GLE 594: An introduction to applied geophysics Electrical Resistivity Methods Fall 2004 Earth Properties and Basic Theory Reading Today : 207-218 Next Lecture : 218-228 1
Introduction Link resistivity (ability of the earth to conduct an electric current) to subsurface structure. Useful because resistivity of earth materials varies by around 10 orders of magnitude. Developed by Conrad Schlumberger (France) and Frank Wenner (United States) in early 20th century. Uses: Archeology, Environmental, Mineral exploration Electricity Basics Voltage V - Electrical potential energy per unit charge [volts] Current i - amount of charge per unit time [amperes] i 1 R V Resistivity R is just a proportionality constant [ohms] R relates current I to voltage I. However, no units of length in this form of Ohm s law. 2
Resistivity Resistance includes length and area We want resisitivity ρ [ohm/m] because - property of the material alone. - no geometry included R ρ L A [ ohm] length resistance area resistance Conductivity σ [siemens/m] or [mhos/m]: 1 [ mhos m] σ ρ It is the ability of the electrical charge to move through the material Electrical Conductivity of Geomaterials Non-conductive minerals Electric field Electric field Hydrated ion mobility Double layer surface conduction Electrolyte Soil (Archie s law) Soil (clays) σ σ el soil Equation ms mg [ ] 0.15TDS[ ] m σ soil aσ nσ + el c el Sr n m L ( 1 n) ΘρgSs Comments TDS: total dissolved salts a 1; m~1-2.4; c~4-5 Θ 10-9 S (for Kaolinite) 3
Non-conductive Minerals 4
Analogous to Heat and Fluid Flow Any solutions you know for one of these flows works for the others with the analogous boundary and initial conditions. (Wang and Anderson 1982) Earth as a Circuit Soils and rocks can be conceptually modeled as a circuit made of a resistor, capacitor, inductor and battery: R C L B Electrodes Resistor R: Capacitor C: Inductor L: Battery B: dissipator of applied energy as heat storage of energy as separation of charges self voltage associated to electromagnetic methods electrokinetics and self-potentials 5
Current Source on Surface Electric potential at distance away from current source on surface given as V(r)ρI/2πr. How? Boundary conditions: 1)As r >, V > 0. 2) V is continuous across any boundary 3) Tangential E continuous across any boundary 4) Normal I continuous across any boundary. 5) Above leads to no vertical current crossing earth-air interface. Current Flow in a Homogeneous and Isotropic Medium Point Current Source: dv ir shell l A dr 2πr 2 Voltage decreases as the inverse of the distance from the current source. Shape of constant voltages are hemispheres for a single point source. V D 2π D dv 2π ( 1) 1 D dr 2 r 2π 1 D 2πD ( 1) 1 r D 6
Two Current Electrodes: Source and Sink Why run an electrode to infinity when we can use it? source sink r source P V source r sink 2πr source V sin k 2πr sink Total Voltage at P: V p V source V sin k 1 2π r source 1 r sink Measurement Practicalities Can t measure potential at single point unless the other end of our volt meter is at infinity. This is inconvenient. It is easier to measure potential difference ( V). This lead to use of four electrode array for each measurement. ρ Resulting measurement given as VV P1 -V P2 ρi/(2π)*(1/r 1-1/r 2-1/r 3 +1/r 4 ). Can be rewritten VρI*G/(2π) where G/2π is the Geometrical Factor of the array. 7
Current density and equipotential lines for a current dipole d fraction total current i f 2 2z π tan 1 d i f 0.5 at i f 0.7 at z d 2 z d Wider spacing Deeper currents Apparent Resistivity Previous expression can be rearranged in terms of resistivity: ρ( V/I) (2π/G). ρ 1 This can be done even when medium is inhomogeneous. Result is then referred to as Apparent Resistivity. ρ 2 Definition:Resistivity of a fictitious homogenous subsurface that would yield the same voltages as the earth over which measurements were actually made. 8
Geometrical Factors Array advantages and disadvantages Array Wenner Schlumberger Dipole-Dipole Advantages 1. Easy to calculate ρ a in the field 2. Less demand on instrument sensivity 1. Fewer electrodes to move each sounding 2. Needs shorter potential cables 1. Cables can be shorter for deep soundings Disadvantages 1. All electrodes moved each sounding 2. Sensitive to local shallow variations 3. Long cables for large depths 1. Can be confusing in the field 2. Requires more sensitive equipment 3. Long Current cables 1. Requires large current 2. Requires sensitive instruments 9