Paul Glover & Emilie Walker Université Laval, Québec, Canada , Q , Matthew Jackson Imperial College, London, UK
� The classical Helmholtz ‐ Smoluchowski equation relates the streaming potential coupling coefficient (SPCC) to li ffi i t (SPCC) t ε ζ f = • zeta potential C ( ( ) ) s η σ + Σ Λ 2 f f s • Pore fluid dielectric permittivity Pore fluid dielectric permittivity • Pore fluid conductivity Pore fluid viscosity • � Developped for capillary tubes � Commonly applied to rocks � However, never been validated for rocks (no measure of zeta potential) � H b lid t d f k ( f t t ti l) � Never even been a theoretical model applied to real rocks � DESPITE most of the theoretical tools being available since 1998 g 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 2/18
� In this presentation: Development of the required D l t f th i d theoretical tools Compilation of a SPCC dataset for p rocks Compilation of a zeta potential dataset for rocks f k Modelling SPCC of rocks as a function of salinity of salinity Modelling ζ of rocks as a function of salinity 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 3/18
SPCC vs. Pore fluid salinity Silica, glass, Silica, glass, sand and sandstone 11 sources Acknowledgments to Jaafar (2009) 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 4/18
Zeta potential vs. Pore fluid salinity salinity Silica, glass, sand and sandstone 7 so rces 7 sources Acknowledgments to Jaafar (2009) 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 5/18
The method is as follows: 1. Calculate the pore fluid conductivity (salinity and temperature) ⎛ ⎞ ) ( ) + ( d d T σ = + + − ⎜ 3 2 2 ⎟ 4 5 T C , d d T d T C C ⎜ ⎟ f f 1 2 3 f + f 1 d C ⎝ ⎠ 6 f Sen and Goode (1992) Sen and Goode (1992) 2. Calculate the pore fluid relative permittivity (salinity and temperature) ( ( ) ) ( ( ) ) ε ε = ε ε + + + + + + + + + + + + 2 3 2 3 T C T C , a a a T a T a T a T a T a T c C c C c C c C c C c C f f o 0 1 2 3 1 f 2 f 3 f Olhoeft (1980) 3 3. Calculate the pore fluid viscosity (salinity and temperature) Calculate the pore fluid viscosity (salinity and temperature) ( ) ( ) ( ) ( ) η = + α + α + α + α m m T C , e e exp T e exp C e exp T C f f 1 2 1 3 2 f 4 3 4 f Phillips et al. (1978) 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 6/18
4. Define the physical chemistry of the double layer K − ( ) + > ⇔ > o - SiOH SiO + H K + + M e > ⇔ > o o SiOH + Me SiOMe + H 5. 5 Calculate or set the pore fluid pH (SiO 2 ‐ H 2 O ‐ CO 2 ) Calculate or set the pore fluid pH (SiO ‐ H O ‐ CO ) ( ) ( ) − − − + − = 3 2 C C C C K K C 2 K K 0 + + + a b w 1 1 2 H H H − − − − = × + × − × + × 16 16 17 2 19 3 K 6.9978 10 5.0178 10 T 2.4434 10 T 7.1948 10 T w Lide (2009); Revil et al. (1999) 6 6. Calculate the Debye screening length and shear plane distance Calculate the Debye screening length and shear plane distance ε ε n = ∑ k T 1 χ = o r b 2 f and χ ζ = 2.4×10 -10 m I Z C d f i i Ν Ν 2 ζ 2 2 i 2000 2000 e I e I f f 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 7/18
7. Calculate the Stern plane potential ( ( ) ) ⎛ ⎛ ⎞ ⎞ − × ε ε Ν + ⎡ ⎡ − ⎤ ⎤ 3 3 10 pH pH + + + pH H 8 10 8 10 k T k 10 K C C C C C 10 ⎜ ⎟ 2 k T r o b Me f ⎢ ⎥ a b f ϕ = b ln ⎜ ⎟ d Γ ⎢ ⎥ o 3 ⎜ ⎟ e I 2 e K ⎣ ⎦ − s f ⎝ ⎠ Revil and Glover (1997; 1998) 8. Calculate the zeta potential ( ( ) ) ζ ζ = ϕ ϕ − χ χ χ χ exp exp ζ ζ d d d d Revil and Glover (1997; 1998) 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 8/18
Σ = Σ + Σ + Σ EDL Prot Stern 9. Calculate the surface conductance s s s s β Γ o e K C s s Me f Σ Σ = ⎛ Stern s ⎞ ( ) 2 3 ⎛ ⎞ − − × ε ε Ν + ⎡ − ⎤ 3 pH ⎜ + + + ⎟ pH 8 10 k T 10 K C ⎜ C C C 10 ⎟ − r o b Me f ⎢ a b f ⎥ pH + + ⎜ ⎟ 10 K K C − ⎜ ⎟ Me f Γ o ⎢ ⎥ ⎜ ⎜ ⎟ ⎟ I 2 e K ⎣ ⎦ − ⎜ ⎟ f s ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠ ⎛⎡ ⎤ ⎛ − ⎞ 1 3 ⎛ ⎞ − ⎛ ⎞ + pH ⎜⎢ ( ) ⎥ ⎜ 10 C K ⎟ − ⎜ ⎟ ⎜ ⎟ Σ = + f Me − + EDL pH R ⎜⎢ B C B 10 S 1 ⎜ ⎟ ⎥ + + ⎜ ⎟ s f ⎜ ⎟ Na H Γ o ⎜ ⎜ ⎟ ⎟ 2 e K ⎜ ⎜ ⎝ ⎝ ⎠ ⎠ ⎢ ⎢ ⎝ ⎝ − ⎠ ⎠ ⎥ ⎥ s s ⎝ ⎝ ⎠ ⎠ ⎣ ⎣ ⎦ ⎦ ⎝ ⎝ ⎞ ⎡ ⎤ + ⎛ ⎞ 1 3 ⎛ ⎞ ⎛ − ⎞ ( ( ) ) + pH ⎟ ⎢ ⎜ ⎟ ⎥ 10 C K − ⎜ ⎟ pH pK ⎜ f Me ⎟ + − B C B 10 S 1 ⎟ f ⎢ ⎥ ⎜ ⎟ − − ⎜ ⎜ ⎟ ⎟ f ⎜ ⎜ ⎟ ⎟ Cl OH Γ Γ o ⎜ ⎜ ⎟ ⎟ ⎟ ⎟ 2 2 e e K − K ⎝ ⎝ ⎠ ⎠ ⎢ ⎢ ⎥ ⎥ ⎝ ⎝ ⎠ ⎠ s ⎝ ⎠ ⎣ ⎦⎠ − × 3 ε ε Ν ( ( ) ) 2 10 k T = r o b − − − R = × 3 ε ε Ν + pH + pH pK S 8 10 k T C 10 10 − w C + + pH r o b f C 10 10 f f Revil and Glover (1997; 1998) 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 9/18
10. Calculate the SPCC ε ζ Δ d V = = f ( ) C Δ η σ + Σ s P d 4 m F f f f f s Glover and Déry (in press) � Fundamental constants ( k b and N A etc.). � Environmental conditions ( T etc.). � Fluid parameters (salinity, pH, pK w , pK 1 and pK 2 etc.). � Rock microstructure parameters ( F, m, φ , d etc.). � Rock-fluid interface parameters, i.e., the electro-chemical parameters associated with surface adsorption reactions ( pK me , pK – etc.). 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 10/18
Value or Parameter Symbol Units Reference range Model variables o C Temperature T 25 Experimental condition 10 ‐ 5 – 3.98 Pore fluid salinity C f mol/L Varied between limits Pore fluid pH pH 6 ‐ 8 ( ‐ ) Varied between limits F Fundamental constants d t l t t ε o 8.854 × 10 ‐ 12 Dielectric permittivity in vacuo F/m Lide (2009) 1.381 × 10 ‐ 23 Boltzmann’s constant k b J/K Lide (2009) 1.602 × 10 19 1 602 × 10 ‐ 19 Charge on an electron Charge on an electron e e C C Lide (2009) Lide (2009) 6.02 × 10 +23 Avagadro’s number N /mol Lide (2009) Fluid parameters Added acid concentration Added acid concentration C a C a variable variable mol/L Calculated from pH mol/L Calculated from pH Added base concentration C b variable mol/L Calculated from pH β s 5 × 10 ‐ 9 m 2 /s/V Revil and Glover (1997) Surface mobility Reaction constant carbonisation pK 1 7.53 ( ‐ ) Wu et al. (1991) 1 Reaction constant dehydrogenisation pK 2 10.3 ( ‐ ) Wu et al. (1991) 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 11/18
Symbol Value or Parameter Units Reference range Rock parameters 2 × 10 ‐ 4 Grain size (diameter) d m St Bee's SST (Jaafar et al., 2009) = − φ m log F log Cementation exponent m 1.86 ( ‐ ) Calculated Formation factor F 19.80 ( ‐ ) St Bee's SST (Jaafar et al., 2009) φ Porosity 0.19 ( ) ( ‐ ) St Bee's SST (Jaafar et al., 2009) ( f ) Rock/fluid interface parameters sites/m 2 Adjusted to fit data Γ s o 1 × 10 +19 Surface site density Adjusted to fit data Adj t d t fit d t Bi di Binding constant for cation f i pK me 7.1 ( ‐ ) (sodium) adsorption on quartz Adjusted to fit data Disassociation constant for pK ( ‐ ) 7.5 ( ‐ ) dehydrogenisation of SiOH dehydrogenisation of SiOH χ ζ 2.4 × 10 ‐ 10 Shear plane distance m Revil and Glover (1997) Σ s Surface conduction (protonic) Prot 2.4 × 10 ‐ 9 S Revil and Glover (1997) β s β s 5 × 10 ‐ 9 m 2 /s/V Revil and Glover (1997) Surface mobility y / / ( ) 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 12/18
SPCC vs. Pore fluid salinity Silica, glass, sand and sand and sandstone 3 different pHs 4 different grain 4 different grain sizes General properties of properties of the SPCC database and absolute values are well described Grain size can b be extremely t l important 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 13/18
Zeta potential vs. Pore fluid salinity Silica glass Silica, glass, sand and sandstone 3 different pHs 3 different pHs Database measurements are very are very scattered Highly sensitive to changes in pH 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 14/18
5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 15/18
Individual modelling suggests Individual modelling suggests that the operating pH is low (about pH 5.5). 5. Individual 6. Conclusions 2. Database 3. Theory 4. Plenary 1. Introduction 16/18
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