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

Taha SochiTaha Sochi

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 a function of both 3. 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.

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

Modelling the Flow Modelling the Flow inin

Porous MediaPorous Media

MethodologiesMethodologies1. Continuum (e.g. Darcy): closed form, easy to 1. 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, low 2. 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 detailed computationally demanding, requires detailed description of pore space.description of pore space.4. Network Modelling: accounts for physics at 4. Network Modelling: accounts for physics at pore level, computationally affordable, good pore level, computationally affordable, good compromise.compromise.

Network Modelling: Capillary FlowNetwork Modelling: Capillary 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 FlowA set of equations representing the capillaries A set of equations representing the capillaries and satisfying mass conservation should be and 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 pressure guess 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 guess : starting with initial guess for flow rate, iterate considering the effect of for flow rate, iterate considering the effect of local pressure and viscosity variation.local pressure and viscosity variation.

Modelling Porous MediumModelling Porous MediumObtain 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 analytically or numerically for cylinder, by using analytically or numerically for cylinder, by using equivalent radius: equivalent radius:

4/18

GR

eq

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

NetworksNetworks

Voxel image extracted from Voxel image extracted from real sample or geologically-real sample or geologically-simulated sample.simulated sample.

Numerical Numerical network network models.models.

Start with known value or initial guess for flow Start 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 when Iterate until convergence is achieved when specified error tolerance in total specified error tolerance in total QQ between two between two consecutive iteration cycles is reached.consecutive iteration cycles is reached.

Network Modelling StrategiesNetwork Modelling Strategies

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.

1. Time-Independence1. Time-Independence

2. Viscoelasticity (Steady-State)2. Viscoelasticity (Steady-State)1. 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.

3. Starting with initial guess for the flow 3. Starting with initial guess for the flow rate and using numerical technique, the rate and using numerical technique, the pressure drop as a function of the flow pressure 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 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.

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 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:

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.injection from reservoir.

Modelling Yield StressModelling Yield Stress

The substance before yield is assumed to be The 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 spanning No element yields unless it is part of a spanning path bridging the inlet to the outlet.path bridging the inlet to the outlet.

1. Yield stress value is usually obtained by 1. Yield stress value is usually obtained by extrapolation.extrapolation.

2. Before yield, the pressure may not be 2. Before yield, the pressure may not be well-defined. well-defined.

3. Yield is highly dependent on the actual 3. 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

1. The conventional percolation applies only 1. The conventional percolation applies only to homogeneous elements.to homogeneous elements.

2. The network elements cannot yield 2. The network elements cannot yield independently.independently.

3. The pure percolation approach ignores 3. 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?

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 by Find the path of minimum yield pressure by increasing the yield pressure continuously. increasing the yield pressure continuously.

Assumption: yield pressure of a number of Assumption: 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.

2. Path of Minimum Pressure (PMP):2. Path of Minimum Pressure (PMP):Find the path of minimum yield pressure by Find the path of minimum yield pressure by finding the minimum yield pressure needed finding the minimum yield pressure needed to reach each node. to reach each node.

Numerical ResultsNumerical ResultsBoth IPM and PMP give lower values than Both 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

1. The assumption that 1. The assumption that The yield pressure of The yield pressure of an ensemble of serially-connected bonds is the an 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 dynamic 2. The effect of tortuosity and dynamic effects of the global pressure field are effects of the global pressure field are ignored.ignored.

Tests, Validations & ComparisonsTests, Validations & Comparisons

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 are 2. Qualitative: all trends of behaviour are reasonable.reasonable.

Sample Experimental ResultsSample Experimental ResultsSadowskiEllis

ParkEllis

Al-FarissHerschel-Bulkley

ChaseBingham

General ConclusionsGeneral Conclusions

* 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 thoroughly * Viscoelasticity and thixotropy were thoroughly examined.examined.

* The model was validated in several cases.* The model was validated in several cases.

Relevant PublicationsRelevant Publications1. Taha Sochi, Martin J. Blunt. Pore-scale network modeling of 1. Taha Sochi, Martin J. Blunt. Pore-scale network modeling of

Ellis and Herschel–Bulkley fluids. Journal of Petroleum Ellis and Herschel–Bulkley fluids. Journal of Petroleum Science and Engineering.Science and Engineering.

2. Taha Sochi. Pore-scale modeling of viscoelastic flow in 2. 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 Press1. Taha Sochi. Single-Phase Flow of Non-Newtonian Fluids in 1. 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 in Modeling the Flow of Yield-Stress Fluids in

Porous Media.Porous Media.

Thank YouThank You

Questions?Questions?