Modelling the Flow of Viscoelastic Fluids in Porous Media

1. Modelling the Flow of ViscoelasticModelling the Flow of Viscoelastic
Fluids in Porous MediaFluids in Porous Media
Pore Scale Modelling ConsortiumPore Scale Modelling Consortium
Imperial College LondonImperial College London
Taha Sochi & Martin BluntTaha Sochi & Martin Blunt

2. ViscoelasticityViscoelasticity
A dual nature of the substance behaviour byA dual nature of the substance behaviour by
showing signs of both viscous fluids and elasticshowing signs of both viscous fluids and elastic
solids.solids.
In its simple form, viscoelsticity can beIn its simple form, viscoelsticity can be
modelled by combining Newton’s law formodelled by combining Newton’s law for
viscous fluids (stressviscous fluids (stress ∝∝ rate of strain) withrate of strain) with
Hook’s law for elastic solids (stressHook’s law for elastic solids (stress ∝∝ strain) .strain) .

3. Upper Convected MaxwellUpper Convected Maxwell
This approach is adopted by UCM modelThis approach is adopted by UCM model
which accounts for frame-invariance:which accounts for frame-invariance:
ττ Stress tensorStress tensor
λλ11 Relaxation timeRelaxation time
µµοο Low-shear viscosityLow-shear viscosity
γγ Rate-of-strain tensorRate-of-strain tensor
γττ o
µλ −=+
∇
1

4. Oldroyd-BOldroyd-B
ττ Stress tensorStress tensor
λλ11 Relaxation timeRelaxation time
λλ22 Retardation timeRetardation time
µµοο Low-shear viscosityLow-shear viscosity
γγ Rate-of-strain tensorRate-of-strain tensor
This is another simple viscoelastic modelThis is another simple viscoelastic model



 +−=+
∇∇
γγττ 21
λµλ o

5. Features of Viscoelastic BehaviourFeatures of Viscoelastic Behaviour
Time-dependency
Intermediate plateau
Strain
hardening
in-situ
convergence-convergence-
divergence withdivergence with
time of fluidtime of fluid
beingbeing
comparable withcomparable with
time of flowtime of flow
DelayedDelayed
response &response &
relaxationrelaxation
Dominance ofDominance of
extension overextension over
shear at highshear at high
flow rateflow rate

6. Modelling Flow in Porous MediaModelling Flow in Porous Media
For a capillary:For a capillary: Pcq ∆= .
Flow rate = conductanceFlow 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=

7. Modelling Flow in Porous MediaModelling Flow in Porous Media
For a network of capillaries, a set ofFor a network of capillaries, a set of
equations representing the capillariesequations representing the capillaries
and satisfying mass conservationand satisfying mass conservation
should be solved simultaneously toshould 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 pressurean initial guess, solve for the pressure
iteratively, updating the viscosity afteriteratively, updating the viscosity after
each cycle, until reaching convergence.each cycle, until reaching convergence.

8. 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 theflow, start with an initial guess for the
flow rate and iterate, considering theflow rate and iterate, considering the
effect of the local pressure and viscosityeffect of the local pressure and viscosity
variation due to converging-divergingvariation due to converging-diverging
geometry, until convergence is achieved.geometry, until convergence is achieved.
This approach is adopted by Tardy andThis approach is adopted by Tardy and
Anderson using a modified Bautista-Anderson using a modified Bautista-
Manero model which is based on theManero model which is based on the
Fredrickson and Oldroyd-B models.Fredrickson and Oldroyd-B models.

9. Tardy-Anderson AlgorithmTardy-Anderson Algorithm
1. Since the converging-diverging1. Since the converging-diverging
geometry is important for viscoelasticgeometry is important for viscoelastic
flow, the capillaries should be modelledflow, the capillaries should be modelled
with contraction.with contraction.
2. Each capillary is2. Each capillary is discretized in the flowdiscretized in the flow
direction and a discretized form of thedirection and a discretized form of the
flow equations is used assuming a priorflow equations is used assuming a prior
knowledge of stress & viscosity at inlet.knowledge of stress & viscosity at inlet.

10. Tardy-Anderson AlgorithmTardy-Anderson Algorithm
3. Starting with an initial guess for the3. 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 thethe 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 whole4. The pressure field for the whole
network is then found iteratively untilnetwork is then found iteratively until
convergence is achieved.convergence is achieved.

11. Thank YouThank You
Questions?Questions?