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 rs, 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.