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.