A New Friction Factor Correlation A New Friction Factor Correlation for Laminar and Single-Phase Fluid for Laminar and Single-Phase Fluid

Flow through Fractured RocksFlow through Fractured Rocks

K. Nazridoust, G. Ahmadi, and D.H. SmithK. Nazridoust, G. Ahmadi, and D.H. Smith

Department of Mechanical and Aeronautical EngineeringDepartment of Mechanical and Aeronautical Engineering Clarkson University, Potsdam, NY 13699-5725 Clarkson University, Potsdam, NY 13699-5725

National Energy Technology LaboratoryNational Energy Technology LaboratoryU.S. Department of Energy, Morgantown, WV 26507-0U.S. Department of Energy, Morgantown, WV 26507-0

CT Scanning Procedures of Fractured RocksCT Scanning Procedures of Fractured Rocks- Geometric Features of Fractures- Geometric Features of Fractures

Single Phase Flows through FracturesSingle Phase Flows through Fractures

- Velocity and pressure contours- Velocity and pressure contours

Gas-Liquid FlowsGas-Liquid Flows- Water Flooding in Oil Saturated Fractures- Water Flooding in Oil Saturated Fractures

Conclusions Conclusions

OutlineOutline

C.T. Scanning of C.T. Scanning of Fractured RocksFractured Rocks

0.5 mm

HD-250 Medical C.T. ScannerHD-250 Medical C.T. Scanner

Pore Space RenderingPore Space Rendering

OMNI-X High Resolution Industrial ScannerOMNI-X High Resolution Industrial Scanner

Source Source

Detector Detector

Rock sample inRock sample inthe pressure the pressure

vessel vessel

OMNI-X Scanner - Penn StateOMNI-X Scanner - Penn State

Healed Natural FractureHealed Natural Fracture

Open Artificial FractureOpen Artificial Fracture

Induced FractureInduced Fracture

Fractures TopologyFractures Topology

Sample diameter is 25 mm. Inset size is 5x5 mm.

aper

ture

length

Extracting Digital FractureExtracting Digital Fracture

Fracture/SectionsFracture/Sections

C.T. Scan Images240 Micron Resolution

Fracture SectionsFracture Sections

Fracture SectionsFracture Sections

No-slip Wall

Inlets

ParallelParallel Plate Model, Laminar FlowPlate Model, Laminar Flow

ijj

2

ii

jji U

xxP

x

1U

xUU

t

0Ux j

j

3i

ii

H

QL12P

1LLe

iP1P

2

3

Q)1(L

HP2f

CHH .avg

TortuosityTortuosity

For ith passage :

Friction FactorFriction Factor

Average aperture height Average aperture height

Governing EquationsGoverning Equations

ContinuityContinuity

MomentumMomentum

TortuosityTortuosity

Fracture Section

Avg. Aperture Height, Havg. (m)

Std. Deviation (m)

Avg. – Std. Deviation (m)

Tortuosity

Section (a) 606 302 3040.1457

Section (b) 573 296 2770.1705

Section (c) 590 304 2820.1513

Section (d) 637 325 3120.1533

.avg1 HH

Frequency – Passage Height DistributionFrequency – Passage Height Distribution

Pressure for different flow rates, Section (a) - AirPressure for different flow rates, Section (a) - Air

Pressure for different flow rates, Section (a) - WaterPressure for different flow rates, Section (a) - Water

Velocity Magnitude, Section (a) - AirVelocity Magnitude, Section (a) - Air

Air Water

Pressure DropPressure Drop

adcb PPPP

bavgcavgaavgdavg HHHH

acdb

HRe

96f

QRe

H

Friction Factor for Laminar Flow between Parallel Friction Factor for Laminar Flow between Parallel PlatesPlates

Friction Factor for Laminar Flow in FracturesFriction Factor for Laminar Flow in Fractures

100Re , Re25.01Re

144f HH

H

687.0

Friction FactorFriction Factor

Friction FactorFriction Factor

Pressure Drop Ratio - AirPressure Drop Ratio - Air

Pressure Drop Ratio - WaterPressure Drop Ratio - Water

Two-Phase FlowsTwo-Phase FlowsWater-OilWater-Oil

WaterWater OilOil

Volume Fraction during Water FloodingVolume Fraction during Water Flooding

Shaded region is the fracture Shaded region is the fracture opening which is made transparent opening which is made transparent so that the flow can be observed. so that the flow can be observed. White regions are rock. The White regions are rock. The contours are shown on a plane contours are shown on a plane through the fracture.through the fracture.

Velocity Magnitude Contours During Velocity Magnitude Contours During Water-Oil Flow on a Plane across FractureWater-Oil Flow on a Plane across Fracture

Volume Fraction of Oil During Water-Oil Flow on a Volume Fraction of Oil During Water-Oil Flow on a Plane across FracturePlane across Fracture

Computational Grid – 3D – 37mmComputational Grid – 3D – 37mm

Volume Fraction of OilVolume Fraction of Oil

Two-Phase Air-WaterTwo-Phase Air-Water Flows though a Flows though a

Multi-Branch Fracture Multi-Branch Fracture

Natural Multi-Branch FracturesNatural Multi-Branch Fractures

Velocity Magnitude Velocity Magnitude ContoursContours

Air Volume Air Volume Fraction ContoursFraction Contours

Air-Water Flow in a Multi-Branch FractureAir-Water Flow in a Multi-Branch Fracture

Air Volume Air Volume Fraction ContoursFraction Contours

Air-Water Flow in a Multi-Branch FractureAir-Water Flow in a Multi-Branch Fracture

Water Volume Fraction Water Volume Fraction Contours on a PlaneContours on a Plane

Air-Water Flow in a Multi-Branch FractureAir-Water Flow in a Multi-Branch Fracture

Velocity Magnitude Velocity Magnitude Contours on a PlaneContours on a Plane

Air-Water Flow in a Multi-Branch FractureAir-Water Flow in a Multi-Branch Fracture

The computer simulation technique is capable of

capturing the features of the flow through the

fracture.

The simulation results are in qualitative agreement

with the parallel plate model.

The newly proposed empirical equation for fracture

friction factor provides reasonably accurate

estimates for the pressure drops in fractures for

range of Reynolds numbers less than 100.

A significant portion of the fracture pressure drop

occurs in the areas with smallest passage aperture.

ConclusionsConclusions

The order of the magnitude of the pressure in

various sections of the fracture is consistent with

the number of passages with smallest aperture that

are present in those sections.

The tortuosity of the fracture passage is an

important factor and needs to be included in the

parallel plate model.

ConclusionsConclusions