Download Structure and evolution of the northern Oman margin: gravity and

TECTONOPHYSICS
ELSEVIER
Tectonophysics 279 (1997) 253-280
Structure and evolution of the northern Oman margin: gravity and
seismic constraints over the Zagros-Makran-Oman collision zone
R Ravaut a'b'*, R. Bayer a, R. Hassani
a.l, D. Rousset c, A.
A1 Yahya'ey d
"Laboratoire de Gdophysique et Tectonique, UMR CNRS 5573, ISTEEM, Universit~ de Montpellier H,
Montpellier 34095, Cedex 5, France
h Laboratoire de Tectonophysique, UMR CNRS 5571, ISTEEM, Universitd de Montpellier 11, Montpellier 34095, Cedex 5, France
Jeune Equipe d'hnagerie Gdophysique, IPRA, University de Pau et des Pays de I'Adour, Pau, France
,I Ministry of Petroleum and Minerals, Directorate General (~f Minerals. P.0 Box 551, Postal Code 113, Muscat, Oman
Accepted 5 May 1997
Abstract
The obduction process in Oman during Late Cretaceous time, and continental-to-oceanic subduction along the ZagrosMakran region during the Tertiary are consequences of the Arabian-Eurasian collision, resulting in construction of
complex structures composed of the Oman ophiolite belt, the Zagros continental mountain belt and the Makran subduction
zone with its associated accretionary wedge. In this paper, we jointly interpret Bouguer anomaly and available petroleum
seismic profiles in terms of crustal structures. We show that the gravity anomaly in northern Oman is characterized by
a high-amplitude negative-positive couple. The negative anomaly is coincident with Late Cretaceous (Fiqa) and Tertiary
(Pabdeh) foreland basins and with the Zagros-Oman mountain belts, whereas the positive anomaly is correlated to the
ophiolite massifs. The Bouguer anomaly map indicates the presence of a post-Late Cretaceous sedimentary basin, the
Sohar basin, centred north of the Batinah plain.
We interpret the negative/positive couple in terms of loading of the elastic Arabian lithosphere. We estimate the
different Cretaceous-to-Recent loads, including topography, ophiolite nappes, sedimentary fill and the accretionary prism
of the Makran trench. A new method, using Mindlin's elastic plate theory, is proposed to model the 2D deflection of the
heterogeneous elastic Arabian plate, taking into account boundary conditions at the ends of the subducted plate. We show
that remnant ophiolites are isolated from Tethyan oceanic lithosphere in the Gulf of Oman by a continental basement ridge,
a NW prolongation of the Saih-Hatat window. Loading the northward-limited ophiolite blocks explains the deflection of
the Fiqa foredeep basin.
West of the Musandam Peninsula, the Tertiary Pabdeh foredeep is probably related to the emplacement of a 8-km-thick
tectonic prism located on the Musandam Peninsula and in the Strait of Hormuz. Final 2D density models along profiles
through the Oman mountain belt and the Gulf of Oman are discussed in the framework of Late Cretaceous obduction of
the Tethys and synchronous subduction and exhumation of the Oman margin.
Keywords: flexure; elasticity; gravity; obduction; Oman; ophiolites
*Corresponding author. Tel.: +33 67 14 36 02; fax: +33 67 14 36 03; e-mail: [email protected]
I Present address: Laboratoire de G6ologiede l'Ing6nieur, d'Hydrogrologie et de Prospectiongrophysique. Universit6de Liege, B~timent
B I9, SART-TILMAN,4000 Liege, Belgium.
0040-1951/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved.
PII S 0 0 4 0 - 1 9 5 1 ( 9 7 ) 0 0 1 2 5 - X
254
P Ravaut et al./Tectonophysics 279 (1997) 253-280
1. Introduction
2. Geological setting
The northern Oman mountain range (Fig. 1) is
mainly characterized by the Cretaceous ophiolite
belt of 600 km long, from the Strait of Hormuz
to the Indian Ocean. High topography, up to 3000
m in the Jebel-Akhdar, composed of Palaeozoic to
Tertiary rocks, is the other major feature of the northeastern Arabian mountain range (Fig. 1). If previous
studies on ophiolites led to a better understanding
of fast-spreading ridge mechanisms, they have been
mainly focused on the obduction process as a stacking of oceanic material sheets (Glennie et al., 1973;
Nicolas, 1988; Michard et al., 1994).
Obduction is accompanied by subduction and
rapid exhumation of the Arabian continental margin and by formation of foredeep basins in external
zones. The obduction event was followed by a Tertiary compressive regime in relation with ArabianEurasian convergence. This period is characterized
by building of continental mountain belts in Zagros
and Oman and creation of foredeep basins. The closing process is still active along the Zagros belt and
the Makran subduction zone.
This tectonic evolution has built the actual northern Oman margin whose deep structures remain almost unknown. Are the ophiolites massifs northward-rooted? How did the convergence process
model the passive northern Oman margin? And how
can we explain the preserved foreland basins? Foreland basins are evidences of the large-scale deformations of the lithosphere due to loading during
the convergence process. An elastic behaviour is
classically adopted to estimate the deflection due to
loading in mountain belt (Karner and Watts, 1983;
McNutt et al., 1988). This approach is used in this
paper, considering that the proposed models will
give effective elastic properties for a more complex
mechanical behaviour of the Arabian lithosphere
(McNutt et al., 1988). Gravity and seismic data are
merged over the Zagros-Makran-Oman to constrain
the geometry of sediment deposits and ophiolite
nappes. Then, 2D elastic models are developed to
explain observed deflection of the Arabian lithosphere due to loading effects (ophiolites, topography
and/or sedimentary loads). Finally, the consequences
of our present-day structural model for the tectonic
evolution of the region are discussed.
Northern Oman was a Tethyan passive continental
margin initiated by pulsed rifting beginning in Permian time. The formation of the Hawasina and Hamrat-Duru basins and opening of the Tethys ocean followed in Triassic time (B6chennec et al., 1988; Pillevuit, 1993). The compressive regime, associated with
Arabian-Eurasian convergence, started during the
Late Cretaceous. It corresponds to a low-angle intraoceanic subduction initiated at mid-oceanic ridge in
the neo-Tethyan domain (Nicolas, 1988). Evidence
of intra-oceanic thrusting includes the high-temperature (HT) metamorphism in the ophiolitic sole during
the Cenomanian (95-90 Ma, Hacker, 1994). In the
Campanian (80 Ma), southward migration of nappes
(ophiolites, Hawasina and Hamrat-Duru sediments)
reached the margin (Glennie et al., 1973; Lanphere,
1981; Boudier et al., 1985; Montigny et al., 1988).
This stage was accompanied by downflexing and
thrusting of the margin, with the deposition of the
Muti formation in a first foredeep basin. Simultaneously, the Arabian continental margin was subducted
northward, as it is shown by the high-pressure (HP)
metamorphism in the Saih-Hatat window (20-23
kbar, 500°C) (Wendt et al., 1993; Searle et al., 1994)
and was quickly exhumed as indicated by the unconformity of Maastrichtian oceanic transgressive
sediments (70 Ma) onto the metamorphic unit. Uncertainties on the Early Cretaceous ages obtained
by the 4°Ar/39Ar method (Montigny et al., 1988; El
Shazly and Lanphere, 1992) do not allow to consider an older high-pressure event in our geodynamic
framework (Searle et al., 1994). Last motions of
nappes associated with exhumation are recorded in
the subsequent Fiqa foredeep basin, filled by nappe
detritus.
From the Maastrichtian to Early Miocene, subsidence and transgression occurred in northern Oman,
with filling of the Sohar basin in the Oman embayment (Fig. 1). During this epoch, compressive
tectonic events were recorded in the Musandam area
with the formation of a third foredeep basin (Pabdeh)
in the Sarjah area (Brown, 1971; Ricateau and Rich6,
1980; Searle, 1985; Searle, 1988). During the Early
Miocene-Pliocene, thrusts were reactivated in the
whole range with large foldings and uplifts (Carbon
et al., 1996). These structures are still preserved in
P. Ravaut et al./Tectonophysics 279 (1997) 253-280
255
a)
b)
Iran
_ Arabian
Gulf
~.
Makran
25°N
"11
m
U.A.E.
lbri
23°N
I
[Autochthonous
] • basement
Hawasina-Hamrat Duru
allochthonous unit
~
--
Maastrlchtlan and Teftiar~
~
[
J. Salakh
J. Ja'alan"
Ophiolile nappes
I
00kin
[
_
Oman
I
I
54°E
56°E
".,
%
\
58°E
Fig. 1. G e o d y n a m i c framework (a) and simplified geological map (b) of the northern O m a n area (modified from Coleman, 1981, and
D. Carbon, pers. commun., 1996). Ophiolite massifs: 1 = Ibra; 2 = Maqsad; 3 = Muscat; 4 = Rustaq; 5 = Haylayn; 6 = Bahia; 7 =
Wuqbah; 8 = Salahi; 9 = Fizh; 10 = Aswad.
256
P. Ravaut et al./Tectonophysics 279 (1997) 253-280
3100 i
-
Stations / 5 I~
4.00
3,75
3,50
3.25
3.00
2,75
ZSO
2,25
2,00
1.75
1.50
1.25
1.00
0.75
0.50
0,25
0,00
300
400
500
600
700
Fig. 2. (a) Density of stations on the studied area. Origin: Petroleum Development Oman (onshore Oman), Bureau Gravim6trique
international (Gulf of Oman: ships and derived satellite altimetry data, Sandwell and Smith, 1992), onshore Iran (Dehghani and Makris,
1984), Open University (northern Oman mountains, Shelton, 1984), Laboratoire de G6ophysique et Tectonique-ISTEEM (northern Oman
mountains, Ravaut et al., 1993, and this paper). (b) Bouguer anomaly map of the Zagros-Makran-Oman area. Gravity data at each
station have been projected on a regular 2.5 × 2.5 km size mesh using a krieging interpolation method. Bouguer anomaly was calculated
in the IGSN71 reference system using the GRS67 ellipsoid formula; topographic corrections have been applied in a radius of 167 km
using a 2600 kg/m 3 density reduction for relief. For recent stations, IGSN71 reference gravity base is Seeb Airport, g = 978923.59-4-0.05
regal (Shelton, 1984). Contour interval: 10 reGal. Projection system: Universal Transverse Mercator (UTM); meridian origin: 57°E.
257
P. Ravaut et al./Tectonophysics 279 (1997) 253-280
A,
B,
C,
0
-20
-40
-60
-80
-100
-120
-140
-160
300
400
500
X UTM (km)
Fig. 2 (continued).
600
700
rnGal
258
P Ravaut et al./Tectonophysics 279 (1997) 253-280
the present-day Oman mountains (3000 m in JebelAkhdar).
Pliocene deformation is characterized by continental collision in the Zagros mountains (Bird,
1978). In Oman, compressive and extensive structures coexist (Carbon et al., 1996). N - S continental
subduction beneath the Zagros belt and oceanic subduction beneath Makran control the tectonic and
sedimentary evolution of the area: the deflection of
the Arabian platform allows the creation of a N W SE foreland basin in the Arabian Gulf, dipping to the
northeast. In the Strait of Hormuz, west of the Zendan fault, this deflection becomes E - W and Recent
sediments are tilted to the east. East of this fault,
subduction of the Tethys is associated with deflection of oceanic basement, active sedimentation, and
formation of the Makran accretionary prism (White
and Ross, 1979).
3. Bouguer anomaly map
For several decades, gravity surveys have been
carried out in Oman for petroleum exploration.
Petroleum Development Oman (PDO) collected data
at about 45,000 stations from surveys realized between 1955 and 1969. These data were gathered
south of the northern Oman mountains and on the
Batinah coastal plain. They are supplemented by
around 900 measurements coveting the coastal plain
and the ophiolite massifs (Manghnani and Coleman,
1981; Shelton, 1984). In 1992, we mapped the gravity field in the Jebel-Akhdar area (Ravaut et al.,
1993) and this survey was extended eastward and
westward in 1993, resulting in 1012 new stations
with a mean spacing of 4 km. All the observed gravity values have been tied to the International Gravity
System Network 1971 (IGSN71) from Seeb Airport
base (Shelton, 1984). For these data, the topographic
corrections were computed using Hammer's table
and 1:100.000 ° topographic maps up to 10 km. The
amplitude of this correction reaches a maximum of
20 mGal in the Jebel-Akhdar area with a standard deviation of 1 mGal for the mountain area (Rey, 1989).
In order to map the gravity anomaly over the
Z a g r o s - M a k r a n - O m a n zone, gravity profiles from
cruises (White and Ross, 1979) and Iranian surveys (Dehghani and Makris, 1984) were extracted
from the Bureau Gravim6trique International (BGI)
data bank. The marine data were processed with
SEAVALID software (Adjaout and Sarrailh, 1992)
by adjusting the cross-over errors observed at intersecting ship tracks.
In all, the surveys represent 52,077 stations
(Fig. 2a). We computed free-air anomaly using the
Geodetic Reference System 1967 ellipsoid formula.
In the Gulf of Oman, 30 km offshore, the data were
densified by adding the gridded (5 minx 5 min) freeair data calculated by Sandwell and Smith (1992)
from sea altimetric measurements. Bouguer anomalies (Fig. 2b) were mapped using gridded (2.5 km
x 2.5 kin) values after applying topographic corrections up to 167 km using the ETOPO5 digital elevation model from the Defense Mapping Agency
(USA), with 2600 kg/m 3 for relief density. On land,
the Bouguer anomaly standard deviations are mainly
related to station elevation uncertainties and topographic corrections, and are estimated to be 2.5 reGal.
The map exhibits negative anomalies on continental domains (Iran and Arabia) and positive anomalies
on the oceanic Gulf of Oman. A spectacular minimum is observed in the Fiqa and Pabdeh foredeeps
( - 6 0 to - 1 2 0 mGal). Toward the Strait of Hormuz,
this gravity minimum assumes an arcuate shape and
is connected northward to the negative anomaly coinciding with the Zagros belt (Snyder and Barazangi,
1986). From the Musandam promontory to Muscat,
the negative trend is bordered by high anomalies
(80 to 150 mGal) located on ophiolite outcrops, except between Muscat and Sohar where anomalies are
shifted northward under the Batinah plain.
The Oman coast line is roughly the limit between a high 'ophiolite anomaly' and the negative
anomaly ( - 5 0 mGal) located eastward on the Maastrichtian-to-Recent Sohar basin. The 90-120 reGal
Bouguer values in the Gulf of Oman combine effects
of the oceanic lithosphere and of the sediments recently deposed and piled into the accretionary prism
of the Makran trench (White and Ross, 1979) as it
is attested by the relative minimum along the Iranian
Makran coast (Fig. 2b).
4. Elastic response of Arabian lithosphere to
Cretaceous-to-Recent loading
In continental thrust belts, the Bouguer anomaly
is classically used to analyze large deformations of
P. Ravaut et al./Tectonophysics 279 (1997) 253-280
the lithosphere. It gives information about rheological behaviour of the continental lithosphere and of
applied forces. Elastic plate models are favoured by
many authors to explain the deformation of continental lithosphere (Karner and Watts, 1983; Royden,
1988; McNutt et al., 1988; Kruse and Royden, 1994;
Lyon-Caen and Molnar, 1989). In our study, the
asymmetric gravity low, extending from the Zagros
to northeast Oman, may be interpreted as evidence
of the elastic deflection of the Arabian lithosphere
and its Moho. The actual deformation is a consequence of complex superposition of topographic
loads (Zagros and Oman mountain belts), internal
density heterogeneities, and associated subsurface
loads (ophiolite nappes of Oman) and bending moments and/or vertical shear forces acting at the ends
of the subducted Arabian plate beneath the Zagros
range and Makran area.
We now estimate the active loads, inherited from
previous tectonic events or related to present-day
convergence, and foredeep geometry, inferred from
gravity data and seismic exploration results. Syn-topost obduction sedimentary basins (Fiqa, Pabdeh
and Gulf of Oman) geometry may be defined
by seismic data and well-log information. They
are representative of the deflection of the Arabian
lithosphere during compressive events. Correcting
Bouguer anomaly for basins aids in modelling the
ophiolite massifs and their associated loads.
259
°~
o
Fig. 3. Simplified 3D view of the 2D elastic model of the
Arabian lithosphere.Symbolparametersare given in Table 1.
or large deformations are now observed in the Gulf
of Oman, excepted along the Makran trench. Therefore, we bound the elastic plate 200 km north of
the main Zagros thrust fault (MZT). In Makran, the
ends correspond to the Iranian Lut block, north of
the Makran prism (Fig. 1). The southern and western
limits of our model are characterized by free boundary conditions and chosen far enough (3000 km) to
avoid edge effects on the studied area. The eastern
border located at the Owen fracture zone is a free
boundary.
4.1. The model (Fig. 3)
The classical equilibrium equation of a 2D thin
elastic plate with variable elastic thickness, caused
by vertical forces F (in N/m 2 ) and overlying an
inviscid fluid of density Pm (kg/m 3) is given by:
A [ D A ( w -- w0)] + p m g ( w -- wo) = F
(1)
where D is the flexural rigidity (N m), w (m) is
the lateral displacement normal to the plate (deflection) and A is the Laplacian operator. In our case,
w is a deflection calculated with respect to an initial
reference surface w0. Eq. 1 is solved using Mindlin
plate theory and a finite elements formulation (see
Appendix A). The mechanical conditions at the ends
of the subducted plate are simulated, at each node
with a 10 km spacing, by vertical shear forces, Q
(N), and bending moments M (N m). No earthquakes
4.2. Geometrical constraints on sedimentary basins
and ophiolite massifs
4.2.1. Syn-to-post obduction sedimentary structures:
(a) The Fiqa and Pabdeh basins. The numerous
seismic profiles were merged during petroleum exploration of the Late Cretaceous Fiqa and TertiaryQuaternary Pabdeh basins and several isopach maps
have already been published (Searle, 1983; Ross et
al., 1986; Patton and O'Connor, 1988; Boote et al.,
1990; Warburton et al., 1990). The bottom of the
Fiqa basin is marked by an onlap above the Muti
formation whereas Maastrichtian-Paleocene overlying series form the top (Fig. 4a). The thickness of
the southern Fiqa basin is detailed by adding results of ten unpublished seismic lines from Ministry
of Petroleum and Minerals of Oman (MPMO). The
260
P. Ravaut et al. / Tectonophysics 279 (1997) 253-280
3500 m thick Fiqa deposits are characterized by an
asymmetric shape and are eastward-limited by the
syn-obduction thrust nappes (Fig. 4b). The maximum
depth is in front of the highest gravity anomalies created by ophiolite blocks. The location of the Lekhwair
forebulge has been estimated at 100 km from the allochthonous wedge (Fig. 1) (Boote et al., 1990).
The so-called 'Pabdeh basin', including all the
post-obduction sediments in our terminology, is not
superimposed on the Fiqa foredeep. Maximum thickness (Fig. 4c), up to 4000 m, is shifted northward
at west of the Musandam Peninsula. The lower part
of this section is mainly related to intense folding
and thrusting during the Early Tertiary (Ricateau and
Rich6, 1980; Searle, 1983; Dunne et al., 1990). The
upper part corresponds to gently northward-dipping
sediments in agreement with the actual subduction
geometry. The Pabdeh basin is bordered by the 2000
m Musandam-Jebel-Akhdar mountains but its variation in depth is not correlated with the mean altitude
of the neighbouring relief.
The N - S extension of both foredeeps is quite
surprising in the frame of a N - S convergence context
between Arabia and Iran since the Late Cretaceous.
(b) The Sohar basin and the oceanic basin. The
Sohar basin was identified from previous academic
and petroleum marine seismic surveys in the Gulf
of Oman (MPMO, unpublished data). It is associated with a negative gravity anomaly (Fig. 2b). A
5000 m thick sequence of Maastrichtian-to-Recent
sediments is locally observed in available wells. The
basin geometry is constrained by sixteen unpublished petroleum seismic lines integrated with well
log data and seismic cross-sections from White and
Ross (1979) (Fig. 5a).
We focus on the N W - S E line 15 carried out during the 1975 French CEPM (Consortium d'Etude
Petroli6re des Marges) project which has been reprocessed on a length of 180 km (Fig. 5b). The Sohar
basin exhibits a thick sedimentary sequence whose
bottom coincides with an energetic reflector at 4 to
5 s (two way traveltime, TWT) depth. Two stages
of deposit may be identified limited by important
reflectors: a 2.5 s TWT thick carbonate unit from
Maastrichtian to Late Miocene and a syn-tectonic
compressive epoch from Late Miocene to Recent.
Eastward, the observed basement rise may correspond to extension of the Oman ophiolite nappes
or of the metamorphic Permo-Triassic continental
margin. In the deepest part of the Oman Gulf, gently
dipping layers are associated with the downflexed
oceanic plate along the Makran trench and are folded
at the front of the accretionary wedge.
The two main reflectors described on the profile
are identified on all seismic lines including profiles
published by White and Ross (1979) and were digitized. Well-log velocity records (Fig. 6a) were used
to convert the TWT seismic reflectors to depth. We
have extrapolated northward the sediments thickness
behind the Makran front, in order to avoid border
effects during sedimentary stripping of the Bouguer
anomaly. The basement reflector is prolonged in
the subduction direction (N-006 °) with a 5 ° slope
deduced from earthquake distributions (Byrne and
Sykes, 1992) and from OBS seismic profiles (Niazi,
1980). The sedimentary cover is limited southward
by ophiolite outcrops of Oman. The resulting isopach
map (Fig. 5c) exhibits at Muscat a N-W-elongated
basement ridge separating the Sohar basin from the
oceanic domain of the Gulf of Oman. Along the
Makran coast line, the sedimentary column of the
prism reaches a maximum thickness of 13 km.
4.2.2. The ophiolite massifs
The wavelength and amplitude of anomaly created by the sedimentary cover partly hides the gravity effect of the ophiolite nappes. Stripping these
effects requires knowledge of the density distribution
in sediments. In the Oman foreland, bulk rock measurements realized during previous work (Manghnani and Coleman, 1981; Shelton, 1984) indicate a
2400 kg/m 3 density for Fiqa sediments and a 2550
kg/m 3 density for Tertiary units. In the Gulf of
Oman, well-log density records (Fig. 6b) are used to
define a more refined density-depth relationship by
adjusting data by the following exponential law:
P = Pb -- 4'0(Pb -- Pf)exp (-~/x)
(2)
where Ph is the matrix density (Pb = 2670
(kg/m3), pf is the fluid density (pf = 1000 kg/m3),
4'o is the near surface porosity (4'o = 0.5) and )~ is
the characteristic depth (,k = 6000 m). A stripped
map representing the effect of deep structures and of
the ophiolites (Fig. 7a) is obtained by removing the
sediment contribution to the Bouguer anomaly map.
Ophiolite gravity highs are enhanced to a maximum
P. Ravaut et aL /Tectonophysics 279 (1997) 253-280
W
0
2
3
4
a
3000
2950
2900
2850
2800
E 2750
2700
2650
2600
Meters
2550
ABOVE
m
-500
-1500-
-1000
-2000 -
-I~00
-2500 -
-2000
-~K)O0 -
-2500
-3500 -
<3000
BELOW
.350O
2500
2450
250
300
350
400
450
500
550
600
X UTM (krn)
Fig. 4. (a) An example of seismic profile crossing the foreland domain in Oman, from Boote et al. (1990). S(
the foreland of the northern Oman mountains, from published documents (Patton and O'Connor, 1988; Boote
seismic data from the Ministry of Petroleum and Minerals of Oman (MPMO) are integrated into previous re
deposits over the foreland of the northern Oman mountains from published documents (Patton and O'Connor,
unpublished seismic data from the Ministry of Petroleum and Minerals of Oman (MPMO) are integrated into !c
pp. 261-264
E
L,,,~,.~ ~
"~--~,~.~..: .....
- ~,
-
! ......
SKM
!-
3OOO
2950
2900
2850
2800
~
2750
2700
2650
2600
Meters
g
ABOVE
.1000 -15C0 -21~0 -;tS,00 `3000-3500 BELOW
.500
-5~30
.1000
.t500
.2500
-3000
-350O
2550
2500
2450
250
300
350
400
450
X UTM (kin)
~e Fig. 5a for location. (b) Isopach map of Fiqa deposits over
et al., 1990; Warburton et al., 1990). Southward, unpublished
~sults. Contour interval is 500 m. (c) Isobath map of Tertiary
1988; Boote et al., 1990; Warburton et al., 1990). Southward,
,revious results. Contour interval is 500 m.
500
550
600
265
P. Ravaut et al./Tectonophysics 279 (1997) 2 5 3 - 2 8 0
Arabian Gulf
Makran
7
>-
BM-A1
BM-B1
350
400
450
500
X UTM (km)
550
600
650
700
Fig. 5. (a) Location of seismic profiles and wells used for sedimentary structural evaluation of basins. Lines 1 to 13 are from MPMO,
lines 14 and 15 are from CEPM, lines w l 6 to w19 are from White and Ross (1979), line b-3a is from Boote et al. (1990). Stars and
squares represent OBS profiles off the Makran coastline (Niazi, 1980). Name of wells: B-1 = Barka-1; F-1 = Fujairah-1; B M - A I =
Batinah Marine A l; BM-B1 = Batinah Marine B1.
of 155 mGal and are seaward shifted. These anomalies are globally still separated from the oceanic
positive domain (220 reGal) by a relative minimum.
In consequence, we assume that the preserved ophiolites along the Oman coast are not connected to
Tethyan oceanic lithosphere of the Gulf of Oman,
except for the Muscat ophiolites.
Anomalies created by the ophiolites were analytically extracted from the stripped map (Fig. 7a). A
low-order polynomial regional anomaly was computed and removed to obtain the ophiolite contribution to the regional field (Fig. 7b). 2D density models along three profiles crossing the northern Oman
margin (Fig. 7c) confirm the rootless character of
the ophiolite blocks as it was already proposed by
Manghnani and Coleman ( 1981) and Shelton (1984).
West of Sumail Gap, nappes are northward-dipping
beneath the Batinah coast. A maximum depth of 10
km is observed on the Rustaq-Haylayn block.
A simple model consisting of an equivalent horizontal stratum is chosen to calculate the mass by
surface unit using the following relationship:
cr = f
Apo(z)dz ,~ Ag/27rG
(3)
where cr is the surface density contrast, h is the thickness of the ophiolite blocks, Apo(z) is the density
contrast of ophiolites with Arabian crust, Ag is the
ophiolites anomaly and G the gravitational constant
(see Table 1 for numerical values).
4.3. Applications
4.3.1. The obduction flexure
The loads associated with the ophiolite anomaly
(Fig. 7b) is defined in Eq. 1 by F0 = ~rg, where
g is the gravitational acceleration. The resulting deflection is accompanied in the foreland domain by
266
P. Ravaut et al./Tectonophysics 279 (1997) 253-280
m
~=~ii
o
e-
'~.=
H
e'~
'
~°~
~ - c - 5 q ,~
..
f
f
267
P. Ravaut et al. / Tectonophysics 279 (1997) 253-280
Meters
o
-lOOO
-2000
-3000
-4ooo
-5ooo
-6ooo
-7000
-8ooo
-9000
t>-
-10000
-11oo0
-120o0
-13000
-14000
-15000
-16ooo
-17000
400
450
500
550
600
650
700
750
X UTM (km)
Fig. 5 (continued). (c) Isopach map of Maastrichtian-to-Recent sediments in the Gulf of Oman. Location of data used for interpolation is
given in (a).
infilling materials acting through a vertical force
Fi = g P i ( t v - w0), where Pi is the density of the
infilling material. Table 1 shows numerical values
of the parameters used in Eq. 1. In the foreland
area, the pre-obduction elevation of the plate is supposed to be null, i.e. w0 = 0. As a first step, the
rigidity is assumed constant over the plate. The theoretical deflection is calculated for various D-values
ranging from 1018 N m to 1025 N m and results are
illustrated on an E - W profile (Fig. 8a and b). An
homogeneous elastic plate assumption is not realistic
to explain the Fiqa foredeep basin. Our preferred values are 5.1023 N m for the external Arabian platform
and 1022 N m over the northern Oman belt and the
coastal plain (Fig. 8a and b). The effective elastic
rigidity under the Oman platform is close to Snyder
and Barazangi's estimation (Snyder and Barazangi,
1986) given for the Arabian foreland basin (1023
N m). The calculated deflection is in agreement with
the observed Fiqa sediment distribution (Fig. 4b).
Moreover, the preserved Lekhwair bulge is roughly
coincident with our computed bulge at 100 km from
the frontal folded-faulted zone. High-to-low effective rigidity variation, from external zones toward the
northern Oman Mountains, is explained by a relative
gently dipping basement (and Moho) that steepens
abruptly.
4.3.2. F l e x u r e related to T e r t i a n - t o - R e c e n t
c o m p r e s s i v e events
The Late Tertiary structural evolution of the
Oman and Zagros thrust belts has resulted in building of important relief (up to 3000 m in Jebel-Akhdar
and 2000 m in Musandam and Zagros). The corresponding load in the right hand side of Eq. 1 is
represented by F r = (hr + ( w - w , ) ) p , g where hr
is altitude of the relief and Pc the density of the
infilling material and topography. The initial deflec-
268
P. Ravaut et al./ Tectonophysics 279 (1997) 253-280
'
I
'
ee4
i
'
i
44
'
41'
_..
i
'
i
'~
41"
-.:.
0
_
~
@
,41.
r
,
0
--
,
I
,
I
0
0
0
0
0
0
,
I
0
0
0
0
0
0
.
I
,
0
0
0
~
0
0
0
~"
|
q~ea
(w)
t
'
I
'
c>
I
'
I
I
i
i
I
I
o
E
v
14=
I
o
,
I
~
,
I
~
I
=
m
I
~
r-"
I
,
I
,
I
,
I
I
~
o
m
o
~
o~
(Ukl) Lp,deo
~
~
u
_o
P. Ravaut et al./Tectonophysics 279 (1997) 253-280
aa
269
b
o
~"U
~.=- d.
.
~-~ °
x
llllllO~
o
oo
o
g
o<[
g
b
A
v
x
llll||llll~l|llllll~
v
;.=
270
(9
E
P. Ravaut et al./ Tectonophysics 279 (1997) 253-280
A,.,
52
36
20
4
I
2
g
6
10
14
140
200
260
320
km
8O
2O
0
g
'
. . . .
'
"
I
i
i
i
,
2
6
14
140
200
260
320
km
o~-
~,_l
>~-:-:-,
,
,
,
,
I_1 t_.=
i
f
j
1 1
2
6
.8
14
20
80
140
200
km
(c)
Fig. 7 (continued). (c) 2D ophiolites models along three profiles
through the Oman mountain range. Observed anomalies are
drawn from the residual map in (b). Density of ophiolites is 3070
kg/m3 (Manghnani and Coleman, 1981; Shelton, t984).
tion w0 is assumed to be null over the continental
part of the Arabian plate. It is poorly constrained for
the Arabian margin and the oceanic Gulf of Oman
and it will be further discussed. In the continental area, the elastic properties estimated from the
study of remnant obduction flexure are conserved.
In Zagros, an additional weak zone (D = 1022 N m)
is considered northward of the MZT as it was already proposed by Snyder and Barazangi (1986).
The rigidity adopted for the oceanic domain in the
Gulf of Oman (D = 5.1023 N m ) is in agreement
with its Cretaceous age (e.g. Watts and Daly, 1981).
In Oman, topographic loads (Fig. 10a) represent
a maximum deflection of 1000 m in front of the
Jebel-Akhdar mountains (Fig. 9a). A clear shift is
observed between the location of calculated maximum depth and the zone of maximum Tertiary
subsidence (Fig. 4c), which attests of the presence of
additional subsurface loads in Musandam Peninsula
and Strait of Hormuz in relation with intense nappe
thrusting from Early Tertiary to Miocene (Ricateau
and Rich6, 1980; Searle, 1983, 1985, 1988; Dunne
et al., 1990; Michaelis and Pauken, 1990). Moreover,
in Zagros, the calculated Moho deepening is 6 km
beneath the MZT, not enough with respect to the 15
km crustal thickening predicted by published gravity
models (Bird, 1978; Snyder and Barazangi, 1986).
Obviously, the geometry of the subducted oceanic
plate under the Makran is not the consequence of the
topographic loads.
The load corresponding to the Makran accretionary prism is applied on the plate (Fig. 9a). The
load associated to the prism of thickness tp is Fp =
tpg(ps - P w ) in Eq. 1, where Pw is the density of
the water, p~ is the density of sediments and tp is
the sedimentary thickness. We add an additional load
F m = tmg(ps - Pw) related to a thick sedimentary
sequence (tm = 8 kin), the so-called 'Musandam
tectonic prism', over the Musandam Peninsula and
the Strait of Hormuz region (Fig. 10a). Finally, in
Makran, vertical shear force Qm = 1.5 x 1017 N and
bending moment Mm = 5 x 102o N m, are applied at
nodes of the mesh defining the northern limit of the
subducted plate. Shear force Qz = 5 x 1016 N and
bending moment M: = 5 x 102o N m are adopted in
Zagros. These conditions are roughly similar, for a
2D modelling, to the ones adopted for the Himalayas
or Apennines mountain belt (Lyon-Caen and Mol-
P Ravaut et al./Tectonophysics 279 (1997) 253-280
271
Table 1
Definition of symbols used in the text
Symbol
Definition
Unit
Value and comments
LU
depth after deflection
initial depth before deflection
flexural rigidity
m
m
Nm
computed deflection is (w - w0)
m
m
kg/m 3
kg/m 3
kg/m 3
kg/m 3
kg/m 3
kg/m 3
kg/m 3
kg/m 3
kg/m 3
kg/m 3
Qz, Qm
crustal thickness
elastic thickness of the lithosphere
density of water
upper mantle density
lower crust density
upper crust density
density contrast of ophiolites, ophiolite density
density of infilling material (sediments)
density of Fiqa formation (gravity correction)
density of Tertiary-to-Recent formation (foredeep basin)
matrix density (density/depth law in Sohar basin)
fluid density (density/depth law in Sohar basin)
gravity anomaly
length characteristic (density/depth law in Sohar basin)
surface porosity (density/depth law in Sohar basin)
thickness of Makran accretionary p r i s m
thickness of Musandam tectonic prism
vertical shear force at ends of the subducted plate
Mz, Mm
bending moment at ends of the subducted plate
Nm
WO
D
hc
he
Pw
Pm
Pint"
Pc
Apo, Po
Pi
Psi
Ps2
Pb
Pf
Ag
~bo
tp
tm
D = 5.1023 for external zones;
D = 1022 for internal zones
1000
3300
2900
2670
400-3070
2500
2400
2550
2670
1000
mGal
m
m
m
N
6000
0.5
Makran: Qm = 1.5"1017
Zagros: Qz = 5.1016
M a k r a n : Mm = 5"1020
Zagros: Mz = 5.1020
nar, 1989; Kruse and Royden, 1994). They represent
the effect of the subducting slab behind the application point on the elastic Arabian plate. Fig. 9b and
Fig. 10b show the final predicted lithospheric flexure
for the Tertiary to Recent times.
5. Consequences on density models of the
northern Oman margin
The deflection model does not simply explain the
regional trend of the stripped anomaly (Fig. 7a).
Density modelling is realized to clarify the deep
structures of oceanic crust and the Arabian margin
whose thinning is superimposed on the deflection.
2D models are proposed along three profiles crossing
the northern Oman mountains and the Gulf of Oman
(Fig. 11). The sedimentary structures are constrained
by isopach maps (Fig. 4b and c) and ophiolite bodies
were previously modeled (Fig. 6c). From seismic
refraction results in Saudi Arabia (Mooney et al.,
1985; Mokhtar and A1 Saeed, 1994), a reference
thickness of 38 km and a crust/upper-mantle density
contrast of 400 kg/m 3 were adopted for the Precambrian Arabian crust. Beneath the continental foreland
domain and oceanic part of the Gulf of Oman, the
Moho shape was approximated by calculated deflection summed for obduction and Tertiary loading
(Fig. 8a, Fig. 9b).
Preliminary density models have shown the rootless character of the ophiolite massifs (Fig. 7c)
whose loading is consistent with flexure of the Fiqa
foreland basin. The ophiolite nappes are probably
separated from the oceanic domain by a continental
ridge previously identified on the seismic profiles.
For the Gulf of Oman, a 6-km-thick oceanic crust
is chosen from field work evidences on Oman ophiolites (Nicolas, 1988). Such a thickness is not sufficient to explain the low values, near 220 mGals,
of the stripped anomaly for a typical oceanic lithosphere. The poor available geophysical constraints
involve large undetermination for deep density solutions: an oceanic crustal thickening by a factor 2, two
P. Ravaut et al./Tectonophysics 279 (1997) 253 280
272
3000
2950
2900
2850
2800
~
J
E'2750
2700
I
2650
2600
Km
....i]
m
ABOVE
O O(;
.O ".'~0 •
O CO
-! CO-
-O
"2CO"
-'..£O
•2 5 0 .
-~ CO
.3 O0 •
*2 50
•3 50 •
-3 03
BELOW
2550
2500
-3 ~x~
2450
250
300
350
400
450
600
550
600
X UTM (km)
Fig. 8. (a) Theoretical deflection caused by ophiolites loading. D = 5 - l023 N m for Arabian platform and D = 1022 N m for internal
part of the northern Oman mountain range.
oceanic lithospheres superimposed during the obduction process and/or northward extension of continental crust beneath the Makran accretionary wedge.
Results of seismic experiments off the Makran coast
(Niazi, 1980) have revealed a 6.7-kin-thick 'volcanic
basement' and allow us to reject the first assumption.
The regional increase of the stripped anomaly
from the Oman mountains to the oceanic domain
is interpreted in terms of thinning of the Arabian
continental crust. For profiles A and B (Fig. 11), we
propose to extend the thinned Arabian continental
crust to 50 km off the Makran coastline. Northward,
the proposed model, corresponding to the low-angle
obduction hypothesis, is poorly constrained. The 2D
models indicate a continuity of the Saih-Hatat window along the 'seismic basement ridge' separating
the Sohar basin from the oceanic domain, and the
external character of the ophiolites massifs (Ibra to
Fizh, Fig. 1) with respect to the continental ridge.
North of Muscat, extension of the continental margin
P Ravaut et al./Tectonophysics 279 (1997) 253-280
I
I
I
b
0
......
-iii!!i ......
J
0
",
,
I
40
80
Distance (km)
a
'~ "~ ~. !
'1
120
~
,
~'~
160
Fig. 8 (continued). (b) Observed Fiqa thickness (full circle line)
along profile 1-1' (see location in (a)) is compared to calculated
deflection for rigidities varying from 1020 to 1024 Nm. Optimal
model corresponds to thick line.
along profile C is more reduced than on profiles A
and B over the Batinah plain.
6. Discussion
The agreement between the predicted and the observed deflections for the obduction stage involves a
flexural rigidity decreasing from the foreland area to
the internal domain. The low rigidity in the internal
zone may be the result of crustal basement thickening of the Arabian margin, consecutive to folding and
thrusting during the obduction process. It may also
reflect the initial elastic contrasts between a rigid
Arabian shield and a weak Permo-Triassic margin.
The Tertiary-to-Recent deflection takes into account the major structural features:
(a) The 4-km-thick Tertiary-to-Recent N - S Pabdeh foredeep basin (Fig. 9c) is explained by: (1) a
corner effect of the sediments piled in the Makran
prism near the Strait of Hormuz; (2) subduction
along Zagros-Makran; and (3) a 8-km-thick tectonic prism over the Musandam Peninsula and the
Strait of Hormuz. In this area, compressive events
have been described during the Tertiary and are
characterized by intense folding and reverse faulting
273
(Ricateau and Rich& 1980; Searle, 1988; Dunne et
al., 1990). Such a tectonic prism is in agreement with
the upthrust Mesozoic platform observed on seismic
profiles (Ross et al., 1986; Michaelis and Pauken,
1990).
(b) On the Zagros belt, bending moment and shear
force allow to tilt the Arabian Moho from about 1°
near the Zagros folded belt to 5 ° at the MZT, and to
predict a 15 km maximum crustal thickening.
(c) Over the Makran, the chosen boundary conditions and the prism loading lead to a deflection in
agreement with the geometry of the upper Campanian reflector (Fig. 9c). Over the oceanic domain,
the predicted deflection is consistent with the depth
of the upper Campanian basement, considering an
initial depth w0 = 2 km for this marker.
The 2 - 4 km amplitude of the deflection calculated over the Batinah plain and the northern Oman
mountains is mainly a result of bending of the subducted lithosphere and accretionary wedge of the
Makran. However, such a subsidence is not observed
in Oman since Campanian time. It indicates that the
elastic assumption is too simple to fully explain the
deformation of the northern Oman region. Decreasing amplitude of the deflection could be obtained for
instance by decoupling the continental part from the
oceanic part of the Arabian lithosphere. Moreover,
compressive events in northern Oman since the Late
Tertiary have resulted in reactivation of a low-angle
thrust ramp, crustal shortening and building of the
northern Oman mountains (Carbon et al., 1996).
The density models have revealed a lateral variation of the continental margin structures from the
Muscat area to Musandam Peninsula. This difference may be partly inherited from variations in extensional process during the Permo-Triassic Tethyan
opening. Lateral variations of the models may also
reflect the different conditions of the subduction and
exhumation of the margin in the Late Cretaceous,
as it is attested by restricted outcropping of the HP
rocks within the Saih-Hatat window.
The rootless character of the ophiolite nappes and
the large seaward extension of the Arabian margin
may be understood in the frame of the tectonic
evolution of the northern Oman region during the
Late Cretaceous:
(a) At the Coniacian-Santonian stage (90-80 Ma)
intra-oceanic obduction occurred, with thrusting and
274
t£ Ravaut el al. ITectonophysics 279 (1997) 253 280
Z~
~
E
~
~
a~
x
._ = II
~_Z
g
g
g
g
g
~
=-~
~
o
•
~N
~'~ =..~
."a
r
~ -!=_ °
g
x
=-E
.,-a := = ~
~ = ~
~°~
a . . . . . . . . . . .
IIIi~~D3
. . . . . . . . . .
~.~
P. Ravaut et al. / Tectonophysics 279 (1997) 253-280
II
I1'
E
._o
v
c-
4
0
a
Observed
Computed
(D
-
-
......... i-,,,.,,,i
0
""
. . . . . . . . . i, . . . . . . . . i . . . . . . . , , i , , . r , . . . , I , , ,
40
80
t
1
...... i....I
120
Distance (km)
III
........
, .........
, .........
, .........
, .......
"111'
2
275
lowing a schema already described by (Chemenda et
al., 1996) for the Saih-Hatat window. Such a mechanism implies detachment of the ophiolite nappes
from obducted oceanic lithosphere and a southeast
sliding of nappes as for the Ibra-Maqsad massifs.
North of the ridge, a symmetrical sliding of the remnant Tethyan oceanic lithosphere may be invoked as
it is proved by field observations (Michard, 1983).
In the Batinah region, the large horizontal motion
of nappes from the continental ridge may be due to
a flat thrust ramp emerging near the present frontal
wedge of the Oman mountains. The present isostatic
balance between ophiolite loads and the Fiqa foredeep which is not significantly eroded, indicates that
the present-day ophiolite nappes are representative
of syn-obduction loading.
E
v
e.O_
.,._,
O
7. Conclusions
8
10
14
,,,,,,,,,~
0
-
Computed
.........
50
~........
100
' " ~
, ~ , , , . . . . . '?"
150
200
Distance (km)
Fig. 9 (continued). (c) T h e theoretical Tertiary deflection (W--Wo)
is c o m p a r e d to o b s e r v e d thickness o f the p o s t - L a t e C r e t a c e o u s
s e d i m e n t s o f the P a b d e h f o r e d e e p basin (profile 11-11') a n d o f
the M a k r a n a c c r e t i o n a r y p r i s m (profile II1-111'). We a s s u m e
that w0 = 0 for profile 11-11'. K n o w l e d g e o f depth after
deflection w d e p e n d s o n a s s u m p t i o n a b o u t initial d e p t h before
deflection, i.e. bathymetry of the margin and Oman abyssal
plain. Discrepancybetween theoretical and observedcurves from
a distance of 100 km on profile 111-I11' may be explained
by an initial depth increase of the bathymetry of 2 km in the
continental-oceanic transition zone.
buckling of Hawasina sediments in front of the overthrust body. At land, platform carbonates are thrust
and deformed in a thick tectonic prism, the 'MuscatMusandam High' (Boote et al., 1990; Rabu et al.,
1993). Consecutive erosion of this relief feed the
first Muti foredeep basin.
(b) During the Campanian-Maastrichtian (80-70
Ma), the evolution of the continental basement ridge
was similar to Saih-Hatat. This structure corresponds
to a continental slab subducted and exhumed fol-
In this paper, we have shown that the Bouguer
anomaly map on the northern Oman mountains is
characterized by a negative-positive couple as in
many collision belts. The large-scale gravity low
over the Zagros-Oman mountains and foreland has
been interpreted in terms of elastic flexure of the
Arabian lithosphere as attested by the Fiqa and Pabdeh foredeep basins. The Late Cretaceous deflection
in the Fiqa foredeep has been explained by the loading effect, during obduction, of the rootless ophiolite
nappes. The Tertiary-to-Recent compressive events
explain the important deflection observed in the Pabdeh basin and Makran trench. We have demonstrated
that deflection in the Pabdeh foredeep is related
to subsurface sedimentary loading, probably in the
Early Tertiary, over the Musandam Peninsula and the
Strait of Hormuz. Detailed studies on this area based
on an accurate foredeep stratigraphy will allow to
restore the compressive history of the MusandamHormuz area.
Gravity and seismic studies have revealed the existence of the Sohar basin and the complexity of the
northern Oman margin geometry. The presence of
a continental basement ridge prolonging the SaihHatat window must be confirmed by complementary
seismic studies.
We have proposed density cross-sections of the
northern Oman margin which are characterized by
a large northward extension of the continental crust.
276
P. Ravaut et al./Tectonophysics 279 (1997) 253-280
Zagros fold belt
a)
Front of Musandam
Tectonic Prism
Oman Mountains
Makran accretionary
prism
Fig. 10. (a) Mesh plot showing the topography and additional Musandam prism loading. (b) Mesh plot showing 3D total deflection of the
Arabian plate related to Tertiary-to-Recent loading and boundary conditions on the subducted plate.
The low value of the Bouguer anomaly in the oceanic
domain and the aerial character of a large part of the
Makran prism indicate that a subducted Arabian
plate over Makran must include a doubled oceanic
lithosphere or a more extended continental crust.
Here also, deep seismic soundings are necessary to
solve this problem.
P. Ravaut et al./Tectonophysics 279 (1997) 253-280
277
A
A
.
230
.
.
.
!
Calculated
I 90
110
7r~
E
30
10
5,0
5
10
15
20
25
,~
30
35
40
45
6~
180
240
300
360
-120
B
290-
210
12 ~
400
B'
-
"~
170-
E
~2,0 -
1030706
12
18
vE
24
30
26
42
48
54
140
2~ c
E
I'~°
1
70
200
260
3::~
300
440
500
C,
m
Lithospheric Mantle
I~
Continental Crust
~0
Oceanic Crust
Ophiolites
E
[~
Cretaceous Sediments
Tertiary to recent
Sediments
Water
Oceanic or lower
continental crust
Fig. 11. Density models along three profiles crossing the northern Oman region and the Gulf of Oman (see location of profiles in Figs. 2
and 7a). The observed anomaly has been corrected for sediment effect. Geometry of ophiolites massifs is taken from models of Fig. 7c.
Density contrast at continental and oceanic Moho discontinuity is chosen as 400 kg/m 3.
P Ravaut et al./Tectonophysics 279 (1997) 253 280
278
Acknowledgements
where ~ = (~, ~))T c I R 5 stands for associated virtual curvatures
and virtual shear strains:
This work was made possible with the support
of the Ministry of Petroleum and Minerals of Oman
and we are very grateful to M. Mohammed Kassim, director general of Minerals and M. Hilal A1
Azri, deputy director general of Minerals. We also
thank Petroleum Development Oman for the release
of gravity data, and TOTAL S.A for the access to
CEPM data. BGI helped us to control the validity of offshore gravity measurements. R. Buck, M.
Diament, R.G. Coleman, E Boudier, R. Russo and
E Lucazeau are kindly acknowledged for their critics and constructive reviews of the manuscript. D.
Carbon is particularly thanked for his collaboration.
Appendix
(r = IM, Q)T = ((M~, M,., M,,.). (Qr, Qv)] T
and is related to e through the constitutive relationships cr = Ds,
where for an isotropic elastic material:
o=i: ,¸ ol witho,- 12,,h3_ li l 1_oO]
hE: :]
A. 1. Mindlin plate equilibrium equations
Let A be the domain of interest:
A = {(x. y , z ) 6 1 R 3 / ( x , y) 6 f2 C I R 2, z 6 [ - h / 2 , h/2]}
where h = h(x. y) is the plate thickness and let u 6 I R 3 be the
generalized displacements
vector u = (w, 0) T = (w, G , 0,.) 7
where w is the lateral displacement normal to the f2-plane and
G, 0,. the normal rotations in the x z and yz-planes. The body
forces are designed by b and here, the only non-zero component
of b is the lateral one:
Apgwe,
e=(l,0,0)
T
where F is the lateral loading per unit area and - A p g w is the
restoring pressure associated with a depression w of the plate in
the underlying fluid.
For any virtual generalized displacement fi = (t~. O)T the
principle of virtual work can be expressed by:
fo U' dn+fcamwi'Tedn=fc rfiredn
0
2
and D, = - -
A. Finite elements formulation
Instead of a fully 3D theory, where (and without special
procedure) the stiffness matrix can be ill-conditioned for small
thickness, we use the Mindlin (or Reissner) plate theory which is
closer to the 3D case (and so more accurate) than the classical
Kirchhoff thin plate, since transverse shear strains are taken into
account.
We briefly recall the expression of the virtual work principle
which is a variational form of the local equilibrium equations
more suitable for finite elements implementation. For more details, see for example Owen and Hinton (1980) (the slight difference in our work is the presence of an additional term in the
stiffness matrix for taking into account the hydrostatic pressure
acting on the bottom of the plate).
b= Fe-
c~ c I R 5 is the vector of bending moments and shear forces:
v is the Poisson ratio, E the young modulus, G the shear modulus
and ~ a shear correction term (o~ = 1.2).
A.2. Finite elements implementation
In a finite elements representation, the displacements and
strains may be expressed by:
nix, y) -- Z
N,(x,y)ui
s(x,y)=ZBi(x,y)ui
t
i
where ui is the vector of nodal displacements, Ni the shape functions and Bi the strain-displacement matrix (containing shape
functions and their derivatives).
Writing in a similar way the virtual displacements and strains,
and noting that the virtual work is true for any virtual displacement t~i, we have for each node i:
£S/od~+£a,g,~,N,ed~:£FN,ed~
Introducing the constitutive law we can write for each element f2 e of the mesh:
m[
K eU I l j : -
F<
i
1
nd
j=l,nd
where nd is the number of nodes in the element f2e and K eli are
given by:
KI,j =
/o
~ (Bid D ~B j + A p g N i N j e ® e ) d ~
and
F~ = f~2,, FNied~2
The K'/-values form the local stiffness matrix K e whose
J
dimensions are 3nd x 3nd. The global stiffness matrix and the
P. Ravaut et al. / Tectonophysics 2 79 (1997) 253-280
global loading vector are then obtained by assembling, respectively, all submatrix K ~ and all vectors F e.
Elements used must be at least quadratic to obtain accurate
solutions. Good results are performed with 6-node triangular elements or 9-node quadrangular elements. In addition, for better
resolution we use a pre-processor associated with a mesh generator which performs a local mesh refinement such that element
sizes are adjusted taking into account the loading distribution
and/or its gradient.
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