This document discusses modeling viscoelastic flow in porous media. It first describes linear and non-linear viscoelasticity models under small and large deformations. It then discusses continuum and pore-scale approaches to modeling viscoelastic flow, noting advantages and disadvantages of each. Numerical methods like finite element and network modeling are presented as ways to solve the governing equations. Network modeling involves discretizing time and simulating flow using a time-independent network model that accounts for past history through effective local time-dependent viscosity.

1. PERM Group Imperial College LondonPERM Group Imperial College London
Viscoelastic Flow in PorousViscoelastic Flow in Porous
MediaMedia
Taha Sochi & Martin BluntTaha Sochi & Martin Blunt
RheologyRheology
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 the Flow in Porous MediaModelling the Flow in Porous 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.
• A. Garrouch:A. Garrouch: A Viscoelastic Model for PolymerA Viscoelastic Model for Polymer
Flow in Reservoir Rocks, SPEFlow in Reservoir Rocks, SPE 54379, 1999.54379, 1999.
Description of the behaviourDescription of the behaviour
of viscoelastic materialsof viscoelastic materials
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
of viscoelastic materialsof viscoelastic materials
under large deformation.under large deformation.
ExamplesExamples
A.A.Upper ConvectedUpper Convected
Maxwell Model:Maxwell Model:
@@ Not of primary interest to us.Not of primary interest to us.
@@ 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 Approaches:1. Continuum Approaches:
These approaches are basedThese approaches are based
on extending the modifiedon extending the modified
Darcy’s Law for the flow ofDarcy’s Law for the flow of
non-Newtonian viscous fluidsnon-Newtonian viscous fluids
in porous media to includein porous media to include
elastic effects.elastic effects.
2. Pore-Scale Approaches:2. Pore-Scale Approaches:
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.
UpsUps && DownsDowns
@ The most direct approach.@ The most direct approach.
@ Closest to analytical solution.@ Closest to analytical solution.
@ Requires pore-space description.@ Requires pore-space description.
@ Very hard to implement.@ Very hard to implement.
@ Huge computational cost.@ Huge computational cost.
@ Serious convergence difficulties.@ Serious convergence difficulties.
These approaches are basedThese approaches are based
on solving the governingon solving the governing
equations of the viscoelasticequations of the viscoelastic
flow over the void space offlow over the void space of
the porous medium:the porous medium:
A. Numerical Methods:A. Numerical Methods:
B. Network Modelling:B. Network Modelling:
ExamplesExamples
A. GogartyA. Gogarty 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
B. Garrouch 1999:B. Garrouch 1999:
( ) n
n
P
K
q −
−
∇





= 1
1
1
β
αλ
φ
φ PorosityPorosity
α Model parameterModel parameter
λ Relaxation timeRelaxation time
n Behaviour index inBehaviour index in
mediamedia
β Model parameterModel parameter
Finite Element, Finite Volume,Finite Element, Finite Volume,
Finite Difference and SpectralFinite Difference and Spectral
methods are prominentmethods are prominent
examples of the numericalexamples of the numerical
methods that could be usedmethods that could be used
to solve the governingto solve the governing
equations.equations.
Governing Equations:Governing Equations:
1. Continuity: to conserve1. Continuity: to conserve
mass.mass.
2. Momentum.2. Momentum.
3. Energy: if energy exchange3. Energy: if energy exchange
occurs (non-isothermaloccurs (non-isothermal
flow).flow).
4. State: i.e. constitutive4. State: i.e. constitutive
equation such as UCM toequation such as UCM to
relate stress to shear rate.relate stress to shear rate.
UpsUps && DownsDowns
@ Relatively easy to implement.@ Relatively easy to implement.
@ 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.
@ Some models may resist such@ Some models may resist such
a formulation.a formulation.
The idea of this approach isThe idea of this approach is
to use a time-independentto use a time-independent
network model to simulatenetwork model to simulate
the viscoelastic flow bythe viscoelastic flow by
Discretising over time:Discretising over time:
** The flow in the network elements isThe flow in the network elements is
considered Poiseuille’s as soon as anconsidered Poiseuille’s as soon as an
effective total viscosity, which is localeffective total viscosity, which is local
time-dependent and accounts for thetime-dependent and accounts for the
local shear and normal stresses, islocal shear and normal stresses, is
obtained.obtained.
** The past history is taken intoThe past history is taken into
account by storing the requiredaccount by storing the required
information from the past runs intoinformation from the past runs into
relevant vectors.relevant vectors.
** The main challenge is to obtain aThe main challenge is to obtain a
time-dependent viscosity functiontime-dependent viscosity function
from the constitutive equation.from the constitutive equation.
** Another possibility is to account forAnother possibility is to account for
the normal elastic stresses bythe normal elastic stresses by
converging-diverging geometry. Thisconverging-diverging geometry. This
geometry may be simple (to avoidgeometry may be simple (to avoid
numerical techniques) and time-numerical techniques) and time-
dependent (to account for time-dependent (to account for time-
dependent effects).dependent effects).
PlanPlan
1. Scan the pressure line.1. Scan the pressure line.
2.2. For each pressure point,For each pressure point,
scan the time line generatingscan the time line generating
the time-independentthe time-independent
rheology at that instantrheology at that instant
considering the past history.considering the past history.
3.3. Simulate the flow using theSimulate the flow using the
time-independent networktime-independent network
model.model.
4.4. Obtain the flow rate,Obtain the flow rate, QQ, as a, as a
function of pressure drop,function of pressure drop, PP,,
and time,and time, tt..
(after Xavier Lopez)(after Xavier Lopez)