Fracture/Conduit Flow
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Transcript of Fracture/Conduit Flow
Fracture/Conduit Flow
Fractured rock (NSW Australia)
Motivation
Karst http://research.gg.uwyo.edu/kincaid/Modeling/wakulla/wakcave2.jpg
~100 m
~11 m3 s-1
White Scar, England; photo by Ian McKenzie, Calgary, Canada
These data and images were produced at the High-Resolution X-ray Computed Tomography Facility of the University of Texas at Austin
Basic Fluid Dynamics
Momentum
• p = mu
Viscosity
• Resistance to flow; momentum diffusion
• Low viscosity: Air
• High viscosity: Honey
• Kinematic viscosity:
Reynolds Number
• The Reynolds Number (Re) is a non-dimensional number that reflects the balance between viscous and inertial forces and hence relates to flow instability (i.e., the onset of turbulence)
• Re = v L/• L is a characteristic length in the system• Dominance of viscous force leads to laminar flow (low
velocity, high viscosity, confined fluid)• Dominance of inertial force leads to turbulent flow (high
velocity, low viscosity, unconfined fluid)
Re << 1 (Stokes Flow)
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford. 519 pp.
Separation
Eddies, Cylinder Wakes, Vortex Streets
Re = 30
Re = 40
Re = 47
Re = 55
Re = 67
Re = 100
Re = 41Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford. 519 pp.
Eddies and Cylinder WakesS
.Go
kaltu
n
Flo
rida
Inte
rna
tion
al U
nive
rsity
Streamlines for flow around a circular cylinder at 9 ≤ Re ≤ 10.(g=0.00001, L=300 lu, D=100 lu)
Eddies and Cylinder Wakes
Streamlines for flow around a circular cylinder at 40 ≤ Re ≤ 50.(g=0.0001, L=300 lu, D=100 lu) (Photograph by Sadatoshi Taneda. Taneda 1956a, J. Phys. Soc. Jpn., 11, 302-307.)
S.G
oka
ltun
Flo
rida
Inte
rna
tion
al U
nive
rsity
L
Flowuax
yz
Poiseuille Flow
Poiseuille Flow
• In a slit or pipe, the velocities at the walls are 0 (no-slip boundaries) and the velocity reaches its maximum in the middle
• The velocity profile in a slit is parabolic and given by:
x = 0 x = a/2
u(x)
• G can be due to gravitational acceleration (G = g in a vertical slit) or the linear pressure gradient (Pin – Pout)/L
2
2
22x
aGxu
Poiseuille Flow
• Maximum
• Average
x = 0 x = a/2
u(x)
2
2
22x
aGxu
2
max 22
aGu
2max 123
2a
Guuaverage
Poiseuille Flow
S.GOKALTUNFlorida International University
Kirchoff’s Current Law
• Kirchoff’s law states that the total current flowing into a junction is equal to the total current leaving the junction.
II22 II33
node
II11 flows into the node
II22 flows out of the node
II33 flows out of the node II11 = = II22 + + II33
Gustav Kirchoff was an 18th century German mathematician
II11
• Ohm’s law relates the flow of current to the electrical resistance and the voltage drop
• V = IR (or I = V/R) where: – I = Current– V = Voltage drop– R = Resistance
• Ohm’s Law is analogous to Darcy’s law
• Poiseuille's law can related to Darcy’s law and subsequently to Ohm's law for electrical circuits.
• Cubic law:
2
12
1a
L
Puave
AuQ ave
Adx
dhKQ
aaL
PQ 2
12
1
L
PaQ
12
3
12
3aK
A = a *unit depth
Fracture Network
5645342312 PPPPPP
563412 QQQ
4523 QQ
2312 2QQ
56
563
56
45
453
45
34
343
34
23
233
23
12
123
12
1212
2
12
12
2
12
L
Pa
L
Pa
L
Pa
L
Pa
L
Pa
54 lu
-216 lu -
900 lu
Q12
Q34
Q56
P
P12
P23
P34
Q23
Q45P45
P56
108 lu
36 lu
Matrix Form
02 2323
1212
K
L
PK
L
P
02 3434
2323
K
L
PK
L
P
02 4545
3434
K
L
PK
L
P
02 5656
4545
K
L
PK
L
P
P
L
PL
PL
PL
PL
P
LLLLL
KK
KK
KK
KK
0
0
0
0
2000
0200
0020
0002
56
56
45
45
34
34
23
23
12
12
5645342312
5645
4534
3423
2312
5645342312 PPPPPP
Back Solution
• Have conductivities and, from the matrix solution, the gradients– Compute flows
• Also have end pressures– Compute intermediate pressures from Ps
1212 K
L
PQ
a
Hydrologic-Electric AnalogyPoiseuille's law corresponds to the Kirchoff/Ohm’s Law for electrical circuits, where pressure drop Δp is replaced by voltage V and flow rate by current I
I12
I23
I56
I45
ΔP12
ΔP23
ΔP34
ΔP45
ΔP56
I23
I45
R
VI 2max 22
aL
PV
KR
1
I34
0.66 0.11 0.111.0 0.14 0.141.8 0.18 0.194.1 0.27 0.287.2 0.36 0.3743.0 0.87 0.92
ReQ (lu3/ts)
Kirchoff’sLBM
Q = 0.11 lu3/ts Q = 0.11 lu3/ts
Kirchoff LBM
5645342312 PPPPPP
Entry Length Effects
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford. 519 pp.
Eddies
Re = 93.3 mm x 2.7 mm
3 mm
2 m
m
Bai, T., and Gross, M.R., 1999, J Geophysical Res, 104, 1163-1177
Serpa, CY, 2005, Unpublished MS Thesis, FIU F
low
‘High’ Reynolds Number
• Single cylinder, Re ≈ 41
Taneda, J. Fluid Mech. 1956. (Also Katachi Society web pages)
y = 0.29x + 0.00
R2 = 1.00
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
3.0E-03
3.5E-03
4.0E-03
0.0E+00 2.0E-03 4.0E-03 6.0E-03 8.0E-03 1.0E-02 1.2E-02 1.4E-02
HEAD GRADIENT
FL
UX
(m
/s)
Non-linear
Non-curving cross joint
0.250
0.255
0.260
0.265
0.270
0.275
0.280
0.285
0.290
0.295
0.1 1.0 10.0 100.0
REYNOLDS NUMBER
HY
DR
AU
LIC
CO
ND
UC
TIV
ITY
(m
/s)
Poiseuille Law Non-linear
Non-curving cross joint
Darcy-Forschheimer Equation
• Darcy:
• +Non-linear drag term:
pa qqqk
pqk
Apparent K as a function of hydraulic gradient
• Gradients could be higher locally• Expect leveling at higher gradient?
0
5
10
15
20
25
30
35
40
1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03
Hydraulic Gradient
Hyd
rau
lic C
on
du
ctiv
ity
(m s
-1)
0.001 0.01 0.1 1 10 100 1000Approximate Reynolds Number
Darcy-Forchheimer Equation
= 1
Streamlines at different Reynolds Numbers
• Streamlines traced forward and backwards from eddy locations and hence begin and end at different locations
Re = 152
K = 20 m/s
Re = 0.31
K = 34 m/s
Future• Gray scale as hydraulic conductivity,
turbulence, solutes