Fracture/Conduit Flow
Motivation
Fractured rock (NSW Australia)
Karst
http://research.gg.uwyo.edu/kincaid/
Modeling/wakulla/wakcave2.jpg
~11 m3 s-1
~100 m
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 = 41
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford
University Press, Oxford. 519 pp.
Eddies and Cylinder Wakes
S.Gokaltun
Florida International University
Streamlines for flow around a circular cylinder at 9 ≤ Re ≤ 10.(g=0.00001, L=300 lu, D=100 lu)
Eddies and Cylinder Wakes
S.Gokaltun
Florida International University
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.)
Poiseuille Flow
y
z
u
x
a
L
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:
2

G   a 
2
ux  
x



2    2 

• G can be due to gravitational
acceleration (G = g in a vertical
slit) or the linear pressure gradient
(Pin – Pout)/L
u(x)
x=0
x = a/2
Poiseuille Flow
2


G  a 
2
ux  
x



2    2 

• Maximum
umax
G a

 
2  2 
2
u(x)
• Average
2
G 2
uaverage  umax 
a
3
12 
x=0
x = a/2
Poiseuille Flow
S.GOKALTUN
Florida 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.
I1
Gustav Kirchoff
was an 18th
century German
mathematician
I1 flows into the node
I2 flows out of the node
I3 flows out of the node
node
I2
I1 = I2 + I3
I3
• 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.
uave
1 P 2

a
12  L
Q  uave A
1 P 2
Q
aa
12  L
A = a *unit depth
• Cubic law:
a P
Q
12 L
3
dh
QK
A
dx
3
a
K
12 
36 lu
Fracture Network
P12
Q12
900 lu
Q23 54 lu
P
P23
Q34
P34
108 lu
Q45
P  P12  P23  P34  P45  P56
Q12  Q34  Q56
Q23  Q45
Q12  2Q23
P45
a12 P12 2a23 P23

12  L12
12  L23
3
Q56
P56
3
a34 P34 2a45 P45 a56 P56



12  L34
12  L45 12  L56
3
-216 lu -
3
3
Matrix Form
P23
P12
K12  2
K 23  0
L
L
P23
P34
2
K 23 
K 34  0
L
L
P34
P45
K 34  2
K 45  0
L
L
P45
P56
2
K 45 
K 56  0
L
L
 K12
 0

 0

 0
 L12
 2 K 23
2 K 23
0
0
L23
P  P12  P23  P34  P45  P56
0
 K 34
K 34
0
L34
0
0
 2 K 45
2 K 45
L45
 P12 
 L 
 12 
0   P23   0 
0   L23   0 
  P   
0   34    0 
 L34
 
 K56   P   0 
 45 
L56   L  P 
 P45 
 56 
 L56 
Back Solution
• Have conductivities and, from the matrix
solution, the gradients
P12
– Compute flows
Q
K12
L
• Also have end pressures
– Compute intermediate pressures from Ps
Hydrologic-Electric Analogy
Poiseuille'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
aP  P12  P23  P34  P45  P56
Vmax
I12
ΔP12
I23
 
P a 2

2 L 2
R
I34
ΔP34
I45
I45
I56
ΔP56
V
R
I23
ΔP23
ΔP45
I
Q = 0.11 lu3/ts
Q = 0.11 lu3/ts
Kirchoff
LBM
Re
0.66
1.0
1.8
4.1
7.2
43.0
1
K
Q (lu3/ts)
LBM
Kirchoff’s
0.11
0.11
0.14
0.14
0.18
0.19
0.27
0.28
0.36
0.37
0.87
0.92
Entry Length Effects
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford
University Press, Oxford. 519 pp.
Eddies
Serpa, CY, 2005, Unpublished MS Thesis, FIU
Bai, T., and Gross, M.R., 1999, J
Geophysical Res, 104, 1163-1177
Flow
3 mm
3.3 mm x 2.7 mm
Re = 9
‘High’ Reynolds Number
Taneda, J. Fluid Mech.
1956. (Also Katachi
Society web pages)
• Single cylinder, Re ≈ 41
Non-curving cross joint
4.0E-03
Non-linear
y = 0.29x + 0.00
R2 = 1.00
3.5E-03
2.5E-03
2.0E-03
1.5E-03
1.0E-03
Non-curving cross joint
5.0E-04
Poiseuille Law
Non-linear
0.295
0.0E+00
0.0E+00
2.0E-03
4.0E-03
6.0E-03
8.0E-03
HEAD GRADIENT
0.290
1.2E-02
1.0E-02
HYDRAULIC CONDUCTIVITY (m/s)
FLUX (m/s)
3.0E-03
1.4E-02
0.285
0.280
0.275
0.270
0.265
0.260
0.255
0.250
0.1
1.0
10.0
REYNOLDS NUMBER
100.0
Darcy-Forschheimer Equation

• Darcy:
k
q  p
• +Non-linear drag term:

k
q  a q q  p
Apparent K as a function of hydraulic gradient
Approximate Reynolds Number
0.001
0.01
0.1
1
10
100
1000
Hydraulic Conductivity (m s-1)
40
35
30
t=1
25
20
15
Darcy-Forchheimer Equation
10
5
0
1.E-09
1.E-08
1.E-07
1.E-06
1.E-05
Hydraulic Gradient
• Gradients could be higher locally
• Expect leveling at higher gradient?
1.E-04
1.E-03
Streamlines at different
Reynolds Numbers
•
Re = 0.31
Re = 152
K = 34 m/s
K = 20 m/s
Streamlines traced forward and backwards from eddy locations and hence
begin and end at different locations
Future
• Gray scale as hydraulic conductivity,
turbulence, solutes