PERM Group Imperial College LondonPERM Group Imperial College London

Viscoelastic Flow in Porous Viscoelastic Flow in Porous MediaMedia

Taha Sochi & Martin BluntTaha Sochi & Martin Blunt

RheologyRheology1. 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 the Flow in Porous Modelling the Flow in Porous MediaMedia

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, SPE Flow in Reservoir Rocks, SPE 54379, 1999.54379, 1999.

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

ExamplesExamplesA.A.Upper Convected Upper 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 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 Approaches:1. Continuum Approaches:

These approaches are based These approaches are based on extending the modified on extending the modified Darcy’s Law for the flow of Darcy’s Law for the flow of non-Newtonian viscous fluids non-Newtonian viscous fluids in porous media to include in 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 based These approaches are based on solving the governing on solving the governing equations of the viscoelastic equations of the viscoelastic flow over the void space of flow over the void space of the porous medium:the porous medium:

A. Numerical Methods:A. Numerical Methods:

B. Network Modelling:B. Network Modelling:

ExamplesExamplesA. Gogarty A. 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

B. Garrouch 1999:B. Garrouch 1999:

n

n

PKq

1

11

Porosity Porosity Model parameterModel parameterRelaxation timeRelaxation timenBehaviour index in Behaviour index in media media Model parameterModel parameter

Finite Element, Finite Volume, Finite Element, Finite Volume, Finite Difference and Spectral Finite Difference and Spectral methods are prominent methods are prominent examples of the numerical examples of the numerical methods that could be used methods that could be used to solve the governing to 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. constitutive 4. 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 is The idea of this approach is to use a time-independent to use a time-independent network model to simulate network model to simulate the viscoelastic flow by the viscoelastic flow by Discretising over time:Discretising over time:

* * The flow in the network elements is The flow in the network elements is considered Poiseuille’s as soon as an considered Poiseuille’s as soon as an effective total viscosity, which is local effective total viscosity, which is local time-dependent and accounts for the time-dependent and accounts for the local shear and normal stresses, is local shear and normal stresses, is obtained. obtained.

** The past history is taken into The past history is taken into account by storing the required account by storing the required information from the past runs into information from the past runs into relevant vectors.relevant vectors.

* * The main challenge is to obtain a The main challenge is to obtain a time-dependent viscosity function time-dependent viscosity function from the constitutive equation.from the constitutive equation.

* * Another possibility is to account for Another possibility is to account for the normal elastic stresses by the normal elastic stresses by converging-diverging geometry. This converging-diverging geometry. This geometry may be simple (to avoid geometry 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).

PlanPlan1. Scan the pressure line.1. Scan the pressure line.

2.2. For each pressure point, For each pressure point, scan the time line generating scan the time line generating the time-independent the time-independent rheology at that instant rheology at that instant considering the past history.considering the past history.

3. 3. Simulate the flow using the Simulate the flow using the time-independent network time-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)