[GRL]
Supporting Information for
The Formation of Graben Morphology in the Dead Sea Fault, and its
Implications
𝐙𝐯𝐢 𝐁𝐞𝐧 − 𝐀𝐯𝐫𝐚𝐡𝐚𝐦𝟏,𝟐 , 𝐑𝐞𝐠𝐢𝐧𝐚 𝐊𝐚𝐭𝐬𝐦𝐚𝐧𝟐,∗
1
Department of Earth Sciences, Tel Aviv University, P.O. Box 39040, Tel Aviv
6997801, Israel
2
Dr. Moses Strauss Department of Marine Geosciences, University of Haifa, 199 Aba
Khoushy Ave, Haifa 3498838, Israel, reginak@research.haifa.ac.il
Contents of this file
Supporting Text S1 : Methods, References
Supporting Table S1
1
1. Methods
1.2 Modeling Setup
Our geometry presents the settings at the Lower Jordan Valley at the DSF region (Figure
1a). A contribution of the shear motion at the transform fault to the developed morphology
is considered to be negligible [e.g. Garfunkel, 1997; Wdowinski and Zilberman, 1996;
Matmon et al., 2014]. Therefore, the geometry is modeled in 2D by rectangular segment of
300 km length and 150 km height (Figure 2a). Horizontal layers of the uniform thickness
present a 2 km thick pre-rift sedimentary layer, 18 km thick upper crustal layer of wet
quartzite rheology, 15 km thick lower crust of plagioclase- dominated rheology, and a 45
km thick strong mantle of dry olivine rheology. Considering the higher water content of the
asthenosphere, rheological parameters of wet (i.e. 500 ppm H/Si) olivine are used in the
lowermost layer of the weak mantle from 80 km to 150 km depth [Brune et al., 2012]. A
small circular weak defect is incorporated in the bottom part of the upper crust (Figure 2a).
1.3 Model Formulation
The problem is solved using thermo-mechanical modeling approach [Brune et al., 2012;
Regenauer-Lieb et al., 2006]. Arbitrary Lagrangian-Eulerian (ALE), implicit FEM model is
designed to solve the system of the following non-linear coupled conservation equations:
(a) Momentum equation:
−∇ ∙ 𝜎 = 𝜌𝒈
(b) Energy equation:
𝜌𝑐𝑝 𝐷𝑡 − ∇ ∙ (𝑘∇𝑇) − 𝜏𝑖𝑗 𝜀𝑖𝑗̇ − 𝜌𝐴 = 0
(c) Continuity equation:
𝐷𝑇
𝜕𝜌
𝜕𝑡
+ ∇ ∙ (𝜌𝒖) = 0
(1)
(2)
(3)
2
is Cauchy stress tensor, 𝜌 is material density, g is gravity acceleration, T is
where
temperature, 𝑐𝑝 is heat capacity at constant pressure, 𝑘 is thermal conductivity, 𝜏𝑖𝑗 is the
deviatoric stress tensor component, 𝜀𝑖𝑗̇ is strain rate tensor component, 𝐴 is radioactive heat
production, 𝐷/𝐷𝑡 is material time derivative, 𝒖 is the local velocity vector.
The deviatoric stress tensor component, 𝜏𝑖𝑗 , is defined by:
𝜏𝑖𝑗 = 𝜎𝑖𝑗 + 𝑝𝛿𝑖𝑗
(4)
1
Where 𝑝 = − 𝑡𝑟𝑎𝑐𝑒(𝜎𝑖𝑗 ) is the trace of the Cauchy stress tensor, or the pressure.
3
The strain rate tensor is defined in terms of the velocity gradient, assuming a small strains
approximation:
1
𝜀𝑖𝑗̇ = 2 (∇𝒖 + ∇𝒖𝑇 )
(5)
The total strain rate is decomposed onto the elastic (e), viscous (v), plastic (pl), and thermal
expansion (th) components:
𝜀̇𝑖𝑗 = 𝜀̇𝑖𝑗 (𝑒) + 𝜀̇𝑖𝑗 (𝑣) + 𝜀̇𝑖𝑗 (𝑝𝑙) + 𝜀̇𝑖𝑗 (𝑡ℎ)
(6)
Elastic flow rule in Eq.6 is presented by:
1+𝜈 𝐷𝜏̃
𝑖𝑗
𝜀̇𝑖𝑗 (𝑒) = (
where
𝐷𝜏̃
𝑖𝑗
𝐷𝑡
𝐸
𝐷𝑡
𝜈 𝐷𝑝
+ 𝐸 𝐷𝑡 )
is Jaumann objective derivative of the deviatoric stress tensor.
The viscous creep strain rate in Eq.6 is modeled by the dislocation creep law with power
law stress-dependent viscosity:
𝑛−1
𝜀̇𝑖𝑗 (𝑣) = 𝐵𝜏𝐼𝐼
exp(−
𝑄+𝑝𝑉
𝑅𝑇
)𝜏𝑖𝑗
(7)
3
where 𝜏𝑖𝑗 is the deviatoric stress tensor component, 𝜏𝐼𝐼 is effective differential stress,
(𝜏𝐼𝐼 = (3𝐽2 )0.5 = (3/2𝜏𝑖𝑗 𝜏𝑖𝑗 )0.5), 𝐽2 is the second invariant of deviatoric stress tensor, Q is
activation energy, T is temperature, B is a pre-exponentional constant, p is pressure, V is
activation volume, R is gas constant, n is a power constant. Presence of volumetric
deformations evokes the following correction for material density:
𝜌 = 𝜌0 [1 − 𝛼𝜃𝑒𝑞 + 𝑝/𝐾]
(8)
where 𝜌0 is density at reference temperature 𝑇0 and zero pressure, K is bulk modulus, 𝛼 is
isotropic coefficient of thermal expansion, 𝜃𝑒𝑞 the equilibrium temperature change (𝜃𝑒𝑞 =
𝑇 − 𝑇0 ) of adiabatic expansion or contraction.
Plastic strain rate in Eq.6 is modeled by:
𝜕𝑄
𝜀̇𝑖𝑗 (𝑝𝑙) = 𝜆̇ 𝜕𝜎
(9)
𝑖𝑗
Where 𝜆̇ is the plastic multiplier and Q the plastic flow potential. The yield function (F) is
presented by the von-Mises yield criteria [Kaus and Podladchikov, 2006]:
𝐹 = 𝜏𝐼𝐼 − 𝜎𝑦𝑠 (𝜀𝑒𝑝 )
(10)
where 𝜎𝑦𝑠 (𝜀𝑒𝑝 ) is yield stress with linear isotropic softening. Effective plastic strain, 𝜀𝑒𝑝 , at
any time t accumulated since the beginning of the deformations is defined as:
𝑡
2
𝜀𝑒𝑝 = ∫0 𝜀̇𝑒𝑝 𝑑𝑡, when 𝜀̇𝑒𝑝 = √3 𝜀̇(𝑝𝑙) ∙ 𝜀̇(𝑝𝑙)
(11)
Isotropic thermal stress-free strain rate in Eq.6 is modeled by:
𝜀̇𝑖𝑗 (𝑡ℎ) = 𝛼
𝐷𝜃𝑒𝑞
𝐷𝑡
𝛿𝑖𝑗
(12)
The model has been designed in Comsol Multiphysics simulation environment and is
available in 2D and 3D formulations.
4
1.4 Initial and Boundary Conditions
Initial temperature distribution is laterally uniform and corresponds to a steady-state
continental geotherm [Turcotte and Schubert, 2002] with surface temperature T=300K and
with surface heat flow of 38𝑚𝑊/𝑚2 [Ben-Avraham et al., 1978]. It is calculated from zero
to 100 km depth (Figure 2a), reflecting the settings, where the thermal lithosphereasthenosphere boundary is located at 20 km below the chemical lithosphere-asthenosphere
boundary (at 80 km) [Brune et al., 2012]. Temperature at 100 km depth (T=1600K) is
extrapolated constantly to the bottom boundary of the model (Figure 2a). During the
simulations a constant temperature is prescribed for the top and bottom boundaries of the
model, and with zero heat flux conditions adopted for the lateral boundaries.
The model has a free top surface, and free-slip boundary condition at the left and bottom
boundaries of the setup (Figure 2a). One-sided extension is modeled by a constant
horizontal velocity applied for the right vertical boundary of the setup. To model a rift
generation, a small defect with low yield strength is incorporated in the bottom part of the
upper crust (Figure 2a). Initial localized faults originated some time after the extension
started.
1.5 Input Data
Input data is presented in Supplementary Table 1. Most of the parameters are adopted from
[Brune et al., 2012] (see also the references therein), while the rheological data for the
5
viscous strain rate for the 'normal' lower crust are adopted from [Petrunin and Sobolev,
2008]. Initial strength of the strong crust is prescribed as 𝜎𝑦𝑠0 = 0.6𝐺𝑃𝑎 (Eq.10), within a
range suggested by [Kaus and Podladchikov, 2006]. Strong strain softening is modeled by
decreasing the yield strength, 𝜎𝑦𝑠 , linearly in the isotropic softening regime. When the
effective plastic strain reaches 𝜀𝑒𝑝 = 0.8, 𝜎𝑦𝑠 drops to 10% of the initial value. At 𝜀𝑒𝑝 >
0.8, 𝜎𝑦𝑠 remains constant. Syn-rift sediments with real sedimentation regimes are deposited
within the rift valley. Specifically, evaporites are deposited with 2000 m/Ma rate
[Garfunkel, 1997] between ca. 3.85 Ma and 3.4 Ma, while the post-evaporitic sediments are
deposited with 300 m/Ma rate [Garfunkel, 1997; Inbar, 2012] at the later times. Slight
eastward gradient in the sediments deposition is prescribed to reduce the slope of the
sediment free surface within the rift and to keep it closer to the horizontal direction.
Rheological parameters for the sediments and also for the evaporites are prescribed
identical to those in the upper crust (we don’t aim to study the flow behavior of evaporites).
However, density of the pre-rift Hatzeva sediments is taken as 2390 𝑘𝑔 ∙ 𝑚−3, of the synrift evaporits as 2150 𝑘𝑔 ∙ 𝑚−3, and of the post-evaporitic sediments as 2280 𝑘𝑔 ∙ 𝑚−3, in
agreement with [Ben-Avraham et al., 2008].
6
2. References
Ben-Avraham, Z., Z. Garfunkel, and M. Lazar (2008), Geology and evolution of the
southern Dead Sea fault with emphasis on subsurface structure, Annu. Rev. Earth
Planet. Sci., 36, 357-387.
Ben-Avraham, Z., R. Haenel, and H. Villinger (1978), Heat flow through the Dead Sea
rift, Marine Geol., 28, 253-269.
Brune, S., A. A. Popov, and S. V. Sobolev (2012), Modeling suggests that oblique
extension facilitates rifting and continental break-up, J. Geophys. Res., 117, B08402.
Inbar, N. (2012), The Evaporitic Subsurface Body of Kinnarot Basin: Stratigraphy, Structure,
Geohydrology, PhD thesis, Tel-Aviv University, Tel-Aviv.
Garfunkel, Z. (1997), The history and formation of the Dead Sea basin, in The Dead Seathe lake and its setting, Oxford Monographs of Geol. and Geophys., vol. 36, edited by
T.M. Niemi et al., pp.36–56.
Kaus, B.J.P., and Y.Y. Podladchikov (2006), Initiation of localized shear zones in
viscoelastoplastic rocks, J. Geophys. Res., 111, B04412.
Matmon, A., D. Fink, M. Davis, S. Niedermann, D. Roo, and A. Frumkin (2014),
Unraveling rift margin evolution and escarpment development ages along the Dead Sea
fault using cosmogenic burial ages, Quaternary Research, 82, 281–295.
Petrunin, A.G., and S.V. Sobolev (2008), Three-dimensional numerical models of the
evolution of pull-apart basins, Phys. Earth Planet. Inter., 171, 387–399.
Regenauer-Lieb, K., R. F. Weinberg, and G. Rosenbaum (2006), The effect of energy
feedbacks on continental strength, Nature, 442(6), 67-70.
7
Turcotte, D. L., and G. Schubert (2002), Geodynamics, New York, ed. 2, Cambridge Univ.
Press.
Wdowinski, S., and E. Zilberman (1996), Kinematic modeling of large-scale structural
asymmetry across the Dead Sea Rift, Tectonophysics, 266, 187–201.
8
Parameter
Upper
Lower
crust
crust
Density, 𝜌, 𝑘𝑔 ∙ 𝑚−3
2700
2850
3300
3300
Poisson ratio
0.25
0.25
0.25
0.25
Young's modulus, 𝐸, 𝐺𝑃𝑎
76.2
100
175
175
0.6
0.6
-
90%
90%
90%
-15.4
-15.56
-15.05
3
3.5
3.5
356
530
480
0
13
10
2.7
3
3
Von Mises initial strength, 0.6
Strong mantle
Weak
mantle
𝐺𝑃𝑎
Maximum plastic strength 90%
softening
Pre-exponential
for
constant -28
dislocation
creep,
log(𝐵), 𝑃𝑎−𝑛 ∙ 𝑠 −1
Power law exponent for 4.0
dislocation creep, n
Activation
energy
dislocation
for 223
creep,
𝑄, 𝑘𝐽 𝑚𝑜𝑙 −1
Activation
volume
dislocation
for 0
creep,
𝑉, 10−6 𝑚3 𝑚𝑜𝑙 −1
Coefficient
of
thermal 2.7
expansion, 𝛼, 10−5 𝐾 −1
9
Radiogenic
heat 1.3
0.2
0
0
1200
1200
1200
2.5
3.3
3.3
production, 𝐴, 10−6 𝑊𝑚−3
Isobaric
heat
capacity, 1200
𝑐𝑝 , 𝐽 𝑘𝑔−1 𝐾 −1
Thermal
conductivity, 2.5
𝑘, 𝑊𝐾 −1 𝑚−1
Table S1. Input data
10