Non-Newtonian Flows in Porous Media: upscaling problems

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1 6/26/8 4th Cargèse Summer School, 208 Non-Newtonian Flows in Porous Media: upscaling problems Davit Y., Zami-Pierre F.,2, de Loubens R.2 and Quintard M. Institut de Mécanique des Fluides de Toulouse (IMFT) Université de Toulouse, CNRS-INPT-UPS, Toulouse FRANCE 2 Total, CSTJF, Avenue Larribau, 6408 Pau, France /27

2 Objective/Outline Motivation: flow of polymer solutions, question about heuristic models in Res. Engng Upscaling Introduction (generalized Stokes) Transition Induced anisotropy, effect of disorder, effect of size of the UC,... Further problems: exclusion zone, viscoelastic Conclusions 2/27

3 Multi-Scale Analysis L Reservoir-Scale η-region V ω-region lη lβ lω σ-phase Darcy-Scale β-phase Sequential multi-scale pattern Used in DRP, Res. Engng, Hydro., etc... Objectives of macro-scale theories: lσ Smoothing operator. Macro. Macro Macro variables, Eqs & BCs Micro-macro link Determination of Effective Properties V Pore-Scale Pore-Scale Darcy-Scale *( ψ )=0 ( )=0 =g(x) ψ ==g*(x) Needs Scale Separation: lβ,lσ REV? REV? REV? L (process dependent) 3/27

4 Multi-Scale Analysis: Upscaling Techniques Form of the equations? averaging and TIP (Marle, Gray, Hassanizadeh, ) averaging and closure (Whitaker, ) homogenization (Bensoussan et al., Sanchez-Palencia, Tartar, ), also closure stochastic approaches (Dagan, Gelhar,...) Effective properties calculations? Assuming the form of Eqs: interpret experiments or DNS Upscaling with closure (averaging, homogenization, stochastic): provides local Unit Cell problems Many Open Problems: High non-linearities, Strong couplings, Evolving pore-scale structure,... 4/27

5 A simple introduction to upscaling with closure Macro c x DNS Tomography Reconstruction Geostatistics... Closure: aver. x Micro bx x Macro-scale Equation Effective property 5/27

6 Flow of a non-newtonian fluid Case of Generalized Stokes equation Pore-Scale problem (Re~0) µ/µ0 00 Rheology: plateau + power law Carreau cross-fluid γ / γ c Upscaling: (vol. aver. ψβ =εβ ψψβ β with εβ=vβ/v)? 6/27

7 Typical local (over a REV) features velocity Remark (far from BCs) Pressure dev. 30 viscosity 7/27

8 Upscaling flow of a nonnewtonian fluid Averaging (vol. aver. ψβ =εβ ψψβ β with εβ=vβ/v) macro Closure? +... micro Problem must be solved for each value of ψvβ β! 8/27

9 Closure? Tentatively: Under several constraints: scale separation, far from BCs,... Problem must be solved for each value of vvβ β! 9/27

10 A classical story: the linear case and Darcy s law (see Sanchez-Palencia, Whitaker,.) Closure (any solution is a linear combination of elementary solutions for ψvβ β=ei for i=,2,3) over a UC! Macro-Scale equation and effective properties Darcy s law: Intrinsic permeability: Important: Proof of symmetry of K0 requires periodicity! 0/27

11 Calculations of the permeability Case of diffusion problem: e.g., permeability, effective diffusion 3 possibilities Initial closure problem Transformation of closure problem into ~Stokes with source term and periodic pressure and velocity permeameters : noperiodicity Making image periodic? I: Percolation problem II: Loss of anisotropy III: potentially various bias See discussion in Guibert et al., 205 I II thin layers + periodicity Eff. Medium.. III /27

12 Calculations over non-periodic images classical permeameters All methods have bias 2 ψvx β 0 P Kxy Kyx y Note: minimal bias if large sample and anisotropy along the axis See discussion in: Manwart et al. 2002; Piller et al. 2009; Guibert et al., 205;... P Bamberger P 2 x P 2/27

13 Non-Linear Case: NonNewtonian Fluid Fluid rheology PLCO µ/µ0 00 plateau + power law Carreau cross-fluid Representation as a deviation from Darcy s law 03 γ / γ c No generic closure independent of fluid velocity! Generic macroscale law: kn, P (rotation matrix ): depend on ψvβ β (modulus and orientation) 3/27

14 Test cases 2D Needs very fine grid! Clashach Bentheimer HPC center EOS-Calmip: Typically: 08 mesh cells 05 cores hours often limited to ~ mm3! 4/27

15 Resolution with OpenFoam FVM with OpenFOAM (SIMPLE, second-order scheme) Use of HPC, calculations up to 00 millions mesh elements a total of hours of CPU time. Conform orthogonal hexahedral elements. Multi-criteria grid convergence study = OK. 5/27

16 Starts in a few narrow constrictions Scaling for transition? ) -n U ( Properties of porescale fields (PDFs) Transition: k Computations allows to analyze various features: k (apparent) n Results k n= Newt onian regime non-newt onian regime Uc FL U FL ψ. FL = intrinsic fluid average 6/27

17 Structure of the Velocity Field y backflow z newtonian non-newtonian Normalized pdf ~similar between Newtonian and nonnewtonian flow! Not valid for pdf of ψpβ β 7/27

18 Transition Scaling A A2 C C2 B B2 P P2 ψu c FL (a ) k v s ψu FL µm.s (b ) k v s U Zami-Pierre et al., 205 8/27

Impact of Domain Size n v β ~y 6 σ= 0 σ = 0.2 n = v β α θ 2 0 2 θ=22 σ= 0 σ = 0.

19 Impact of Domain Size n v β ~y 6 σ= 0 σ = 0.2 n = v β α θ θ=22 σ= 0 σ = p β -ρ βg β 22 ~x kn α (degree) L L Anisotropy induced by non-linear behavior decreases with L for disordered media Effective property variance decreases with L 9/27

Impact of Domain Size θ=22 6 B m ed iu m : k n 4 2 B medium: α B medium: β 0 20 40 kn Rotation Angles ( ) Bentheimer Req

20 Impact of Domain Size θ=22 6 B m ed iu m : k n 4 2 B medium: α B medium: β kn Rotation Angles ( ) Bentheimer Req (pore size) 0 3 L 60 2 L/Req Disorder no anisotropy induced by non-linearity if L large enough! L/Req 20/27

21 Impact of disorder and velocity Aσ=0 Aσ=0.05 Aσ=0.0 Aσ=0.20 Aσ=0.30 B Angle α of P kn Aσ=0 Aσ=0.05 Aσ=0.0 Aσ=0.20 Aσ=0.30 B 0 00 U U 2/27

22 Practical Consequences Eng. Practice: apparent Darcy s law Discussion: Fitting parameter (rock dependent) P=I for all vvβ β if isotropic disordered media and REV ( need tests for various sizes)! Apparent permeability ~ scales with (K0)½ classical scaling may introduce artificial dependence upon parameters such as porosity: versus Description of transition near the critical velocity may not be well described by an apparent viscosity (no observed angle in the apparent permeability in the case of PLCO) 22/27

23 Further upscaling Depletion layer treated as an effective BC L Reservoir-Scale η-region V ω-region Pore-Scale lη lβ lω σ-phase Zami-Pierre et al., 207 see Chauveteau (982), Sorbie & Huang (99) (double-layer model) Darcy-Scale β-phase lσ V effective BC SubPore-Scale cont. DLVO zone model 23/27

24 Further upscaling Viscoelastic fluids upper convected Derivative Rheological models FENE-P: 24/27

Example of results: De et al., soft matter, 208 Steady-state! Deborah number:...also Weissenberg number De= 0.00 0.

25 Example of results: De et al., soft matter, 208 Steady-state! Deborah number:...also Weissenberg number De= Normal stress along average flow direction -see previous discussion on apparent permeability, etc - elastic turbulence? 25/27

26 Further perspectives: N-momentum equations, multi-component aspects,... Superfluid: 2 momentum equations complex behavior macroscale model? Polymer solution as multi-component systems: Mechanical segregation, degradation (bio., mech.) Model? Momentum balances: diffusion theory or N-momentum equations Composition: Continuous models or PBM (population balance model),... pdf see Allain et al. (200, 203, 205), Soulaine et al. (205, 207) mol. weight 26/27

27 Conclusions Upscaling tells that this is not always possible to separate in an apparent Darcy s law permeability and viscosity Specific anisotropy effects Simplifications arise for disordered media Various results published in the literature for various rheology: power-law (...), Ellis and Herschel Bulkley fluids (Sochi & Blunt, 2008), Yield-Stress Fluids (Sochi, 2008), etc Additional problems: retention effects, Inaccessible Pore Volume (IPV), mobile/immobile effects Perspectives: viscoelastic, multicomponent, coupling with other transport problems (transport of species, heat transfer, etc ),... 27/27

This is an author-deposited version published in: Eprints ID : 15932

This is an author-deposited version published in: Eprints ID : 15932 Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited