1. The document discusses viscoelastic flow in porous media, including linear and non-linear models of viscoelasticity. 2. It describes continuum and pore-scale approaches to modeling viscoelastic flow, noting advantages and limitations of each. 3. Network modeling is presented as an example pore-scale approach, with the Tardy-Anderson algorithm provided as a specific technique for solving the network flow equations iteratively.

1. PERM Group Imperial College LondonPERM Group Imperial College London
Viscoelastic Flow in PorousViscoelastic Flow in Porous
MediaMedia
Taha Sochi & Martin BluntTaha Sochi & Martin BluntRheologyRheology
1. Linear Viscoelasticity:1. Linear Viscoelasticity:
τ Stress tensorStress tensor
λ1 Relaxation timeRelaxation time
t TimeTime
µο Low-shear viscosityLow-shear viscosity
γ Rate-of-strain tensorRate-of-strain tensor
Berea networkBerea network Sand pack networkSand pack network
Modelling Flow inModelling 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 ofP. Tardy & V. Anderson: Current Modelling of
Flow Through Porous Media.Flow Through Porous Media.
Description of the behaviourDescription of the behaviour
under small deformation.under small deformation.
ExamplesExamples
A. Maxwell Model:A. Maxwell Model:
γ
τ
τ o
t
µλ −=
∂
∂
+ 1
B. Jeffreys Model:B. Jeffreys Model:






∂
∂
+−=
∂
∂
+
tt
o
γ
γ
τ
τ 21
λµλ
λ2 Retardation timeRetardation time
2. Non-Linear Viscoelasticity:2. Non-Linear Viscoelasticity:
Description of the behaviourDescription of the behaviour
under large deformation.under large deformation.
ExamplesExamples
A.A.Upper ConvectedUpper Convected
Maxwell Model:Maxwell Model:
@@ Characterises VE materials.Characterises VE materials.
@@ Serves as a starting point forServes as a starting point for
non-linear models.non-linear models.
γττ o
µλ −=+
∇
1
Upper convected timeUpper convected time
Derivative of the stress tensorDerivative of the stress tensor
∇
τ
( ) vvv ∇⋅−⋅∇−∇⋅+
∂
∂
=
Τ
∇
τττ
τ
τ
t
v Fluid velocityFluid velocity
∇v Velocity gradient tensorVelocity gradient tensor
B. Oldroyd B Model:B. Oldroyd B Model:




 +−=+
∇∇
γγττ 21
λµλ o
∇
γ
( ) vvv ∇⋅−⋅∇−∇⋅+
∂
∂
=
Τ
∇
γγγ
γ
γ
t
Upper convected timeUpper convected time
Derivative of the rate-of-strainDerivative of the rate-of-strain
tensortensor
1. Continuum Approach:1. Continuum Approach:
This is based on extending theThis is based on extending the
modified Darcy’s Law for themodified Darcy’s Law for the
flow of non-Newtonian viscousflow of non-Newtonian viscous
fluids in porous media tofluids 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 theThis is based on solving the
governing equations of thegoverning equations of the
viscoelastic flow over the voidviscoelastic flow over the void
space. The prominent examplespace. The prominent example
of this approach is networkof this approach is network
modelling:modelling:
ExampleExample
GogartyGogarty et alet al 1972:1972:
( )
[ ]mapp
q
K
q
P −
+=∇ 5.1
243.01||
µ
∇P Pressure gradientPressure gradient
q Darcy velocityDarcy velocity
µapp Apparent viscosityApparent viscosity
K PermeabilityPermeability
m Elastic correction factorElastic correction factor
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)
ViscoelasticityViscoelasticity
Dual nature of substanceDual nature of substance
behaviour by showing signsbehaviour by showing signs
of both viscous fluids andof both viscous fluids and
elastic solids.elastic solids.
Features ofFeatures of
ViscoelastcViscoelastc
BehaviourBehaviour
1. Time Dependency:1. Time Dependency:
2. Strain Hardening:2. Strain Hardening:
3. Intermediate Plateau:3. Intermediate Plateau:
Due to delayed response andDue to delayed response and
relaxation.relaxation.
Due to dominance of extensionDue to dominance of extension
over shear at high flow rate.over shear at high flow rate.
Due to convergence-divergenceDue to convergence-divergence
geometry with time of fluidgeometry with time of fluid
being comparable to time ofbeing 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)( == µcc
),( Pcc µ=
),,( tPcc µ=
For a network of capillaries, a setFor a network of capillaries, a set
of equations representing theof equations representing the
capillaries and satisfying masscapillaries and satisfying mass
conservation should be solvedconservation should be solved
simultaneously to produce asimultaneously 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, solveStarting with an initial guess, solve
for the pressure iteratively, updatingfor the pressure iteratively, updating
the viscosity after each cycle, untilthe viscosity after each cycle, until
convergence is achieved.convergence is achieved.
For the steady-state flow, start withFor the steady-state flow, start with
an initial guess for the flow rate andan initial guess for the flow rate and
iterate, considering the effect of theiterate, considering the effect of the
local pressure and viscositylocal pressure and viscosity
variation due to converging-variation due to converging-
diverging geometry, until achievingdiverging geometry, until achieving
convergence.convergence.
ExampleExample
Tardy-Anderson Algorithm:Tardy-Anderson Algorithm:
1. Since the converging-diverging1. Since the converging-diverging
geometry is important for viscoelasticgeometry is important for viscoelastic
flow, the capillaries should beflow, the capillaries should be
modeled with contraction.modeled with contraction.
2. Each capillary is discretized in the2. Each capillary is discretized in the
flow direction and a discretized formflow direction and a discretized form
of the flow equations is usedof the flow equations is used
assuming a prior knowledge of stressassuming a prior knowledge of stress
& viscosity at inlet.& viscosity at inlet.
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.