INGENIERÍA Y CIENCIAS APLICADAS: MODELOS MATEMÁTICOS

MEMORIAS DEL XII CONGRESO INTERNACIONAL DE
MÉTODOS NUMÉRICOS EN INGENIERÍA Y CIENCIAS APLICADAS
CIMENICS'2014
ISLA DE MARGARITA, VENEZUELA, 24 al26 de marzo de 2014 .
INGENIERÍA Y CIENCIAS APLICADAS:
MODELOS MATEMÁTICOS Y
COMPUTACIONALES
Editores
E. DÁVILA, J. DEL RÍO, M. CERROLAZA
Instituto Nacional de Bioingeniería
Universidad Central de Venezuela
R. CHACÓN
Universidad de Los Andes
SOCIEDAD VENEZOLANA DE
MÉTODOS NUMÉRICOS EN INGENIERÍA
Facultad de Ingeniería, Universidad Central de Venezuela
.
IW
INGENIERÍA Y CIENCIAS APLICADAS:
MODELOS MATEMÁTICOS Y COMPUTACIONALES
Primera edición: marzo de 20 14
SVMNl
© 2014 Los Editores
(Q 2014
Diseño de la portada:
Vannessa Duarte/Liseth Valencia
Instituto Nacional de Bioingeniería
Universidad Central de Venezuela
Solicitud de ejemplares a:
Sociedad Venezolana de Métodos Numéricos en Ingeniería
Universidad Central de Venezuela
Caracas, Venezuela
S' +58 (0)212 285 .2827 1 285.9608 1286.8094/286.4534
Impresión :
Miguel Ángel García & Hijos, Caracas, Venezuela
La figura en la portada del libro es cortesía de M. Valera, J . Guevara y J. León, y corresponde a una
batimetría del Lago de Valencia considerando la velocidad y dirección del viento alrededor del lago.
Nada de este libro puede ser reproducido, almacenado en un sistema de información mecánico o
electrónico, fotocopiado, grabado o transmitido sin la autorización escrita de la SVMNI o de los editores.
ISBN: 978-980-7161-04-6
Ingeniería y ciencias aplicadas: modelos
matemáticos y computacionales
ORGANIZADORES
K Dávila, M. Cerrolaza
Instituto Nacional de Bioingeniería
Universidad Central de Venezuela, Caracas, Venezuela
+58 (0)212 285.2827 1285.9608 1 286.80941 286.4534
Email: [ever!íng.davila¡miguel. ce!Tolaza} @inabio .edu. ve
R. Chacón
Universidad de Los Andes, Mérida, Venezuela
+58 (0)274 240.2644 12645
Email: rdehacon@),ula.ve
w
w
COMITÉ ORGANIZADOR LOCAL
G. Uzcátegui (Presidenta)
V. Duarte, J. Del Río, J. Vivas, L. Valencia, E. Salinas
Universidad Central de Venezuela
N. González
Universidad de Los Andes
COMITÉ CIENTÍFICO
M . Aliabadi, Queen Aim)' Col!ege, UK
G. Larrazábal, Texas Un iversity, USA
P. Delage, Eco/e Nationale des Ponts et Chaussees . Francia
J. Rincón, Universidad del Zu/ia, Venezuela
E. AJarcón, Universidad Politécnica de l'viadrid, Espaí'ía
R. Callarotti, Instituto Venezolano de In vestigaciones Científicas, Venezuela
L. Nallim, Universidad Nacional de Salta, Argentina
E . Fance!Io, Universidad Federal de Santa Catarina, Brasil
M .C. Rivara, Universidad de Chile, Chile
A. Maure, Universidad Nacional del Cuyo, Argentina
S. Buitrago, Universidad Católica Andrés Bello, Venezuela
E. Rank, Technícal University ofMunich, Alemania
V . Griffiths, Colorado School of.Mines, USA
M. E. Elberg, Universidad de Los Andes, Venezuela
M . Doblaré, Universidad de Zaragoza, Esparza
J. Sulem, Eco/e Nationale des Ponts et Chaussees. Francia
G . Buscaglia, Universidad de Sao Paulo, Brasil
A. Larreteguy, Universidad Argentina de la Empresa, Argentina
M. Martínez, Universidad Central de Venezuela, Venezuela
A. Salvadori, Universidad de Brescia, Italia
PATROCINADORES
Universidad Central de Venezuela, UCV
Consejo de Desarrollo Científico y Humanístico, CDCH-UCV
FONACIT
ÍNDICE
pág.
CONFERENCIAS INVITADAS (Cl)
Advanced computer methods in electromagnetic problems with applications to
grounding analysis
Af Casteleiro, F. Navarrina, f . Colominas, J. París, X Nogueira
Effective penneability of carbonate reservoirs using the random finite element mcthod
D. V. Griffiths, J. Pa iboon, .J Huang. G. Fenton
Development of subject-specific models with material properties and boundary
conditions derived from medica! imaging
MC. HaBa Tho. TT. Dao, S. Bensamoun
Analysis of potentials in crystalnetworks by means of nonlinear optimization techniques
15
33
41
R. Meziat
MEC.Á.NICA DE SÓLIDOS Y MATERIALES {MS.M)
Diagnóstico de fallas en la estructnra del rotor de un generador eléctrico
P. Roa, J. García
Fatigue lite and dynamic behavior of submarine pipelines subjected io slug t1ow and
vortex-induced vibrations
B. Bossio, A. Blanco, E. Casanova
Absorción de energía en el colapso axial de tubos concéntricos de metal expandido
H. Borges.
Graciano. G. Martínez
Modeling and simulation in a component of an "air dril!" planter
G. Bowges, Af l'vfedina
Ellipsoidal impulse responses in fractured media
P. Contreras, L. Rincón, J. Bwgos
Análisis de grietas radiales en elementos cilíndricos usando el método de elementos
de contorno 2D
P. Teixeira, Jf González
Modelamiento bidimensional elástico de materiales bifásicos a escala mesoscópica
H. D 'Armas. G. Martínez. L Llanes
Evaluación de la condición " columna fue11.e viga débil" en pórticos de concreto
reforzado mediante anál isis estáticos no lineales
A. Marinil/i
Diseño y análisis de un tren de atenizaje triciclo para avión no tripulado utilizando
elementos finitos
O González, G. Afartinez,
Graciano
Force fabric and macroscopic friction in three-dimensional granular materials
V. Urdaneta, J. Petit. X García, E. lYledina
Finite dil:1'erence modeling of rupture propagation under velocity dependent and
thermal weakening processes
A. Nieves, O. Rojas, S Day
e
e
MECÁNICA DE f"LUJDOS (MFj
Testing of openfoam capabilities f(Jr laminar, turbulent and two-phase t1ow models
C. Montilla. A . Blanco, L. Rojas
7
13
19
25
31
37
45
51
57
63
INGENIERÍA \' CIENCIAS APLiCADAS o MODELOS MATEMJÍ'rJCOS Y COMPUTACIONALES
E. Dávila. J. De! Río. M. Cerrolaza, H. Chm:ún (Editores) ~d 20i4 SVI\1"NI
Todos Jos derechos rese1vados
ELLIPSOIDAL llVlPlJLSE RESPONSES IN FRACTlJRED ,\fEDIA
Pedro Contreras
pcontreras@ula. ve
Departamento de Física y Centro de Física Fundamental. lJLA. Mérida-Venezuela.
Luís Rincón
lrincon(iijula.ve
Departamento de Química, Facultad de Ciencias, ULA. Mérida-Venezuela .
.José Burgos
joseburgos@ula. ve
Universidad Nacional Experimental Sur del Lago, Zulia y Departamento de Física, ULA
Abstract. A1ultiple vertical fracture sets, combined with horizontal fine layering produce an
equivalent medium of orthorhombic or monoclinic symmelly. This is particularZv importan! in
F acture reservoir characterization. Fractured reservoirs are azimutha! anisotropic ¡,vith respect
to elastic-'vvave propagation. In this work we introduce an el!ipsoidal approximation for
monoc!inic media that is ab!e to characterized _!i-actured medía near the vertical axis of
,1ymmetry. The procedure is basicallv two-fold. First, we estímate phase velocities near the
vertical axis using an expansion of the slowness. Secondly. the phase velocities are used to build
the group velocities near the vertical axis. We particular/y establish that for monoclinic media
el/ipsoidal jimctions in the phase domain correspond to ellipsoidal fimctions in the group
domain. Finalry, in order to validate the approximation, the P. S1 ami S 2 e/lipsoidal impulse
re.1ponses are comparedfor difj'erent polar angles with the exact responses obtained by solving
numericallv the eigenvectors problem jf-om the Christojjél equation. Ewmples are shown Jór
monoclinic media, and are validated showing results fi·om a previous work fór orthorhombic
media. .Ihe whole procedure is validjór homogeneous media.
Keywords: Monoclinic symme!Jy, Orthorhombic symmetry, Anisotropy, Elastic wave
propagation, Christoffel equation.
l.
JNTRODlJCTION
An orthorhombic model describes a layered medium fractured in two orthogonal directions
with azimuthal anisotropic dependence of the group and phasevelocities. Wave propagation in
orthorhombic media has been extensively studied (see [1,2] among otbers) Numerical methods
MSM·-261NGEN1ER[A Y CIENCIAS 1\PLl C'ADAS: MODELOS MATEM_t\TICOS Y COMPUTACIO NALES
such as finite diffcrences and ray-tracing using weak anisotropic [3] and elliptical approximation
[4] have contributed to the modeling of wave-front in orthorhombic media. Racently Tsvankin
[3] stressed out the importance ofhavi ng different approximations to the group wave velocities in
order to perform velocity analysis in fracture slructures.
A monoclinic model describes two sets ofveJiicalnon-corrugated fractures with a horizontal
symmetry plane. Mon oclinic media bave azimuthal anisotropic depcndence of the group and
phase velocities. There is abundant geological (in situ) [3,7] ev idence of multi ple fracture sets,
which CotToborate the importance of monoclinic models in seismic reservoir characterization;
however, velocity analysis, and parameter estimation for monoclinic media is a highly
challenging task [5] dueto the large number of e las tic constants involved.
This paper introduces a mathematical treatment based of the solution of the Christoffel
equation in ten11S of the slowness vector expansion around a vertical axis of symmetry. [t is
important to choose a coordina te frame where the mathematical description of wave propagation
has the simples! fom1 [5]. For instance the equation for the eigenvectors in terms ofthe slowness
and the conespondent eigenva lues are
F := [Gu.- 6¡¡]U1 =O,
U is the polarization vector of a plane wave,ó';¡ is the Kronecker's delta, and Gi s the symmetric
Christoffel matrix, p represents an horizonta l slowness vector, and q represents tbe vertical
slowness vector.
For a horizontal re±1ector the condition is p1 = p 2 = O; tberefore, the zero offset horizontal
events in orthorhombic and mon oclinic media are obtained by substituting the correspondents
values of q, and the derivatives q 1
'
= !.'l
= - !'.!.; where i = 1,2,
i!pi
F3
and
1'3
is the derivative of
. .
¡¡zq
.
. e
.
b
h
F respect to q. Tl1e secon d ord er d envat1ves
q i¡· = -8 a contam !Iltorrnatwn a out t e
'
Pi PJ
byperbolic move-out. The derivatives of fourth order q ;¡·kl
,
= --~
ap;ap 1arkap,
contain infonnation
about the non-hyperbolic move-out [3]. Finally, the slowness can be written as
(1)
From symmetry considerations the deriva ti ves of first q,¡ lo, and third order q,ijk 1o, are equal to
zero for both types of symmetry. The derivatives of fourth order are taken to be zero since nonhyperbolic move out is neglected. Eq. ( l) yields suitable expressions for the vertical slowness q in
the orthorhombic and the monoclinic cases for the three propagation modes; namelyP, 5 1 and 5 2 .
In voigt's notation the elastic tensor for the monoclinic structure has the form [1 ,2,5]
eMN--
C21
C12
Czz
c23
c31
Cn
C33
e
0
o
o
o
c16
Cz6
c13
o
o
c36
o o
o o e"
Cz6
o o c36 )
C44
C4s
o
C1s
Css
o
O
o
c66
INGENIERÍA Y C:TENCJAS APLICADAS: MODELOS MATE MÁ T!COS Y COMPUTACIONALES MSM -27
Orthorhombic symmetry is dcscnbed by nine elastic constants (with e 16 = e 26 = e 36 = C~ 5 =
0). lt has two vertical planes of symmetry [XZ], and (YZ]_ There is not general solution for the
Christoffel equation fór orthorhombic symmetry; however, .in both symmetry planes are possible
analytical solutions_ For other angles, a mmierical solurion is given for each mode by the roots of
the characteristical polynomial [2,4] .
Monoclinic symmetry is described by thirteen elastic constants, 'but if e45 becomes zero, ihen
ihe matrix becomes diagonal for vertical propagation and the horizontal axes coincide with the
polarization directions ofthe vertical traveling shear wave [1,5]. Therefore, monoc!inic symmetry
is described by twelve elastic constants_ There is not general solution for the Christoffcl equation
for monoclinic symmetry; the solution always requires a numerical treatment for each
propagation mode. Thc eigenvalue method through the characteristical polynomial gives the
phase velocities and the eigenvector method for U yields the group velocities [2,5,6].
2.
ELLlPSOIDAL PHASE VELOCITIES (WAVE-FRONTS) NEAR THE VERTICAL
AXIS
Eq.(l) yields the ellipsoidal phase velocities (wave-f.ronts) near the vertical axis [4,5 ];
howcvcr, the con·espondent derivation is long, and we refer for a detailed description of. the
method to look at appendix of Ref. [5]. A shear mode separation always occurs in fracture media,
where two shear velocities cxists_ The substitution ofq,;¡ lo in Eq.(l) using a slowness vector
p = (Pt- p 2 , q) , anda unitary phase vector ñ = (sin 8; 1 cos 8; 2 , sin 8; 1 sin 8; 2 , cos 8il) yields the
square of the phase velocities W; = p (where p is the density). The calculation for the
monoclinic media near the vertical axis gives the following expression
V/
W i
. 82il ("'i,nnw
82 + 2"'i,nmo
- n
e + ¡;ui.nmo
•· ez)
= wi 'z COS ezil + Sll1
VV11
COS Í2
VV12
$!11 rliz COS i2
V22
Sll1 i2 •
(2)
Eg .(2) represents an ellipsoid in the phase doma in for each elastic mode i = P, 51 and 52 . For
waves traveling in the vertical direction: WP.z = ~' wsl.z = 1 , and wsz.z = - 1 -_
fi:,,
.jc55
¡e;:;
For the longitudinal mode P, thc square of tbe horizontal NMO velocitics for the vertical
symmetry planes [11] and [22] are
(e33- e44)(el3 + 2Cuess + c33ess) + e'f6ce33- ess)
P.nmo
(·e·33
=
Wru¡
L
¡.v:P,nmo _ ((l3-
f22 l
-
-
C'44 )(C'33
-
C'55 )
'
ess)(ef3 + 2e23e44 + e33e44) + e}6(C.n- e44)
(e:;3- e44)(e:;3- ess)
Outside the symmetry planes the NMO velociiy is given by the [12] term
Pnmo
wr1z]
= e36
[(2ess- e13)- e3'l(e13 + e23 +eH+ ess)J
.
.
(C33 - C44)(e33 - ess)
For the shear modc 51> the square ofthe NMO velocities are:
¡.v:Sl,nmo [11]
-
+ C-s] 2
e-u + [Cl3
e .. _ e>.. , WS1,nmo
[1z1
33
55
-
-
+ e-s]
e326
e16 + e,6
. [e13
- e + -----'-'-:--- 66 c. _ e _·
e. . _ e , , ¡.v:S!,nmo
rzz1
33
· 55
Finally, for the shear mode 52 the square ofthe NMO velocities are:
33
·So
MSM-28 INGEN IERÍA Y CIENCI ~ S \ PLICADAS: MODELOS MATEMÁTICOS Y COMPUTACIONALES
3.
ELLIPSOIDAL GROUP VELOCITIES (lJ\lPULSE RESPONSES) NEAR THE
VERTICAL AXES
[n order to generate expressions in monoclinic media for group velocities within thc
ellipsoidal approximation at small group polar angles 0 11 , and arbitrary azimuthal group
angles 0 21 , we rely on a transfonnation using in Ref. [8] for VT! media, and in Ref. [4] for
orthorhombic media. These transfonmltions are proposed and used here as ícJUows:
w-u
group,l'nonoclinic '
0 11 are the polar, and 0 21 are the azimuthal group velocities angles. Tben, ellipsoidal group
velocities near the vertical axis are ellipsoids in the group domain. The inverse of the square
of the group velocity (impulse response) obtained from "Eq. (2)" for each propagation mode
(i , P, 51 , 5 2 ) is
.- u
_ cos 0[1
Wgr·oup -
\Jlfi,z
_.
2
+ sm 0;1
(cos 0f2
¡.yi ,nmo
fu]
i
2
+
2 sin 0 12 cos 0;z
w;i.nnw
[Izj
sin 0r2 )
+ w;i.nmo
'
(3)
[22]
i
where W9·r-oup = p~,:oup·
4.
NUMERICAL EXAJVIPLES
The validation of our approximation is performed on the cracked Greenhom shale case of
ortborhombic symmetry, but by aggregating a non-perpendicular fi·acture se t, according to the
Muir-Schoenberg theory [7,9] , the elastic matrix then, becomes monoclinic. The values of the
elastic constants are: cll = 336.6, c12 = 117.3, c13 = 103.3, c22 = 310.0, c23 = 92.3, c33 =
223 .9, C44 = 49 .1, C55 = 54.0, C66 = 94.6, C16 = 30.0, C26 = 30.0 and C36 = 10.0. As in
Ref. [9] we ha ve chosen Ct¡ to be dimensionless, and the density p has been set to unity.
The exact group velocity (impulse response) can be found numerically with the expression
[2,6]
where CiJkl are the elastic constants in the usual notation, a1 represents the eigenvectors that
corresponden! to the unit polarization vcctors, f! 1 correspond to the normal direction of the wavefront, Eq.(4) will be numerica!ly solved in arder to compare with Eq.(3)
Figure (1) shows horizontal slices of the 3-D impulse responses for each wave propagatíon
mode at different polar angles in the monoclinic case. Jt can be observecl that the exact group
velocity (red color) and the ellipsoidal group velocities (blue color) are almost the same for polar
angles with small vertical aperture. As the polar angle 0 11 increases their separation al so
increases and the ellipsoidal approximation begins to deteriorate. The maximum approximation
error is reached ata polar angle of 90° (not shown here). Figure (2) shows horizontal slices of the
INGENIERÍA \ CIENCIAS APLICADAS: MODELOS MATEMA TICOS Y COMPUTACIONALES MSM-29
3-D impulse responses for each propagation mode at different polar angles Qllt in the
orthorhombic case_
15 10 S
O
S
10 lS
; :n;:~sJo :U ::~0~
15 10 5
o
5 10 15
15 lQ 5
o
-S
10 l.S
5
ü
S 10 1 5
15 10 S
Q
5
10 15
15 10 S
O
~-
10 15
15 10
;. · :~t:~3sJo :u:(~"~
15 10 .S O S 10 15
(3)
15 10 S
O
5
10 l S
Figure 1: Azimuthal view of each wave-mode impulse response for different po lar angles 0liin
the monoclinic case. Red color represents the exact solutions from Eq.(4), blue color represents
the ellipsoidal approximation of .Eq.(3 ).
lt can be observed that exact (red co lor) and ellipsoidal (blue color) group velocities are almost
the same at the [XY] symmetry plane for Q)li = 5°, but at the polar angle 0 11 of 15° the elliptical
approximation of the shear modes begin to deteriorate, as it was shown in Ref[4]; bence, the
horizontal velocity is not well reproduce for large values of 0 11 .
5.
CONCLUSIONS
The impulse responses of al! diflerent modes of wave propagation in monoclinic med ia
near the vertical axis are ellipsoids Figure ( l). This implies that the ideas depicted in [4, 8] are
applicable to monoclin ic media as well. On the other hand, it is shown that the horizontal Nom1al
M ove-Out velocities for horizontal reflectors are ellipscs for monoclinic media in agreement with
the results of [5]. lt is found that the oti diagonal NMO velocities are con!rolled in monocl inic
media for three elastic parame ters: the longitudinal mode w1~;]m"is controlled by C36 • the
. con t ro 11 ed by [ , all( j tb e t ransversa
.
1 ,,r:'2 mol:\ e vv¡
rATSZ ,nmo.IS
t ransversa1 ,,<'1 mo d·e ruSl,nmo
vv ¡
!S
16
112
121
controlled by C26 . By making these three elastic constants to be zero, tb e orthorhombic case of
Ref[4] is reproduced as in Figure (2).
Therefore, we particularly establish that for monoclinic media ellipsoidal functions in the
phase domain correspond to ellipsoidal functions in the group domain. The ellipsoidal
approximation is therefore a simple but a powerful device to reproduce elastic group velocities in
fractured media accurately near the vertical axis.
A.cknowledgements
We thank Dr. V. Grechka for stimulating discussions. We also acknowledge discussions with
Profesores D. Gutíerrez, R. Almeida and L. Seijas from the University of Los Andes. This
research was supponed by the Grant CDCHTA-ULA C-1851 -13-05-B .
MSM0-30 INGENIERÍA Y CIENCIAS APLICADAS: MODELOS MATEMATJCOS Y COMPUTACIONALES
P-wave
u,
'"
~
lg~
!~ .
o
5 1-w a •.te
u
5
1 510 5
o
5
!Q~
!~ :(]A
5 1015
1510 5
~
"'«
e
:i~s
1510 5
o
5 1015
5 1015
Sl - wa\1€1
P-wav"i!
"'
o
52-wav e
iR~
!~ ~"
1510 5 O 5 lDlS
5 2 - W;;!'.IS
iQr:~J
lQr:~J
·!~
iJ
I~
o~
'.
"1!
1510 S O S 1 0 15
8
1.510 5
o
5 10 lS
Figure 2: Azimuthal view of each wave-mode impulse response for different polar angles
0 1 ;in the orthorhombic case reproduced according to Ref[4]. Red color represents the exact
solutions from Eq. (4), blue color represents the ellipsoidal approximation ofEq.(3).
REFERENCBS
fl ]. Musgrave, M. Ci}'Stal Acoustics, Holden-Day, 1970.
[2]. Helbig, K . Foundations of anisotropic for exploration geophysics, Pergamon Press HGE
Vol. 22, 1994.
[3]. Tsvankin l. & Grccllka V. Seismology of azimuthal!y anisotropic media ami seismic
fracture characterization, Society of Exploration Geophysicists. Geophysical Rcferences
Series. Vol. 17,2011.
[4].Contreras P., Klie H. & Michelena R. Estimation of elastic constants from ellipsoidal
velocities in orthorhombic media, 68'11 Ann. i nterna/. Mtg. Soc. Expl. Geophys. pp. 1491 1494, 1998.
(5]. Grechka V., Contreras P. & Tsvankin I. Inversion of normal move-out for monoclinic
media. Geophysical Prospecting. Vol. 48, pp. 577-602, 2000.
[6].Zhou B. & Grcenhalgh S. On the computation of elastic group velocities for a general
anisotropic medium. J. Geophys. Eng. pp. 205-215,2004.
[7J.Schoenberg M. & Helbig K. Modelling elastic behavior in a vertical fracture earth,
53,.dAnn. Meeting ofthe European Soc. c!f'E-">:p/. Geophys. pp. 288-289, 1995.
[8]. Byum S. Seismic parameters for media with elliptical velocíties dependencies.
Geophysics. Vol. 47, pp. !62 1- 1626, 1982.
[9).Dellinger, J. Anisotropic seismic wave propagation. Ph.D.Thesis, Stanford University,
1991.