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