Microscale Modelling of the Frequency Dependent Resistivity of Porous Media
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1 Preseted at the COMSOL Coferece 2008 Haover Microscale Modellig of the Frequecy Deedet Resistivity of Porous Media J.Volkma, N.Klitzsch, O.Mohke ad R.Blaschek Alied Geohysics ad Geothermal Eergy, E.ON Eergy Research Ceter,, RWTH-Aache Uiversity
2 Cotet motivatio theory ad modellig idea goverig equatios model verificatio first arameter studies outlook
3 Motivatio I geohysics: R( f ) ϕ R( f ) = Sectral Iduced Polarisatio (SIP) (=Imedace Sectroscoy)
4 Motivatio SIP-roerties: R( f), ϕ ( f) R (emirical relatios) structural roerties (ore radius, ier surface area, ) hydraulic coductivity State of the Art: exerimetal results emirical models (equivalet circuits: Cole-Cole etc.) theoretical models describig simle ore systems (M&M, SNP, ) -> umerical simulatio required
5 IP-Effect catios boud by egative surface charges ore sace costrictio is equivalet to io selective membrae small ores=active zoe large ores=assive zoe
6 Modellig Parameters modellig with Comsol- Multihysics (FEM-Software) 2D axial symmetric model for cylidrical 3D roblem sequece of smaller ad larger ores alied alteratig voltage M&M: differet mobilities for aios ad catios i the smaller ores : μ μ
7 Modellig Parameters modellig with Comsol- Multihysics (FEM-Software) 2D axial symmetric model for cylidrical 3D roblem sequece of smaller ad larger ores alied alteratig voltage M&M: differet mobilities for aios ad catios i the smaller ores : μ μ ω t = Excess cocetratio of catios at 2 π
8 Goverig Equatios (time domai) Equatios for aio (idex ) ad catio (idex ) movemet drive by diffusio ad migratio i a exteral electric field accordig to Marshall ad Madde (1959): (Z-I) (Z-II) (Z-III) C= DΔ C+ μc U t C = DΔC C U t F Δ U= ( C C) ε [ μ ] =cotiuity equatio for =cotiuity equatio for C C =Poisso s equatio for the otetial U Eistei relatio: D = μ kt B e
9 Goverig Equatios (time domai) Equatios for aio (idex ) ad catio (idex ) movemet drive by diffusio ad migratio i a exteral electric field accordig to Marshall ad Madde (1959): (Z-I) (Z-II) (Z-III) Eistei relatio: C= DΔ C+ μc U t C = DΔC C U t F Δ U= ( C C) ε D = μ kt B e [ μ ] Costats: D μ F k B e T ε diffusio coefficiets io mobilities Faraday s costat Boltzma s costat elemetary charge temerature ermittivity
10 Time-Harmoic Aroach Determiatio of frequecy deedet quatities if siusoidal voltage is alied ->assumtio of harmoic time deedece This meas: C = c + c e i ω t 0 c, c, u C = c + c e i ω t 0 with c = c = u = 0 U = i t u e ω c, c = cost 0 0 Further assumtios: small electric field ad excess-cocetratios -> quadratic terms eglected equal cocetratios without alied voltage: 0 0 c = c = c
11 Goverig Equatios (frequecy domai) Liearised equatios accordig to Marshall ad Madde (1959): (F-I) (F-II) (F-III) iωc = DΔ c + μ c u F Δ u = ( c c) ε [ μ ] iωc = D Δc c u Eistei relatio: D = μ kt B e equatios are o loger time deedet c, c, u cotai iformatio about amlitude ad hase more efficiet calculatio of frequecy deedet quatities
12 Alicatio Modes / Boudary Coditios Alicatio Modes: cotiuity equatios ( ) -> Electrokietic Flow (chekf) Poisso s equatio ( ) -> Electrostatics (emes) Boudary Coditios: u : axial symmetry u c, c c c : axial symmetry : axial symmetry equatios solved for frequecies f = i i t rocedure: I I e ϕ ω = e -> R = U -> I, Hz R( f) ϕ ( f) R
13 Alicatio Modes / Boudary Coditios Alicatio Modes: cotiuity equatios ( ) -> Electrokietic Flow (chekf) Poisso s equatio ( ) -> Electrostatics (emes) Boudary Coditios: u : zero charge / symmetry u c, c c c : isulatio / symmetry : isulatio / symmetry equatios solved for frequecies f = i i t rocedure: I I e ϕ ω = e -> R = U -> I, Hz R( f) ϕ ( f) R
14 Alicatio Modes / Boudary Coditios Alicatio Modes: cotiuity equatios ( ) -> Electrokietic Flow (chekf) Poisso s equatio ( ) -> Electrostatics (emes) Boudary Coditios: u c = c = = ± i cost 0 0 u c, c equatios solved for frequecies f = i i t rocedure: I I e ϕ ω = e -> R = U -> I, Hz R( f) ϕ ( f) R
15 Verificatio - 1D Blue: L1=1µm L2=1µm Red: L1=1.5µm L2=0.5µm Gree: L1=0.5µm L2=1.5µm 1:assive zoe 2:active zoe Crosses: 1D Comsol model Solid lie: aalytical solutio accordig to Marshall ad Madde
16 Verificatio - 3D Crosses: 2D axial symmetric model Solid lie: 3D model
17 Results 3D Red (basic model): ore legth 1µm, ore radius 0.1µm (small ores) ud 1µm (large ores) Blue: scaled geometry x10 Gree: scaled geometry x100
18 Results 3D -> f ( ϕ ) ( s) mi cf. Titov et al. (2002): f ( ϕ ) l mi 2-2 l2 legth of the arrow ores Deedece of the hase miimum o the scale factor (with regard to the basic model)
19 Coclusio Aroach verified by: 1. qualitatively good agreemet betwee modelled ad exerimetal results 2. agreemet betwee 2D axial symmetric model ad 3D model 3. agreemet betwee 1D model ad aalytical solutio accordig to Marshall ad Madde 4. agreemet betwee the results of frequecy deedet calculatios ad those of time deedet calculatios (Blaschek ud Hördt, 2007)
20 Outlook studies of the ifluece of 1. geometric roerties (legths, radii) 2. electrolyte roerties (mobilities, cocetratios) more realistic model - IP as a surface effect For this urose: set surface charges? set mobilities close to the ore wall? set a cocetratio rofile close to the ore wall? Problem: harmoic fuctios vs. costat surface quatities
21 Outlook Reduced aio mobility at the ore walls: -> Problem: may mesh-elemets
22 Refereces 1. Marshall, D.J. ad Madde, T.K., Iduced Polarizatio, a study of its causes, Geohysics, 24 (4),, (1959) 2. Blaschek,, R. ad Hördt,, A., Numerical modelig of the IP-effect at the ore scale, 4th Iteratioal Symosium o Three-Dimesioal Electromagetics,, Freiberg, Germay, (Setember 27-30, 2007) 3. Titov,, K., Komarov,, V., Tarasov,, A. ad Levitski,, A., Theoretical ad exerimetal study of time domai-iduced iduced olarizatio i water- saturated sads, Joural of Alied Geohysics, 50, (2002)
23 Thaks for your attetio!
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