Neuville, A.

Schmittbuhl, J.

Toussaint, R.

Earth-prints Repository

Euro-conference 
“Rock Physics and Geomechanics”
on Natural hazards : 
Thermo-hydro-mechanical processes in rocks
Erice, Italy
Septembre 2007
• Permanent 
• Laminarity
• Lubrication => locally smooth surface
* Velocity
* Hydraulic flow
• Incompressibility : 
Equation to be solved (2D) :
vP r
r
∆=∇ η










=
0
),,(
),,(
zyxv
zyxv
v y
x
r
Stokes :
Phdzvq
h
∇−== ∫
rrr
η12
3
0)),(.( 3 =∇∇ Pyxh rr
Local parabolic law
Local cubic law
0. =∇ ur
r
ENERGY CONSERVATION
• Conduction
• Convection
• Assumptions :
* Tw constant and invariant
*  => In plane convection
* Normal to plane conduction
* Lubrication 
• Local temperature law
T∇−=
rr λϕ TTv ∆=∇⇒ χ
rr
.
Tvcrρ










=
0
),,(
),,(
zyxv
zyxv
v y
x
r
( )022442 ),(
),(
azaza
yxh
yxfTT w ++=−⇒
Hydro-thermal coupling in a rough fracture
Amélie Neuville, Renaud Toussaint, Jean Schmittbuhl
Institut de Physique du Globe de Strasbourg
Contact : Amelie.Neuville@eost.u-strasbg.fr
EXAMPLE : GEOTHERMAL EXPLOITATION
The context of study is an area where stream lines and isobars would be flat if 
the fracture were plane. For example, it could take place in the framed area 
between injection and pumping wells. Then we will see how the roughness of 
the fracture modifies the hydraulic flux and the thermalization as well.
T0 Tw
Tw
Injection Pumping Fracture plane
z
Impact of roughness on 
the fluid thermalization ? 
Fracture 
network
Rough fracture 
.
z
Dipolar circulation
in homogeneous media
Stream 
lines
Expected
length 
of thermalization
We aim at estimating the 
characteristic length of 
thermalization      which  
depends on the fracture 
morphology
refl
EQUIVALENT PROBLEM AT COARSENED SCALE
Reference
with flat fracture
Obtained
with rough fracture
T0
Tw
New
length of
thermalization ?
x
)(xT
BIBLIOGRAPHY
[1] Batrouni G.G., Hansen A. , and Schmittbuhl J., 2002, Elastic response of rough surfaces in partial contact. Europhys. Lett., 60(5):724-730
[2] Méheust Y., Schmittbuhl J., 2001, Geometrical heterogeneities and permeability anisotropy of rough fractures. J. Geophys. Res. 106 (B2), 2089.2102.
[3] Méheust Y., Schmittbuhl J., 2003, Scale effects related to flow in rough fractures. PAGEOPH 160 (5.6), 1023.1050
[4] Neuville A., Toussaint R., Schmittbuhl J., ''Hydro-thermal flows in a rough fracture'', in preparation
[5] Schmittbuhl J., Schmitt F., Scholz, C., 1995, Scaling invariance of crack surfaces. J. Geophys. Res. 100, 5953-5973. 
[6] Turcotte D.L. and Schubert G., 2002, Geodynamics, 2nd ed. (Cambridge University Press), especially p.262.264
2D-TEMPERATURE LAW
• Energy conservation
• Boundary flow
∫
∫
=
h
h
dzzyxv
dzzyxTzyxv
yxT
),,(
),,(),,(
),(
2/h
w
z
T
∂
∂
−= λϕ
0)(),(
2
. 2 =−+∇ wTTNuyxh
Tu χ
rr
Convection Conduction
T,u,h r
with
12.4
17
70
≈=
−
=
ref
wNu
ϕ
ϕ
h
TTw
ref
−
= λϕ
Nusselt number
where
Equation to be solved (2D)
depend on (x,y)
Averaging 
over thickness
Properties of fluid :
Density [kg/m3] : ρ Specific heat capacity [ J/kg/K] : c
Conductivity [W/m/K] : λ Diffusivity [ m2/s] : χ = λ / (ρc)
ABSTRACT
Heat exchange during laminar flow is studied at the fracture scale on the 
basis of the Stokes equation. The aperture is obtained from a self-affine 
geometrical model shown to be a realistic description of a natural fracture. 
We study the influence of the fracture roughness on the heat flux through the 
fracture sides when a cold fluid is injected and we estimate at which distance 
the thermal equilibration between the fluid and the rock temperature is 
reached. We show that at a coarse grained scale, the basic equation for heat  
flux is identical to the one for parallel plates, but with a different 
characteristic thermal length. Statistical computations and comparisons with 
flat parallel plates are made for the hydraulic and thermal results. The 
hydraulic aperture of rough fractures can be higher or lower than the one of 
parallel plates having the same mean mechanical aperture (or geometrical 
aperture) : the aspect ratio of the fracture appears to be an important 
parameter. Our model also shows that the fracture roughness induces 
channeling effects in hydraulic and thermal flows. Although fracture 
roughness is shown to induce a large variability of behaviors, the thermal 
equilibration is often reached with a higher characteristic thermal length than 
for a system with parallel plates having the same hydraulic aperture. A 
boundary element model describing how the fracture elastically closes is used 
to introduce the study of the hydro-thermo mechanical coupling of a rough 
fracture.
Various realizations
Various RMS (Root Mean Square of the aperture)
Same mean aperture <h>
Various aspect ratios
STATISTICAL RESULTS WITH
1
h1(x,y)
2a
Aperture grid
No connected pressure knots
Boundary pressure knots
Other pressure knots
ELASTIC CLOSURE OF THE FRACTURE :
HYDRO-MECHANICAL COUPLING
METHOD
As reference case, the fracture is modeled by two parallel plates 
which are separated by a distance h with a pressure P0 at the inlet and 
PL at the outlet.
h ?
Equivalence at 
coarsened scale  ? 








−−=−
ref
wwref l
xTTTT exp)( 0
Nu
Peh
Nu
uhlref == χ2
2
χ2
uhPe ⋅=Peclet number
.
z
BOUNDARY CONDITIONS
dy)y,x(u
dy)y,x(T)y,x(u
)x(T
y
y
l
x
l
x
∫
∫
=
1D-TEMPERATURE :
)y,x(h
dz)z,y,x(v
)y,x(u )y,x(h
∫
=
r
r • Discretization
• Systems solved with the biconjugate gradient method
• Use of dimensionless variables : h*, q*, T*, …
0)(8)()(
,
,
21,1,,,1,1, =−+−+− −+−+ wjiji
jijiji
y
jijiji
x TT
h
NuaTTuTTu χ
0)()()()(
,1,5.0,
3
,1,5.0,
3
,,1,5.0
3
,,1,5.0
3
=−+−+−+−
−
−++−−++ jijijijijijijijijijijiji PPhPPhPPhPPh
FINITE DIFFERENCES NUMERICAL SCHEME
2a
2a
0)(2. 2 =−+∇ wTTNuhTu
χrr
0)),(.( 3 =∇∇ Pyxh
rr
Apertures
Temperatures, Hydraulic flux 
a : Aperture mesh size
h∆
h
xx∆
[ ]( ) [ ]( )yxhgyxhg ∆∆∆=∆∆∆ ,,,, λλλλ ζζ
ROUGH FRACTURE : SELF AFFINE APERTURE (e.g. Ref. [5] )
Where g gives the probability to get a height 
difference ∆h between two coordinate points 
separated by [∆x,∆y].
Property in spatial domain
ς−−
≈
1)(~ kkh
Aperture :
2D Fourier transform : )(~ kh
ln ( |k| )
ζ : Roughness exponent
ζ ≈ 0.8
),( yxh
HYDRAULIC FLOW
Tw
Twz
x
2/hq
r
2/hq −
r
v
r
h
RESULTS ILLUSTRATION
Aperture h*(x,y) Hydraulic flow q*(x,y) ( )*Tln−( )*Tln−2D-Temperature
1D-Temperature
y/a
y/a
y/a
y/a
x/a x/a x/a x/a
z
.
(0.97 , 0.99)( lref* , <hth*> )
1.00<hh*>
( 0.05 , 1)( RMS , <h>)
Characteristics
(0.98 , 1.00)( lref* , <hth*> )
0.98<hh*>
( 0.15 , 1)( RMS , <h>)
(1.09 , 1.02)( lref* , <hth*> )
0.94<hh*>
( 0.25 , 1)( RMS , <h>)
(0.61 , 0.89)( lref* , <hth*> )
0.82<hh*>
( 0.35 , 1)( RMS , <h>)
(3.36 , 1.35)( lref* , <hth*> )
1.02<hh*>
( 0.35 , 1)( RMS , <h>)
1 mm
2 mm
3 mm
4 mm
5 mm
D
1
2
h1(x,y)h2(x’,y’)
d(x”,y”)
D
D
1st block :
Infinitely rigid
Rough
2nd block :
Elastically deformable
Flat
D=0 mm
1 Contact point
D=1.72 mm
116 Contact points
D=3.72 mm
422 Contact points
S
am
e
 ap
ertu
re
 featu
res
Mask :
Contact zones
Computation of
f : Forces, where contact
d : Displacement, where no contact
Use of Elastostatic Green function for half space
f > 0
Everywhere
?
EndYesNo : Change mask
Initial contact zones 
= 
Cut through the rough surface
MECHANICS : METHOD (Ref. [1] )
HYDRO-COUPLING (ongoing)
APERTURE CLOSURE
• Implications of fracture roughness on hydraulic properties
* Channeling 
* Aspect ratio influence on the hydraulic aperture:
○ On average :
○ With high variability
* Possible explanation :
○ Parallel channels, perpendicular to the flow (Lx≥Ly) 
○ Parallel Channels, parallel to the flow (Lx<Ly) 
• Implications of fracture roughness on thermal properties
* Channeling 
* On average inhibited heating of the fluid
* Definition of a thermal aperture for a coarse grained equivalence
 Ly>Lx : Enhanced hydraulic flow
 Ly=Lx : Slightly inhibited hydraulic flow
 Ly<Lx : Inhibited hydraulic flow
• Improvement of boundary conditions
* Mechanics :  variable contact zone (fracture closure)
* Variations of the host rock temperature
* Fracture geometry : Anisotropy of the aperture (geological 
anisotropic features)
• Application to real apertures
• Which Fourier modes are dominating the flows ?
• Using ρ(T) dependence allowing natural convection
• Lattice Boltzmann implementation
* Full Navier Stokes equation
* 3D-aspect
* Time dependence
PROSPECTS
CONCLUSION 
1
2
3
4
5
6
7
8
9
1 1.1 1.20.4 0.5 0.6 0.7 0.8 0.9
0.2 0.3 0.4 0.5 0.60 0.1
lref*
hh* 
Dimensionless hydraulic aperture
RMS / <h>
1
1.1
1.2
0.4
0.5
0.6
0.7
0.8
0.9
hh*  Dimensionless hydraulic aperture
PL
3∆P/4
∆P/2
∆P/4
P0
1
200
400
600
800
1000
1200
1400
1600
1800
2000
200
400
600
800
1000
1200
1400
1600
1800
2000
y/a
1000 0 00
200
400
600
800
1000
1200
1400
1600
1800
2000
1000x/a x/a x/a x/a
h(x,y)
Contact
lref* for parallel plates 
separated by hh* :
lref* = (hh*)4
Enhanced hydraulic flow
Higher permeability
HYDRAULIC APERTURE
THERMAL LENGTH
RANGE OF VALIDITY
• ρ constant 
Low thermal dilatation coefficient
• z-Conduction >> (x,y)-Conduction
Re >> 1.2
• x-Conduction << x-Convection
Re >> 0.43
• Lubrication
Re << 102
• No heat source because of viscosity
Re << 4.104 for ∆T=100 C
3
0
2
1014 −<<
−
= .)TT(c
uEc
w
Other aspect ratio ; case requiring a 
pressure and temperature boundary 
conditions further away along x
( )
x
th
ref L
P
Nu
hl ∆⋅
⋅⋅
=
χη24
4
• lref for rough apertures is 
determined by computing the slope 
of   vs x.
• Then it is possible to define a 
thermal aperture hth equivalent at 
coarse grained scale with :
( )Tln−
Using the parallel plates reference 
solutions,
Slightly inhibited hydraulic flow
Slightly lower permeability
Inhibited hydraulic flow
Lower permeability
P
Lqh x
xh ∆
><−>=< η123
Isobars
2D-Temperature
x/a
y/a
e-5
e-15
e-10
e-20
ε
u : Velocity averaged on the aperture height
Contactx/a
y/a
On average, lref* for rough 
apertures is superior to
lref* for parallel plates 
with the same hydraulic 
aperture :
The “thermalization” is 
inhibited
=>For a rough aperture, 
<h> ≠ hh* 
=> For a rough aperture, 
hh* ≠ hth* 
i :                for rough fracture
ii : Model line of  i
iii : Model line for parallel plates
separated by <hh*>=1
( )*Tln−
(hh*)4
0 < (RMS/<h>) ≤ 0.1
0.1 < (RMS/<h>) ≤ 0.2
0.2 < (RMS/<h>) ≤ 0.3
0.3 < (RMS/<h>) ≤ 0.4
0.4 < (RMS/<h>) ≤ 0.5
0.5 < (RMS/<h>) ≤ 0.6
0.6 < (RMS/<h>) ≤ 0.7
hh*=1 ; RMS=0
RMS : Root Mean Square of the aperture
<h>  : mean aperture


Hydro-thermal flows in a rough fracture

http://www.earth-prints.org/bitstream/2122/2716/1/27_Neuville.pdf

Hydro-thermal flows in a rough fracture

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