Modelling the Flow of Viscoelastic Modelling the Flow of Viscoelastic

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

ViscoelasticityViscoelasticity

A dual nature of the substance behaviour by A dual nature of the substance behaviour by showing signs of both viscous fluids and elastic showing signs of both viscous fluids and elastic solids.solids.

In its simple form, viscoelsticity can be In its simple form, viscoelsticity can be modelled by combining Newton’s law for modelled by combining Newton’s law for viscous fluids (stress viscous fluids (stress ∝∝ rate of strain) with rate of strain) with Hook’s law for elastic solids (stress Hook’s law for elastic solids (stress ∝∝ strain) . strain) .

Upper Convected MaxwellUpper Convected Maxwell

This approach is adopted by UCM model This approach is adopted by UCM model which accounts for frame-invariance:which accounts for frame-invariance:

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

This is another simple viscoelastic modelThis is another simple viscoelastic model

+−=+

∇∇

γγττ21

λµλo

Features of Viscoelastic BehaviourFeatures of Viscoelastic BehaviourTime-dependency

Intermediate plateau

Strain

hardening

in-situ

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

Modelling Flow in Porous MediaModelling Flow in Porous Media

For a capillary:For a capillary: Pcq ∆= .

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

1. 1. Newtonian fluidNewtonian fluid:: constant)( == µcc

2. 2. Viscous non-Viscous non-NewtonianNewtonian::

),( Pcc µ=

3. 3. Viscoelastic fluidViscoelastic fluid:: ),,( tPcc µ=

Modelling Flow in Porous MediaModelling Flow in Porous MediaFor a network of capillaries, a set of For a network of capillaries, a set of equations representing the capillaries equations representing the capillaries and satisfying mass conservation and satisfying mass conservation should be solved simultaneously to should be solved simultaneously to produce a consistent pressure field:produce a 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.

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.

Modelling Flow in Porous MediaModelling Flow in Porous Media

3. 3. Viscoelastic fluidViscoelastic fluid: for the steady-state : for the steady-state flow, start with an initial guess for the flow, start with an initial guess for the flow rate and iterate, considering the flow rate and iterate, considering the effect of the local pressure and viscosity effect of the local pressure and viscosity variation due to converging-diverging variation due to converging-diverging geometry, until convergence is achieved. geometry, until convergence is achieved. This approach is adopted by Tardy and This approach is adopted by Tardy and Anderson using a modified Bautista-Anderson using a modified Bautista-Manero model which is based on the Manero model which is based on the Fredrickson and Oldroyd-B models. Fredrickson and Oldroyd-B models.

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

Thank YouThank You

Questions?Questions?