This document discusses modelling non-Newtonian fluid flow through porous media. It defines Newtonian and non-Newtonian fluids, describes various rheological models including time-independent, viscoelastic, and time-dependent behaviors. It also discusses different modelling methodologies like continuum, bundle of capillaries, and network modelling approaches. Network modelling accounts for physics at the pore level and is computationally affordable. The document outlines strategies for modelling time-independent, viscoelastic, and time-dependent non-Newtonian fluid flow through pore networks.

1. Pore-Scale Modelling of Non-NewtonianPore-Scale Modelling of Non-Newtonian
Flow in Porous MediaFlow in Porous Media
Taha SochiTaha Sochi

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

3. 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 a function of both3. Time-dependent: strain rate is a 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.

4. RheologyRheology
ofof
Non-Newtonian FluidsNon-Newtonian Fluids

5. 1. Time-Independent1. Time-Independent

6. 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

7. 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 +=

8. 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

9. 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

10. 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

11. 3. Time-Dependent3. Time-Dependent

12. 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’’’’

13. 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

14. Modelling the FlowModelling the Flow
inin
Porous MediaPorous Media

15. MethodologiesMethodologies
1. Continuum (e.g. Darcy): closed form, easy to1. Continuum (e.g. Darcy): closed form, easy to
use, no computational cost, simplistic,use, no computational cost, simplistic,
inappropriate for transition/dynamic effects.inappropriate for transition/dynamic effects.
2. Bundle of Capillaries: easy to use, low2. Bundle of Capillaries: easy to use, low
computational cost, simplistic.computational cost, simplistic.
3. Numerical: difficult, closest to analytical,3. Numerical: difficult, closest to analytical,
computationally demanding, requires detailedcomputationally demanding, requires detailed
description of pore space.description of pore space.
4. Network Modelling: accounts for physics at4. Network Modelling: accounts for physics at
pore level, computationally affordable, goodpore level, computationally affordable, good
compromise.compromise.

16. Network Modelling: Capillary FlowNetwork Modelling: Capillary 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=

17. Network FlowNetwork Flow
A set of equations representing the capillariesA set of equations representing the capillaries
and satisfying mass conservation should beand satisfying mass conservation should be
solved simultaneously:solved simultaneously:
1.1. Newtonian fluidNewtonian fluid: solve once and for all since: solve once and for all since
conductance is known in advance.conductance is known in advance.
2.2. Viscous non-NewtonianViscous non-Newtonian: starting with initial: starting with initial
guess for viscosity, solve for the pressureguess for viscosity, solve for the pressure
iteratively, updating viscosity after each cycle.iteratively, updating viscosity after each cycle.
3.3. Fluid with memoryFluid with memory: starting with initial: starting with initial
guess for flow rate, iterate considering theguess for flow rate, iterate considering the
effect of local pressure and viscosity variation.effect of local pressure and viscosity variation.

18. Modelling Porous MediumModelling Porous Medium
Obtain 3-dimensional image of the pore space.Obtain 3-dimensional image of the pore space.
Build a topologically-equivalent network.Build a topologically-equivalent network.
Account for non-circularity, when calculatingAccount for non-circularity, when calculating QQ
analytically or numerically for cylinder, by usinganalytically or numerically for cylinder, by using
equivalent radius:equivalent radius:
4/1
8






=
π
G
Req
where the conductance,where the conductance, GG, is found empirically, is found empirically
from numerical simulation.from numerical simulation.

19. NetworksNetworks
Voxel image extracted fromVoxel image extracted from
real sample or geologically-real sample or geologically-
simulated sample.simulated sample.
NumericalNumerical
networknetwork
models.models.

20. Start with known value or initial guess for flowStart with known value or initial guess for flow
parameters, solve the pressure field.parameters, solve the pressure field.
Simulating FlowSimulating Flow
Update the effective viscosity using pseudo-Update the effective viscosity using pseudo-
Poiseuille definition.Poiseuille definition.
Obtain total flow rate & apparent viscosity.Obtain total flow rate & apparent viscosity.
Iterate until convergence is achieved whenIterate until convergence is achieved when
specified error tolerance in totalspecified error tolerance in total QQ between twobetween two
consecutive iteration cycles is reached.consecutive iteration cycles is reached.

21. Network Modelling StrategiesNetwork Modelling Strategies

22. 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.
1. Time-Independence1. Time-Independence

23. 2. Viscoelasticity (Steady-State)2. Viscoelasticity (Steady-State)
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.

24. 3. Starting with initial guess for the flow3. Starting with initial guess for the flow
rate and using numerical technique, therate and using numerical technique, the
pressure drop as a function of the flowpressure drop as a function of the flow
rate is found for each capillary.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.

25. There are three major cases:There are three major cases:
1. Flow of strongly shear-dependent fluid in1. Flow of strongly shear-dependent fluid in
medium which is not very homogeneous:medium which is not very homogeneous:
3. Time-Dependence3. Time-Dependence
a. Difficult to track fluid elements in pores anda. Difficult to track fluid elements in pores and
determine their shear history.determine their shear history.
b. Mixing of fluid elements with various shearb. Mixing of fluid elements with various shear
history in individual pores.history in individual pores.
Very difficult to model because:Very difficult to model because:

26. 2. Flow of shear-independent or weakly shear-2. Flow of shear-independent or weakly shear-
dependent fluid in porous medium:dependent fluid in porous medium:
Apply single time-dependent viscosity functionApply single time-dependent viscosity function
to all pores at each instant of time and henceto all pores at each instant of time and hence
simulate time development.simulate time development.

27. 3. Flow of strongly shear-dependent fluid in very3. Flow of strongly shear-dependent fluid in very
homogeneous porous medium:homogeneous porous medium:
a. Define effective pore shear rate.a. Define effective pore shear rate.
b. Use very small time step to find viscosity inb. Use very small time step to find viscosity in
the next instant assuming constant shear.the next instant assuming constant shear.
c. Find change in shear and hence makec. Find change in shear and hence make
correction to viscosity.correction to viscosity.
Possible problems: edge effects in case ofPossible problems: edge effects in case of
injection from reservoir.injection from reservoir.

28. Modelling Yield StressModelling Yield Stress

29. The substance before yield is assumed to beThe substance before yield is assumed to be
fluid with very high viscosity.fluid with very high viscosity.
No element yields unless it is part of a spanningNo element yields unless it is part of a spanning
path bridging the inlet to the outlet.path bridging the inlet to the outlet.

30. 1. Yield stress value is usually obtained by1. Yield stress value is usually obtained by
extrapolation.extrapolation.
2. Before yield, the pressure may not be2. Before yield, the pressure may not be
well-defined.well-defined.
3. Yield is highly dependent on the actual3. Yield is highly dependent on the actual
shape of the pore space.shape of the pore space.
4. Yield may depend on the porous medium.4. Yield may depend on the porous medium.
DifficultiesDifficulties

31. 1. The conventional percolation applies only1. The conventional percolation applies only
to homogeneous elements.to homogeneous elements.
2. The network elements cannot yield2. The network elements cannot yield
independently.independently.
3. The pure percolation approach ignores3. The pure percolation approach ignores
the dynamic aspects of the pressure field.the dynamic aspects of the pressure field.
Is it Percolation?Is it Percolation?

32. 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 pressure byFind the path of minimum yield pressure by
increasing the yield pressure continuously.increasing the yield pressure continuously.
Assumption: yield pressure of a number ofAssumption: 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.
2. Path of Minimum Pressure (PMP):2. Path of Minimum Pressure (PMP):
Find the path of minimum yield pressure byFind the path of minimum yield pressure by
finding the minimum yield pressure neededfinding the minimum yield pressure needed
to reach each node.to reach each node.

35. Numerical ResultsNumerical Results
Both IPM and PMP give lower values thanBoth IPM and PMP give lower values than
the network model:the network model:
Boundaries Threshold Yield Pressure (Pa)
Lower Upper Network 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

36. 1. The assumption that1. The assumption that The yield pressure ofThe yield pressure of
an ensemble of serially-connected bonds is thean ensemble of serially-connected bonds is the
sum of their yield pressures is not evident.sum of their yield pressures is not evident.
Failure of IPM & PMP (MTP)Failure of IPM & PMP (MTP)
2. The effect of tortuosity and dynamic2. The effect of tortuosity and dynamic
effects of the global pressure field areeffects of the global pressure field are
ignored.ignored.

37. Tests, Validations & ComparisonsTests, Validations & Comparisons

38. Code Validation
1. Quantitative: Newtonian, Bingham, Boger,1. Quantitative: Newtonian, Bingham, Boger,
and low-shear regimes.and low-shear regimes.
2. Qualitative: all trends of behaviour are2. Qualitative: all trends of behaviour are
reasonable.reasonable.

39. Sample Experimental ResultsSample Experimental Results
Sadowski
Ellis
Park
Ellis
Al-Fariss
Herschel-Bulkley
Chase
Bingham

42. General ConclusionsGeneral Conclusions

43. * Ellis & Herschel-Bulkley fluids were modelled* Ellis & Herschel-Bulkley fluids were modelled
and implemented.and implemented.
* 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 developed and employed.were developed and employed.
* Steady-state viscoelastic algorithm was* Steady-state viscoelastic algorithm was
developed and implemented.developed and implemented.
* Viscoelasticity and thixotropy were* Viscoelasticity and thixotropy were
thoroughly examined.thoroughly examined.
* The model was validated in several cases.* The model was validated in several cases.

44. Relevant PublicationsRelevant Publications
1. Taha Sochi, Martin J. Blunt. Pore-scale network modeling of1. Taha Sochi, Martin J. Blunt. Pore-scale network modeling of
Ellis and Herschel–Bulkley fluids. Journal of PetroleumEllis and Herschel–Bulkley fluids. Journal of Petroleum
Science and Engineering.Science and Engineering.
2. Taha Sochi. Pore-scale modeling of viscoelastic flow in2. Taha Sochi. Pore-scale modeling of viscoelastic flow in
porous media using a Bautista–Manero fluid.porous media using a Bautista–Manero fluid.
International Journal of Heat and Fluid Flow.International Journal of Heat and Fluid Flow.
Papers In PressPapers In Press
1. Taha Sochi. Single-Phase Flow of Non-Newtonian Fluids in1. Taha Sochi. Single-Phase Flow of Non-Newtonian Fluids in
Porous Media.Porous Media.
2. Taha Sochi.2. Taha Sochi. Computational Techniques for Modeling Non-Computational Techniques for Modeling Non-
Newtonian Flow in Porous Media.Newtonian Flow in Porous Media.
3. Taha Sochi.3. Taha Sochi. Modeling the Flow of Yield-Stress Fluids inModeling the Flow of Yield-Stress Fluids in
Porous Media.Porous Media.

45. Thank YouThank You
Questions?Questions?