PERM Group Imperial College LondonPERM Group Imperial College London

Viscoelastic Flow in Porous Viscoelastic Flow in Porous MediaMedia

Taha Sochi & Martin BluntTaha Sochi & Martin BluntRheologyRheology1. Linear Viscoelasticity:1. Linear Viscoelasticity:

Stress tensorStress tensorRelaxation timeRelaxation time

t TimeTimeLow-shear viscosityLow-shear viscosity

Rate-of-strain tensorRate-of-strain tensor

Berea networkBerea network Sand pack networkSand pack network

Modelling Modelling Flow in Flow in

Porous MediaPorous Media

ReferencesReferences• R. Bird, R. Armstrong & O. Hassager: DynamicsR. Bird, R. Armstrong & O. Hassager: Dynamics

of Polymeric Liquids, Vol. 1, 1987.of Polymeric Liquids, Vol. 1, 1987.

• P. Carreau, D. De Kee & R. Chhabra: Rheology ofP. Carreau, D. De Kee & R. Chhabra: Rheology of

Polymeric Systems, 1997.Polymeric Systems, 1997.

• W. Gogarty, G. Levy & V. Fox: W. Gogarty, G. Levy & V. Fox: ViscoelasticViscoelastic

Effects in Polymer Flow Through Porous MediaEffects in Polymer Flow Through Porous Media,,

SPE 4025, 1972.SPE 4025, 1972.

• V. Anderson, J. Pearson & J. Sherwood: V. Anderson, J. Pearson & J. Sherwood:

Oscillation Superimposed on Steady Shearing, Oscillation Superimposed on Steady Shearing,

Journal of Rheology Vol. 50, 2006.Journal of Rheology Vol. 50, 2006.

• P. Tardy & V. Anderson: Current Modelling of P. Tardy & V. Anderson: Current Modelling of

Flow Through Porous Media.Flow Through Porous Media.

Description of the behaviour Description of the behaviour under small deformation.under small deformation.

ExamplesExamplesA. Maxwell Model:A. Maxwell Model:

γττ ot

1

B. Jeffreys Model:B. Jeffreys Model:

tt o

γγττ 21

Retardation timeRetardation time

2. Non-Linear Viscoelasticity:2. Non-Linear Viscoelasticity:

Description of the behaviour Description of the behaviour under large deformation.under large deformation.

ExamplesExamplesA.A.Upper Convected Upper Convected Maxwell Model:Maxwell Model:

@ @ Characterises VE materials.Characterises VE materials.

@ @ Serves as a starting point for Serves as a starting point for

non-linear models.non-linear models.

γττ o

1

Upper convected timeUpper convected timeDerivative of the stress tensorDerivative of the stress tensor

τ

vvv

τττττt

v Fluid velocityFluid velocityvVelocity gradient tensorVelocity gradient tensor

B. Oldroyd B Model:B. Oldroyd B Model:

γγττ 21 o

γ

vvv

γγγγγt

Upper convected timeUpper convected timeDerivative of the rate-of-strain Derivative of the rate-of-strain tensortensor

1. Continuum Approach:1. Continuum Approach:

This is based on extending the This is based on extending the modified Darcy’s Law for the modified Darcy’s Law for the flow of non-Newtonian viscous flow of non-Newtonian viscous fluids in porous media to fluids in porous media to include elastic effects.include elastic effects.

2. Pore-Scale Approach:2. Pore-Scale Approach:

UpsUps & & DownsDowns

@ Easy to implement.@ Easy to implement.

@ No computational cost.@ No computational cost.

@ No account of detailed physics@ No account of detailed physics

at pore level.at pore level.

This is based on solving the This is based on solving the governing equations of the governing equations of the viscoelastic flow over the void viscoelastic flow over the void space. The prominent example space. The prominent example of this approach is network of this approach is network modelling:modelling:

ExampleExampleGogarty Gogarty et alet al 1972: 1972:

mapp qKq

P 5.1243.01||

PPressure gradientPressure gradientqDarcy velocityDarcy velocityappApparent viscosity Apparent viscosity

KPermeabilityPermeabilitymElastic correction Elastic correction factorfactor

UpsUps & & DownsDowns

@ Modest computational cost.@ Modest computational cost.

@ No serious convergence issues.@ No serious convergence issues.

@ Requires pore-space description.@ Requires pore-space description.

@ Approximations required.@ Approximations required.

(After Xavier Lopez)(After Xavier Lopez)

ViscoelasticitViscoelasticityyDual nature of substance Dual nature of substance

behaviour by showing signs behaviour by showing signs of both viscous fluids and of both viscous fluids and elastic solids.elastic solids.

Features of Features of Viscoelastc Viscoelastc BehaviourBehaviour

1. Time Dependency:1. Time Dependency:

2. Strain Hardening:2. Strain Hardening:

3. Intermediate Plateau:3. Intermediate Plateau:

Due to delayed response and Due to delayed response and relaxation.relaxation.

Due to dominance of extension Due to dominance of extension over shear at high flow rate.over shear at high flow rate.

Due to convergence-divergence Due to convergence-divergence geometry with time of fluid geometry with time of fluid being comparable to time of being comparable to time of flow.flow.

1. Newtonian Fluid:1. Newtonian Fluid:

2. Viscous non-Newtonian:2. Viscous non-Newtonian:

3. Viscoelastic Fluid:3. Viscoelastic Fluid:

constant)( cc

),( Pcc

),,( tPcc

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

Network ModellingNetwork Modelling

Pcq .

Flow rate = conductance × Pressure drop

For a capillary:For a capillary:

1. Newtonian Fluid:1. Newtonian Fluid:

2. Viscous non-Newtonian:2. Viscous non-Newtonian:

3. Viscoelastic Fluid:3. Viscoelastic Fluid:

Solve once and for all.Solve once and for all.

Starting with an initial guess, solve Starting with an initial guess, solve for the pressure iteratively, updating for the pressure iteratively, updating the viscosity after each cycle, until the viscosity after each cycle, until convergence is achieved.convergence is achieved.

For the steady-state flow, start with For the steady-state flow, start with an initial guess for the flow rate and an initial guess for the flow rate and iterate, considering the effect of the iterate, considering the effect of the local pressure and viscosity local pressure and viscosity variation due to converging-variation due to converging-diverging geometry, until achieving diverging geometry, until achieving convergence. convergence.

ExampleExampleTardy-Anderson Algorithm:Tardy-Anderson Algorithm:1. Since the converging-diverging 1. Since the converging-diverging geometry is important for viscoelastic geometry is important for viscoelastic flow, the capillaries should be flow, the capillaries should be modeled with contraction.modeled with contraction.

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

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.