## Direct Current Resistivity

A DC resistivity survey is ultimately an electromagnetic phenomenon and is therefore governed by Maxwell’s equations. However, the fact that the ground is energized with a time invariant direct current allows us to use a much simpler model.

### Deriving the DCR Equations

We can start from time domain differential form of the Ampere-Maxwell equation equation (equation (5) on Ampere-Maxwell)

(326)$\boldsymbol{\nabla} \times \mathbf{h} = \mathbf{j}_{total} + \frac{\partial \mathbf{d}}{\partial t},$

where $$\mathbf{h}$$ is the magnetic field, $$\mathbf{j}_{total}$$ is the total current in the system, and $$\mathbf{d}$$ is the electric displacement. We can divide up $$\mathbf{j}_{total}$$ as follows

(327)$\mathbf{j}_{total} = \sigma\mathbf{e} + \mathbf{j}_{source},$

which states that the total current densisty can be divided into the current in the ground ($$\sigma\mathbf{e}$$ due to Ohm’s law) and the current in the source wires ($$\mathbf{j}_{source}$$). Since we are in the steady state case, $$\frac{\partial \mathbf{d}}{\partial t}=0$$. Using that assumption and substituting (327) into (326) we obtain

(328)$\boldsymbol{\nabla} \times \mathbf{h} - \sigma\mathbf{e} = \mathbf{j}_{source}.$

$\boldsymbol{\nabla \times} \mathbf{e} = \mathbf{0}.$

In other words it is a potential field. In particular, this means that

(329)$\mathbf{e} = -\boldsymbol{\nabla}\phi,$

where $$\phi$$ is the electric potential. This allows us to write (328) as

(330)$\boldsymbol{\nabla} \times \mathbf{h} + \sigma\boldsymbol{\nabla}\phi = \mathbf{j}_{source}.$

Taking the divergence of both sides of (330) gives the governing equation for DC resistivity

(331)$\boldsymbol{\nabla} \cdot \sigma\boldsymbol{\nabla}\phi = \boldsymbol{\nabla}\cdot\mathbf{j}_{source}.$