Midterm results Average mark 73.7% (29.5 / 40) Median mark 30 / 40

Electrical Methods Resistivity is a rock property – quantifies resistance to current flow Range of rock resistivities is very large Widely used in mineral exploration, and in geoenvironmental applications Widely used in boreholes in the oil and gas industry

Resistivity method Is an example of a “controlled source” method Requires direct electrical contact with the ground (electrodes, spools of wire, …) Labour intensive, but better resolution than gravity or magnetics

Resistivity method

Resistance and resistivity Resistivity: An inherent rock property, controls resistance to current flow Resistance: Simply a ratio between voltage drop and current flow in an electric circuit (i.e., across a rock sample): Resistance depends on geometrical factors, as well as resistivity: Resistance: When we pass a current, I through a sample or through the ground, and measure the resultant voltage drop V , we are actually measuring the total \resistance", R of the rock(s). Resistance is simply obtained from such an experiment using the relation (4.1) Resistivity: The total resistance, R in a given experiment is a function of i) geometrical factors and ii) the \resistivity", of the target. The former is only related to the experimental conguration, while the latter is an inherent property of the target (i.e., of the rock). For example, if we use a cylindrical core sample as the target and pass the current along the length of the sample, the total resistance is proportional to the length, l and inversely proportional to the cross-sectional area, A so that (4.2) where is the internal resistivity of the sample. Thus we may measure the resistivity using (4.3) provided we know the geometrical quantity (A/l). The resistivity may also be dened as the resistance between opposite faces of a unit cube of the target material. Solving for resistivity:

Resistivity and conductivity Resistivity: An inherent rock property, controls resistance to current flow – measures resistance for a unit cube. Conductivity: The inverse of resistivity (i.e., the ease with which current flows) Resistance: When we pass a current, I through a sample or through the ground, and measure the resultant voltage drop V , we are actually measuring the total \resistance", R of the rock(s). Resistance is simply obtained from such an experiment using the relation (4.1) Resistivity: The total resistance, R in a given experiment is a function of i) geometrical factors and ii) the \resistivity", of the target. The former is only related to the experimental conguration, while the latter is an inherent property of the target (i.e., of the rock). For example, if we use a cylindrical core sample as the target and pass the current along the length of the sample, the total resistance is proportional to the length, l and inversely proportional to the cross-sectional area, A so that (4.2) where is the internal resistivity of the sample. Thus we may measure the resistivity using (4.3) provided we know the geometrical quantity (A/l). The resistivity may also be dened as the resistance between opposite faces of a unit cube of the target material.

Ohm’s law The current, I distributes itself across the cross-sectional area, A of the sample. We define the current density, Ohm’s law states , thus Resistance: When we pass a current, I through a sample or through the ground, and measure the resultant voltage drop V , we are actually measuring the total \resistance", R of the rock(s). Resistance is simply obtained from such an experiment using the relation (4.1) Resistivity: The total resistance, R in a given experiment is a function of i) geometrical factors and ii) the \resistivity", of the target. The former is only related to the experimental conguration, while the latter is an inherent property of the target (i.e., of the rock). For example, if we use a cylindrical core sample as the target and pass the current along the length of the sample, the total resistance is proportional to the length, l and inversely proportional to the cross-sectional area, A so that (4.2) where is the internal resistivity of the sample. Thus we may measure the resistivity using (4.3) provided we know the geometrical quantity (A/l). The resistivity may also be dened as the resistance between opposite faces of a unit cube of the target material.

Ohm’s law in “continuous media” Ohm’s law states From calculus: In 3-D:

Injecting current at a single electrode Earlier, for a sample: Geometrical term How do we do this from the surface, for ground measurements? Start with a single current electrode …

Injecting current at a single electrode Current, I is distributed over a half-sphere, hence Using Ohm’s law or

Injecting current at a single electrode Integrating Solving for resistivity: Geometrical term

Current flow in the ground

Field electrode arrays Apply the single electrode formula for each of the four electrode combinations:

Geometrical term

Field electrode arrays The use of the Wenner array implies that everytime we wish to either move to a new position laterally, or change the array spacing, a, we would have to move all four electrodes. Since moving electrodes in the eld takes time, other arrays are used for specic applications. Geometrical term

Field electrode arrays Geometrical term (for the gradient array, we usually require L – x > 3l ) The gradient array is a conguration in which a large current electrode spacing is used (in Figure 4.5b) we usually require L The Gradient array is often used for “lateral profiling” (see later) In the next array, the Schlumberger array we use x = 0

Field electrode arrays Geometrical term The Schlumberger array is a variation on the gradient array, in which the potential electrodes remain at the centre of the spread, and the current electrodes spacing is varied. For the Schlumberger array we may use equation (4.26) with x = 0, so that the apparent resistivity is given by The Schlumberger array is commonly used for “depth sounding” (see later)

Field electrode arrays Geometrical term For surveys in which both the lateral position of the array and the array spacing are varied, the dipole-dipole array is common, for which The dipole-dipole array is commonly when both the lateral position, and spacing are varied (see later)

Next lecture: Types of resistivity surveys There are several variations on resistivity surveys: A “lateral profile” aims to locate anomalies, along a line or on a map A “depth profile” aims to construct a vertical profile of subsurface resistivities and depths A combination of the above, which aims to construct a “pseudo-section” or a “real section”