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Charge distribution 2

Basic equations

There are two basic concepts used to derive equations suitable for describing galvanic surveys.

The divergence of current density, J, must equal the negative time-rate-of-change of free charges within the volume. We need consider only one point source of current, I, at location r_s, so at any position r, the following equation holds. By using a delta function, the equation states that everywhere except at the point of the source, the divergence of current density should be zero. So, in the adjacent figure, in which box should divergence NOT equal zero?

The second requirement is a 3D version of Ohm's law, . Sigma is conductivity (the inverse of resistivity) and E is the electric field with units of Volts per metre.

Additional details needed for developing forward modelling equations are given within the tutorial on general inversion methodology. These details are not needed for a basic understanding of resistivity surveying.

Boundary conditions

When current flows from one material across the interface into a second material, two conditions must hold. Recall that the field vectors describing electric field E and current density J can be decomposed into two components, one that is tangent to a surface and another that is perpendicular, or normal, to the surface. The two boundary conditions are:

Tangential components of electric field E must be continuous, and
The normal component of current density J must be continuous.

Consequenses of Boundary Conditions

There are several important effects that result from these two boundary conditions. Three of relevance to understanding of resistivity are:

Since the normal component of J is continuous across a boundary where conductivity changes, the normal component of the E field must be NOT equal. Here is the effect illustrated:

This change in electric field is related to charges building up at an interface between the regions with differing conductivity. In fact, the charge density will be related to the ratio of the two conductivities.

If then a negative charge exists on the boundary within region 1.

. Charges accumulate in this way at boundaries between regions of differing conductivity. Potentials measured at the surface (the data acquired in a resistivity survey) are a direct consequence of this charge distribution that arises within the ground when current is forced to flow using a high power transmitter. In addition, this concept forms the basis of forward modelling equations which are used to calculate the response to a model of the geology that is composed of a great many small cells, each with it's own, constant conductivity. This is the subject of the forward modelling page in the tutorial on general inversion methodology.

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