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Modelling the Flow of non-Newtonian Modelling 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

DefinitionDefinitionof of

Newtonian & Non-Newtonian FluidsNewtonian & Non-Newtonian Fluids

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 both 2. 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.

RheologyRheologyOfOf

Non-Newtonian FluidsNon-Newtonian Fluids

Time-IndependentTime-Independent

Time-DependentTime-Dependent

ViscoelasticViscoelastic

Thixotropic vs. ViscoelasticThixotropic vs. Viscoelastic

Time-dependency of viscoelastic arises Time-dependency of viscoelastic arises because response is not instantaneous.because response is not instantaneous.

Time-dependent behaviour of thixotropic Time-dependent behaviour of thixotropic arises because of change in structure.arises because of change in structure.

Network ModellingNetwork Modelling&&

Flow SimulationFlow Simulation

Network ModellingNetwork ModellingObtain 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 calculating Account for non-circularity, when calculating QQ from analytical expression for cylinder, by using from analytical expression for cylinder, by using equivalent radius: equivalent radius:

4/1

8

=

πG

Req

where the conductance, where the conductance, GG, is found empirically , is found empirically from numerical simulation.from numerical simulation.

Start with initial guess for effective viscosity, and Start with initial guess for effective viscosity, and hence solve the pressure field. hence solve the pressure field.

Flow SimulationFlow Simulation

Update effective viscosity using analytical Update 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 when Iterate until convergence is achieved when specified tolerance error in total specified tolerance error in total QQ between two between two consecutive iteration cycles is reached.consecutive iteration cycles is reached.

Network Modelling Network Modelling OfOf

Time-Independent FluidsTime-Independent Fluids

Combine the pore space description of the Combine 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 analytical The bulk rheology is used to derive analytical expression for the flow in simplified pore expression for the flow in simplified pore geometry. geometry.

Examples: Herschel-Bulkley & Ellis models. Examples: Herschel-Bulkley & Ellis models.

Network Modelling StrategyNetwork Modelling Strategy

This is a general time-independent modelThis is a general time-independent model

ττ StressStressττοο Yield stressYield stress

CC Consistency factorConsistency factor γγ Strain rateStrain ratenn Flow behaviour index Flow behaviour index

Herschel-BulkleyHerschel-Bulkley

n

oCγττ +=

This is a shear-thinning modelThis is a shear-thinning model

ττ StressStressµµοο Zero-shear viscosityZero-shear viscosity

γγ Strain rateStrain rateττ1/21/2 Stress at Stress at µµο ο / 2/ 2αα Indicial parameter Indicial parameter

EllisEllis

1

21

1−

+

= α

/

o

ττ

γμτ


Time-Dependent FluidsTime-Dependent Fluids

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:


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 various b. 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:

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 function Apply single time-dependent viscosity function to all pores at each instant of time and hence to all pores at each instant of time and hence simulate time development.simulate time development.

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 make c. Find change in shear and hence make

correction to viscosity.correction to viscosity.

Possible problems: edge effects in case of Possible problems: edge effects in case of injection from reservoir & long CPU time.injection from reservoir & long CPU time.

GodfreyGodfrey

This is suggested as a thixotropic modelThis is suggested as a thixotropic model

)1(

)1()(''

'

/''

/'

λ

λ

µµµµ

t

t

i

e

et−

−

−∆−

−∆−=

µµ ViscosityViscositytt Time of shearing Time of shearing

µµii Initial-time viscosityInitial-time viscosity

∆∆µµ’’ & & ∆∆µµ’’ ’’ Viscosity deficits associated Viscosity deficits associated with time constants with time constants λλ’’ && λλ’’’’

Stretched Exponential ModelStretched Exponential Model

This is a general time-dependent model This is a general time-dependent model

)1)(()( / st

iiniet λµµµµ −−−+=

µµ ViscosityViscositytt Time of shearing Time of shearing

µµii Initial-time viscosityInitial-time viscosity

µµinin Infinite-time viscosityInfinite-time viscosity

λλss Time constantTime constant


Viscoelastic FluidsViscoelastic Fluids

There are mainly two effects to model:There are mainly two effects to model:


1. Time dependency:1. Time dependency:

Apply the same strategy as in the case of Apply the same strategy as in the case of time-dependent fluid after modelling the time-dependent fluid after modelling the transient state.transient state.

Network Modelling StrategyNetwork Modelling Strategy2. 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 each divergence, with the contribution of each component being unquantified and highly component being unquantified and highly dependent on pores actual shape, it is difficult dependent on pores actual shape, it is difficult to predict the share of each especially when to 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 the depending on porous medium, to find the contribution of each as a function of flow rate. contribution of each as a function of flow rate.

Upper Convected MaxwellUpper Convected Maxwell

This is the simplest and apparently the This is the simplest and apparently the second most popular modelsecond most popular model

τ τ Stress tensorStress tensor

λλ11 Relaxation timeRelaxation time

µµοο Low-shear viscosityLow-shear viscosity

γ γ Rate-of-strain tensorRate-of-strain tensor

γττo

µλ −=+∇

1

Oldroyd-BOldroyd-B

τ τ Stress tensorStress tensor

λλ11 Relaxation timeRelaxation time

λλ22 Retardation timeRetardation time

µµοο Low-shear viscosityLow-shear viscosity

γ γ Rate-of-strain tensorRate-of-strain tensor

+−=+∇∇

γγττ21

λµλo

This is the second in simplicity and This is the second in simplicity and apparently the most popular modelapparently the most popular model

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.

Future WorkFuture Work

Implementation of time-dependent Implementation of time-dependent strategystrategy

Possible implementation of viscoelastic Possible implementation of viscoelastic effects.effects.

AcknowledgementsAcknowledgements

Pore Scale Modelling Consortium Pore Scale Modelling Consortium

Schlumberger Research Centre Schlumberger Research Centre

Thank YouThank You

Questions?Questions?

Appendixes Appendixes

Analytical Expressions Analytical Expressions for for

Volumetric Flow RateVolumetric Flow Rateinin

Cylindrical DuctCylindrical Duct

Herschel-Bulkley

το C n Herschel parameters

L Tube length

∆P Pressure difference

τw ∆PR/2L Where R is the tube radius

( )

+

++

−++−−

∆= +

nnnPL

CQ oowoow

own

n

/11/12)(2

/13)(8

223

/1

11 ττττττττπ

Newtonian: το = 0 n = 1

Power law: το = 0 n ≠ 1

LC

PRQ

8

. 4∆= π

Bingham: το ≠ 0 n = 1

nn

PLC

RQ

L

R

n

nn

111

213

4/1

4

8. ∆

=

−

+π

+

−∆=

44

3

1

3

41

8

.

w

o

w

o

LC

PRQ

ττ

ττπ

EllisEllis

µµοο τ τ1/21/2 αα Ellis parametersEllis parameters

R R Tube radiusTube radius

L L Tube lengthTube length

∆∆P P Pressure dropPressure drop

∆+

+∆=−1

2/1

4

23

41

8

8α

ταµ L

PR

L

PRQ

o

Convergence Convergence &&

Tests Tests

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.

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.

Testing the Code

1. Newtonian & Bingham quantitatively verified.

3. All results are qualitatively reasonable:

2. Comparison with previous code gives

similar results.

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