Non-Newtonian Flow in Porous Media

1. PERM Group Imperial College LondonPERM Group Imperial College London
Non-Newtonian Flow in Porous MediaNon-Newtonian Flow in Porous Media
Taha Sochi & Martin BluntTaha Sochi & Martin Blunt
Non-Newtonian FluidNon-Newtonian Fluid
Shear stress is not proportionalShear stress is not proportional
to shear rate:to shear rate: ττ ≠≠ µγµγ
Three groups of behaviour:Three groups of behaviour:
1.1. Time-independent: shear rate dependsTime-independent: shear rate depends
only on instantaneous stress.only on instantaneous stress.
2.2. Time-dependent: shear rate is functionTime-dependent: shear rate is function
of magnitude & duration of shear.of magnitude & duration of shear.
3.3. Viscoelastic: Shows partial elasticViscoelastic: Shows partial elastic
recovery on removal of stress.recovery on removal of stress.
Herschel ModelHerschel Model
τ = ττ = τοο ++ CCγγnn
ττ Shear stressShear stress
ττοο Yield stressYield stress
CC Consistency factorConsistency factor
γγ Shear rateShear rate
nn Flow behaviour indexFlow behaviour index
Herschel classes:Herschel classes:
Flow rate in cylindrical tube:Flow rate in cylindrical tube:
( ) 





+
+
+
−
+
+
−
−





=
+
1/n1
τ
1/n2
)τ(τ2τ
1/n3
)τ(τ
ττ
ΔP
L
C
8π
Q
2
oowo
2
ow
ow
3
1/n
n
11
ττοο CC nn Herschel parametersHerschel parameters
LL Tube lengthTube length
∆∆PP Pressure differencePressure difference
ττww ∆∆PR/2LPR/2L ((R =R = tube radius)tube radius)
Berea networkBerea network Sand pack networkSand pack network
(after Xavier Lopez)(after Xavier Lopez)
Comparison with Single TubeComparison with Single Tube
Fluid with Yield StressFluid with Yield Stress
3mm3mm
Equivalent single tubeEquivalent single tube
2.5mm2.5mm
Equivalent single tubeEquivalent single tube
Network ModellingNetwork Modelling
1.1. Obtain 3D image of the pore space.Obtain 3D image of the pore space.
2.2. Build a topologically-equivalentBuild a topologically-equivalent
network in terms of pore sizes,network in terms of pore sizes,
shapes and connectivity.shapes and connectivity.
3.3. Account for non-circularity, whenAccount for non-circularity, when
calculatingcalculating QQ for cylinder, by usingfor cylinder, by using
equivalent radius:equivalent radius:
4/1
8






=
π
G
Req
4.Start with initial guess for viscosity4.Start with initial guess for viscosity
in each network element.in each network element.
5.Invoke conservation of volume and5.Invoke conservation of volume and
hence solve the pressure field.hence solve the pressure field.
6.Update viscosity using Herschel6.Update viscosity using Herschel
expression with pseudo-Poiseuilleexpression with pseudo-Poiseuille
definition.definition.
7.Recompute the pressure using the7.Recompute the pressure using the
updated viscosities.updated viscosities.
8.Iterate until convergence is8.Iterate until convergence is
achieved when specified toleranceachieved when specified tolerance
error in totalerror in total QQ is reached.is reached.
9.Obtain total flow rate and apparent9.Obtain total flow rate and apparent
viscosity.viscosity.
Experimental ResultsExperimental Results
Chase:Chase: 9 datasets for Bingham9 datasets for Bingham
aqueous solution of Carbopolaqueous solution of Carbopol
941 in column of glass beads.941 in column of glass beads.
Sample:Sample:
Al-Fariss:Al-Fariss: 16 datasets for waxy &16 datasets for waxy &
crude oils in 2 packed beds ofcrude oils in 2 packed beds of
sand. Data is found inconsistent.sand. Data is found inconsistent.
Sample:Sample:
Why Network Yields FirstWhy Network Yields First
Future WorkFuture Work
1.Modelling viscoelasticity.1.Modelling viscoelasticity.
2.Including more physics, e.g.2.Including more physics, e.g.
adsorption & wall exclusion.adsorption & wall exclusion.
3.Modelling 2-phase flow with3.Modelling 2-phase flow with
two non-Newtonian fluids.two non-Newtonian fluids.
ReferencesReferences
• Skelland A. Non-Newtonian Flow and Heat Transfer.Skelland A. Non-Newtonian Flow and Heat Transfer.
• M. BluntM. Blunt et al.et al. Detailed Physics, PredictiveDetailed Physics, Predictive
Capabilities and Macroscopic Consequences forCapabilities and Macroscopic Consequences for
Pore-Network Models of Multiphase Flow.Pore-Network Models of Multiphase Flow.
• Sorbie K. Polymer-Improved Oil Recovery.Sorbie K. Polymer-Improved Oil Recovery.
• Lopez X. Pore-Scale Modelling of Non-NewtonianLopez X. Pore-Scale Modelling of Non-Newtonian
Flow.Flow.
• Valvatne P. Predictive Pore-Scale Modelling ofValvatne P. Predictive Pore-Scale Modelling of
Multiphase Flow.Multiphase Flow.
• G. ChaseG. Chase et al.et al. Incompressible Cake Filtration of aIncompressible Cake Filtration of a
Yield stress Fluid.Yield stress Fluid.
• T. Al-FarissT. Al-Fariss et al.et al. Flow of Shear-Thinning LiquidFlow of Shear-Thinning Liquid
With Yield Stress Through Porous Media.With Yield Stress Through Porous Media.