Ve poster 2006
1
PERM Group Imperial College London PERM Group Imperial College London Viscoelastic Flow in Porous Viscoelastic Flow in Porous Media Media Taha Sochi & Martin Blunt Taha Sochi & Martin Blunt Rheology Rheology 1. Linear 1. Linear Viscoelasticity: Viscoelasticity: Stress tensor Stress tensor Relaxation time Relaxation time t Time Time Low-shear viscosity Low-shear viscosity Rate-of-strain Rate-of-strain tensor tensor Berea network Berea network Sand pack network Sand pack network Modelling the Flow in Modelling the Flow in Porous Media Porous Media References References • R. Bird, R. Armstrong & O. Hassager: R. Bird, R. Armstrong & O. Hassager: Dynamics Dynamics of Polymeric Liquids, Vol. 1, 1987. of Polymeric Liquids, Vol. 1, 1987. • P. Carreau, D. De Kee & R. Chhabra: P. Carreau, D. De Kee & R. Chhabra: Rheology of Rheology of Polymeric Systems, 1997. Polymeric Systems, 1997. • W. Gogarty, G. Levy & V. Fox: W. Gogarty, G. Levy & V. Fox: Viscoelastic Viscoelastic Effects in Polymer Flow Through Porous Effects in Polymer Flow Through Porous Media Media, SPE 4025, 1972. SPE 4025, 1972. • A. Garrouch: A. Garrouch: A Viscoelastic Model for A Viscoelastic Model for Description of the Description of the behaviour of behaviour of viscoelastic materials viscoelastic materials under small deformation. under small deformation. Examples Examples A. Maxwell Model: A. Maxwell Model: γ τ τ o t 1 B. Jeffreys Model: B. Jeffreys Model: t t o γ γ τ τ 2 1 Retardation time Retardation time 2. Non-Linear 2. Non-Linear Viscoelasticity: Viscoelasticity: Description of the Description of the behaviour of behaviour of viscoelastic materials viscoelastic materials under large deformation. under large deformation. Examples Examples A. A. Upper Convected Upper Convected Maxwell Model: Maxwell Model: @ @ Not of primary interest Not of primary interest to us. to us. @ @ Characterises VE Characterises VE materials. materials. @ @ Serves as a starting Serves as a starting point for point for non-linear models. non-linear models. γ τ τ o 1 Upper convected time Upper convected time Derivative of the stress Derivative of the stress tensor tensor τ v v v τ τ τ τ τ t v Fluid velocity Fluid velocity v Velocity gradient Velocity gradient tensor tensor B. Oldroyd B Model: B. Oldroyd B Model: γ γ τ τ 2 1 o γ v v v γ γ γ γ γ t Upper convected time Upper convected time Derivative of the rate- Derivative of the rate- of-strain tensor of-strain tensor 1. Continuum Approaches: 1. Continuum Approaches: These approaches are These approaches are based on extending the based on extending the modified Darcy’s Law modified Darcy’s Law for the flow of non- for the flow of non- Newtonian viscous Newtonian viscous fluids in porous media fluids in porous media to include elastic to include elastic effects. effects. 2. Pore-Scale 2. Pore-Scale Approaches: Approaches: Ups Ups & & Downs Downs @ Easy to implement. @ Easy to implement. @ No computational cost. @ No computational cost. @ No account of detailed @ No account of detailed physics physics at pore level. at pore level. Ups Ups & & Downs Downs @ The most direct approach. @ The most direct approach. @ Closest to analytical @ Closest to analytical solution. solution. @ Requires pore-space @ Requires pore-space description. description. @ Very hard to implement. @ Very hard to implement. @ Huge computational cost. @ Huge computational cost. @ Serious convergence @ Serious convergence These approaches are These approaches are based on solving the based on solving the governing equations of governing equations of the viscoelastic flow the viscoelastic flow over the void space of over the void space of the porous medium: the porous medium: A. Numerical A. Numerical Methods: Methods: B. Network B. Network Modelling: Modelling: Examples Examples A. Gogarty A. Gogarty et al et al 1972: 1972: m app q K q P 5 . 1 243 . 0 1 P Pressure gradient Pressure gradient q Darcy velocity Darcy velocity app Apparent viscosity Apparent viscosity K Permeability Permeability m Elastic Elastic correction factor correction factor B. Garrouch 1999: B. Garrouch 1999: n n P K q 1 1 1 Porosity Porosity Model parameter Model parameter Relaxation time Relaxation time n Behaviour index Behaviour index in media in media Model parameter Model parameter Finite Element, Finite Finite Element, Finite Volume, Finite Volume, Finite Difference and Spectral Difference and Spectral methods are prominent methods are prominent examples of the examples of the numerical methods that numerical methods that could be used to solve could be used to solve the governing the governing equations. equations. Governing Equations: Governing Equations: 1. Continuity: to 1. Continuity: to conserve conserve mass. mass. 2. Momentum. 2. Momentum. 3. Energy: if energy 3. Energy: if energy exchange exchange occurs (non- occurs (non- isothermal isothermal flow). flow). 4. State: i.e. 4. State: i.e. constitutive constitutive equation such as equation such as UCM to UCM to relate stress to relate stress to shear rate. shear rate. Ups Ups & & Downs Downs @ Relatively easy to @ Relatively easy to implement. implement. @ Modest computational cost. @ Modest computational cost. @ No serious convergence @ No serious convergence issues. issues. @ Requires pore-space @ Requires pore-space description. description. @ Approximations required. @ Approximations required. @ Some models may resist @ Some models may resist such such a formulation. a formulation. The idea of this The idea of this approach is to use a approach is to use a time-independent time-independent network model to network model to simulate the simulate the viscoelastic flow by viscoelastic flow by Discretising over time: Discretising over time: * * The flow in the network The flow in the network elements is considered elements is considered Poiseuille’s as soon as an Poiseuille’s as soon as an effective total viscosity, effective total viscosity, which is local time-dependent which is local time-dependent and accounts for the local and accounts for the local shear and normal stresses, is shear and normal stresses, is obtained. obtained. * * The past history is taken The past history is taken into account by storing the into account by storing the required information from the required information from the past runs into relevant past runs into relevant vectors. vectors. * * The main challenge is to The main challenge is to obtain a time-dependent obtain a time-dependent viscosity function from the viscosity function from the constitutive equation. constitutive equation. * * Another possibility is to Another possibility is to account for the normal elastic account for the normal elastic stresses by converging- stresses by converging- diverging geometry. This diverging geometry. This geometry may be simple (to geometry may be simple (to avoid numerical techniques) avoid numerical techniques) and time-dependent (to account and time-dependent (to account for time-dependent effects). for time-dependent effects). Plan Plan 1. Scan the pressure 1. Scan the pressure line. line. 2. 2. For each pressure For each pressure point, scan the time point, scan the time line generating the line generating the time-independent time-independent rheology at that rheology at that instant considering the instant considering the past history. past history. 3. 3. Simulate the flow Simulate the flow using the time- using the time- independent network independent network model. model. 4. 4. Obtain the flow Obtain the flow rate, rate, Q Q , as a function , as a function of pressure drop, of pressure drop, P P , , and time, and time, t t . . (after Xavier (after Xavier Lopez) Lopez)
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Transcript of Ve poster 2006
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)
