Pore-Scale Modelling of Non-Newtonian Pore-Scale Modelling of Non-Newtonian Flow in Porous MediaFlow in Porous Media

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

Supervisor: Prof. Martin BluntSupervisor: Prof. Martin Blunt

Taha SochiTaha Sochi

Definition of Newtonian & non-Newtonian fluidsDefinition of Newtonian & non-Newtonian fluidsOutlineOutline

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 behaviour Modelling yield stress behaviour

Modelling viscoelastic flowModelling viscoelastic flow

Results, recommendations & acknowledgementsResults, recommendations & acknowledgements

Questions and discussionsQuestions and discussions

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.

3. Time-dependent: strain rate is function of both 3. 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.

RheologyRheologyofof

Non-Newtonian FluidsNon-Newtonian Fluids

1. Time-Independent1. Time-Independent

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

StressStressZero-shear viscosityZero-shear viscosityStrain rateStrain rateStress at Stress at Indicial parameter Indicial parameter

A. EllisA. Ellis

1

21

1

α

/

o

ττ

γμτ

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

B. Herschel-BulkleyB. Herschel-Bulkley

no C

2. Viscoelastic2. Viscoelastic

Convergence-Convergence-divergence with divergence with

time of fluid time of fluid being being

comparable with comparable with time of flowtime of flow

Delayed Delayed response & response & relaxationrelaxation

Dominance of Dominance of extension over extension over shear at high shear at high

flow rateflow rate

Time-dependency

Strain

hardening

Intermediate plateau

A. Upper Convected MaxwellA. Upper Convected Maxwell

This is the simplest and most popular This is the simplest and most popular modelmodel

Stress tensorStress tensorRelaxation timeRelaxation timeLow-shear viscosityLow-shear viscosityRate-of-strain tensorRate-of-strain tensor

γττ o

1

B. Oldroyd-BB. Oldroyd-B

Stress tensorStress tensorRelaxation timeRelaxation timeRetardation timeRetardation timeLow-shear viscosityLow-shear viscosityRate-of-strain tensorRate-of-strain tensor

γγττ 21 o

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

3. Time-Dependent3. Time-Dependent

A. GodfreyA. GodfreyThis is suggested as a thixotropic modelThis is suggested as a thixotropic model

)1(

)1()(''

'

/''

/'

t

ti

e

et

ViscosityViscositytt Time of shearing Time of shearingiiInitial-time viscosityInitial-time viscosity’’’’ ’’ Viscosity deficits Viscosity deficits associated associated with time constants with time constants ’’ ’’’’

B. Stretched Exponential ModelB. Stretched Exponential Model

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

)1)(()( / st

iini et

ViscosityViscositytt Time of shearing Time of shearingiiInitial-time viscosityInitial-time viscosityininInfinite-time viscosityInfinite-time viscosityssTime constantTime constant

Viscoelastic vs. ThixotropicViscoelastic vs. ThixotropicTime-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.

Modelling the Flow Modelling the Flow inin

Porous MediaPorous Media

Capillary FlowCapillary Flow

For a capillary:For a capillary: Pcq .

Flow rate = conductance Flow rate = conductance × Pressure × Pressure dropdrop

1. 1. Newtonian fluidNewtonian fluid:: constant)( cc

2. 2. Viscous non-NewtonianViscous non-Newtonian:: ),( Pcc

3. 3. Fluid with MemoryFluid with Memory:: ),,( tPcc

Network FlowNetwork Flow

For a network of capillaries, a set of For a network of capillaries, a set of equations representing the capillaries and equations representing the capillaries and satisfying mass conservation should be satisfying mass conservation should be solved simultaneously to produce a solved 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.

Network FlowNetwork Flow

3. 3. Fluid with memoryFluid with memory: for the steady-: for the steady-state viscoelastic flow, start with an state 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 local considering the effect of the local pressure and viscosity variation due to pressure and viscosity variation due to converging-diverging geometry, until converging-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 pressure an initial guess, solve for the pressure iteratively, updating the viscosity after iteratively, updating the viscosity after each cycle, until reaching convergence.each cycle, until reaching convergence.

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.

Network Modelling StrategyNetwork Modelling Strategy

The main networks used in this study are the The main networks used in this study are the sand pack and Berea of sand pack and Berea of Øren and co-workers.Øren and co-workers.

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 uniform fluid 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 complex in this assessment instead of more complex and realistic model such as non-uniform bundle and realistic model such as non-uniform bundle or cubic network because of simplicity which is or cubic network because of simplicity which is a big advantage to see the hidden features. a big advantage to see the hidden features.

* Good results are obtained for both sand pack * Good results are obtained for both sand pack and Berea. and Berea.

Results: Network-Bundle of TubesResults: Network-Bundle of TubesSand pack= 0.0Pa

Sand pack= 1.0Pa

Berea= 0.0Pa

Berea= 1.0Pa

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 and Bulkley 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 highly behaviour for yield-stress fluid is highly dependent on the distribution and realisation dependent on the distribution and realisation because of the random nature of the network because of the random nature of the network generation. Therefore no firm conclusion can be generation. Therefore no firm conclusion can be reachedreached

Results: Random-Regular NetworksResults: Random-Regular NetworksCubic-Sand pack= 0.0Pa

Cubic-Sand pack= 1.0Pa

Cubic-Berea= 0.0Pa

Cubic-Berea= 1.0Pa

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-Bulkley The main sources of failure for Herschel-Bulkley are experimental errors and imperfect are experimental errors and imperfect modellingmodelling of yield stress phenomenon. of yield stress phenomenon.

Results: ExperimentalResults: ExperimentalSadowski

Ellis

ParkEllis

Al-FarissHerschel-Bulkley

ChaseBingham

Network Modelling Network Modelling ofof

Yield Stress FluidsYield Stress Fluids

What is Yield Stress ?What is Yield Stress ?The stress at which the substance starts The stress at which the substance starts flowing.flowing.

The substance is assumed to be solid The substance is assumed to be solid below its yield stress and fluid above.below its yield stress and fluid above.

DifficultiesDifficulties1. The yield stress value is usually obtained 1. 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 the 3. The yield is highly dependent on the actual shape of the pore space and its fine actual shape of the pore space and its fine details. This is compromised by modelling details. 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 flow the 4. While in the case of bulk and tube flow the yield stress is a property of the fluid, in the yield stress is a property of the fluid, in the case of porous media it may depend on the case of porous media it may depend on the porous media as well.porous media as well.

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 Find the path of minimum yield pressure connecting the inlet to the outlet by connecting the inlet to the outlet by increasing the yield pressure continuously. increasing the yield pressure continuously. The threshold yield pressure is the value at The 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 of A. The yield pressure of a number of serially-connected bonds is the sum of their serially-connected bonds is the sum of their yield pressures.yield pressures.B. Backtracking is allowed.B. Backtracking is allowed.

Predicting Network Threshold Yield PressurePredicting Network Threshold Yield Pressure2. Path of Minimum Pressure (PMP):2. Path of Minimum Pressure (PMP):Find the path of minimum yield pressure Find the path of minimum yield pressure connecting the inlet to the outlet by finding connecting the inlet to the outlet by finding the minimum yield pressure needed to the minimum yield pressure needed to reach each node. The threshold yield reach each node. The threshold yield pressure is the minimum of the values pressure 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 of A. The yield pressure of a number of serially-connected bonds is the sum of their serially-connected bonds is the sum of their yield pressures.yield pressures.B. Backtracking is B. Backtracking is notnot allowed. allowed.

ResultsResults

Random networks: IPM and PMP agree in Random networks: IPM and PMP agree in most cases. When they disagree, PMP gives most cases. When they disagree, PMP gives higher values. The reason is backtracking is higher 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 time PMP is more efficient in terms of CPU time and memory.and memory.

Cubic networks: IPM and PMP agree in all Cubic networks: IPM and PMP agree in all cases investigated. This is due to less cases investigated. This is due to less likelihood of backtracking as it involves likelihood of backtracking as it involves more tortuous path.more tortuous path.

ResultsResultsBoth IPM and PMP give lower values than Both 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

ResultsResultsCubic 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

Network Modelling Network Modelling ofof

Viscoelastic FluidsViscoelastic Fluids

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

Tardy AlgorithmTardy Algorithm1. Since converging-diverging geometry 1. Since converging-diverging geometry is important for viscoelastic flow, the is important for viscoelastic flow, the capillaries should be modelled with capillaries should be modelled with contraction.contraction.

2. Each capillary is 2. Each capillary is discretized in the flow discretized in the flow direction and a discretized form of the direction and a discretized form of the flow equations is used assuming a prior flow equations is used assuming a prior knowledge of stress & viscosity at inlet.knowledge of stress & viscosity at inlet.

Tardy AlgorithmTardy Algorithm

3. Starting with an initial guess for the 3. 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 the the 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 whole 4. The pressure field for the whole network is then found iteratively until network is then found iteratively until convergence is achieved.convergence is achieved.

Some ResultsSome ResultsSand 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

Some ResultsSome ResultsSand 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

Results, Recommendations Results, Recommendations &&

AcknowledgementsAcknowledgements

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.

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.

AcknowledgementsAcknowledgementsMartinMartinPore-Scale Modelling ConsortiumPore-Scale Modelling ConsortiumSchlumbergerSchlumbergerSchlumberger Cambridge Research CentreSchlumberger Cambridge Research CentreValerie Anderson and John CrawshawValerie Anderson and John CrawshawBill and GeoffBill and GeoffStaff and Students in Imperial CollegeStaff and Students in Imperial CollegeFamily and friendsFamily and friends

Thank YouThank You

Questions?Questions?