This document outlines pore-scale modelling of non-Newtonian flow in porous media. It defines Newtonian and non-Newtonian fluids, and describes various rheological models including time-independent, viscoelastic, and time-dependent behaviors. Methods for modelling flow in porous networks include combining pore space descriptions with fluid rheology models. Results show good agreement between models and experiments for some fluids like Ellis fluids, but mixed results for Herschel-Bulkley fluids due to experimental errors and yield stress approximations. Network models are dependent on pore structure details for yield stress fluids.

1. Pore-Scale Modelling of Non-NewtonianPore-Scale Modelling of Non-Newtonian
Flow in Porous MediaFlow in Porous Media
Pore Scale Modelling ConsortiumPore Scale Modelling Consortium
Imperial College LondonImperial College London
Supervisor: Prof. Martin BluntSupervisor: Prof. Martin Blunt
Taha SochiTaha Sochi

2. Definition of Newtonian & non-Newtonian fluidsDefinition of Newtonian & non-Newtonian fluids
OutlineOutline
Rheology of non-Newtonian fluidsRheology of non-Newtonian fluids
Modelling the flow in porous media in generalModelling the flow in porous media in general
Modelling time-independent flowModelling time-independent flow
Modelling yield stress behaviourModelling yield stress behaviour
Modelling viscoelastic flowModelling viscoelastic flow
Results, recommendations &Results, recommendations &
acknowledgementsacknowledgementsQuestions and discussionsQuestions and discussions

3. DefinitionDefinition
ofof
Newtonian & Non-Newtonian FluidsNewtonian & Non-Newtonian Fluids

4. NewtonianNewtonian:: stress is proportional to strain rate:stress is proportional to strain rate:
τ ∝ γτ ∝ γ
Non-NewtonianNon-Newtonian: this condition is not satisfied.: this condition is not satisfied.
Three groups of behaviour:Three groups of behaviour:
1. Time-independent: strain rate solely depends on1. Time-independent: strain rate solely depends on
instantaneous stress.instantaneous stress.
3. Time-dependent: strain rate is function of both3. Time-dependent: strain rate is function of both
magnitude and duration of stress.magnitude and duration of stress.
2. Viscoelastic: shows partial elastic recovery on2. Viscoelastic: shows partial elastic recovery on
removal of deforming stress.removal of deforming stress.

5. RheologyRheology
ofof
Non-Newtonian FluidsNon-Newtonian Fluids

6. 1. Time-Independent1. Time-Independent

7. This is a shear-thinning modelThis is a shear-thinning model
ττ StressStress
µµοο Zero-shear viscosityZero-shear viscosity
γγ Strain rateStrain rate
ττ1/21/2 Stress atStress at µµοο / 2/ 2
αα Indicial parameterIndicial parameter
A. EllisA. Ellis
1
21
1
−






+
= α
/
o
τ
τ
γμ
τ

8. This is a general time-independent modelThis is a general time-independent model
ττ StressStress
ττοο Yield stressYield stress
CC Consistency factorConsistency factor
γγ Strain rateStrain rate
nn Flow behaviour indexFlow behaviour index
B. Herschel-BulkleyB. Herschel-Bulkley
n
o
Cγττ +=

9. 2. Viscoelastic2. Viscoelastic
Convergence-Convergence-
divergence withdivergence with
time of fluidtime of fluid
beingbeing
comparable withcomparable with
time of flowtime of flow
DelayedDelayed
response &response &
relaxationrelaxation
Dominance ofDominance of
extension overextension over
shear at highshear at high
flow rateflow rate
Time-dependency
Strain
hardening
Intermediate plateau

10. A. Upper Convected MaxwellA. Upper Convected Maxwell
This is the simplest and most popularThis is the simplest and most popular
modelmodel
ττ Stress tensorStress tensor
λλ11 Relaxation timeRelaxation time
µµοο Low-shear viscosityLow-shear viscosity
γγ Rate-of-strain tensorRate-of-strain tensor
γττ oµλ =+
∇
1

11. B. Oldroyd-BB. Oldroyd-B
ττ Stress tensorStress tensor
λλ11 Relaxation timeRelaxation time
λλ22 Retardation timeRetardation time
µµοο Low-shear viscosityLow-shear viscosity
γγ Rate-of-strain tensorRate-of-strain tensor






+=+
∇∇
γγττ 21 λµλ o
This is the second in simplicity andThis is the second in simplicity and
popularitypopularity

12. 3. Time-Dependent3. Time-Dependent

13. A. GodfreyA. Godfrey
This is suggested as a thixotropic modelThis is suggested as a thixotropic model
)1(
)1()(
''
'
/''
/'
λ
λ
µ
µµµ
t
t
i
e
et
−
−
−∆−
−∆−=
µµ ViscosityViscosity
tt Time of shearingTime of shearing
µµii Initial-time viscosityInitial-time viscosity
∆∆µµ’’ && ∆∆µµ’’’’ Viscosity deficits associatedViscosity deficits associated
with time constantswith time constants λλ’’ && λλ’’’’

14. B. Stretched Exponential ModelB. Stretched Exponential Model
This is a general time-dependent modelThis is a general time-dependent model
)1)(()( / st
iini
et λ
µµµµ −
−−+=
µµ ViscosityViscosity
tt Time of shearingTime of shearing
µµii Initial-time viscosityInitial-time viscosity
µµinin Infinite-time viscosityInfinite-time viscosity
λλss Time constantTime constant

15. Viscoelastic vs. ThixotropicViscoelastic vs. Thixotropic
Time-dependency of viscoelastic arisesTime-dependency of viscoelastic arises
because response is not instantaneous.because response is not instantaneous.
Time-dependent behaviour of thixotropicTime-dependent behaviour of thixotropic
arises because of change in structure.arises because of change in structure.

16. Modelling the FlowModelling the Flow
inin
Porous MediaPorous Media

17. Capillary FlowCapillary Flow
For a capillary:For a capillary: Pcq ∆= .
Flow rate = conductanceFlow rate = conductance × Pressure× Pressure
dropdrop
1.1. Newtonian fluidNewtonian fluid:: constant)( == µcc
2.2. Viscous non-Viscous non-
NewtonianNewtonian::
),( Pcc µ=
3.3. Fluid with MemoryFluid with Memory:: ),,( tPcc µ=

18. Network FlowNetwork Flow
For a network of capillaries, a set ofFor a network of capillaries, a set of
equations representing the capillaries andequations representing the capillaries and
satisfying mass conservation should besatisfying mass conservation should be
solved simultaneously to produce asolved simultaneously to produce a
consistent pressure field:consistent pressure field:
1.1. Newtonian fluidNewtonian fluid: solve once and for all: solve once and for all
since conductance is known in advance.since conductance is known in advance.

19. Network FlowNetwork Flow
3.3. Fluid with memoryFluid with memory: for the steady-: for the steady-
state viscoelastic flow, start with anstate viscoelastic flow, start with an
initial guess for the flow rate and iterate,initial guess for the flow rate and iterate,
considering the effect of the localconsidering the effect of the local
pressure and viscosity variation due topressure and viscosity variation due to
converging-diverging geometry, untilconverging-diverging geometry, until
convergence is achieved.convergence is achieved.
2.2. Viscous non-NewtonianViscous non-Newtonian: starting with: starting with
an initial guess, solve for the pressurean initial guess, solve for the pressure
iteratively, updating the viscosity afteriteratively, updating the viscosity after
each cycle, until reaching convergence.each cycle, until reaching convergence.

20. Network ModellingNetwork Modelling
ofof
Time-Independent FluidsTime-Independent Fluids

21. Combine the pore space description of theCombine the pore space description of the
medium with the bulk rheology of the fluid.medium with the bulk rheology of the fluid.
The bulk rheology is used to derive analyticalThe bulk rheology is used to derive analytical
expression for the flow in simplified poreexpression for the flow in simplified pore
geometry.geometry.
Network Modelling StrategyNetwork Modelling Strategy
The main networks used in this study are theThe main networks used in this study are the
sand pack and Berea ofsand pack and Berea of Øren and co-workers.Øren and co-workers.

22. Results: Network-Bundle of TubesResults: Network-Bundle of Tubes
* A comparison was made for Herschel-Bulkley* A comparison was made for Herschel-Bulkley
fluid between random networks and a uniformfluid between random networks and a uniform
bundle of tubes to assess the model.bundle of tubes to assess the model.
* The uniform bundle of tubes model was used* The uniform bundle of tubes model was used
in this assessment instead of more complexin this assessment instead of more complex
and realistic model such as non-uniformand realistic model such as non-uniform
bundle or cubic network because of simplicitybundle or cubic network because of simplicity
which is a big advantage to see the hiddenwhich is a big advantage to see the hidden
features.features.
* Good results are obtained for both sand pack* Good results are obtained for both sand pack
and Berea.and Berea.

23. Results: Network-Bundle of TubesResults: Network-Bundle of Tubes
Sand pack
το = 0.0Pa
Sand pack
το = 1.0Pa
Berea
το = 0.0Pa
Berea
το = 1.0Pa

24. Results: Random-Regular NetworksResults: Random-Regular Networks
* A comparison was also made for Herschel-* A comparison was also made for Herschel-
Bulkley fluid between the random networks andBulkley fluid between the random networks and
their cubic equivalent (similar distribution,their cubic equivalent (similar distribution,
coordination number, permeability and porosity).coordination number, permeability and porosity).
* The analysis revealed that the cubic network* The analysis revealed that the cubic network
behaviour for yield-stress fluid is highlybehaviour for yield-stress fluid is highly
dependent on the distribution and realisationdependent on the distribution and realisation
because of the random nature of the networkbecause of the random nature of the network
generation. Therefore no firm conclusion can begeneration. Therefore no firm conclusion can be
reachedreached

25. Results: Random-Regular NetworksResults: Random-Regular Networks
Cubic-
Sand pack
το = 0.0Pa
Cubic-
Sand pack
το = 1.0Pa
Cubic-Berea
το = 0.0Pa
Cubic-Berea
το = 1.0Pa

26. Results: ExperimentalResults: Experimental
Good results are obtained for Ellis.Good results are obtained for Ellis.
Mixed results are obtained for Herschel-Bulkley.Mixed results are obtained for Herschel-Bulkley.
The main sources of failure for Herschel-BulkleyThe main sources of failure for Herschel-Bulkley
are experimental errors and imperfectare experimental errors and imperfect
modellingmodelling of yield stress phenomenon.of yield stress phenomenon.

27. Results: ExperimentalResults: Experimental
Sadowski
Ellis
Park
Ellis
Al-Fariss
Herschel-Bulkley
Chase
Bingham

28. Network ModellingNetwork Modelling
ofof
Yield Stress FluidsYield Stress Fluids

29. What is Yield Stress ?What is Yield Stress ?
The stress at which the substance startsThe stress at which the substance starts
flowing.flowing.
The substance is assumed to be solidThe substance is assumed to be solid
below its yield stress and fluid above.below its yield stress and fluid above.

30. DifficultiesDifficulties
1. The yield stress value is usually obtained1. The yield stress value is usually obtained
by extrapolation and this limits the accuracy.by extrapolation and this limits the accuracy.
2. Before yield, the pressure is not well-2. Before yield, the pressure is not well-
defined if substance is regarded as solid.defined if substance is regarded as solid.
3. The yield is highly dependent on the3. The yield is highly dependent on the
actual shape of the pore space and its fineactual shape of the pore space and its fine
details. This is compromised by modellingdetails. This is compromised by modelling
the throats with regular cylindrical ducts.the throats with regular cylindrical ducts.
4. While in the case of bulk and tube flow4. While in the case of bulk and tube flow
the yield stress is a property of the fluid, inthe yield stress is a property of the fluid, in
the case of porous media it may depend onthe case of porous media it may depend on
the porous media as well.the porous media as well.

31. Predicting Network Threshold Yield PressurePredicting Network Threshold Yield Pressure
1. Invasion Percolation with Memory (IPM):1. Invasion Percolation with Memory (IPM):
Find the path of minimum yield pressureFind the path of minimum yield pressure
connecting the inlet to the outlet byconnecting the inlet to the outlet by
increasing the yield pressure continuously.increasing the yield pressure continuously.
The threshold yield pressure is the value atThe threshold yield pressure is the value at
which the outlet is first reached.which the outlet is first reached.
Assumptions:Assumptions:
A. The yield pressure of a number ofA. The yield pressure of a number of
serially-connected bonds is the sum of theirserially-connected bonds is the sum of their
yield pressures.yield pressures.
B. Backtracking is allowed.B. Backtracking is allowed.

33. Predicting Network Threshold Yield PressurePredicting Network Threshold Yield Pressure
2. Path of Minimum Pressure (PMP):2. Path of Minimum Pressure (PMP):
Find the path of minimum yield pressureFind the path of minimum yield pressure
connecting the inlet to the outlet by findingconnecting the inlet to the outlet by finding
the minimum yield pressure needed tothe minimum yield pressure needed to
reach each node. The threshold yieldreach each node. The threshold yield
pressure is the minimum of the valuespressure is the minimum of the values
obtained for the nodes at outlet.obtained for the nodes at outlet.
Assumptions:Assumptions:
A. The yield pressure of a number ofA. The yield pressure of a number of
serially-connected bonds is the sum of theirserially-connected bonds is the sum of their
yield pressures.yield pressures.
B. Backtracking isB. Backtracking is notnot allowed.allowed.

35. ResultsResults
Random networks: IPM and PMP agree inRandom networks: IPM and PMP agree in
most cases. When they disagree, PMP givesmost cases. When they disagree, PMP gives
higher values. The reason is backtracking ishigher values. The reason is backtracking is
allowed in IPM but not in PMP.allowed in IPM but not in PMP.
PMP is more efficient in terms of CPU timePMP is more efficient in terms of CPU time
and memory.and memory.
Cubic networks: IPM and PMP agree in allCubic networks: IPM and PMP agree in all
cases investigated. This is due to lesscases investigated. This is due to less
likelihood of backtracking as it involveslikelihood of backtracking as it involves
more tortuous path.more tortuous path.

36. ResultsResults
Both IPM and PMP give lower values thanBoth IPM and PMP give lower values than
the network model, e.g. for sand pack:the network model, e.g. for sand pack:
Boundaries Threshold Yield Pressure (Pa)
Lower Upper Actual IPM PMP
0.0 1.0 80.94 53.81 54.92
0.0 0.9 71.25 49.85 51.13
0.0 0.8 61.14 43.96 44.08
0.0 0.7 56.34 38.47 38.74
0.0 0.6 51.76 32.93 33.77
0.0 0.5 29.06 21.52 21.52

37. ResultsResults
Cubic network resembling Berea:Cubic network resembling Berea:
Boundaries Threshold Yield Pressure (Pa)
Lower Upper Actual IPM PMP
0.0 1.0 128.46 100.13 100.13
0.0 0.9 113.82 89.34 89.34
0.0 0.8 100.17 81.07 81.07
0.0 0.7 87.36 69.89 69.89
0.0 0.6 75.21 58.59 58.59
0.0 0.5 61.25 47.31 47.31

38. Network ModellingNetwork Modelling
ofof
Viscoelastic FluidsViscoelastic Fluids

39. Steady-State Viscoelastic FlowSteady-State Viscoelastic Flow
A sensible strategy for modelling theA sensible strategy for modelling the
steady-state viscoelastic flow is to startsteady-state viscoelastic flow is to start
with an initial guess for flow rate andwith an initial guess for flow rate and
iterate, considering the effect of the localiterate, considering the effect of the local
pressure and viscosity variation due topressure and viscosity variation due to
converging-diverging geometry, untilconverging-diverging geometry, until
convergence is achieved.convergence is achieved.
This approach is adopted by Tardy usingThis approach is adopted by Tardy using
a modified Bautista-Manero model whicha modified Bautista-Manero model which
is based on the Fredrickson and Oldroyd-is based on the Fredrickson and Oldroyd-
B models.B models.

40. Tardy AlgorithmTardy Algorithm
1. Since converging-diverging geometry1. Since converging-diverging geometry
is important for viscoelastic flow, theis important for viscoelastic flow, the
capillaries should be modelled withcapillaries should be modelled with
contraction.contraction.
2. Each capillary is2. Each capillary is discretized in the flowdiscretized in the flow
direction and a discretized form of thedirection and a discretized form of the
flow equations is used assuming a priorflow equations is used assuming a prior
knowledge of stress & viscosity at inlet.knowledge of stress & viscosity at inlet.

41. Tardy AlgorithmTardy Algorithm
3. Starting with an initial guess for the3. Starting with an initial guess for the
flow rate and using iterative technique,flow rate and using iterative technique,
the pressure drop as a function of thethe pressure drop as a function of the
flow rate is found for each capillary.flow rate is found for each capillary.
4. The pressure field for the whole4. The pressure field for the whole
network is then found iteratively untilnetwork is then found iteratively until
convergence is achieved.convergence is achieved.

42. Some ResultsSome Results
Sand PackSand Pack BereaBerea
Sand PackSand Pack BereaBerea
Effect of convergence-divergenceEffect of convergence-divergence Effect of convergence-divergenceEffect of convergence-divergence
Effect of number of slicesEffect of number of slices Effect of number of slicesEffect of number of slices

43. Some ResultsSome Results
Sand PackSand Pack BereaBerea
Sand PackSand Pack BereaBerea
Newtonian caseNewtonian case
Newtonian caseNewtonian case
Effect of kinetic parameterEffect of kinetic parameter Effect of kinetic parameterEffect of kinetic parameter

44. Results, RecommendationsResults, Recommendations
&&
AcknowledgementsAcknowledgements

45. General Results and ConclusionsGeneral Results and Conclusions
* Network model was extended to include Ellis* Network model was extended to include Ellis
and Herschel-Bulkley models.and Herschel-Bulkley models.
* Network-bundle and random-regular* Network-bundle and random-regular
comparisons revealed sensible trends.comparisons revealed sensible trends.
* Reasonable agreement with experiments was* Reasonable agreement with experiments was
obtained.obtained.
* Yield stress was investigated & 2 algorithms* Yield stress was investigated & 2 algorithms
were implementedwere implemented
* Tardy algorithm for steady-state viscoelastic* Tardy algorithm for steady-state viscoelastic
flow was investigated and implemented.flow was investigated and implemented.
* Viscoelasticity and thixotropy were studied.* Viscoelasticity and thixotropy were studied.

46. Recommendations for Future WorkRecommendations for Future Work
* Including more physics in the model.* Including more physics in the model.
* Implementing 2-phase of 2 non-Newtonians.* Implementing 2-phase of 2 non-Newtonians.
* Extending yield stress analysis.* Extending yield stress analysis.
* Considering more complex void-space.* Considering more complex void-space.
* Fully investigating the Tardy algorithm.* Fully investigating the Tardy algorithm.
* Investigating the effect of converging-* Investigating the effect of converging-
diverging geometry on yield stress.diverging geometry on yield stress.
* Developing and implementing other transient* Developing and implementing other transient
and steady-state viscoelastic algorithms.and steady-state viscoelastic algorithms.
* Developing and implementing time-dependent* Developing and implementing time-dependent
thixotropic algorithms.thixotropic algorithms.

47. AcknowledgementsAcknowledgements
MartinMartin
Pore-Scale Modelling ConsortiumPore-Scale Modelling Consortium
SchlumbergerSchlumberger
Schlumberger Cambridge Research CentreSchlumberger Cambridge Research Centre
Valerie Anderson and John CrawshawValerie Anderson and John Crawshaw
Bill and GeoffBill and Geoff
Staff and Students in Imperial CollegeStaff and Students in Imperial College
Family and friendsFamily and friends

48. Thank YouThank You
Questions?Questions?