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

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

Quick introduction to polymer flooding Outline of talk 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 = η s + η 0 η s 1 + λ 1 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: Simulated resistance factors Shear thickening Polymer degradation RF = RF Q = Δp(Q) Δp water (Q) Δp

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 changing reservoir conditions, e.g., permeability, porosity, temperature etc. Such a model is needed in order to give quantitative answers to important questions 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), τ el 2) Polymer residence time (time of observation), τ res Define the ratio: N De = τ el τ res The onset of shear thickening is believed to happen at a critical value of N De, when τ el τ res The relaxation time is related to a translational diffusion coefficient as follows: Notation: τ el = 2R h 2 D t, D t = kt 6πη s R h, R h = 3 10πN A 1 3 η Mw 1 3 - R h : Hydrodynamic radius (equivalent, spherical size of the polymer coil in solution) - [η]: Polymer intrinsic viscosity - M w : Polymer molecular weight - N A : Avogadros number

Shear thickening model The residence time is computed using a Kozeny-Carman approach (Lake, 1989): τ res = L p = 12 1 φ, v p φ γ c γ c = 4v p R p By setting the ratio of time scales equal to a specific number, N De an expression for the critical shear rate, γ c If we define λ 2 = 1 γ c, we can show: λ 2 = 1 N 3 φ η s η M w De 10 1 φ RT 1, we can obtain The elongational contribution to the total pressure drop is computed in terms of: η el = 1 + λ 2 γ x 2 The model can be used to capture variations in shear thickening with both fluid and reservoir parameters m+n x 2 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 large amounts 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: dm w dt Notation: = f rup M w, f rup = r deg τ αd 2M β d w R p f rup : fraction of ruptured molecules (probability of fracture) r deg τ α_d = r deg η γ αd, α d > 1: «critical» shear stress for chain rupture 2/R p specific surface area M w β d 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 C pol = 1500 ppm Ambient temperature, constant salinity (synthetic sea water) One phase RF = RF Q = Δp polymer Δp water

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 experimental conditions: 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 were kept 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.