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