Bollettino di Geofisica Teorica ed Applicata Vol. 57, n. 3, pp. 233-246; September 2016
DOI 10.4430/bgta0178
Numerical modelling of gravitational sinking of anhydrite
stringers in salt (at rest)
S. Li1 and J. Urai2
1
2
College of Petroleum Engineering, China University of Petroleum, Beijing, China
Structural Geology, Tectonics and Geomechanics, RWTH Aachen University, Germany
(Received: April 8, 2015; accepted: June 15, 2016)
ABSTRACT
A large number of salt bodies contain layers of anhydrite material which is generally
referred to as “stringers”. The movement and deformation of embedded anhydrite
bodies are processes which are not yet fully understood. It is observed that stringers
tend to sink towards the bottom of salt bodies at velocities highly dependent on the
mechanical properties of both salt and anhydrites, with given density contrast between
salt and denser anhydrites. The rheological differences between salt and the embedded
anhydrites are a major issue, contributing to the complexity of the problem. On a
geological timescale, the salt behaves as a Newtonian or a power-law fluid. The
anhydrite stringers present elastic or brittle properties under certain conditions. Finite
Element Modelling (FEM) has been employed in this study by using the FEM package
ABAQUS (SIMULIA, Dassault Systems) in order to numerically simulate the sinking
of an anhydrite stringer embedded in the salt. Furthermore, numerical modelling of
isolated anhydrite stringers in salt at rest is compared with observations of stringers in
seismic data. FEM simulation of the anhydrite stringer sinking and the gravitational
sinking of anhydrite blocks embedded in the salt will be studied and demonstrated with
two different methods of rheology, respectively. The study results indicate that sinking
velocity is closely related to several factors, including the viscosity, the thickness of the
stringer, as well as the density contrast between stringer and salt for a given viscosity.
The results also prove that anhydrite stringer fragments do not sink significantly over
the geological timescale if the halite is deformed by non-Newtonian viscosity. But,
when Newtonian viscosity is dominant, the fragments are likely to sink hundreds
of metres through the Zechstein salt during a few Ma. In conclusion, the modelling
of the sinking of anhydrite or anhydrite inclusions provides an important scope for
understanding the movement and deformation of embedded stringers.
Key words: stringer, rheology, Newtonian, power-law, sinking velocity, FEM.
1. Introduction
Anhydrite body inclusions are common in salt diapirs. However, the movement and deformation
of those embedded anhydrite or anhydrite bodies are not yet fully understood. Evaporites containing
thick layers of anhydrite (“stringers”) have been explored by many researchers (Weinberg, 1993;
van der Bogert, 1997, 1998; Chemia and Koyi, 2008; Chemia et al., 2008, 2009). These studies
© 2016 – OGS
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Li and Urai
demonstrate that, due to the density contrast between salt and denser anhydrite or anhydrite,
the stringers tend to move downwards in the salt at velocities which are highly dependent on
the mechanical properties of the salt and the geometrical properties of the embedded stringers.
In contrast to the Newtonian or power-law fluid behavior of the halite, the stringers can exhibit
elastic to brittle behavior or fold and flow at much higher viscosity than the surrounding salt. In
these studies, the rise and fall of a dense layer in salt diapirs is modelled in a numerical way, and
the upward transport of inclusions in Newtonian and power-law salt diapirs is also explored. Some
reports focus on the in situ stress model of the stringer in salt. Physical models were used to study
the deformation of intra-stringers, which is related to halokinesis and inversion tectonics (Eleman,
1997). In addition, some research on the deformation of salt diapirs has been published. Examples
include the following: mechanics of active salt diapirism (Schultz-Ela and Jackson, 1993), a
numerical model set up for the initiation of salt diapirs with frictional overburdens (Podladchikov
et al., 1993; Poliakov et al., 1993), the modelling of the salt flow by overburden (Koyi, 1996),
the shaping of the salt diapirs explored by Koyi (1998), the salt minibasins investigated by Ings
and Beaumont (2010), the effective viscosity of rocksalt discussed by van Keken et al. (1993),
whose research is also concerned with the program of radioactive waste management. Physical
models (Koyi, 2001) and numerical models (Chemia et al., 2008, 2009) were applied to study the
whole process in which anhydrite blocks are entrained by salt and then descend as a whole within
the structure. Numerical models were used to quantify the descent rate of entrained anhydrite
blocks within a salt diapir (Koyi, 2001). Other related issues were also systematically studied,
such as the effects of viscosity [Newtonian and nonlinear: Chemia et al. (2008)], position of the
anhydrite layer (Chemia and Koyi, 2008), the different rising rate of salt diapirs in connection
with the inclusion of anhydrite layers/blocks and their descent within a salt structure (Chemia et
al., 2009), the effect of size/aspect ratio and orientation of denser blocks on sinking rate and mode
(Burchardt et al., 2011, 2012a, 2012b).
In this paper, the sinking of stringers or stringer fragments in salt blocks or domes is discussed.
The researchers investigate how the sinking velocity of stringers is influenced by material
conditions and geometrical properties through numerical simulation. For the validation of the
numerical modelling approach, two sets of simulations have been applied. First in Li et al. (2008)
and Li (2013), relations between strain rate and stress obtained from simulation and experimental
measurement were compared with each other. The relation between strain rate and stress can be
obtained by either simulation or experimental measurement. For the former method, Finite Element
Modelling (FEM) is used to simulate the cylindrical salt body under uniaxial compression. As for
the latter method, the relation between strain rate and stress is obtained by varying and measuring
rheological parameters (Urai and Spiers, 2007).
Second, researchers tested a simulation of the flow of a power-law fluid between two flat plates.
The numerical solution for the flow velocity is compared to the analytical solution of Turcotte and
Schubert (1982). Finally, comparison of simulated power-law creep with a theoretical result was
completed in Li (2013). In the study, a sinking velocity formula [i.e., Stokes law: Lamb (1994)]
was used to evaluate the numerical results in the model in which the velocity of a ball sinking in
a steady-state Newtonian fluid (n=1) was simulated.
In recent years, the deformation mechanisms and rheology of rock salt have been investigated
in a large number of studies, including laboratory experiments and microstructural investigations
(Carter and Hansen, 1983; Heard and Ryerson, 1986; Urai et al., 1986; Wawersik and Zeuch,
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Numerical modelling of gravitational sinking of anhydrite stringers in salt Boll. Geof. Teor. Appl., 57, 233-246
1986; Aubertin et al., 1989; Senseny et al., 1992; Carter et al., 1993; Peach and Spiers, 1996;
Weidinger et al., 1997; Cristescu and Hunsche, 1998; Spiers and Carter, 1998; Hunsche and
Hampel, 1999; Martin et al., 1999; Ter Heege et al., 2005a, 2005b; Urai and Spiers, 2007; Urai
et al., 2008). During dynamic recrystallization at geological strain rate, salt is in the transition
regime where both dislocation creep, related to Newtonian flow, and solution-precipitation creep,
related to non-Newtonian flow, operate (Urai et al., 2008). If the dislocation creep and pressure
solution act in parallel, the steady-state flow of a rock salt can be expressed as the summation of
the strain rate from dislocation creep processes and the strain rate from pressure solution (Urai et
al., 2008). In power-law creep, the strain rate is related to the flow stress using the equation:
(1)
.
where ε is the strain rate, Δσ=σ1-σ3 is the differential stress, and A=A0 exp(-Q/RT) is the viscosity
of the salt. Within the viscosity described above, A0 is a material-dependent parameter, Q is the
specific activation energy, while R is the gas constant (R=8.314 Jmol-1k-1) and T is the temperature.
The other mechanism is solution-precipitation creep:
.
(2)
The strain rate is dependent on the strain size D, Δσ=σ1-σ3 is the differential stress, and
B=B0 exp(-Q/RT)(1/TDm) is the viscosity of the salt. Within the viscosity described above, B0 is
a material-dependent parameter, Q is the specific activation energy, while R is the gas constant
(R=8.314 J/mol-1 k-1), and T is the temperature. The order m influences strain rate dependent on
the grain size.
Fig. 1 summarizes low-temperature laboratory data for a wide range of halite based on
experiments by Sandia, BGR, Utrecht University, and other laboratories. The dashed line
represents an extrapolation of the dislocation creep with n=5 (30-50 °C). The solid lines are roomtemperature solution-precipitation creep laws for different grain sizes. During active salt tectonics
(differential stress in the order of a few MPa as shown by subgrain size piezometry), the two kinds
of rheology have similar strain rates, and both are interpreted to be important in coarse-grained
diapiric salt (Schléder and Urai, 2005). In this study, numerical experiments were designed to be
sensitive to salt rheology, geometry, and density of stringer or fragment. The purpose and interest
of the research is to investigate the sinking velocity of stringers as influenced by salt rheology,
density contrast, and geometrical properties. In recent years, Koyi (2001) studied the effect of
thickness on descending rate. The experimental results indicated that thicker blocks sink faster in
the Newtonian salt. Chemia et al. (2008, 2009) studied the effect of differences in the rheology of
salt. The same geometry as that of the Gorleben salt diapir and the anhydrite layers was simulated
to study the interaction between these two rock units. These studies investigated the effect of
including different salt rheology within the same model in order to simulate the different subgroups
of Zechstein formation. In our study, we changed the value of thickness, density contrasts, and
the rheology of the salt within which the embedded stringer descends. A generic model and a case
from Zechstein salt in the northern Netherlands were chosen. Cases obtained from seismic data
tend to produce results which later can be compared with observations in real fields. In this way,
more insights are likely to be provided for further studies. Different from Chemia et al. (2008),
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Li and Urai
Fig. 1 - Differential stressstrain rate diagram for rock
salt (modified after Schléder
and Urai, 2005).
the salt tectonics in our models is not active and the stringer sinks after salt tectonics. The study
is similar to the research of Burchardt et al. (2011, 2012a, 2012b), who used extensive numerical
modelling to study sinking anhydrite blocks within a Newtonian salt body. In our model, the
stringer is brittle rather than viscous, and salt flow presents the property of both Newtonian and
non-Newtonian fluid. In this way, the stringers tend to sink due to gravitational loading and the
density contrast between itself and surrounding salt. The relation between sinking velocity and
material or geometrical properties can be clearly observed.
2. Methods and models
2.1. Gravitational sinking of a single stringer in salt block (a generic model)
As for the simplest, generic model for the sinking of an anhydrite stringer embedded in a
salt body, we have chosen a setup consisting of a single rectangular stringer inside a rectangular
salt block (Fig. 2). The geometry of the model is described by the width and height of the salt
body, which are respectively set to be 8 and 4 km in all of the following models; h stands for the
thickness of the stringer, w is the width, and D is the initial depth, which is measured from the
top of the stringer. The model consists of a salt block, the geometry of which is 8 km wide and 4
km high.
For the case of Newtonian flow, the rheology is presented with A0=4.70×10-10 Pa-1s-1, Q=24530
J/mol, m=3, n=1, D=0.01 m, R=8.314 J/(mol·k) (Spiers et al., 1990). For the case of non-Newtonian
flow, the rheology is presented with A0=1.82×10-39 Pa-5s-1, Q=32400 J/mol, n=5, R=8.314 J/(mol·k)
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Numerical modelling of gravitational sinking of anhydrite stringers in salt Boll. Geof. Teor. Appl., 57, 233-246
Fig. 2 - Sinking of a stringer embedded in a rectangular salt block.
(Schoenherr et al., 2007). The surface temperature is set to be 50 oC and, at the depth of 4 km, the
temperature is set to be 140 oC (i.e., the lower limit of the model). 2D plane strain conditions are
applied. Using this model, we observed that the sinking velocity of the central point in the stringer
depends on the following factors: a) thickness and aspect ratio of the stringer, b) the density
contrast between the stringer and the surrounding salt.
2.2. Gravitational sinking of 30-to-80 m-thick Zechstein intra-salt stringers
The concept is based on seismic observations from the Zechstein salt (van Gent et al., 2011;
Strozyk et al., 2012). During salt tectonics, many Zechstein intra-salt stringer fragments appear
due to breakage, the thickness of which ranges from 30 to 80 m. The fragments are located in the
upper part of the salt section. The width of the fragments ranges from 100 to 1000 m (Fig. 3). The
Mesozoic is the main phase of salt tectonics and tectonics stopped from the Paleogene (Geluk,
2000, 2005). In our study, adaptive remeshing techniques in FEM were applied (Li et al., 2012a,
2012b). The models calculate the sinking velocity and trajectory of fragment geometries for the
two different kinds of salt rheology after salt tectonics stopped. The calculation results can be
very useful. On the one hand, we can evaluate the relation between sinking velocity of stringer
fragments and viscosity for the two different kinds of salt rheology. On the other hand, we can
also evaluate the relation between sinking velocity and the geometrical properties of the stringer
fragment. Besides, by comparing the modelling results with the observations of seismic data, salt
rheology can be further studied to facilitate other research (Li et al., 2012b).
Modelling and calculation involve two major steps (Li et al., 2012b). First, a vertical
displacement of the top salt boundary should be applied to simulate the down-building of
overburden sediments and formation of a salt pillow (Li et al., 2012a). Second, gravitational
loading is applied to the salt section after tectonics and the sinking velocity and trajectory of
stringer fragments are surveyed.
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Li and Urai
Fig. 3 - 3D seismic data (left) and 2D seismic profile (right) with interpreted, physically isolated Zechstein 3 stringer
fragments embedded in the deformed salt sections, northern Netherlands (by courtesy of NAM).
Two models with different dominant salt rheology are applied: model A is performed with
pressure-solution creep (n=1), and model B is performed with dislocation creep (n=5; Table 1).
In addition, the geometry and configuration of stringer fragments in the two models are also
different. We compared the sinking velocity of fragments in two models which have different salt
deformation mechanisms. Table 1 shows the rheological and mechanical properties of both the
salt layer and the stringer fragments used for the two models.
Fig. 4 - Model width is 18 km and initial model thickness 1 km. A: initial model setup before salt tectonics without
differential stresses.
Table 1 - Material parameters for anhydrite and rock salt [following the two different deformation mechanisms at
50° C (i.e., lower limit of the model)].
Salt A: pressure-
solution creep
(Newtonian flow)
Salt B: dislocation creep
(Non-Newtonian flow)
Anhydrite blocks
A0
4.70×10-10 Pa-1 s-1
1.802×10-39 Pa-5 s-1
-
Q [J/mol]
24530
53920
-
R [J/(mol·k)]
8.314
8.314
-
m,n
3/1
0/ 5
-
ρ/density [kg/m ]
2200
2200
2900
E/Youngs Modulus [GPa]
10
10
40
υ/ Poisson’s ratio
0.4
0.4
0.3
Reference
Spiers et al., 1990
Hunsche and Hampel, 1999
Sayers, 2008
238
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Numerical modelling of gravitational sinking of anhydrite stringers in salt Boll. Geof. Teor. Appl., 57, 233-246
A simplified model of the Zechstein salt section (Fig. 7) was applied. This model included
seven stringer fragments which are elongated, broken, and isolated. The width of each fragment
is chosen from 100, 600, and 1000 m, while thickness is chosen from 30, 60, and 80 m. All
fragments are embedded in a rectangular salt section 1 km thick and 18 m long, which is a plane
strain model based on seismic observations from van Gent et al. (2011). The boundary condition
for the model is displacement equals zero when it is perpendicular to the bottom and both sides
of the model. The top of the salt surface is fixed in x and y direction after salt tectonics lasting
for 4 Ma. When salt tectonics stops, a model with a time span of 60 Ma is performed to simulate
the Zechstein salt at rest in the Tertiary. Table 2 is a summarization which presents the geometric
parameters of stringer fragments and salt section as well as the time duration of each experimental
phase.
Table 2 - Geometrical and temporal model setup.
Width of salt body
18000 m
Height of salt body 1000 m
Stringer fragment thicknesses
30, 60, 80 m
Stringer fragment widths
100, 600, 1000 m
Amplitude down-building
300 m
Duration salt tectonics
4 Ma
Duration salt at rest
60 Ma
3. Results
3.1. Gravitational sinking of a simple stringer in salt block
3.1.1. Stringer thickness
First we investigate the influence of the variations in the thickness, and therefore mass, of the
stringer. In this part, the width of the stringer w is a constant value and the height of the stringer
h is various. The boundary conditions are zero displacement perpendicular to the boundary at the
bottom and at the sides. The material properties of the stringer are Young’s modulus E=40 GPa,
Poisson ratio υ=0.3, and density ρ=2400 kg/m3 (Sayers, 2008). This geostatic state is used as the
initial stress condition for the next step.
Fig. 5 - Sinking velocity of the stringer vs. stringer thickness. The variation in stringer thickness (140, 160, 180, 200,
220, and 240 m) results in aspect ratios of 8.6, 7.5, 6.7, 6.0, 5.4, and 5.0 respectively due to the constant length of the
stringer (1200 m). a) The salt rheology is Newtonian (n=1); b) the salt rheology is non-Newtonian (n=5).
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Li and Urai
Fig. 5 illustrates the relation between the sinking velocity and the thickness of the stringer,
separately for n=1 and n=5. The stringer thickness h is 140, 160, 180, 200, 220, and 240 m, and
therefore the aspect ratio is 8.6, 7.5, 6.7, 6.0, 5.4, and 5.0, respectively. The variation of the size,
and thereby the mass, of the stringer influences the differential stress Δσ linearly. The sinking
velocity of the stringer are determined by the strain rate on the differential stress when n=1 and
5th order relation when n=5. The displacement and velocity of the stringer have the same trend as
the strain rate. Moreover, from the result, we can know that the stringer with the same size in both
n=1 and n=5 rheological salt has similar differential stress Δσ distribution around it because of
the same mass, and the peak differential stress around the stringer changes from 100 to 150 kPa
with various stringer size.
3.1.2. Density contrast
The second parameter influencing the sinking velocity we discuss is the density contrast
between the salt and the stringer. A single stringer 1200 m wide and 160 m thick is located in the
salt. The material properties of the stringer are Young’s modulus E=40 GPa, Poisson ratio υ=0.3,
and density ρ=2400 kg/m3. The salt density is 2040 kg/m3, and the stringer density is 2200, 2300,
2400, 2500, 2600, 2700, and 2800 kg/m3, respectively. The initial location of the stringer is in the
centre of the salt block.
Fig. 6 - Sinking velocity of the stringer vs. stringer density. The variation of stringer density is 2200, 2300, 2400, 2500,
2600, 2700, and 2800 kg/m3. a) The salt rheology is Newtonian (n=1); b) the salt rheology is non-Newtonian (n=5).
The results of the simulations confirm the expectations. The numerical solution shows that for
Newtonian fluid flow (n=1), the sinking velocity of the stringer in the salt follows a linear relation
depending on the density contrast (Fig. 6a). For non-Newtonian fluid flow (n=5), the sinking
velocity of the stringer follow a nonlinear relation with 5th power order (Fig. 6b). The variation of
the density influences the differential stress Δσ in the salt linearly. The strain rate varies from the
density contrast due to the relation έ=A(Δσ)n and έ=B(Δσ).
3.2. Gravitational sinking of 30-to-80 m-thick Zechstein 2 intra-salt stringers
Figs. 7a and 8a show the initial model setup with the stringer fragments Nos. 1-7 in salt section.
Figs. 7b and 8b show that the stringer fragments rotate during salt tectonics when Newtonian law
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Fig. 7 - Results of Model A
using salt rheology of n=1.
Model width is 18 km and initial model thickness 1 km: a)
initial model setup before salt
tectonics without differential
stresses; b) Model A during salt
tectonics (-62 Ma); c-g) Model
A after salt tectonics (-60, -40,
-20 Ma, and today).
Fig. 8 - Results of Model B
(right) using salt rheologies of
n=5. Model width is 18 km and
initial model thickness 1 km: a)
initial model setup before salt
tectonics without differential
stresses; b) Model B during salt
tectonics (-62 Ma); c-g) Model
B after salt tectonics (-60, -40,
-20 Ma, and today).
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Li and Urai
dominates, while the fragments in the salt with n=5 (Model B; Figs. 8b and 8c) almost do not
sink. Furthermore, we detect average sinking velocity and differential stresses around the stringer
fragments in both Model A and Model B, depending strongly on the geometry and location of the
fragments in the salt body (Table 3). Finally, the bending deformation of stringer fragments can
be clearly seen in Model A during the sinking process and the bending deformation also increases
the internal stress of the stringer fragment.
From Table 3, we can figure out the relation between the sinking velocity and the thickness
of the stringer, both for Model A (n=1) and Model B (n=5). Here average sinking velocity in the
central point of a stringer fragment is investigated because of rotation or deformation. And the
sinking velocity of a stringer fragment will decrease when it gets close to the bottom of the salt
section.
The variation of the size and mass of the stringer linearly influences the differential stress,
and the velocity of the stringer is determined by the strain rate, depending on the thickness of the
stringer in the salt which is linearly dependent on the differential stress when n=1 and has nonlinear
relation when n=5. The differential stress of a stringer fragment keeps stable and does not change
with salt rheology. The stringer fragments Nos. 3, 4, and 5 share the same width of 1000 m, but
have different thicknesses of 30, 60, and 80 m, respectively. For Model A, the average sinking
velocities of the three stringer fragments are 28.4, 49.8, and 101.7 m/Ma separately, while the
differential stresses around stringers are 133, 146 and 154 kPa. For Model B, the average sinking
velocities are 0.45, 0.72, and 0.90 m/Ma, while the differential stresses around the stringers are
131, 146, and 154 kPa. Moreover, the stringer fragments Nos. 1, 4, and 6 have the same width of
80 m but different thicknesses of 100, 500, and 1000 m, respectively. For Model A, the average
sinking velocities are 40.5, 75.8, and 101.7 m/Ma, while the differential stresses around stringers
are 136, 148, and 154 kPa. In Model B, the average sinking velocities are 0.63, 0.84, and 0.90
m/Ma, while the differential stresses around stringers are 132, 147, and 154 kPa.
Furthermore, the stringer fragments Nos. 2, 6, and 7 have the same width of 600 m but different
thicknesses of 30, 60, and 80 m, respectively. For Model A, the average sinking velocities are
27.4, 49.8, and 75.8 m/Ma, while the differential stresses around stringers are 134, 138, and
Table 3 - Results of average sinking velocity, sinking displacement and differential stress of the stringer fragments 1-7
in Model A with the rheology n=1 (pressure-solution creep) and Model B with the rheology n=5 (dislocation creep).
Model A
(n=1)
Model B
(n=5)
Stringer
fragment
No.
(see Fig. 3a)
Fragment
width/
thickness
[m]
Average
sinking
velocity
[m/M]
Sinking
displacement
in 10 Ma
[m]
Max. differential
stress around
stringer
[×103 Pa]
Average
sinking
velocity
[m/M]
Sinking
displacement
in 60 Ma
[m]
Max. differential
stress around
stringer in salt
[×103Pa]
1
100/80
40.5
405
136.0
0.63
37.8
132.0
2
600/30
27.4
274
134.0
0.44
26.4
130.0
3
1000/30
28.4
284
133.0
0.45
27.0
131.0
4
1000/80
101.7
1017
154.0
0.90
54.0
154.0
5
1000/60
49.8
498
146.0
0.72
43.2
146.0
6
600/80
75.8
758
148.0
0.84
50.4
147.0
7
600/60
49.8
49.8
138.0
0.63
37.8
136.0
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Numerical modelling of gravitational sinking of anhydrite stringers in salt Boll. Geof. Teor. Appl., 57, 233-246
148 kPa. For Model B, the average sinking velocities are 0.44, 0.63, and 0.84 m/Ma, while the
differential stresses around stringers are 130, 136, and 147 kPa. Here one thing we should pay
attention to is that the linear relation is not obvious in this example for Model A (n=1), because
the movement of stringer fragments with large width and thickness is sometimes restricted by the
boundary effect.
4. Discussion
Simulation is used to evaluate the effect of model geometry and material parameters on the
movement of anhydrite stringers which are embedded in salt. Whether using the generic model
or the model based on the Zechstein salt, results indicate that the sinking velocity of the stringer
varies linearly with parameter A and parameter B [A=A0 exp(-Q/RT), B=B0 exp(-Q/RT)(1/TDm)],
both of which are derived from the flow law έ=A(Δσ)n and έ=B(Δσ). However, any change in
the differential stress Δσ leads to either linear change in the sinking velocity for Newtonian
rheology (n=1) or nonlinear change for power law rheology (n>1) (the differential stress Δσ is
caused by varying the size of the stringer or the density contrast between the stringer and the salt).
In this case, the non-linear change of the sinking velocity follows a power law with the same
exponent n. Density and geometry change of the stringer is related to its own gravity that affects
the differential stress Δσ.
The calculation result of the single stringer in the generic model shows that the velocity of the
sinking anhydrite stringer is greater than 1000 m/Ma, which indicates that if the salt follows the rule
of Newtonian rheology and the viscosity B is 10-20 Pa·s, the stringer will get close to the bottom
of the salt body in a time range of about 10 Ma. The velocity of the sinking anhydrite stringer is
similar to that of the anhydrite blocks which are embedded in Newtonian salt (Burchardt et al.,
2011). Besides, the velocity of stringer fragments varies from 15 to 100 m/Ma if anhydrite blocks
are embedded in Zechstein salt at rest (effective stress decreases after relaxation) which follows
the Newtonian law and has a salt viscosity of around 10-19 Pa·s. All of the stringer fragments would
get close to the base of the salt section within a time span of the entire Tertiary. However, it is
observed in seismic data (van Gent et al., 2011; Strozyk et al., 2012) that some stringers are still
located close to the top or in the upper part of the salt body after a longer time (~ 60 Ma). This
discrepancy suggests that the long-term rheological behavior of the salt is, at least in some cases,
quite different from what we expected from the rheological parameters which are obtained in
the laboratory. When the salt follows power-law rheology, the sinking velocities are rather slow.
The explanation for this situation is that due to the gravitational sinking stringer in long-term
rheological salt, the peak differential stress around the stringer is 100-150 kPa, which leads to a
rather low strain rate of around 10-20-10-19 s-1. It is clear that stringers do not significantly sink over
geologic times.
Other issues discussed here include that the initial location (depth) of the stringer also has a
big impact on the salt flow and its own descent (Chemia et al., 2008). In our study, this factor
seems to be not as important as salt rheology, density contrast, and geometrical properties of the
stringer, because the method employed in this study is gravitational sinking stringer in salt at rest
(after tectonics) since we focus on the relation between sinking velocity and factors including
salt rheology, density contrast, and geometrical properties of the stringer due to gravitational
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Li and Urai
loading. Salt tectonic process or the movement of salt diapir can have a direct impact on the
rise and fall of stringers or stringer fragments. The complex model which includes salt tectonic
movement and gravitational loading could be taken into account in further research of movement
and deformation of embedded stringers.
5. Conclusion
This study has demonstrated that salt rheology and model geometry of stringers (fragments)
are the dominant factors affecting the sinking velocity of anhydrite stringers in the generic model
or anhydrite blocks in Zechstein salt at rest (the stress in Zechstein salt is released after tectonic
movement). The study has also indicated that the sinking velocity of the stringer linearly changes
with the rheological parameters A, B, and n. Moreover, as is shown in the results of the generic
model, the factors affecting sinking velocity at a given rheology are stringer thickness and density
contrast. Thickness and width of stringer fragment are also factors that control sinking velocity
in Zechstein salt. The velocity changes nonlinearly for power-law rheology (n>1) and changes
linearly for Newtonian rheology when the value of stress Δσ changes. In this case, the nonlinear
change of the sinking velocity follows a power law with the same exponent n.
In contrast to the Newtonian rheology leading to high sinking speed (>15~100 m/Ma), the
power-law rheology results in stringers presents no significant sinking over the geologic timescale,
which is consistent with the seismic data observations. These large differences in sinking velocity
are due to the very low differential stress around the stringers, on the order of 0.1 MPa, after salt
tectonics stopped. Finally, we can conclude that the simulation of gravitational sinking of a simple
anhydrite stringer in the generic model and anhydrite blocks in rock salt based on observations
from natural settings enables a profound understanding of movement and deformation of embedded
stringers or fragments.
Acknowledgements. We thank the Ministry of Oil and Gas of the Sultanate of Oman and Petroleum
Development Oman LLC (PDO) for granting permission to publish the results of this study; Nederlands
Aardolie Maatschappoj BV for providing the seismic data; and S. Abe, H. van Gent, F. Strozyk, and L.
Reuning for their technical assistance on our research of sinking stringers. The research is also funded by
the startup project of China University of Petroleum, Beijing (n. 2462014YJRCOM) and supported by the
Science Foundation of China University of Petroleum, Beijing (n. C20/60).
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Corresponding author: 246
Shiyuan Li
College of Petroleum Engineering, China University of Petroleum
18 Fuxue Road, Changping, Beijing 102249, China
Phone:; fax:; e-mail: [email protected]