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
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