Current Flow - a simple electrical circuit with a battery, ammeter, resistor, and a voltmeter.

Ohm’s Law - electrical current (i) is equal to voltage (V) divided by resistance (R).

Resistivity - a material property which is equal to resistance times area divided by length (ohm-m)

Electrical potential around an isolated electrode in a homogeneous, isotropic earth is radially symmetric. Voltage depends on current and resistance. Current density decreases with depth. Depth of penetration of current increases with distance between current electrodes.

Resistivity is determined from Ohm's law using the potential difference (voltage) between two electrodes for a known current and correcting for the geometrical pathway of current through the earth (i.e., distances between the current and potential electrodes).

Single - current refracts or bends at an interface much like seismic waves. Current flowing from low/high resistivity into high/low resistivity bends away/towards the interface. Current preferentially flows in the least resistive layer.

Depth of current penetration is higher/lower when the lower layer is lower/higher resistivity.

Apparent Resistivity - Resistivity is determined from Ohm's law using the potential difference (voltage) between two electrodes for a known current. At small spacing between electrodes, apparent resistivity is close to the resistivity of the upper layer. With increased space between electrodes more current passes through the lower layer and apparent resistivity changes. Apparent resistivity never reaches the resistivity of the lower layer because some current always travels through the upper layer. Changes in apparent resistivity with electrode spacing depend on depth to the interface and the contrast in resistivity.

Multiple interfaces - If the middle layer has an intermediate resistivity, it may be hard to detect. If the middle layer has a higher or lower resistivity than the other two layers, it can be seen with an expanding spread survey where apparent resistivity will increase and then decrease or vice versa.

Constant Spread Traverse - there are reversals in apparent resistivity as each electrode crosses the contact boundary. Reversals in apparent resistivity are caused by changes in the current distribution as the electrodes are moved. The reversals in apparent resistivity can be used to locate the contact.

Expanding Spread Traverse - there may also be reversals in apparent resistivity as electrodes move across the contact boundary. However, the pattern may not be as obvious because more than one electrode may cross the boundary between measurements.

Dipping contacts also produce apparent resistivity reversals but they are not as pronounced.

Electrode Configurations

Wenner - spacing between electrodes is the same. Easy mathematics. However, more work in the field because all electrodes have to be moved between each measurement. Moreover, it is more likely to be influenced by lateral variations in resistivity.

Schlumberger - spacing between potential electrodes is smaller than current electrodes. Easier to use in the field but spacing between potential electrodes must be increased periodically as the current electrodes are moved farther apart in order to get a potential difference large enough to measure.

Dipole-Dipole - current and potential electrodes are separated from each other by a large distance. Used for very deep (kilometers) profiling.

Electrical Resistivities varies with porosity, pore fluid salinity, and clay content (1 (wet clay) to 10,000 (low porosity igneous/metamorphic rock) Ohm-m).

Curve fitting - Apparent Resistivity versus electrode spacing falls on a series of master curves when they are normalized by resistivity of the upper layer and the depth to the interface. Each master curve has a different contrast in resistivity across the interface. An unknown curve can be overlain and matched to a master curve. The depth index, resistivity index, and k value can be used to determine the resistvities and depth to the interface.

Iterative computer programs - ERSolve takes an initial guess for number of layers, thicknesses, and resistivities, it then iteratively adjusts these values until the computed apparent resistivity curve matches the observed curve.

Equivalence and Suppression - Electrical resistivity analysis is not unique. More than one model may give an acceptable fit to the data (equivalence). It is also possible that some layers that are thin or have a small contrast in resistivity will not be resolved (suppression).