PERM Group Imperial College LondonPERM Group Imperial College London

Non-Newtonian Flow in Porous MediaNon-Newtonian Flow in Porous Media

Taha Sochi & Martin BluntTaha Sochi & Martin Blunt

Non-Newtonian FluidNon-Newtonian FluidShear stress is not proportional Shear stress is not proportional to shear rate: to shear rate: ≠≠

Three groups of behaviour:Three groups of behaviour:1.1. Time-independent: shear rate depends Time-independent: shear rate depends

only on instantaneous stress.only on instantaneous stress.

2.2. Time-dependent: shear rate is function Time-dependent: shear rate is function of magnitude & duration of shear.of magnitude & duration of shear.

3.3. Viscoelastic: Shows partial elastic Viscoelastic: Shows partial elastic recovery on removal of stress.recovery on removal of stress.

Herschel ModelHerschel ModelCCnn

Shear stressShear stressYield stressYield stressC C Consistency factorConsistency factorShear rateShear ratenn Flow behaviour index Flow behaviour index

Herschel classes:Herschel classes:

Flow rate in cylindrical tube:Flow rate in cylindrical tube:

1/n1τ

1/n2)τ(τ2τ

1/n3)τ(τττ

ΔPL

C8πQ

2

oowo2

owow

3

1/n

n11

C C nn Herschel parameters Herschel parametersL L Tube lengthTube lengthP P Pressure differencePressure differenceww PR/2L PR/2L ((R =R = tube radius) tube radius)

Berea networkBerea network Sand pack networkSand pack network

(after Xavier Lopez)(after Xavier Lopez)

Comparison with Single TubeComparison with Single Tube

Fluid with Yield StressFluid with Yield Stress

3mm3mm

Equivalent single tubeEquivalent single tube

2.5mm2.5mm

Equivalent single tubeEquivalent single tube

Network ModellingNetwork Modelling1.1. Obtain 3D image of the pore space.Obtain 3D image of the pore space.

2.2. Build a topologically-equivalent Build a topologically-equivalent network in terms of pore sizes, network in terms of pore sizes, shapes and connectivity.shapes and connectivity.

3.3. Account for non-circularity, when Account for non-circularity, when calculating calculating QQ for cylinder, by using for cylinder, by using equivalent radius:equivalent radius:

4/18

GReq

4.Start with initial guess for viscosity 4.Start with initial guess for viscosity in each network element.in each network element.

5.Invoke conservation of volume and 5.Invoke conservation of volume and hence solve the pressure field.hence solve the pressure field.

6.Update viscosity using Herschel 6.Update viscosity using Herschel expression with pseudo-Poiseuille expression with pseudo-Poiseuille definition.definition.

7.Recompute the pressure using the 7.Recompute the pressure using the updated viscosities.updated viscosities.

8.Iterate until convergence is 8.Iterate until convergence is achieved when specified tolerance achieved when specified tolerance error in total error in total QQ is reached.is reached.

9.Obtain total flow rate and apparent 9.Obtain total flow rate and apparent viscosity.viscosity.

Experimental ResultsExperimental ResultsChase:Chase: 9 datasets for Bingham 9 datasets for Bingham aqueous solution of Carbopol aqueous solution of Carbopol 941 in column of glass beads.941 in column of glass beads.

Sample:Sample:

Al-Fariss:Al-Fariss: 16 datasets for waxy & 16 datasets for waxy & crude oils in 2 packed beds of crude oils in 2 packed beds of sand. Data is found inconsistent. sand. Data is found inconsistent.

Sample:Sample:

Why Network Yields FirstWhy Network Yields First

Future WorkFuture Work1.Modelling viscoelasticity.1.Modelling viscoelasticity.2.Including more physics, e.g. 2.Including more physics, e.g. adsorption & wall exclusion.adsorption & wall exclusion.3.Modelling 2-phase flow with3.Modelling 2-phase flow with two non-Newtonian fluids.two non-Newtonian fluids.

ReferencesReferences• Skelland A. Non-Newtonian Flow and Heat Transfer.Skelland A. Non-Newtonian Flow and Heat Transfer.• M. Blunt M. Blunt et al. et al. Detailed Physics, PredictiveDetailed Physics, Predictive Capabilities and Macroscopic Consequences for Capabilities and Macroscopic Consequences for Pore-Network Models of Multiphase Flow.Pore-Network Models of Multiphase Flow.• Sorbie K. Polymer-Improved Oil Recovery.Sorbie K. Polymer-Improved Oil Recovery.• Lopez X. Pore-Scale Modelling of Non-NewtonianLopez X. Pore-Scale Modelling of Non-Newtonian Flow.Flow.• Valvatne P. Predictive Pore-Scale Modelling ofValvatne P. Predictive Pore-Scale Modelling of Multiphase Flow.Multiphase Flow.• G. Chase G. Chase et al. et al. Incompressible Cake Filtration of a Incompressible Cake Filtration of a Yield stress Fluid.Yield stress Fluid.• T. Al-Fariss T. Al-Fariss et al. et al. Flow of Shear-Thinning LiquidFlow of Shear-Thinning Liquid With Yield Stress Through Porous Media.With Yield Stress Through Porous Media.