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Shear and dilatational wave velocities for unsaturated soils
ARTICLE IN PRESS Soil Dynamics and Earthquake Engineering 29 (2009) 946–952
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Shear and dilatational wave velocities for unsaturated soils Enrico Conte , Renato Maria Cosentini, Antonello Troncone ` della Calabria, 87036 Rende, Cosenza, Italy Dipartimento di Difesa del Suolo, Universita
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 June 2008 Received in revised form 30 October 2008 Accepted 11 November 2008
The primary objective of this study is for presenting some simple-to-use expressions relating the shear and dilatational wave velocities (VS and VP) to some physical and constitutive parameters of unsaturated soils. To this purpose, a simplified formulation is developed using the theory of linear poroelasticity in conjunction with some constitutive parameters widely used in geotechnical engineering. The derived expressions are of practical interest in view of the fact that they could be employed for evaluating the involved soil parameters from VS and VP measurements by in-situ or laboratory geophysical tests. & 2008 Elsevier Ltd. All rights reserved.
keywords: Wave velocity Shear wave Dilatational wave Unsaturated soils
1. Introduction Wave propagation in fluid-containing porous media has received great attention in recent years, because of its importance in geotechnical engineering for determining soil properties from the results of field or laboratory geophysical tests. Owing to the fact that in these tests the soil is usually subjected to small amplitude excitations, the material parameters obtained in this way are commonly referred to the undisturbed state of the soil. In addition, linear elasticity is the most frequently used constitutive assumption to predict the mechanical behaviour of the soil. In this context, Foti et al. [1] proposed a novel procedure for evaluating the porosity of saturated soils from measured shear and dilatational wave velocities. This procedure is very useful in practice, especially in the case of coarse-grained soils in which undisturbed sampling can only be achieved using sophisticated techniques such as freezing. The basic theory for wave propagation in porous media saturated by a single compressible fluid was originally formulated by Biot [2,3]. He demonstrated that a single shear wave and two dilatational waves can propagate in an infinite, homogeneous, isotropic poroelastic medium as opposed to the case of a oneconstituent elastic medium in which only one dilatational wave and one shear wave propagate. Since the development of Biot’s theory, many attempts have been made to present more general models for the analysis of wave propagation though saturated soils [4–12]. Corresponding author at: Dipartimento di Difesa del Suolo, Universita` della Calabria, Ponte P. Bucci, Cubo 41b 87036 Rende, Cosenza, Italy. Tel.: +39 0984 496524; fax: +39 0984 496526. E-mail address: [email protected] (E. Conte).
0267-7261/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2008.11.001
Unsaturated soils are more complicated multiphase systems comprising a solid skeleton and pores filled by more than one fluid (air and water). A simplification is usually made when the soil is very close to saturation. In such circumstances, in fact, air contained in the pores is present in the form of occluded bubbles that can be considered as one with pore-water. As a consequence, an accepted assumption is to retain the formal structure of Biot’s theory, while introducing an appropriate bulk modulus for the water–air mixture [4,13]. Brutsaert [14] was perhaps the first to develop a comprehensive model for describing wave propagation through a porous medium saturated by two fluids. Extensions to this model were afterwards presented by several authors [15–18]. Further contributions were provided by Garg and Nayfeh [19] and Wei and Muraleetharan [20] using the mixture theory, and Tuncay and Corapcioglu [21] who employed a volume-averaging technique. These studies showed the existence of three different dilatational waves and one shear wave whose propagation velocities are frequency-dependent (dispersive waves) due to the relative movement of the fluids with respect to the solid phase, which causes a viscous dissipation of energy. While all these formulations are highly valuable from a theoretical point of view, their use is generally unsuitable for engineering purposes because some of the involved material parameters are difficult to determine experimentally, especially in in-situ conditions. In this paper, a simplified formulation is developed with the primary object of presenting simple-to-use expressions for the shear and dilatational wave velocities (VS and VP) in unsaturated soils excited by small perturbations. These expressions are of practical interest in view of the fact that they could be readily adopted for evaluating some soil parameters from VS and VP measurements by in-situ or laboratory tests. The formulation is based on the elasticity theory in conjunction with the constitutive
ARTICLE IN PRESS E. Conte et al. / Soil Dynamics and Earthquake Engineering 29 (2009) 946–952
relationships proposed by Fredlund and Morgenstern [22] to describe the volume change in unsaturated soils, and Darcy’s law to express the flow of the fluids through the soil. Three different conditions are analysed: (a) water and air move separately with respect to the solid phase; (b) one fluid flows and the other is at rest and (c) no relative motion occurs between the solid and the fluid phases.
947
convention is used. On the other hand, when the soil approaches a dry condition, the coefficients of water volume change go to zero. Following Fredlund and Rahardjo [23], Eq. (1a) can also be formulated in a linear elasticity form in which the net total stress and matric suction are considered as stress-state variables. This leads to the equation [24] dr ¼ Ede w dpw I ð1 wÞ dpa I
2. Basic assumptions Unsaturated soils are made up of three constituents: soil skeleton, water and air, which in the context of mixture theory are viewed as three continua simultaneously occupying the same regions of space. In this study, wave propagation in such materials is analysed under the assumptions of infinitesimal strains, isothermal conditions, solid particle incompressibility, isotropic linearly elastic soil skeleton and validity of Darcy’s law for describing the flow of air and water through the soil. Gravity force, chemical and electric effects are neglected. In addition, the medium is assumed to be infinite and homogeneous so that reflection, refraction, scattering and mode conversion phenomena are also ignored. Due to these simplified assumptions, the formulation presented in the following sections is approximate, but of practical interest from an engineering viewpoint. The sign convention of continuum mechanics is used, according to which the normal stresses are positive in tension. Moreover, the symbol p is adopted for pore pressures instead of the usual u to avoid confusion with the displacements.
where r is the total stress tensor and e is the strain tensor of the solid skeleton, E is the tensor of the elastic moduli, I stands for the identity tensor and w is a parameter that is related to the volume change coefficients of the solid phase by the equation
w ¼ ms2 =ms1
dv ¼ ms1 dðsm þ pa Þ ms2 dðpa pw Þ
de ¼ E1 dr þ dnw ¼
dna ¼ ma1 dðsm þ pa Þ ma2 dðpa pw Þ
(1c)
where ev is the volumetric strain of the solid structure, ms1 and ms2 denote the associated coefficients of volume change with respect to a change in the mean net total stress, (sm+pa), and in matric suction (papw) (which are the stress-state variables usually adopted to describe the behaviour of the unsaturated soils), nw is the volume w fraction of water, mw 1 and m2 are the coefficients of water volume a change, n is the volume fraction of the air phase, and ma1 and ma2 are the respective volumetric coefficients. Moreover, sm is the mean total stress (i.e., sm ¼ (sx+sy+sz)/3, where sx, sy and sz are the total normal stresses in x, y and z directions), pw and pa are the porewater and pore-air pressures, respectively. Continuity requirement for the volume change of the three phases leads to the following equations relating these coefficients: ms1
ma1
(2a)
a ms2 ¼ mw 2 þ m2
(2b)
¼
mw 1
þ
As highlighted by Fredlund and Morgenstern [22], when the pores are completely filled by water (saturated soils) the volume change coefficients of this phase and those of the solid one are replaced by the coefficient of volume change mv of the soil skeleton, and Eqs. (1a) and (1b) reduced to the compressibility form relationship commonly adopted in geotechnical engineering, that is dev ¼ mvd(sm+pw), in which the above-specified sign
3K s
w 3K s
dpw I þ
1w dpa I 3K s
(5a)
I dr þ cww dpw þ cwa dpa
(5b)
I dr þ caw dpw þ caa dpa
(5c) s
1
where E denotes the inverse of E, and K is the bulk modulus of the soil skeleton. The other coefficients appearing in Eqs. (5a)–(5c) are
cw ¼
mw 1 ms1
(6a)
ca ¼
ma1 ms1
(6b)
wcw
cww ¼
cwa ¼ (1b)
cw
3K s
ca
dna ¼
(1a)
w dnw ¼ mw 1 dðsm þ pa Þ m2 dðpa pw Þ
(4)
After performing a series of algebraic operations, Eqs. (3), (1b) and (1c) can be cast in the form
3. Constitutive equations The formulation developed in this paper utilizes the constitutive relationships introduced by Fredlund and Morgenstern [22] to describe the volume change of unsaturated soils. These relationships can be expressed in differential form as follows:
(3)
caw ¼
caa ¼
K
s
þ
s w ms1 mw 2 m2 m1 s m1
(6c)
s w ð1 wÞcw ms1 mw 2 m2 m1 s s m K 1
(6d)
ms1 ma2 ms2 ma1 ms1
(6e)
wca Ks
þ
ð1 wÞca ms1 ma2 ms2 ma1 ms1 Ks
(6f)
Eqs. (5a)–(5c) are similar in form to those presented by Loret and Khalili [25]. However, the constitutive parameters appearing in Eqs. (6a)–(6f) are different from those considered by these latter authors. It is also relevant to note that for an isothermal and reversible process in which dissipation does not occur, the following equation can be written [11,25]: dHs ¼ e dr þ nw dpw þ na dpa
(7)
where Hs is the complementary free energy for the solid skeleton. As a result, the skeleton state equations are
e¼
qH s ; qr
nw ¼
qHs ; qpw
na ¼
qHs qpa
(8)
from which the following relationships can be obtained: de ¼
q2 Hs q2 H s q2 H s dr þ dpw I þ dp I 2 qpw qr qpa qr a qr
dnw ¼
q2 Hs q2 Hs q2 Hs I dr þ 2 dpw þ dp qrqpw qpw qpa a q pw
(9a)
(9b)
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q2 Hs q2 Hs q2 Hs I dr þ dpw þ 2 dpa qrqpa qpa qpw q pa
dna ¼
(9c)
By comparing these relationships to Eqs. (5a)–(5c), it can be noted that the coefficients in these latter equations correspond to the second-order partial derivatives appearing in Eqs. (9a)–(9c) that are the terms of the Hessian matrix of Hs. Owing to the symmetry of this matrix, the following identity holds [24]: mw 1
¼
ms2
(10)
Moreover, the change in the volume fraction of water can also be expressed as dmw
w
dn ¼
rw
dpw n Kw w
(11a)
w
where m is the water mass content per unit volume of the soil, rw is the mass density of water, and Kw is the bulk modulus of water. This equation derives from the relationship dMw ¼ d(rwVw), in which Mw and Vw are the mass content and the volume of water, respectively. Likewise, for the air phase we have dna ¼
dma
ra
na
dpa Ka
(11b)
where ma, ra, and Ka stand for the above-defined parameters referred to air. Lastly, the mass balance equations are invoked. Under the hypothesis of small amplitude perturbations, these equations are [11] dmw
rw dma
ra
w
w
s
¼ n divðu u Þ
(12a)
¼ na divðua us Þ
(12b)
where the symbol ‘‘div’’ indicates the divergence operator, and us, uw and ua are the displacement fields for the solid, water and air phases, respectively. Substituting Eqs. (11a) through (12b) into (1b), (1c) and (3), and after performing a series of algebraic operations, the following equations can be obtained [24], which are the stress– strain equations adopted in the present study
r ¼ Gðgrad us þ gradT us Þ þ HI div us þ nw ½wL þ ð1 wÞCI div uw þ na ½wC þ ð1 wÞNI div ua
(13)
pw ¼ W div us nw L div uw na C div ua
(14)
pa ¼ M div us nw C div uw na N div ua
(15)
W ¼
KwA þ nw L þ na C D
N¼
C¼
K ½n
ms1
þK
w
ðms1 mw 2
s w a w w a a w s D ¼ ðms1 mw 2 m2 m1 Þðn K þ n K Þ þ n n m1
(16c)
D s w K w K a ðms1 mw 2 m2 m1 Þ D
ms2 mw 1 Þ
(16i)
nSK
as ¼
(16j)
1 2nSK SK
in which n is Poisson’s ratio and G is the shear modulus of the soil skeleton. They are related to ms1 by the equation [23] ms1 ¼
3ð1 2nSK Þ 2ð1 þ nSK ÞG
(16k)
It is worth mentioning that r, pw and pa in Eqs. (13)–(15) are viewed as incremental quantities referred to their initial values (before excitation occurs).
4. Governing equations The equations governing wave propagation in porous media are the linear momentum equation expressing the equilibrium of all forces acting on an elementary volume of medium, and the equations of the motion for the fluid phases. The former is s
w
a
€ ra na u € ¼0 € rw nw u div r rs ð1 nÞu
(17)
where rs is the mass density of the solid phase, n is the soil porosity and the dot notation indicates time derivatives. Moreover, the following identities can be written: na ¼ nð1 Sr Þ
(18a)
nw ¼ nSr
(18b)
where Sr is the degree of saturation of the soil. Darcy’s law is usually adopted to describe the motion of water and air through the soil. For an isotropic medium, by neglecting the gravity forces, this law is [11] w
s
_ u _ Þ¼ nw ðu
s
kw
rw g ka
ra g
w
€ Þ ðgrad pw þ rw u
(19)
a
€ Þ ðgrad pa þ ra u
(20)
in which kw and ka are the coefficients of permeability for the water and air phases, respectively, and g is the gravity acceleration. Taking advantage of the constitutive relationships (13) through (16k) and after performing some algebraic manipulations, Eqs. (17), (19) and (20) become 2
ðH þ 2G þ nw W þ na MÞgrad div us Gðgrad div us r us Þ þ nw ½wL þ ð1 wÞC nw L na Cgrad div uw þ na ½wC þ ð1 wÞN na N nw Cgrad div ua þ
(16b)
K B þ nw C þ na N D
w
(16h)
(16a)
s w K w ½na ms1 þ K a ðms1 mw 2 m2 m1 Þ L¼ D a
s w B ¼ nw ma1 þ K w ðms1 mw 2 m2 m1 Þ
a
a
M¼
(16g)
_ u _ Þ¼ na ðu
where the symbol ‘‘grad’’ indicates the spatial gradient, the superscript T denotes the transpose and the coefficients appearing in them take the form H ¼ 2as G wW ð1 wÞM
a s w s w A ¼ na mw 1 þ K ðm1 m2 m2 m1 Þ
ðnw Þ2 rw g w ðna Þ2 ra g a €s ¼ 0 _ u _ sÞ þ _ u _ s Þ ð1 nÞrs u ðu ðu kw ka
(21)
nw Wgrad div us þ ðnw Þ2 Lgrad div uw þ na nw Cgrad div ua
(16d)
ðnw Þ2 rw g w €w ¼ 0 _ u _ s Þ nw rw u ðu kw
(22)
na M grad div us þ nw na C grad div uw þ ðna Þ2 N grad div ua (16e)
(16f)
ðna Þ2 ra g a €a ¼ 0 _ u _ s Þ na ra u ðu ka
(23)
where r2 is the Laplacian operator. It is relevant to note that when w the soil is completely saturated (Sr ¼ 1, ms1 ¼ ms2 ¼ mw 1 ¼ m2 ),
ARTICLE IN PRESS E. Conte et al. / Soil Dynamics and Earthquake Engineering 29 (2009) 946–952
some terms appearing in these equations take an indeterminate form. By performing the limit of these terms for Sr approaching unity, Eqs. (21)–(23) reduce to the following equations: Kw grad div us 2Gð1 þ as Þ þ ð1 nÞ2 n Gðgrad div us r2 us Þ þ ð1 nÞK w grad div uw þ
n2 rw g w €s ¼ 0 _ u _ s Þ ð1 nÞrs u ðu kw
(24)
ð1 nÞK w nr g w _ u _ s Þ rw u €w ¼ 0 grad div us þ K w grad div uw w ðu kw n (25)
which coincide with those presented by Coussy [11] for a porous medium saturated by one fluid, provided that the solid particles are incompressible. In the above equations, the effects of inertial coupling between the fluid and solid phases are ignored.
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and c2 ¼ V 2S ¼
G ð1 nÞrs
(33b)
This means that a transverse wave (S-wave) propagates in the plane tangent to the wavefront with velocity VS. On the other hand, substituting Eq. (29) into Eqs. (26)–(28) and after multiplication by n leads to the following system written in compact form: 0 s 1 u€ n B wC ðK c2 MÞ@ u€ n A ¼ 0 (34) a u€ n in which the matrices K and M are 2 6 K¼4
H þ 2G þ nw W þ na M
nw ½wL þ ð1 wÞC nw L na C
na ½wC þ ð1 wÞN nw C na N
nw W
ðnw Þ2 L
nw na C
na M
nw na C
ðna Þ2 N
3 7 5
(35a) 2
5. Wave velocities for unsaturated soils The discontinuity propagation theory [11,26] is employed here to obtain explicit relationships of the wave velocities for unsaturated soils. In this theory, a wave is considered as a singular surface of the second-order (displacement and velocity are continuous across it, but this does not hold for acceleration) propagating with a finite speed that depends on the properties of the soil. For the sake of completeness, three different conditions are analysed: (a) water and air move separately with respect to the solid phase; (b) one fluid flows with respect to the solid phase and the other is at rest and (c) no relative motion occurs between the solid and the fluid phases, which is equivalent to considering the porous medium as a closed (undrained) system. In the condition denoted as (a), wave propagation is governed by Eqs. (21) through (23). Applying the jump operator [[ ]] to these equations and taking advantage of Hadamard’s compatibility conditions [11] lead to the following equations: s
6 M¼4
s
s
w
a
w
s
w a
w
a 2
a
2 a
a
€ n þ n n C½½n u € n þ ðn Þ N½½n u € n c n ra ½½u € ¼ 0 n M½½n u
€ s ¼ u€ sn n þ u€ st t; ½½u
€ w ¼ u€ w €w ½½u n n þ ut t;
€ a ¼ u€ an n þ u€ at t ½½u
ða2p1 a2p2 a2p3 þ A1 a2p3 þ A2 a2p2 A3 a2p1 A0 Þ ¼ 0
s
½G c2 ð1 nÞrs u€ t ¼ 0
(30)
w
(31)
ðc2 nw rw Þu€ t ¼ 0 a
ðc2 na ra Þu€ t ¼ 0
(32)
from which it results that w a u€ t ¼ u€ t ¼ 0
(33a)
(37a)
where a2p1 ¼ a2p2 ¼ a2p3 ¼
H þ 2G þ nw W þ na M ð1 nÞrs nw L
rw na N
ra
(37b)
(37c)
(37d)
A1 ¼
nw W½wL þ ð1 wÞC nw L na C ð1 nÞrs rw
(37e)
A2 ¼
na M½wC þ ð1 wÞN nw C na N ð1 nÞrs ra
(37f)
A3 ¼
(29)
where t is the unit vector lying in the plane tangent to the wavefront. Substituting these latter into Eqs. (26)–(28) and multiplied by t, we obtain
By solving
ðV 2P Þ3 ða2p1 þ a2p2 þ a2p3 ÞðV 2P Þ2 þ ða2p1 a2p2 þ a2p1 a2p3 þ a2p2 a2p3 þ A1 þ A2 A3 ÞV 2P
(28) in which n is the unit vector normal to the discontinuity surface (wavefront) and c is the speed of propagation. In addition, the ¨ s]], [[u ¨ w]] and [[u ¨ a]] can be decomposed in the discontinuities [[u form
(36)
where ‘‘det’’ denotes the determinant, and c ¼ V2P. Eq. (36) leads to the following cubic equation in V2P:
(27) a
(35b)
na ra
2
(26) € n þ ðnw Þ2 L½½n u € n þ nw na C½½n u € n c2 nw rw ½½u € ¼ 0 nw W½½n u
0
3 0 0 7 5
detðK c2 MÞ ¼ 0
w
a
n rw
0
0 w
It is worth noting that owing to Eq. (10), K is a symmetric matrix [24]. According to Eq. (34), three different dilatational waves (P-waves) can propagate along the normal direction to the wavefront. Their velocities are the solutions of the equation
€ n þ nw ½wL þ ð1 wÞC nw L na C½½n u € n ðH þ G þ nw W þ na MÞ½½n u € n þ ½G c2 ð1 nÞrs ½½u € ¼ 0 þ na ½wC þ ð1 wÞN nw C na N½½n u
ð1 nÞrs 0
A0 ¼
nw na C 2
rw ra ½wL þ ð1 wÞC nw L na Cnw na MC ð1 nÞrs rw ra ½wC þ ð1 wÞN nw C na Nnw na WC þ ð1 nÞrs rw ra
(37g)
(37h)
From Eqs. (33b) and (37a) as well as from the expressions presented below for the cases (b) and (c), positive and negative roots are obtained for VS and VP. They correspond to the wave propagation velocities in the positive and negative directions, respectively. For the purposes of the present study, only the positive roots are considered. In case (b), when water motion is prevented with respect to the solid phase (uwus ¼ 0), the governing Eqs. (21)–(23) reduce to
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½H þ 2G þ nw wL þ nw ð1 wÞC þ na ðM nw CÞ grad div us Gðgrad div us r2 us Þ þ na ½ðwC þ ð1 wÞNÞ na N grad div ua þ
ðna Þ2 ra g a _ u _ s Þ ½ð1 nÞrs þ nw rw u €s ¼ 0 ðu ka
(38)
ðna Þ2 ra g a €a ¼ 0 _ u _ s Þ ra na u ðu ka
In this case, it can be shown the existence of one shear wave propagating with velocity (39)
Proceeding as above, the application of the discontinuity propagation method leads to the existence of one shear wave and two different dilatational waves, the propagation velocities of which are V 2S ¼
G ð1 nÞrs þ nw rw
KwA KaB 2 € ¼0 þ ð1 wÞ grad div u Gðgrad div u r uÞ ru 2Gð1 þ as Þ þ w D D
(45)
na ðM nw CÞgrad div us þ ðna Þ2 Ngrad div ua
Taking into account the constitutive relationships (13) through (16k), Eq. (44) becomes
V 2S ¼
G
(46)
r
and a single dilatational wave with velocity V 2P ¼
ð2Gð1 nSK Þ=ð1 2nSK ÞÞ þ ðdP1 =dP2 Þ
r
where (40)
2 a s 2 w s dP1 ¼ K w K a ðw2 þ mw 2 =m1 Þ þ n½w K ð1 Sr Þ þ ð1 wÞ K Sr =m1
(48a)
and V 2P ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 2 ðbp1 þ bp2 Þ ðbp1 bp2 Þ2 þ 4Bo 2
s 2 s dP2 ¼ nðSr K a þ ð1 Sr ÞK w Þðw2 þ mw 2 =m1 Þ þ n Sr ð1 Sr Þ=m1
(41a)
(48b)
(41b)
It is relevant to note that when the pores are completely filled by air (dry soil, Sr ¼ 0) or water (saturated soil, Sr ¼ 1), the shear wave velocity is again provided by Eq. (46) in which the mass density of the soil is
where 2
bp1 ¼
2
bp2 ¼
(47)
H þ 2G þ na ðM nw CÞ þ nw wL þ nw ð1 wÞC ð1 nÞrs þ nw rw
r ¼ ð1 nÞrs þ nra
(49a)
or
na N
(41c)
ra
r ¼ ð1 nÞrs þ nrw
(49b)
respectively. On the other hand, Eq. (47) reduces to na ½wC þ ð1 wÞN na NðM nw CÞ B0 ¼ ½ð1 nÞrs þ nw rw ra
(41d)
Similarly, when water flows and air is at rest (uaus ¼ 0) the expressions for VS and VP are V 2S
¼
G ð1 nÞrs þ na ra
(42)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðc2p1 þ c2p2 Þ ðc2p1 c2p2 Þ2 þ 4C o 2
(43a)
where c2p1 ¼
c2p2 ¼
H þ 2G þ nw ðW na CÞ þ na wC þ na ð1 wÞN ð1 nÞrs þ na ra nw L
rw
C0 ¼
nw ½wL þ ð1 wÞC nw LðW na CÞ ½ð1 nÞrs þ na ra rw
(43b)
(43c)
(43d)
Lastly, for the condition denoted as (c), when no relative motion between the phases occurs (ua ¼ uw ¼ us ¼ u), the governing field equation is € ¼0 div r ru
(44)
in which
r ¼ ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra
(44a)
ð2Gð1 nSK Þ=ð1 2nSK ÞÞ þ ðK a =nÞ ð1 nÞrs þ nra
(50)
ð2Gð1 nSK Þ=ð1 2nSK ÞÞ þ ðK w =nÞ ð1 nÞrs þ nrw
(51)
or V 2P ¼
and V 2P ¼
V 2P ¼
depending on whether the soil is dry or fully saturated. This latter equation coincides with that proposed by Foti et al. [1] (Eq. (17) in their paper) for evaluating the porosity of saturated soils from Vp and Vs measurements. It should be noted that the solution obtained in this section accounts for no dissipation of energy associated with the viscous flow induced by the relative motion between the fluids and the solid. The analysis of these phenomena and their effects on the wave velocity is beyond the scope of the present study. Nevertheless, it should be mentioned that at low frequencies the relative motion between the fluid and solid phases can be ignored, whereas at very high frequencies the fluids are free to move through the porous medium without dissipation [14]. In the light of these considerations, the equations derived in the present study can be referred to these latter conditions. Miura et al. [12] conducted an interesting study on the frequency-dependent characteristics of the waves in various saturated soils from clay to soft rocks. The study highlighted that the coefficient of permeability is the main soil parameter controlling the wideness of the low-frequency range. In particular, the smaller is the coefficient of permeability, the wider the range. Considering the permeability characteristics of the saturated soils and the frequency content of most geophysical tests in which the maximum frequency is of the order of some hundred kHz [10], it can be assumed that during these tests the soil is excited at low frequencies [1,2,12]. This assumption should be even more accepted for unsaturated soils that are generally characterized
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by much smaller values of the coefficient of permeability than the saturated soils. Owing to these considerations, Eqs. (46) and (47), which were derived for the low-frequency range, may be used in the field of geotechnical engineering for the characterization of the soils. In this connection, Eq. (47) can be cast in a more convenient form by introducing some additional simplifications. To this purpose, it is relevant to note that Eq. (3) is similar to the wellknown relationship first proposed by Bishop [27]
r ¼ r¯ pa I þ wðpa pw ÞI
#
Appendix A The expressions of VP and VS for unsaturated soils as obtained for the low-frequency range, are, respectively
3ð1 2nSK ÞS2r þ nSr ð1 Sr Þ½K w Sr þ K a ð1 Sr Þ 2ð1 þ nSK ÞG 2ð1 n Þ " # Gþ 1 2nSK 3ð1 2nSK ÞS2r þ n2 Sr ð1 Sr Þ n½K a Sr þ K w ð1 Sr Þ mw 2 SK 2ð1 þ n ÞG SK
V 2P ¼
unsaturated soils, have been derived. These expressions are of practical interest from an engineering viewpoint, considering the fact that they could be adopted for evaluating some material properties of unsaturated soils when VP and VS measurements from field or laboratory geophysical tests are available. Owing to the fact that these tests are based on the application of a small perturbation to the soil, the material parameters obtained in this way are commonly referred to the initial or undisturbed state of the soil.
(52)
in which r¯ is the effective stress tensor, and w is a parameter ranging between 0 and 1, which is often replaced by Sr [20,28–30] as an approximation. Using this latter simplification and considering Eq. (16k), Eq. (47) becomes "
"
K a K w mw 2
ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra
# 3ð1 2nSK ÞS2r þ nSr ð1 Sr Þ½K w Sr þ K a ð1 Sr Þ 2ð1 þ nSK ÞG 2ð1 nSK Þ " # Gþ 1 2nSK 3ð1 2nSK ÞS2r n½K a Sr þ K w ð1 Sr Þ mw þ n2 Sr ð1 Sr Þ 2 SK 2ð1 þ n ÞG K a K w mw 2
V 2P ¼
ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra
(53)
(A.1)
where G can be obtained inverting Eq. (46) in which r is expressed by Eq. (44a), that is
and
G ¼ ½ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra V 2S
V 2S ¼
(54)
Finally, under the reasonable assumptions that the air mass density is ignored (raffi0) and the bulk modulus of air is negligible with respect to that of water (Ka/Kwffi0), the expression for V2P simplifies as follows (see Appendix A): V 2P ¼
2ð1 nSK Þ 2 V þ 1 2nSK s
S2r ½ð1 nÞrS þ Sr nrw mw 2
3ð1 2nSK ÞS2r 2ð1 þ nSK ÞV 2s
þ
nSr ½ð1 nÞrS þ Sr nrw Kw
(55) In these latter equation, rw, rS and K may be considered as physical constants that assume standard values, and VP and VS can be measured using laboratory or field tests. As a result, Eq. (55) represents a relationship in which the unknown variables are the porosity n, Poisson’s ratio nSK, degree of saturation Sr, and the coefficient of volume change for the water phase. This latter w is defined as mw 2 ¼ qn /q(papw) owing to Eq. (2b), and can be experimentally obtained from a soil–water characteristic curve [23]. From an engineering viewpoint, Eq. (55) can be employed for evaluating the soil properties appearing in it from VS and VP measurements by geophysical tests, such as cross-hole or downhole tests. In particular, Eq. (55) can be readily inverted to obtain mw whose evaluation is not generally a simple operation, 2 especially in in-situ conditions [31]. The resulting expression is mw 2 ¼
V 2P ¼
Moreover, considering the values that generally characterize the bulk modulus of water, the bulk modulus of air and the mass density of air (Kw ¼ 2.25 106 kPa, Ka ¼ 1.45 102 kPa, and ra ¼ 0.001 t/m3), it may be assumed that Ka is negligible with respect to Kw (i.e., Ka/Kwffi0), and raffi0. As a result, Eq. (A.3) takes the form 2ð1 nSK Þ " Gþ 1 2nSK nð1 Sr Þ V 2P ¼
Kw
nS2r ð1 Sr Þ Kw # 3ð1 2nSK ÞS2r n2 Sr ð1 Sr Þ w m2 þ 2ð1 þ nSK ÞG ðK w Þ2
ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra (A.4)
which reduces to the simpler equation 2ð1 nSK Þ Gþ" 1 2nSK V 2P ¼
6. Concluding remarks
ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra
(A.3)
ð1 2nSK ÞS2r 3 1 nSr þ w ½ð1 nÞrS þ Sr nrw 2ð1 þ nSK ÞV 2S ð1 2nSK ÞV 2P 2ð1 nSK ÞV 2S K
Obviously, Eq. (55) can also be inverted to evaluate Poisson’s ratio of the solid skeleton, degree of saturation or porosity of the soil, once the other parameters are known. Owing to the fact that in the geophysical tests the soil is excited by very small levels of energy, the parameters obtained in this way are commonly referred to the initial or undisturbed state of the soil.
(A.2)
" # Ka 3ð1 2nSK ÞS2r nSr ð1 Sr Þ w w þ m ½K Sr þ K a ð1 Sr Þ 2 2ð1 þ nSK ÞG Kw ðK w Þ2 2ð1 nSK Þ " # Gþ 1 2nSK n 3ð1 2nSK ÞS2r n2 Sr ð1 Sr Þ þ ½K a Sr þ K w ð1 Sr Þ mw 2 w 2 SK 2ð1 þ n ÞG ðK Þ ðK w Þ2
#
(56)
G ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra
In these expressions, G is the shear modulus and nSK is Poisson’s ratio of the soil skeleton, mw 2 denotes the coefficient of water volume change with respect to a change in matric suction, Kw is the bulk modulus of water and Ka is that of air, n is the porosity and Sr is the degree of saturation of the soil, rs, rw and ra are the mass density of the solid phase, water phase and air phase, respectively. By dividing both the numerator and denominator of the second term on the right-hand side of Eq. (A.1) by (Kw)2, yields
w
"
951
S2r mw 2
3ð1 2nSK ÞS2r 2ð1 þ nSK ÞG
ð1 nÞrs þ Sr nrw
!# þ
nSr Kw
(A.5)
in which the shear modulus G can be evaluated from Eq. (A.2) as Theoretical expressions relating VP and VS to the soil parameters that describe the volume changes experienced by
G ¼ ½ð1 nÞrs þ Sr nrw V 2S
(A.6)
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Finally, after substituting Eq. (A.6) into Eq. (A.5), the expression for VP can be written as follows: V 2P ¼
2ð1 nSK Þ 2 V þ 1 2nSK S
S2r ½ð1 nÞrs þ Sr nrw mw 2
3ð1 2nSK ÞS2r 2ð1 þ nSK ÞV 2S
þ
nSr ½ð1 nÞrs þ Sr nrw Kw
(A7)
References [1] Foti S, Lai G, Lancellotta R. Porosity of fluid-saturated porous media from measured seismic wave velocities. Ge´otechnique 2002;52(5):359–73. [2] Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Lower frequency range. Journal of Acoustic Society of America 1956;28(2):168–78. [3] Biot MA. Theory of propagation of elastic waves in a fluid-saturated porous solid II. Higher frequency range. Journal of Acoustic Society of America 1956;28(2):179–91. [4] Ishihara K. Approximate forms of wave equations for water-saturated porous materials and related dynamic modulus. Soils and Foundations 1970;10(4): 10–38. [5] Zienkiewicz OC, Chang CT, Bettess P. Drained, undrained, consolidating and dynamic behaviour assumptions in soils. Ge´otechnique 1980;30(4):385–95. [6] Bowen RM. Compressible porous media models by use of the theory of mixtures. International Journal of Engineering Science 1982;20(6):697–735. [7] Pride SR, Gangi AF, Morgan FD. Deriving the equations of motion for porous isotropic media. Journal of Acoustic Society of America 1992;92(6):3278–90. [8] Gajo A. The effects of inertial coupling in the interpretation of dynamic soil tests. Ge´otechnique 1996;46(2):245–57. [9] Wilmanski K. Thermomechanics of continua. Berlin: Springer; 1998. [10] Santamarina JC, Klein KA, Fam MA. Soils and waves. New York: Wiley; 2001. [11] Coussy O. Poromechanics. Chichester, UK: Wiley; 2004. [12] Miura K, Yoshida N, Kim Y-S. Frequency dependent property of waves in saturated soil. Soils and Foundations 2001;41(2):1–19. [13] Tamura S, Tokimatsu K, Abe A, Sato M. Effects of air bubbles on B-value and P-wave velocity of a partly saturated sand. Soils and Foundations 2002;42(1): 121–9. [14] Brutsaert W. The propagation of elastic waves in unconsolidated unsaturated granular mediums. Journal of Geophysical Research 1964;69(2):243–57.
[15] Gray WG. General conservation equations for multi-phase systems: 4. Constitutive theory including phase change. Advance in Water Resource 1983;6:130–40. [16] Berryman JG, Thigpen L, Chin RCY. Bulk elastic wave propagation in partially saturated porous solids. Journal of Acoustic Society of America 1988;84(1): 360–73. [17] Santos JE, Douglas J, Corbero J, Lovera OM. A model for wave propagation in a porous medium saturated by a two-phase fluid. Journal of Acoustic Society of America 1990;87(4):1439–48. [18] Lo W-C, Majer E, Sposito G. Wave propagation through elastic porous media containing two immiscible fluids. Water Resources Research 2005;41: 1–20. [19] Garg SK, Nayfeh AH. Compressional wave propagation in liquid and/or gas saturated elastic porous media. Journal of Applied Physics 1986;60(9): 3045–55. [20] Wei C, Muraleetharan KK. A continuum theory of porous media saturated by multiple immiscible fluids: I. Linear poroelasticity. International Journal of Engineering Science 2002;40:1807–33. [21] Tuncay K, Corapcioglu MY. Body waves in poroelastic media saturated by two immiscible fluids. Journal of Geophysical Research 1996;101(B11): 25,149–59. [22] Fredlund DG, Morgenstern NR. Constitutive relations for volume change in unsaturated soils. Canadian Geotechnical Journal 1976;13(3):261–76. [23] Fredlund DG, Rahardjo H. Soil mechanics for unsaturated soils. New York: Wiley; 1993. [24] Cosentini RM. La propagazione delle onde elastiche in mezzi porosi non saturi. PhD thesis, Universita` Mediterranea di Reggio Calabria, Italy, 2006. [25] Loret B, Khalili N. A three-phase model for unsaturated soils. International Journal for Numerical and Analytical Methods in Geomechanics 2000;24: 893–927. [26] Lancellotta R. Geotecnica. Bologna, Zanichelli, 2004. [27] Bishop AW. The principle of effective stress. Teknisk Ukeblad 1959;106(39): 859–63. [28] Bishop AW, Blight GE. Some aspects of effective stress in saturated and partly saturated soils. Ge´otechnique 1963;13(3):177–97. [29] Jommi C. Remarks on the constitutive modelling of unsaturated soils. In: Proceedings of the first international workshop on unsaturated soils. Trento, Italy; 2000. p. 139–53. [30] Gallipoli D, Gens A, Sharma R, Vaunat J. An elasto-plastic model for unsaturated soil incorporating the effects of suction and degree of saturation on mechanical behaviour. Ge´otechnique 2003;53(1):123–35. [31] Fredlund DG. The 1999 R.M. Hardy Lecture: the implementation of unsaturated soil mechanics into geotechnical engineering. Canadian Geotechnical Journal 2000;37:963–86.
Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
Shear and dilatational wave velocities for unsaturated soils Enrico Conte , Renato Maria Cosentini, Antonello Troncone ` della Calabria, 87036 Rende, Cosenza, Italy Dipartimento di Difesa del Suolo, Universita
a r t i c l e in f o
a b s t r a c t
Article history: Received 18 June 2008 Received in revised form 30 October 2008 Accepted 11 November 2008
The primary objective of this study is for presenting some simple-to-use expressions relating the shear and dilatational wave velocities (VS and VP) to some physical and constitutive parameters of unsaturated soils. To this purpose, a simplified formulation is developed using the theory of linear poroelasticity in conjunction with some constitutive parameters widely used in geotechnical engineering. The derived expressions are of practical interest in view of the fact that they could be employed for evaluating the involved soil parameters from VS and VP measurements by in-situ or laboratory geophysical tests. & 2008 Elsevier Ltd. All rights reserved.
keywords: Wave velocity Shear wave Dilatational wave Unsaturated soils
1. Introduction Wave propagation in fluid-containing porous media has received great attention in recent years, because of its importance in geotechnical engineering for determining soil properties from the results of field or laboratory geophysical tests. Owing to the fact that in these tests the soil is usually subjected to small amplitude excitations, the material parameters obtained in this way are commonly referred to the undisturbed state of the soil. In addition, linear elasticity is the most frequently used constitutive assumption to predict the mechanical behaviour of the soil. In this context, Foti et al. [1] proposed a novel procedure for evaluating the porosity of saturated soils from measured shear and dilatational wave velocities. This procedure is very useful in practice, especially in the case of coarse-grained soils in which undisturbed sampling can only be achieved using sophisticated techniques such as freezing. The basic theory for wave propagation in porous media saturated by a single compressible fluid was originally formulated by Biot [2,3]. He demonstrated that a single shear wave and two dilatational waves can propagate in an infinite, homogeneous, isotropic poroelastic medium as opposed to the case of a oneconstituent elastic medium in which only one dilatational wave and one shear wave propagate. Since the development of Biot’s theory, many attempts have been made to present more general models for the analysis of wave propagation though saturated soils [4–12]. Corresponding author at: Dipartimento di Difesa del Suolo, Universita` della Calabria, Ponte P. Bucci, Cubo 41b 87036 Rende, Cosenza, Italy. Tel.: +39 0984 496524; fax: +39 0984 496526. E-mail address: [email protected] (E. Conte).
0267-7261/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2008.11.001
Unsaturated soils are more complicated multiphase systems comprising a solid skeleton and pores filled by more than one fluid (air and water). A simplification is usually made when the soil is very close to saturation. In such circumstances, in fact, air contained in the pores is present in the form of occluded bubbles that can be considered as one with pore-water. As a consequence, an accepted assumption is to retain the formal structure of Biot’s theory, while introducing an appropriate bulk modulus for the water–air mixture [4,13]. Brutsaert [14] was perhaps the first to develop a comprehensive model for describing wave propagation through a porous medium saturated by two fluids. Extensions to this model were afterwards presented by several authors [15–18]. Further contributions were provided by Garg and Nayfeh [19] and Wei and Muraleetharan [20] using the mixture theory, and Tuncay and Corapcioglu [21] who employed a volume-averaging technique. These studies showed the existence of three different dilatational waves and one shear wave whose propagation velocities are frequency-dependent (dispersive waves) due to the relative movement of the fluids with respect to the solid phase, which causes a viscous dissipation of energy. While all these formulations are highly valuable from a theoretical point of view, their use is generally unsuitable for engineering purposes because some of the involved material parameters are difficult to determine experimentally, especially in in-situ conditions. In this paper, a simplified formulation is developed with the primary object of presenting simple-to-use expressions for the shear and dilatational wave velocities (VS and VP) in unsaturated soils excited by small perturbations. These expressions are of practical interest in view of the fact that they could be readily adopted for evaluating some soil parameters from VS and VP measurements by in-situ or laboratory tests. The formulation is based on the elasticity theory in conjunction with the constitutive
ARTICLE IN PRESS E. Conte et al. / Soil Dynamics and Earthquake Engineering 29 (2009) 946–952
relationships proposed by Fredlund and Morgenstern [22] to describe the volume change in unsaturated soils, and Darcy’s law to express the flow of the fluids through the soil. Three different conditions are analysed: (a) water and air move separately with respect to the solid phase; (b) one fluid flows and the other is at rest and (c) no relative motion occurs between the solid and the fluid phases.
947
convention is used. On the other hand, when the soil approaches a dry condition, the coefficients of water volume change go to zero. Following Fredlund and Rahardjo [23], Eq. (1a) can also be formulated in a linear elasticity form in which the net total stress and matric suction are considered as stress-state variables. This leads to the equation [24] dr ¼ Ede w dpw I ð1 wÞ dpa I
2. Basic assumptions Unsaturated soils are made up of three constituents: soil skeleton, water and air, which in the context of mixture theory are viewed as three continua simultaneously occupying the same regions of space. In this study, wave propagation in such materials is analysed under the assumptions of infinitesimal strains, isothermal conditions, solid particle incompressibility, isotropic linearly elastic soil skeleton and validity of Darcy’s law for describing the flow of air and water through the soil. Gravity force, chemical and electric effects are neglected. In addition, the medium is assumed to be infinite and homogeneous so that reflection, refraction, scattering and mode conversion phenomena are also ignored. Due to these simplified assumptions, the formulation presented in the following sections is approximate, but of practical interest from an engineering viewpoint. The sign convention of continuum mechanics is used, according to which the normal stresses are positive in tension. Moreover, the symbol p is adopted for pore pressures instead of the usual u to avoid confusion with the displacements.
where r is the total stress tensor and e is the strain tensor of the solid skeleton, E is the tensor of the elastic moduli, I stands for the identity tensor and w is a parameter that is related to the volume change coefficients of the solid phase by the equation
w ¼ ms2 =ms1
dv ¼ ms1 dðsm þ pa Þ ms2 dðpa pw Þ
de ¼ E1 dr þ dnw ¼
dna ¼ ma1 dðsm þ pa Þ ma2 dðpa pw Þ
(1c)
where ev is the volumetric strain of the solid structure, ms1 and ms2 denote the associated coefficients of volume change with respect to a change in the mean net total stress, (sm+pa), and in matric suction (papw) (which are the stress-state variables usually adopted to describe the behaviour of the unsaturated soils), nw is the volume w fraction of water, mw 1 and m2 are the coefficients of water volume a change, n is the volume fraction of the air phase, and ma1 and ma2 are the respective volumetric coefficients. Moreover, sm is the mean total stress (i.e., sm ¼ (sx+sy+sz)/3, where sx, sy and sz are the total normal stresses in x, y and z directions), pw and pa are the porewater and pore-air pressures, respectively. Continuity requirement for the volume change of the three phases leads to the following equations relating these coefficients: ms1
ma1
(2a)
a ms2 ¼ mw 2 þ m2
(2b)
¼
mw 1
þ
As highlighted by Fredlund and Morgenstern [22], when the pores are completely filled by water (saturated soils) the volume change coefficients of this phase and those of the solid one are replaced by the coefficient of volume change mv of the soil skeleton, and Eqs. (1a) and (1b) reduced to the compressibility form relationship commonly adopted in geotechnical engineering, that is dev ¼ mvd(sm+pw), in which the above-specified sign
3K s
w 3K s
dpw I þ
1w dpa I 3K s
(5a)
I dr þ cww dpw þ cwa dpa
(5b)
I dr þ caw dpw þ caa dpa
(5c) s
1
where E denotes the inverse of E, and K is the bulk modulus of the soil skeleton. The other coefficients appearing in Eqs. (5a)–(5c) are
cw ¼
mw 1 ms1
(6a)
ca ¼
ma1 ms1
(6b)
wcw
cww ¼
cwa ¼ (1b)
cw
3K s
ca
dna ¼
(1a)
w dnw ¼ mw 1 dðsm þ pa Þ m2 dðpa pw Þ
(4)
After performing a series of algebraic operations, Eqs. (3), (1b) and (1c) can be cast in the form
3. Constitutive equations The formulation developed in this paper utilizes the constitutive relationships introduced by Fredlund and Morgenstern [22] to describe the volume change of unsaturated soils. These relationships can be expressed in differential form as follows:
(3)
caw ¼
caa ¼
K
s
þ
s w ms1 mw 2 m2 m1 s m1
(6c)
s w ð1 wÞcw ms1 mw 2 m2 m1 s s m K 1
(6d)
ms1 ma2 ms2 ma1 ms1
(6e)
wca Ks
þ
ð1 wÞca ms1 ma2 ms2 ma1 ms1 Ks
(6f)
Eqs. (5a)–(5c) are similar in form to those presented by Loret and Khalili [25]. However, the constitutive parameters appearing in Eqs. (6a)–(6f) are different from those considered by these latter authors. It is also relevant to note that for an isothermal and reversible process in which dissipation does not occur, the following equation can be written [11,25]: dHs ¼ e dr þ nw dpw þ na dpa
(7)
where Hs is the complementary free energy for the solid skeleton. As a result, the skeleton state equations are
e¼
qH s ; qr
nw ¼
qHs ; qpw
na ¼
qHs qpa
(8)
from which the following relationships can be obtained: de ¼
q2 Hs q2 H s q2 H s dr þ dpw I þ dp I 2 qpw qr qpa qr a qr
dnw ¼
q2 Hs q2 Hs q2 Hs I dr þ 2 dpw þ dp qrqpw qpw qpa a q pw
(9a)
(9b)
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q2 Hs q2 Hs q2 Hs I dr þ dpw þ 2 dpa qrqpa qpa qpw q pa
dna ¼
(9c)
By comparing these relationships to Eqs. (5a)–(5c), it can be noted that the coefficients in these latter equations correspond to the second-order partial derivatives appearing in Eqs. (9a)–(9c) that are the terms of the Hessian matrix of Hs. Owing to the symmetry of this matrix, the following identity holds [24]: mw 1
¼
ms2
(10)
Moreover, the change in the volume fraction of water can also be expressed as dmw
w
dn ¼
rw
dpw n Kw w
(11a)
w
where m is the water mass content per unit volume of the soil, rw is the mass density of water, and Kw is the bulk modulus of water. This equation derives from the relationship dMw ¼ d(rwVw), in which Mw and Vw are the mass content and the volume of water, respectively. Likewise, for the air phase we have dna ¼
dma
ra
na
dpa Ka
(11b)
where ma, ra, and Ka stand for the above-defined parameters referred to air. Lastly, the mass balance equations are invoked. Under the hypothesis of small amplitude perturbations, these equations are [11] dmw
rw dma
ra
w
w
s
¼ n divðu u Þ
(12a)
¼ na divðua us Þ
(12b)
where the symbol ‘‘div’’ indicates the divergence operator, and us, uw and ua are the displacement fields for the solid, water and air phases, respectively. Substituting Eqs. (11a) through (12b) into (1b), (1c) and (3), and after performing a series of algebraic operations, the following equations can be obtained [24], which are the stress– strain equations adopted in the present study
r ¼ Gðgrad us þ gradT us Þ þ HI div us þ nw ½wL þ ð1 wÞCI div uw þ na ½wC þ ð1 wÞNI div ua
(13)
pw ¼ W div us nw L div uw na C div ua
(14)
pa ¼ M div us nw C div uw na N div ua
(15)
W ¼
KwA þ nw L þ na C D
N¼
C¼
K ½n
ms1
þK
w
ðms1 mw 2
s w a w w a a w s D ¼ ðms1 mw 2 m2 m1 Þðn K þ n K Þ þ n n m1
(16c)
D s w K w K a ðms1 mw 2 m2 m1 Þ D
ms2 mw 1 Þ
(16i)
nSK
as ¼
(16j)
1 2nSK SK
in which n is Poisson’s ratio and G is the shear modulus of the soil skeleton. They are related to ms1 by the equation [23] ms1 ¼
3ð1 2nSK Þ 2ð1 þ nSK ÞG
(16k)
It is worth mentioning that r, pw and pa in Eqs. (13)–(15) are viewed as incremental quantities referred to their initial values (before excitation occurs).
4. Governing equations The equations governing wave propagation in porous media are the linear momentum equation expressing the equilibrium of all forces acting on an elementary volume of medium, and the equations of the motion for the fluid phases. The former is s
w
a
€ ra na u € ¼0 € rw nw u div r rs ð1 nÞu
(17)
where rs is the mass density of the solid phase, n is the soil porosity and the dot notation indicates time derivatives. Moreover, the following identities can be written: na ¼ nð1 Sr Þ
(18a)
nw ¼ nSr
(18b)
where Sr is the degree of saturation of the soil. Darcy’s law is usually adopted to describe the motion of water and air through the soil. For an isotropic medium, by neglecting the gravity forces, this law is [11] w
s
_ u _ Þ¼ nw ðu
s
kw
rw g ka
ra g
w
€ Þ ðgrad pw þ rw u
(19)
a
€ Þ ðgrad pa þ ra u
(20)
in which kw and ka are the coefficients of permeability for the water and air phases, respectively, and g is the gravity acceleration. Taking advantage of the constitutive relationships (13) through (16k) and after performing some algebraic manipulations, Eqs. (17), (19) and (20) become 2
ðH þ 2G þ nw W þ na MÞgrad div us Gðgrad div us r us Þ þ nw ½wL þ ð1 wÞC nw L na Cgrad div uw þ na ½wC þ ð1 wÞN na N nw Cgrad div ua þ
(16b)
K B þ nw C þ na N D
w
(16h)
(16a)
s w K w ½na ms1 þ K a ðms1 mw 2 m2 m1 Þ L¼ D a
s w B ¼ nw ma1 þ K w ðms1 mw 2 m2 m1 Þ
a
a
M¼
(16g)
_ u _ Þ¼ na ðu
where the symbol ‘‘grad’’ indicates the spatial gradient, the superscript T denotes the transpose and the coefficients appearing in them take the form H ¼ 2as G wW ð1 wÞM
a s w s w A ¼ na mw 1 þ K ðm1 m2 m2 m1 Þ
ðnw Þ2 rw g w ðna Þ2 ra g a €s ¼ 0 _ u _ sÞ þ _ u _ s Þ ð1 nÞrs u ðu ðu kw ka
(21)
nw Wgrad div us þ ðnw Þ2 Lgrad div uw þ na nw Cgrad div ua
(16d)
ðnw Þ2 rw g w €w ¼ 0 _ u _ s Þ nw rw u ðu kw
(22)
na M grad div us þ nw na C grad div uw þ ðna Þ2 N grad div ua (16e)
(16f)
ðna Þ2 ra g a €a ¼ 0 _ u _ s Þ na ra u ðu ka
(23)
where r2 is the Laplacian operator. It is relevant to note that when w the soil is completely saturated (Sr ¼ 1, ms1 ¼ ms2 ¼ mw 1 ¼ m2 ),
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some terms appearing in these equations take an indeterminate form. By performing the limit of these terms for Sr approaching unity, Eqs. (21)–(23) reduce to the following equations: Kw grad div us 2Gð1 þ as Þ þ ð1 nÞ2 n Gðgrad div us r2 us Þ þ ð1 nÞK w grad div uw þ
n2 rw g w €s ¼ 0 _ u _ s Þ ð1 nÞrs u ðu kw
(24)
ð1 nÞK w nr g w _ u _ s Þ rw u €w ¼ 0 grad div us þ K w grad div uw w ðu kw n (25)
which coincide with those presented by Coussy [11] for a porous medium saturated by one fluid, provided that the solid particles are incompressible. In the above equations, the effects of inertial coupling between the fluid and solid phases are ignored.
949
and c2 ¼ V 2S ¼
G ð1 nÞrs
(33b)
This means that a transverse wave (S-wave) propagates in the plane tangent to the wavefront with velocity VS. On the other hand, substituting Eq. (29) into Eqs. (26)–(28) and after multiplication by n leads to the following system written in compact form: 0 s 1 u€ n B wC ðK c2 MÞ@ u€ n A ¼ 0 (34) a u€ n in which the matrices K and M are 2 6 K¼4
H þ 2G þ nw W þ na M
nw ½wL þ ð1 wÞC nw L na C
na ½wC þ ð1 wÞN nw C na N
nw W
ðnw Þ2 L
nw na C
na M
nw na C
ðna Þ2 N
3 7 5
(35a) 2
5. Wave velocities for unsaturated soils The discontinuity propagation theory [11,26] is employed here to obtain explicit relationships of the wave velocities for unsaturated soils. In this theory, a wave is considered as a singular surface of the second-order (displacement and velocity are continuous across it, but this does not hold for acceleration) propagating with a finite speed that depends on the properties of the soil. For the sake of completeness, three different conditions are analysed: (a) water and air move separately with respect to the solid phase; (b) one fluid flows with respect to the solid phase and the other is at rest and (c) no relative motion occurs between the solid and the fluid phases, which is equivalent to considering the porous medium as a closed (undrained) system. In the condition denoted as (a), wave propagation is governed by Eqs. (21) through (23). Applying the jump operator [[ ]] to these equations and taking advantage of Hadamard’s compatibility conditions [11] lead to the following equations: s
6 M¼4
s
s
w
a
w
s
w a
w
a 2
a
2 a
a
€ n þ n n C½½n u € n þ ðn Þ N½½n u € n c n ra ½½u € ¼ 0 n M½½n u
€ s ¼ u€ sn n þ u€ st t; ½½u
€ w ¼ u€ w €w ½½u n n þ ut t;
€ a ¼ u€ an n þ u€ at t ½½u
ða2p1 a2p2 a2p3 þ A1 a2p3 þ A2 a2p2 A3 a2p1 A0 Þ ¼ 0
s
½G c2 ð1 nÞrs u€ t ¼ 0
(30)
w
(31)
ðc2 nw rw Þu€ t ¼ 0 a
ðc2 na ra Þu€ t ¼ 0
(32)
from which it results that w a u€ t ¼ u€ t ¼ 0
(33a)
(37a)
where a2p1 ¼ a2p2 ¼ a2p3 ¼
H þ 2G þ nw W þ na M ð1 nÞrs nw L
rw na N
ra
(37b)
(37c)
(37d)
A1 ¼
nw W½wL þ ð1 wÞC nw L na C ð1 nÞrs rw
(37e)
A2 ¼
na M½wC þ ð1 wÞN nw C na N ð1 nÞrs ra
(37f)
A3 ¼
(29)
where t is the unit vector lying in the plane tangent to the wavefront. Substituting these latter into Eqs. (26)–(28) and multiplied by t, we obtain
By solving
ðV 2P Þ3 ða2p1 þ a2p2 þ a2p3 ÞðV 2P Þ2 þ ða2p1 a2p2 þ a2p1 a2p3 þ a2p2 a2p3 þ A1 þ A2 A3 ÞV 2P
(28) in which n is the unit vector normal to the discontinuity surface (wavefront) and c is the speed of propagation. In addition, the ¨ s]], [[u ¨ w]] and [[u ¨ a]] can be decomposed in the discontinuities [[u form
(36)
where ‘‘det’’ denotes the determinant, and c ¼ V2P. Eq. (36) leads to the following cubic equation in V2P:
(27) a
(35b)
na ra
2
(26) € n þ ðnw Þ2 L½½n u € n þ nw na C½½n u € n c2 nw rw ½½u € ¼ 0 nw W½½n u
0
3 0 0 7 5
detðK c2 MÞ ¼ 0
w
a
n rw
0
0 w
It is worth noting that owing to Eq. (10), K is a symmetric matrix [24]. According to Eq. (34), three different dilatational waves (P-waves) can propagate along the normal direction to the wavefront. Their velocities are the solutions of the equation
€ n þ nw ½wL þ ð1 wÞC nw L na C½½n u € n ðH þ G þ nw W þ na MÞ½½n u € n þ ½G c2 ð1 nÞrs ½½u € ¼ 0 þ na ½wC þ ð1 wÞN nw C na N½½n u
ð1 nÞrs 0
A0 ¼
nw na C 2
rw ra ½wL þ ð1 wÞC nw L na Cnw na MC ð1 nÞrs rw ra ½wC þ ð1 wÞN nw C na Nnw na WC þ ð1 nÞrs rw ra
(37g)
(37h)
From Eqs. (33b) and (37a) as well as from the expressions presented below for the cases (b) and (c), positive and negative roots are obtained for VS and VP. They correspond to the wave propagation velocities in the positive and negative directions, respectively. For the purposes of the present study, only the positive roots are considered. In case (b), when water motion is prevented with respect to the solid phase (uwus ¼ 0), the governing Eqs. (21)–(23) reduce to
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E. Conte et al. / Soil Dynamics and Earthquake Engineering 29 (2009) 946–952
½H þ 2G þ nw wL þ nw ð1 wÞC þ na ðM nw CÞ grad div us Gðgrad div us r2 us Þ þ na ½ðwC þ ð1 wÞNÞ na N grad div ua þ
ðna Þ2 ra g a _ u _ s Þ ½ð1 nÞrs þ nw rw u €s ¼ 0 ðu ka
(38)
ðna Þ2 ra g a €a ¼ 0 _ u _ s Þ ra na u ðu ka
In this case, it can be shown the existence of one shear wave propagating with velocity (39)
Proceeding as above, the application of the discontinuity propagation method leads to the existence of one shear wave and two different dilatational waves, the propagation velocities of which are V 2S ¼
G ð1 nÞrs þ nw rw
KwA KaB 2 € ¼0 þ ð1 wÞ grad div u Gðgrad div u r uÞ ru 2Gð1 þ as Þ þ w D D
(45)
na ðM nw CÞgrad div us þ ðna Þ2 Ngrad div ua
Taking into account the constitutive relationships (13) through (16k), Eq. (44) becomes
V 2S ¼
G
(46)
r
and a single dilatational wave with velocity V 2P ¼
ð2Gð1 nSK Þ=ð1 2nSK ÞÞ þ ðdP1 =dP2 Þ
r
where (40)
2 a s 2 w s dP1 ¼ K w K a ðw2 þ mw 2 =m1 Þ þ n½w K ð1 Sr Þ þ ð1 wÞ K Sr =m1
(48a)
and V 2P ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 2 ðbp1 þ bp2 Þ ðbp1 bp2 Þ2 þ 4Bo 2
s 2 s dP2 ¼ nðSr K a þ ð1 Sr ÞK w Þðw2 þ mw 2 =m1 Þ þ n Sr ð1 Sr Þ=m1
(41a)
(48b)
(41b)
It is relevant to note that when the pores are completely filled by air (dry soil, Sr ¼ 0) or water (saturated soil, Sr ¼ 1), the shear wave velocity is again provided by Eq. (46) in which the mass density of the soil is
where 2
bp1 ¼
2
bp2 ¼
(47)
H þ 2G þ na ðM nw CÞ þ nw wL þ nw ð1 wÞC ð1 nÞrs þ nw rw
r ¼ ð1 nÞrs þ nra
(49a)
or
na N
(41c)
ra
r ¼ ð1 nÞrs þ nrw
(49b)
respectively. On the other hand, Eq. (47) reduces to na ½wC þ ð1 wÞN na NðM nw CÞ B0 ¼ ½ð1 nÞrs þ nw rw ra
(41d)
Similarly, when water flows and air is at rest (uaus ¼ 0) the expressions for VS and VP are V 2S
¼
G ð1 nÞrs þ na ra
(42)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðc2p1 þ c2p2 Þ ðc2p1 c2p2 Þ2 þ 4C o 2
(43a)
where c2p1 ¼
c2p2 ¼
H þ 2G þ nw ðW na CÞ þ na wC þ na ð1 wÞN ð1 nÞrs þ na ra nw L
rw
C0 ¼
nw ½wL þ ð1 wÞC nw LðW na CÞ ½ð1 nÞrs þ na ra rw
(43b)
(43c)
(43d)
Lastly, for the condition denoted as (c), when no relative motion between the phases occurs (ua ¼ uw ¼ us ¼ u), the governing field equation is € ¼0 div r ru
(44)
in which
r ¼ ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra
(44a)
ð2Gð1 nSK Þ=ð1 2nSK ÞÞ þ ðK a =nÞ ð1 nÞrs þ nra
(50)
ð2Gð1 nSK Þ=ð1 2nSK ÞÞ þ ðK w =nÞ ð1 nÞrs þ nrw
(51)
or V 2P ¼
and V 2P ¼
V 2P ¼
depending on whether the soil is dry or fully saturated. This latter equation coincides with that proposed by Foti et al. [1] (Eq. (17) in their paper) for evaluating the porosity of saturated soils from Vp and Vs measurements. It should be noted that the solution obtained in this section accounts for no dissipation of energy associated with the viscous flow induced by the relative motion between the fluids and the solid. The analysis of these phenomena and their effects on the wave velocity is beyond the scope of the present study. Nevertheless, it should be mentioned that at low frequencies the relative motion between the fluid and solid phases can be ignored, whereas at very high frequencies the fluids are free to move through the porous medium without dissipation [14]. In the light of these considerations, the equations derived in the present study can be referred to these latter conditions. Miura et al. [12] conducted an interesting study on the frequency-dependent characteristics of the waves in various saturated soils from clay to soft rocks. The study highlighted that the coefficient of permeability is the main soil parameter controlling the wideness of the low-frequency range. In particular, the smaller is the coefficient of permeability, the wider the range. Considering the permeability characteristics of the saturated soils and the frequency content of most geophysical tests in which the maximum frequency is of the order of some hundred kHz [10], it can be assumed that during these tests the soil is excited at low frequencies [1,2,12]. This assumption should be even more accepted for unsaturated soils that are generally characterized
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by much smaller values of the coefficient of permeability than the saturated soils. Owing to these considerations, Eqs. (46) and (47), which were derived for the low-frequency range, may be used in the field of geotechnical engineering for the characterization of the soils. In this connection, Eq. (47) can be cast in a more convenient form by introducing some additional simplifications. To this purpose, it is relevant to note that Eq. (3) is similar to the wellknown relationship first proposed by Bishop [27]
r ¼ r¯ pa I þ wðpa pw ÞI
#
Appendix A The expressions of VP and VS for unsaturated soils as obtained for the low-frequency range, are, respectively
3ð1 2nSK ÞS2r þ nSr ð1 Sr Þ½K w Sr þ K a ð1 Sr Þ 2ð1 þ nSK ÞG 2ð1 n Þ " # Gþ 1 2nSK 3ð1 2nSK ÞS2r þ n2 Sr ð1 Sr Þ n½K a Sr þ K w ð1 Sr Þ mw 2 SK 2ð1 þ n ÞG SK
V 2P ¼
unsaturated soils, have been derived. These expressions are of practical interest from an engineering viewpoint, considering the fact that they could be adopted for evaluating some material properties of unsaturated soils when VP and VS measurements from field or laboratory geophysical tests are available. Owing to the fact that these tests are based on the application of a small perturbation to the soil, the material parameters obtained in this way are commonly referred to the initial or undisturbed state of the soil.
(52)
in which r¯ is the effective stress tensor, and w is a parameter ranging between 0 and 1, which is often replaced by Sr [20,28–30] as an approximation. Using this latter simplification and considering Eq. (16k), Eq. (47) becomes "
"
K a K w mw 2
ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra
# 3ð1 2nSK ÞS2r þ nSr ð1 Sr Þ½K w Sr þ K a ð1 Sr Þ 2ð1 þ nSK ÞG 2ð1 nSK Þ " # Gþ 1 2nSK 3ð1 2nSK ÞS2r n½K a Sr þ K w ð1 Sr Þ mw þ n2 Sr ð1 Sr Þ 2 SK 2ð1 þ n ÞG K a K w mw 2
V 2P ¼
ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra
(53)
(A.1)
where G can be obtained inverting Eq. (46) in which r is expressed by Eq. (44a), that is
and
G ¼ ½ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra V 2S
V 2S ¼
(54)
Finally, under the reasonable assumptions that the air mass density is ignored (raffi0) and the bulk modulus of air is negligible with respect to that of water (Ka/Kwffi0), the expression for V2P simplifies as follows (see Appendix A): V 2P ¼
2ð1 nSK Þ 2 V þ 1 2nSK s
S2r ½ð1 nÞrS þ Sr nrw mw 2
3ð1 2nSK ÞS2r 2ð1 þ nSK ÞV 2s
þ
nSr ½ð1 nÞrS þ Sr nrw Kw
(55) In these latter equation, rw, rS and K may be considered as physical constants that assume standard values, and VP and VS can be measured using laboratory or field tests. As a result, Eq. (55) represents a relationship in which the unknown variables are the porosity n, Poisson’s ratio nSK, degree of saturation Sr, and the coefficient of volume change for the water phase. This latter w is defined as mw 2 ¼ qn /q(papw) owing to Eq. (2b), and can be experimentally obtained from a soil–water characteristic curve [23]. From an engineering viewpoint, Eq. (55) can be employed for evaluating the soil properties appearing in it from VS and VP measurements by geophysical tests, such as cross-hole or downhole tests. In particular, Eq. (55) can be readily inverted to obtain mw whose evaluation is not generally a simple operation, 2 especially in in-situ conditions [31]. The resulting expression is mw 2 ¼
V 2P ¼
Moreover, considering the values that generally characterize the bulk modulus of water, the bulk modulus of air and the mass density of air (Kw ¼ 2.25 106 kPa, Ka ¼ 1.45 102 kPa, and ra ¼ 0.001 t/m3), it may be assumed that Ka is negligible with respect to Kw (i.e., Ka/Kwffi0), and raffi0. As a result, Eq. (A.3) takes the form 2ð1 nSK Þ " Gþ 1 2nSK nð1 Sr Þ V 2P ¼
Kw
nS2r ð1 Sr Þ Kw # 3ð1 2nSK ÞS2r n2 Sr ð1 Sr Þ w m2 þ 2ð1 þ nSK ÞG ðK w Þ2
ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra (A.4)
which reduces to the simpler equation 2ð1 nSK Þ Gþ" 1 2nSK V 2P ¼
6. Concluding remarks
ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra
(A.3)
ð1 2nSK ÞS2r 3 1 nSr þ w ½ð1 nÞrS þ Sr nrw 2ð1 þ nSK ÞV 2S ð1 2nSK ÞV 2P 2ð1 nSK ÞV 2S K
Obviously, Eq. (55) can also be inverted to evaluate Poisson’s ratio of the solid skeleton, degree of saturation or porosity of the soil, once the other parameters are known. Owing to the fact that in the geophysical tests the soil is excited by very small levels of energy, the parameters obtained in this way are commonly referred to the initial or undisturbed state of the soil.
(A.2)
" # Ka 3ð1 2nSK ÞS2r nSr ð1 Sr Þ w w þ m ½K Sr þ K a ð1 Sr Þ 2 2ð1 þ nSK ÞG Kw ðK w Þ2 2ð1 nSK Þ " # Gþ 1 2nSK n 3ð1 2nSK ÞS2r n2 Sr ð1 Sr Þ þ ½K a Sr þ K w ð1 Sr Þ mw 2 w 2 SK 2ð1 þ n ÞG ðK Þ ðK w Þ2
#
(56)
G ð1 nÞrs þ Sr nrw þ ð1 Sr Þnra
In these expressions, G is the shear modulus and nSK is Poisson’s ratio of the soil skeleton, mw 2 denotes the coefficient of water volume change with respect to a change in matric suction, Kw is the bulk modulus of water and Ka is that of air, n is the porosity and Sr is the degree of saturation of the soil, rs, rw and ra are the mass density of the solid phase, water phase and air phase, respectively. By dividing both the numerator and denominator of the second term on the right-hand side of Eq. (A.1) by (Kw)2, yields
w
"
951
S2r mw 2
3ð1 2nSK ÞS2r 2ð1 þ nSK ÞG
ð1 nÞrs þ Sr nrw
!# þ
nSr Kw
(A.5)
in which the shear modulus G can be evaluated from Eq. (A.2) as Theoretical expressions relating VP and VS to the soil parameters that describe the volume changes experienced by
G ¼ ½ð1 nÞrs þ Sr nrw V 2S
(A.6)
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E. Conte et al. / Soil Dynamics and Earthquake Engineering 29 (2009) 946–952
Finally, after substituting Eq. (A.6) into Eq. (A.5), the expression for VP can be written as follows: V 2P ¼
2ð1 nSK Þ 2 V þ 1 2nSK S
S2r ½ð1 nÞrs þ Sr nrw mw 2
3ð1 2nSK ÞS2r 2ð1 þ nSK ÞV 2S
þ
nSr ½ð1 nÞrs þ Sr nrw Kw
(A7)
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