Stable-unstable flow of geothermal fluids in fractured rock

$: Stable-unstable flow of geothermal fluids in fractured rock$
Stable–unstable flow of geothermal fluids in fractured rock

T. GRAF1 AND R. THERRIEN2

1Center of Geosciences, Georg-August-University Gottingen, Goldschmidtstraße 3, 37077 Gottingen, Germany;2Departement de Geologie et Genie Geologique, Universite Laval, Ste-Foy, Quebec, G1K 7P4, Canada

ABSTRACT

Density-driven geothermal flow in 3-D fractured rock is investigated and compared with density-driven haline flow.

For typical matrix and fracture hydraulic conductivities, haline flow tends to be unstable (convecting) while geother-

mal flow is stable (non-convecting). Thermal diffusivity is generally three orders of magnitude larger than haline dif-

fusivity and, as a result, large heat conduction diminishes growth of geothermal instabilities while low mass diffusion

enables formation of unstable haline ‘fingering’ within fractures. A series of thermal flow simulations is presented to

identify stable and unstable conditions for a wide range of hydraulic conductivities for matrix and fractures. The clas-

sic Rayleigh stability criterion can be applied to classify these simulations when fracture aperture is very small. How-

ever, the Rayleigh criterion is not applicable when the porous matrix hydraulic conductivity is very small, because

stabilizing fracture–matrix heat conduction is independent of matrix hydraulic conductivity. In that case, the numeri-

cally estimated critical fracture conductivity is nine orders of magnitude larger than the theoretically calculated criti-

cal fracture conductivity based on Rayleigh theory. The numerical stability analysis presented here may be used as a

guideline to predict if a geothermal system in 3-D fractured rock is stable or unstable.

Key words: density, fracture, geothermal, numerical model, stable, unstable

Received 21 April 2008; accepted 25 November 2008

Corresponding author: T. Graf, Center of Geosciences, Georg-August-University Gottingen, Goldschmidtstraße 3,

37077 Gottingen, Germany.

Email: [email protected]. Tel: +49-551-39 7919. Fax: +49-551-39 9379.

Geofluids (2009) 9, 138–152

INTRODUCTION

Understanding the circulation of deep groundwater is essen-

tial to address issues related to nuclear waste disposal, seawa-

ter intrusion, and geothermal energy production.

Groundwater flow can be driven by spatial density differ-

ences that can result from increases in salinity (haline flow)

and/or temperature (geothermal flow). When, for example,

a fluid of high density overlies a less dense fluid, the system

is potentially unstable and density-driven flow may take

place, thereby enhancing fluid convection and increasing, for

example, geothermal energy productivity (Farvolden et al.

1988; Kolditz & Clauser 1998; Hurter & Schellschmidt

2003; Graf & Therrien 2005; Bataille et al. 2006). Here, we

will use the term ‘stable’ when referring to non-convecting

systems and ‘unstable’ when referring to convecting systems.

Fractures can highly disturb the convective pattern of

density-driven flow (Murphy 1979; Shikaze et al. 1998;

Valliappan et al. 1998; Bachler et al. 2003; Graf & Therrien

2007a,b). Fractures are high-permeability structures located

within low-permeability rock, and can therefore permit

density-driven convective flow. In that case, unstable ‘fin-

gers’ form where solutes (or thermal energy) are transported

at a rate that is several orders of magnitude larger than that

resulting from diffusion (or conduction) alone. Conversely,

fractures may also dissipate fingers by creating large haline

(or geothermal) gradients between the fracture and the sur-

rounding rock matrix. In that case, diffusion (or conduc-

tion) may reduce the growth of fingers and stabilize

groundwater flow.

Clearly, the magnitude of fracture and matrix permeabil-

ity controls whether a system is convecting (unstable) or

non-convecting (stable). The greater the permeability of

fracture and rock matrix, the greater the potential of con-

vective flow. If fractures are absent or if fracture permeabil-

ity is very low relative to matrix permeability, it can be

expected that the Rayleigh criterion (Rayleigh 1916) can

be used as the stability criterion. Otherwise, the Rayleigh

criterion fails and numerical models have to be used to

simulate density-driven groundwater flow, and to verify

whether a system is stable or unstable. Here, we will apply

the Rayleigh criterion to fractured systems where fracture

Geofluids (2009) 9, 138–152 doi: 10.1111/j.1468-8123.2008.00233.x

� 2009 Blackwell Publishing Ltd

permeability is very small, such that the system can be con-

sidered to be homogeneous.

Numerical simulations of haline convection in 2-D homo-

geneous porous media (Elder 1967; Wooding 1969; Voss &

Souza 1987; Schincariol et al. 1994; Wooding et al. 1997;

Oldenburg & Pruess 1998; Simmons et al. 1999, 2002) and

in 2-D fractured porous media (Graf & Therrien 2007a)

have indicated that: (i) distinct fingers grow near the salt

source; (ii) at early simulation times (<1 year), the number

of fingers is high and distinct convection develops; and (iii)

at later times (>1 year), fingers coalesce to a single dense

plume and convection is less apparent.

Kolditz (1995) numerically investigated 3-D geothermal

flow at the Soultz-sous-Forets site in France, but without

accounting for density variations. Simulations presented by

Kolditz (1995) suggest that high-permeability fractures are

the dominant transport path for geothermal energy and

that conductive heat flow from the fractures into the sur-

rounding rock matrix eliminates thermal gradients at the

fracture–matrix interface.

Bachler et al. (2003) used an analytical model to study the

impact of fracture zones on hydrothermal convection in the

Rhine Graben. They found that temperature anomalies typi-

cally follow fracture zones, suggesting the presence of con-

vection cells with rotation axes normal to the fractures.

Bachler et al. (2003) concluded that low- and high-tempera-

ture anomalies correspond to the downwelling and upwell-

ing regions of the convection cells, respectively. Bachler

et al. (2003) also undertook a 3-D numerical study of ther-

mal convection, which confirmed their analytical results.

The goal of the present study was to investigate further

density-driven geothermal flow in 3-D fractured porous

rock. We used the HydroGeoSphere numerical model

(Therrien & Sudicky 1996; Therrien et al. 2008) to com-

pare convective flow patterns (fingering) of haline and geo-

thermal flow. Furthermore, the role of fracture and matrix

permeability in the generation and dissipation of geothermal

instabilities was investigated. We identify conditions leading

to geothermal convective flow within the fracture plane, and

determine whether fracture–matrix conduction suppresses

unstable flow. Other questions considered here include the

following: Does the number of geothermal fingers change

with time? Is the number of fingers high at early times, and

do fingers coalesce at later times? Does geothermal density-

driven convection also develop in the rock matrix?

NUMERICAL MODEL

The HydroGeoSphere model

HydroGeoSphere is a 3-D variable-density, saturated–

unsaturated groundwater flow, multi-component solute

transport, and heat transfer model for fractured porous

media, and is based on the FRAC3DVS model (Therrien

& Sudicky 1996). The HydroGeoSphere model applies the

control volume finite element (CVFE) method to the flow

equation, the Galerkin finite element method with full

upstream weighting to solute transport and heat transfer

equations (Therrien & Sudicky 1996; Graf & Therrien

2005). It is assumed that 2-D fracture elements and 3-D

matrix elements share common nodes in the 3-D grid.

Thus, heads, concentrations, and temperatures are assumed

to be identical along the fracture–matrix interface.

For haline flow simulations, the flow equation is coupled

with the solute transport equation, and for the case of geo-

thermal flow, the flow equation is coupled with the heat

transfer equation. This coupling arises because density vari-

ations cause nonlinearities in the flow equation. In both

cases, the coupled system of equations is solved by the

Picard iteration.

The HydroGeoSphere model applies the first level of the

Oberbeck–Boussinesq (OB) approximation (Oberbeck

1879; Boussinesq 1903; Holzbecher 1998; Kolditz et al.

1998) to discretize groundwater flow, solute transport and

heat transfer equations. The OB assumption reflects the

degree to which density variations are accounted for. Level

1 of the OB approach considers density effects only in the

buoyancy term of the momentum equation (Darcy equa-

tion) and neglects density in the governing equations. This

assumption is generally correct because spatial density vari-

ations are commonly minor relative to the absolute density

value (Murphy 1979; Evans & Raffensperger 1992; Kol-

ditz et al. 1998; Bachler et al. 2003). The OB assumption

is, however, not valid when density variations are large.

One example where the OB assumption is not valid is

given in Straus & Schubert (1977), who studied natural

convection of water in thick geothermal layers, where tem-

perature differences of 345 K cause fluid density variations

by a factor of 2. In another example, Jupp & Schultz

(2000, 2004) have examined hydrothermal convection cells

in a porous medium where fluid density varies by a factor

of 25. In comparison, fluid density varies by a factor of

1.25 in the simulations presented here, and the level 1 OB

approximation is valid.

The spatiotemporally discretized matrix equations are

solved using the WATSIT iterative solver package for gen-

eral sparse matrices (Clift et al. 1996) and a conjugate gra-

dient stabilized (CGSTAB) acceleration technique (Rausch

et al. 2005). A more detailed description of the model can

be found in Therrien et al. (2008). Governing equations

for groundwater flow, solute transport, and heat transfer

are presented below.

Governing equations

The following three equations describe 3-D haline and

geothermal variable-density flow in porous media (Bear

1988; Holzbecher 1998):

Flow of geothermal fluids in fractured rock 139

� 2009 Blackwell Publishing Ltd, Geofluids, 9, 138–152

�r � �vf g � Cfluid ¼ SSoh0

otð1Þ

�r � Jsolute � Csolute ¼ �oc

otð2Þ

�r � Jheat � Cheat ¼ �~cð ÞboT

otð3Þ

where Equations (1), (2), and (3) are for fluid flow, sol-

ute transport, and heat transfer, respectively. The diver-

gence operator is given by �[L)1], / is the dimensionless

matrix porosity, v[L T)1] is the average fluid velocity,

SS[L)1] is specific storage, h0[L] is freshwater head, t[T]

is time, c[)] is relative solute concentration, q[M L)3] is

fluid density, �~cð Þb M L�1 T �2H�1� �

is bulk heat capacity,

and T[Q] is absolute temperature. Sources and sinks are

denoted by C, and the solute mass flux and thermal

energy flux are represented by Jsolute and Jheat, respec-

tively, with:

Jsolute ¼ Jadvection þ Jdispersion þ Jdiffusion ð4Þ

Jheat ¼ Jconvection þ Jconduction ð5Þ

The specific storage coefficient accounts for the com-

pressibility of both matrix and fluid, and it is defined as

(Frind 1982):

SS ¼ �0g �m þ ��flð Þ ð6Þ

where q0[M L)3] is reference density, g [L T)2] is gravita-

tional acceleration, am[M)1 L T2] is matrix compressibility

and afl[M)1 L T2] is fluid compressibility.

The fluid flow, solute transport and heat transfer conti-

nuity equations for 2-D discrete fractures are written as

(Therrien & Sudicky 1996; Holzbecher 1998):

�r � vfr� �

� Cfrfluid þ qnjIþ � qnjI� ¼ S fr

S

ohfr0

otð7Þ

�r � J frsolute � Cfr

solute þ XnjIþ � XnjI� ¼ocfr

otð8Þ

�r � J frheat � Cfr

heat þ KnjIþ � KnjI� ¼ �~cð ÞloT fr

otð9Þ

where the last two terms on the left-hand side in each

equation denote normal components of fluid flux, solute

mass flux and heat exchange across the fracture–matrix

interfaces I + and I ). Specific storage in the fracture,

S frS L�1� �

, can be derived from Equation (6) by assum-

ing that the fracture is incompressible, such that

am ¼ 0, and by setting its porosity to 1 (Graf & Therrien

2005):

S frS ¼ �0g�fl ð10Þ

Constitutive equations

Fluid velocity

For variable-density flow conditions, the average fluid

velocity, vi, is a function of both the freshwater head,

h0[L], and relative fluid density, qr ¼ (q/q0) ) 1[)]. Fluid

velocities for matrix and fracture are given by Darcy’s law

(Bear 1988):

vi ¼ �K 0

ij

�

�0

�

oh0

oxjþ �r�j

� �i; j ¼ 1; 2;3 ð11Þ

vfri ¼ �K fr

0

�0

�fr

oh0

oxjþ �fr

r �j cos’

� �i; j ¼ 1;2 ð12Þ

where i and j are spatial indices, l0[M L)1T)1] is reference

viscosity, l and lfr[both M L)1T)1] are fluid viscosity in

matrix and fracture, respectively, xj [L] is space, gj [)] is an

indicator for flow direction with gj ¼ 0 in horizontal

directions and gj ¼ 1 otherwise, and u[)1�] is the incline

of the fracture, with u ¼ 0� for a vertical fracture and

u ¼ 90� for a horizontal fracture. The freshwater hydraulic

conductivities, K0ij and K fr

0 L T�1� �

, of both media are

given by (Bear 1988):

K0ij ¼

�ij�0g

�0

ð13Þ

K fr0 ¼ð2bÞ2�0g

12�0

ð14Þ

where jij[L2] is the permeability of the porous medium,

g[L T)2] is gravitational acceleration, and (2b) [L] is frac-

ture aperture.

Fluid density

If haline flow is simulated, fluid density is only a function

of relative solute concentration c, and it is calculated

with:

� ¼ �0 þo�

oc� c ð15Þ

where ¶q/¶c ¼ (qmax ) q0)/(1 ) 0)[M L)3] is the haline

expansion coefficient. Equation (15) is linear and implies

that freshwater density at c ¼ 0 is q ¼ q0, and that salt-

water density at c ¼ 1 is q ¼ qmax.

If geothermal flow is simulated, fluid density is only a

function of fluid temperature T, and calculated using

(Holzbecher 1998):

140 T. GRAF & R. THERRIEN


where TC [Q] and T [Q] are temperatures in Celsius and

Kelvin, respectively, with T ¼ TC + 273.15.

Fluid viscosity

If haline flow is simulated, the model uses relative

concentration (c ¼ C/C0) as the primary unknown. In

that case, the absolute concentration C is undetermined

and, thus, fluid viscosity has to be assumed to be

constant:

� ¼ �0 ð17Þ

If geothermal flow is simulated, fluid viscosity is only a

function of fluid temperature T, and calculated using (Hol-

zbecher 1998):

Summary of equations for haline and geothermal flow

When simulating haline flow, the HydroGeoSphere model

solves for coupled groundwater flow and solute transport,

and does not consider heat transfer. Flow velocities in the

fracture and the rock matrix are given by Darcy’s law.

Fluid density is calculated using a linear relationship

between density and solute concentration, and fluid viscos-

ity is assumed to be constant.

Here, relative concentration is chosen as the transport

variable because absolute concentrations are a priori

unknown, and fluid viscosity has to be assumed to be con-

stant (Equation 17). This assumption of constant viscosity

for haline flow is a limitation of the results presented here,

because fluid viscosity can change by a factor of 1.5

between freshwater and saltwater. Further studies will

explore the impact of that assumption.

When simulating geothermal flow, HydroGeoSphere

solves for coupled groundwater flow and heat transfer, and

does not consider solute transport. Flow velocities in the

fracture and the rock matrix are given by Darcy’s law.

Table 1 identifies the equations that describe haline and

geothermal flow.

Numerical formulation of buoyancy

HydroGeoSphere simulates variable–density flow in the 3-D

porous matrix and in the 2-D fracture. We use 3-D eight-

node (block) elements to represent the porous rock matrix,

and 2-D three-node (triangular) elements to represent the

non-planar fracture. The numerical formulation of buoyancy

in both matrix and fracture elements is presented below.

Buoyancy vector in 3-D porous matrix elements

Upon discretizing the groundwater flow equation (1) in

the 3-D rock matrix using the CVFE method, the nodal

entries of the buoyancy vector in matrix element e,

ge[L2T)1], are given by

geI ¼

ZV e

K 0ij

�0

��e��e

r �owe

I

ozdV e I ¼ 1; . . . ; 8 ð19Þ

where Ve[L3] is the volume of matrix element e,

��e M L�1T�1� �

is average viscosity in e, ��er �½ � is average

� ¼1000 � 1� TC�3:98ð Þ2

503570 �TCþ283

TCþ67:26

� for 0�C � TC � 20�C

996:9 � 1� 3:17� 10�4 TC � 25ð Þ � 2:56� 10�6 TC � 25ð Þ2�

for 20�C < TC � 175�C

1758:4þ 1000 � T �4:8434� 10�3 þ T 1:0907� 10�5 � T � 9:8467� 10�9 � �

for 175�C < TC � 300�C

8>><>>:

ð16Þ

� ¼1:787� 10�3 � exp �0:03288þ 1:962� 10�4 � TC

�� TC

�for 0�C � TC � 40�C

10�3 � 1þ 0:015512 � TC � 20ð Þð Þ�1:572 for 40�C < TC � 100�C0:2414 � 10 ^ 247:8= TC þ 133:15ð Þð Þ � 10�4 for 100�C < TC � 300�C

8<: ð18Þ

Table 1 Equations used to simulate haline and geothermal flow in fractured

porous rock. Reference is made to the equation number in the text.

Equation Haline flow Geothermal flow

Governing equations

Groundwater flow 1 and 7 1 and 7

Solute transport 2 and 8 NA

Heat transfer NA 3 and 9

Constitutive equations

Flow velocity 11 and 12 11 and 12

Fluid density 15 16

Fluid viscosity 17 18

NA, not applicable.



relative density in e, and weI �½ � is the value of the 3-D

approximation function in e at node I. With usual 3-D

approximation functions for regular 3-D block elements

(cf. Istok 1989), Equation (19) can be integrated for all

eight nodes such that the elemental buoyancy vector

becomes

ge¼K0ij

�0

�e �er �

Lx �Ly

4� �1 �1 �1 �1 1 1 1 1f gT ð20Þ

where Lx and Ly[L] are edge lengths of matrix block e in

x- and y-direction, respectively.

Buoyancy vector in 2-D fracture elements

According to Frind (1982) and Graf & Therrien (2005),

nodal entries of the buoyancy vector in fracture element

(fe), gfe[L2 T)1], are calculated as

g feI ¼

ZAfe

K fr0

�0

��fe��fe

r � cos’owfe

I

o�zdAfe I ¼ 1;2;3 ð21Þ

where Afe[L2] is the surface area of fracture element (fe),

��fe M L�1 T�1� �

is the average viscosity in fe, ��fer �½ � is the

average relative density in fe, wfeI �½ � is the value of the 2-D

approximation function in fe at node I, and �z [L] is the

local z-axis of fe. Upon integration in Equation (21), Graf

& Therrien (2007c,2008b) have derived the following

form of the elemental buoyancy vector for 2-D triangular

fracture elements:

gfe ¼ K fr0

�0

��fe��fe

r � cos’1

2

�x3 � �x2

�x1 � �x3

�x2 � �x1

8<:

9=; ð22Þ

where �xI [L] is the value of the local x-coordinate of

node I.

GEOTHERMAL FLOW IN FRACTURED ROCK

Comparison between haline and geothermal flow in

fractured rock

A 2-D test problem of forced convection

A simplified test problem is presented here to highlight

physical differences between haline and geothermal flow

in fractured rock. The flow system is shown in Fig. 1

and consists of a 2-D domain of dimensions 8 m ·10 m. Forced convection (imposed hydraulic gradient

¶h0/¶x ¼ 0.0625) is assumed to be the main lateral dri-

ving force of salt and heat, while density-driven flow is

neglected. An open fracture is located along the x-axis,

in the flow direction, and it is embedded in a low-per-

meability porous rock matrix. The fracture inlet is

assigned Dirichlet boundary conditions, either C ¼ C0 or

T ¼ T0, while all other boundary conditions are assumed

to be zero-dispersive or zero-conductive flux boundaries

(i.e. ¶C/¶n ¼ 0 and ¶T/¶n ¼ 0, where n is the direc-

tion normal to the domain boundary). Simulation

parameters are given in Table 2, and grid spacing is

chosen to satisfy stability criteria formulated by Weathe-

rill et al. (2008).

Results shown in Fig. 1A indicate that haline transport is

controlled by forced convection (advection) in the fracture

and that fracture–matrix diffusion plays a minor role. Con-

versely, thermal transport is dominated by both fracture–

matrix conduction, and conduction within the matrix

(Fig. 1B).

The fundamental difference between haline and geother-

mal flow is that solutes are mainly transported by advective

transport whereas in this example, heat is mainly trans-

ported by heat conduction through the rock formation.

0 . 9

0 . 7

0 . 5

0 . 3

0 . 1

0 . 9

0 . 7

0 . 5

0 . 3

0 . 1

T T = 0

Rock matrix

Fracture

Flow

T T / 0

x - distance (m)

y-

dis

tan

ce (

m)

1 0 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

9

10

0.1

0.3

0.5

0.7

0.9

C C = 0

Rock matrix

Fracture

Flow

C C / 0

x - distance (m)

y-

dis

tan

ce (

m)

1 0 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

9

10

0.1

0.3

0.5

0.7

0.9

(A) (B)

Fig. 1. Comparison of 2-D haline (A) and geothermal (B) forced convection in fractured rock. Simulation parameters are given in Table 2.



Haline diffusivity is typically three orders of magnitude

smaller than thermal diffusivity. Thus, fracture–matrix con-

duction is 1000 times faster than fracture–matrix diffusion.

As a result, thermal gradients between fracture and matrix

equilibrate much faster than haline gradients.

A 3-D test problem of free convection

A second test problem is introduced to compare haline

and thermal free convection in 3-D fractured rock.

Changes of density are fully accounted for (Equations 15

and 16) such that density-driven flow resulting from free

convection is the main driving force.

The model domain for the 3-D test problem is a cubic

box having a side length equal to 10 m. The box repre-

sents a porous matrix and contains a single non-planar

fracture. The fracture is represented by nine planar facets

as shown in Fig. 2, and it is discretized by triangular frac-

ture elements using the technique presented by Graf &

Therrien (2008a). Fracture geometry and facet vertex

locations are given by Graf & Therrien (2008b). Simula-

tion parameters are given in Table 2. The 3-D grid spac-

ing for the haline and the thermal convection simulation

is equal to 0.1 m and 0.143 m, respectively. Numerical

stability analyses have indicated that the spacing used is

appropriate for both haline and thermal convection

simulations.

Two simulations of the 3-D test problem are carried

out: (i) free haline convection; and (ii) free geothermal

convection. Lateral boundaries of both simulations are

assumed to be impermeable and top and bottom bound-

aries are assigned a constant hydraulic head h0 ¼ 0; the

initial condition for flow is h0 ¼ 0. The total simulation

time is 10 years in all cases.

In the first simulation, a constant concentration c ¼ 1 is

assigned to the top boundary, and all other boundaries are

assigned a zero-dispersive flux boundary condition. The

initial concentration is c ¼ 0. In Equation (15), the maxi-

mum density (qmax) is assumed to be equal to

1200 kg m)3, giving the relative density of +0.2 in Equa-

tions (11) and (12).

In the second simulation, a constant temperature

T ¼ 250�C is assigned to the bottom boundary, and all

other boundaries are assigned a zero-conductive flux

boundary condition. Water density at 250�C is about

800 kg m)3, giving the relative density of )0.2 in Equa-

tions (11) and (12). The initial temperature is T ¼ 0�C.

The model does not consider phase changes of water such

that water is assumed to be liquid at 0�C. An additional

simulation with the initial condition T ¼ 1�C, not pre-

sented here, showed that the results are not sensitive to

the initial temperature. Effects associated with the density

maximum at 4�C were also not observed.

Density-driven forces and flow parameters are identical

in both simulations. The two simulations, however, differ

in the transport behavior of species: (i) low haline diffusi-

vity and constant fluid viscosity; and (ii) high thermal

diffusivity and variable fluid viscosity. This difference in

transport behavior is exhibited in Fig. 3. The figure shows

isohalines and isotherms in the fracture at different times.

Clearly, convective (unstable) groundwater circulation

Table 2 Model parameters used for the 2-D and 3-D test problems.

Parameter Value

Reference density* (q0) 1000 kg m)3

Reference viscosity� (l0) 1.124 · 10)3 kg m)1 sec)1

Acceleration due to gravity (g) 9.80665 m sec)2

Tortuosity* (s) 0.1

Matrix permeability*� (jij) 10)15 m2

Matrix porosity§ (/) 0.35

Matrix longitudinal dispersivity* (al) 0.1 m

Matrix transverse dispersivity* (at) 0.005 m

Fracture dispersivity*– (afr) 0.1 m

Fracture aperture*� (2b) 0.05 mm

Aqueous diffusion coefficient* (Dd) 5 · 10)9 m2 sec)1

Specific heat of rock** (~cs) 1030 J kg)1 K)1

Specific heat of water** (~cl) 4184 J kg)1 K)1

Thermal conductivity of rock�� (ks) 2.45 W m)1 K)1

Thermal conductivity of water�� (kl) 0.6 W m)1 K)1

Density of rock**�� (qs) 2650 kg m)3 K)1

*cf. Shikaze et al. (1998).

�cf. Kolditz et al. (1998).�Kept constant unless indicated otherwise.§cf. Frind (1982).–cf. Therrien & Sudicky (1996).**cf. Oldenburg & Pruess (1998).��cf. Bolton et al. (1996).

x - direction (m)

02

46

810

y - direction (m)0 2 4 6 8 10

z -

dir

ecti

on

(m

)

0

2

4

6

8

10

5

1

4

9

8

76

3

2

1

2

3

4

5

6

7

8

9

10

Fig. 2. Geometry of the triangulated non-planar fracture for comparison of

haline and thermal convection. Circled numbers are facet IDs and numbers

in italic are facet vertex IDs.



establishes in the case of haline flow, whereas the thermal

flow regime remains stable and convection is absent. The

difference in convective pattern is due to different diffusi-

vities. Low haline diffusivity allows for the existence of

large haline fracture–matrix gradients, thus facilitating the

formation of distinct fingers within the fracture plane.

The number of haline fingers is large at early times, and

the fingers coalesce at later times. On the other hand,

large thermal diffusivity evens out thermal fracture–

matrix gradients, thereby dissipating convection in the

fracture and stabilizing the flow regime. This result is in

agreement with previous findings by Kolditz (1995) and

Bachler et al. (2003).

Fluid viscosity is assumed to be constant for haline flow,

and haline fingers are produced by density variations. In

contrast, fluid viscosity is variable for thermal flow. Visco-

(E)

(F)

(B)

(A)

(C)

0.5 year

(D)

1 year

2 year

0.5 year

1 y

2 yearr

C C / 0

C C / 0

C C / 0

Temperature (°C)

z -

dir

ecti

on

(m

)

x - direction (m)

y - direction (m)

Temperature (°C)

z -

dir

ecti

on

(m

)

x - direction (m)

y - direction (m)

Temperature (°C)

z -

dir

ecti

on

(m

)

x - direction (m)

y - direction (m)

z -

dir

ecti

on

(m

)

x - direction (m)

y - direction (m)

z -

dir

ecti

on

(m

)

x - direction (m)

y - direction (m)

z -

dir

ecti

on

(m

)

x - direction (m)

y - direction (m)

Fig. 3. Comparison of 3-D haline (A–C) and geothermal (D–F) free convection in fractured rock. Density-driven forces and flow parameters are identical.



sity decreases close to the domain bottom (where the

buoyancy effect is large), thereby increasing hydraulic

conductivity and potentially facilitating the generation

of fingers. Nevertheless, varying fluid viscosity in the case

of thermal flow did not appear to facilitate the generation

of thermal fingers (Fig. 3 D-F). This is an intriguing result,

and it is due to the high thermal diffusivity that stabilizes

thermal flow.

The sensitivity of haline flow results to variations of the

parameters listed in Table 2 has been studied by Graf &

Therrien (2007a). The heat transfer parameters listed in

Table 2 are typical for plutonic rock, and are not subject

to major variations. Therefore, the geothermal flow results

presented here can be expected to be insensitive to varia-

tions of heat transfer parameters.

Results presented in this section and in Figs 1 and 3

suggest that fracture and matrix permeability control

whether a geothermal flow regime is stable (convective cir-

culation is absent) or unstable (convective circulation

exists). For example, large fracture permeability leads to

large flow velocities in the fracture, thereby enhancing con-

vective circulation and reducing the relative importance of

fracture–matrix conduction. While Graf & Therrien

(2007c, 2008b) have studied haline flow in the model

domain of Fig. 2, the present paper focuses on stable–

unstable geothermal flow in the same model domain for

different permeability ratios. These simulations are dis-

cussed in the following section.

Stable–unstable geothermal flow in fractured rock

Numerical analysis of stability

A series of 3-D geothermal flow simulations using the

model domain shown in Fig. 2 is simulated. In each simu-

lation, matrix and fracture permeability are modified, and

each simulation is classified as ‘unstable’ or ‘stable’

depending on whether convective circulation occurred or

not, respectively. Circulation was detected visually by

inspecting the velocity field in the fracture. Convection

occurs if a circular pattern in the flow field is observed.

Table 3 summarizes stable/unstable simulations, and pro-

vides corresponding matrix and fracture conductivity that

have been calculated using Equations (13) and (14).

Hydraulic conductivities equal to 10)50 and 10)25 m sec)1

both correspond to the same scenario where the rock

matrix is essentially impermeable. However, simulations

using both values are shown for completeness, to cover a

wide range of values for the matrix.

Figure 4 shows that some geothermal flow simulations

(where each simulation is labelled as sim in the text) are

unstable with distinct convective patterns (e.g. sim 6, 9, 11,

12, 14), while others are stable with undisturbed horizontal

isotherms (e.g. sim 5, 8, 18, 20, 21). Figure 5 shows the sta-

bility behavior of each simulation listed in Table 3 as a func-

tion of fracture and matrix conductivity. The grey curve

shown in Fig. 5 separates the plot area into an area of stable

simulations and an area of unstable simulations. Crossing

the grey curve from the stable field to the unstable field rep-

resents the onset of instability. We applied the Rayleigh cri-

terion to address whether the position of the vertical and

horizontal segments of the grey curve can be predicted.

A modified form of sim 18 (stable) was simulated where

the system was perturbed by increasing the initial tempera-

ture on facet vertex 5 by 10%, as previously done by

Weatherill et al. (2004) for a haline convection problem.

That perturbation does not trigger convection, suggesting

that the generation of fingers is only attributed to the pres-

ence of a buoyancy force that is larger in facets 6 and 7

(vertical) than in facets 8 and 9 (inclined).

In heterogeneous (e.g. fractured) media, the classical Ray-

leigh criterion fails to predict the onset of unstable flow. The

reason is that the 2-D fracture is located within a thermally

conducting medium. Therefore, a Rayleigh criterion for the

fracture-only situation can not be applied. However, if a very

small fracture aperture is assumed, presence of the fracture

may be neglected, and homogeneity can be assumed. In that

case, the thermal Rayleigh number for the porous matrix,

Ra [)], can be formulated as (Nield & Bejan 1999):

Table 3 Summary of stable/unstable geothermal flow simulations and cor-

responding matrix and fracture hydraulic conductivities.

Simulation ID

Matrix–K0ij

(m sec)1)

Fracture–K fr0

(m sec)1)

Fracture

aperture

(mm)

Stable simulations

2 10)50 72.706 10

3 10)50 290.826 20

5 10)50 454.415 25

8 10)25 454.415 25

15 10)10 454.415 25

16 10)8 454.415 25

17 10)8 491.496 26

18 10)8 530.030 27

20 10)8 290.826 20

21 10)8 7.27 · 10)7 0.001

23 10)10 530.030 27

Unstable simulations

1 10)50 744.514 32

4 10)50 654.358 30

6 10)50 570.019 28

7 10)25 654.358 30

9 10)25 570.019 28

10 10)10 570.019 28

11 10)5 570.019 28

12 10)5 491.496 26

13 10)5 418.789 24

14 10)5 290.826 20

19 10)8 570.019 28

22 10)5 7.27 · 10)7 0.001

The IDs of simulations whose results are presented in Fig. 4 are highlighted(in bold and italic).



Ra ¼ � � g � � � � � DTð Þ �H� �Dth

ð23Þ

where b [Q)1] is thermal expansion coefficient, j [L2] is

permeability, DT [Q] is temperature difference between

top and bottom of the model domain, H [L] is domain

height, and Dth[L2 T)1] is thermal diffusivity of the porous

matrix (Nield & Bejan 1999).

The Rayleigh criterion states that unstable flow begins if

Ra is larger than the critical Rayleigh number Rac. In a

homogeneous (e.g. unfractured) system with boundary

conditions described above, we have Rac ¼ 3 (Nield &

sim 6 t = 0.3 year sim 12 t = 0.1 year

sim 14 t = 0.1 yearsim 9 t = 0.3 year

sim 11 t = 0.1 year sim 5,8,18,20,21 t = 0.1 year

(E)

(F)

(B)

(A)

(C)

(D)

z -

dir

ecti

on

(m

)

Temperature (°C)

x - direction (m)

y - direction (m)

z -

dir

ecti

on

(m

)

Temperature (°C)

x - direction (m)

y - direction (m)

z -

dir

ecti

on

(m

)

Temperature (°C)

x - direction (m)

y - direction (m)

z -

dir

ecti

on

(m

)

Temperature (°C)

x - direction (m)

z -

dir

ecti

on

(m

)

Temperature (°C)

x - direction (m)

y - direction (m)

z -

dir

ecti

on

(m

)

Temperature (°C)

x - direction (m)

y - direction (m)

y - direction (m)

Fig. 4. Selected scenarios of stable-unstable geothermal flow in 3-D fractured rock. Convection cells and thermal fingers develop in unstable scenarios (e.g.

sim 6, 9, 11, 12, 14; A–E), while stable scenarios are characterized by undisturbed horizontal isotherms (e.g. sim 5, 8, 18, 20, 21; F). Associated fracture and

matrix conductivities are given in Table 3.



Bejan 1999). We used Equation (23) to calculate the

matrix permeability that corresponds to Ra ¼ 3. With

Equations (16) and (18), we calculated water density and

viscosity at 125�C, which is the average between initial

temperatures on top (0�C) and bottom (250�C) of the

model domain. The thermal expansion coefficient at

125�C is b ¼ 10)3 K)1 (Holzbecher 1998). The critical

matrix permeability for Ra ¼ 3 was obtained as

1.78 · 10)14 m2, and the critical matrix hydraulic conduc-

tivity is 7.43 · 10)7 m sec)1. Therefore, the vertical grey

line shown in Fig. 5 represents the Rayleigh criterion

Ra > Rac to predict the onset of unstable flow.

Likewise, if a very low matrix hydraulic conductivity is

assumed, unstable flow occurs only in the fracture. If the

rock matrix is neglected for the moment, the criterion

Ra ¼ 3 gives the critical fracture aperture as

3.97 · 10)4 mm [where fracture permeability ¼ (2b)2/

12], and the critical fracture hydraulic conductivity is

5.49 · 10)7 m sec)1. However, the critical fracture

hydraulic conductivity obtained from plotting numerical

results (Fig. 5) is 550 m sec)1, exceeding the theoretically

calculated conductivity by nine orders of magnitude. The

very high fracture conductivity required to trigger unstable

flow is due to stabilizing conductive heat flux from the

fracture into the low-permeability rock matrix. Only an

increase of fracture conductivity by the factor 109 leads to

destabilizing convection in the fracture that is larger than

stabilizing fracture–matrix conduction.

Clearly, hydraulic conductivities of matrix and fracture

control whether thermal convection in the fracture occurs

(Fig. 5). When matrix conductivity increases and fracture

conductivity is very small (crossing the vertical line of

Ra ¼ 3 in Fig. 5), thermal convection occurs in the

porous matrix but not in the fracture. The reason is that:

(i) fracture velocities are too small for convection to occur,

and (ii) the Rayleigh criterion (Ra > Rac ¼ 3) determines

the onset of convection in the matrix. For example, the

convective pattern shown in Fig. 4 for sim 14 represents

convection in the matrix, not in the fracture. Interestingly,

the high-temperature zone (for sim 14), whose center is

located at x 7 m, y 8 m, z 3 m, is therefore a cross-

section of a thermal finger that is located in the matrix and

that is rising across the fracture.

Conversely, increasing fracture conductivity at small

matrix conductivity (crossing the horizontal line in Fig. 5)

leads to thermal convection only within the fracture plane.

For example, for sim 6 and sim 9, convection in the matrix

is absent because the vertical line of Ra ¼ 3 has not been

crossed. Therefore, the results of sim 6 and sim 9 are virtu-

ally identical, and only show thermal convection in the

fracture (Fig. 4).

Unstable geothermal flow in 3-D fractured rock

In this section, we discuss the formation of fingering in

unstable geothermal flow systems. Fig. 6 presents results of

unstable geothermal flow of sim 12. Isotherms in the frac-

ture are shown in Fig. 6A–C. Clearly, unstable flow with

distinct fingering develops in the fracture. The number of

fingers decreases from 6 (0.05 year) to 4 (0.1 year) to 2–3

(0.15 year), which is in agreement with results of haline

fingering. Comparison of isotherms at early times

(Fig. 6A,D) indicates that the high-permeability fracture

acts as a trigger to form unstable fingers. Interestingly,

geothermal instabilities also grow in the porous matrix as

illustrated in Fig. 6E,F. The reason is that sim 12 plots on

the right of the Ra ¼ 3-line in Fig. 5, thus falling into an

area where unstable flow is initiated by large matrix con-

ductivity.

0

200

400

600

800

1E-60 1E-50 1E-40 1E-30 1E-20 1E-10 1E + 00

Stable Unstable

Clay Granite Dolomite Limestone Sandstone

Basalt Schist

Tuff

Matrix - (m sec–1)K ij 0

Fra

ctu

re -

(m

sec

–1)

K 0 fr

Fra

ctu

re a

per

ture

(m

m)

30

10

15

20

0

25

Ra

=3

6 9 11

14

12 18 5 8

20

21

4

1

7

10 19

16 13

22

3

2

23

15 17

Fig. 5. Stability diagram showing fracture

hydraulic conductivity vs. matrix hydraulic con-

ductivity. Here, the stability index is shown by

symbol ‘�’ when a simulation was classified as

stable and by symbol ‘·’ when it was classified

as unstable. The results show the dependence of

stability/instability on hydraulic conductivity of

both fracture and matrix. The grey curve indi-

cates the approximate locus of minimum critical

fracture–K0fr or matrix–K0

ij required to produce

unstable geothermal flow. The IDs of simulations

whose results are presented in Fig. 4 are high-

lighted (in bold and italic).



Results presented in Fig. 5 suggest that, in low-perme-

ability fractured rock (clay, granite), thermal convection

will only occur for extremely large fractures of aperture

>27 mm, as shown by the horizontal line in Fig. 5. On the

other hand (as demonstrated by Graf & Therrien 2008b),

haline convection happens at much smaller fracture

(E)

(F)

(B)

(A)

(C)

(D)

0.05 year 0.05 year

0.1 year

0.15 year 0.15 year

0.1 year

Temperature (°C)

z -

dir

ecti

on

(m

)

x - direction (m)

y - direction (m)

Temperature (°C)

z -

dir

ecti

on

(m

)

x - direction (m)

y - direction (m)

Temperature (°C)

z -

dir

ecti

on

(m

)

x - direction (m)

y - direction (m)

Temperature (°C)

z -

dir

ecti

on

(m

)

x - direction (m)

y - direction (m)

Temperature (°C)

z -

dir

ecti

on

(m

)

x - direction (m)

y - direction (m)

Temperature (°C)

z -

dir

ecti

on

(m

)

x - direction (m)

y - direction (m)

Fig. 6. Results of geothermal convection in 3-D fractured rock (sim 12) with isotherms 25�C to 225�C (interval 25�C). Figures (A–C) show isotherms in the

fracture, indicating that the number of thermal fingers decreases with time. Figures (D–F) show isotherms in the porous matrix along three cross-sections,

indicating that geothermal instabilities also grow in the porous matrix as illustrated in (E) and (F).



apertures. It can therefore be concluded that thermal

convection in fractured rock can only be established in very

high-permeability fracture zones.

The finding of unstable thermal fingering in the porous

matrix (Fig. 6E,F) contrasts with the results of unstable

haline flow in 3-D fractured rock (Graf & Therrien

2008b). In the latter case, fingers are restricted to the frac-

ture, and low fracture–matrix diffusion is the key process

to enable haline fingering in the fracture.

SUMMARY AND CONCLUSIONS

In this study, we investigate stable–unstable geothermal

flow in 3-D fractured rock. Results are compared with

haline flow, and difference in haline and thermal diffusivi-

ties is shown to be the reason for different behavior of

haline and thermal flow. For given matrix and fracture

hydraulic conductivities, haline flow tends to be unstable

(convecting) while thermal flow is stable (non-convecting).

The reason is that thermal diffusivity is generally three

orders of magnitude larger than haline diffusivity. Thus,

low diffusion enables formation of unstable ‘fingering’

while large conduction diminishes growth of instabilities.

A series of stable–unstable thermal flow simulations was

then carried out where hydraulic conductivity of matrix

and fracture vary over a wide range. Simulations indicate

that the classic Rayleigh criterion can be applied when frac-

ture aperture is very small. Conversely, the Rayleigh crite-

rion fails when the porous matrix is assumed to be

impermeable because stabilizing fracture–matrix conduc-

tion is independent of matrix hydraulic conductivity. The

numerically estimated critical fracture conductivity is nine

orders of magnitude larger than the theoretically calculated

critical fracture conductivity based on Rayleigh theory.

Simulations of a selected scenario of unstable thermal

flow (sim 12) illustrate that unstable ‘fingers’ form in the

fracture (Fig. 6). At early simulation times, the number of

fingers is high and distinct convection develops while at

later times, fingers coalesce and convection is less apparent.

This result is in agreement with findings of prior free con-

vective flow studies in homogeneous and heterogeneous

media (Wooding 1969; Wooding et al. 1997; Simmons

et al. 1999, 2002; Graf & Therrien 2008b).

Another result of the sim 12 simulation is that unstable

thermal fingers also grow within the porous matrix. This

outcome contrasts with the results of unstable haline flow

in 3-D fractured rock (Graf & Therrien 2008b) where

the presence of fingers is restricted to the fracture. In that

case, low haline fracture–matrix diffusion prevents finger

growth in the matrix but enables haline fingering in the

fracture.

In summary, results presented here indicate that:

(1) Low haline diffusivity facilitates the formation of dis-

tinct fingers in a fracture.

(2) Large thermal diffusivity evens out thermal fracture–

matrix gradients, thereby stabilizing the flow in a frac-

ture.

(3) Fracture and matrix permeability control whether a

geothermal flow regime is stable or unstable.

(4) The classic Rayleigh criterion can be applied in frac-

tured rock when fracture aperture is very small.

(5) The Rayleigh criterion fails in fractured rock when the

porous matrix is assumed to be impermeable.

(6) In unstable geothermal flow regimes, numerous ‘fin-

gers’ form at early times. The number of fingers

decreases with time. This outcome is in agreement

with results of haline flow.

(7) Thermal convection in fractured rock can only be

established in very high-permeability fracture zones.

(8) In unstable thermal 3-D flow regimes, thermal fingers

grow in the fracture and porous matrix. This outcome

contrasts with the results of 3-D unstable haline flow

where fingers only form in a fracture.

The results presented here are also applicable to larger

spatial domains because the scale considered here (10 m)

allows for simulating all relevant processes (convection,

conduction, fracture–matrix interaction) of thermal flow.

On a larger scale, the convective pattern may be different,

but the effect of thermal convection within the fracture,

relative to fracture–matrix conduction, can be expected to

be identical to those at the scale of this study.

This study has shown that it is important to analyze

haline and geothermal convection within a 3-D framework.

Clearly, numerically estimated stability criteria of 2-D sys-

tems are not applicable to 3-D. The numerical stability

analysis presented here may help to predict if other scenar-

ios of thermal flow in 3-D fractured rock are stable or

unstable. This classification could aid to predict the long-

term efficiency and productivity of geothermal systems.

While the present study and the study by Graf & Therrien

(2008b) focus on haline and thermal convection in

fractured rock, respectively, the interaction between thermal

and haline (thermohaline) convection in fractured rock

remains unexplored. Because temperature and salt have

different diffusivities, thermohaline convection is commonly

termed ‘double-diffusive convection’ (DDC) (Pritchard &

Richardson 2007). DDC has been studied in: (i) homoge-

neous porous media (Evans & Nunn 1989; Fournier 1990;

Oldenburg et al. 1995; Oldenburg & Pruess 1998; Beji

et al. 1999); (ii) anisotropic porous media (Tyvand 1980);

(iii) oceans (Stern 1960; Turner 1979; Schmitt 1994); (iv)

laboratory experiments (Yoshida et al. 1987; Turner 1995);

and (v) groundwater wells (Love et al. 2007), where

DDC is properly documented and understood (Brandt &

Fernando 1995).

Double-diffusive convection occurs in many systems

including geothermal reservoirs, waste disposal, groundwa-

ter contamination, chemical transport in packed-bed reac-



tors, grain-storage installations, food processing and others.

Recently, double-diffusive natural convection in porous

media has received considerable attention in the context of

numerous potential applications. Fournier (1990) has

hypothesized that DDC plays a major role in the dynamics

of a geothermal reservoir. Fournier (1990) has pointed out

that ‘few geochemists and reservoir engineers involved with

the exploitation of geothermal resources appear to be aware

of this phenomenon’ and that ‘reservoir engineers should

keep in mind that double-diffusive convection may greatly

influence results of tracer tests and the behavior of injected

waste fluids’. Fournier’s hypothesis has been confirmed

later by Oldenburg & Pruess (1998) who performed

numerical simulations in the Salton Sea Geothermal System

in Southern California, showing that DDC in geothermal

reservoirs is a key process.

However, many questions remain to be answered about

the nature of DDC in active geothermal systems. Little is

known about the size and shape of individual DDC cells

that may form in rocks of different porosities or in frac-

tured rocks, or about the conditions under which indivi-

dual cells may form and persist.

Double-diffusive convection may explain layered convec-

tion and the occurrence of distinct fluid types in geothermal

reservoirs, where different convection cells contain locally

well-mixed fluids (Fournier 1990; Oldenburg & Pruess

1998). However, the presence of fractures has been

neglected in previous studies of DDC, although fractured

geothermal reservoirs are systems where DDC is likely to

control the transport of thermal energy. Therefore, DDC in

fractured rock will be the subject of our future studies.

ACKNOWLEDGEMENTS

We thank Ontario Power Generation (OPG), the Nuclear

Waste Management Organization (NWMO), and the Natu-

ral Sciences and Engineering Research Council of Canada

(NSERC) for financial support of this project. Author TG

wishes to acknowledge Martin Sauter (Georg-August-

University Gottingen) for providing travel funds. We thank

the editorial board of Geofluids (Steve Ingebritsen, Martin

Appold, Peter Nabelek) and two anonymous reviewers for

giving detailed comments that considerably improved the

manuscript.

NOMENCLATURE

Latin letters

(2b) [L] Fracture aperture

A [L2] Surface area

~c [L2 T)2Q)1] Specific heat

c [)] Relative solute concentration

C [M L)3] Absolute solute concentration

Dd [L2 T)1] Aqueous diffusion coefficient

Dth [L2 T)1] Thermal diffusivity

g [L T)2] Acceleration due to gravity

h0 [L] Equivalent freshwater head

H [L] Domain height

I+ I) [)], Fracture–matrix interface

J Variable flux

k [M L T)3Q)1] Thermal conductivity

K0ij [L T)1] Freshwater hydraulic conductivity of porous

matrix

K fr0 [L T)1] Freshwater hydraulic conductivity of fracture

Lv [L] Geometry of porous matrix element v ¼ x,y,z

q [M L)3 T)1] Fluid flux

Ra [)] Rayleigh number

Rac [)] Critical Rayleigh number

SS [L)1] Specific storage

t [T] Time

T [Q] Absolute temperature in Kelvin

TC [Q] Relative temperature in centigrade

v [L T)1] Linear flow velocity

w [)] Approximation function

Greek letters

afl [M)1 L T2] Fluid compressibility

afr [L] Fracture dispersivity

al [L] Matrix longitudinal dispersivity

am [M)1 L T2] Matrix compressibility

at [L] Matrix transverse dispersivity

b [Q)1] Thermal expansion coefficient

C variable Sources and sinks

gj [)] Indicator for flow direction

j [L2] Permeability

K [M T)3] Convective–dispersive–conductive heat flux

l [M L)1 T)1] Fluid viscosity

q [M L)3] Fluid density

qr [)] Relative fluid density

s [)] Factor of tortuosity

/ [)] Matrix porosity

u [1�] Fracture incline

X [M M)1 T)1] Advective–dispersive–diffusive solute flux

Sub- and superscripts

0 [)] Reference fluid

b [)] Bulk

e [)] Porous matrix element

fe [)] Fracture element

fr [)] Fracture

i, j [)] Spatial indices

I [)] Nodal index

l [)] Liquid phase

n [)] Normal direction

s [)] Solid phase

Special symbols

¶ [)] Partial differential operator



D [)] Difference

� [L)1] Divergence operator

REFERENCES

Bachler D, Kohl T, Rybach L (2003) Impact of graben-parallel

faults on hydrothermal convection–Rhine Graben case study.Physics and Chemistry of the Earth, 28, 431–41.

Bataille A, Genthon P, Rabinowicz M, Fritz B (2006) Modeling

the coupling between free and forced convection in a verticalpermeable slot: Implications for the heat production of an

Enhanced Geothermal System. Geothermics, 35, 654–82.

Bear J (1988) Dynamics of Fluids in Porous Media. Elsevier, New

York.Beji H, Bennacer R, Duval R, Vasseur P (1999) Double-diffusive

natural convection in a vertical porous annulus. Numerical HeatTransfer, Part A Applications, 36, 153–70.

Bolton EW, Lasaga AC, Rye DM (1996). A model for the kineticcontrol of quartz dissolution and precipitation in porous media

flow with spatially variable permeability: Formulation and exam-

ples of thermal convection. Journal of Geophysical Research, 101(B10), 22157–87.

Boussinesq VJ (1903) Theorie Analytique de la Chaleur [chapter2.3], 2nd edn. Gauthier-Villars, Paris.

Brandt A, Fernando HJS (1995) Double-diffusive Convection, Geo-physical Monograph 94, American Geophysical Union, Wash-

ington, DC.

Clift SS, D’Azevedo F, Forsyth PA, Knightly JR (1996) WATSIT-1 and WATSIT-B Waterloo Sparse Iterative Matrix Solvers,User’s Guide with Developer Notes for Version 2.0.0, (8). Uni-

versity of Waterloo, Waterloo, Canada.

Elder JW (1967) Transient convection in a porous medium. Jour-nal of Fluid Mechanics, 27, 609–23.

Evans GE, Nunn JA (1989) Free thermohaline convection in sedi-

ments surrounding a salt column. Journal of GeophysicalResearch, 94, 2707–16.

Evans DG, Raffensperger JP (1992) On the stream function for

variable-density groundwater flow. Water Resources Research,

28, 2141–5.

Farvolden RN, Pfannkuch O, Pearson R, Fritz P (1988) Region12: Precambrian Shield. In: (eds Back W, Rosenshein JS, Seaber

PR), v.O-2: pp. 101–114. Hydrogeology. The Geology of North

America, Boulder, CO.

Fournier RO (1990) Double-diffusive convection in geothermalsystems: the Salton Sea, California, geothermal system as a likely

candidate. Geothermics, 19, 481–96.

Frind EO (1982) Simulation of long-term transient density-depen-dent transport in groundwater. Water Resources Research, 5,

73–88.

Graf T, Therrien R (2005) Variable-density groundwater flow and

solute transport in porous media containing nonuniform dis-crete fractures. Advances in Water Resources, 28, 1351–67.

Graf T, Therrien R (2007a) Variable-density groundwater flow

and solute transport in irregular 2D fracture networks. Advancesin Water Resources, 30, 455–68.

Graf T, Therrien R (2007b) Coupled thermohaline groundwater

flow and single-species reactive solute transport in fractured

porous media. Advances in Water Resources, 30, 742–71. Erra-tum 2008: Advances in Water Resources, 31, 1765.

Graf, T, Therrien R (2007c) Benchmarking three-dimensional var-

iable-density flow in fractured porous media. Geological Society

of America, Annual Meeting, Abstracts with Programs, Denver,CO, USA, 39, 348.

Graf T, Therrien R (2008a) A method to discretize non-planar frac-tures for 3D subsurface flow and transport simulations. Interna-tional Journal for Numerical Methods in Fluids, 56, 2069–90.

Graf T, Therrien R (2008b) A test case for the simulation of

three-dimensional variable-density flow and solute transport indiscretely-fractured porous media. Advances in Water Resources,31, 1352–63.

Holzbecher EO (1998) Modeling Density-driven Flow in PorousMedia. Springer Verlag, Berlin.

Hurter S, Schellschmidt R (2003) Atlas of geothermal resources in

Europe, Geothermics, 32, 779–87.

Istok J (1989) Groundwater Modeling by the Finite ElementMethod. American Geophysical Union, Washington.

Jupp TE, Schultz A (2000) A thermodynamic explanation for

black smoker temperatures. Nature, 403, 880–3.

Jupp TE, Schultz A (2004) Physical balances in subseafloor hydro-thermal convection cells. Journal of Geophysical Research, 109,

B05101, 1–12.

Kolditz O (1995) Modelling flow and heat transfer in fracturedrocks: Conceptual model of a 3-D deterministic fracture net-

work. Geothermics, 24, 451–70.

Kolditz O, Clauser C (1998) Numerical simulation of flow and

heat transfer in fractured crystalline rocks: application to the hotdry rock site in Rosemanowes (U.K.). Geothermics, 27, 1–23.

Kolditz O, Ratke R, Diersch H-JG, Zielke W (1998) Coupled

groundwater flow and transport: 1. Verification of variable-

density flow and transport models. Advances in Water Resources,21, 27–46.

Love AJ, Simmons CT, Nield DA (2007) Double-diffusive con-

vection in groundwater wells. Water Resources Research, 43,W08428, doi: 10.1029/2007WR006001.

Murphy HD (1979) Convective instabilities in vertical fractures

and faults. Journal of Geophysical Research, 84, B11, 6121–30.

Nield DA, Bejan A (1999) Convection in Porous Media. SpringerVerlag, New York.

Oberbeck A (1879) Ueber die Warmeleitung der Flussigkeiten bei

Berucksichtigung der Stromung infolge von Temperaturdifferen-

zen. Annalen der Physik und Chemie, 7, 271–92.Oldenburg CM, Pruess K (1998) Layered thermohaline convec-

tion in hypersaline geothermal systems. Transport in PorousMedia, 33, 29–63.

Oldenburg CM, Pruess K, Lippmann MJ (1995) Heat and masstransfer in hypersaline geothermal systems. Proceedings of the

World Geothermal Congress, 18–31 May 1995, Vol. 3, 1647–

52. International Geothermal Association, Florence Italy, andLawrence Berkeley Laboratory Report LBL-35085.

Pritchard D, Richardson CN (2007). The effect of temperature-

dependent solubility on the onset of thermosolutal convection

in a horizontal porous layer. Journal of Fluid Mechanics, 571,59–95.

Rausch R, Schafer W, Therrien R, Wagner C (2005) Introductionto Solute Transport Modelling, Gebruder Borntraeger, Berlin.

Rayleigh JWS (1916) On convection currents in a horizontal layerof fluid when the higher temperature is on the under side. Philo-sophical Magazine, Series 6, 92, 529–46.

Schincariol RA, Schwartz FW, Mendoza CA (1994) On the gener-ation of instabilities in variable-density flows. Water ResourcesResearch, 30, 913–27.

Schmitt RW (1994) Double diffusion in oceanography. AnnualReview of Fluid Mechanics, 26, 255–85.

Shikaze SG, Sudicky EA, Schwartz FW (1998) Density-dependent

solute transport in discretely-fractured geologic media: is predic-

tion possible? Journal of Contaminant Hydrology, 34, 273–91.



Simmons CT, Narayan KA, Wooding RA (1999) On a test casefor density-dependent groundwater flow and solute transport

models: The Salt Lake problem. Water Resources Research, 35,

3607–20.

Simmons CT, Pierini ML, Hutson JL (2002) Laboratory investi-gation of variable-density flow and solute transport in unsatu-

rated-saturated porous media. Transport in Porous Media, 47,

215–44.Stern ME (1960) The ‘‘salt-fountain‘‘ and thermohaline convec-

tion. Tellus, 12, 172–5.

Straus JM, Schubert G (1977) Thermal convection of water in a

porous medium: Effects of temperature- and pressure-dependentthermodynamic and transport properties. Journal of GeophysicalResearch, 28, 325–33.

Therrien R, Sudicky EA (1996) Three-dimensional analysis of vari-

ably saturated flow and solute transport in discretely-fracturedporous media. Journal of Contaminant Hydrology, 23, 1–44.

Therrien R, McLaren RG, Sudicky EA, Panday SM (2008) Hydro-

geo-sphere – a three-dimensional numerical model describingfully- integrated subsurface and surface flow and solute trans-

port. Universite Laval, University of Waterloo, Canada.

Turner JS (1979) Buoyancy Effects in Fluids, Cambridge University

Press, Cambridge.Turner JS (1995) Laboratory models of double-diffusive processes.

In: Double-diffusive Convection (eds Brandt A, Fernando HJS),

pp. 334. Geophysical Monograph 94, American Geophysical

Union, Washington, DC.

Tyvand PA (1980) Thermohaline instability in anisotropic porousmedia. Water Resources Research, 16, 325–30.

Valliappan S, Wang W, Khalili N (1998) Contaminant transport

under variable density flow in fractured porous media. Interna-tional Journal for Numerical and Analytical Methods in Geome-chanics, 22, 575–95.

Voss CI, Souza WR (1987) Variable density flow and solute trans-

port simulation of regional aquifers containing a narrow fresh-water-saltwater transition zone. Water Resources Research, 23,

1851–66.

Weatherill D, Simmons CT, Voss CI, Robinson NI (2004) Test-

ing density- dependent groundwater models: two-dimensionalsteady state unstable convection in infinite, finite and inclined

porous layers. Advances in Water Resources, 27, 547–62.

Weatherill D, Graf T, Simmons CT, Cook PG, Therrien R, Rey-

nolds DA (2008) Discretizing the fracture–matrix interface tosimulate solute transport. Ground Water, DOI: 10.1111/

j.1745-6584.2007. 00430.x

Wooding RA (1969) Growth of fingers at an unstable diffusinginterface in a porous medium or Hele-Shaw cell. Journal ofFluid Mechanics, 39, 477–95.

Wooding RA, Tyler SW, White I (1997) Convection in groundwa-

ter below an evaporating salt lake: 2. Evolution of fingers andplumes. Water Resources Research, 33, 1219–28.

Yoshida J, Nagashima H, MA WJ (1987) A double-diffusive lock-

exchange flow with small density difference. Fluid DynamicsResearch, 2, 205–15.



Stable-unstable flow of geothermal fluids in fractured rock

Documents

Transcript of Stable-unstable flow of geothermal fluids in fractured rock