HighHigh--Resolution XResolution X--ray Computed ray Computed Tomography of Tomography of MacroporousMacroporous KarstKarstfor Permeability Measurement and for Permeability Measurement and
NonNon--DarcianDarcian Flow via Lattice Flow via Lattice Boltzmann ModelsBoltzmann Models
Mike SukopMike Sukop11, Haibo Huang, Haibo Huang22, , Kevin CunninghamKevin Cunningham33, and , and Peter AlvarezPeter Alvarez11
1Florida International University2University of Science and Technology of China3United States Geological Survey, Ft. Lauderdale
Objectives
• Compute K of highly macroporous karst rock • Compute K under different hydraulic
gradients (different Re). Expect reduced apparent K at high Re due to eddy head dissipation
Introduction• LBM is a mesoscopic method based on the
scale between molecular dynamics and familiar continuum approaches
• A particle stream-and-collide perspective with interparticle forces is adequate for most simulations
• LBMs handle complex geometries well
Kinetic Theory• Complete set of position (x) and momentum (p)
coordinates for all particles gives dynamical state of system
• Together with classical mechanics, allows prediction of future states
• However, this level of description is not possible• Use a statistical description: focus on the
distribution function of the “state” of molecules
( )tf ,,px
LBM BasicsLBM Basics
0 11
22
33
44
5566
77 88
D2Q9
e1
e2
e3
e4
e5e6
e7 e8
f1
f2
f3
f4
f5f6
f7 f8
f0
Histogram view of the distribution function, f.
Lattice Unit, lu
f1f5
1 2 5 a
f
f2 f3
f4
3 4 6 7 8f6 f7 f8
f0
0
02468
10121416
0 2 4 6 8 10 12 14 16 18a
f
D3Q19
StreamingStreaming ( ) ( )tftttf aaa ,, xex =Δ+Δ+
Single Relaxation Time BGK (Bhatnagar-Gross-Krook) Approximation
( ) ( )⎥⎦
⎤⎢⎣
⎡−•+•+= 2
2
4
2
2 23
2931)(
cccwf a
eqa
uueuexx aaρ
( ) ( ) ( ) ( )[ ]τ
tftftfttfeq
aaaaa
,,,1, xxxex −−=+Δ+
Streaming
Collision (i.e., relaxation towards equilibrium)
Collision and streaming steps must be separated if solid boundaries present (bounce back boundary is a separate collision)
• wa are 4/9 for the rest particles (a = 0), • 1/9 for a = 1, 2, 3, 4, and • 1/36 for a = 5, 6, 7, 8.
• τ relaxation time (viscosity and diffusion)• c speed on lattice (1 lu /time step)
-40 -30 -20 -10 0 10 20 30 40-2
0
2
4
6
8
10
12
14
16x 10
-3 Red-open-symbols=Simulation V-profile, Blue-line=Poiseuille
g = 0.00001 lu ts-2
τ = 1 tsDiameter = 63 lu
Poiseuille flow in a circular pipe
ux
r
( )22
4raGux −=
μa
r
-20 -15 -10 -5 0 5 10 15 20-1
0
1
2
3
4
5x 10-4 Red-open-symbols=Simulation V-profile, Blue-line=Poiseuille
( ) ( )
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
−+⎟⎠⎞
⎜⎝⎛−
∂∂−= ∑
∞
=13
22 cos
cosh
cosh141
21,
k
k
k
k
k
k
x az
ab
ay
aza
xpzyu α
α
α
αμ
( ) ...,2,1,2
12 =−= kkkπα Papanastasiou, T. C., Georgiou, G. C. , Alexandrou, A. N.,
(2000). Viscous fluid flow. CRC Press, Boca Raton, p.259
For square, a = b
g = 0.000001 lu/ts2
τ = 1 tsHeight = 31 lu
a
-a
y
z b-b
Poiseuille flow in a rectangular duct
ux
x
Why use LBM in macroporous karst context?
• Easy to incorporate complex geometry
• Transition to higher Reynolds numbers
Why use LBM in macroporous karst context?
• Easy to incorporate complex geometry
• Transition to higher Reynolds numbers
Re = 88
Data and Tools• Karst data scales
– 0.0003 to 0.3 m high-resolution CT scans– 0.002 to 30 m borehole imagery– 1 to 1000 m cave diver sonic rangefinder data
• Medium simulation required for borehole and rangefinder data
• LBM integrative tool– Compute K at multiple scales– Assess non-Darcy potential and impacts
• T > 1,600,000 ft2 d-1• 7,000 gpm wells
Renken, R.A., Dixon, J., Koehmstedt, J., Lietz, A.C., Ishman, S., Marella, R.L., Telis, P., Rogers, J., and Memberg, S., 2005, Impact of Anthropogenic Development on Coastal Ground-Water Hydrology in Southeastern Florida, 1900-2000: Reston, Va., U.S. Geological Survey Circular 1275, 77 p.
T>10
5 m2 d-
1(K
~ 0.
04 m
s-1 )
4-7 m3s-1
Geology to model parameters
GSA Special Paper 404, 2006
Characterization of Aquifer Heterogeneity Using Cyclostratigraphy and Geophysical Methods in the Upper Part of the Karstic Biscayne Aquifer, Southeastern Florida By Kevin J. Cunningham, Janine L. Carlson, G. Lynn Wingard, Edward Robinson, and Michael A. Wacker U.S. Geological SurveyWater-Resources Investigations Report 03-4208
Example Data Set
Data and image produced at the High-Resolution X-ray Computed Tomography Facility of the University of Texas at Austin
Burrow porosity in Miami Limestone barrier bar deposited during the last interglacial
(maximum unit thickness ~ 1m)
Photo: Mike Wacker/USGS
8- and 16-bit slices
ML-184
ML-185
• With 0.8 mm slice spacing, 401 slices = 321 mm
1 mm
0.8
mm
1024 pixels = 278 mm
0.271 mm/pixel
Data and image produced at the High-Resolution X-ray Computed Tomography Facility of the University of Texas at Austin
Thresholding (<75/255 →pore)
ML-184
Gray 75
Data and image produced at the High-Resolution X-ray Computed Tomography Facility of the University of Texas at Austin
601x601x1203 (pixels) 282x282x282 (pixels and lattice units)
Bulk of Sample and Experimental Cube
~7.5
cm
~22 million cells. Limit set by memory of computer and code.
Data produced at the High-Resolution X-ray Computed Tomography Facility of the University of Texas at Austin
Data produced at the High-Resolution X-ray Computed Tomography Facility of the University of Texas at Austin
Velocity Magnitude
Darcy’s Law
• q flux• k permeability• ρ density• μ viscosity• p pressure• g gravity
p∇−=μ1kq
h∇−= Kq• h head (= p/ρg) • K hydraulic
conductivity (LT-1)
h∇−=μρgkq
Biscayne Aquifer SamplesFPL-Q3a-1 FPL-Q3a-2 G3837-18 G3837-22 C100-Q5e-1 ML-01 kbm-1
Porosity0.16 0.23 0.29 0.29 0.50 0.64 0.81
z-axis Hydraulic Conductivity (m/s)
0 0.35 0.47 2 18 34 53
27.75 cm
9.1 cmLBM-measured subsamples
Hydraulic Conductivity
Values
Freeze and Cherry (1979) Groundwater, Prentice-Hall
Maximum k from air permeability laboratory measurements
(Cunningham et.al., 2006)
Darcy-Forschheimer Equation
• Darcy:
• +Non-linear drag term:
pa −∇=+ qqqkμ
p−∇=qkμ
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-03Hydraulic Gradient
Hyd
raul
ic C
ondu
ctiv
ity (m
s-1)
0.001 0.01 0.1 1 10 100 1000Approximate Reynolds Number
Darcy-Forchheimer Equation
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
Conclusions
• LBM can measure permeabilities outside the range routinely accessible to laboratory measurements
• LBM can assess magnitude of departure from Darcy flow