfor larger
permeabilities.Althoughpermeabilityaffectsthe magnitude
of attenuation, the location of the peak of 1/Q does not
change.
We should note that although the experimental and
standard inelastic solid. We inspect a similar trend of
Murphy's[1982]experimentalresults.Thispart of theattenu- theoreticalpeaksin the attenuationof P 1 do not quitematch,
ation wasanalyzedby Winldcr and Nttr [1982],and Murphy Y/n ½ta/.[1992]concludedthat the peak is stronglydependent
½ta/.[1986].As suggested
by Y/n ½ta/.[1992],we subtractthis on the history of saturation. Their experimental results
part of the attenuationto obtain the attenuationdue to the showedthat attenuation peak for imbibition shiftstoward a
higherwater saturation around 97%.
relativemovementof solid and fluid phases.
In Figures 15-16 we use the inversequality factor, I/Q,
instead of the attenuation coefficient. Quality factor is a
Conclusions
measureof the energylost during one period of oscillation.
The inversequality factor is calculatedfrom
We analyzed the characteristics of body waves in
poroelastic porous media saturated by two immiscible
1 2c•i
(32)
Newtonian
fluids.
We
demonstrated
the existence
of an
additional compressionalwave arising becauseof a second
Attenuation of S at a frequencyof 500 Hz can be seenin fluid phasein the pores.The first and secondcompressional
Figure 15.The match is reasonablefor water saturationsless wavesare analogousto the fast and slowcompressionalwaves
than0.90. Figure 16showsthe attenuationof P1 asa function in Biot's theory.We showedthat the third compressionalwave
20
.....
Experimental
Numerical
Data
Results
16O.O&
--
........
•
o.o3
-
e- 0.02
-
500 Hz
1000
Hz
1500
2000
Hz
Hz
,
12-
!
o
,
8!
o
,,
.--
ß
,,
,,
,
.
•ß O.O1 -
--:-::::3--:':::'::':::"'i
............... ,.o
0.00
.............
0.0
Water
o
o,o
of S.
' o.½
--I
o.I•
o.,•
•o
saturation
Comparison of predicted and experimental
attenuationof S (500 Hz).
Figure 13. Effect of water saturationon the attenuation Figure 15.
coefficient
I '
0.2
Water
saturation
25,156
TUNCAY
AND CORAPCIOGLU:
WAVE PROPAGATION
4O
IN POROUS MEDIA
In the microscale,the grainswill be assumedto be linearly
elastic and the fluids
P1
.....
30-
Experiment• [•ato (571--647
K=9 0 10-•
m
10
m
Hz)
K=5.0
K=2.0
are Newtonian.
The coefficients
of
macroscopicconstitutiverelationswill be expressed
in terms
of measurablequantities. Momentum transferterms will be
formulatedin terms of intrinsic and relative permeabilities
assumingthe validity of Darcy's law.
10-•m=
Microscopic Constitutive Relations, Mass and
Momentum Balance Equations
/,
In this studythe compressibleporousmedium consistsof
compressiblesolid grains and two immiscibleviscouscompressiblefluids.Then the constitutiverelationsare givenby
o
10-
....;--'•.. ...................
-;-.,
. _•_• ..........
o
0.2
o.o
;ø'1
Woter
xs- KsV'u
sI+ Os u•+ (Vu)r
i
0.4-
0.6
0.•8
soturotion
Figure 16. Effect of the intrinsicpermeabilityon attenuation
of P 1 (600 Hz).
is associatedwith the pressuredifference between the fluid
phasesand dependent on the slope of capillary pressuresaturation relation. We numerically examined the effect of
water saturation and frequency on the phasevelocitiesand
attenuation coefficientsof body waves for Massilion sandstonesaturatedby air and water phases.Our numericalresults
showedthat presenceof a gasphasesignificantlyreducesthe
phasevelocitiesof the first and secondcompressionalwaves.
The third compressionalwave has the lowest phasevelocity
andhighestattenuationcoefficientwhichmakeit verydifficult
to observe,if not impossible.We also examinedthe effect of
water saturation
V'u•
1
on the attenuation
coefficients.
whereus,xs,Ks, Gs,andI are the displacement,
incremental
stresstensor,bulk modulus,shearmodulusof the solidphase,
andtheunit tensor,respectively.The superscript
T denotesthe
transposeof a tensor.We assumethat both fluid phasesare
Newtonian
with constitutive
relations
37'
Vi
ti---Pi
I+Di[•7•.
+•7v).r
--2•
I] i=1,2 (A2)
wherevi, xi, Pi, and Di are the velocity,incrementalstress
tensor,incrementalporefluid pressure,andshearviscosityof
fluid phasei, respectively.
In (A2) thebulk viscosityof fluids
isassumed
to benegligible.The stateequationsof fluidphases
are assumed to be in the form of
1•-dP/ 1dPi
The attenua-
Kidt
i=1,2
(A3)
@idt
tion coefficientof the first compressionalwave is very small
when the medium is saturatedby air only. It increasesas the
water saturationincreasesand exhibits a peak at a very high
whereKi., @i,andPi arethebulk modulus,massdensity,and
pressureof phasei, respectively.The massbalanceequations
water
of fluid phasesare expressedas
saturation.
The attenuation
of the rotational
wave is
similar exceptfor the presenceof a peak. We comparedthe
resultswith the experimental data available in the literature.
Although the phasevelocity of the first compressionaland
rotational wavesare well predictedby the theory, there is a
discrepancybetweenthe experimentaland theoreticalvalues
of attenuationcoefficients.However, it waspreviouslyshown
that the drainage processused in many laboratory studies
results in heterogeneousgas distribution in the samples
[Domenico, 1974, 1976]. Since we assume isotropy and
homogeneityin the calculations,some of the discrepancies
betweenour model results and the experimentaldata can be
associatedwith Domenico's conclusions.Yin ½t al [1992]
showedthat the peak in the attenuation coefficient of P1 is
stronglydependenton the history of saturation.Their experimentalresultsdemonstratedthat attenuationpeakfor imbibition shifts toward a higher water saturation around 97%.
Thesecommentsmay explain the discrepanciesbetweenthe
modelresultsandexperimentaldata for theattenuationcoeffidents.
Appendix' Derivation of Governing Equations
1d@•=_V.•.
i-1,2
@idt
where ui is the displacementof the fluid phasei from a
referenceposition, that is, incremental displacement.We
continuewith the momentumbalanceequationin terms of
incremental
fluids.
The
macroscale
mass
and momentum
balanceequationsand constitutiverelationswill be obtained
by volumeaveragingthe correspondingmicroscaleequations.
stresses and velocities as
aS'
V•cj-Os•
j-s,1,2
(AS)
We canrewrite(A5) by usingthemassbalanceequations(A4)
as
) j-s,1,2 (A6)
V"Cj
- O(OY5')+V-(oy5.5.
at
Macroscopic Constitutive Relations
Our next stepis to obtain macroscopicconstitutiverelations by averagingthe microscopicrelations over a representativeelementaryvolume [Bear, 1972].Volume averaging
of (A1) yields
In this appendixwe apply the volumeaveragingtechnique
to obtain the governingequationsof wave propagationin a
linearly elasticporousmedium saturatedby two immiscible
Newtonian
(A4)
,
a:
sdV-K, '(ets•
) +l
• . u/ndAI
+
(A?)
r- ,+
4 /-,,2
TUNCAY AND CORAPCIOGLU:
WAVE PROPAGATION
where
IN POROUS MEDIA
25,157
To explore the constitutive relations associatedwith the
volume
changes,
westartbyintroducing
•j
-
u,-•
--
,
1
-11jPj•tr(11j•.)
- Kj(11/V'•.+A11j)
j-s,1,2 (A17)
is a second-order tensor with zero trace. Since there is no mass
exchangebetweenthe phases,the velocityof the interfaceis
equal to the velocity of a point at the interface, that is,
material surface.By employingthe volume averagingrules,
the integralin (A7) can be expressedas
i- 1, 2
Equation(A17) doesnotcontainanyrotationaldeformations.
For an elastic porous medium saturatedby a singlefluid,
Pride et al. [1992] showedthat
(Ps
-P)
V.us - -11s•
(A9)
Pr
(A18)
whereKrrisdefinedasthe"frame"or "drained"
bulkmodulus.
where the superscript( 0 ) refersto the referenceconfiguration. Since the displacementsare assumedto be small, by
Weassume
thatin caseof twofluidsP'-r
isgivenby
definition
•j.Vaj ~ 0. Thenvolume
averaged
constitutive
relationsfor the solid phasecan be expressedas
[
Then, we can rewrite Eq. (A18) as
4 i-l, 2
+(1-•)P:)
V'u• - -11•(P•-$,P,-(1-•)P2)
- ($,P•
(A20)
+O,11,V•+11•7•)
r- 11,V'•
I+
where•s is the intrinsicaveragedincrementalstressof the
We can express the change in volume fraction of the
nonwettingphasefrom (A18) as
solid phase. Similarly, the volume averaged constitutivc
relationsfor fluid phasesare
=
=
0
(A21)
%•i' Ki (11iV'ui+A ai) !
2
and the changein volumefractionof the wettingphaseas
.
0
A112
A((1-•)(1-11))
= (1-•)(1-11)-(1
_•o)(1-11s
)
4..v,t
2.ar)d4
(A12)
(A22)
-
Under the small deformationsassumption,the interfacesof
Sincewe havealreadyassumed
that AS• andAasaresmall,
the phasesare not allowedto experiencelargedeformations.
Thenby usingEqs.(AS) and(A12), andassuming
0u]& >>
VrVU
i, we canwrite
OK;).
J#' 0t
theproduct
ofthese
terms
canbeneglected.
Equations
(A16),
(Al7), and (A20) can be rewritten in the matrix form, and
solutions
for theunknowns,
A11
s, AS•,]Ys,•, and•2 are
(A13) obtainedby invertingthe coefficientmatrix as
A11s
- b,V'•+
b=V.u•+
baV'•
(A23)
We must note that here is a jump in the stressesof the
immiscible fluids at the fluid-fluid
interface because of the
presenceof interfacialtensionand curvatureof the interface.
Assumingsmoothpressurevariationswithin the averaging
-11•P•- allU.u•+a12U.
u!+al•7.u2
volume, we can write
•* - •2'"V•w(•)
(A14)
-(1-11s)(1-•)•2
' a,,V'•+
anV.u'•+
aa3V.•
2 (A27)
alsoknown as capillarypressureis assumedto be a function
ofSk(saturation
ofthenonwetting
phase)
only.Fromnowon
fluid phase1 will be consideredas the nonwettingphaseand
fluidphase2 asthewettingphase.S• is relatedto thevolume
fractionsby
$,.-
1
i- 1,2
(A15)
Then Si+S2=1. Notingthatfluidpressures
weworkwithare
(A25)
(A26)
whereP• andP2 arethcintrinsic
avcraged
pressures.
Pcap,
11i
(A24)
AS, -
where
ai•,bi,andci areconstants
expressed
interms
ofmaterial
parameters(see(4)-(10)).
Our next step is the evaluation of the solid matrix's shear
modulusGw We assume
thatall shearresistance
of theporous
mediumis providedby the solidmatrix only. This uncouples
theshear
deformation
of all phases,
thatis,Kij=Jij=0.
If an
externalshearstress•e isappliedto thematerial,wecanwrite
--D
--D
--D
--
the incrementalpressures,
as a first-orderapproximationwe
can write
• - •22
' dPex'AS,
(A16)
where
•jDisthedeviatoric
stress
ofphase
j. Inother
words,
providedthat changein saturation,AS• is small.
the fluidsare viscousbut the mechanicalshearresponseof the
25,158
TUNCAY
AND
CORAPCIOGLU:
WAVE
porousmedium is provided by the solid matrix only. Fluid
viscosities
will
be taken
into consideration
later when we
discussthe momentumtransfersbetweenthe phases.We can
rewrite the constitutiverelations by introducing (A28) into
(A25) and definitionsof incrementalstresstensorof phases1
and 2 into (A26) and (A27).
PROPAGATION
IN POROUS
MEDIA
Substitutionof the constitutiverelations(equations(A29)(A31)) andtheinteractionterms(equations(A35) and(A36))
in the averagedmomentumbalanceequations(A34) yield(1)(3). Equations(1)-(3) are the governingequationsfor lowfrequencywavepropagationin a poroelasficmediumsaturat-
edbytwoimmiseible
fluidswithunknowns:
u,, u•, andu-2.
References
+
(A30)
('•1)'S1(1•s)•l'(•21V"•S
+•22V'•11
+•23V'V3)I
(A31)
Bear,J., Dynamicsoœtloids
in porousmedia,AmericanElsevier,New
York, 1972.
Berryman,J.G., Confu'raation
of Biot'stheory,Appl Phys.Lett., 37,
382-384, 1980.
Biot, M.A., Theory of propagationof elasticwave in a fluid saturated
poroussolid,I, Low frequencyrange,J Acoust.$oc. Am., 2•,
168-178, 1956a.
Biot, M.A., Theoryof propagationelasticwavesin a fluid saturated
poroussolid,II, Higherfrequencyrange,J. Acoust.Soc.Am., 2&
169-191, 1956b.
Macroscopic Momentum Balance Equations
Brutsaert,W., The propagationof elasticwavesin unconsolidated
unsaturatedgranularmediums,J. Cmophys.Res., 69, 243-257,
Employing the volume averaging theorems,the volume
averageof (A6) is obtainedas
•0i5>
•v.
c<oiS.
vi>)+--•!
O•5.
C
•-u)
.•idA
Ot
,,
1964.
Brutsaert,W., and J.N. Luthin, The velocityof soundin soilsnearthe
surfaceasa functionot'themoisturecontent,J. Cmophys.
Res.,69,
643-652, 1964.
Chattopadhyay,
A., andR.K. De, Lovewavetypesin a porouslayer
(A32)
with irregularinterface,I•t. J. E•g ScZ,21, 1295-1303,1983.
Corapcioglu,
M.Y., andK. Tuncay,Propagation
of wavesin porous
media, in Advatwcsi• PocousMedi•, 3, edited by M.Y.
Corapcioglu,361-440,Elsevier,Amsterdam,1996.
Deresicwiez,H., The effectof boundarieson wave propagationin a
liquid-filledporoussolid,II, Love wavesin a porouslayer,
The •te•al on the left-hand side of (A32) vanishess•
the
Sei•aol. Soc.Am., 51, 51-59, 1961.
velocityof the •tefface is •ual to the velocityof a po•t at
Deresicwiez,
H., The effectof boundarieson wavepropagationin a
•e •tefface, that is, no mms exchangebetweenthe phases.
liquid-filledporoussolid,IV, Surfacewavesin a half-space,
We assume •at the se•nd te•
on the left-hand side can be
S½ismol.
Soc.Am., $• 627-638,1962.
Deresicwiez,
H., The effectof boundarieson wavepropagationin a
neglectedunder •e small derogation assumption.Avenge
liquid-filled
poroussolid,V!, Lovewavesin a doublesurface
layer,
velodtiesanddispla•ments for all phasesa• •lat• by
Bu•. $eismol $oc. Am., 54, 417-423, 1964
Deresiewicz,H., The effectof boundarieson wave propagationin a
liquid-filledporoussolid,IX, Love wavesin a porousinterrod
.5..aaA
j,g3s,1,2
stratum, Bull. Seim•ol. $o½.Am., 55, 919-923, 1965.
Domenico, S.N., Effectsofwater saturation of sandreservoirsencased
S•ce we a• •terest• • the 1ow-f•quencywave propagain shales,Geophy•i½•29, 759-769, 1974.
tion, that is, characteristicleng• of the micros•pic •ale is Domenico,S.N., Effectof brine-gasmixtureon velocityin an unconsolidatedsandreservoir,Geophyd½• 41, 882-894, 1976.
•aller than the wavelength,the displa•ents ap•afing •
Elliott, S.E., and B.F. Wiley, Compressionalvelocitiesof partially
•e •tegr•d • (A33) •n be assum• to be •nstant. Emsaturated,unconsolidatedmuds, Geophy•i½•40, 949-954, 1975.
plo•ent of •e ave•g•g •les aM substitutionof (A33) •
Cmrg,S.K., and A.H. Nayœeh,Compressionalwave propagationin
(A32)•
l•ea•ation of <0j> • •e •sult•g •uations
yield
liquidand/orgassaturatedelasticporousmedia,]. Appl. Phys.,60,
3045-3055, 1986.
Gregory,A.R., Fluid saturationeffectson dynamicelasticproperties
of sedimen•ry rocks, GeophyMc441, 895-921, 1976.
Jones,J.P., Rayleigh wavesin a porous, elastic,saturatedsolid,
Aeou•t. $o½.Am., 33, 959-962, 1961.
Miksis,M.,J., Effectsof con•ct line movementon the dissipationof
wavesin partiallysaturatedrocks,J Geophy•.Re•., 93, 6624-6634,
O)•-•'(xj)+•fxj'ajdA
j• i-s,
1,2(A34)
Momentum
1988.
Transfer
Mochizuld,S., Attenuationin partially saturatedrocks,]. Creophy•.
Re•., 87, 8598-8604, 1982.
Due to the complexity of the pore-scaleproblem, we
approximatetheinteractiontermsby assumingthe validityof
Darey's law. Sincethe theoryis for low-frequencywaves,the
assumptionof laminar flow is a reasonableone.Then,
Murphy, W.F., Effectsof partial water saturationon attenuationin
Massilionsandstone
and Vycor porousglass,J. A½ou•t.$oc.Am.,
71, 1458-1468, 1982.
Murphy, W.F., Acousticmeasures
of partialgassaturationin tight
sandstones,
]. Geophy•.Re•., 89, 11549-11559,1984.
Murphy, W.F., K.W. Winlder, and R.L. Kleinberg,Acousticrelaxation in sedimendaryrocks:Dependenceon grain con•cts and
fluid saturation,Geophy•ic4$1, 757-766,1986.
Nagy,P. B., L. Adler,andP.B.Bonner,Slowwavepropagationin airfilled porousmaterialsand natural rocks,Appl. Phys.Lett.,
(l-a$)•g•!(V,_
V) (A35)
2504-2509, 1990.
(A36)Nagy,
P. B., Slowwavepropagationin air-filledpermeablesolids,
A½ouel.$o½.Am., 93, 3224-3234, 1993.
where K is the intrinsic permeability of the porousmedium
andka is therelativepermeabilityof phasei.
Plona, T.J., Observationof a secondbulk compressional
wave in a
porousmedium at ultrasonicfrequencies,AppI. Phys. Lett.,
259-261, 1980.
TUNCAY
AND CORAPCIOGLU:
WAVE PROPAGATION
Pride,S.R., A.F. Gangi, and F.D. Morgan, Derivingthe equationsof
motion for isotropicmedia, ]. Acoust. Soc. Am., 88, 3278-3290,
1992.
Santos, J.E., J.M. Corbero, and J. Douglas, Static and dynamic
behaviorof a porous solid, ]. Acoust. Soc. Am., 87, 1428-1438,
1990.
Tajuddin,M., Rayleighwavesin a poroelastichalf-space,
So½.Am., 7•,, 682-684, 1984.
Tuncay, K., Wave propagation in singl• and double-porosity
deformableporousmedia saturatedby multiphasefluids, Ph.D.
Dissertation,Tex. A&M Univ., College Station, 1995.
Van Gcnuchten,M. T., A closedform equation for predictingthe
hydraulicconductivityof unsaturated
soils.SoilS•i. So½.Am. ],
44, 892-898, 1980.
IN POROUS MEDIA
25,159
Winkler, K.W., and A. Nur, Seismicattenuation:Effectof porefluids
and frictionalsliding,Geophysics,47, 1-15, 1982.
Wyckoff, R.D., and H.G. Botset, The flow of gas-liquidmixture
throughunconsolidatedsands,Physics,7, 325-345, 1936.
Yin, C.S., M.I. Batzle, and B.J. Smith, Effectsof partial liquid/gas
saturation on extensionalwave attenuation in berea sadstone,
Geopl•ys.R•. L•tt., lP, 1399-1402,1992.
M. Yavuz Corapcioglu,Departmentof Civil Engineering,
Texas
A&M University,CollegeStation,TX 77843-3136.
K. Tuncay,Facultyof Engineering,
Izmir Instituteof Technology,
Gaziosmanpasa
Bulvari,No.16, Cankaya,Izmir, Turkey.
(ReceivedMarch 9, 1995;revisedJuly 1, 1996;
accccptedJuly 11, 1996.)