ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2008

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 551

Modeling internal deformation ofsalt structures targeted forradioactive waste disposal

ZURAB CHEMIA

ISSN 1651-6214ISBN 978-91-554-7281-8urn:nbn:se:uu:diva-9279

"...Though we are justices and doctors, and churchmen, MasterPage, we have some salt of our youth in us..."

W. Shakespeare

Dedicated to:salt of the earth...

List of Papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Chemia, Z., Koyi, H., and Schmeling, H. (2008) Numerical modellingof rise and fall of a dense layer in salt diapirs. Geophysical JournalInternational, 172(2):798–816

II Chemia, Z. and Koyi, H. (2008) The control of salt supply on entrain-ment of an anhydrite layer within a salt diapir. Journal of StructuralGeology, 30(9):1192–1200

III Chemia, Z., Schmeling, H., and Koyi, H. (2008) The effect of the saltviscosity on future evolution of the Gorleben salt diapir. Tectonophysics,submitted

Reprints were made with permission from the publishers.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Rock Salt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Processes of diapir growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Dense inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 The evolution history of the Gorleben diapir . . . . . . . . . . . . . . 151.6 Long-term safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Numerical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Fundamental principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . . 182.1.3 Conservation of composition . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Symmetric boundary condition . . . . . . . . . . . . . . . . . . . . . . . . 203 Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1 Rocksalt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Anhydrite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Overburden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Sedimentation technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Summary of the Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1 Paper I: Rise and fall of a dense layer . . . . . . . . . . . . . . . . . . . . 255.1.1 Modeling concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2 Paper II: The parameters that influence salt supply . . . . . . . . . . 305.2.1 Modeling concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3 Paper III: Evolution of the Gorleben salt diapir . . . . . . . . . . . . 345.3.1 Modeling concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.3.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

List of Figures

1.1 Global distribution of basins containing salt structures . . . . . . 131.2 Modes of diapir piercement . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Line drawing of the Gorleben salt diapir . . . . . . . . . . . . . . . . . 16

4.1 Sediment aggradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.1 The initial geometric condition . . . . . . . . . . . . . . . . . . . . . . . . 265.2 Regions of the diapir piercement . . . . . . . . . . . . . . . . . . . . . . 275.3 Snapshots of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4 Illustrated areas of the diapir and entrained anhydrite . . . . . . . 315.5 Shapes of passive diapirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.6 Salt supply and entrainment of the anhydrite layer . . . . . . . . . 335.7 The initial geometry of the model for Gorleben diapir. . . . . . . 355.8 Evolution of the models for different salt rheologies . . . . . . . . 365.9 Internal displacement field of the diapir (PF-Model) . . . . . . . . 37

1. Introduction

1.1 ObjectivesFor the last 40 years, scientists and engineers have been searching for geologi-cally suitable repositions for radioactive waste. Salt layers and structures havebeen targets as repositories for hazardous waste [e.g. Gorleben and Morslebensalt diapirs in Germany; WIPP site in USA and Anloo, Gasselte (Drenthe) andWinschoten (Groningen) in Netherlands]. Currently only one repository in saltis in use, the Waste Isolation Pilot Plant (WIPP) where long-lived radioactivewaste is buried in deep salt beds (40 kilometres east Carlsbad, New Mexico).However, many other salt structures are targets for waste storage (e.g. Gor-leben diapir in Germany).

Tectonic stability of a salt structure is a significant factor in evaluating itssuitability as a repository (e.g. hydrocarbon storage, waste disposal, etc.). Toevaluate safety of a repository many different scenarios have been consid-ered such as: suberosion and diapirism, brine migration, thermo-mechanicalfractures, flooding of a disposal mine, large brine inclusions, diapirism to thebiosphere, solution mining etc (e.g. onshore disposal committee, 1989). Manyof these scenarios have been applied to Gorleben salt diapir, which is currentlyused as an intermediate storage facility and was targeted as a future final repos-itory for high-grade radioactive waste. However, the influence of dense inclu-sions, anhydrite layer, which is present within the Gorleben salt diapir, wasnot considered as a possible disturbance of the repository. Based on analogueand numerical models Koyi [2001] suggested that Gorleben diapir might beactive internally due to presence of a dense anhydrite layer (blocks) within it.According to Koyi’s [2001] models, the denser blocks entrained by the diapir,sink within the externally inactive Gorleben diapir. Indications for movementof the anhydrite blocks comes from acoustic emission measurement that haverecorded displacement on the boundary between rock salt and the anhydriteblocks [Spies and Eisenblätter, 2001].

Understanding the structural evolution of an initially tabular salt layer withintercalated anhydrite layer of high density and high viscosity and investiga-tion of the parameters that influence its development are required for evalua-tion of salt structures as a "safe repository". Salt tectonics is therefore criticalfor radioactive waste storage in rock salt in many parts of the globe.

First, a short introduction to the modern understanding of the salt tectonicsis presented below, followed by geology of the Gorleben diapir, which hasbeen used as a general guideline in this work. Second, the applied modeling

11

procedures, together with the theoretical background for the modeling are ex-plained. Finally a brief summary of each paper including conclusions is given.

1.2 Rock SaltHalite, a colorless or white mineral sometimes tinted by impurities that iscommonly known as "rock salt" is found in beds as evaporites. The term "rocksalt" is used to include all rock bodies composed primarily of halite [Jackson,1997a,b]. Salt naturally occurs in immense deposits, and occasionally in sur-face deposits in arid areas as the mineral halite. Salt deposits are widespread.There are major salt basins in the different parts of the world (Fig. 1.1). Anumber of these salt deposits are mined for halite.

Salt has mechanical properties different from those of most clastic and car-bonate rocks. Under unusually high strain rates, salt fractures like most otherrocks. However, under typical geologic strain rates, salt flows like a fluid inthe subsurface and at surface [Weijermars et al., 1993]. Salt is also relativelyincompressible so it is less dense than most carbonates and all moderately tofully compacted siliciclastic rocks. Salt rheology and incompressibility makeit inherently unstable under a wide range of geologic conditions. Diapirs are amajor type of salt structures resulting from tectonic deformation. Bedded saltdeposits are nearly horizontal, although some contain fault zones and otheranomalies. Salt beds range in thickness from a few tens of meters to severalthousands of meters. The differential compaction of the sediments that coversalt may produce instabilities and flows within the salt layer that may result insalt domes or diapirs.

Furthermore, salt is also an effective conductor of heat, elevating the ther-mal maturity of rocks above salt structures and cooling rocks that lie belowor adjacent to salt bodies. The modern investigation of the thermo-mechanicalbehavior of salt started in the mid-1930’s and has acquired considerable tech-nical depth and sophistication. This has been largely due to anticipated useof domal or bedded salt deposits as ideal for storage of petroleum products,sites for special radioactive wastes repositories, and to expanded need for hy-drocarbon storage caverns. In addition, interest in salt tectonics comes fromthe oil industry because many of the world’s great hydrocarbon provinces liein salt basins (e.g., Gulf of Mexico, Persian Gulf, North Sea, Lower CongoBasin, Campos Basin, and Pricaspian Basin). Moreover, evaporites provideeconomic resources of potash salts, sodium salts, gypsum, sulfur, borates, ni-trates, and zeolites [Warren, 1999].

1.3 Processes of diapir growthModern interpretations of salt tectonics consider differential loading as thedominant force driving salt flow. For most of the past 70 years, the prevailingview of salt tectonics was that its mechanics were dominated by salt buoy-

12

Figure 1.1: Equal-area Mollweide projections showing global distribution of basinscontaining salt structures (black areas). Basins containing only undeformed salt areomitted. (a) Global distribution of basins. (b) Detail of European basins. Symbols: ADSouth Adriatic; ZS Zechstein; PY Pripyat; DD Dnepr-Donetz; PC Pricaspian; EA EastAlpine; TV Transylvanian; CR Carpathian; MN Moesian; CK Cankiri; HP Haymana-Polatli; JU Jura; RH Rhodanian; LG Ligurian; BL Balearic; BE Betic; AL Atlas;LV Levantine; MS Messinian; AG Agadir; SF Safi; BR Berrechid; ER Essaouira; LULusitanian; GQ Guadalquivir; MA Maestrat; EB Ebro; CT Cantabrian-West Pyrenees;AQ Aquitaine; TY Tyrrhenian; SC Sicilian; IO Ionian; SZ Suez; DS Dead Sea; PAPalmyra; CL Cicilia-Latakia; AM Amadeus [Hudec and Jackson, 2007].

13

ancy [Jackson, 1995, 1997a,b]. The overburden layers were envisioned as adense fluid having negligible yield strength and diapirism was explained byRayleigh-Taylor instabilities. However, the concept of a fluid overburden fellout of support after workers began to recognize the importance of roof strengthas a control on diapir growth. Although, one of the principal resisting forcesopposing slat flow, the strength of the overburden, or its critical thickness wasreported in early works [e.g., Biot, 1965] it became favor in the late 1980s.

(a) Reactive piercement (b) Active piercement

(c) Erosional piercement (d) Ductile piercement

Figure 1.2: Modes of diapir piercement, shown in schematic cross sections. The over-burden is brittle except in (d) [Hudec and Jackson, 2007].

Diapirs are no longer pictured like giant lava lamps rising through a yieldingoverburden. Instead, the position and shape of viscous salt bodies are seen todepend on how the brittle, mechanically competent overburden deforms. Saltstructures are defined as tectonically active when they rise due to differentialloading created by the overburden layers and/or by regional tectonics [Jack-son and Vendeville, 1994; Koyi, 1998; Hudec and Jackson, 2007]. A diapirmay become inactive (and stops rising) when the driving forces (differentialloading, regional tectonics) cease, salt layer is depleted or a strong lid coversthe diapir. However, buried salt can remain static in the subsurface for tens ofmillions of years. Buried salt may rise as a diapir if: (i) the regional extensionweaken and thin the overburden, which makes room for a reactive diapir to rise(Fig. 1.2a), (ii) The flaps of thin overburden lift, rotate, and shoulder aside, asthe diapir forcibly breaks through them by active diapirism (Fig. 1.2b), (iii)Erosion may remove the roof of the diapir (Fig. 1.2c). All these processesmay occur at various times during the growth of a single salt structure. Di-apirs having fluid overburdens may rise by ductile thinning of the diapir roof[Nettleton, 1934; Talbot et al., 1991, Fig. 1.2d].

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1.4 Dense inclusionsInclusions of denser rocks are common in salt diapirs. The size and lithologyof such entrained inclusions varies from place to place. Sedimentary, volcanicand even some plutonic inclusions (several kilometers in diameter) character-ize many of the diapirs in the Zagros fold-thrust belt [Kent, 1979]. Many of theinclusions are related to the original salt deposits or are older [Gansser, 1992].On Hormuz Island, inclusions of white rhyolites and trachytes frequently oc-cur as irregular masses up to one kilometer long [Gansser, 1992]. Many saltdiapirs contain varying amounts of other evaporite rocks, especially anhydriteor its hydrated form, gypsum, and/or non-evaporite rocks. Most non-haliteinclusions in salt are originally interbedded with salt (e.g. Zechstein forma-tion). During the diapiric processes, these inclusions can be transported bysalt flow. The fact that the salt can carry denser blocks close to the surface wasan enigma until late 1990s. Whereas the external dynamics of the diapir wasintensively studied, few researchers have focused on internal dynamics of thesalt induced by intercalated layers.

The ability of salt diapirs to lift large inclusions of dense rocks generallydepends on rising velocity of salt and negative buoyancy of the inclusions.Denser inclusions can be lifted if salt in the diapir rises faster than the inclu-sions sink. Sinking velocity of the inclusions is controlled by the effective vis-cosity of the salt, which is a function of the rheological properties, strain rateand temperature of the salt [Weinberg, 1993]. An example of a diapir whichhas entrained dense blocks is the Gorleben diapir in Germany [Bornemann,1991]. Geophysical and subsurface data confirm that large blocks of denseranhydrite are present at relatively shallow depth within the Gorleben salt di-apir [Richter-Bernburg, 1980]. These anhydrite blocks, deformed within thediapir, form a key horizon in the Zechstein salt sequence [Bornemann, 1982;Zirngast, 1991].

1.5 The evolution history of the Gorleben diapirThe Gorleben salt diapir, which roughly trends NE-SW, is about 14km longand up to 4km wide. It contains different cycles of the Zechstein salt formation(z2, z3 and z4, Fig. 1.3). The base of the Zechstein lies at about 3.1 to 3.3kmbelow the surface. The stratigraphy and the evolution history of the Gorlebendiapir is well explored by four exploratory wells and two shafts that are drilledinto the Gorleben diapir, in addition to more than forty shallow wells thatexplored overlying beds and the salt dissolution zone at the base of the caprock[Bornemann, 1982, 1991].

It is suggested that the salt structures from the Gorleben diapir evolvedabove an elevation of the basement and the total primary thickness of theZechstein salt in that area is estimated to be about 1.1 to 1.4km [Zirngast,1996]. Halokinesis was initiated in the Early Triassic to Middle Triassic, anda thick salt pillow formed during Keuper (Late Triassic) and Jurassic time

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q

Anhydrite

Z2 salt

BoreholeZ3 salt

Z4 salt Level of plant survey

NW SE

su +sm

su + sm

kru

kru

kro kr0

so + mkj0 - Wd

t t

1000m

Figure 1.3: Line drawing of a northwest-southeast profile of the Gorleben salt di-apir in northwest Germany showing entrained segments of dense anhydrite withina diapir. Symbols: q-Quaternary, t-Tertiary, kro-Upper Cretaceous, kru-Lower Creta-ceous, so+m-Upper Bunter, su+sm-Buntsandstein. Modified after Bornemann [1991].

[Zirngast, 1991; Jaritz, 1993]. The diapiric stage was reached in Late Jurassicand Early Cretaceous and the diapir continued into the Tertiary. Maximum riserates are calculated as 0.08 mm a−1 for the Late Cretaceous and less than 0.02mm a−1 in Miocene to recent times [Zirngast, 1996]. The internal structure ofthe Gorleben diapir is very complex, including vertical or steeply inclined andoverturned fold axes. A very remarkable feature is the main anhydrite, whichforms a key horizon and is folded within the diapir. The main anhydrite is animportant layer with respect to nuclear waste storage because of its size anddensity. It is about 80m thick and denser (ca. 3000 kg m−3) than the salt (ca.2200 kg m−3).

1.6 Long-term safetyFinal disposal of radioactive waste is planned in deep geological formations(Fig. 1.3). Radioactivity form the waste decreases in time due to the radioac-tive decay. Nevertheless, in case of long-living nuclides the radiation after100 000 years will still requires the waste to be isolated from the biosphere.Therefore, long-term periods up to one million years and even more have tobe considered. Long-term safety analyses are performed to determine the ra-diological effects of the waste on the biosphere for the next one million years[Federal Institute for Geosciences and Natural Resources (BGR)].

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2. Numerical modeling

All the mathematical sciences are founded on relations be-tween physical laws and laws of numbers, so that the aimof exact science is to reduce the problems of nature to thedetermination of quantities by operations with numbers.

James Clerk Maxwell

2.1 Fundamental principlesAll of Computational Fluid Dynamics, in one form or another, is based on thefundamental governing equations of fluid dynamics, continuity, momentum,and energy equations. They are the mathematical statements of three funda-mental physical principles upon which all of fluid dynamics is based.

We model the dynamics of salt tectonics driven by differential loading andcompositionally induced density variations using a two dimensional finite dif-ference code (FDCON modified after H. Schmeling). This code solves equa-tions for an initially horizontally layered multi-composition system. Differ-ent chemical compositions (i.e. layers) may be assigned different densitiesand rheologies. The dynamics of such flow can be described by the equa-tions of conservation of mass, momentum and composition [e.g. Weinbergand Schmeling, 1992].

2.1.1 Conservation of massThe law of conservation of mass states that the mass of a closed system ofsubstances will remain constant, regardless of the processes acting inside thesystem [Antoine Lavoisier, 1785]. According to conservation of mass, reac-tions and interactions, which change the properties of substances, leave un-changed their total mass. The time rate of change of density of the given fluidelement as it moves through space is given as:

∂ρ

∂ t+∇(ρu) = 0 (2.1)

where ρ is density, t is time, and u is fluid velocity.

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Using the Boussinesq approximation density differences are sufficientlysmall to be neglected, except where they appear in terms multiplied by theacceleration due to gravity. Thus for a Boussinesq fluid (ρ = constant) equa-tion 2.1 can be written as

∇u = 0 (2.2)

2.1.2 Conservation of momentumThe conservation of momentum is a fundamental concept of physics alongwith the conservation of mass. The conservation of momentum states that,within some problem domain, the amount of momentum remains constant;momentum is neither created nor destroyed, but only changed through theaction of forces as described by Newton’s laws of motion [Landau and Lifsic,1987]. Momentum is conserved in all three physical directions at the sametime. Consequently the force balance for an incompressible fluid undergoingflow in two dimensions can be defined by adding the volume, surface andinertial forces together and equating their sum to zero.

We neglect the inertial force associated with the acceleration of a fluid.This assumption is appropriate for the slow motion of very viscous or highPrandtl number fluids. Salt behaves as a highly viscous fluid on geologicaltime scale [Weijermars et al., 1993]. The viscosity of salt is about 1017 −1019 Pa s [Hunsche and Hampel, 1999; Spiers et al., 1990; Urai et al., 1986].With assumption of incompressibility the equation of motion is given by

0 =−∇P+∂τ ji

∂x j−ρg (2.3)

where τ ji is the deviatoric stress tensor and can be expressed in terms of ve-locities using the constitutive law

τi j = 2ηkεi j (2.4)

where ηk is the viscosity of the k-th component and εi j is the strain-rate tensordefined as

εi j ≡∂εi j

∂ t=

12

(∂ui

∂x j+

∂u j

∂xi

)(2.5)

To specify the density in the buoyancy term of the momentum equationwe take the mean of the densities of the different materials weighted by theirconcentrations

ρ = ∑k

ckρk (2.6)

where ck and ρk is the concentration and density of the kth chemical compo-nent.

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Introducing the stream-function ψ as

u =−∂ψ

∂ z∂x∂ψ

and using constitutive law with assumption of two dimensionality for theequation of motion we obtain

4∂ 2η

∂x∂ z∂ 2ψ

∂x∂ z+(

∂ 2η

∂ z2 −∂ 2η

∂x2

)(∂ 2ψ

∂ z2 −∂ 2ψ

∂x2

)=−g∑

kρk

∂ck

∂x(2.7)

where x and z are horizontal and vertical coordinates respectively. Given thechemical concentrations ck(x,z, t), the biharmonic equation (2.7) is solved nu-merically.

In equation (2.7), η is the effective viscosity defined at boundaries betweenthe different materials, where mixture can occur. In mixtures of different com-positions, an effective viscosity may be defined by assuming that the totalstrain rate is given as the weighted sum of the strain rates in each component,or assuming that the total stress is given as the weighted sum of the stressesin each component. Accordingly, the effective viscosity would be given as theharmonic or the arithmetic mean of the viscosities of the components. As it isnot possible to specify which of these means is to be preferred in a complexflow field [see Schmeling et al., 2008, for a comparison of different averagingschemes], the effective viscosity of a mixture is defined here as the geometricmean as:

logη = ∑k

ck logηk (2.8)

2.1.3 Conservation of compositionThe conservation of composition is conveniently defined as

∂ck

∂ t+u∇ck = 0 (2.9)

where ck is the concentration of the k-th chemical component. At any location,the concentrations of the different components add up to 1, i.e.

nmat

∑k=1

ck = 1 (2.10)

where nmat is the number of components of the problem. This definition al-lows consideration of the chemical mixtures of different materials as well asimmissible fluids. If the fluid is immissible, ck is equal to 1 only at the posi-tions which are occupied by the material k, otherwise it is 0.

The conservation of composition is solved using a marker approach. Theadvantage of the marker approach is that it introduces no numerical diffusion.Following any fluid particle on its flow path, equation (2.9) may be written in

19

the Lagrangian reference system(∂ck

∂ t

)particle

= 0 (2.11)

The equation (2.11) implies that we have to solve for the flow path equation(∂x∂ t

)particle

= u (2.12)

and have to ensure that the ckvalues remain constant at x(t)particle.To solve (2.12), model domain is filled with closely spaced marker points

initially distributed on a rectangular grid with small random displacement.These fluctuations, which have maximum amplitude of a half marker grid dis-tance, are also responsible for the initial perturbations of the horizontal in-terfaces between materials of different density. Each marker is assigned itsparticular chemical composition, i.e. ck(xm), where xm(t) is the position of amarker at the time t. Equation (2.12) is then solved for all markers by a com-bination of fourth-order Runge-Kutta integration with a predictor-correctorstep.

2.2 Symmetric boundary conditionIn simulations, we assume that symmetric state exists on the boundary. Thistreatment of the boundary condition corresponds to the physical assumptionthat, on the two sides of boundary, the same physical processes exist. Thevariable values at the same distance from the boundary at the two sides arethe same. The function of such a boundary is that of a mirror that reflects allthe fluctuations generated by the simulation region. Instead of simulating thewhole structure, we set the appropriate boundary conditions and reduce theproblem size.

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3. Rheology

In geology and geophysics, the term rheology refers to the study of mechanicalproperties and their role in the deformation and the flow of the materials thatform the Earth. We can describe the Earth’s rheology mathematically in manyways, but it is always a function of intrinsic, i.e., material parameters, andextrinsic, i.e., environmental parameters [Park, 1989]. The rheology of eachmaterial used in numerical calculations is described bellow.

3.1 RocksaltMany studies have focused on experimental deformations of the steady stateflow properties and processes of natural rocksalt [e.g., Hunsche and Hampel,1999; Carter et al., 1993; Urai et al., 1986]. The sets of the steady-state datafit well a power law creep relation:

ε = A∗ ·σn · exp(−H∗/RT ) (3.1)

where ε is the steady-strain creep rate, σ = (σ1−σ3) is the stress differenceof the maximum and minimum principal stresses, T is absolute temperature,A∗ is a material constant, n power law exponent, and H∗ = E∗ + PV is theactivation enthalpy. E∗ is the activation energy, V is the activation volume, Ris the universal gas constant (8.32 J mol−1), and P is the (lithostatic) pressure.

Numerical models presented in this study are based on an approach wherepower-law ductile creep describes deformation behavior of the rock. The vis-cosity depends non-linearly on deviatoric stress, which is given as:

η = A · exp(

H∗

RT

)· τ1−n

II (3.2)

where A =(

3n+1

2 ·A∗)−1

is a pre-exponential constant, and τII is the 2nd in-variant of the stress tensor.

Although salt compositions vary from one formation to another, the rheo-logical behavior is qualitatively similar. The domal salt at Gorleben, for ex-ample, has significantly low water content (ca 0.1–0.2%) and creeps slowly[Fairhurst, 2002]. Yet, flow laws essentially similar to that in equation (3.2)can be used to study internal dynamics of the Gorleben diapir.

In some of the simulations, we can assume that the rocksalt behaves asNewtonian fluid [e.g. wet salt, Weijermars et al., 1993]. The viscosity of wet

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salt is virtually independent of stress or strain rate. Wet salt therefore has noyield strength. Data from field observations and rock mechanics both suggestthat natural faulting in wet rock salt is rare. Thus, disregarding temperature de-pendency of the viscosity, the deformation behaviour of rock can be describedby equation (3.2) when H∗ = 0 and n = 1.

3.2 AnhydriteThe deformation experiments on anhydrite rocks showed that the strength ofthe fine-grained anhydrite is strongly temperature and strain-rate dependentabove 300◦C whereas coarse grained anhydrite rock remains relatively insen-sitive to strain-rate and temperature even at 450◦C. Although anhydrite hasa high strength at room temperature, fine grained aggregates weaken rapidlyabove temperatures of 100◦C to 200◦C, depending on grain size and strainrate [Muller et al., 1981]. We model anhydrite with a power-law rheology in-dependent of temperature and a low value for the exponent, i.e. n = 2 [Mulleret al., 1981].

η = A · τ1−nII (3.3)

where A =(

3n+1

2 ·A∗)−1

and A∗ = 3.08×10−27Pa−n s−1.Since the modeling approach is based on the mechanics of continua rather

than fracture mechanics, the brittle rheology of anhydrite cannot be achievedin numerical models. We model anhydrite with an effective viscosity rangingbetween 1019−1021 Pa s.

3.3 OverburdenThe sedimentary overburden rocks simulated in all the models are assignedhigh viscosity (effective viscosity ranges between 1023 and 1025 Pa s), becausemost sedimentary rocks (clastic or carbonate) behave as non-Newtonian mate-rials or as brittle solids, rather than ductile fluids, depending on factors such astheir depth of burial, etc. [Vendeville and Jackson, 1992]. Our assumption ofa very high viscosity keeps ductile deformation in the overburden extremelysmall [Rönnlund, 1989; Hughes and Davison, 1993].

We also use high viscosity overburden to simulate postdiapirc state of theGorleben diapir. In analogue models often the postdiapiric state is simulatedby a critical thickness of the overburden [Biot, 1965; Koyi et al., 2001]. Alter-natively, sufficiently stiff overburden can be used to simulate inactive diapirs[Biot, 1965]. Since the thickness of the overburden for Gorleben diapir is wellknown and it is reported that the diapir is externally inactive we used stiffoverburden approach. However, lower viscosity contrast between the overbur-den and the salt, which causes viscous drag along their boundaries with thesalt is considered as well.

22

4. Sedimentation technique

Albert Einstein is reputed to have warned his son HansAlbert against becoming a river engineer, saying that thephysics of sediment transport was too complex . . .

Philip A. Allen, 1997

Sedimentation is a process of deposition of a solid material from a state ofsuspension or solution in a fluid (usually air or water). It is one of the im-portant parameters that drive salt tectonics. The importance of sedimentationrate and its impact on the diapir evolution has been studied previously and it isshown that sedimentation rate can control the rate of rise of a diapir and its ge-ometry [Vendeville et al., 1993; Vendeville and Jackson, 1992; Koyi, 1998].We introduce here a technique of sedimentation which makes it possible togradually down-build salt diapir and investigate the effect of sedimentation onthe evolution of the diapir.

Two different modes of sediment aggradation (constant and variable) areused in the simulations. Sediments accumulate on the top of the model with aprescribed rate, se. The initial position of the sediment surface is given by thetop of the pre-kinematic overburden, zs0 (Fig. 4.1a). Above the initial modelsurface, background markers are assumed with a low viscosity and a low den-sity. During sedimentation, the position of the surface of sediment layer, zs,rises with the prescribed rate, zs = zs0 + se, thereby thickening the overbur-den (Fig. 4.1b). If the salt surface lies below zs, then the salt is also coveredwith sediments (Fig. 4.1c). During the simulation, sediments are deposited ateach time step between the previous model surface at time, t−dt, zs(t−dt),and the new sedimentation surface, zs(t) [but only where zs(t− dt) is belowzs(t)]. This is done by assuming all background markers, whose positions arebetween zs(t−dt) and zs(t) with sedimentary rock properties.

In the variable sedimentation mode, the sedimentation rate varies accordingto the evolution of the crest of the diapir. During the simulation, the velocity atwhich the crest of the diapir rises (vc) is determined. A quarter of this velocityis assigned to the sedimentation rate, so the surface of the sediment layergrows depending on the evolution of the crest of the diapir.

zs = zs0 +∫ t

0

14

vcdt (4.1)

23

The choice of the factor 1/4 in the variable mode of the sedimentation ismanually calibrated to form a columnar diapir. The quarter of the rising ve-locity of the crest of the diapir seems to be a good approximation for the for-mation of the diapir whose crest remains uncovered and possess a columnarshape.

Initial Salt Layer

pre-kinematic overburden layers

Low viscosity backgraund material

zs0

(a) Initial configuration

Initial Salt Layer

zs0

pre-kinematic overburden layers

zs

Aggraded sediments

Diapiric rise > Sedimentation rate

Salt Layer

(b) After sediment deposition

Initial Salt Layer

zs0

pre-kinematic overburden layers

zs

Diapiric rise < Sedimentation rate

Salt Layer

Aggraded sediments

(c) Burial of the diapir

Figure 4.1: Line drawing of gradually down-build salt diapir by sediment aggradation.

24

5. Summary of the Papers

This chapter presents a brief summary of the papers included in this the-sis. Each paper is summarized, describing the main objectives, the systemof methods used, the main results and conclusions.

5.1 Paper I: Rise and fall of a dense layerPaper I addresses the issue of presence of dense inclusions within salt diapirs.In this paper, we used the results of 2-D numerical calculations to quantita-tively analyze the parameters (sedimentation rate, salt viscosity, width of theperturbation, and stratigraphic position of dense layers within a salt layer) thatgovern the rise and fall of dense blocks (anhydrite) within salt diapirs. Modelspresented in this study are not scaled to any particular salt diapir. Neverthe-less, the Gorleben salt diapir (with an intercalated dense anhydrite layer) hasbeen used as a general guideline for the modeling.

5.1.1 Modeling conceptMany investigators numerically modeled different aspects of salt diapirism[e.g., Woidt, 1978; Schmeling, 1987; Romer and Neugebauer, 1991]. How-ever, few studies have addressed the problem of down-building and multi-compositional salt structures.

A two-dimensional box (4× 4km) with a grid resolution 161× 161, rep-resenting the model domain, consisted of a salt layer (1040m thick) simu-lating the Zechstein salt formation, an anhydrite layer (80m thick) embed-ded within the salt layer, a pre-kinematic overburden layer (160m thick), andan arbitrary material simulating water or air (Fig. 5.1). The dense anhydritelayer (ρa = 2900 kg m−3) is initially embedded as a horizontal layer within aviscous salt. The anhydrite layer is assigned non-Newtonian rheology [Kirbyand Kronenberg, 1987]. During the experiments effective viscosity of the an-hydrite ranges between 1019 and 1021Pa s. The overburden, representing apre-kinematic layer, is placed on the salt layer apart from the perturbation toinitiate salt flow (Fig. 5.1). A pre-kinematic perturbation (400 or 800m wide)is initiated at the left-hand side of the upper boundary of the salt layer as atrigger for diapirism. The perturbation is down-built by aggradation of non-Newtonian sediments (n = 4, constant temperature) at a rate that is variedsystematically.

25

Several parameters (sedimentation rate, salt viscosity, perturbation widthand stratigraphic position of the anhydrite layer) are varied systematically tounderstand their influence on entrainment of the anhydrite layer whereas oth-ers are kept constant.

t=0.00 Ma

Length (km)

Dep

th(k

m)

0 0.5 1 1.5 2 2.5 3 3.5 4 −4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Salt Layer

ABCD

Anhydite layer

Pre-kinematic overburden layer

Low viscous background material

Figure 5.1: Line drawing showing the initial geometric condition and the points (A, B,C and D), which are monitored throughout model evolution. The box has no-slip at thebottom, free-slip at the top and reflective side boundaries. The box can be consideredas showing the right hand half of a symmetric structure.

5.1.2 SummaryThe evolution of the modeled diapirs depends on the parameters that vary dur-ing the simulations. The rate of rise of the diapir increases as the overburdenaggrades and increases the driving pressure (differential pressure) on the un-derlying salt layer. An increase in differential loading accelerates salt flow andyields to entrainment of the anhydrite layer. However, high sedimentation rateduring the early stages of the diapir evolution bury the initial perturbation andprevents formation of a diapir. As a result, the anhydrite layer sinks within theburied salt layer.

Sedimentation rate and viscosity of salt are the main parameters controllingthe burial of the diapir. Diapirs of low-viscosity salt (1016 Pa s) survive forhigh sedimentation rates (e.g. 15 mm a−1) whereas increasing the salt viscos-ity requires a decrease of the sedimentation rate for the diapir to rise withoutbeing buried (Fig. 5.2). Increase of salt viscosity by an order of magnitudeand decrease of sedimentation rate by the same order often result in diapirs ofsimilar geometry.

26

Diapirs down-built by slow sedimentation rate are characterized by a slowvertical growth, because the salt extrudes and spreads laterally faster thanit ascends vertically. In regions of slow vertical salt movement (e.g. withinthe overhang), entrained blocks may sink. In contrast, where vertical flowdominates, the anhydrite blocks are entrained and dragged up to higher lev-els. When the overhang widens faster than the overburden thickens, the pre-kinematic overburden is bent and sinks. The entrained anhydrite segments arecarried sideways into the wide overhang. Additional sediments cover the crestof the diapir forming mini-basins that segment the diapir in its later stages. Themini-basins push the anhydrite blocks down as they sink into and segment thediapir.

Numerical models with salt viscosity 1017 Pa s predict that diapirs can po-tentially grow with an embedded anhydrite layer, and deplete their sourcelayer in a very short time ranging from three to as little as 0.7 Myr. That thenatural diapirs typically grow more slowly and for much longer time than theabove predictions, clearly suggesting that salt does not flow freely and con-tinuously throughout the growth history of the diapir. However, as it is shownhere, many parameters such as sedimentation rate, salt viscosity, perturbationwidth and the stratigraphic location of the anhydrite layer can retard or speedup the evolution of the diapir.

1017

1018

1019

10−1

100

101

Sedd

imen

tati

on r

ate

(mm

/a)

Zone of no diapirism

Zone of diapirism

Viscosity (Pa s)

Figure 5.2: Logarithmic plot showing regions where the diapir is piercing for sedi-mentation range 0.1− 5 mm a−1 and viscosity range 1016− 1019 Pa s. Circles showthe models where piercement takes place and the squares models where the diapir isburied. Symbols are data points and the contour defines region of piercement

27

5.1.3 ConclusionsModel results show that the combined effect of four parameters (sedimen-tation rate, salt viscosity, perturbation width, and stratigraphic location of adense layer) shape a diapir and the mode of entrainment rather than the netvalue of each parameter individually.

I Sedimentation rate: Fast sedimentation suppresses the rate of diapiricrise. Dense embedded layers are likely to sink within the salt layer leftbeneath the thick overburden before forming a diapir. However, sedi-mentation rate slower than the rate of diapiric rise, can down-build adiapir that entrains and deforms the anhydrite layer (Fig. 5.3).

II Salt viscosity: For the same sedimentation rate, increasing viscosity ofthe salt layer decreases the rise rate of the diapir and reduces the amount(volume) of the anhydrite layer transported into the diapir. An anhy-drite layer sinks easily into less viscous salt, whereas salt that is moreviscous can prevent sinking of the anhydrite layer for longer time. Fur-thermore, a more viscous salt provides a larger viscous drag, and hence,assists in entrainment and carrying upward of the anhydrite layer. Modelresults show that viscous salt (> 1018Pa s) is capable of carrying sepa-rate blocks of the anhydrite layer to relatively higher stratigraphic levels(Fig. 5.3).

III Perturbation width: Wide perturbation allows significantly larger vol-umes of salt supply to the stem of the diapir and can more easily entrainan embedded anhydrite layer than narrower diapirs. (Fig. 5.3).

IV Stratigraphic location: The presence and stratigraphic location of ananhydrite layer significantly alters the time evolution of the diapir (de-celerates the diapiric growth rate by 17 percent). In addition, an anhy-drite layer embedded in the middle of a salt layer is entrained moreeffectively than a layer embedded within the upper or lower half of thesalt layer (Fig. 5.3).

V The anhydrite layer is entrained as long as rise rate of the diapir exceedsthe descent rate of the denser anhydrite layer. However, entrained blocksinevitably sink back if the rise rate of the diapir is less than the rate ofdescent of the anhydrite layer or the diapir is permanently covered by astiff overburden in case of high sedimentation rates.

VI Our model results support earlier studies that diapirs containing denserblocks can be active internally even though they might be consideredinactive externally. The Gorleben diapir in Germany, which has beenconsidered as a repository for radioactive waste maybe such an exam-ple.

28

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

−2.5

−2

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−1

−0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

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−2

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−1

−0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

−2.5

−2

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−1

−0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

−2.5

−2

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−1

−0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

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−2

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−1

−0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

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−2

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−1

−0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

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−2

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−1

−0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

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−2

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−1

−0.5

0

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Dep

th (

km)

Dep

th (

km)

Dep

th (

km)

Dep

th (

km)

t = 7.8 Ma t = 1.26 Ma t = 1.26 Ma

t = 10.1 Ma t = 10.1 Ma t = 10.1 Ma

t = 1.26 Ma t = 1.26 Ma t = 1.27 Ma

t = 6.07 Ma t = 6.07 Ma t = 6.07 Ma

Length (km) Length (km) Length (km)

sedimentation rate

salt viscosity

perturbation width

stratigraphic location

4 mm/a 2 mm/a 1 mm/a

1.e17 Pa s 1.e18 Pa s 5.e18 Pa s

400 m 600 m 800 m

800 m 480 m 200 m

Figure 5.3: Snapshots of models where the sedimentation rate, salt viscosity, pertur-bation width and the stratigraphic location of the anhydrite layer is varied.

29

5.2 Paper II: The parameters that influence salt supplyAnalogue and numerical models confirm that dense block can be entrained bydiapiric flow if the rate of diapiric rise is greater than the rate of decent of thedense entrained blocks [Weinberg, 1993; Koyi, 2001; Chemia et al., 2008].The influence of four parameters (sedimentation rate, viscosity of salt, strati-graphic location of the anhydrite layer within the salt layer, and the perturba-tion width) described in previous section strongly influence the style and theamount of entrainment of dense inclusions into a diapir [Paper I]. To investi-gate the entrainment of a denser layer, which depends on multiple parameters,we combine these parameters into a single parameter, which is the rate of saltsupply (volume/area of the salt that is supplied to the diapir with time, i.e. cu-mulative flow of salt from the salt layer to the diapir). In this study, we inves-tigate in detail the influence of these four parameters on rate and the amountof salt supply during the development of a diapir down-built from a salt layerwith an initially intercalated dense anhydrite layer. Using two-dimensionalnumerical calculations, a whole range of values for these four parameters areexplored.

5.2.1 Modeling conceptThe results presented in this paper are based on models, which are describedin Paper I. In this article, we extend the range of the parameters and recast re-sults in terms of salt supply. At different stages of the diapir growth, the areaof the diapir and the area of the entrained anhydrite layer are calculated. Thecontour comprising the chemical composition of the salt layer defines the areaof the diapir. The contour closes at the level of the perturbation, which mightsink or rise during the model evolution and the area is calculated above thisdynamic level (Fig. 5.4). Similarly, we define that the anhydrite is entrainedif it is carried upward into the diapir above the perturbation level. The closedcontour comprising the chemical composition of the anhydrite layer above theperturbation level defines the amount of the entrained anhydrite layer (Fig.5.4). If the anhydrite layer is disrupted then the cumulative area of the en-trained anhydrite is considered sum of the areas of the closed contours abovethe perturbation level (Fig. 5.4).

In two dimensional numerical experiments, the parameter cumulativepercentage entrainment (E) is conveniently defined as E = (Aat /Aa0)× 100,where Aat is the area of anhydrite layer which has moved above a referenceline (set at the perturbation level) at any given time t, and Aa0 is theoriginal area of the anhydrite layer at t = 0 (Fig. 5.4). Similarly, cumulativepercentage salt supply S is defined as S = (Ast /As0)× 100, where Ast is thearea of salt layer which has moved above a reference line (area of the diapir,Fig. 5.4) at any given time t, and As0 is the original area of the salt layerbeneath the surface at t = 0, (Fig. 5.4).

30

0 0.5 1 1.5 2 2.5 3 3.5 4-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0Area of the diapir

Area of the entrained anhydrite layer

Length (km)

Dep

th (

km)

Figure 5.4: Illustrated areas of the diapir and the entrained anhydrite layer used forcalculated salt supply and entrainment

5.2.2 OverviewThe geometry of an active diapir is molded by six parameters: rate of salt sup-ply, dissolution, sediment accumulation, erosion, extension, and shortening[Koyi, 1998]. The rate of salt supply is the rate at which salt flows from thesource layer into the diapir and contributes to its growth. In a system wherelateral movement (extension and shortening) is insignificant, salt supply andsedimentation rate govern the evolution of a down-built diapir [Vendeville andJackson, 1993; Koyi, 1998]. If the rate of salt supply is less than the rate ofsediment accumulation, upward-narrowing diapirs form (Fig. 5.5a). In con-trast, upward-widening diapirs form when the salt supply is greater than therate of sediment accumulation and columnar diapirs form when the rate ofsalt supply is equal to the rate of sediment accumulation [Fig. 5.5, Jacksonand Talbot, 1991; Vendeville and Jackson, 1993; Talbot, 1995; Koyi, 1998].

5.2.3 ConclusionsWe conclude that salt supply controls the amount of entrainment, distributionof initially interbedded dense blocks/layers within a diapir, and dictates theinternal structure of the diapir. With the range of parameters used in our mod-els, entrainment of the anhydrite layer, which has been inevitable, was directlygoverned by the rate of salt supply (Fig. 5.6). On the other hand, the entrainedanhydrite segments sink within the diapir when salt supply decreases dramat-ically or ceases entirely. The rate of sinking at this stage depends strongly onthe viscosity of the salt; anhydrite segments sink slower within more viscoussalt.

31

(a) Diapir rise > Aggradation rate (b) Diapir rise = Aggradation rate

(c) Diapir rise < Aggradation rate

Figure 5.5: The cross-sectional shapes of passive diapirs are tied to the relative ratesof net diapir rise (salt rise minus erosion and dissolution) and sediment aggradation.Modified from Giles and Lawton (2002).

Although the viscosity of salt significantly alters salt supply, down-builtdiapirs grow in height at essentially similar rates. The variation in salt supplyresults in the formation of overhangs of different width. However, the effectof the viscosity variation on rate of rise of the crest of the diapir is negligible.Salt supply is governed by parameters (sedimentation rate, viscosity of saltperturbation width, and stratigraphic location of the anhydrite layer), whichcontrol the internal and external evolution of the diapir.

32

0 1 2 3 4 5 6 70

20

40

60

80

100

Time (Myr)

Cum

ulat

ive

salt

supp

ly, S

(%)

1 mm/a2 mm/a0.5 mm/a0.25 mm/a0.1 mm/a0.05 mm/a

(a) Effect of the sedimentation rate

0 0.5 1 1.5 2 2.50

20

40

60

80

100

Time (Myr)

Cum

ulat

ive

entr

ainm

ent,

E (%

) 2 mm/a1 mm/a0.5 mm/a0.25 mm/a0.1 mm/a0.05 mm/a

0 5 10 15 20 25 30 350

20

40

60

80

100

Time (Myr)

Cum

ulat

ive

salt

supp

ly, S

(%) 5×1017 Pa s

1018 Pa s

1017 Pa s

5×1018 Pa s

1019 Pa s

(b) Effect of the salt viscosity

0 5 10 15 20 250

20

40

60

80

100

Time (Myr)

Cum

ulat

ive

entr

ainm

ent,

E (%

) 1017 Pa s

5×1017 Pa s

1018 Pa s

5×1018 Pa s

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

Time (Myr)

Cum

ulat

ive

salt

supp

ly, S

(%)

Pw=400 m

Pw=500 m

Pw=600 m

Pw=800 m

Pw=1200 m

Pw=1600 m

(c) Effect of the perturbation width

0 0.5 1 1.5 2 2.50

20

40

60

80

100

Time (Myr)

Cum

ulat

ive

entr

ainm

ent,

E (%

)

Pw=1600 m

Pw=1200 m

Pw=800 m

Pw=500 m

Pw=400 m

0 2 4 6 8 100

20

40

60

80

100

Time (Myr)

Cum

ulat

ive

salt

supp

ly, S

(%) NO Anhy.

La = 800 m

La = 200 m

La = 480 m

(d) Effect of the stratigraphy

0 2 4 6 8 100

20

40

60

80

100

Time (Myr)

Cum

ulat

ive

entr

ainm

ent,

E (%

)

La = 800 m

La = 480 m

La = 200 m

Cumulative EntrainmentSalt Supply

Figure 5.6: Effect of the sedimentation rate, salt viscosity, perturbation width (Pw)and stratigraphic location of the anhydrite layer (La) on cumulative salt supply andentrainment of the anhydrite layer.

33

5.3 Paper III: Evolution of the Gorleben salt diapirThe Gorleben diapir, which has been targeted for radioactive waste disposal,contains large blocks of anhydrite. Numerical models that depict the geometri-cal configuration of the Gorleben diapir are used to understand internal struc-ture of diapir caused by movement of the anhydrite blocks for various saltrheologies. The Gorleben salt is modeled with rheological parameters chosenfrom laboratory experiments on different salt formations to test which rheol-ogy is able to activate previously inactive anhydrite blocks and if the mobilityof anhydrite blocks can disturb any repository within the diapir.

5.3.1 Modeling conceptIn this study, we use documented configuration of the internal structure ofthe Gorleben diapir as the initial setup of the models (Fig. 1.3). The com-plex internal stratigraphy of the Gorleben diapir is simplified so that only theMain Anhydrite layer is used in the modeling (Fig. 5.7). The Main Anhydritelayer is modeled as an average 80m thick, (density = 2900 kg m−3) embed-ded within the Zechstein salt formation. In all models an effective viscosity ofanhydrite is between 1019 and 1021 Pa s.

Overburden layers are unified and given corresponding depth-independentdensity (2600 kg m−3). The sedimentary overburden rocks simulated in allthe models are assigned high viscosity (effective viscosity ranges between1023 and 1025Pa s). However, few models are deployed with lower viscositycontrast between the overburden and the salt to elucidate the importance ofthe overburden layers.

The different cycles of the Zechstein salt z2, z3 and z4 represent the saltlayer above the basement (Figs 1.3 and 5.7). The Zechstein salt formation ismodeled as a power law salt using the currently available material parametersfor different salt formation [Table 5.1, Kirby and Kronenberg, 1987]. A tem-perature distribution within the model linearly varies with depth (at the surface0◦C and 140◦C at the bottom). A mean value of the density for rock salt is as-sumed to be 2200 kg m−3 [Landolt-Boernstein, 1982]. For comparison, wemodel Newtonian salt with an average viscosity ranging between 1017 and1018Pa s [Urai et al., 1986; Spiers et al., 1990; Carter et al., 1993; Van Kekenet al., 1993].

5.3.2 SummaryIf the rheology of the salt from the Gorleben diapir is to be approximatedto Newtonian salt with viscosity of 1017Pa s, then the anhydrite layer withinthe diapir will sink with a maximum average velocity of ca 5 mm a−1. Thisobservation suggests that if the average effective viscosity of salt ranges from1017 to 1018 Pa s [e.g., Urai et al., 1986; Spiers et al., 1990; Hunsche andHampel, 1999] diapirs with intercalated denser layer must be internally activeeven though the surface of the diapir does not rise. The fact that the internal

34

0 1000 2000 3000 4000 5000 6000−4000

−3500

−3000

−2500

−2000

−1500

−1000

−500

0t = 0 Myr

Dep

th (

m)

Lenght (m)

OverburdenAnh

ydrit

e La

yer

Salt

a b c

Figure 5.7: The initial geometry of the model. The temperature distribution within thebox linearly varies with depth at the surface 0◦C and 140◦C at the bottom. Points a, band c define passive markers used to monitor deformation of the plant survey.

movement of the diapir is due to the presence of the anhydrite layer is verifiedin the model where the anhydrite layer was removed from the initial setup,but all the other parameters were kept the same. In this model salt remainedinactive (maximum vertical velocity 9.6× 10−5 cm a−1) both externally andinternally for more than 5 million years.

Based on the movement of passive markers, it is suggested that for the saltmodeled with rheological parameters of the Paradox Formation (weakest), thewestern margin of the potential repository (adjacent to larger anhydrite blocks,point a) will sink with an average rate of 3.2 mm a−1 during a subsequent timeperiod of 0.1 Ma. The eastern margin of the potential repository (point c) willsink with a rate of 0.2 mm a−1 within the same time. Unlike marginal points ofthe potential repository, the mid-point (point b) rises with a rate of 0.73 mma−1. The deformation rate of the potential repository is sufficiently reducedif the effective viscosity of salt is high (models: AI, SD, RD, and PB, Fig.5.8). This observation suggests that if the average effective viscosity of saltranges from 1019 to 1020 Pa s the diapirs with intercalated denser layer mustdeform with a rate that is not significant. It should be noted that the predictedfast deformation rates (weak PF-rheology) lie in the range to be observed bygeodetic measurements.

Depending on size and orientation of the anhydrite layer, deformation pat-tern is significantly different within the diapir (Fig. 5.9). Horizontal blockssink much slower than vertical blocks. Furthermore, the rate of descent differson different sides of the diapir. Descent rate, mode and pattern depend on ver-tical and horizontal location and orientation of the anhydrite blocks within the

35

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0 PF: t = 0.010 Myr

Dep

th (

km)

(a)

PF: t = 0.047 Myr

(b)

PF: t = 0.208 Myr

(c)

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0 AI: t = 1.038 Myr

Dep

th (

km)

(d)

AI: t = 2.597 Myr

(e)

AI: t = 4.805 Myr

(f)

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0 VD: t = 0.011 Myr

Dep

th (

km)

(g)

VD: t = 0.104 Myr

(h)

VD: t = 1.003 Myr

(i)

0 1 2 3 4 5 6−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0 N17: t = 0.010 Myr

Dep

th (

km)

Length (km)

(j)

0 1 2 3 4 5 6

N17: t = 0.104 Myr

Length (km)

(k)

0 1 2 3 4 5 6

N17: t = 0.274 Myr

Length (km)

(l)

Figure 5.8: Evolution of the models for different salt rheology. (a-c) Paradox For-mation (PF), (d-f) Avery Island (AI), (g-i) Vacherie Dome (VD), (j-l) Newtonian saltwith viscosity 1017 Pa s.

36

0 1 2 3 4 5 6−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

deformedundeformed

displacementDep

th (

km

)

Length (km)

Figure 5.9: Internal deformation of the diapir (PF-Model) shown after 85 kyr. Arrowsdefine velocity vectors at t = 85 kyr.

diapir. In addition, in models with variable salt viscosity descent rate dependson the rheology of the salt, which is in direct contact with the anhydrite layer.

5.3.3 ConclusionsNumerical models show that salt structures with intercalated dense layers areinternally active. We have shown that the mobility of anhydrite blocks dependson the effective viscosity of salt which has to be lower than a threshold valueof around 1018− 1019 Pa s. During the post-depositional stage, the effectiveviscosity may fall below this threshold (e.g. due to changing temperature ormigrating/diffusing water). The internal deformation of the salt diapir by thedescending blocks increases with decrease in effective viscosity of salt. Un-like salt viscosities beyond the values 1020 Pa s, for the common range of theeffective viscosity of salt (1017−1020 Pa s) relatively high rate of descend isobserved for the intercalated dense layers. The mobility of these dense layersdirectly influences any repository within the diapir. However, the rate of defor-mation that the repository undergoes is strongly dependent on salt rheology.

37

Tabl

e5.

1:M

odel

Para

met

ers:

A∗

isa

pre-

expo

nent

ial

cons

tant

,H∗ -

activ

atio

nen

thal

py,

n-po

wer

-law

expo

nent

,T t−

T bis

tem

pera

ture

atto

pan

dbo

ttom

,res

pect

ivel

y.η

min

,ηm

axan

dη

are

min

imum

,max

imum

and

arith

met

icm

ean

ofth

esa

ltvi

scos

ityoc

curr

ing

inm

odel

s,re

spec

tivel

y.

Mod

elN

atur

alR

ocks

alt

A∗

nH∗

T t−

T bE

ffec

tive

visc

osity

ofsa

lt[P

as]

Com

men

ts[P

a−n

s−1 ]

[kJ

mol−

1 ][◦

C]

ηm

inη

max

η

PFPa

rado

xFo

rmat

ion

6.91

83E

-16

1.39

28.8

00-

140

2.29

E16

2.29

E18

2.6E

18PF

-Col

dPa

rado

xFo

rmat

ion

6.91

83E

-16

1.39

28.8

00

8.6E

171.

9E19

1.09

E19

PF-N

APa

rado

xFo

rmat

ion

6.91

83E

-16

1.39

28.8

00-

140

7.14

E16

3.2E

199.

7E18

No

anhy

drite

AI

Ave

ryIs

land

1.44

54E

-32

4.10

33.6

0-14

05.

9E18

1.7E

231.

29E

20A

I-O

V21

Ave

ryIs

land

1.44

54E

-32

4.10

33.6

0-14

01.

7E18

2.7E

225.

2E19

Ove

rbur

den

visc

os-

ityis

1021

Pas

AI-

OV

22A

very

Isla

nd1.

4454

E-3

24.

1033

.60-

140

6.2E

181.

7E22

8.9E

19O

verb

urde

nvi

scos

-ity

is10

22Pa

sV

DV

ache

rie

Dom

e4.

1687

E-1

62.

2262

.90

0-14

05.

7E16

1.0E

211.

05E

19R

DR

icht

onD

ome

2.23

87E

-32

5.01

82.3

00-

140

2.9E

191.

0E25

4.3E

20SF

Sala

doFo

rmat

ion

1.54

88E

-35

4.90

50.2

00-

140

4.0E

191.

0E25

4.2E

20PB

Perm

ian

Bas

in4.

6774

E-3

04.

5072

.00

0-14

02.

9E19

9.8E

244.

0E20

N17

New

toni

an10

171

-0-

140

--

-N

18N

ewto

nian

1018

1-

0-14

0-

--

38

6. Concluding Remarks

In order to assess the potential importance of sinking or mobility of entrainedblocks within a diapir, it is important to understand how an anhydrite layerwas initially entrained into a diapir. Two time-scales have to be considered:the time scale of diapiric growth t1 (e.g. by down-building) and the time scaleof sinking of the blocks, t2. Keeping all material parameters the same, entrain-ment of dense blocks requires that t1 < t2, i.e. during diapiric rise, the blocksdo not sink faster than the diapir grows. The time scale of the diapiric growth(t1) is governed by four main parameters (sedimentation rate, salt viscosity,perturbation width and stratigraphic location of dense layer), which are sys-tematically studied in this thesis.

In a system where lateral movement (extension and shortening) is insignif-icant, keeping all material parameters constant, the condition for entrainmentof dense blocks by a down-built diapir is that the sedimentation rate should beless than a maximum value smax, so the diapir remains uncovered and it has tobe greater than a minimum sedimentation rate smin, otherwise no entrainmentwill take place. Therefore, sedimentation rate within the range where the con-dition smax < se < smin is fulfilled is required for the entrainment. Dependingon the evolution history of diapirs modeled in this thesis, we distinguishedthree ranges of sedimentation rate:

I High sedimentation rates (> 4 mm a−1), which bury the initial pertur-bation and results in an immediate sinking of the embedded anhydritelayer within the salt layer. In this case, salt supply to the perturbation iszero and the anhydrite layer is not entrained.

II Intermediate sedimentation rates (3–0.5 mm a−1) that are initially lessthan the rate at which the crest of the diapir is growing so that the crestof the diapir remains uncovered and provides highest salt supply. Mod-els with intermediate sedimentation rate are characterized by high per-centage entrainment (80-90 percent).

III Slow sedimentation rates (0.25–0.05 mm a−1) that segments the diapirdue to formation of mini-basins and reduces salt supply. Entrainment ofthe anhydrite layer in these models is limited (less than 30 percent).

Down-building and formation of a diapir is a slow process (c.f. the timescale of 10–100 Myr) and requires relatively stiff salt rheology (effective vis-cosity > 1019 Pa s) for efficient entrainment of dense blocks at geologicallyreasonable rates. However, during the post-depositional stage, when the saltsupply to the diapir ceases and the diapir becomes inactive, entrained seg-ments inevitably sink back into the diapir. As the time scale of sinking ofdense blocks (t2), controls the mobility of the blocks, the condition t2 > t1

39

has implicitly been taken as a guarantee for the safety of the stability of anyrepository built within a diapir where dense blocks exist. However, as it isshown in this thesis if the effective viscosity of salt falls below the thresholdvalue of around 1018− 1019 Pa s, the mobility of anhydrite blocks might in-fluence any repository within a diapir. Moreover, redistribution of stresses atboundary between salt rocks with distinctly different creep properties (e.g., ifthe salt layers within Zechstein Formation (z2, z3 and z4) from the Gorlebendiapir differ in rheology) can control the direction and the rate of sinking ofthe dense blocks.

An accurate monitoring of the current deformation within the Gorleben di-apir will help us understand advanced diapirism of evaporites with a layeredrheology. The structural evolution of a formerly tabular salt layer in which ananhydrite layer of high density and high viscosity is entrained is a key for un-derstanding the internal deformation of any repository built in a diapiric struc-ture. The development of such structure must reliably be forecasted within thenext one million years.

40

7. Summary in Swedish

I den här avhandlingen används resultat från numeriska modeller för att visaatt externt inaktiva saltstrukturer, vilka utgör potentiella förvarsplatser för ra-dioaktivt avfall, skulle kunna vara internt aktiva på grund av närvaro av inretunga lager eller block.

I de tre uppsatser som avhandlingen bygger på används saltdiapiren i Gor-leben, vilken diskuterats i samband med den slutgiltiga förvaringen av högak-tivt radioaktivt avfall, som en allmän referens.

I de första två uppsatserna presenteras systematiska studier över deparametrar som kontrollerar utvecklingen av en saltdiapir och vad somhänder om den innehåller ett tungt anhydritlager. Reslutat från de numeriskamodellerna visar att inkorporering av ett tungt anhydritlager i en saltdiapirberor av fyra parametrar: sedimentationshastighet, saltets viskositet,störningens bredd, samt den stratigrafiska lokaliseringen av det tungalagret. Den kombinerade effekten av dessa fyra parametrar, vilka hardirekt betydelse för salttillförselns hastighet (volym/yta förhållandet av dettillförda saltet som funktion av tiden), kommer att bestämma diapirens formoch sättet på vilket inkorporeringen av det tunga lagret sker. Saltdiapirer,"nedåtbyggda" med högviskösa sedimentära enheter har potentialen attväxa med ett inbäddat anhydritlager och utarma källan (salttillförselnupphör). När saltkällan snabbt minskar eller upphör helt och hållet kommeremellertid det inbäddade anhydritlagret/segmentet att börja sjunka inomdiapiren. I inaktiva diapirer är detta ofrånkomligt och starkt beroende avreologin hos det salt som är i direktkontakt med anhydritlagret. Under detpost-depositionella stadiet, och om saltets effektiva viskositet är undertröskelvärdet 1018−1019 Pa s, skulle anhydritblockets mobilitet kunna tänkaspåverka ett slutförvar inne i diapiren. Daipirens interna deformation somfunktion av de nedåtgående anhydritblocken kommer emellertid att minskasnär saltets effektiva viskositet ökar.

Resultaten i avhandlingen visar att saltstrukturer som innehåller tunga ochviskösa lager/block med stor sannolikhet kommer att undergå interna defor-mationsprocesser när de tunga lagren börjar sjunka inne i diapiren. Blockensstorlek och orientering bestämmer deformationsmönstret. Applicering av re-sultaten från modelleringarna på Gorlebendiapiren visar att sjunkhastighetenhos det interna anhyritlagret skiljer sig på olika sidor av diapiren vilket an-tyder icke symmetrisk deformation. Av detta kan man dra slutsatsen att ensjunkning av anhydritblocken inne i Gorlebendiapiren kommer att initiera ettinre, icke symmetriskt flöde.

41

Acknowledgments

I have crossed paths with several colleagues since the beginning of my studiesand would not have been able to carry out all of this work alone. I would liketo express my deepest gratitude to all of those who have been involved andsupported this work.

Especially, I would like to mention the supervisor of this work, ProfessorHemin Koyi, who introduced me to this field of geology and offered me thepossibility to do research. He shaped my learning with his valuable geologicalinsight and I would sincerely like to express my deep feelings towards him.I would also like to acknowledge Professor Harro Schmeling, whose usefulsuggestions have helped to give this project its current shape, as he was my co-supervisor and the author of the FDCON numerical code used in this research.Without his valuable contribution, this work would not have been possible.Many thanks also go to Professor Christopher Talbot for raising my interestin Global Tectonics and for patiently answering all of my questions related tosalt tectonics.

I would like to thank the people who shared their valuable knowledge andinspired me in the very beginning of my PhD research. I would like to ex-press my gratitude to Jacek Gutowski who kindly introduced me to basics ofgeology, Stanislaw Burliga for sharing his knowledge about salt tectonics.

I would like to extend deep gratitude to my former colleague in Uppsala anddear friend Faramarz Nilfouroushan, who in the early years of my postgradu-ate studies helped me greatly to really get started. I thank Faramarz for proofreading this thesis. Thank you to Örjan Amcoff, who kindly translated thesummary of the thesis into Swedish, and to Taher Mazloomian, Anna Caller-holm and Leif Nyberg who helped me to print the thesis and who providedtechnical assistance critical for completing the project.

I am thankful to Murad Odisharia and Guram Odisharia, who paved my wayinto science. They were instrumental in my coming to Uppsala University as aMaster student. I studied under the supervision of Sergej Zilintikevich and theco-supervision of Igor Ezau. I would like to express my profound gratitude tothem for inspiring me and helping to define my research topic.

I feel everlasting gratitude for lecturers and professors of the Tbilisi StateUniversity, where I enrolled with encouragement from my physics teacherJimsher Qajaia and my mathematics teacher Tengo Chemia. They instilled inme from an early age the desire and skills necessary to obtain a Universitydegree. I greatly enjoyed collegiate life at Tbilisi University, where I madelifelong friends that I have always missed during my stay in Uppsala.

43

Uppsala is a place that gives great opportunity to meet people from all overthe world. I have to extend my thanks to those people that I met during mystay in Uppsala: Albert Mihranyan , Giorgi Metrevely, Davit Khoshtaria, JuanDiego Martín, Jose Luis Perez, Petros Mouyiasis, Stela Tagliati, Tomas Björk-lund and Caroline Burberry. I would like to convey thanks to all of you foryour valued support and friendship.

Next, I wish to thank the complete collegium: Sadoon Morad, Hans An-nersten, Hans Harryson, Peter Lazor, Håkan Sjöström, Ulf Andersson, HowriMansurbeg, Khalid Al-Ramadan, Erik Ogenhall, Sofia Winell, Zuzana Konop-kova, Kristina Zarins, Osama Helal, Lijam Hagos Zemichael, Tuna Eken,Majid Beiki, David Gee , Christoph Hieronymus, Juliane Hübert, Nazli Is-mail, Nina Ivanova, Thomas Kalscheuer, Artem Kashubin, Hesam Kazemeini,Alireza Malehmir, Peter Schmidt, Hossein Shomali, Arnaud Pharasyn andFrances Mary Deegan. All personnel in the department of Earth Sciences,who, during these past several years, in innumerable ways helped me to carryout this work. I also thank VR, The Swedish Research Council for providingthe necessary support for this research.

Finally, special and dearest thanks to my family and my girlfriend who lightup my life and who have given me something else to think about during thesepast four years.

44

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