Jour nal of Theoretical and Applied Mechanics , Sofia, V ol. 47 No . 2 (2017) pp . 34-60
DOI: 10.1515/jtam-2017-0009
INFLUENCE OF POR OELASTICITY ON THE 3D SEISMIC
RESPONSE OF COMPLEX GEOLOGICAL MEDIA
F R A N K W U T T K E 1 , P E T I A D I N E V A 2 ,
I O A N N A - K LEONIKI F O N T A R A 3 ∗ ,
1 Institute of Applied Geoscience, Kiel Univer sity , Germany
2 Institute of Mechanics, Bulgarian Academy of Sciences, Sofia, Bulgaria
3 Institute of Civil Engineering, T echnical Univer sity of Berlin, Germany
[Recei ved 21 April 2017. Accepted 26 June 2017]
A B S T R A C T : Elastic wa ve propagation in 3D poroelastic geological media
with localized heterogeneities, such as an elastic inclusion and a canyon is
in vestig ated to visualize the modification of local site responses under consid-
eration of water saturated geomaterial. The extended computational en viron-
ment herein de v eloped is a direct Boundary Inte gral Equation Method (BIEM),
based on the frequency-dependent fundamental solution of the gov erning equa-
tion in poro-visco elastodynamics. Bardet’ s model is introduced in the analysis
as the computationally ef ficient viscoelastic isomorphism to Biot’ s equations
of dynamic poroelasticity , thus replacing the two-phase material by a com-
plex v alued single-phase one. The potential of Bardet’ s analogue is illustrated
for lo w frequenc y vibrations and all simulation results demonstrate the depen-
dency of w av e field de veloped along the free surface on the properties of the
soil material.
K E Y W O R D S : 3D seismic wav e propagation; saturated soil, surface relief, elas-
tic inclusion, Boundary Integral Equation Method (BIEM).
1 . I NTR ODUCTION
T o date, it has proven impossible or technically e xpensi v e to consider important me-
chanical characteristics of geological materials as anisotropy , poroelasticity , inelas-
ticity , inhomogeneity and heterogeneity in design codes, e. g. CEN Eurocode 8 [1].
This is due to se veral reasons, the primary one being the sheer complexity of the me-
chanical models, describing the seismic scenarios that may be dev eloped at a giv en
geographical location. Propagation of seismic w a ves through heterogeneous and in-
homogeneous geological structures causes reflection, refraction, dif fraction and scat-
tering phenomena, that are dif ficult to quantify .
Dif ferent types of heterogeneities, such as b uried cracks, ca vities, inclusions, sur -
face and subsurf ace topography , sedimentary basins complicate the ov erall picture
∗ Corresponding author e-mail: [email protected]
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Influence of Poroelasticity on the 3D Seismic Response of ... 35
generating lar ge amplification, as well as spatial v ariations in the seismic motions.
In particular , the porous structure containing fluids modify the w av e field beha viour
drastically . All these ha v e important repercussions in the analysis of lar ge infras-
tructure such as dams, bridges, pipelines, tunnels, etc. The literature is quite rich
in terms of results obtained for 2D wa ve scattering problems by dif ferent types of
heterogeneities in homogeneous [2-6], discrete inhomogeneous [7] or continuously
inhomogeneous continua [9-10]. The three-dimensional nature of the seismic motion
doesn’ t simplify the modelling and it is also not taken into consideration in the design
codes. Our attention in this work focuses on modelling of 3D seismic w av e propa-
gation in a heterogeneous poroelastic re gion, considering the influence of porosities,
densities and topographical peculiarities on the seismic site response and its change
in contrast to non-porous media. The boundary integral equation method (BIEM) is a
well-kno wn computational technique with discussed in the literature advantages for
solution of seismic wa ves in a comple x infinite continuum, see Manolis et al . [11]. In
what follo ws, we will present a short state of the art, concerning BIEM for solution
of 3D seismic wa ves in saturated, poro-elastic soils. Generally , the BIEM adv an-
tages are as follo ws: ( a ) reduction of the model dimensionality; ( b ) fast solution at
selected internal points in terms of boundary-only information and without recourse
to domain discretization; ( c ) high numerical accuracy since quadrature technique is
directly applied to the boundary inte gral equation, which in turn is an exact math-
ematical statement of the problem under consideration; ( d ) a priori satisfaction of
Sommerfeld’ s radiation condition at infinity . The BIEM requires two basic ingredi-
ents in its formulation, namely a reciprocal relation and a fundamental solution. The
fundamental solution plays the ke y role in describing a gi ven boundary-v alue prob-
lem (BVP) by a system of boundary integral equations (BIE’ s), based on the recip-
rocal theorem. The comprehensi ve state-of-the-art re views for dynamic fundamental
solutions, deriv ed for the porous media and their subsequent incorporation within
coupled BIE’ s can be found in Schanz [12], Gatmiri and Kamalian [13], Gatmiri and
Nguyen [14], Se yrafian et al . [15] and Gatmiri and Eslami [16]. Application-type
examples, that serve as benchmarks can also be found in the literature, see for in-
stance Theodorakopoulos and Besk os [17], Dominguez [18] and Albers et al . [19].
In Schanz [8, 12] v arious formulations in poroelasticity together with analytical and
numerical methods of solutions, associated with these formulations are discussed.
W av e scattering in poroelastic media is in vestigated by Y amamoto and Kitahara
[20], using BIEM. Reflections of plane harmonic wa v es in a fluid-filled poroelastic
half-space are discussed in Lin et al . [21] via analytical approach. Hasheminejad and
A v azmohammadi [22] in vestigated the w av e dif fraction by two circular ca vities in a
poroelastic medium. Nenning and Schanz [23] studied wa v e propagation problems
in a poroelastic medium with finite element method (FEM) by introducing infinite
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36 F . W uttke, P . Dine v a, I.-K. Fontara
elements for representing far field beha viour . BIEM results for solution of 3D elas-
todynamic problems, treating the media as a monophasic pure elastic continuum can
be found in Gonsalv es et al . [24], Sanchez-Sesma and Luzon [25], T adeu et al . [26],
Galis et al . [27], Chaillat et al . [28], Fu et al . [29], Grasso et al . [30], Chaillat and
Bonnet [31], Liu et al . [32], Niu and Dra vinski [33].
The a v ailable studies on scattering of elastic wa v es in a three-dimensional poroe-
lastic medium, approximated by a two-phase model are strongly limited and without
focus of the site response ef fects. Related publication are: Zimmerman and Stern [34]
studied the scattering of plane P- wa ves in a poroelastic space with spherical inclu-
sions by BIEM; Zhao and Han [35] in vestigated the scattering of Rayleigh-w a ves by a
hemi-spherical saturated alluvial v alley by F ourier -Bessel series expansion method;
Liu et al . [36] in vestigated the scattering of plane SV w a ves through spherical in-
clusions; Cheng and Detournay [37] presented a unified BIEM formulation for the
solution of quasi-static, anisotropic poroelasticity; Schanz [8] dev eloped 3-D time
domain BEM formulation for wa ve propagation in poroelastic solids via con v olution
quadrature method; Ding and Jiang [38], Ding et al . [39] proposed a time domain
BIEM for dynamic analysis of saturated porous media, subjected to external forces;
Ba et al . [40] studied 3-D scattering of obliquely incident transv erse shear SV -wa v es
through an alluvial v alley , embedded in a fluid-saturated, poroelastic layered half-
space; Liu et al . [41] presented an indirect BIEM for 3D scattering in a fluid saturated
poroelastic half-space.
In a series of papers, Bardet and Morochnik [42-43] proposed an equiv alent vis-
coelastic model, describing the dynamic response of saturated poroelastic materials,
that obey Biot’ s theory . The phase velocity and attenuation of the longitudinal and
shear wa ves in soils are described by a viscoelastic isomorphism, related to Biot’ s
material constants in such a way , that the wa ve numbers of poroelastic model accord
complex w av e numbers in the viscoelastic model. This approach has been v erified by
solution of the 3D problem for scattering of plane compression wa ves by a spherical
poroelastic inhomogeneity in [43], where the authors sho w that viscoelastic isomor -
phism solutions accord identical to the original poroelastic problem solution within
the specific range of parameters examined.
The brief literature re view sho ws that there is still limited number of BIEM results
for synthesis of 3D seismograms in saturated, poroelastic soils containing dif ferent
types of heterogeneities such as layers, cavities, inclusions or free surf ace relief. This
fact moti v ates the authors to consider the 3D elastodynamic problem, taking into
consideration the soil poroelasticity by the usage of Bardet viscoelastic model and
the numerical technique of boundary inte gral equations. This work is a continuation
of the pre vious authors’ results to model 2D plane wa v e scattering problems in in-
homogeneous and heterogeneous geological media [7,10,11] via BIEM, no w adding
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Influence of Poroelasticity on the 3D Seismic Response of ... 37
ef fects of the soil poroelasticity and three-dimensional nature of the seismic motion.
The paper is structured as follo ws: Section 2 gi v es an o v ervie w about the com-
putational model, basing on both the viscoelastic isomorphism to Biot’ s equations of
dynamic poroelasticity presented by the Bardet model and the direct BIEM for 3D
wa ve scattering in a heterogeneous poroelastic half-space. Next, Section 3 discusses
results from solution of two illustrati ve numerical examples for an elastic inclusion,
embedded in a poroelastic half-space and a can yon along the free surface of a poroe-
lastic half-space, subjected to plane time-harmonic P wa ve, both under consideration
of modified site responses. Finally , concluding remarks are gi v en in Section 4.
2 . C OMPUT A TION AL MODEL
2.1. T HE VISCOELASTIC ISOMORPHISM T O B I OT ’ S E Q U A T I O N S O F D Y N A M I C
P O R O - ELASTICITY – B ARDET MODEL
The most widely-used model, describing the propagation of elastic wa ves in a porous
medium has been proposed by Biot [44] for lo w-frequency re gimes, considered here
and by Biot [45] for high-frequencies. The representati v e elementary v olume (REV)
of a solid-fluid system with v olume V consists of an isotropic elastic porous skeleton
with porosity n defined as n = V pore /V , where V pore is the pore volume. The linear
Biot’ s model in the frame of continual theory is v alid if the wa v elength ˜
λ is lar ge in
comparison with the characteristic radius of the pores r . In low-frequenc y regime,
the Biot’ s model can be used, when ˜
λ  L  r , while at high frequencies the Biot’ s
model is correct, when ˜
λ ≈ L  r , where L is the characteristic size of the REV .
The used terminology for poroelastic materials is as follo ws:
• Solid phase (grains, solid material structure) with the following notations for
the elastic b ulk module K g and the density ρ g .
• Fluid with the follo wing notations for the bulk module K f and the density of
fluid ρ f .
• Dry porous solid structure (all pores are filled with gas phase/air) with the
follo wing material properties: elastic bulk module K dry and density ρ dry , de-
fined as ρ dry = (1 − n ) ρ g . The solid-fluid system density is defined as ρ sat =
ρ dry + nρ f = (1 − n ) ρ g + nρ f .
Biot’ s equations of motion in the case of time-harmonic excitation with frequenc y
ω in respect to solid displacements u i ( i = x, y , z ) and fluid displacements U i are as
follo ws:
µ ∆ u i + [( λ + µ + Q 2 /R ) e S
, i + Qε f
, i ] + ω 2 ( ρ 11 u i + ρ 12 U i )=0 ,
(1)
( Qe S
, i + R ε f
, i ) + ω 2 ( ρ 21 u i + ρ 22 U i )=0 .
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38 F . W uttke, P . Dine v a, I.-K. Fontara
Here: the dynamic mass Biot’ s coef ficients are ρ 11 = (1 − n ) ρ g + n ( τ α − 1) ρ f ,
ρ 12 = ρ 21 = − n ( τ α − 1) ρ f , ρ 22 = nτ α ρ f , τ α = 1 + τ r (1 − n ) /n , τ r = 0 . 5 for
sphere, whereas the range of τ r for other ellipsoids is between 0 and 1, see Ma vko et
al . [46]; ∆ is the Laplacian operator; e s = e s
ii = u i, i and ε f = ε f
ii = U i, i , λ , µ are
Lame constants. It is stated that summation on repeating indices and comma denotes
partial dif ferentiation. The shear stif fness of the porous material is captured by the
solid skeleton and is not af fected by fluid saturation. Due to this f act, both dry and
saturated materials ha v e the same shear modulus, µ dry = µ sat and hitherto denoted
by just µ . The Biot’ s elastic constants Q , R , and P are as follows:
Q =
nK g  1 − n − K dry
K g 
 1 − n − K dr y
K g
+ n K g
K f  ;
R = K g n 2
 1 − n − K dr y
K g
+ n K g
K f  ;
P = 3(1 − ν dry )
1 + ν dry
K dry + Q 2
R .
(2)
The solution for a plane time-harmonic wa ve, satisfying the Biot’ s equation of
motion was obtained in Bardet [43] and Lin et al . [21]. F ollo wing the Biot’ s wa ve
equation, three mathematical solutions hav e been identified, corresponding to shear
wa ve S, transmitted through the solid sk eleton, the first fast dilatational P wa ve and
second slo w dilatational P wa v e. The corresponding wa ve phase v elocities are com-
plex and frequenc y dependent, hence they correspond to dissipati ve and dispersi ve
wa ves. In a series of publications, Bardet [43] discussed the applicability of the
viscoelastic beha viour equi v alent to Biot’ s model [44] of dynamic poroelasticity in
the lo w frequency range. As a matter of fact, Bardet [43] proposed a single-phase
viscoelastic material representation for saturated, poroelastic soils. At first, the vis-
coelastic material constants are complex-v alued with the real and imaginary part,
expressed via Biot’ s coef ficients Q , R , P , based on the viscoelastic isomorphism.
Therefore, the model proposed by Bardet [43] describes the dissipativ e wa v e be-
ha viour with a visco-elastic K elvin-V oigt model. The follo wing e xpressions describe
the complex-v alued phase velocities of visco-elastic materials, follo wing the equiv-
alence between the dynamic responses of the poro-elastic and visco-elastic solids of
K elvin-V oigt rheological model.
For a K elvin-V oigt model, the comple x wa ve numbers are defined as
(3) k 2
S = ω 2
( C ∗
S ) 2 , k 2
P = ω 2
( C ∗
P ) 2 ,
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Influence of Poroelasticity on the 3D Seismic Response of ... 39
where the complex-v alued phase velocities C ∗
S and C ∗
P are
(4) ( C ∗
S ) 2 = C 2
S (1 − iω ξ S ) and ( C ∗
P ) 2 = C 2
P (1 − iω ξ P ) .
Here: C 2
S = µ/ρ and C 2
P = ( λ + 2 µ ) /ρ are the real parts of the elastic P- and S-wa ve
velocities, ξ S and ξ P are the corresponding attenuation coef ficients, that represent the
small hysteretic damping ratios for P- and S-wa ve v elocities in geomaterial, λ and µ
are the real parts of the complex v alued Lame constants, i is the imaginary unit. In
the lo w frequency range when ω ξ P  1 and ω ξ S  1 , Eqn. (3) can be approximated
as
(5) k S ≈ ω (1 + i 0 . 5 ω ξ S ) /C S and k P ≈ ω (1 + i 0 . 5 ω ξ P ) /C P .
Bardet (1992) defines the equi valent ef fectiv e phase velocities and the correspond-
ing attenuation coef ficients, as follows:
C P = p ( P + 2 Q + R ) /ρ sat ; C S = p µ/ρ sat , (6)
ξ P = ρ sat
b  Q + R
P + 2 Q + R
nρ f
ρ sat  2 ; ξ S = ρ sat
b  nρ f
ρ sat  2 , (7)
b = n 2 g ρ f / ˆ
k , g is the gra vity acceleration; ˆ
k is the hydraulic conducti vity .
(8) λ sat = λ dry + Q 2
R = 3 ν dry
1 + ν dry
K dry + Q 2
R ; µ sat = µ dry = µ ,
Morochnik and Bardet [42] obtained the approximated Eqns. (6)–(8) in the case
of ω ρ sat /b  1 . Due to the fact, that the hydraulic conducti vity ˆ
k for soils has
small v alues (for example for sand ˆ
k = 10 − 6 ÷ 10 − 4 ) , this condition is fulfilled for
lo wer frequencies, more specifically for the frequencies encountered in earthquake
engineering problems. Eqns. (6)–(8) sho w that the equi v alent viscous model consid-
ers the soil compression stif fening (bulk modulus), associated with the induced pore
pressure during the seismic loading, which causes an increase of λ sat in comparison
with λ dry , hence increases the v alue of the longitudinal wa ve phase v elocity in case
of saturated soils.
The applicability of Bardet [43] viscoelastic isomorphism to the solution of seis-
mic wa ve propagation problems in a fluid-saturated half-space is discussed in Dine v a
et al . [47]. The solutions for free-field motion in a homogeneous poroelastic half-
space under incident longitudinal P-wa ve, obtained by the two-phase Biot’ s model
in Lin et al . [21] and those obtained by the one-phase viscoelastic Bardet model are
identical for lo w frequencies, see Dine va et al . [47]. The limitations of the Bardet
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40 F . W uttke, P . Dine v a, I.-K. Fontara
model are: (a) it is v alid for the lo w frequency diapason, discussed abo v e; (b) it can-
not account for the second (slo w) longitudinal wa v e, because it describes a one-phase
viscoelastic material; (c) it is not possible to account for the boundary conditions of
pore fluid pressure, since one-phase viscoelastic material is considered. Bardet model
shares the same range of applicability dictated by Biot’ s theory , because it is recov-
ered from the original Biot’ s model. Regardless of the abo ve mentioned limitations,
the main adv antage of the Bardet’ s model is that it can be easily incorporated in the
existing BIEM softw are, av oiding the complicated fundamental solution of the gov-
erning equation in Biot’ s model.
Figure 1 sho ws the sensitivity of the P- and S-w av e v elocity for dry and saturated
sandstone to the porosity . Results are plotted for v alues of Poisson’ s ratio, ranging
Fig. 1. P- and SV -wa v e v elocity v ariation with porosity for pure elastic, dry and saturated soil
by Bardet (1992) model.
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Influence of Poroelasticity on the 3D Seismic Response of ... 41
between 0.1 and 0.4. For the illustration are used the reference geological material
properties, gi ven in Lin et al . [21]: K g = 36000 MPa; ρ g = 2650 kg/m 3 ; K f =
2000 MPa; ρ f = 1000 kg/m 3 . The dry b ulk modulus can be calculated from the
relationship, deri v ed from e xperimental data for poroelstic standstone, gi v en in Gal.
et al . (1998) [48] and discussed in Lin et al . (2005): K dry = K cr + (1 − n/n cr )( K g −
K cr ) , where K cr = 200 MPa and n cr = 0 . 36 . The bulk modulus for a dry porous
material is equal to zero at a ‘critical’ porosity , in which the percentage of pore space
in a unit v olume becomes too lar ge to form a sustainable solid frame.
As can be seen from Fig. 1, the wa v e v elocity decreases with increasing the poros-
ity . This behaviour is more pronounced in the case of P-w a ves, for which the wa v e
velocity can e xpress 89% decrease, comparing with the velocity for small porosity
v alues. Observ e that the dif ference between the P-wa v e v elocity for saturated and dry
soil is small for lo w values of porosity . Howe ver , this is no longer true for high val-
ues of porosity (n > 0.3), where the wa ve v elocity of saturated and dry soil can dif fer
significantly up to 67%. For dry materials and for lo w porosity v alue, the results are
nearly the pure elastic case. Note also, that the w a v e v elocity decreases by increasing
the Poisson’ s ratio v alues, whereas this trend is eliminated by increasing the porosity .
2.2. P R O B L E M F O R M U L A T I O N : G O V E R N I N G E Q U A T I O N A N D B O U N D A RY C O N -
DITIONS
Consider in Fig. 2, a 3D finite local geological heterogeneity Ω 1 , such as a canyon
or an elastic inclusion with arbitrary geometry and the follo wing material properties:
density ρ 1 , Lame constants λ 1 and µ 1 , phase velocities C (Ω 1 )
p = p ( λ 1 + 2 µ 1 ) /ρ 1
and C (Ω 1 )
S = p µ 1 /ρ 1 . The boundary of the inclusion is denoted by Γ Ω 1 = Γ ∪ Γ f f ,
where Γ f f is the free-surface of the heterogeneity and Γ is the interf ace between Ω 1
and Ω 0 (the dotted line in Fig. 2). The finite geological heterogeneity Ω 1 is situated
in a homogeneous poro-elastic half-space Ω 0 with material characteristics porosity n ,
phase velocities C (Ω 0 )
p , C (Ω 0 )
S and the corresponding attenuation coef ficients ξ (Ω 0 )
P ,
ξ (Ω 0 )
S , computed by expressions (6) and (7). An incident time-harmonic plane P/SV -
wa ve with frequenc y ω impinges on the heterogeneity . Inside the range of Ω 1 (when
consider elastic inclusion) the scattered wa ve field u sc
i , t sc
i is equal to the total wa ve
field u i , t i . Outside the range of Ω 1 , the total wa ve field u i , t i is equal to the sum
of the scattered by the heterogeneity Ω 1 wa ve field u sc
i , t sc
i and the free field wa ve
motion u f f
i , t f f
i in Ω 0 . The free field wa ve motion is defined as the w a ve field in the
half-space without any heterogeneity . The solution for the free field wa ve motion
can be found in Achenbach [49] and Omidv ar et al . [50]. The e xpressions for the
displacement and traction free-field motion are gi ven in the Appendix. The gov erning
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42 F . W uttke, P . Dine v a, I.-K. Fontara
Fig. 2. Scattering of plane time-harmonic wa ve by a 3D heterogeneity in a poroelastic half-
space.
equation in frequency domain for l = 1 or l = 0 is
(9)  ( C Ω l
P ) 2 − ( C Ω l
S ) 2  u i,ij + ( C Ω l
P ) 2 u j,ii + ω 2 u j = 0 i, j = x, y , z .
Note, that in the case of l = 0 , the phase velocities of longitudinal and shear
wa ves are comple x-valued, due to the viscoelastic beha viour of the saturated soil.
The boundary conditions are as follo ws:
• at z = 0 ; along the free surface, tractions are equal to zero
t k = 0 for k = x, y , z ,
• along the rest of the heterogeneity’ s sides (when it is elastic inclusion), i.e.
along the interface boundary ( x, y , z ) ∈ Γ , between the elastic inclusion Ω 1
and the half-space Ω 0 , the follo wing conditions should be satisfied:
u (Ω 1 )
k ( x, y , z , ω )
inside Ω 1
= u f f , (Ω 0 )
k ( x, y , z , ω ) + u sc , (Ω 0 )
k ( x, y , z , ω )
outside Ω 1
(10)
t (Ω 1 )
k ( x, y , z , ω )
inside Ω 1
= − [ t f f , (Ω 0 )
i ( x, y , z , ω ) + t sc , Ω 0
i ( x, y , z , ω )] .
outside Ω 1
(11)
Finally , in the semi-infinite half-space Ω 0 , the Sommerfeld’ s radiation condition
is satisfied at infinite.
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Influence of Poroelasticity on the 3D Seismic Response of ... 43
2.3. D IRECT BIEM F O R 3 D W A V E S C A T T E R I N G I N A P O R O E L A S T I C H A L F - S PAC E
The abov e boundary-v alue problem (BVP) is here reformulated via boundary inte gral
equations (BIE’ s) using Betti’ s reciprocal theorem in conjunction with the frequency
dependent fundamental solution for 3D wa ve motion, see Dominguez [18]. What
follo ws is a presentation of the BIEM formulation of the formulated in Section 2.2
boundary-v alue problems in case that the local 3D heterogeneity is an elastic inclu-
sion (for example, alluvial basin/foundation) or a free surf ace canyon.
L OCAL SITE HETER OGENEITY IN FORM OF AN ELASTIC INCLUSION
The elastic inclusion is situated in Ω 1 with boundaries Γ Ω 1 . The follo wing boundary
integral equation holds for the e xternal domain with boundary Γ ext = Γ ∪ Γ a , where
Γ a is the free surface in Ω 0 outside Γ f f
(12) c ij  u j ( χ , ω ) − u ff
j ( χ , ω )  = Z
Γ ext
U ∗ (Ω 0 )
ij ( χ , ξ , ω )  t j ( ξ , ω ) − t f f
j ( ξ , ω )  d Γ ext ( ξ )
− Z
Γ ext
P ∗ (Ω 0 )
ij ( χ , ξ , ω )  u j ( ξ , ω ) − u f f
j ( ξ , ω )  d Γ ext ( ξ ) ; χ ∈ Γ ext .
Inside the range of elastic inclusion Ω 1 with boundary Γ Ω 1 = Γ ∪ Γ f f , the follow-
ing boundary integral equation holds:
(13) c ij u j ( χ , ) = Z
Γ Ω 1
U ∗ (Ω 1 )
ij ( χ , ξ , ω ) t j ( ξ , ω ) d Γ Ω 1 ( ξ )
− Z
Γ Ω 1
P ∗ (Ω 1 )
ij ( χ , ξ , ω ) u j ( ξ , ω ) d Γ Ω 1 ( ξ ); ξ ∈ Γ Ω 1 .
where: χ and ξ are the position v ectors of the source and field point, respecti vely; c ij
is the jump term depending on the surface geometry at the collocation point; U ∗ (Ω 0 )
ij ,
U ∗ (Ω 1 )
ij is the fundamental solution for displacement in domains Ω 0 and Ω 1 , respec-
ti vely; P ∗
ij ( χ , ξ , ω ) = σ ∗
ij k ( χ , ξ , ω ) n k ( ξ ) is the corresponding traction fundamental
solution; σ ∗
ij k = C ij mn ∂ U ∗
mk /∂ ξ n = − C ij mn ∂ U ∗
mk /∂ χ n , n k are the components of
the outward pointing unit normal v ector; C ij k l is the stif fness tensor . The expression
for the fundamental solution can be found in Dominguez [18]. The unkno wns in the
system of BIE’ s (12), (13) are the total displacement along free surf ace ( z = 0) , and
the total wa ve field (displacement and traction) along the interface surf ace Γ . The
numerical realization of this problem, ho wev er , is in respect to the scattered wa v e
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44 F . W uttke, P . Dine v a, I.-K. Fontara
field, thus leading to higher accuracy , due to the well-kno wn fact that the scattered
wa ve field satisfies the Sommerfeld’ s radiation condition at infinite. Here, we will
describe shortly the discretization procedure follo wing Reinoso et al . [51]. After
discretization of the boundary inte gral equations for the e xternal zone with boundary
Γ a ∪ Γ , (outside the inclusion), (see, Fig. 2), the follo wing matrix form of the alge-
braic equations with respect to the scattered displacement vector u s = u α
s + u Γ
s and
traction vector t s = t α
s + t Γ
s is obtained
(14) H 0 u s − G 0 t s = 0 ⇒ H 0 ( u α
s + u Γ
s ) − G 0 ( t α
s + t Γ
s )=0 .
Here, u α
s , u Γ
s , t α
s , t Γ
s are scattered displacement and traction v ectors along boundaries
Γ a and Γ , H 0 , G 0 are the corresponding influence matrices in Ω 0 along Γ a ∪ Γ . F or
the total traction vector along boundary Γ a can be written
(15) t α
total
= t α
s
scattered
+ t α
f f = 0 ⇒ t α
s
scattered
= − t α
f f .
Ha ving in mind Eqn. (15), the matrix equation Eqn. (14) can be written in the form
(16) H 0 ( u α
s + u Γ
s ) − G 0 t Γ
s = G 0 t Γ
f f ; in Ω 0 .
Note, that the suf fix “f f” in the field v ariables stands for the free field motion,
while the suf fix “s” for the scattered wa v e field.
After discretization of the boundary inte gral equation for the internal zone Ω 1
with boundary Γ f f ∪ Γ , (inside the inclusion), (see, Fig. 2), the matrix form of the
algebraic equations with respect to the scattered (equals to the total) displacement
and traction vector is obtained in Eqn. (17), ha ving in mind, that u Γ
1 = u Γ
0 − u Γ
f f and
t Γ
1 = − t Γ
0 = − t Γ
s − t Γ
f f along Γ .
H 1 [ u f f
1 + u Γ
1 ] − G 1 [ t f f
1 + t Γ
1 ]=0
H 1 [ u f f
1 + u Γ
s + u Γ
f f ] − G 1 [ − t Γ
f f − t Γ
s ] = 0 in Ω 1
(17)
H 1 u f f
1 + H 1 u Γ
s + G 1 t Γ
s = − H 1 u Γ
f f − G 1 t Γ
f f
Where: H 1 , G 1 are the influence matrices in Ω 1 along boundaries Γ f f ∪ Γ ; u f f
1 , t f f
1
are the displacement and traction vector , correspondingly , inside Ω 1 along boundary
Γ f f .
Equation (16) for the e xternal zone Ω 0 and Eq. (17) for the internal zone Ω 1 form
the follo wing system (18) of linear algebraic equations AX = B :
(18)  H 0 H 1 0 − G 0
H 1 H 1 H 1  



u a
s
u Γ
s
u f f
1
t Γ
s




=  − G 0 t a
f f − H 1 u Γ
f f − G 1 t Γ
f f 
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Influence of Poroelasticity on the 3D Seismic Response of ... 45
The unkno wns are scattered displacement vectors u f f
1 , u Γ
s , u a
s along boundaries Γ f f ,
Γ , Γ a , correspondingly and scattered traction vector t Γ
s along boundary Γ , while u Γ
f f
and t Γ
f f are the kno wn vectors of displacement and traction for the free field motion
along Γ , presented in the Appendix.
L O C A L S I T E H E T E RO G E N E I T Y I N F O R M O F A C A N Y O N
In this case, the follo wing boundary inte gral equation along the surface Γ can yon =
Γ ∪ Γ a (see Fig. 2) describes the problem
(19) c ij  u j ( χ , ω ) − u ff
j ( χ , ω )  = Z
Γ canyon
U ∗
ij ( χ , ξ , ω )  t j ( ξ , ω )
0 − t f f
j ( ξ , ω )  dS ( ξ )
− Z
Γ canyon
P ∗
ij ( χ , ξ , ω )  u j ( ξ , ω ) − u f f
j ( ξ , ω )  dS ( ξ )
After discretization of BIE (Eqn. 19), the follo wing algebraic system with respect
to the total displacement vector u (3 N × 1) is obtained:
H u sc = G t sc → H ( u − u f f ) = G ( t − t f f ) consider t = 0
H [3 N × 3 N ] u (3 N × 1) = H [3 N × 3 N ] u f f (3 N × 1)
− G [3 N × 3 N ] t f f (3 N × 1)
(20)
The matrices G and H are the final global influence matrices. The y are full
matrices of size 3 N × 3 N , where N is the number of global boundary nodal points.
The numerical solution of both BVPs follo ws the standard BIEM collocation
procedure. In particular , the surface boundary is discretized into quadratic trian-
gle isoperimetric (six-node) boundary elements, using continuous polynomial ap-
proximations for the boundary geometry , the displacement and the traction vectors.
T wo types of integrals are obtained after discretization of BIE, depending on whether
or not the radial distance r between the source and recei ver points is zero: (a) at
r 6 = 0 the integrals are re gular , there are no singularities, solution is numerical;
(b) at r → 0 there are two types of singularities: ( i ) the displacement-based ker -
nels R U ∗
ij ( x, ξ , ω ) t j ( ξ , ω ) dξ e xhibit a weak singularity of type O (1 /r ) and these
integrals are solv ed by appropriate quadrature rule explained in Dominguez (1993);
(ii) the traction-based kernels R P ∗
ij ( x, ξ , ω ) u j ( ξ , ω ) dξ e xhibit a strong singularity of
the type O (1 /r 2 ) and these type of integrals are solv ed via application of the well-
kno wn rigid body motion concept. After inte grating both weak and strong singulari-
ties in the integrands, computing all inte grals and satisfying the prescribed boundary
conditions, an algebraic system of equations is obtained, rearranged and solved with
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46 F . W uttke, P . Dine v a, I.-K. Fontara
respect to the unkno wn displacements and tractions in terms of the prescribed v alues,
all in the frequenc y domain. In general, BIEM mesh discretization issues, such as
mesh density is controlled by the solution of the follo wing test example, demonstrat-
ing the verification of the applied numerical algorithm. In the case of equal material
characteristics of the half-space and the elastic inclusion, the solution of the abo ve
formulated BVP’ s accord the free field solution for a 3D wa v e propagation in an
elastic homogenous half-space, subjected to incident time-harmonic wa ve.
(a)
(b)
Fig. 3. (a) A pure elastic heterogeneity AB C D A 1 B 1 C 1 D 1 or (b) a canyon with prism
geometry 800 × 800 × 800 m, embedded in a poroelastic half-space under incident time-
harmonic P-wa ve. The additional discretization area along the free surface is presented
by 8 squares T 1 N 1 E 1 A 1 , N 1 N A 1 B 1 , N T B 1 M 1 , B 1 M 1 C 1 M , C 1 F 1 M G 1 , C 1 F 1 D 1 F ,
D 1 F E G , E 1 E A 1 D 1 , all with size of 800 m.
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Influence of Poroelasticity on the 3D Seismic Response of ... 47
3 . N UMERICAL SIMULA TIONS AND SITE RESPONSE STUDIES
In this section, we apply the foregoing discussed computational scheme to solv e the
defined two boundary-v alue problems. All numerical examples, that follow serv e
to visualize the wa ve field modification under assumption of porous systems and to
demonstrate the potential of the proposed simple visco-elastic mechanical model and
accompanied numerical tool to deal with wa ve propagation in spatial 3D heteroge-
neous poroelastic geological media. As explained before, the follo wing two types of
3D site heterogeneities are analysed: a can yon and an elastic inclusion.
S I T E R E S P O N S E S T U DY – COMBINED EF FECT OF TOPOGRAPHY AND POR OELAS -
TICITY
Consider a finite can yon embedded in a poroelastic soil half-space. An incident time-
harmonic plane P-wa ve with frequenc y ω is propagating vertically in the plane x – z
with incidence angle θ = π / 2 according to axis O x . The used BIEM mesh consists
of 576 triangle quadratic isoperimetric boundary elements, along the canyon’ s sides
(a) (b)
(c) (d)
Fig. 4. Displacement amplitudes | u z | versus x/a ( a = 400 m) along the free surf ace (at
line y = 0) of a poro-elastic half-space, containing a canyon under v ertical incident time-
harmonic P-wa ve with frequenc y f = 6 . 4 Hz for: (a) dry soil with Poisson ratio ν = 0 . 4 ;
(b) saturated soil with Poisson ratio ν = 0 . 4 ; (c) dry soil with Poisson ratio ν = 0 . 1 ; and (d)
saturated soil with Poisson ratio ν = 0 . 1 .
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48 F . W uttke, P . Dine v a, I.-K. Fontara
(a) (b)
(c) (d)
Fig. 5. Displacement amplitudes | u x | v ersus x/a ( a = 400 m), along the free surface (at
line y = 0) of a poroelastic half-space, containing a canyon under time-harmonic P-wa ve
propagating along axis O z with frequency f = 6 . 4 Hz for: (a) dry soil with Poisson ratio
ν = 0 . 4 ; (b) saturated soil with Poisson ratio ν = 0 . 4 ; (c) dry soil with Poisson ratio ν = 0 . 1 ;
and (d) saturated soil with Poisson ratio ν = 0 . 1 .
and along the free surface out of the can yon, respectiv ely . The number of the global
boundary nodes is N = 1201 , and the algebraic system after discretisation is of size
3603 (see Fig. 3).
Figures 4a, b, c, d present the displacement amplitudes | u z | due to time-harmonic
vertically incident P-w av e with frequency 6.4 Hz versus ratio x/a ( a = 400 m) along
the free surface z = 0 on the line y = 0 . The Poisson’ s ratio is ν dry = 0 . 4 or 0.1
and the porosity is between 0.1 and 0.35. The obtained results are for dry (Figs. 4a,
c) and saturated (Figs. 4b, d) poro-elastic sandstone with Poisson’ s ratio equal to 0.4
and 0.1. The analogous results for the displacement amplitudes | u x | are sho wn in
Figs. 5a, b, c, d.
Displacement amplitudes | u z | along the canyon’ s bottom AB C D (see Fig. 3) at
frequency f = 6 . 4 Hz of time-harmonic vertical incident P-w av e propagating in
poroelastic soil with porosity n = 0 . 1 and n = 0 . 3 are sho wn in Figs, 6–7, cor-
respondingly , for the cases of (a) dry and (b) saturated soil. The frequenc y f =
6 . 4 Hz corresponds to the follo wing value of the non-dimensional frequenc y Ω =
( ω a ) / ( π C S ) = (2 a ) λ S = 5 , where λ S is the wa velength of the shear wa ve and
2 a = 800 m, 2 a is the cube’ s size.
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Influence of Poroelasticity on the 3D Seismic Response of ... 49
(a)
(b)
Fig. 6. Displacement amplitude | u z | along the canyon situated in a poroelastic half-space
with porosity n = 0 . 1 for: (a) dry; (b) saturated soil material, under time-harmonic P-wa ve
propagating along the axis O z with frequenc y f = 6 . 4 Hz.
Figures 4-7 demonstrate the influence of porous media on the seismic signals,
computed at the surface of a can yon resting on a poroelastic half-space. The first
observ ation is that for almost all analyzed porosities, the displacement amplitudes of
saturated soil are smaller than those for dry soils. This is due to the fact, that the
increasing pore pressure induced by the passage of seismic wa ves tends to stif fen the
porous soil/rock skeleton and hence, to reduce the appearing deformation. The shear
strength of Biot’ s porous material is provided by the solid sk eleton and it is assumed
to be unaf fected by the fluid saturation, which implies µ sat = µ dry .
On the other hand, the Lame coefficient λ changes to λ sat = λ dry + Q 2 /R , and
thus indicates a strengthening of the porous skeleton, due to dilatation deformation,
follo wed by an increase of the phase velocity of longitudinal P-wa ve. In particular ,
the P-wa ve characteristics are modified significantly by the change from dry to sat-
urated geomaterial. Additional damping ef fect occurs due to the viscous type of the
fluid and the matrix beha viour . Howe ver , for lo w porosity value n = 0 . 1 there is
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50 F . W uttke, P . Dine v a, I.-K. Fontara
(a)
(b)
Fig. 7. Displacement amplitude | u z | along the canyon, situated in a poroelastic half-space
with porosity n = 0 . 3 for: (a) dry; (b) saturated soil material, under time-harmonic P-wa ve
propagating along the axis O z with frequenc y f = 6 . 4 Hz.
almost no dif ference between the seismic response in case of dry and saturated soil.
In this case, the dry frame strength increases and becomes a pure elastic modulus.
In consequence, it is obvious that the seismic response changes significantly , when
we consider saturated, porous geological media. It must be noticed, that the abov e
conclusions are in agreement with the ef fects visible in Fig. 1(a).
S I T E R E S P O N S E S T U DY – C O M B I N E D E FF E C T O F E L A S T I C H E T E RO G E N E I T Y A N D
POR OELASTICITY
In what follo ws, we in vestig ate the seismic field in a poroelastic half-space, contain-
ing an elastic inclusion. The geometry of the finite elastic inclusion AB C D A 1 B 1 C 1 D 1
(see Fig. 3) is rectangular with size 10 × 10 m along the free surface ( z = 0) and
with depth equal to 3 m. The inclusion is surrounded by the top side A 1 B 1 C 1 D 1 ,
bottom side AB C D and the follo wing four sides A 1 AB 1 B , B 1 B C 1 C , C 1 C D 1 D
and D 1 D A 1 A embedded in a poroelastic soil half-space. Note, that for the case of
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Influence of Poroelasticity on the 3D Seismic Response of ... 51
the elastic inclusion the top side A 1 B 1 C 1 D 1 e xists compared with the pre vious case
of the canyon. The used BIEM mesh consists of 24 triangle quadratic boundary ele-
ments (50 global nodes) for the inclusion and 52 triangle quadratic boundary elements
(117 global nodes) along the free surface boundary (Fig. 2) outside the inclusion. The
algebraic system of equations, obtained after discretization of the boundary integral
equations Eqns. (12)–(13) is of size 501.
The porelasticity of the half-space is described again by the Bardet model, hav-
ing the same ke y material characteristics as those gi ven abo v e. The material prop-
erties of the elastic inclusion are considered as an elastic foundation with the fol-
lo wing material characteristics: ρ = 2400 . 0 kg/m 3 , λ = 0 . 12100000 E+11 N/m 2 ,
µ = 0 . 18150000 E+11 N/m 2 , ν = 0 . 2 , C P = 4490 . 73 m/s, C S V = 2750 . 0 m/s. An
incident time-harmonic plane P-wa ve with frequenc y ω is propagating in the plane
O xz with incident angle θ = π / 2 , according to axis O x .
Frequency dependent synthetic seismograms | u x | , | u y | , | u z | for observer point
B 1 with coordinates (2.5, -2.5, 0) along the free surface of the elastic foundation,
rested on dry or saturated half-space, subjected to incident P-wa v e with incident angle
θ = π / 4 with respect to the axis O x are presented in Figs, 8 a, b, c. The Poisson’ s
ratio is ν dry = 0 . 4 and the porosity takes the v alue of n = 0 . 1 or 0.34, while the
foundation’ s depth is equal to 3 m or 2.5 m.
From Figs. 8 a, b and c it can be seen, that the seismic response is very sensiti ve to
the foundation depth and this ef fect depends on the frequency of the seismic e xcita-
tion. For high v alues of the foundation depth ( d = 3 m), the displacement distribution
curve with frequency v ariation is jagged compared with the lo wer v alue of d = 2 . 5 m,
that leads to a smooth displacement distrib ution. Moreov er and much more important
for practical design, the ef fect of the influence of the foundation depth on the seismic
signal, recorded at free surface becomes e ven stronger for higher v alues of porosity
in both cases, for dry and saturated soils.
It is sho wn, that the seismic signals depend strongly on the poroelastic beha viour
of the soil. High v alues of porosity can significantly modify not only quantitativ ely
b ut also qualitati vely the seismic response. Pronounced oscillating beha viour of the
seismic signal is observed for high frequenc y v alues. Furthermore, significant dif fer-
ences can be noticed at the seismic signals when they are recorded for a saturated and
dry geological profile.
The obtained results sho w that not in all cases the saturated case leads to lower
displacement amplitude. The seismic response in saturated domains depends also on
additional factors, such as the frequency of the seismic e xcitation, the geometry of
the heterogeneity and other ke y factors of the seismic scenario.
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52 F . W uttke, P . Dine v a, I.-K. Fontara
(a)
(b)
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Influence of Poroelasticity on the 3D Seismic Response of ... 53
(c)
Fig. 8. Displacement amplitudes at point B 1 (2.5,-2.5,0) along the elastic foundation of
depth d = 3 m or d = 2 . 5 m, rested in dry and saturated half-space with Poisson’ s ratio
ν dry/saturated = 0 . 4 and porosity n = 0 . 1 or n = 0 . 34 versus frequenc y of incident P-wa ve
with incident angle θ = π / 4 : (a) amplitude | u x | ; (b) amplitude | u y | ; (c) amplitude | u z | .
4 . C ONCLUSIONS
This work in vestigates 3D seismic w av e propagation problems in poroelastic geo-
materials to analyze the resulting seismic site ef fects in a homogeneous fluid satu-
rated half-space with elastic heterogeneities and surface topography , based on the
viscoelastic approximation of the Biot’ s model. It is also sho wn, that the Bardet one-
phase viscoelastic model is able to approximate and simulate the dynamic beha viour
of poroelastic media well and deli ver an e xcellent base for more realistic site studies.
The proposed numerical technique benefits from combining ef ficiently the adv an-
tages of the viscoelastic isomorphism with those of the BIEM itself, when it comes
to handling of 3D wa ve propagation and scattering problems in semi-infinite hetero-
geneous poroelastic domains. The simulation results re v eal that the seismic field in
a poroelastic half-space with a 3D heterogeneity is sensiti ve to the stif fness and the
Poisson’ s ratio of the solid skeleton of saturated soil. The elastic b ulk module K dry
is much greater than the fluid b ulk module K f ( K dry  K f ) at lo w porosities. Such
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54 F . W uttke, P . Dine v a, I.-K. Fontara
a material is close to the elastic solid dominated case, where the dry frame strength
is high and the fluid has no ef fect on the solid-fluid system behaviour . The great-
est influence of porosity (associated with the fluid phase) can be observed for high
porosity v alues where K dry ≈ K f . Geomaterial for which is satisfied the condition
K dry  K f is a fluid dominated material, where the portion of pore space is too large
to form sustainable dry frame. More sensiti ve to the Poisson’ s ratio v ariations are the
poroelastic materials, for which is satisfied the condition K dry ≥ K f . Poisson’ s ratio
represents the consolidation status of the solid-skeleton, ha ving in mind that its lo wer
v alues concerns consolidated solid skeleton, while higher v alues unconsolidated one.
The numerical results sho w that the presence of water in the poroelastic material
results in damping of the seismic wa ves. This phenomenon has been e xperimentally
prov ed by Badiey et al . [52] and Oqushwitz [53]. The physical reasons for this ef fect
may be (a) dissipation due to the viscoelastic beha viour of the skeleton or dissipation
because of the fluid flo w; (b) hardening ef fect, since the presence of fluid in the soil
acts as material ‘stif fener’. This ef fect can be e xplained by the used Bardet model.
The pore pressure induced by the passage of seismic wa ves resists the compression
and stif fens the rock. The shear strength of Biot’ s porous material is provided by
the solid skeleton (i.e. the matrix) and it is unaffected by the fluid saturation, which
implies µ sat = µ dry . The Lame coef ficient λ , ho wev er , changes to λ sat = λ dry +
Q 2 /R , indicating a strengthening ef fect.
At this point, we stress that insofar as the Bardet model for one-phase materials is
concerned, it cannot take into account the pore pressure boundary conditions along
existing boundaries. Thus, there are limitations when the aforementioned model is
applied to seismic analysis of poroelastic media, but it is still useful for engineering
practice. More specifically , the viscoelastic-poroelastic similarity is useful for e x-
tracting approximate solutions, that can be used as benchmark cases to calibrate the
accuracy of more adv anced (in terms of constitutiv e laws) computational methods,
applied to wa ve propagation problems in fluid-saturated soils.
The simulation results re veal, that the seismic wa ve field in a poroelastic hetero-
geneous half-space is a complex result of the mutual play of dif ferent key f actors,
such as the type and the characteristics of the seismic load, the wa v e-heterogeneity
interaction, the type of heterogeneity and the poroelastic properties of the soil. The
proposed approach has the potential to be extended and applied for dynamic soil-
structure interaction problems, in volving poroelastic media.
A C K N O W L E D G E M E N T
The second author ackno wledges the support of the Bilateral Bulgarian Greek (BG)
Project-based Personnel Program between B AS and A UTH and B AS project DCD–
4/03.05.2017.
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Influence of Poroelasticity on the 3D Seismic Response of ... 55
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A PPENDIX : F R E E FI E L D M OT I O N
Consider the wa ve propagating in the plane x 0 − z 0 with incident angle θ in respect
to the axis O x 0 . The position of the plane x 0 − z 0 in respect to the reference coordi-
nate system is defined by the angle θ h between the reference plane x − y and plane
Fig. 9. Position of the w a v e propagation plane x 0 − z 0 where the P-w a v e is propagating under
incident angle θ in respect to the axis O x 0 , θ h is the angle between the reference plane x − y
and plane x 0 − y 0 .
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Influence of Poroelasticity on the 3D Seismic Response of ... 59
x 0 − y 0 . The displacement components in the reference coordinate system O xy z are
denoted by ( u x u y u z ) T , while in the system O x 0 y 0 z 0 the displacement components
are denoted by ( u 0
x u 0
y u 0
z ) T . The relation between displacement components in both
coordinate systems is as follo ws:
(A1) 

u x
u y
u z

 = T 

u 0
x
u 0
y
u 0
z

 = 

cos θ h − sin θ h 0
0 0 1
sin θ h cos θ h 0

 

u 0
x
u 0
y
u 0
z


(A2) x 0 = x cos θ h + z sin θ h ; z 0 = z .
Note, that in the case θ h = 0 , x 0 = x , y 0 = y , z 0 = z and the wa ve propagates in
the plane x − z , where


u x
u y
u z

 = 

u 0
x
u 0
y
u 0
z

 ; T = 

100
001
010

 .
Displacement of the free-field motion at any point of the half-space can be ob-
tained by the follo wing expression in the case θ h = 0
u x = A in
P  k
k P  e ( − ν z − ik x ) + A in
P RP P  k
k P  e ( ν z − ik x 1 )
(A3)
+ A in
P RP S  iν 0
k P  e ( ν 0 z − ik x 1 )
u y = 0 (A4)
u z = A in
P  − iν
k P  e ( − ν z − ik x 1 ) + A in
P RP P  iν
k P  e ( ν z − ik x 1 )
(A5)
+ A in
P RP S  − k
k P  e ( ν 0 z − ik x 1 ) ,
where
k = k P cos θ ; ν = ik P sin θ ; ν 0 = q k 2 − k 2
S ;
(A6)
k P = ω
C P
; k S = ω
C S
; ∆( k ) = (2 k 2 − k 2
S ) 2 − 4 k 2 ν ν 0
(A7) RP P = − (2 k 2 − k 2
S ) 2 + 4 k 2 ν ν 0
∆( k ) ; RP S = − 4 ik ν (2 k 2 − k 2
S )
∆( k )
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60 F . W uttke, P . Dine v a, I.-K. Fontara
T raction of the free-field motion can be obtained by the follo wing expression:
(A8) t f f =




[ χu f f
x,x + λu f f
y ,y + λu ff
z ,z ] n x + µ ( u f f
x,y + u f f
y ,x ) n y + µ ( u f f
x,z + u f f
z ,x ) n z
µ ( u f f
x,y + u f f
y ,x ) n x + [ χu f f
y ,y + λu ff
x,x + λu f f
z ,z ] n y + µ ( u f f
y ,z + u ff
z ,y ) n z
µ ( u f f
x,z + u f f
z ,x ) n x + µ ( u ff
y ,z + u ff
z ,y ) n y + [ χu ff
x,x + λu f f
y ,y + λu ff
z ,z ] n z




with χ = ( λ + 2 µ ) .
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Why institutions use Plag.ai for originality review, entry 81

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