Stratabound Rayleigh convection observed in a 4D ... · Stratabound Rayleigh convection observed in...

Stratabound Rayleigh convection observed in a 4Dhydrothermal reactive transport model based on theregional geological evolution of Allermohe (Germany)

M. KUHN1 ,* AND A. GUNTHER2 ,�

1TU Hamburg-Harburg, Water Management and Water Supply, Hamburg, Germany; 2TU Braunschweig, Department of

Environmental Geology, Braunschweig, Germany

ABSTRACT

We investigated stratabound Rayleigh convection as a means of transport for leaching of solutes from salt diapirs,

their dissipation into the Rhaetian sandstone aquifer and subsequent precipitation of anhydrite with resulting

cementation. Reactive transport modelling has been conducted in the context of the recent structure of the Aller-

mohe site and its palaeogeological development. Resulting flow fields depict large (km-scale) stratabound Ray-

leigh convection. However, our simulations show the vicinity of the Allermohe well as a potential area of

anhydrite dissolution in contrast to field observations, with precipitation concentrated elsewhere.

Key words: Allermohe, hydrochemical modelling, numerical modelling, palaeohydrology, SHEMAT, solute

transport, structural geological modelling

Received 29 September 2005; accepted 16 January 2007

Corresponding author: Michael Kuhn, RWTH Aachen University, Applied Geophysics, Lochnerstrasse 4–20, 52056

Aachen, Germany.

Email: [email protected]. Tel: +49 241 80 94831. Fax: +49 241 80 92132.

Geofluids (2007) 7, 301–312

INTRODUCTION

The Allermohe, well located south-east of Hamburg, Ger-

many (Fig. 1), was recently considered for the installation

of a major geothermal heating station (GHS). The target

aquifer for hot water was the Rhaetian sandstone, which

has fed the GHS of Neustadt-Glewe (100 km east of Ham-

burg) with up to 120 m3 of water per hour since 1995.

To reach the Rhaetian sandstone, the depth of the Aller-

mohe, 1 bore was increased to 3300 m in 1997, so that it

now taps a 70-m-thick sandstone aquifer with a tempera-

ture of 125�C. Although the temperature and thickness of

the aquifer are ideal for geothermal energy production, the

porosity, originally as high as 20%, is now largely occluded

by secondary anhydrite cement. As a result, the extractable

amount determined by a pumping test in 1998 is only

3 m3 h)1, and is too low for an economical use of the

resource (Baermann et al. 2000).

It is clear that the thermal environment is affected by

the presence of salt domes, and that the temperature field

is strongly coupled to the groundwater flow field as well as

to chemical reactions of salt dissolution. It is therefore use-

ful to investigate the interplay between those processes to

test conceptual hypotheses concerning anhydrite cementa-

tion. Intense research has been conducted in the past two

decades on fluid flow, heat transfer and mechanisms of salt

transport in aqueous solution.

The starting point for many of these investigations has

been the work of Hanor (1987), who presented geochemi-

cal and physical evidence for the existence of density inver-

sions in the Gulf of Mexico Coast pore fluids driven by

large-scale convective fluid flow with flow rates greater

than 1 m day)1. Hanor (1987) qualifies this conclusion by

noting that this fluid overturn must take place along a net-

work of interconnected sands because it is otherwise not

possible to drive large volumes of fluid at high velocity

across the existing shale beds.

*Present address: RWTH Aachen University, Applied Geophysics,

Lochnerstrasse 4–20, 52056 Aachen, Germany.

�Present address: Federal Institute for Geosciences and Natural

Resources, Stilleweg 2, 30655 Hannover, Germany.

Geofluids (2007) 7, 301–312 doi:10.1111/j.1468-8123.2007.00182.x

� 2007 The Authors

Journal compilation � 2007 Blackwell Publishing Ltd

Sharp et al. (2001) also studied the Gulf of Mexico

basin, a region penetrated by thousands of boreholes that

have produced extensive data sets, to resolve the question

of how observed high levels of sediment diagenesis can

occur in basins dominated by low-permeability shales. It

has been shown by Sharp et al. (2001) that diffusive trans-

port of diagenetic reactants cannot account for required

mass fluxes but that density-driven free convection can. As

shown by Bjorlykke et al. (1988), low-permeability argilla-

ceous strata in basins may retard or prevent such convec-

tion. In contrast, Sharp et al. (2001) support the concept

of free convection in the vicinity of salt domes. They have

shown that cross-formational solute transport due to con-

vection is feasible in areas with relatively thin shale layers

possessing permeabilities at the upper end of the expected

range of shale hydraulic conductivities, and can explain

field-based observations of salinity inversions. In their

study, heterogeneity in shale permeability was modelled as

conduits to enhance the potential for free convection. Wil-

liams & Ranganathan (1994) also applied conduits with

high vertical permeabilities of 10)14 m2, in combination

with simulation of dewatering processes, to explain thermal

anomalies around salt structures. Their simulations show

that the absence of conduits with sufficiently high vertical

permeability precludes the formation of thermal plumes

due to convection. However, they admit that there is no

knowledge on how wide, deep and permeable such con-

duits are or even how common or uncommon they might

be. Moreover, in regions with laterally extensive interbed-

ded shales and sands, the prerequisites formulated by Wil-

liams & Ranganathan (1994) may be difficult to satisfy.

Evans et al. (1991) point out that in salt dome environ-

ments groundwater is driven largely by density gradients

rather than by hydraulic head gradients. Those density gra-

dients arise from thermal effects due to the high thermal

conductivity of the salt relative to the surrounding rock,

and from varying salinities, due to dissolution of the salt

itself. Thermally driven convection near salt domes presents

a simple and plausible transport mechanism provided the

thermal effects are not overwhelmed by salinity effects

(Evans & Nunn 1989). The numerical model of Evans

et al. (1991) was able to predict upward flow, needed to

explain salinity inversion around salt structures in the Gulf

of Mexico, as shown by Hanor (1987), only when regional

groundwater salinity was relatively homogeneous and high.

The modelling approach presented in this study focused

solely on the Rhaetian aquifers without directly including

the adjacent strata because: (1) Allermohe region is charac-

terized by massive shales with thickness of up to 1000 m

both above and below the Rhaetian aquifers; (2) detailed

information about water composition and rock properties

are scarce at these depths; (3) convection is likely to have

been able to proceed without inputs from or outputs to

the adjacent shales; and (4) there is little definitive evi-

dence for the existence of fluid conduits that would have

allowed cross-formational flow. Although the conceptual

model is simple, the structural and numerical model is

sophisticated. The intention of the 4D numerical studies

(3D spatial and a geological structure-evolution through

time) using the simulation programme SHEMAT (Bartels

et al. 2003) is to:

• present a new technique of loose coupling between

structural geology and reactive transport modelling to

create a 4D numerical model taking into account chan-

ging topographic boundaries over geological time scales;

• emphasize the probability of stratabound Rayleigh con-

vection, without cross-formational flow, in structures

with strong relief close to salt domes;

• assess the hypothesis of Lenz et al. (1997), with regard

to stratabound Rayleigh convection, that the cementa-

tion of the Rhaetian reservoir is a result of leaching of

solutes from the neighbouring salt domes and subse-

quent precipitation of anhydrite.

Previous numerical simulations of circulation around salt

domes were two-dimensional (2D). We have developed a

3D model because 2D simulations of the Allermohe site

are not able to represent the flow field and temperature

distribution accurately. The geometry around the Aller-

mohe Well, in the vicinity of two salt structures, is too

complex to be approximated in 2D. Unlike previous stud-

ies, chemical reactions are implemented not only via trans-

port boundary conditions that reflect NaCl dissolution and

precipitation, but through a chemical module in SHEMAT

specifically designed for the numerical solution of mineral

reactions in highly saline solutions (Kuhn 2003).

4000

3500

3500

300025

00

2000

3500

MeckelfeldDiapir

AW

Cross-Section 1

Cross-Section 2

Cross-

Section 3

Cross-Section 4

ReitbrookDiapir

53°30' N10°00' E

53°24' N10°10' E

Lowermost CretaceousUpper JurassicMiddle JurassicPemian Salt

Major Normal FaultSyncline AxisAnticline AxisBoundary of Salt Domes

3 km

Fig. 1. Geological subcrop map (Lower Cretaceous and younger units

uncovered) of the study area (TK 2526, sheet Allermohe, Ehlers 1993,

modified after Baldschuhn et al. 2001). Isolines: depth contours of base of

Keuper horizon (b.s.l.). Also shown: locations of cross-sections and position

of the Allermohe well (AW).

302 M. KUHN & A. GUNTHER


Journal compilation � 2007 Blackwell Publishing Ltd, Geofluids, 7, 301–312

GEOLOGICAL SETTING AND HISTORY

The study site is located SE of Hamburg (Germany)

between 53�24¢–53�30¢N and 10�00¢–10�10¢E (Fig. 1).

The study area is concurrent with part no. 2526 of the

geological map of Hamburg (Allermohe sheet, 1:25,000;

Ehlers 1993). The Allermohe well has been drilled at the

position 53�28¢N and 10�06¢E in the centre of the NE

quadrant of the map sheet (Fig. 1).

More than 100 deep boreholes have been drilled within

the area of Allermohe during hydrocarbon and iron ore

exploration. As a result, the subsurface is very well charac-

terized to the depth of the Lower Jurassic Formation. The

occurrence, thickness and character of older stratigraphic

successions, like the Rhaetian sandstone, investigated here

can only be deduced from deep boreholes outside the

study area and from seismic investigations (Frisch 1993).

The Rhaetian sandstone is the topmost unit of the Keuper

and will therefore be represented in the following by the

Lias depth base (Fig. 2). The area of Allermohe is situated

at the SE margin of the central North German Basin. A

prominent NE–SW striking and SE dipping normal fault

cuts the centre of the Allermohe area (Fig. 1, between the

salt structures). This structure was active during Mesozoic

times (Jaritz 1973), probably due to isostatic compensation

in an extensional tectonic setting, and was reactivated in

Early Tertiary times. From Late Triassic times (Middle

Keuper) until the end of the Mesozoic, extensional normal

faulting caused the uplift of the so-called ‘Hamburg Block’

and the construction of the ‘Quickborn Swell’ (both out-

side the Allermohe map sheet). Salts from the Zechstein

(Upper Permian) and Rotliegend (Lower Permian) accu-

mulated above the faults. The mobilization of the Rotlieg-

end salt raised the overlying Zechstein in some areas and

even partly pierced this formation. This led, most probably,

to the intrusion of Early Permian salts into the diapirs of

Meckelfeld and Reitbrook. The base of the Zechstein is

located at an average depth of 4700–4900 m below mean

sea level (b.s.l.). However, due to halokinesis (deformation

of halite by flowage) the Zechstein base was steeply

inclined and even uplifted up to 4300 m b.s.l. in some

areas. Salt migration resulted in a lowering of the Zech-

stein base to depths of some 5700 m b.s.l. north of the

Meckelfeld salt dome (Frisch 1993). In the SW quadrant

of the Allermohe map sheet, the Meckelfeld salt dome

evolved in the Jurassic, during Dogger times, and at the

eastern margin the Reitbrook salt dome was emplaced a lit-

tle bit earlier, in the Jurassic during Lias times. The upper

surface of the Meckelfeld salt dome is oval and slightly

NE–SW striking with an area of approximately 19 km2

(Fig. 1). Its maximum extent is 7 km · 4 km and it is

hosted within the surrounding Upper Cretaceous forma-

tions. The actual vertical extent of the Meckelfeld diapir is

approximately 3000 m, and it is situated on top of the

major Permian saliniferous residuals (Zechstein, Rotlieg-

end; Frisch 1993). The western upper surface of the Reit-

brook salt dome extends to the eastern margin of the

Allermohe map sheet. It is also ovoid in shape and elon-

gates in the NW–SE direction (Fig. 1). The Reitbrook dia-

pir measures 3–4 km in diameter at its maximum extent.

The salt dome surface has been uplifted to a level of

850 m b.s.l. The overlying Cretaceous and Cenozoic layers

are not pierced by the Reitbrook diapir.

GEOLOGICAL MODEL

Allermohe 3D structure

The 3D model of Allermohe has been obtained by digit-

izing and attributing georeferenced lithological and struc-

tural contour lines of major stratigraphic units from the

‘Tectonic Atlas of Northwest Germany’ (Baldschuhn et al.

2001). To construct surfaces from the data, triangulated

irregular networks (TIN) were computed from the georef-

erenced lithological data and converted to grids with pixel

sizes of 250 m · 250 m and 50 m in depth. Additional

structural features, such as the traces of faults and fold

axes, were incorporated as breaklines for the TIN construc-

tions. The final 3D GIS model measures 11 km · 11 km

and has a depth of 6000 m. The size of the modelling area

was chosen based on analogy with a study by Williams &

Ranganathan (1994) of a setting similar to the North Ger-

man basin in which salt structures are located 10–30 km

apart from each other. The potential Rhaetian reservoir

sandstone is located at the top of the Keuper horizon

(Fig. 2).

Because the structural setting of the Allermohe area is

affected by the intrusion of the two salt domes, Meckelfeld

and Reitbrook, the topography of the Keuper layer is

highly influenced by halokinetic processes. In general, as

seen from Fig. 2, topological relief of the formations

increases with increasing depth. In the SW quadrant of the

area, the Meckelfeld salt dome occurs almost completely

within the study area (Fig. 1). The base of the Keuper lies,

Topography

Tertiary

Upper CretaceousLower CretaceousDoggerLiasKeuperUpper BuntsandsteiLower BuntsandsteiZechstein

Meckelfeld diapir

0

1

2

3

4

5

km b.s.l.

Fig. 2. Structural configuration of Allermohe region measuring 11 km ·11 km with a depth of 6000 m (no vertical exaggeration) and obtained by

digitizing georeferenced structural contour lines of major stratigraphic units

from the ‘Tectonic Atlas of Northwest Germany’ (Baldschuhn et al. 2001).

Compare with Table 1 for petrography.

4D hydrothermal reactive transport model of Allermohe 303



depending on its location, between 1800 m and 4500 m

b.s.l.

Cross-sections cut out of the 3D model

From the GIS-based structural grid data (Fig. 2), cross-sec-

tions of any desired orientation (Fig. 1) through the 3D

model can be derived using the program SECTION (Gun-

ther 2003a) to illustrate the complex architecture around

the Allermohe well. Cross-section 1 from west to east,

ending in the Reitbrook salt dome, is displayed in Fig. 3.

The main stratigraphic units (ages and petrography given

in Table 1) are the Cenozoic, Upper and Lower Creta-

ceous, Dogger, Lias, Keuper, Upper and Lower Buntsand-

stein and the Zechstein layer. The diapir, constituted from

Zechstein salt, pierces the overlaying formations from the

Buntsandstein to the Dogger, whereas stratigraphic layers

from the Lower Cretaceous upward are domed up. The

pre-Cretaceous layers descend smoothly away from the

Reitbrook diapir and start to ascend slightly again at a dis-

tance of approximately 6 km to the so-called Hamburg

block, which lies outside the Allermohe map sheet.

The WNW–ESE cross-section 2 (Fig. 4) crosses the

Allermohe well. The stratigraphy is similar to that of cross-

section 1, with the exception that the pre-Cretaceous layers

do not ascend towards the Hamburg block but remain

more or less at the same level. Cross-section 3 (Fig. 5) is

oriented SW to NE and runs through both salt domes.

The pre-Cenozoic layers descend away from the diapirs

Cenozoic (Pleistocene + Tertiary)

Upper CretaceousReitbrook

Zechstein

Dogger

LiasKeuper

Buntsandstein

5 km

W E

Buntsandstein

5 km5 km

W

Fig. 3. Cross-section 1 of the 3D model (compare with Fig. 1 for location;

no vertical exaggeration). The Rhaetian sandstone is located at the top of

the Keuper horizon.

[Correction made 27 June 2007: high resolution figure inserted]

Table 1 Stratigraphic column (TK 2526, sheet Allermohe, Ehlers 1993).

Stratigraphy Petrography (high to low probability) Thickness ca. (m) Time ca. (Myr)

Pleistocene Sand, silt, clay 0–250 0.01–1.7

Tertiary Sand, silt, clay, limestone, shale, sandstone 585–1200 1.7–66

Upper Cretaceous Marlstone, limestone, chalk 290–1000 66–98

Lower Cretaceous Shale, marlstone, sandstone 50–220 98–140

Dogger Shale, marlstone, sandstone 0–900 160–185

Lias Shale 0–1000 185–210

Keuper Shale, marlstone, sandstone, salt, anhydrite, gypsum 250–1000 210–230

Upper Buntsandstein Shale, marlstone, sandstone, salt, dolomite, anhydrite 500–650 245–247

Lower Buntsandstein Sandstone, shale 500–700 247–250

Zechstein Salt, anhydrite, limestone 100–3800 250–270

Rotliegend Shale, sandstone, salt, volcanics Not known 270–290

The Rhaetian sandstone is located at the top of the Keuper horizon.


Upper Cretaceous Reitbrook

Zechstein

DoggerLias

Keuper

Buntsandstein

5 km

WNW ESEAllermöhe

5 km


no vertical exaggeration).



Upper Cretaceous

Reitbrook

Zechstein

Dogger

Lias

Keuper

Buntsandstein

Meckelfeld

5 km

SW NE

Buntsandstein

5 km5 km







and reach a minimum between them. The formations are

more steeply inclined into the Meckelfeld salt dome than

the Reitbrook salt dome. The deeper layers of the Bunt-

sandstein abut the Reitbrook diapir almost horizontally.

The distance between the diapirs at depth is around 5 km.

Cross-section 4 (Fig. 6) runs NW from the Reitbrook dia-

pir. Characteristic features here are the border between the

Lias and the Keuper, which is nearly horizontal, and the

Buntsandstein layers, which do not show large relief along

this section. The last cross-section shown here runs S–N

(Fig. 7) through the Allermohe well. This cross-section

cuts neither the Meckelfeld diapir nor the Reitbrook salt

dome. The Keuper layer descends slightly towards the cen-

tre of the cross-section and increases in thickness at its

southern and northern borders.

Geological restoration sequence

Previous studies indicate the importance of the structural

geometry on anhydrite transport mechanisms (Kuhn 2004)

and led to investigation of this specific, historic geological

restoration sequence (palinspastic reconstruction). This has

been done to assess stratabound thermal-free convection as

a possible fluid flow driving process in the Rhaetian sand-

stone aquifer. Free convection provides a means of trans-

port for the precipitation of anhydrite but has to be

studied in the context of the evolving structural and litho-

logical geometry resulting from salt tectonics. The basis for

the 3D restoration sequence through time is the structural

model of the study area (Fig. 2). The recent 3D structure

of the Allermohe location is evolved backwards to Keuper

times based on the available geological data. The work has

been done by applying the program RESTORE which

implements a structural modelling algorithm to analyse the

vertical kinematic history from geological cross-section data

(Gunther 2003b). The principle here is the step-by-step

(sequential) restoration of the geological horizons to pre-

deformation and/or presedimentary conditions. For each

restoration sequence all geological horizons were moved

upwards according to vertical shear vectors that were

derived from flattening (straightening) the uppermost hori-

zon of the particular sequence to a fixed elevation datum

(in this case 0 m b.s.l.). All lower stratigraphic horizons to

be restored are moved upwards accordingly. Palaeostruc-

tural models can be derived for a specific geological hori-

zon by restoring it incrementally, by flattening all higher

stratigraphic horizons successively (see Fig. 8 for restor-

ation schema). In this simple restoration approach, both

compaction and isostatic compensation processes have

been neglected. The restoration sequence has been applied

on the Lias depth base because this is equivalent to the

Rhaetian layer at the top of the Keuper. Fig. 9 displays the

shape of the Lias base for different geological times, with

the Meckelfeld salt dome in the foreground and the Reit-

brook diapir in the background. The 4D historic restor-

ation sequence (Fig. 9) is used to examine whether

changing topography of the Rheatian aquifer caused free


Upper CretaceousReitbrook

Zechstein

Lias

Keuper

Buntsandstein

5 km

NW SE

Zechstein

Buntsandstein

5 km5 km





Upper Cretaceous

Zechstein

Dogger

Lias

Keuper

Buntsandstein

5 km

S NAllermöhe

ZechsteinBuntsandstein

5 km5 km

N




Original bedsRestored beds

Target bed Template bed Shear vectors

Fig. 8. Restoration schema.





convection-induced flow fields to vary significantly over

geological time, and to determine when and where anhyd-

rite precipitation was most likely in the vicinity of the

Allermohe well.

Numerical model numerical code and process coupling

The Simulator for HEat and MAss Transport (SHEMAT,

Bartels et al. 2003) has been chosen to carry out the

numerical investigations, because with its graphical user

interface ‘Processing SHEMAT’ (Kuhn & Chiang 2003) it

is an easy-to-use, general purpose reactive transport code

for a wide variety of thermal and hydrogeological problems

in two and three dimensions. SHEMAT solves coupled

problems involving fluid flow, heat transfer, species trans-

port and chemical water–rock interaction. It is a finite dif-

ference code that solves the flow and transport equations

on a Cartesian grid. The ‘IAPWS Industrial Formulation’

(Wagner et al. 2000) is the equation of state used for

water. In SHEMAT, the different flow, transport and reac-

tion processes can be selectively coupled. Flow and heat

transport are coupled in that the fluid parameters density,

viscosity, compressibility, thermal conductivity and thermal

capacity are functions of temperature and pressure. Flow

and salt transport are coupled via fluid density implemen-

ted by a linear approximation. Fluid density is of particular

interest and its accurate representation generally vital for

modelling fluid flow in the vicinity of salt structures. Both

thermal effects and varying salinities may induce buoyancy-

driven free convection. SHEMAT’s chemical speciation

module (Kuhn 2003) is a modification of the geochemical

modelling code PHRQPITZ (Plummer et al. 1988). It

permits calculation of geochemical reactions in brines and

other highly concentrated electrolyte solutions using the

Pitzer virial-coefficient approach for ion activity correc-

tions. Reaction-modelling capabilities include calculation of

aqueous speciation and mineral-saturation as well as disso-

lution and precipitation of mineral phases. For a detailed

description of equations incorporated in the program and

code verification the interested reader is referred to the

book ‘Numerical Simulation of Reactive Flow in Hot

Aquifers’ edited by Clauser (2003).

Numerical model set-up

A detailed conceptual model is a prerequisite for numerical

modelling. A conceptual model is qualitative but consistent

with all known geological and hydrological data. A numer-

ical model based on the conceptual model can be used to

test the conceptual model’s quantitative viability. The

Allermohe region is characterized by massive shales, with

up to a 1000-m thickness, above and below the Rhaetian

aquifer. Detailed information about water composition and

rock properties are scarce for the required depth range.

Furthermore, there is ongoing debate in the literature

about cross-formational fluid conduits within argillaceous

settings (Williams & Ranganathan 1994; Sharp et al.

2001). Therefore, we have chosen to run a sophisticated

structural and numerical model based on a simple concep-

tual model (Fig. 10). The focus is solely on the Rhaetian

aquifer itself and the potential of stratabound Rayleigh

convection as a fluid flow driving process that could have

led to the observed anhydrite cementation. The Rhaetian

layer has been extracted from the 3D structure and

assigned a uniform thickness of 250 m. The 3D models

were then overlain and underlain with distinct geological

units representing massive shales known to be above and

below the Rhaetian sandstone (Fig. 10). The resulting

depth range of the numerical model is from 1750 m to

4750 m b.s.l. so the salt domes do not impinge on the

model’s top boundary. Simulations are transient with

respect to fluid flow, heat transfer, species transport and

B: Tertiary structure D: Late Cretaceous structure

C: Early Cretaceous structure E: Middle Jurassic structure

A: Recent structure

570 to 200940 to 5701310 to 9401680 to 13102050 to 1680

2420 to 20502790 to 24203160 to 27903530 to 31603900 to 3530

Meters b.s.l.

Fig. 9. Restoration sequence of the Lias depth base representing the Rhae-

tian top; the foreground displays the Meckelfeld salt dome and the Reit-

brook diapir is situated in the background. The shape of the Lias depth

base is shown for recent times (A), Tertiary (B), Early Cretaceous (C), Late

Cretaceous (D) and Middle Jurassic times (E).

Meckelfelddiapir

Reitbrockdiapir

Shale: k = 10-16 m2 φ = 0.005

Shale

Rhaetian sandstone: k = 4.0x10-13 m2, φ = 0.07

Constant temperature: 60°C

Heat flux: 0.06 W m-2

Depth1750 m

Depth4750 m

3000 mThickness

250 m

Fig. 10. Conceptual model of the Rhaetian layer, embedded into massive

shale layers, extracted from the 3D geological model (Fig. 2).




chemical reactions and were conducted for a time period

of 50,000 years for each of the 3D models. The reactive

transport was not coupled in time to the structural evolu-

tion of the model region, but discrete structural configura-

tions at discrete times were modelled. The geologic history

(generic time dimension) from Early Cretaceous times on

was studied in five 3D reactive transport models (Fig. 9A–

E) to investigate varying palaeostructural constraints in the

Rhaetian sandstone. The individual geological 3D struc-

tures of the entire restoration sequence have been trans-

ferred into reactive transport models by discretizing the

modelled areas into cells measuring 250 m · 250 m hori-

zontally with 50-m heights. This means the Rhaetian sand-

stone is discretized by five cells in the vertical direction.

The resulting five models consist of 96 000, 67 200,

59 200, 59 200 and 52 800 cells for the recent structure,

Tertiary, Late Cretaceous, Early Cretaceous and Dogger

times respectively. No-flow, non-reactive simulation runs

were conducted using the model of the recent structure to

determine the purely conductive temperature profile at the

Allermohe well. The numerical models were constructed to

fit the conceptual model (Fig. 10) and major rock types

deduced from the explanations to the geological map of

Hamburg (Ehlers 1993, Table 1). Hence, rock properties

were applied, as listed in Table 2, for rock thermal capacity

(qCP(r)), rock thermal conductivity (kr), porosity (U) and

permeability (k). Rock properties of the Rhaetian sandstone

were obtained from mineralogical investigations of the ori-

ginal Allermohe core material. However, no measurements

of the rock thermal conductivity were available. Therefore,

we applied values of 1.3, 2.5 and 4.5 W m)1 K)1 for the

recent model structure to cover an appropriate range for

sandstones. The models representing the geologic past

(Fig. 9B–E) were performed only with a rock thermal con-

ductivity of 1.3 W m)1 K)1. The initial water composition

was chosen in accordance with formation water analysis

(Table 3). Numerical simulations were performed with a

constant basal heat flow of 0.06 W m)2, corresponding to

published data within the Atlas of Geothermal Resources

of Europe (Hurter & Haenel 2002). At the upper bound-

ary, a constant temperature (e.g. 60�C for the recent struc-

ture) varied according to the burial depth of the particular

model (e.g. 1750 m for the recent structure) and the geo-

thermal gradient. The lateral boundaries of the model are

closed to heat transfer (perfectly insulated). All six bound-

aries are impermeable, defining a closed box with no

sources or sinks for the pore fluid. The initial hydraulic

heads were defined on the basis of burial depths ranging

from 4500 m for the recent structure to 1650 m for the

Dogger structure. We exclude regional flow based on the

findings of Muller & Papendieck (1975), who inferred

stagnant water conditions in the formations of the Rhaet

from chemical and isotopic analyses of the formation

waters.

We assume that calcium and sulphate and sodium chlor-

ide are available as anhydrite and halite, respectively, from

the salt structures only. The chemical calculations are based

on the ‘equilibrium’ assumption that reaction rate is very

fast compared with other processes involved. This neglects

reaction kinetics, consistent with the fact that the satura-

tion length of anhydrite is much less than the dimensions

of the model cells (Bartels et al. 2002). The Rhaetian sand-

stone is initially assumed to be totally free of anhydrite.

The aim is to determine anhydrite precipitation patterns

for the entire area of the Allermohe map sheet and to

investigate whether precipitation of anhydrite may occur at

the Allermohe well due to anhydrite dissolution from the

diapirs and transport of calcium and sulphate through the

regional structures.

RESULTS OF REACTIVE TRANSPORTMODELLING

Recent structure

Simulated temperature can be compared with down hole

temperature survey conducted before the pumping test in

1998 to assess whether the simulated thermal field is rea-

sonable; few other deep data are available for comparison

with the numerical results. Both purely conductive and

coupled flow and heat transfer models show reasonable

agreement (Fig. 11). Free thermal convection results from

an unstable distribution of fluid density because of tem-

perature or salinity differences. When a porous medium is

heated from below, cool, dense fluid is at the top and hot,

less dense fluid at the bottom of the system. This situation

is gravitationally unstable, because cool fluids tend to sink

and hot fluids tend to rise, eventually leading to free ther-

mal convection. This instability is intensified by the flow

towards the diapirs, where the water dissolves salt and

becomes more dense, then sinks between the diapirs. For

the recent Allermohe structure (Fig. 9A) free thermal

Table 2 Rock properties applied for numerical simulations with data from

Grober et al. (1963), Mercer et al. (1982) and Neuzil (1994).

Rock type

q CP(r)

(MJ m)3 K)1)

kr

(W m)1 K)1)

U

(–)

k

(m2)

Shale 2.04 1.28 0.005 1.0E)16

Salt 1.95 6.1 0.0001 0.0

Rhaetian sandstone 1.6 1.3/2.5/4.5 0.07 0.4E)12

Table 3 Composition of the Allermohe formation water (Lenz et al. 1997)

with total dissolved solids of 218 g l)1, density 1.146 g l)1 and pH 5.4.

Ion

(mg l)1)

K+ Na+ Ca2+ Mg2+ Cl) SO2�4 HCO�3

1250 75 000 6690 1300 132 200 465 240




convection occurs and the evolved flow field turns clock-

wise. Starting from the western border of the Reitbrook

salt dome (Fig. 12) the main flow is directed south, down

the relief of the Rhaetian layer. Approximately 2000 m

south of Reitbrook, flow turns west towards the Meckel-

feld diapir and, ca. 1500 m north of the Meckelfeld diapir,

the main flow direction is guided north down to the deep-

est point of the formation (Fig. 12, see also Fig. 1). Com-

ing out of the trough, the brine flows upwards towards the

Allermohe well and backwards to the Reitbrook diapir.

Classic Rayleigh convection theory was developed in the

context of an infinite, permeable horizontal layer bounded

at the top and bottom by isothermal, impermeable forma-

tions (Lapwood 1948). The dimensionless Rayleigh num-

ber (Ra), indicating the tendency of a system towards free

convection, is based on the ratio of buoyant forces that

drive fluid flow to the viscous forces inhibiting fluid move-

ment. The present model scenario differs from the classic

Rayleigh scenario in that here the lower boundary condi-

tion is constant heat flux instead of constant temperature,

and the convection occurs on an inclined rather than on a

vertical plane. No analytical solution has yet been deduced

for a system as heterogeneous and complex as that of

Allermohe. Therefore, any calculation of Ra, after either,

e.g. Ingebritsen & Sanford (1998) for the purely thermally

driven case of free convection, or Nield (1968) for a sali-

nity-driven case, is not strictly valid. However, determina-

tion of Ra, here following Ingebritsen & Sanford (1998),

because no salinity gradients were initially implemented in

the numerical model, results in values between 188 and

652 for rock thermal conductivities of the Rhaetian sand-

stone from 4.5 to 1.3 W m)1 K)1 respectively. All values

are well above the critical Ra of 4p2 and fluid flow convec-

tion patterns are identical for all three applied rock thermal

conductivities. Although the theoretical Ra can only very

roughly indicates the tendency of the Allermohe system

towards free convection, the high Ra numbers calculated

suggest that free convection is likely to have occurred,

which is consistent with the numerical results. Wood &

Hewett (1982) have also studied free thermal convection

in inclined layers. They realized that fluid motion takes the

form of closed loops, which are located across the vertical

extent of permeable layers, with fluid motion up dip along

the warmer, lower boundary and down dip along the

cooler, upper boundary. This kind of free thermal convec-

tion is not possible within our model because thickness of

the Rhaetian sandstone is insufficient. Our results show

that stratabound convection in inclined layers is possible

with down-flow regions laterally separated from the

up-flow regions. The clockwise direction of the thermal

convection flow field results from the thermal effects of the

salt domes, which owe to their high thermal conductivity

relative to the surrounding rock (Table 2). Around shallow

parts of a salt dome, temperature iso-surfaces bulge up, so

that the surrounding rock is heated by the salt structure.

Around deeper parts, iso-surfaces bulge down and the salt

dome cools surrounding rock masses. The Rhaetian sand-

stone is located around the deeper parts of the Reitbrook

and Meckelfeld diapirs (Fig. 2). Thus, the NW quadrant

(Fig. 1), due to its distance from the salt structures, is the

hottest area in the Rhaetian sandstone. Water migrates up-

dip (cross-section 2, Fig. 4) towards the Allermohe well,

cools in the vicinity of the Reitbrook diapir, and sinks

along cross-section 3 (Fig. 5) into deeper parts of the

Rhaetian sandstone between the salt diapirs. We can con-

clude that the geometry is the first-order control on ther-

mal patterns and as a consequence the clockwise turning

direction of the flow field. The temperature depth profile

of the coupled numerical simulation at the position of the

Allermohe well shows an excellent fit for the entire inter-

section between field observations and numerical computa-

Fig. 11. Temperature survey from 1998 compared with results of numer-

ical simulations based on the Rhaetian layer extracted from the 3D struc-

ture of the Allermohe site (Figs 2 and 10).

Fig. 12. Plan view of the modelled area (Allermohe map sheet) for the

recent structure (Fig. 9A) showing the Allermohe well (black square) and

the Meckelfeld (SW) and Reitbrook (E) diapirs. The arrows display the Dar-

cian flow field from the 3D simulation. Black arrows represent areas where

anhydrite precipitation in the Rhaetian sandstone exceeds 10 mol m)3 after

a simulated time period of 50,000 years.





tions from 1750 m to 3250 m depth, including the depth

range of the Rhaetian sandstone from 3000 m to 3250 m

(Fig. 11). Although free Rayleigh convection is observed,

the temperature profile still looks conduction dominated,

with a slightly increased gradient compared with the purely

conductive reference model. The black arrows in Fig. 12

indicate areas anhydrite precipitation amounts exceeding

10 mol m)3 due to transport of solutes from the salt struc-

tures and subsequent precipitation of anhydrite. This

threshold value was chosen to delineate areas of precipita-

tion. To entirely occlude the initial porosity of the Rhae-

tian sandstone, about 1500 mol m)3 of anhydrite is

required (determined from rock samples). Anhydrite is a

retrograde dissolving mineral that increases in solubility

with decreasing temperature. As shown by Kuhn et al.

(2002), anhydrite solubility is far more sensitive to tem-

perature than it is to salinity for the investigated ranges of

60–180�C, and 3.0–5.5 mol l)1 NaCl respectively. For any

given temperature, anhydrite solubility is almost constant

over this salinity range. Hence, temperature governs an-

hydrite precipitation and dissolution in our numerical si-

mulations. Anhydrite in thermodynamic equilibrium with

respect to the brine at the salt domes is going to precipi-

tate when flow is downwards, following the shape of the

Rhaetian aquifers, and conversely will dissolve where flow

is ascending. In the simulation, significant amounts of

anhydrite are present only in the southern parts of the

investigated area, not around the Allermohe well. Flow

towards the location of the Allermohe well is directed

upwards along the Rhaetian sandstone, leading to

decreased temperatures and lowering the probability of

precipitation. It is unlikely that excess of calcium and sul-

phate leached from the salt dome would reach the location

of the Allermohe well unless free thermal convection

occurred in the anti-clockwise direction. Temporal develop-

ment of the sodium chloride and calcium concentrations at

the Allermohe site, based on the recent structure, is shown

in Fig. 13. The system develops from initial conditions to

equilibrium within approximately 20,000 years with flow

velocities around 1 m per year (Fig. 12). Whereas tempera-

ture approaches the observed value of 125�C and calcium

concentration remains almost constant, the sodium chlo-

ride concentration increases from the observed amount of

around 3000 mmol l)1 to values above 5000 mmol l)1.

These results can be interpreted in at least two different

ways: (1) a horizontal convection cell does exist, consistent

with the agreement of measured and simulated tempera-

ture profile and Ca concentration. However, the sodium

chloride contents of the formation water may not be gov-

erned by thermodynamic equilibrium in combination with

a permeable dome-aquifer contact zone as assumed in the

simulations. If this contact zone were less permeable, due

to salt sealing the pore space, the transport of sodium and

chloride from the dome into the aquifer could be diffusion

controlled. (2) A stratabound Rayleigh convection cell

does not exist, and the processes of halite dissolution and

precipitation are adequately reproduced in the model. In

this case the fluid flow conditions are not driven by strat-

abound convection, but rather by cross-formational flow.

Cross-formational flow would be facilitated by over-pres-

sured deeper formations and faults and fractures that allow

fluid flow into overlying units.

Restoration sequence (geological history)

As stated before, the evolving flow field within a geologi-

cal system strongly depends on the geometry. The palaeo-

structural geological systems (Fig. 9) were investigated in

the same manner as the recent 3D model (see above). In

none of the cases did anhydrite precipitation occur within

the vicinity of the Allermohe well. The oldest investigated

geological structure is the one representing Dogger times

(e.g. the Dogger base lying flat at 0 m b.s.l.) with the

Lias layer being the uppermost stratigraphic unit. Diapi-

rism had already led to the formation of the salt domes

of Meckelfeld and Reitbrook. The Rhaetian formation dis-

plays only a weak relief in which little free thermal con-

vection evolves. This is in accordance with a comparably

low Rayleigh number for the system of 72 (for a rock

thermal conductivity of 1.3 W m)1 K)1). No significant

precipitation of anhydrite is observed and the results are

not shown here. The next generic structure is the one

representing Early Cretaceous times with the Dogger layer

being the uppermost stratigraphic unit (Fig. 14). The

Rhaetian layer has stronger relief compared with Dogger

times, resulting in a Rayleigh number of 176 (for a rock

thermal conductivity of 1.3 W m)1 K)1). Fluid flow can

be observed throughout the entire area and in the vicinity

of the Allermohe well. However, subsequent anhydrite

precipitation (black arrows) is sparse. The model shows

no active flow field that would be able to transport suffi-

cient solutes from the salt domes to the Allermohe well.

110

120

130

140

150

160

170

0 10 000 20 000 30 000 40 000 50 000

Time (years)

Ca

(mm

ol l

–1),

Tem

per

atu

re (

°C)

2000

2500

3000

3500

4000

4500

5000

5500

6000

NaC

l (m

mo

l l–1

)

Calcium

Temperature

Sodium chloride

Fig. 13. Variation of temperature, sodium chloride and calcium content at

the Allermohe well from initial conditions to a simulated time period of

50,000 years within the recent 3D structure (Fig. 9A).




Fig. 15 exhibits the situation in Late Cretaceous times,

when the Lower Cretaceous stratigraphic unit was the

uppermost layer of the 3D structure. Compared with

Early Cretaceous times (Fig. 14) the flow field has chan-

ged significantly, in such that the main reactive area is

situated in the NW of the map sheet. Fluid flow is

enhanced due to a larger Rayleigh number (here 225 for

1.3 W m)1 K)1) that owes to increasing topography of

the Rhaetian sandstone. The Allermohe well is situated in

the centre of a small stratabound convection cell in con-

tact with the Reitbrook salt structure. However, anhydrite

precipitation around the Allermohe well is not indicated.

The flow field consists of a number of smaller convection

cells compared with the large cell existing within the

recent structure (Fig. 12). Fig. 16 displays the flow field

resulting from the geologic structure at the Allermohe site

during Tertiary times when the Upper Cretaceous layer

was the topmost stratigraphic unit. The pattern of flow

has not changed much from the previous historic stage

(Fig. 15), but the intensity of flow has increased substan-

tially. This is due to the fact that both formation tem-

perature and formation topography increase with burial

depth and as a consequence viscosity decreases and the

thermal gradient increases respectively. Both effects result

in an increased Rayleigh number, which is now 389 (with

rock thermal conductivity 1.3 W m)1 K)1), enhancing the

probability of free thermal convection and intensifying

fluid flow velocities. Significant amounts of anhydrite pre-

cipitate in the southern part of the Allermohe map sheet,

north of the Meckelfeld diapir and between both salt

structures. A clockwise convection cell again occurs in the

vicinity of the Allermohe well. The NW quadrant exhibits

high flow rates but no precipitation of anhydrite.

CONCLUSIONS

The installation of a GHS at the Allermohe location failed

because the target aquifer, the Rhaetian sandstone, was

unexpectedly and extensively mineralized by anhydrite,

reducing porosity and permeability. Although the tempera-

ture and thickness of the aquifer are well suited for geo-

thermal energy production, the extractable amount of

water is too low for an economical use of the resource.

Reactive transport simulations have been conducted to

help understand the observed cementation. Free strat-

abound Rayleigh convection has been shown to be

Fig. 15. Plan view of the modelled area (Allermohe map sheet) during Late

Cretaceous times (Lower Cretaceous ¼ top of stratigraphy).


Fig. 14. Plan view of the modelled area (Allermohe map sheet) during

Early Cretaceous times (dogger ¼ top of stratigraphy).


Fig. 16. Plan view of the modelled area (Allermohe map sheet) during Ter-

tiary times (Upper Cretaceous ¼ top of stratigraphy).





first-order dependent on the geometry of the geological

structure. The temporal development of sodium chloride

concentration (Fig. 13) at the Allermohe well suggests that

either the simulation of NaCl dissolution at the salt dome–

sandstone contact is not accurate enough, or that fluid-

flow drivers other than stratabound Rayleigh convection

need to be taken into account, including processes like

cross-formational flow. The simulations of the entire 3D

historical restoration sequence show the area around the

Allermohe well to be free of anhydrite precipitation during

the geologic past (Figs 12, 14–16). Instead, precipitation

occurs predominantly in southern parts of the investigated

area. The free thermal convection patterns become more

organized and the intensity of the observed fluid flow

increases substantially with burial depth of the formation.

We find that the hypothesis of Lenz et al. (1997) regard-

ing stratabound Rayleigh convection as the process con-

trolling anhydrite precipitation cannot be proven. Their

assumption that the observed anhydrite cementation at the

location of the Allermohe well may be due to solutes lea-

ched from the salt structures, transported through the sys-

tem, and subsequently precipitated, requires a transport

mechanism different from the one investigated here.

Future work will focus on coupling reactive transport

simulations to more sophisticated palaeostructural models,

such that the evolving material parameters of the aquifer

through time (i.e. permeability, porosity and conductivity)

will be incorporated into the restoration process. More-

over, the influence of both local and regional isostatic

effects on the palaeostructural environments will be mod-

elled. This seems particularly important with respect to

halokinetic influences on the structural geometry of the

Allermohe test site. Next, the incorporation of the faults

are crucial to improve the quality of the structural models,

in such that movement along these at particular geological

time intervals will clearly modify the palaeostructural

geometries and also might initiate or facilitate cross-

formational flow to some extent. Our current structural

modelling approach represents a first approach to loose

coupling of palaeostructural and reactive transport models

to study observed mineralization patterns. For further

simulations it is planned to build a 3D numerical model

from the entire 3D geological GIS model presented here

and to investigate cross-formational flow due to thermo-

haline convection and other probable fluid flow drivers.

The reaction boundaries will be refined to evaluate whether

thermodynamic equilibrium is an accurate description of

the chemical processes or if a kinetic approach needs to be

considered.

ACKNOWLEDGEMENTS

Research work reported here was supported by the German

Federal Ministry for Economic Affairs (BMWi) under grant

032 70 95. Special thanks are given to Martin Appold,

Steve Ingebritsen and Ward Sanford for their highly con-

structive reviews.

REFERENCES

Baermann A, Kroger J, Taugs R, Wustenhagen K, Zarth M(2000) Anhydrite cement in Rhaetian sandstones in southeast-

ern Hamburg: morphologie and structure. Zeitschrift fur ange-wandte Geologie, 46, 138–144 (in German).

Baldschuhn R, Binot F, Fleig S, Kockel F (2001) Tectonic atlas ofnorthwest Germany and the German North Sea sector – struc-

tures, structural development, palaeogeography. GeologischesJahrbuch, Reihe A, Heft 153.

Bartels J, Kuhn M, Schneider W, Clauser C, Pape H, Meyn V,

Lajczak I (2002) Core flooding laboratory experiment validates

numerical simulation of induced permeability change in reser-

voir sandstone. Geophysical Research Letters, 29, 1320. DOI:10.1029/2002GL014901.

Bartels J, Kuhn M, Clauser C (2003) Numerical simulation of

reactive flow using SHEMAT. In: Numerical Simulation ofReactive Flow in Hot Aquifers – SHEMAT and Processing SHE-MAT (ed. Clauser C), pp. 5–74. Springer Publishers, Heidel-

berg.

Bjorlykke K, Mo A, Palm E (1988) Modelling of thermal convec-

tion in sedimentary basins and its relevance to diagenetic reac-tions. Marine and Petroleum Geology, 5, 338–351.

Clauser C (2003) Numerical Simulation of Reactive Flow in HotAquifers – SHEMAT and Processing SHEMAT. Springer Pub-lishers, Heidelberg.

Ehlers J (1993) Geological Map of Hamburg 1:25.000, ExplicationsConcerning Sheet No. 2526 Allermohe. Geologisches Landesamt

Hamburg, Freie Hansestadt Hamburg (in German).Evans DG, Nunn JA (1989) Free thermohaline convection in

sediments surrounding a salt column. Journal of GeophysicalResearch, 94, 413–422.

Evans DG, Nunn JA, Hanor JS (1991) Mechanisms drivinggroundwater flow near salt domes. Geophysical Research Letters,18, 927–930.

Frisch U (1993) Pre-Pleistocene. In: Geological Map of Hamburg1:25.000, Explications Concerning Sheet No. 2526 Allermohe (ed.

Ehlers J), pp. 14–27. Geologisches Landesamt Hamburg, Freie

Hansestadt Hamburg (in German).

Grober H, Erk S, Grigull U (1963) Heat Transfer. Springer Pub-lishers, Berlin (in German).

Gunther A (2003a) SECTION: An AVENUE program for deri-

ving arbitrary oriented vertical cross-sections through grid-based

3D models (http://arcscripts.esri.com/details.asp?dbid¼12710,last access 20 March 2007).

Gunther A (2003b) RESTORE: an AVENUE program for vertical

restoration of geological horizons with grid based data (http://

arcscripts.esri.com/details.asp?dbid¼12650, last access 20March 2007).

Hanor JS (1987) Kilometre-scale thermohaline overturn of pore

waters in the Louisiana Gulf Coast. Nature, 327, 501–503.Hurter S, Haenel R (2002) Atlas of Geothermal Resources in Eur-

ope. European Commission, No. EUR 17811.

Ingebritsen SE, Sanford WE (1998) Groundwater in Geologic Pro-cesses. Cambridge University Press, Cambridge.

Jaritz W (1973) Development of the salt structures of NW Ger-

many. Geologisches Jarhbuch A, 10, 1–77 (in German).

Kuhn M (2003) Chemical equilibrium speciation for brines at

high temperatures and ionic strength. In: Numerical Simulation




of Reactive Flow in Hot Aquifers – SHEMAT and ProcessingSHEMAT (ed. Clauser C), pp. 75–170. Springer Publishers,

Heidelberg.

Kuhn M (2004) Reactive Flow Modeling of Hydrothermal Systems.Lecture Notes of Earth Sciences 103. Springer Verlag, Heidel-berg.

Kuhn M, Chiang W-H (2003) Pre- and post processing with

‘‘Processing SHEMAT’’. In: Numerical Simulation of ReactiveFlow in Hot Aquifers – SHEMAT and Processing SHEMAT (ed.

Clauser C), pp. 75–151. Springer Publishers, Heidelberg.

Kuhn M, Bartels J, Pape H, Schneider W, Clauser C (2002)

Modeling chemical brine-rock interaction in geothermal reser-voirs. In: Water–Rock Interaction (eds Stober I, Bucher K), pp.

147–169. Kluwer Academic Publishers, Dordrecht.

Lapwood ER (1948) Convection of a fluid in a porous media.

Proceedings Cambridge Philosophical Society, 44, 508–521.Lenz G, Horn H, Kellner T (1997) Final Report on Geology and

Drilling of the Bore Hamburg Allermohe 1. Geothermie Neu-

brandenburg GmbH (in German).Mercer JW, Thomas SD, Ross B (1982) Parameters and Vari-

ables Appearing in Repository Siting Models. GeoTrans Inc.,

NUREG/CR-3066, US Nuclear Regulatory Commission,

Washington, DC.Muller EP, Papendieck G (1975) Distribution, genesis and

dynamic of deep-seated waters under special consideration of

the Zechstein. Zeitschrift geologischer Wissenschaften, 3, 167–196

(in German).

Neuzil CE (1994) How permeable are clays and shales? WaterResources Research, 30, 145–150.

Nield DA (1968) Onset of thermohaline convection in a porous

medium. Water Resources Research, 4, 553–560.

Plummer LN, Parkhurst DL, Fleming GW, Dunkle SA (1988) Acomputer program incorporating Pitzer’s equations for calcula-

tion of geochemical reactions in brines: U.S. Geological Survey

Water-Resources Investigations Report 88–4153, 310 p.Sharp JM Jr, Fenstemaker TR, Simmons CT, McKenna TE, Dick-

inson JK (2001) Potential salinity-driven free convection in a

shale-rich sedimentary basin: example from the Gulf of Mexico

basin in south Texas. AAPG Bulletin, 85, 2089–2110.Wagner W, Cooper JR, Dittmann A, Kijima J, Kretzschmar H-J,

Kruse A, Mares R, Oguchi K, Sato H, Stocker I, Sifner O, Tak-

aishi Y, Tanishita I, Trubenbach J, Willkommen Th (2000) The

IAPWS Industrial Formulation 1997 for the thermodynamicproperties of water and steam. Journal of Engineering for GasTurbines and Power, 122, 150–182.

Williams MD, Ranganathan V (1994) Ephemeral and soluteplumes formed by upwelling groundwaters near salt domes.

Journal of Geophysical Research, 99, 15667–15681.

Wood JR, Hewett TA (1982) Fluid convection and mass transfer

in porous sandstones – a theoretical model. Geochimica etCosmochimica Acta, 46, 1707–1713.




Stratabound Rayleigh convection observed in a 4D ... · Stratabound Rayleigh convection observed in...

Documents

Transcript of Stratabound Rayleigh convection observed in a 4D ... · Stratabound Rayleigh convection observed in...