This document discusses modeling the flow of non-Newtonian fluids in porous media. It defines Newtonian and non-Newtonian fluids and describes different types of non-Newtonian behavior including time-independent, time-dependent, and viscoelastic. Network modeling techniques are presented for simulating flow using pore-scale images and representative rheological models. Strategies are discussed for modeling time-independent, time-dependent, and viscoelastic fluids using network modeling approaches. Future work is noted to implement time-dependent modeling strategies and investigate viscoelastic effects.

1. Modelling the Flow of non-NewtonianModelling the Flow of non-Newtonian
Fluids in Porous MediaFluids in Porous Media
Imperial College London & Schlumberger Research CentreImperial College London & Schlumberger Research Centre
Taha Sochi & Martin BluntTaha Sochi & Martin Blunt

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.
2. Time-dependent: strain rate is function of both2. Time-dependent: strain rate is function of both
magnitude and duration of stress.magnitude and duration of stress.
3. Viscoelastic: shows partial elastic recovery on3. Viscoelastic: shows partial elastic recovery on
removal of deforming stress.removal of deforming stress.

4. RheologyRheology
OfOf
Non-Newtonian FluidsNon-Newtonian Fluids

5. Time-IndependentTime-Independent

6. Time-DependentTime-Dependent

7. ViscoelasticViscoelastic

8. Thixotropic vs. ViscoelasticThixotropic vs. Viscoelastic
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.

9. Network ModellingNetwork Modelling
&&
Flow SimulationFlow Simulation

10. Network ModellingNetwork Modelling
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
from analytical expression for cylinder, by usingfrom analytical expression 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.

11. Start with initial guess for effective viscosity, andStart with initial guess for effective viscosity, and
hence solve the pressure field.hence solve the pressure field.
Flow SimulationFlow Simulation
Update effective viscosity using analyticalUpdate effective viscosity using analytical
expression with pseudo-Poiseuille definition.expression with pseudo-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 tolerance error in totalspecified tolerance error in total QQ between twobetween two
consecutive iteration cycles is reached.consecutive iteration cycles is reached.

12. Network ModellingNetwork Modelling
OfOf
Time-Independent FluidsTime-Independent Fluids

13. 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.
Examples: Herschel-Bulkley & Ellis models.Examples: Herschel-Bulkley & Ellis models.
Network Modelling StrategyNetwork Modelling Strategy

14. 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
Herschel-BulkleyHerschel-Bulkley
n
o
Cγττ +=

16. 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
EllisEllis
1
21
1
−






+
= α
/
o
τ
τ
γμ
τ

17. Park

18. Network ModellingNetwork Modelling
OfOf
Time-Dependent FluidsTime-Dependent Fluids

19. 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:
Network Modelling StrategyNetwork Modelling Strategy
a. Difficult to track fluid elements in pores anda. Difficult to track fluid elements in pores and
determine their deformation history.determine their deformation history.
b. Mixing of fluid elements with variousb. Mixing of fluid elements with various
deformation history in individual pores.deformation history in individual pores.
Very difficult to model because:Very difficult to model because:

20. 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:
Network Modelling StrategyNetwork Modelling Strategy
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.

21. 3. Flow of strongly shear-dependent fluid in very3. Flow of strongly shear-dependent fluid in very
homogeneous porous medium:homogeneous porous medium:
Network Modelling StrategyNetwork Modelling Strategy
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 & long CPU time.injection from reservoir & long CPU time.

22. GodfreyGodfrey
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’’’’

23. Stretched Exponential ModelStretched 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

24. Network ModellingNetwork Modelling
OfOf
Viscoelastic FluidsViscoelastic Fluids

25. There are mainly two effects to model:There are mainly two effects to model:
Network Modelling StrategyNetwork Modelling Strategy
1. Time dependency:1. Time dependency:
Apply the same strategy as in the case ofApply the same strategy as in the case of
time-dependent fluid after modelling thetime-dependent fluid after modelling the
transient state.transient state.

26. Network Modelling StrategyNetwork Modelling Strategy
2. Thickening at high flow rate:2. Thickening at high flow rate:
As the flow in porous media is mixed shear-As the flow in porous media is mixed shear-
extension flow due mainly to convergence-extension flow due mainly to convergence-
divergence, with the contribution of eachdivergence, with the contribution of each
component being unquantified and highlycomponent being unquantified and highly
dependent on pores actual shape, it is difficultdependent on pores actual shape, it is difficult
to predict the share of each especially whento predict the share of each especially when
the pore space description is approximate.the pore space description is approximate.
One possibility is to use average behaviour,One possibility is to use average behaviour,
depending on porous medium, to find thedepending on porous medium, to find the
contribution of each as a function of flow rate.contribution of each as a function of flow rate.

27. Upper Convected MaxwellUpper Convected Maxwell
This is the simplest and apparently theThis is the simplest and apparently the
second most popular modelsecond most popular model
ττ Stress tensorStress tensor
λλ11 Relaxation timeRelaxation time
µµοο Low-shear viscosityLow-shear viscosity
γγ Rate-of-strain tensorRate-of-strain tensor
γττ o
µλ −=+
∇
1

28. Oldroyd-BOldroyd-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
apparently the most popular modelapparently the most popular model

29. Discussion
Time-independent: good results for Ellis and
mixed results for Herschel-Bulkley. The main
reason is apparently yield stress.
Time-dependent: strategy developed for
modelling some cases of time-dependency.
Viscoelastic: strategy to be developed for
modelling time-dependency and thickening at
high flow rates in porous media.

30. Future WorkFuture Work
Implementation of time-dependentImplementation of time-dependent
strategystrategy
Possible implementation of viscoelasticPossible implementation of viscoelastic
effects.effects.

31. AcknowledgementsAcknowledgements
Pore Scale Modelling ConsortiumPore Scale Modelling Consortium
Schlumberger Research CentreSchlumberger Research Centre

32. Thank YouThank You
Questions?Questions?

33. AppendixesAppendixes

34. Analytical ExpressionsAnalytical Expressions
forfor
Volumetric Flow RateVolumetric Flow Rate
inin
Cylindrical DuctCylindrical Duct

35. Herschel-Bulkley
το C n Herschel parameters
L Tube length
∆P Pressure difference
τw ∆PR/2L Where R is the tube radius
( ) 





+
+
+
−
+
+
−
−





∆
=
+
nnnP
L
C
Q oowoow
own
n
/11/12
)(2
/13
)(8
223
/1
11 ττττττ
ττ
π

36. Newtonian: το = 0 n = 1
Power law: το = 0 n ≠ 1
LC
PR
Q
8
. 4
∆
=
π
Bingham: το ≠ 0 n = 1
n
n
P
LC
R
Q
L
R
n
n
n
1
11
213
4
/1
4
8
.
∆










=
−
+
π














+





−
∆
=
4
4
3
1
3
4
1
8
.
w
o
w
o
LC
PR
Q
τ
τ
τ
τπ

37. EllisEllis
µµοο ττ1/21/2 αα Ellis parametersEllis parameters
RR Tube radiusTube radius
LL Tube lengthTube length
∆∆PP Pressure dropPressure drop













 ∆
+
+
∆
=
−1
2/1
4
23
4
1
8
8
α
ταµ L
PR
L
PR
Q
o

38. ConvergenceConvergence
&&
TestsTests

39. Convergence
Usually converges quickly (<10 iterations).
Algebraic multi-grid solver is used.
Could fail to converge due to non-linearity.
Convergence failure is usually in the form of
oscillation between 2 values.
Sometimes, it is slow convergence rather than
failure, e.g. convergence observed after several
hundred iterations.

40. To help convergence:
1. Increase the number of iterations.
2. Initialise viscosity vector with single value.
3. Scan fine pressure-line.
4. Adjust the size of solver arrays.

41. Testing the Code
1. Newtonian & Bingham quantitatively verified.
3. All results are qualitatively reasonable:
2. Comparison with previous code gives
similar results.