International PhD Course in HYDROGEOPHYSICS Hydrological geophysical relationships Andrew Binley Lancaster University
Overview In the course we will concentrate on electrical, electromagnetic and radar methods for hydrogeophysical investigations. Here we discuss the known and assumed links between hydrological and electrical properties of the subsurface. We cover: the basic definition of electrical properties; theoretical and empirical relationships; example applications. Acknowledgement: many slides were provided by David Lesmes
Objective Electrical Properties (10-3 Hz to 10 9 Hz) Soil/Rock Properties (Moisture, Salinity, Texture, Permeability)
Approach Laboratory Field ~
Hydrological properties Porosity ratio of pore volume (V p ) to total volume (V t ) φ = V V Moisture content ratio of pore water volume (V w ) to total volume (V t ) θ = p t V V w t Effective saturation ratio of changeable moisture content to total changeable moisture content. S e θ θr = θ θ s r
Hydrological properties Effective saturation a function of the pressure head of the pore fluid, for example in the van Genuchten (1980) model: S e θ θr = θ θ s r = 1 [ ] n 1+ α( ψ ) m 1 S(ψ) 0 ψ
Hydrological properties Hydraulic conductivity controls the rate at which water moves through the porous media is a function of the permeability (k s ), density (ρ w ) and viscosity (μ), : K s = k s ρwg μ Many empirical models exist which relate the permeability to pore size or grain size or surface area. For example the Kozeny-Carman equation k s = 3 φ (1 φ) 2 1 5S por S por = surface area per unit volume of solid
Electrical properties Grain Pore- Solution Surface Electrical Transport = Flow + Storage
Electrical properties 0.001 Hz 1kHz: Four-Electrode 100 Hz 100 MHz: Two-Electrode 10 MHz 1GHz: Transmission Lines L A G ( f ) C( f ) Flow - conductivity σ ε L ( f ) = G( f ) A Storage permittivity L ( f ) = C( f ) A
Conductivity and permittivity are complex variables '' (S/m) ' (S/m) κ' 0.016 0.014 0.012 0.01 1.E-02 1.E-03 1.E-04 1.E-05 1.E+09 1.E+06 1.E+03 1.E+00 Berea Sandstone 0.01M NaCl σ dc 1.E-03 1.E+00 1.E+03 1.E+06 IP Frequency (Hz) 1.E-03 1.E+00 1.E+03 1.E+06 Frequency (Hz) 1.E-03 1.E+00 1.E+03 1.E+06 Frequency (Hz) κ Flow σ Storage κ = σ ωε 0 After Lesmes and Friedman (2005)
Electrical Parameters Soil/Rock Properties Dielectric (radar) Water Content Conductivity Salinity, Texture/ (resistivity, ground conductivity) Lithology Induced polarisation (IP) Spectral induced polarisation * σ f (SIP) ( ) Texture/Lithology, Surface Chemistry Grain/Pore Size?
Relationships Many petrophysical models exist which describe the relationships between geophysical and hydrological properties. Some are semi-empirical and based on geometrical averaging some are purely empirical. Here we concentrate on common models.
Permittivity moisture content Topp et al. (1980) κ = 2 3 3.03 + 9.3θ + 146θ 76.3θ Widely used for time domain reflectometry (TDR) and some radar
Permittivity moisture content 500 MHz Complex refractive index model (CRIM) κ = θ κ + ( φ θ ) κ + (1 φ) w a κ s Mixing model based on individual components 150 MHz κ w = 81, κ a = 1 5 κ s 20 (typically) After West et al. (2003)
Permittivity moisture content CRIM: κ = θ κ + ( φ θ ) κ + (1 φ) w a κ s General mixing model: κ a i Θ i a κ a = Θ κ i Dielectric constant on fraction i Volume of fraction i Limits are: 1 (perpendicular flow), -1 (parallel flow) i
Resistivity/conductivity Archie s empirical law (Archie, 1942) is the most widely used. σ σ = σ 1 F w φ m S n w Formation factor: F σ σ w m = = φ Cementation index: 1.5 m (typically) 3 σ w Saturation index: 1.3 n 2 (typically)
Resistivity/conductivity The cementation index increases as the grains become less spherical (Jackson, 1978) Increasing proportion of platey fragments σ = σ w φ m S n w Formation factor: F σ σ w m = = φ Cementation index: 1.5 m (typically) 3 Saturation index: 1.3 n 2 (typically)
Resistivity/conductivity Archie s law assumes no surface conductivity 10 Rock Conductivity (S/m) 1 0.1 High CEC Low CEC C1 C26 0.01 0.1 1 10 100 Solution Conductivity (S/m) After Lesmes and Friedman (2005)
Resistivity/conductivity Archie s law assumes no surface conductivity log(σ soil ) σ F w σ = + σ surf σ = σ w F Surface conductivity σ surf = BQ v F log(σ w ) (Waxman and Smits, 1968) B Equivalent ionic conductance of the clay exchange cations Q v Effective clay content
Resistivity/conductivity In unsaturated porous media with surface conductivity: σ = σ w φ m S n w m σ = φ S n w σ w + BQ S v (Waxman and Smits, 1968)
Resistivity/conductivity Conductivity is then a function of many hydrological properties: particle geometry Pore water conductivity Texture/particle size σ porosity = φ m S n w σ w + BQ S v saturation We need some way of separating out some of these effects
Induced polarisation (IP) Advantages A direct measure of surface properties of soils/rocks Permeability prediction? Contaminant detection/monitoring? Disadvantages Direct contact and low frequency method - slow Electrode and electromagnetic coupling errors Polarisation mechanisms not fully understood
Induced polarisation - polarisation mechanisms Electric-Double Layer (EDL) Polarisation (Schwarz, 1962) Membrane Polarisation (Marshall & Madden, 1959)
Induced polarisation equivalent circuit σ bulk κ bulk * σ surf ( ω ) σ θ bulk + ' σ surf ( ) ω σ '' surf ( ) ω θ lowfreq σ '' σ bulk surf ' + σ ( ω) surf ( ) ω e.g., Vinegar and Waxman(1984)
Induced polarisation field parameters Phase angle θ = tan 1 ( '' ' ) '' ' σ / σ σ / σ. Perfect frequency effect. σ PFE = 100* ( ω ) ( ) 1 σ ω 0 σ ( ω ) 0 Chargeability m = V max 1 ( t t ) 1 0 t 1 t 0 V ()dt t
Induced polarisation lithology effects 100 1 10 " (ms/m) 0.1 0.01 Borner Knight & Nur B C Para " ( S/cm) 1 0.1 0.01 0.001 B C Perp Taherian 0 1 10 100 1000 0.001 0.0001 0.01 1 Spor (1/μm) d 10 (mm) silts & sands tills Slater and Lesmes (2001)
Permeability prediction Tube/Crack Models Kozeny-Carman k = R af 2 h 1 afs 2 por Grain Models Hazen (1911) Krumbien and Munk (1943) Berg (1970)
Permeability prediction from permittivity Since κ is a function of porosity (or moisture content) we may expect to see a correlation with permeability Saturated Dry Knoll et al. (1995)
Permeability prediction from conductivity/resistivity Similarly, we may expect to see a positive correlation between bulk conductivity and permeability Purvance & Andrcevic (2000)
Permeability prediction from conductivity/resistivity However, because of the surface conductivity effects the observed relationships may be weak or negative Purvance & Andrcevic (2000)
Permeability prediction using IP IP may be able to account for the surface effects in our petrophysical models 1 '' (ms/m) 0.1 0.01 0.001 0 1 10 100 Sp (1/μm) Specific surface area Borner & Schon (1991)
Permeability prediction using IP Kcalc (m/s) Borner & Schon (1991) (modified Kozeny-Carmen model) 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 K calc = a FS c por[ el] 1.E-07 1.E-05 1.E-03 K meas (m/s) Kcalc (m/s) 1.0E-02 1.0E-03 1.0E-04 1.0E-05 1.0E-06 1.0E-07 Slater & Lesmes (2001) Grain size model K = ) calc 0.002( d10[ EL] 1.0E-07 1.0E-05 1.0E-03 K meas (m/s) d sands & sand/clay mixes tills tills sands/silts[1] sands/silts[2]
Permeability prediction using IP The model of Borner & Schon (1991) or Slater & Lesmes (2001) assume a frequency independent imaginary conductivity In some cases this is not valid ρ (Ωm) 30 20 10 ρ φ -25-20 -15-10 φ (mrad) 0-5 0.001 0.01 0.1 1 10 100 1000 10000 Frequency (Hz)
Permeability prediction using IP If we take a single frequency in this case σ (S m -1 ) 0.05 0.04 Eggborough core Eggborough blocks Hatfield blocks 0.03 0.02 10 20 30 40 50 60 Surface area per unit pore volume S por (μm -1 ) Binley et al. (2005)
Permeability prediction using Spectral IP Can we use the spectral properties to estimate permeability? VEC16-1 depth = 17.61 m VEC15-5 depth = 16.07 m VEC7-5 depth = 8.22 m ρ (Ωm) 50 40 30 20 10 0 ρ φ 0.01 0.1 1 10 100 1000 10000 Frequency (Hz) -30-20 -10 φ (mrad) Binley et al. (2005)
Permeability prediction using Spectral IP Cumulative injection volume (ml/g) 0.2 0.16 0.12 0.08 EC16-1 EC15-5 EC7-5 0.04 VEC16-1 depth = 17.61 m VEC15-5 depth = 16.07 m VEC7-5 depth = 8.22 m ρ (Ωm) 0 1000 100 10 1 0.1 0.01 50 40 30 20 10 0 Pore size (μm) 0.01 0.1 1 10 100 1000 10000 ρ φ Frequency (Hz) -30-20 -10 φ (mrad) Binley et al. (2005)
Permeability prediction using Spectral IP The relaxation time is related to the surface area Decreasing τ Relaxation timeτ (s) 1 0.9 0.8 0.7 0.6 0.5 0.4 φ Frequency 0.3 0.2 τ = 7.823 S por -1 r 2 = 0.75 0.1 10 20 30 40 50 60 Surface area per unit pore volume S por (μm -1 ) Binley et al. (2005)
Permeability prediction using Spectral IP The relaxation time is then correlated to the permeability Relaxation timeτ (s) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 τ = 0.37 K 0.26 r 2 = 0.78 0.2 0.1 0.01 0.1 1 10 Vertical hydraulic conductivity K (m d -1 ) Binley et al. (2005)
Permeability prediction using Spectral IP The spectra show sensitivity to saturation and so this must be taken into account in the vadose zone φ (mrad) -15-10 Sample VEC16-1 D 0 = 52.95 μm Legend Graph 1 100% 88% 73% 53% 48% 36% 25% -5 0.01 0.1 1 10 100 1000 Frequency (Hz) Binley et al. (2005)
Unsaturated characteristics using Spectral IP Cumulative injection volume (ml/g) 0.2 0.16 0.12 0.08 0.04 EC16-1 EC15-5 EC7-5 VEC16-1 depth = 17.61 m VEC15-5 depth = 16.07 m VEC7-5 depth = 8.22 m ρ (Ωm) 0 1000 100 10 1 0.1 0.01 50 40 30 20 10 0 Pore size (μm) 0.01 0.1 1 10 100 1000 10000 ρ φ Frequency (Hz) -30-20 -10 φ (mrad) Binley et al. (2005)
Unsaturated characteristics using Spectral IP Can we estimate moisture retention curves using IP spectra? ε G(r) θ(ψ) Frequency R(m) ψ
Summary Empirical and semi-empirical models are available to link hydrological and geophysical properties. Some of these models may be site-specific. We often need to account for sensitivity to various properties (moisture content, pore shape, clay content, etc). Conductivity/resistivity permeability relationships may be limited. IP may help in accounting for surface controlled effects. Spectral IP may offer greater value in constraining hydrological variables.