Effects of Turbulence on Hydraulic Heads and Parameter Sensitivities in Preferential

Ground-Water Flow Layers

Barclay Shoemaker and Eve Kuniansky U.S. Geological Survey

Preferential flow layer

Diffuse flow layer

Project Funded by USGS Ground-Water Resources ProgramKevin Dennehy, Program Manager

Contributors / Collaborators:Kevin Cunningham (USGS)

Joann Dixon (USGS)Keith J. Halford (USGS)

Preferential flow layer

Diffuse flow layer

Streamlines trace out the path of a ‘mass-less’ particle moving within the ground-water flow system.

WHAT IS TURBULENT GROUND-WATER FLOW ?

Fluid Inertial Forces > Viscous Forces

Reynolds numbers indicate whether flow is laminar or turbulent

forcesviscousforcesinertialqdRe ==

μρ

Flow is turbulent when the critical Reynolds number (NRe) is exceeded

turbulentisflowNRe ,Re>

WHAT IS TURBULENT GROUND-WATER FLOW ?

Darcy’s Law is not valid for turbulent flow

Notice turbulencedecreases Specific Discharge, energy is lost to eddies

Δh/ΔxGradient

Specific Discharge

α

Laminar Flow

Turbulent Flow

Dashed line, flow laminar for entire range of graph (Darcian flow)

critical Reynolds number

WHAT IS TURBULENT GROUND-WATER FLOW ?

WHY STUDY TURBULENT FLOW ?•It’s fundamental hydrology

• Could explain most groundwater movement in karst

• Implications for:• Fate of injected waters

• ASR• Wastewater

• Saltwater intrusion• Nutrient loading (from submarine

groundwater discharge)• Contaminate transport

Streamlines trace out the path of a ‘mass-less’ particle moving within the ground-water flow system.

Conduit Flow Process Mode 2 (CFPM2)

CFPM2 Governing Flow Equation

thSW

zhKlam

zyhKlam

yxhKlam

x szzyyxx ∂∂=+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

Traditional MODFLOW with Darcy’s Law and laminar hydraulic conductivity

CFP Mode 2 computes horizontal turbulent flow using turbulent hydraulic conductivity.

thSW

zhKlam

zyhKturb

yxhKturb

x szzyyxx ∂∂=+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

lamadjturb KFK =

Turbulent horizontal hydraulic conductivity (Kturb) is a non-linear function of the Reynolds Number (Re)

after critical Reynolds number (NRe) is exceeded.

11 −− ΔΔ=

kiterkiterturb

critlamkiteradj hK

hKF

porelamcrit dK

lvNh Δ=Δ Re

CFPM2 Turbulent K

Derived by Kuniansky and Halford, 2008

SPECIFIC DISCHARGE, IN METERS PER DAY

0 50 100 150 200 250 300

HYD

RA

ULI

C G

RA

DIE

NT,

UN

ITLE

SS

0.000

0.001

0.002

0.003

0.004

0.005

0.006

Darcy WeisbachDarcy's LawCFPM2 Approach

Critical head difference is exceeded

SPECIFIC DISCHARGE, IN METERS PER DAY

0 50 100 150 200 250 300

HYD

RA

ULI

C G

RA

DIE

NT,

UN

ITLE

SS

0.000

0.001

0.002

0.003

0.004

0.005

0.006

Darcy WeisbachDarcy's LawCFPM2 Approach

Critical head difference is exceeded

CFPM2 Benchmark Testing

Permeameter Data Verify New Turbulence Process for MODFLOW

By Eve L. Kuniansky and others.

(a) Cube 6 Vertical Axis

0

10

20

30

40

CRITICAL VELOCITYMEASUREDSIMULATED

Klam = 48 m/dNRe = 1.44RMS = 0.70 m/d

(b) Cube 6 Horizontal Axis 1

0

10

20

30

40

Klam = 61 m/dNRe = 1.44RMS = 0.35 m/d

(c) Cube 6 Horizontal Axis 2

0

10

20

30

40

0.00 0.25 0.50 0.75 1.00

?H/?L, DIMENSIONLESS

Klam = 36 m/dNRe = 1.44RMS = 0.57 m/d

SPEC

IFIC

VEL

OC

ITY,

IN M

ETER

S PE

R D

AY

CFPM2 Testing

ET Station

Broward CountyMiami-Dade County

Bisc

ayne

Bay

Water Conservation Area No. 3A

Water Conservation Area No. 3B

Everglades National

Park

L-67A

L-67C

L-29

L-31N

C-4

C-6

C-2

0 2 4 6 Kilometers

0 2 4 6 Miles

80°30' 80°20'

26°00'

25°50'

25°40'

CFPM2 APPLICATION TO BISCAYNE AQUIFER

Picture from Kevin Cunningham, USGS

CFPM2 APPLICATION TO BISCAYNE AQUIFER

Hydrogeologic conceptualization of Lake Belt area from:Cunningham and Dixon, written communication

Preferential flow layer

Diffuse flow layer

CFPM2 APPLICATION TO BISCAYNE AQUIFER

A

θ⎟⎟⎠

⎞⎜⎜⎝

⎛=

vgdKlam 32

2

To estimate Klam, one could use the resistance terms in the Darcy-Weisbach equation, limited by effective porosity. Limiting

by effective porosity accounts for the resistance offered by “dead end” voids.

CFPM2 APPLICATION TO BISCAYNE AQUIFER

Derived by Eve Kuniansky, 2008

Table 1. Initial estimates of laminar horizontal hydraulic conductivity for preferential flow layers

Layer Source Data NumberWells

MeanEffectivePorosity(%)

Number ofMeasurementOn cores orimages

MeanVugDiameter(cm)

Laminar HorizontalHydraulic Conductivity(meters per day)

2 DirectMeasurementon cores

23 11.8 240 0.9 200,000

5 DirectMeasurementon cores

6 18.3 65 0.8 300,000

8 MostlyMeasurementsfrom digitalborehole wallimages

22 14.8 438 3.5 5,000,000

Data from Kevin Cunningham and others, 2006

CFPM2 APPLICATION TO BISCAYNE AQUIFER

Critical reynolds numbers are uncertain

Broward CountyMiami-Dade County

Bisc

ayne

Bay

Water Conservation Area No. 3A

Water Conservation Area No. 3B

Everglades National

Park

L-67A

L-67C

L-29

L-31N

C-4

C-6

C-2

Explanation

Flow vector

1

2

3

4

Canal or stream

Turbulence Codes

Flow to right and front are laminar

Flow to right is turbulant

Flow to front is turbulant

Flow to right and front are turbulant

0 2 4 6 Kilometers

0 2 4 6 Miles

Preliminary Run—layer 8

Upper critical Reynolds Number equals 55

Head difference from laminar

elevation due to turbulence

Model parameter

rch hk1 vk1 hk2 vk2 hk3 vk3 hk4 vk4 hk5 vk5 hk6 vk6 hk7 vk7 hk8 vk8 T v1 v2 v3

Com

posi

te-s

cale

d se

nsiti

vity

, uni

tless

0.00

0.01

0.02

0.03

0.04

Scenario 1Scenario 2Scenario 3Scenario 4Scenario 5

CFPM2 APPLICATION TO BISCAYNE AQUIFER

1. Extent of turbulent flow increases with increasing hydraulic conductivity, mean void diameter, groundwater temperature, and decreasing critical Reynolds numbers.

2. When turbulence was active (occurring in about 56% of preferential flow model cells), head differences from laminar elevations ranged from about 18 to +27 cm.

3. The composite-scaled sensitivities of horizontal hydraulic conductivities decreased by as much as 70% when turbulence was essentially removed.

4. This study highlights potential errors in model calculations based on the equivalent porous media assumption, which assumes laminar flow in uniformly distributed void spaces

Summary

Limitations

• Macro-scale simplification of impacts of turbulent flow

• Vast uncertainty in aquifer hydraulic properties and boundaries

• Theory is sound, but applications on systems with uncertainty may produce unreliable predictions

Thanks !For more information,bshoemak@usgs.gov

Picture taken by Eve Kuniansky of field trip to Fish River Cave near Yangshuo, China,

Shoemaker, W. B., K. J. Cunningham, E. L. Kuniansky, and J. Dixon (2008), Effects of turbulence on hydraulic heads and parameter sensitivities in preferential groundwater flow layers, Water Resour. Res., 44, W03501, doi:10.1029/2007WR006601.