A Model for Non-Newtonian Flow in Porous Media at Different Flow Regimes

Outline of talk

• Quick introduction to polymer flooding

• Polymer behaviour in bulk versus porous medium

• Mathematical modeling of polymer flooding:– Challenges

– Some details on implementation in IorCoreSim

• Shear thickening and polymer mechanical degradation

• Experimental background for simulation results

• Some simulation results

• Conclusion

Quick introduction to polymer flooding

• Polymers are macromolecules

• When added to the injection brine increased water phase viscosity

• Can lead to:– Improved sweep

– A more stable displacement (less viscous fingering)

– A more efficient oil production!

Image: http://www.npd.no/en/Publications/Resource-Reports/2014/Chapter-2/

Polymers are complex fluids!

• Polymers are non-Newtonian fluids, i.e., the relationship between shear stress and shear rate is not constant

• The viscosity of EOR polymers depend on many factors

• The behaviour in bulk solution can be qualitatively different from the behaviour in porous media

• Good mathematical models are needed to capture the polymer behaviour in a numerical simulator!

Typical EOR polymers

• Mostly synthetic polymers based on polyacrylamide (PAM)

• Certain biological polymers, e.g., xanthan

• More recently: Associative polymers

xanthan

Image source: http://petrowiki.org/Polymers_for_conformance_improvement

Polymers in bulk solution

• In a rheometer, EOR polymers tend to display Newtonian and shear thinning behaviour

• Can usually be well represented by the empirical Carreau-Yasuda model:

𝜂 = 𝜂𝑠 + 𝜂0 − 𝜂𝑠 ⋅ 1 + 𝜆1 𝛾 𝑥 −𝑛/𝑥

Bulk viscosity of a polymer solution

Shear thinning

Polymers in porous media• In laboratory coreflooding experiments, the additional flow resistance due to

added polymer is quantified by the resistance factor, RF

• Typical behaviour:

𝑅𝐹 = 𝑅𝐹 𝑄 =Δ𝑝(𝑄)

Δ𝑝𝑤𝑎𝑡𝑒𝑟(𝑄)

𝛥𝑝

Simulated resistance factors

Shear thickening

Polymer degradation

Modeling challenges

• The simulator must provide an adequate description of the in-situ polymer rheology:– Onset of the different flow regimes

– Magnitude of viscous and extensional flow resistance

• The model should capture changes in polymer properties as a function of changingreservoir conditions, e.g., permeability, porosity, temperature etc.

• Such a model is needed in order to give quantitative answers to importantquestions concerning field implementation

Polymer model in IORCoreSim

• A new model has recently been implemented in an in-house simulator at IRIS, IorCoreSim

• The model includes:– Newtonian, shear thinning and shear thickening fluid rheology

– Polymer mechanical degradation

– Adsorption and permeability reduction

– Inaccessible pore volume and depletion layers

– Effective salinity model

• Plan for the rest of this talk:– Discuss the shear thickening and degradation parts of the simulation model in more detail

– Show some simulation results

Shear thickening

• Due to elongation of the polymer molecules

• Can happen in e.g.:– Fast transient flows (e.g. capillary tubes with a sharp contraction)

– Quasi steady state flow (e.g. flows with a stagnation point)

– Porous media flow

• Results in an increase in the effective flow resistance (viscosity)

• The elongational contribution to the effective viscosity can be orders of magnitude higher than the shear thinning part

Left image: NGUYEN, Tuan Q.; KAUSCH, Hans-Henning (ed.). Flexible polymer chains in elongational flow: Theory and experiment. Springer Science & Business Media, 2012.Middle and right images: NGUYEN, Tuan Q.; KAUSCH, Hans-Henning. Mechanochemical degradation in transient elongational flow. In: Macromolecules: Synthesis, Order and Advanced Properties. Springer Berlin Heidelberg, 1992.

Shear thickening model• Based on relating two characteristic times of the polymer:

1) Polymer relaxation time (time to relax a deformed polymer chain), 𝜏𝑒𝑙2) Polymer residence time (time of observation), 𝜏𝑟𝑒𝑠

• Define the ratio:

𝑁𝐷𝑒 =𝜏𝑒𝑙𝜏𝑟𝑒𝑠

• The onset of shear thickening is believed to happen at a critical value of 𝑁𝐷𝑒, when 𝜏𝑒𝑙 ≈ 𝜏𝑟𝑒𝑠

• The relaxation time is related to a translational diffusion coefficient as follows:

𝜏𝑒𝑙 =2𝑅ℎ

2

𝐷𝑡, 𝐷𝑡=

𝑘𝑇

6𝜋𝜂𝑠𝑅ℎ, 𝑅ℎ =

3

10𝜋𝑁𝐴

13

⋅ 𝜂 𝑀𝑤

13

• Notation:

- 𝑅ℎ: Hydrodynamic radius (equivalent, spherical size of the polymer coil in solution)

- [𝜂]: Polymer intrinsic viscosity

- M𝑤: Polymer molecular weight

- N𝐴: Avogadros’ number

Shear thickening model

• The residence time is computed using a Kozeny-Carman approach (Lake, 1989):

𝜏𝑟𝑒𝑠 =𝐿𝑝𝑣𝑝

= 12 ⋅1 − 𝜙

𝜙 𝛾𝑐, 𝛾𝑐=

4𝑣𝑝𝑅𝑝

• By setting the ratio of time scales equal to a specific number, 𝑁𝐷𝑒∗ ≈ 1, we can obtain

an expression for the critical shear rate, 𝛾𝑐

• If we define 𝜆2 =1

𝛾𝑐, we can show:

𝜆2 =1

𝑁𝐷𝑒∗ ⋅

3

10⋅

𝜙

1−𝜙⋅𝜂𝑠 𝜂 𝑀𝑤

𝑅𝑇

• The elongational contribution to the total pressure drop is computed in terms of:

𝜂𝑒𝑙 = 1+ 𝜆2 𝛾 𝑥2𝑚+𝑛𝑥2

• The model can be used to capture variations in shear thickening with both fluid and reservoir parameters

Lake, L.W. (1989) - Enhanced oil recovery, Prentice Hall Inc., Old Tappan, NJ

Polymer degradation

• When the chemical bonds of a polymer molecule are broken

• Different types of degradation: biological, chemical, or mechanical

• Mechanical degradation can happen when a polymer molecule is exposed to largeamounts of stress

• Polymer degradation reduces the effective molecular weight of the polymer and, hence, the viscosity

Mechanical degradation model in IORCoreSim

• Polymer molecular weight is updated in each grid block by:

𝑑𝑀𝑤

𝑑𝑡= −𝑓𝑟𝑢𝑝 ⋅ 𝑀𝑤 , 𝑓𝑟𝑢𝑝= 𝑟𝑑𝑒𝑔 ⋅ 𝜏

𝛼𝑑⋅2𝑀𝑤

𝛽𝑑

𝑅𝑝

• Notation:

𝑓𝑟𝑢𝑝: fraction of ruptured molecules (probability of fracture)

𝑟𝑑𝑒𝑔 ⋅ 𝜏𝛼_𝑑

= 𝑟𝑑𝑒𝑔 ⋅ 𝜂 ⋅ 𝛾𝛼𝑑, 𝛼𝑑 > 1: «critical» shear stress for chain rupture

2/𝑅𝑝 specific surface area

𝑀𝑤𝛽𝑑 longer chains have a higher probability of fracturing

Experimental background

• Data from:Stavland, A., Jonsbråten, H.C., Lohne, A., Moen, A., Giske, N.H. (2010) - Polymer Flooding – Flow Properties in Porous Media Versus Rheological Parameters, SPE 131103

• Serial cores (sandstone)

• Polymer injected at various flow rates, and at a concentration of 𝐶𝑝𝑜𝑙 = 1500 𝑝𝑝𝑚

• Ambient temperature, constant salinity (synthetic sea water)

• One phase

𝑅𝐹 = 𝑅𝐹 𝑄 =Δ𝑝𝑝𝑜𝑙𝑦𝑚𝑒𝑟

Δ𝑝𝑤𝑎𝑡𝑒𝑟

Results 1a

• Four different HPAM polymers, with reported molecular weights of 5, 10, 15, and 20 million daltons, all with a hydrolysis degree of 30 %

• Some variations in permeabilty and porosity

Results 1b

• When plotted versus an effective shear rate in porous media (obtained from a capillary bundle model)

• Shear thickening occurs at lower shear rates for higher molecular weight polymers

𝛾 ∝𝑄

𝑘𝜙

Results 2a

• Data for a single polymer type at different permeabilities

• Shear thickening occurs at lower flow rates for lower permeability cores

Results 2b

• 1530 polymer: Degradation at various rates and permeabilities throughout the core

Conclusion

• The model gives good predictions for the tested polymers and experimentalconditions:– Variations in molecular weight

– Variations in permeability

– Progressive degradation as a function of spatial distance

• The parameters in the shear thickening and degradation parts of the model werekept constant among all cases

• Current work:– Look at how the model scales from the lab to the field

– Single well radial model

Acknowledgements

The authors acknowledge the Research Council of Norway and the industry partners; ConocoPhillipsSkandinavia AS, BP Norge AS, Det Norske Oljeselskap AS, Eni Norge AS, Maersk Oil Norway AS, DONG Energy A/S, Denmark, Statoil Petroleum AS, ENGIE E&P NORGE AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS; of The National IOR Centre of Norway for support.