!“* mLic
UCRL-JC-
PREPRINT
Transversely Isotropic ElasticityPoroelasticity Arising from Thin
Layers
andIsotroRic
JamesG.BerrymanLawrence Llvemore National Laboratory, LNennore, California 94550
This paper was prepared for submittal to theProceedings of the Third International Conference
on Theoretical and Computational Acoustics,Newark, New Jersey,
July 14-18, 1997
July1997
DISCLAIMER
This document was prepared as an account of work sponsored by an agency ofthe U mted States Government. Neither the Unitad States Government nor theUmversityof Californianor any of their employees, makes any warraty, exprasaor unplied, or ●ssumes any legal liability or responsibility for the ●ccuracy,comphtteness, or usefubes of any Woinnatiw epperatua, product, or processdisclosed. or represents thet its use would not irdrixtge privately ownad rights.Reference herein to any specific commercial product, process, or sm’vice by tradename, &ademark, manufacturer, or otherwise, doas not necessarilyy constitute orimply its endorsement, recommendation or favoring by the United StatesGovernmentor the University of Califomi* The views and opinions of ●uthoraexpressed herein do not necase@y stateor retlectthosaof the UNtad statesGovernment or the Univamity of “CaMrnia, andshallnotbeuee dforadvardsingor product endorsement ptupee.
-$ ‘. .
..
‘Bansversely Isotropic Elasticity and Poroekwticity
Arising from Thin Isotropic Layers
James G. BenymanLawrence Ljverznore National Laboratory
P. O. Box 808 L-202Livermore, CA 94551-9900
*
..
Abstract
Since the classic work of Postma [1955} and Backus [1962], much has been learned about el~ticconstants in vertical transversely isotropic (VTI) media when the anisotropy is due to fine la,yer-
ing of isot ropic elastic materials. However, new results are still being discovered. For example,
the P-wave anisotropy parameter cll/c= lies in the range ~ < C1l/CW < (~ + 2P) (1/(~ + 2P)),
when the layers are themselves composed of isotropic elastic materials with Lam6 constants ~
and p and the vertical average of the layers is symbolized by (“). The lower bound corrects aresult of Post ma. For porous layers, a connected solid frame forms the basis of the elastic be-havior of a poroelastic medium in the presence of confining forces, while connected pores permita percolating fluid (if present) to influence the mechanical response of the system from wit hln.
For isotropic and anisotropic poroelastic media, we establish general formulas for the behavior
of transversely isotropic poroelasticity arising from laminations of isotropic components. TheBackus averaging method is shown to provide elementary means of constructing general for-mulas. The results for confined fluids are then compared with the more general Gassmann[1951] formulas that must be satisfied by any anisotropic poroelastic medium and found to bein complete agreement. Such results are important for applications to oil exploration usingAVO (amplitude versus offset) since the presence or absence of a fluid component, as well asthe nature of the fluid, is the critical issue and the ways in which the fluid influences seismicreflection data still need to be
1 Introduction
Two primary goals of seismic
better understood.
reflection data processing are: (1) to image geologic structureand (2) to provide information about lithology for interpretation, The process used to achievethe second goaJ is made complex by the fact that the same seismic velocity may result fromseveral different combinations/mixtures of materials in the earth. The resulting questions ofuniqueness make it necessary to explore the possible range of seismic velocities that can occurwithin the set of circumst antes deemed mostly likely to occur in the earth at the site of interest.
Fine horizontal layering (i.e., layers with thickness small compared to the wavelength of theseismic wave) is known to cause vertical transverse isotropy (VTI) - wherein wave speeds varywith angle in such media, but are uniquely determined by the angle from the vertical. Efforts inthis area are represented in the literature by work of Postma [1955], Backus [1962], Berryman[1979], Schoenberg and Muir [1987], Anderson [1989], and many others. There has continuedto be some doubt about the range of anisotropy parameters possible in such media. Here I willcorrect an error of Postma [1955] and show that the P-wave anisotropy parameter Cll /CSScanbe a factor of 2 smaller than previously supposed. I also obtain a simple upper bound on thisparameter in terms of layer elastic parameters.
Then, in order to explore the area of most interesting applications of such results, weconsider percolation phenomena in fluid-saturatd porous media, where two distinct sets ofpercolating continua intertwine. A connected solid frame forms the basis of the elastic behaviorof a poroelastic medium in the presence of external confining forces, while connected porespermit a percolating fluid (if present) to influence the mechanical response of the system fromwit kin.
There is a great deal of current interest in the anisotropy of Earth materials, and especially
1
.
so when there is fluid present in pores and fractures in the Earth. Fluids of economic interestto the oil industry are typically oil, gas, and water, while fluids of interest in environmentalremediat ion applications are generally the same. Erwiroxunentdconcerns often center aroundfluid contaminants which may be in the form of oil or g% or “could be other undesirableorganic materials in ground water. Brines or steam may be used to flush other fluids out of theground, whether for economic purposes or for environmental cleanup. Thus, it is important tounderstand the role of pore fluids in determining effective constants of such materials, and thefine layering or laminate model of earth materials plays a significant role in the analysis.
In this work, I study some simple means of estimating the effects of fluids on elastic con-st ants and in particular we will derive formulas for anisot ropic poroelastic constants using astraightforward generalization of the method of Backus [1962] for determining the effective con-stants of a laminated elastic material. There has been some prior work in this area by Norris[1993], Gurevich and Lopatnikov [1995], among others. One distinction between these earlierapproaches and mine arises from the desire to understand the transition from elastic ardysisto poroelastic whereas the earlier work in this area has started with poroelasticity as given andthen applied a genertization of Backus’ approach to the lamination analysis. Fhmlly, I wantto mention that methods similar to the ones to be presented here could as easily be appliedin the same context to the problem of determining percolation for fluid flow or effective fluidpermeability (Darcy’s constant) and that would be of some interest in these applications aswell.
2 Wave Propagation in Anisotropic Elastic Media
First, I will introduce the notation needed in the later analysis.h tensor notation, the relationship between components of stress ~ij and strain uk,~is given
by
where c~jkl is the adiabatic stiffness tensor, and repeated indices on the right h~d side Of (3)are summed. In (1), Uk is the kth Cartesian component of the displacement vector u, and%k,l= /luk/&cf. Whereas for an isotropic elastic medium the stiffness tensor has the form
depending on only two parameters (the Lam! constants, Aand p), this tensor can have up to 21independent constants for general aaisotropic elastic media. The stiffness tensor has pairwisesymmetry in its indices such that Cijkl = Cjikl and Cijkl = Cijlk) which will be used later to
simplify the resulting equations.The general equation of motion for wave propagation through an anisotropic elastic medium
is given by
@ii = @ij,j = Cijkfuk,lj Y (3)
where iii is the second time derivative of the ith Cartesian component of the displacementvector u and p is the density (assumed constaat ). Equation (3) is a statement that the productof mass times acceleration of a particle is determined by the internal stress force aijJ.
2
.
.
A commonly used simplification of the notation for elastic analysis is given by introducing
the strain tensor, where
f?aj = *(”i,j + ‘j,i) = *(%+3
(4)
Then, using one version of the Voigt convention, in which the pairwise symmetries of the stiffnesstensor indices are used to reduce the number of indices from 4 to 2 using the rules 11 ~ 1,22a2,33- 3,230r32~4,130r 31~5, and120r21x6,1 have
c11 C12 C13
C12 C22 C23
C13 Q3 Cm
2C442CM
2c~
(5)
Although the Voigt convention introduces no restrictions on the stiffness tensor, I have chosento limit discussion to the form in (5), Which is not completely general. Of the 36 coefficients(of which 21 are generally independent), I choose to treat only those cases for which the 12coefficients shown (of Which nine are generally independent ) are nonzero. This form includesall orthorhombic, cubic, bexagon~, ad isotropic systems, Whideexcluding tricli~c, monocli~c,trigonal, and some tetragonal systems, since each of the latter contains additional off-diagonalconstants that may be nonzero. Nevertheless, I restrict the discussion to (5) or to the stillsimpler case of transversely isotropic (TI) materials,
For TI materia.k, c1l = czz s a, C12= b, c13 = c23 - ~, cm ~ C, CM = CM s 1, and css = m.
There is also one further constraint on the constants that a = b+ 2m, following from rotational
symmetry in the z1x2-plane. In such materials, (5) may be replaced by
(711022033
U23(731
abfbafffc
2121
2m
= 9 (6)
in which the matrix has the same symmetry as hexagonal systems and of which isotropic
ellezzeaaezseal
ela
symmetry is a special case (having a = c = ~ + 2P, b = f = A, and Z= m = p).Recall that the equation of motion may be written as
Piii = cijkl~k,lj. (7)
After Fourier transforming in both space and time [i.e., U(X,t) = uexp i(k” x - wt), where kis the wavevector and w is the angular frequency], I find
(@J26ak_ ~jk~kjki)uk = 0, (8)
which providea three equations for the components of displacement ul, w, Wj. These equationscan be solved if and only if the determinant of the coefficients vanishes, which implies
det(~2t$il - Cijkikjkl) = 0. (9)
3
?
The left hand side of this equality would be a perfect square for all values of k? and k: iff+l= O, while the right hand side would be a perfect square if the quantity defined asA=(a-f)(c -1)-( f+l)’ = O. The first case with f + Z = O will virtually never happen
because both f and 1 are normally positive quantities. The second case with A = O can occur
for some types of anisotropic media, but I show later that this cannot occur for finely layeredmed~a. Nevertheless, if A = O, then the dispersion relations of (14) reduce to
(17)
for UP = u+ and
PW:V= Kkf+ k;) = lk2 (18)
for ~.. = U-, showing that the P-wave surface for velocity squared is elliptical if a # c, whilethe SV-wave surface for velocity squared is circular and therefore isotropic. The dispersionrelation (15) shows that the SH-wave surface for velocity squared is always an ellipse as longas / + m. I call A the anellipticity parameter because, if A # O, then the dispersion relationsfor qP- and qSV-waves are anelliptical (i.e., something other than elliptical) in shape.
Phase velocities are obtained as a function of angle from these expressions by first definingthe wavevector angle 6 such that
k = k(sin O&l+ cos (kia). (19)
Then, the phase velocity vector for each type of wave is given in general by
Vph = vPk(sin 651 + cos 053),
where vph z Uik.The group velocity is defined by
(20)
(21)
where the group angle @is determined by
thdjak~hrl~= —.
a~/ak3(22)
One other angle is particularly important, since it is the one that is most easily measured,and that is the angle of particle motion @for a wave passing a particular point in space. Theparticle motion is given by the displacement vector u, so
u = u(sin @l + cos +23), (23)
where
tan@= ul/u3. (24)
5
— .... ..—.— ——w~
.
#
..
3 Averaging Thin Layers for Low Frequency Behavior
Backus [1962] presents an elegant method of producing the effective constants for a thin layeredmedium composed of eit her iostropic or anisot ropic elastic layers. For simplicity, I assume thatthe layers are isotropic, in which case equation (5) becomes
(711C722033
U23
U31
012
(25)
The key idea presented by Backus is that these equations can be rearranged into a form whererapidly varying coefficients multiply slowly varying stresses or strains. By doing so, I arrive atthe following equation
elle22033
U23
(731
e12
which can be averaged essentially by inspection. Equation (26) can be viewed as a Legendretransform of the or&inal equation, to a different set of dependent/independent variables in
which both vectors have components with mixed physical significance, some being stresses and
some being strains. Otherwise these equations are completely equivalent to the original ones.
Performing the layer average,find, using the notation of (6),
< all >
<022>
-<e=>< eza >< e31 >< U12> 1=
(%’%)(r+2A
+2/4
()A
m
while assuming the variation is along the z or 23 direction, I
(*) (k)(W) (XM(&) -(dz)
- f2/c b - fzjc f/c; - f2/c a - f2/c f/c \
f/c ffc -l/c I
which can then be solved to yield the expressions
“=(&Y(&)-y
1/211/21
2m
+4(-)
ell ‘ezz033
Q23
C31
e12
ell ‘
e22
t733
U23
U31
e12,
, (27)
(28)
6
●
✎
b=(+y(+y+,(~)>
() 1 –1c—
= A+2p
‘=(&)(&)-’()
-11= :
P
(29)
(30)
(31)
(32)
and
m=(P). (33)
One very important fact that is known about these equations is that they reduce to isotropicresults with a = c, b = f, and 1 = m if the shear modulus is a constant, regardless of thebehavior of A.
4 Review of Known Inequalities for the Elastic Constants
Since the stress-strain relation (6) is derivable from an energy functional, it is not hard to showthat the matrix must be nonnegative or the material will be mechanically unstable. Nonneg-ativity of the matrice implies that all its principal minors must be nonnegative, which in turnimplies the following inequalities:
a= b+2m>0, c~o,
1>o, m>O,
and
(az - b2)/4m = b + m ~ O, ac-f2~0,
and
[a(a.-2f2)-b(bc- 2f2)]/4m=(b+m)c-f2 >00
(34)
(35)
(36)
(37)
The second inequtity in (36) follows from (37), (35), and the second inequality in (34) andis therefore often omitted from such listings. Similarly, the inequality for a follows from thosefor m and b + m. All of these inequalities must be satisfied regardless of the source of theanisotropy.
The formulas (28)-(33) can be used to derive some very simple relations among the constants.For example,
C>f (38)
7
..
follows directly from (30) and (31), simply noting that A/(A + 2P) < 1 in every layer. Theinequality
is derived directly from the fact that
(39)
(40)
which follows from the fact that the bulk modulus must be nonnegative in each layer so thatA + 2p/3 >0 everywhere. Then, the two shear moduli must satisfy-
l<na
since
follows easily from the well-known Cauchy-Schwartz inequality (CY@2CY= ~~lz ~d @ = l/#112. Equality applies in the Cauchy-Schwartz
(41)
(42)
~ (a2) (@2) by setting
inequalities only when(1 = const x ~, which implies in the present circumstances that p must be constant for / = m.But this is precisely the condition mentioned earlier for the layer equations to be isotropic, soI exclude this case from consideration.
Another inequality can be derived from the formulas obtained for finely layered media. Ishowed earlier that the anellipticity parameter given by
Aa(a-/)(c- 2)-( f+/)2 (43)
has the property that the dispersion relations for both qP- and qSV-waves are simple ellipseswhen A = O and are anelliptic otherwise. Using the results (28)-(33), A is shown to satisfy
A >0 for any fine layered transversely isotropic medium by noting that
(44)
The inequaMy follows again from the Cauchy-Schwartz inequality in another form somewhatmore transparent for the present application (a)2 = ((pa)lfz(a,p)llz } s (P@(~/P). Equ~tyagain applies only when p is identically constant. But, it was mentioned earlier that finely
layered media are isotropic if p does not vary, so A >0 holds for idl such finely layered mediaif their overall constants are anisotropic.
5 Range of the Anisotropy Parameters
I!rom (15), we know that the SH-wave in finely layered VTI media has an elliptical surface forvelocity squared. Furthermore,
m//= (/40 21 (45)
8
.
follows from (32), (33), and (42). So the horizontal shear wave velocity for SH-waves is alwaysgreater than or equal to the vertical velocity. I choose to define the ratio m/1 as the SH-waveanisot ropy parameter, and have the simple universal result that t hls parameter is always greaterthan or equal to unity.
The qP- wave does not always have an elliptical dispersion relation, but it is neverthelessalways true that if k3 = Othen P: = ak~ and if kl = Othen pw$ = ck~. Thus, I may define theP-wave anisotropy parameter to be a/c and seek to determine what the range of this parametermight be. Formula (28) for a may be rewritten as
. ( ~+@)+(&)k$J-’= (A+ 2p)2 - ~2a (46)
which can be rearranged into the convenient and illuminating form
a=(A+2p)- KH*)-(+J](+J1. (47)
This formula is very instructive because the term in square brackets is again in Cauchy-Schwartzform, so this factor is nonnegative. Furthermore, the magnitude of this term depends principallyon the fluctuations in the A Lam4 constant, largely independent of p. Clearly, if ~ = constant,then this factor vanishes identically, regardless of the behavior of p. Large fluctuations in Awill tend to make this term large. If in addition I consider the combination
:-1= [(~+2P&)-1]-[(*)(& )-(*y] (48)
the first bracket on the right hand side is again in Cauchy-Schwartz form showing that it alwaysmakes a positive contribution unless A + 2p = constant, in which case it vanishes. Similarly,the second term always makes a negative contribution unless A = constant, in which case itvanishes.
If the finely layered medium is composed of only two distinct types of isotropic elasticmaterials and they appear in the layering sequence with equal spatial frequency, then I findthat
(49)
This result agrees with Postma [1955] except for an obvious typographical error in the denom-inator of his pubiished formula. This formula shows clearly that if PI = p2 then the P-waveanisotropy parameter is identically equal to unity as expected. Also, if PI # p2 but Al = A2,then (49) implies a/c ~ 1, as we inferred from (48).
Now, I use this formuIa to deduce the smallest possible value of the right hand side of (49).The shear moduli must not be equal (for anisotropy), so without loss of generality I supposethat p2 > pl. Then, the numerator is seen to become negative by taking AZtowards negativevalues and Al ~ +00. The smallest value A2can take is determined by the bulk modulus bound
~2 + #p2 >0. So we may set AQ= - ~P2 in both the numerator and denominator. This choice-
9
*
.
.
ako makes the factor A2+ 2p2 = $PZ as small as possible in the denominator, thus helping tomagnify the effect of the negative numerator as much as possible. The result so far is that
:-’= :(P2LP1) (-A’A-R1-P1)(50)
The parameter Al may vary from – ~pl to plus infinity. At Al = – ~Pl, the second expressionin parentheses is positive, But, this expression is also a monotonically decreasing function of Aland approaches – 1 as Al ~ +00. Thus, the smallest value of the P-wave anisotropy parameteris given by
(51)
This result differs by a factor of 2 from the corresponding result of Postma [1955], the earlierresult being obtained improperly by allowing three of the four elastic constants to vanish andaJso using a physically motivated but unnecessary restriction that both Al and A2 must benonnegative. If we had used the nonnegativity constraint on the ~’s, my result would havechanged to
a1 IJ2-P1>l-=- —
c 2p2 – i’
which is(52) and
the same inequuzity as that found by Postma, butwas in fact improperly obtained.
(52)
his equality differed from that in
As a find point about the formula (47), note that it implies in general that
a<(A+ 2p), (53)
so I have a general upper bound on the P-wave anisotropy parameter stating that
Before concluding this section, note one further identityeter. The general formula can be rearranged to give
(54)
for the P-wave anisotropy pwam-
:-1=4[(-)(&) -(*)(I&)I o (55)
This formula is not in Cauchy-Schwartz form, but is nevertheless probably the simplest formof the result for this anisotropy parameter. In particular, it is easy to see from this form thatif either p = constant or A+ p = constant, then the right hand side vanishes identically. The
first result is well-known and the second haa been known since Postma’s [1955] work to betrue for two-constituent layered media [also see (49)]. The present result generalizes Postma’sobservation in this case.
10
,
.
b
s
6 Significance of Fluctuations
Some further results of the type presented in the preceding section have been found recentlyby Anno [1997] and by Berryman, Grechka, and Berge [1997]. In all cases, it is found thatfluctuations in the Lam6 parameter ~ play a key role in the analysis. In some sense this isinevitable for anisot ropy due to iayering because there are only two independent parameters,J and p, and the first result is that the anisotropy cannot exist if the shear modulus does notfluctuate. Thus, fluctuations in p may be assumed from the outset, and the only question tobe addressed is how fluctuations in J tiect the results.
It is very important to recognize however that poroelastic analysis shows the mechanicaleffect of fluids is negligible on the shear modulus p but not negligible overall and therefore mustbe contained entirely in changes in Lam6 A. This fact provides the motivation to study effectsof layering in poroelastic media cent aining fluids, which is the subject of the following sections.
7 Porous Elastic Materials Containing Fluids
Now I want to broaden my scope and consider materials composing the laminate are not homegeneous isotropic elastic materiak, but rather elastic materials containing voids or pores. Thepores may be either air-filled, or alternativelythey may be partially or fully saturated with aliquid, a gas, or a fluid mixture. For simplicity, I suppose here that the pores are either air-filledor they are fully saturated with some other homogeneous fluid. When the porous layers areair-filled, it is generally adequate to assume that the analysis of the preceding section holds, butwith the new interpretation that the Lam6 parameters are those for the porous elastic mediumin the absence of saturating fluids. The resulting effective constants ~~? and p~~ are then saidto be those for “dry” — or somewhat more accurately “drained” — conditions. These con-st ants are also sometimes called the ‘frame” constants, to distinguish them from the constantsassociated with the solid materials composing the frame, which are often called the ‘grain” or“mineral” constants.
One simplification that arises immediately is due to the fact that the presence of pore fluidshas no mechanical effect on the shear moduli, so pdr = p. There may be other effects on the
shear moduli due to the presence of pore fluids, such as softening of cementing materials orexpansion of interstitial clays, which I call !chemical” effects to distinguish them from the purelymechanical effects to be considered here. We neglect all such chemical effects in the followinganalysis. This means that the lamination analysis for the effective shear moduli (since it isuncoupled from the amdysis involving A) does not change in the presence of fluids. Thus,equations (32) and (33) continue to apply for the poroelastic problem, and we can thereforesimplify our system of equations in order to focus on the parts of the analysis that do changeiri the presence of fluids.
The presence of a saturating pore fluid introduces the possibility of an additional controlfield and an additional type of strain variable. The pressure pf in the fluid is the new fieldparameter that can be controlled. Allowing sufficient time for global pressure equilibration willpermit us to consider pf to be a constant throughout the percolating (connected) pore fluid,while restricting the analysis to quasist atic processes. The change (in the amount of fluid mass
contained in the pores (see Berryman and Thigpen [1985]) is the new type of strain variable,measuring how much of the original fluid in the pores is squeezed out during the compression
11
“.
of the pore volume while including the effects of compression or expansion of the pore fluiditself due to changes in p~. It is most convenient to write the resulting equations in terms ofcompliance rat her than stiffnesses, so the basic equation to be considered takes the form:
I The constants appearing in the matrix on the right hand side will be defined in the following twoparagraphs. It is important to write the equations this way rather than using the inverse relationin terms of the stiffnesses, because the compliance sij appearing in (56) are simply related tothe drained constants Jd, and P&. in the same way they are related in normal elssticit y, whereasthe individual stiffnesses obt ained by inverting the equation in (56) must contain coupling termsthrough the parameters @and 7 that depend on the pore and fluid compliance. Thus, I findthat
(57)
I and
Vdrti12=— —
Edr ‘(58)
where the drained Young’s modulus Ed~ is defined by the second equality of (57) and the drainedPoisson’s ratio is determined by
~&v& = (59)
2(& + ~) “
When the external stress is hydrostatic so a = all = 022 = crw, the equation (56) telescopes
down to
where e = ell + ezz + e~, .kfdr = ~dr + 3P is the drained bulk modulus, a = 1- Kdr/K~ isthe Biot-Wti parameter [Biot and Wfis, 1957] with Km being the bulk modulus of the solidminerals present, and Skempton’s pore-pressure buildup parameter B [Skempton, 1954] is givenby
B=1
1 + KP(l/Ki - l/Km)”(61)
New parameters appearing in (61) are the bulk modulus of the pore fluid Kf and the poremodulus Ifp = a/#K& where #is the porosity. The expressions for a and B can be generalizedslightly by supposing that the solid frame is composed of more than one constituent, in whichcase the Km appearing in the definition of a is replaced by Ks and the Km appearing explicitlyin (61) is replaced by K@ [see Brown and Korringa, 1975; Rice and Cleary, 1976; Berryman and
12
“4-’—’”” —..—..—.,—..—” ..-. .r. ,.. . “,.
—
*
.
,
.
Milton, 1991; Berryman and Wang, 1995]. This is an important additional complication [Bergeand Berryman, 1995], but one that I will not pursue here.
Comparing (56) and (60), I find that
B+-= 3hd,
and
(62)
(63)
With all the constants defined now in terms of measurable quantities, I can continue withthe analysis that generalizes the Backus [1962] approach to computing the layer averages, Itshould be clear at this point that the appropriate Legendre transformed equations are
()ul 1
(722 =–em
(
(E/(1 - I/z) vE/(1 - V2) v/(1 – v) @?/(1 - v)
vE/(1,– V2) J??/(1- ZJ) v/(1 - v) /3E/(1 -v)v/(1 – v) I//(l - v) -(1 - v - 2v2)/(1 - V)E B(1 + 2v)/(1 - v)
~E/(1 -v) pE/(1 -v) p(l + 2v)/(1 - v) - [7 - 2P21?/(1 - ~)] )
()ell
ezzx
(733, (64)
–Pf
with the fast variables on the left and the slow variables (actually constant) in the vector onthe right. Signs have been chosen so the matrix is symmetfic. I have also dropped the subscriptdr from the drained constants v and E in (64) as there shodd be no confusion. Note that the3 x 3 submatrix in the upper left is identical to that in (26) after the change in notation from~,p to E,v is taken into account.
Once I have this equation, the averaging is trivial. If the assumed form of the resulting
equations is taken — in analogy to (6) and consistent with the general structure of the matrixin- (64) — to be
then the resulting rearrangement of these equations is
( H<(711> a-z b-x y z<022> b-x U-X $/ Z
-<em> = y v~~
<~> z Zvw
13
?
(65)
(66)
. . =. ._,.,._ .... .— — -4
where
~ = f2k - 2fgh + cg2 , ~=jk-gh C9- fhck – h2 ~k_h2’ ‘=ck_h2’
and
k h c‘u=——
ck_hz’ ‘=c). _h2’ ‘=-ck_h2”
(67)
(68)
It is not difficult to check that these equations reduce correctly to the earlier ones if I first setg=h=Oand thenletk~O,
Now all the matrix elements appearing in (66) are obtained directly by averaging (64) andtherefore are assumed known. I do not list dl of these relations as they should be clear fromthe expressions already given, but to provide two examples note that
(-)W=2 g+ (-)-(7) and z= ~ . (69)
Given all these equations, it is then straightforward to invert for the desired final expressions:
Ea
()— +3,
= 1–V2
b=()s ‘x’
(70)
(71)
wc =-— (72)
Uw - ~2 ‘
f=cg+hz, (73)
g=hy+kz, (74)
hv= .— (75) ,Uw - ~29
and
k = -~. (76)Uw -
The order in which the computations are done in practice is this: first compute c, h, and k;next compute f and g; then compute z using (67); finally compute a and b.
The results shiw that, whereas transverse isotropy due to layering in elastic materials pro-duces five independent constants (recall that a = b + 2rn in general for transverse isotropy),
transverse isotropy due to layering in poroelastic materials results in eight independent con-stants (a = b + 27n still holds for poroelasticit y as is easily shown from our formulas). When
14
.
performing the averaging based on (64), we see that all the new terms in the matrix depend onaverages of the poroelastic constant /3 which is proportional to the Biot- Willis parameter andtherefore related to effective stress [the relative importance of external and internal loading —
see (60)]. However, only the new diagonal term w depends directly on the bulk modulus Kf of
the pore fluid through ~. It follows that, when I solve for the effective constants, I find that winfluences all these effective constants. So the presence of pore fluid can significantly affect thepressure dependence of such materials, while having little or no effect on the shear response.
This completes the analysis of the constants for transverse isotropy in poroelasticity arisingfrom thin layering of isotropic elastic and porous materials. Next I check that these results areconsistent with known general results for anisot ropic poroelast icity [Gassmann, 1951; Brownand Korringa, 1975].
8 Relations for Anisotropy in Poroelastic Materials
Gassmann [1951] and Brown and Korringa [1975] have considered the problem of obtainingeffective constants for anisotropic poroelastic materials when the pore fluid is confiried withinthe pores. The confinement condition amounts to the constraint that the increment of fluidcontent ~ = O,while the external loading u is changed and the pore-fluid pressure pf is allowed
to respond as necessary and equilibrate.To provide a simple derivation of the Gassmann equation for anisotropic materials, I consider
the anisotropic generalization of (56)
(Hell $11 512 $13 -A
)()
all
eaa = S12 822 S23 +2 (722
es S13 S23 $33 -B3 033 “(77)
-c -/% -P2 -B3 Y -Pf
The shear terms are excluded as before since they do not interact mechanically with the fluideffects. This form is again not completely general in that it includes ort horhombic, cubic,hexagon~, and all isotropic systems, but excludes triclinic, monoclinic, trigonal, and sometetragonal systems that would have some nonzero off-diagonal terms in the full elastic matrix.Also, I have assumed that the material axes are aligned with the spatial axes. But this latterassumption is not significant for the derivation that follows. Such an assumption is importantwhen properties of laminated materials having arbitrary orientation relative to the spatial axesneed to be considered, but I do not treat this more general problem here.
Before proceeding, I want to discuss the significance of the matrix elements appearing in (77)briefly. In the so-called ‘jacketed test,” a porous sample is enclosed in a thin jacketing materialwith a tube into the pore space to permit the fluid to flow freely in or out while maintainingconstant fluid pressure. Then it is sufficient to consider the case with pt = 0. M is possible underthese circumstances, at least in principle, to make 12 independent measurements by varying uii’sand measuring ejj’s and (. h fact memurements of ~~ned elastic comphances ~e commo~ymade in such a manner, but it is less common for the ~i’s to be measured this way. To completethe measurements, a second common test — the s~called ‘unjacketed test” — is performed inwhich a uniform pressure field is, applied to the sample so that all = 022 = CTw= -pj. Then,by making measurements of the ejj ‘S agtin M Pj v~iw a @ of sofid materi~ comPfiances xi
15
.
is measured, given essentially by row sums of the matrix in (77)
~i=~$ij-~~, for z=1,2,3. (78)j=l
These three constants are expected to be directly related to the compliance of the mineral grainscomposing the porous frame; if the frame is microhomogeneous (i.e., containing a single solidconstit utent ), the compliance xi will be the compliance of the mineral composing the framesuch that xi ~i = l/K~, where Km is the bulk modulus of the mineral. If the measurementapparatus is inadequate so that the ~i’s could not be determined directly in the jacketed test,then we see from (78) that they can be determined by combining results from the jacketed andunjacketed measurements on the solid compliance. The remaining constant ~ can again bemeasured (at least in principle) directly in the unjacketed test by making measurements on thechanges in fluid content ~. An idternative to these rather difficult measurements is the confinedtest which I describe next,
If the fluid is confined, then c s O in (77) and pf becomes a linear function of all, 022,033. Eliminating p~ from the resulting equations, I obtain the general expression for the straindependence on external stress under confined conditions:
(3=[(;E37-1(!)’1‘2.”3)1(?3
‘E13(?:o ‘7’)The sij ‘S are fluid-drained constants, while the $ij*‘s are the fluid-confined constants.
The fundamental result (79) was obtained earlier by both Gassmann [1951] and Brown andKorringa [1975], and may be written as
‘iPi, for i,j = 1,2,3.* ..—% = % (80)
-1
This expression is just the anisotropic generalization of the well-known Gammann equation forisotropic, microhomogenmus porous media. Equation (80) has often been written in a slightlydifferent way, by making use of the formulas (78) to eliminate the ~’s in favor of the solid
“and drained compliance. The principal advantage of such an alternative formula is that allconstants appearing explicitly can be obt ained by measurements of porous frame strain, withoutresorting to the more difficult me~urements of changes in pore-fluid content.
Now it is not difficuJt to see that the lamination formulas derived earlier in the papersatisfy these general conditions. This simple test provides one means of checking that I did thelamination ans3ysis correctly and also provides a convenient means of summarizing the results.
9 Conclusions
In the present paper, I have discussed isotropic and anisotropic poroelastic media and estab-lished general formulas for the behavior of transversely isotropic elasticity and poroelasticity
16
arising from laminations of isotropic components. The Backus [1962] averaging method is shownto provide elementary means of constructing general formulas. The results for confined fluidsare then compared with the more general Gassmann [1951] formulas that must be satisfied byany anisot ropic poroelastic medium and found to be in complete” agreement. The transitionfrom analysis of laminations of elastic materials to laminations of poroelastic materials in thepresence of saturating pore fluids follows easily from the simple observation that cert tin vari-ables are quasistatically constant across a layered medium and provides a very intuitive andmathematically transparent approach to obtaining formulas of current interest. Choosing todo the analysis in terms of compliance rather than stiffnesses also proved to be an important
simplification for the poroelastic case.
Such results are especially important for applications to oil exploration using AVO (ampli-tude versus offset ) since the presence or absence of the fluid component, as well as the precisenature of the fluid, is one of the most critical iesues. For this reason, the ways in which thefluid can influence seismic reflection data need to be understood in more’ detail than has beenpossible in the past [Thomsen, 1993; Mukerji and Mavko, 1994].
Acknowledgments.
I thank Patricia Berge and Vladimir Grechka for helpful technical discussions on poroelasticityand anisotropy, and Phil Anno for a preprint of his paper. Work WW’performed under theauspices of the U. S. Department of Energy by the Lawrence I.Jverrnore National Laboratoryunder contract No. W-7405- ENG-48 and supported specifically by the Geosciences ResearchProgram of the DOE Office of Energy Research within the Office of Basic Energy Sciences,Division of Engineering and Geosciences, as part of the Advanced Computational Tdnologyhdtiative.
REFERENCES
D. L. Anderson, Z’heoy oj the Earth, Blackwell, Boston, 1989, pp. 317-319.
P. Anno, Equivalent anisotropy for finely-layered media, in Proceedings of the Project Review,Center for Wave Phenomena, Colomdo School of Mines, Vail, Colorado, May 18-21,1997,pp. 307-312.
G. E. Backus, Long-wave elastic anisotropy produced by horizontal layering, J. Geophgs. Res.,67,4427-4440 (1962).
P. A. Berge and J. G. 13erryman, Realizability of negative pore cornpressibtity in poroelasticcomposites, ASME J. Appl. Mech. 62, 1053-1062 (1995).
J. G. Berryrnan, Long-wave elastic anisotropy in transversely isotropic media, Geophysics 44,896-917 (1979).
J. G. Berryman, V. Grechka, and P. A. Berge, Ana3ysis of Thomsen parameters for finely layeredVTI media, in Proceedings of the SiztpSeveth Annual Meting of the Society of ExplomtionGeophysicists, Dallas, Texas, November 2-7, 1997, in press.
17
J. G. Berryman and G. W. Milton, Exact results for generalized Gassmann’s equations incomposite porous media with two constituents, Geophysics 56, 1950–1960 (1991).
J. G. Berryman and L. Thlgpen, Nonlinear and semilinear dynamic poroelasticity with mi-crostructure, J. Mech. Phys. Solids 33, 97-116 (1985).
J. G. Berryman and H. F. Wang, The elastic coefficients of double-porosity models for fluidtransport in jointed rock, J. Geophys. Res. 100, 24611-24627 (1995).
M. A. Biot, Mechancis of deformation and acoustic propagation in porous media, J. AppL HUN.
33, 1482-1498 (1962).
M. A. Biot and D. G. Willis, The elwtic coefficients of the theory of consolidation, J. AppLMech. 24,594-601 (1957).
R. J. S. Brown and J. Korringa, on the dependence of the elastic properties of a porous rockon the compressibility of a pore fluid, Geophysics 40, 608-616 (1975).
F. Gassmann, Uber die elastizitat poroser medien, Veirteljahrsschrift der NaturforschendenGeseZlsch@ in Ztirich 96, 1-23 (1951).
I B. Gurevich and S. L. Lopatnikov, Velocity and attenuation of elastic waves in finely layeredporous rocks, Geophys. J. Int. 121,933-947 (1995).
I T. Mukerji and G. Mavko, Pore fluid effects on seismic velocity in anisotropic rocks, Geophysics59,233-244 (1994).
IA. N. Norris, Low-frequency dispersion and attenuation in partially saturated rocks, J. Acoust.
Sac. Am. 94,359--370 (1993).
G. W. Postma, Wave propagation in a stratified medium, Geophysics 20,780-806 (1955).
I J. R. Rice and M, P. Cleary, Some basic stress diffusion solutions for fluid-saturated elasticporous media with compressible constituents, Rev. Geophys. 14, 227-241 (1976).
M. Schoenberg and F. Muir, A cidculus for finely layered anisotropic media, Geophysics 64,581-589 (1989).
A. W. Skempton, The pore-pressure coefficients A and 1?, Geotechnique 4, 143-147 (1954).:,
L. Thomsen, Weak anisotropic reflections, in Oflset-Dependent Reflectivity — Theory and Pmc-tice of AVO Analysis, J. P Castagna and M. M. 13ackus (eds.), Society of Exploration
,. Geophysicists, Tulsa, Oklahoma, 1993, pp. 103-111.
J
18
Technical Inform
ation Departm
ent • Lawrence Liverm
ore National Laboratory
University of C
alifornia • Livermore, C
alifornia 94551
