Modelling the Flow of non-Newtonian Modelling the Flow of non-Newtonian

Fluids in Porous MediaFluids in Porous Media

Pore Scale Modelling Consortium Pore Scale Modelling Consortium Imperial College LondonImperial College London

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:

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 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

StressStressYield stressYield stressCC Consistency factorConsistency factorStrain rateStrain ratenn Flow behaviour index Flow behaviour index

Herschel-BulkleyHerschel-Bulkley

n

oC

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

StressStressZero-shear viscosityZero-shear viscosityStrain rateStrain rate

Stress at Stress at Indicial parameter Indicial parameter

EllisEllis

1

21

1

α

/

o

ττ

γμτ

Park

Network Modelling Network Modelling OfOf

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:

Network Modelling StrategyNetwork Modelling Strategy

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 shear b. 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:

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 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:

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 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()(''

'

/''

/'

t

t

i

e

et

ViscosityViscosity

tt Time of shearing Time of shearing

iiInitial-time viscosityInitial-time viscosity

’’’’ ’’ Viscosity deficits Viscosity deficits

associated associated

with time constants with time constants ’’ ’’’’

Stretched Exponential ModelStretched Exponential Model

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

)1)(()( / st

iiniet

ViscosityViscosity

tt Time of shearing Time of shearing

iiInitial-time viscosityInitial-time viscosity

ininInfinite-time viscosityInfinite-time viscosity

ssTime constantTime constant

Network Modelling Network Modelling OfOf

Viscoelastic FluidsViscoelastic Fluids

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 of Apply the same strategy as in the case of time-dependent fluid.time-dependent fluid.

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 most popular This is the simplest and most popular modelmodel

Stress tensorStress tensor

Relaxation timeRelaxation time

Low-shear viscosityLow-shear viscosity

Rate-of-strain tensorRate-of-strain tensor

γττo

1

Oldroyd-BOldroyd-B

Stress tensorStress tensor

Relaxation timeRelaxation time

Retardation timeRetardation time

Low-shear viscosityLow-shear viscosity

Rate-of-strain tensorRate-of-strain tensor

γγττ21

o

This is the second in simplicity and This is the second in simplicity and popularitypopularity

Future WorkFuture Work

Implementation of time-dependent Implementation of time-dependent strategystrategy

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

Thank YouThank You

Questions?Questions?