Steady State Equations

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)\[\boldsymbol{\nabla} \times \mathbf{h} - \sigma\mathbf{e} = \mathbf{j}_{source}.\]

Faraday’s law is also simplified in steady state. It becomes

\[\boldsymbol{\nabla \times} \mathbf{e} = \mathbf{0}.\]

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)\[\boldsymbol{\nabla} \cdot \sigma\boldsymbol{\nabla}\phi = \boldsymbol{\nabla}\cdot\mathbf{j}_{source}.\]