Lecture 5a - Electrical Resistivity Method - Principles

Electrical Methods for Near Surface Investigations

• Electrical methods are widely used for near-surface investigations (top 50 meters).
• Sensitive to water content and related earth properties.
• Focus on DC resistivity, induced polarization, and self-polarization methods.
• Professor Andy Binley is an expert in applying electrical geophysical techniques to environmental problems, especially groundwater.

Basic Principles of DC Resistivity

• Why make electrical measurements of the subsurface?
  • Understanding current flow through geomaterials provides insights into hydrologically relevant properties.

DC Resistivity Defined

• DC stands for direct current: steady current flowing in one constant direction.
• Not alternating current (AC), which changes magnitude and direction.
• Ohm's Law: $V = IR$, where:
  • $V$ is voltage or potential difference
  • $I$ is current
  • $R$ is resistance
• In resistivity method, current is injected into the ground, and surface potential differences are measured.
• Interested in resistivity (a material property) rather than absolute resistance.

Resistivity vs. Resistance

• Resistance depends on the geometry of the material.
• Resistivity accounts for the area and length of the material.
• Resistivity ($\rho$) is related to conductivity ($\sigma$) by: $\rho = \frac{1}{\sigma}$
• High resistivity = low conductivity and vice versa.

Factors Affecting Bulk Resistivity

• Composition of solid particles
• Amount and connectivity of pore space
• Fluids filling pore spaces
• Salinity of fluids
• Temperature
• Temperature is important because electrical resistivity varys with temperature.

Applications of Resistivity Measurements

• Detect changes in geology or soil types.
• Locate the water table.
• Detect pollutants.

Electrical Conductivity of Geomaterials

• Consider subsurface material as solid, liquid, and gas components.
• Solid component (e.g., silica rocks) has negligible conductivity.
• Current flows mostly through the liquid phase (electrolytic current) via dissolved ions.
• Gas in pore spaces is effectively non-conducting.
• Bulk conductivity is controlled by the liquid in pore spaces.
• Electrical methods are effective for assessing variations in water content.

Formation Factor

• Describes how much conductivity decreases when electrolyte is partially replaced by solid rock material.
• $F = \frac{\sigma<em>w}{\sigma</em>e}$, where:
  • $F$ is the formation factor
  • $\sigma_w$ is the conductivity of the electrolyte fluid
  • $\sigma_e$ is the conductivity of the bulk material (earth)
• Alternatively, in terms of resistivity: $F = \frac{\rho<em>e}{\rho</em>w}$, where:
  • $\rho_e$ is the resistivity of the bulk material (earth)
  • $\rho_w$ is the resistivity of the electrolyte fluid
• Formation factor indicates the number, size, and connectivity of pore spaces.

Archie's Law

• Relates bulk conductivity to fluid conductivity, effective porosity, and cementation.
• $\sigma<em>e = \sigma</em>w \cdot \phi^m$, where:
  • $\phi$ is the effective porosity
  • $m$ is the cementation exponent (tortuosity indicator)
• For unsaturated media, Archie's law includes saturation:
  • $\sigma<em>e = \sigma</em>w \cdot \phi^m \cdot S_w^n$, where:
    • $S_w$ is the degree of saturation (0 to 1)
    • $n$ describes how gas is distributed within the pore space

Surface Conductivity

• Conduction over the surface of solid grains.
• Mineral grains have a negative charge on their surface, attracting positive ions (cations).
• Forms a double layer of charge (fixed and diffuse).
• Creates a surface conduction path.

Combined Conductivity Equation

• $\sigma = \sigma<em>{\text{electrolytic}} + \sigma</em>{\text{surface}}$
• At high fluid conductivity, electrolytic conduction dominates (Archie region).
• At low fluid conductivity, surface conduction dominates.

DC Resistivity Measurement

• Map subsurface variations in resistivity or conductivity.
• Inject current into the ground using electrodes.
• Measure potential differences (voltages) at the surface using other electrodes.
• Calculate resistance from current and voltage measurements.
• Convert resistance to resistivity using electrode geometry.

Four-Electrode Arrangement

• Inject current $I$ through current electrodes C1 and C2.
• Measure potential difference $\Delta V$ through potential electrodes P1 and P2.
• Current flows at 90 degrees to equipotential surfaces.

Potential Differences Around an Electrode

• Homogeneous subsurface, widely spaced current electrodes.
• Current flows radially, equipotential surfaces are hemispherical.
• Voltage drop $V$ related to current, resistivity, and geometric factor:
  $V = \frac{I \rho}{2 \pi r}$
• Rearrange to solve for resistivity: $\rho = \frac{2 \pi r V}{I}$
• Simplified: $\rho = k \frac{V}{I}$, where $k$ is the geometric factor ($2 \pi r$).

Non-Ideal Electrode Arrangement

• Electrodes are not sufficiently separated.
• Equipotential surfaces are distorted.
• Current flow is more focused towards the surface.
• More sensitive to resistivity variations near the surface.

Apparent Resistivity

• For a homogeneous subsurface: the resistivity is constant.
• For a realistic subsurface with resistivity variations: use apparent resistivity.
• Apparent resistivity ($\rho_a$) is the resistivity of a homogeneous subsurface that would give the same voltage and current measurements.
• $\rho_a = k \frac{\Delta V}{I}$
• $k$ is the geometric factor, which depends on electrode positions.

Typical Resistivity Values for Geomaterials

• Loose materials (soils, unconsolidated sediments): low resistivities (high water content, well-connected pore space).
• Solid rocks: larger resistivities.
• Clay: low resistivity (ionically active clay minerals).
• Metallic minerals: low resistivities (good conductors, e.g., metallic ores).

Summary

• Electrical current flow in geomaterials is dominated by electrolytic current pathways.
• Importance of porosity, permeability, and fluid chemistry.
• Surface conduction is significant when electrolytic conductivity is very low.
• The resistivity method injects current and measures potential differences to calculate apparent resistivity.
• Apparent resistivity is a broad average over the volume of investigation.
• Conversion from apparent resistivity to actual subsurface resistivity models is covered in the next lecture.