RAYLEIGH WAVES IN SATURATED POROELASTIC MEDIA
G.M. DE SOUSA HENRIQUES VITORINO, J.A. TEIXEIRA DE FREITAS, I.D. MOLDOVAN
Instituto Superior Técnico, Technical University of Lisbon
1. INTRODUCTION
This work addresses the modelling of the propagation of elastic Rayleigh waves in singlephase media (dry soils) and biphasic media (saturated soils).
These basic equations are recalled to define the problem and establish the notation. The
variables are then separated in time and space to apply a time discretization procedure and
obtain a spectral formulation. This discretization can be implemented using periodic (Fourier)
and non-periodic (Wavelet) time approximation bases. The formal solutions of the spectral
problems are obtained to establish the definitions of P- and S-waves. They are combined under
free-surface conditions to derive the equations that define the Rayleigh wave.
The results obtained are presented in both spectral form (space dependent solutions, for
a given frequency) and in transient form (space and time dependent solutions), assuming that
the waves are caused by a seismic signal. The formulation reported here will be added to an
existing finite element program to extend its application to the analysis of the seismic response
of saturated soils.
To simplify the presentation, most supporting information is referred to the dissertation
that is summarized here [6], where details on the sources that have been used are given.
2. SOLID MECHANICS EQUATIONS
In the domain equilibrium condition (1),
and
and
are the stress components,
the acceleration and velocity components, so that
and
the body forces
represent the associated
mass and damping coefficients:
(1)
The boundary equilibrium condition (2) relates the stress components with the applied
forces,
, and
defines the component of the unit outward normal vector:
(2)
The domain compatibility condition (3) relates the strain and displacement components,
and
, respectively, and the boundary compatibility condition (4) defines the prescribed
displacement field:
(3)
(4)
The elasticity condition (5), where
and
are the Lamé constants, relates the stress and
strain fields [1,2]:
(5)
3. APPROXIMATION IN TIME
It is assumed that all variables, and their time derivatives, present in the modelling of the
elastodynamic response of saturated porous media, say the displacement , the velocity
and
the acceleration , are separated in time and space,
(6)
(7)
(8)
where
defines a complete time approximation basis with support
.
The commonly used periodic (or periodically extended) spectral decomposition method is
recovered using a Fourier basis with real frequencies:
(9)
(10)
When a non-periodic basis is used [4,5], for instance a Wavelet basis, the algorithm still
returns pairs of conjugate frequencies. However, they are now complex, that is, with imaginary
parts that induce the necessary numerical damping.
4. BOUNDARY VALUE PROBLEM
The discretization of the time dimension of the equations governing the elastodynamic
response of a saturated poroelastic body generates a set of
fully uncoupled boundary value
problems, which can be stated in matrix form as follows, where subscript
identifies the order
of the spectral decomposition:
(11)
(12)
(13)
(14)
(15)
The stress and strain vectors are organized as follows for single- and biphasic media,
where, in the latter case,
and
;
(16)
;
(17)
are the pore fluid stress and the fluid content. In the definition
of the displacement field:
;
(18)
is the (small) displacement of the solid skeleton and
is the pore fluid seepage
displacement. The explicit form of the body force vector is:
;
(19)
In the Neumann condition (14), vector
collects the components of the (total) surface
forces, and the pressure prescribed on the boundary, and the boundary equilibrium matrix
collects de adequate components of the unit outward normal vector,
(20)
The explicit form of the Dirichlet condition (15) is the following, where
is the normal component of the seepage displacement:
;
(21)
The definition of the elastic stiffness matrix in the constitutive relation (22) is, where
and
represents the Biot coefficients:
,
(22)
5. W AVE EQUATION
The wave equation is obtained combining the domain conditions (11) to (13) to eliminate
the stress and strain terms as independent variables:
(23)
The homogenous form (
) of this system of differential equations can be reduced to
the Helmholtz equation,
(24)
where the (complex) wave numbers
depend on the forcing frequency,
, assuming that the
displacement field, , is derived from scalar potential functions,
. This generates two sets of
solutions for homogeneous soils, namely a P-wave family
and one S-wave family
. In saturated porous media there are two set of P-waves
and one set of S-waves
.
The P-wave solutions are obtained assuming that the displacement field is the gradient of
a potential function
, to yield the following definitions for single- and biphasic media,
;
where
(25)
is a proportionality coefficient and
the gradient vector:
(26)
The wave numbers of the Helmholtz equation (24) are defined as follows for single- and
biphasic media:
(27)
(28)
(29)
This latter equation has two independent roots, thus leading to two independent P-wave
solutions.
The S-wave solution is obtained assuming that the displacement field is the anti-gradient
of potential
:
;
(30)
(31)
The Helmholtz equation and the associated wave numbers are defined as follows,
(32)
(33)
for single-phase media and, for biphasic media:
(34)
(35)
(36)
6. RAYLEIGH WAVES
Rayleigh waves are generally defined as plane waves satisfying free-surface conditions.
They can be modelled by a linear combination of P-waves and S-waves,
(37)
(38)
where the combination weights
and
represent (unknown) generalized amplitudes. The
corresponding definitions for saturated porous media are:
(39)
(40)
Therefore, the free-surface conditions that characterize the propagation of Rayleigh
waves in single-phase medium define an eigen-problem:
(41)
with the following solutions:
(42)
(43)
(44)
(45)
(46)
(47)
The equations above are replaced by the following in saturated porous media, where
:
(48)
(49)
(50)
(51)
(52)
(53)
7. NUMERICAL APPLICATIONS
The results above are used to model the propagation of Rayleigh waves in both
frequency and time domains. In the latter case they are combined to represent the Loma Prieta
earthquake. The results are obtained for a 100mx30m strip of Molsand soil.
To assess the stability of the numerical solutions, the spectral Rayleigh wave models
have been run for three sets of forcing frequencies, namely 10Hz, 100Hz and 250Hz. The (real
part) of the solutions obtained for frequency  = 100Hz (and unit amplitudes, X) are presented
in Figures 1 and 2 for single-phase and biphasic media, respectively.
The values presented in the stress colour scales should be used only to assess relative
values, as the tests are implemented for unit amplitudes, that is, with X=1 in the Rayleigh wave
definitions presented above.
a) Displacement field
b) Stress component
c) Stress component
d) Stress component
Figure 1 – Rayleigh wave in single-phase medium (real part,  = 100Hz)
a) Displacement in the solid phase
b) Seepage displacement field
c) Stress component
d) Stress component
e) Stress component
f)
Fluid pressure
Figure 2 – Rayleigh wave in saturated poroelastic medium (real part,  = 100Hz)
These results show that, in both instances, the free surface conditions are correctly
modelled. They also capture the elliptic movement that typifies the propagation of Rayleigh
waves. The dampening of the displacement in the solid phase of the poroelastic medium, as
compared with the result obtained with the single-phase model, is expected, as it is caused by
the presence of a liquid phase.
The spectral Rayleigh wave solutions are next combined to model the Loma Prieta
seismic signal. A non-periodic (pseudo-spectral) time integration procedure is implemented
using a wavelet time basis to determine, by collocation, the amplitudes of the Rayleigh waves
[3].
The results obtained with the model derived for saturated poroelastic media are shown in
Figures 3 to 5. They are obtained with a relatively weak time basis, with dimension N=32. These
results show that the free surface conditions are adequately modelled throughout the period of
the analysis.
Figure 3 – Displacement in
the solid phase
Figure 4 – Seepage
displacement
t(s)
0.48
0.52
0.56
0.60
0.64
Figure 5 – Propagation of the stress waves in a saturated poroelastic medium
The pattern that is obtained for the propagation of the waves in both solid and fluid
phases of the saturated soil recover the expected movement, with seepage concentrating closer
to the free surface.
It is noted that these results are obtained in a single time step, which corresponds to the
duration of the seismic signal. The quality of the animation of the movement [4] can be
improved by increasing the dimension of the time approximation basis.
8. CLOSURE
Rayleigh waves are defined by a set of differential equations subject to free-surface
conditions. These equations are derived separating the variables in space and time to obtain a
set of problems defined in the frequency domain. The discretization in time is usually
implemented using Fourier series. The variation in space is defined by a set of equations that
model adequately the mechanical behaviour of the medium of propagation.
The combination of these equations leads to the spectral (Helmholtz) definition of the
wave equation. Rayleigh waves are defined, under plane strain conditions, combining P- and Swaves under free-surface conditions, assuming that they propagate in two distinct
homogeneous media, namely single-phase (dry) and biphasic (saturated) media.
The approximation in time is implemented using a system of Wavelets to circumvent the
difficulties inherent to the modelling of non-periodic problems with Fourier series. The modelling
of the propagation of seismic waves is obtained combining this temporal series with the
solutions of the spectral definitions obtained for the Rayleigh waves.
REFERENCES
[1]
ARANTES E OLIVEIRA, E.R., E. BORGES PIRES, D. CAMOTIM, Mecânica dos Meios
Contínuos, Instituto Superior Técnico, Universidade Técnica de Lisboa, Departamento
de Engenharia Civil e Arquitectura, Secção de Mecânica Estrutural e Estruturas, 1991.
[2]
AZEVEDO, A., Mecânica dos Sólidos, Faculdade de Engenharia da Universidade do
Porto, Departamento de Engenharia Civil, Secção deEstruturas, 1996.
[3]
DOYLE, J.F., Wave Propagation in Structures, An FFT-Based Spectral Analysis
Methodology, Springer-Verlag, New York, 1989.
[4]
MOLDOVAN, I.D., Hybrid-Trefftz Elements for Elastodinamic Analysis of Saturated
Porous Media, Tese de Doutoramento, Instituto Superior Técnico, Universidade
Técnica de Lisboa, Departamento de Engenharia Civil e Arquitectura, 2008.
[5]
TEIXEIRA DE FREITAS, J.A., Rayleigh Waves in Saturated Poroelastic Media,
Relatório Interno, Instituto Superior Técnico, Universidade Técnica de Lisboa,
Departamento de Engenharia Civil e Arquitectura, 2009.
[6]
VITORINO, G.M.S.H., Modelação das Ondas Rayleigh, Master’s dissertation,
Department of Civil Engineering and Architecture, Instituto Superior Técnico, Technical
University of Lisbon, 2010.