Modelling the 3D Electromagnetic Wave Equation: Negative Apparent Conductivities and Phase Changes

Abstract

We often use electromagnetic methods in exploration geophysics to map the resistive structure of the subsurface using instruments that work at low induction numbers. These instruments are usually very portable and versatile, they can be used at the surface, mounted on an airplane or placed inside a wellbore. To process the data acquired with these methods, we need to be able to compute the electric and magnetic fields by numerically solving Maxwell’s equations in the low induction-number domain. Previous studies have used the integral form of Maxwell’s equations, however, these approaches only calculate an approximation of the apparent conductivity. In this chapter, we solved Maxwell’s equations on the frequency domain using the finite-difference method with a staggered grid for the electric field and compute the apparent conductivities with post-processing. We show the results of four different examples, and consider sources on the ground, on the air, and in a wellbore. Negative apparent conductivities and phase changes are observed whenever there is a high conductivity contrasts, the induction-number is low and the source is a vertical magnetic dipole. With these conditions, there is a polarity reversal on the imaginary component of the magnetic field. Furthermore, the results indicate that it is crucial to consider the real component on the measurements, which has previously been ignored.

Appendices

Appendix 1

Equation (10) was discretized using an scheme of central finite differences into a regular grid. The secondary electric field is computed using the following finite difference equations

$$\displaystyle \begin{aligned} &\frac{1}{\Delta x \Delta y}(E^{sy}_{i+1,j+\frac{1}{2},k}-E^{sy}_{i+1,j-\frac{1}{2},k}-E^{sy}_{i,j+\frac{1}{2},k}+E^{sy}_{i,j-\frac{1}{2},k}) \\ &- \frac{1}{\Delta y^{2}} (E^{sx}_{i+\frac{1}{2},j+1,k}-2E^{sx}_{i+\frac{1}{2},j,k}+E^{sx}_{i+\frac{1}{2},j-1,k}) \\ &-\frac{1}{\Delta z^{2}}(E^{sx}_{i+\frac{1}{2},j,k+1}-2E^{sx}_{i+\frac{1}{2},j,k}+E^{sx}_{i+\frac{1}{2},j,k-1}) \\ &+\frac{1}{\Delta x \Delta z}(E^{sz}_{i+1,j,k+\frac{1}{2}}-E^{sz}_{i+1,j,k-\frac{1}{2}}-E^{sz}_{i,j,k+\frac{1}{2}}+E^{sz}_{i,j,k-\frac{1}{2}}) \\ &+i\omega\mu\sigma_{i+\frac{1}{2},j,k} E^{sx}_{i+\frac{1}{2},j,k}=-i\omega\mu(\sigma_{i+\frac{1}{2},j,k}-\sigma^{p}_{i+\frac{1}{2},j,k})E^{px}_{i+\frac{1}{2},j,k},{} \end{aligned} $$

(20)

$$\displaystyle \begin{aligned} &\frac{1}{\Delta y \Delta z}(E^{sz}_{i,j+1,k+\frac{1}{2}}-E^{sz}_{i,j+1,k-\frac{1}{2}}-E^{sz}_{i,j,k+\frac{1}{2}}+E^{sz}_{i,j,k-\frac{1}{2}}) \\ & - \frac{1}{\Delta z^{2}} (E^{sy}_{i,j+\frac{1}{2},k+1}-2E^{sy}_{i,j+\frac{1}{2},k}+E^{sy}_{i,j+\frac{1}{2},k-1}) \\ &-\frac{1}{\Delta x^{2}}(E^{sy}_{i+1,j+\frac{1}{2},k}-2E^{sy}_{i,j+\frac{1}{2},k}+E^{sy}_{i-1,j+\frac{1}{2},k}) \\ &+\frac{1}{\Delta x \Delta y}(E^{sx}_{i+\frac{1}{2},j+1,k}-E^{sx}_{i+\frac{1}{2},j,k}-E^{sx}_{i-\frac{1}{2},j+1,k}+E^{sx}_{i-\frac{1}{2},j,k}) \\ &+i\omega\mu\sigma_{i,j+\frac{1}{2},k} E^{sy}_{i,j+\frac{1}{2},k}=-i\omega\mu(\sigma_{i,j+\frac{1}{2},k}-\sigma^{p}_{i,j+\frac{1}{2},k})E^{py}_{i,j+\frac{1}{2},k}{} \end{aligned} $$

(21)

and

$$\displaystyle \begin{aligned} &\frac{1}{\Delta x \Delta z}(E^{sx}_{i+\frac{1}{2},j,k+1}-E^{sx}_{i+\frac{1}{2},j,k}-E^{sx}_{i-\frac{1}{2},j,k+1}+E^{sx}_{i-\frac{1}{2},j,k}) \\ &-\frac{1}{\Delta x^{2}} (E^{sz}_{i+1,j,k+\frac{1}{2}}-2E^{sz}_{i,j,k+\frac{1}{2}}+E^{sz}_{i-1,j,k+\frac{1}{2}}) \\ &-\frac{1}{\Delta y^{2}}(E^{sz}_{i,j+1,k+\frac{1}{2}}-2E^{sz}_{i,j,k+\frac{1}{2}}+E^{sz}_{i,j-1,k+\frac{1}{2}}) \\ &+\frac{1}{\Delta y \Delta z}(E^{sy}_{i,j+\frac{1}{2},k+1}-E^{sy}_{i,j+\frac{1}{2},k}-E^{sy}_{i,j-\frac{1}{2},k+1}+E^{sy}_{i,j-\frac{1}{2},k}) \\ &+i\omega\mu\sigma_{i,j,k+\frac{1}{2}} E^{sz}_{i,j,k+\frac{1}{2}}=-i\omega\mu(\sigma_{i,j,k+\frac{1}{2}}-\sigma^{p}_{i,j,k+\frac{1}{2}})E^{pz}_{i,j,k+\frac{1}{2}}.{} \end{aligned} $$

(22)

To compute conductivities at the edge of each cell, we use a conductivity average with four adjacent cells (see Fig. 17). In our approach, for a regular grid, the conductivity in $\sigma _{i+\frac {1}{2},j,k}$ is defined by

$$\displaystyle \begin{aligned} \sigma_{i+\frac{1}{2},j,k}=\frac{\sigma_{i,j,k} + \sigma_{i,j-1,k} + \sigma_{i,j,k-1} + \sigma_{i,j-1,k-1}}{4}, \end{aligned} $$

(23)

in $\sigma _{i,j+\frac {1}{2},k}$ we have

$$\displaystyle \begin{aligned} \sigma_{i,j+\frac{1}{2},k}=\frac{\sigma_{i,j,k} + \sigma_{i-1,j,k} + \sigma_{i,j,k-1} + \sigma_{i-1,j,k-1}}{4}, \end{aligned} $$

(24)

Fig. 17

Conductivity cells. (a) Conductivity associated to $E_{i+\frac {1}{2},j,k}^{sx}$, (b) Conductivity associated to $E_{i,j+\frac {1}{2},k}^{sy}$, and (c) Conductivity associated to $E_{i,j,k+\frac {1}{2}}^{sz}$

and $\sigma _{i,j,k+\frac {1}{2}}$ is defined as

$$\displaystyle \begin{aligned} \sigma_{i,j,k+\frac{1}{2}}=\frac{\sigma_{i,j,k} + \sigma_{i-1,j,k} + \sigma_{i,j-1,k} + \sigma_{i-1,j-1,k}}{4}. \end{aligned} $$

(25)

In this approach we have used homogeneous Dirichlet boundary conditions [13, 21], where the electric field is equal to zero at the boundaries (Eq. (10)). Other authors have used Neumann boundary conditions [21], where the first derivative of the tangential electric field is equal to zero at the boundaries, then on the boundaries we could use a backward scheme.

Appendix 2

Faraday’s law in differential form is given by

$$\displaystyle \begin{aligned} H_{sx}=-\frac{1}{i\omega\mu}\left( \frac{\partial E_{sz}}{\partial y}-\frac{\partial E_{sy}}{\partial z}\right), \end{aligned} $$

(26)

$$\displaystyle \begin{aligned} H_{sy}=-\frac{1}{i\omega\mu}\left( \frac{\partial E_{sx}}{\partial z}-\frac{\partial E_{sz}}{\partial x}\right) \end{aligned} $$

(27)

and

$$\displaystyle \begin{aligned} H_{sz}=-\frac{1}{i\omega\mu}\left( \frac{\partial E_{sy}}{\partial x}-\frac{\partial E_{sx}}{\partial y}\right). \end{aligned} $$

(28)

These equations are discretized with a staggered grid central finite differences. The magnetic field is located at face’s cells (see Fig. 1) so, we obtain:

$$\displaystyle \begin{aligned} H^{sx}_{i,j+\frac{1}{2},k+\frac{1}{2}}=-\frac{1}{i\omega\mu}\left(\frac{E^{sz}_{i,j+1,k+\frac{1}{2}}-E^{sz}_{i,j,k+\frac{1}{2}}}{\Delta x}-\frac{E^{sy}_{i,j+\frac{1}{2},k+1}-E^{sy}_{i,j+\frac{1}{2},k}}{\Delta z}\right), \end{aligned} $$

(29)

$$\displaystyle \begin{aligned} H^{sy}_{i+\frac{1}{2},j,k+\frac{1}{2}}=-\frac{1}{i\omega\mu}\left(\frac{E^{sx}_{i+\frac{1}{2},j,k+1}-E^{sx}_{i+\frac{1}{2},j,k}}{\Delta z}-\frac{E^{sz}_{i+1,j,k+\frac{1}{2}}-E^{sz}_{i,j,k+\frac{1}{2}}}{\Delta x}\right) \end{aligned} $$

(30)

and

$$\displaystyle \begin{aligned} H^{sz}_{i+\frac{1}{2},j+\frac{1}{2},k}=-\frac{1}{i\omega\mu}\left(\frac{E^{sy}_{i+1,j+\frac{1}{2},k}-E^{sy}_{i,j+\frac{1}{2},k}}{\Delta x}-\frac{E^{sx}_{i+\frac{1}{2},j+1,k}-E^{sx}_{i+\frac{1}{2},j,k}}{\Delta y}\right). \end{aligned} $$

(31)

With this formulation we obtain a second-order accuracy scheme as a consequence of the centered finite-differences stencil used to compute the magnetic field.