The telluric sounding (TS) method was introduced in the 1960s and involves simultaneously recording the horizontal components of electric fields (E) at different sites (Berdičevskij and Keller 1965; Yungul 1966). The measured E can be affected by galvanic distortion, which is caused by gradients in electrical conductivity associated with near-surface heterogeneities (Chave and Smith 1994). The accumulation of charge at conductivity boundaries strongly alters E (Groom and Bailey 1989). This local distortion of E can be described by a real-valued second-rank tensor D that relates the electric field measured at a local (E\(_M\)) and regional (E\(_B\)) site according to (Chave and Jones 2012)
$$\begin{aligned} {\mathbf{E}}_M = {\mathbf{D }}{{\mathbf{E }}}_B \end{aligned}$$
(1)
where
$$\begin{aligned} {\mathbf{D }} = \begin{bmatrix} D_{xx}&D_{xy} \\ D_{yx}&D_{yy} \end{bmatrix}. \end{aligned}$$
(2)
In this case, D is relative to axes X and Y, typically north and east. Introducing a rotation matrix R(\(\theta\)),
$$\begin{aligned} {\mathbf{R }}(\theta ) = \begin{bmatrix}\cos \theta&\sin \theta \\ -\,\sin \theta&\cos \theta \end{bmatrix}, \end{aligned}$$
(3)
and the transpose \({\mathbf {R^T}}\)(\(\theta\)), the value of D relative to axes \(X^\prime\) and \(Y^\prime\) (which are rotated \(\theta ^\prime\) clockwise from north and east) is given by the matrix D\(^\prime\) (Lilley 2015)
$$\begin{aligned} \begin{bmatrix}D'_{xx}&D'_{xy} \\D'_{yx}&D'_{yy} \end{bmatrix} = {\mathbf{R }}(-\theta ')\begin{bmatrix}D_{xx}&D_{xy} \\D_{yx}&D_{yy} \end{bmatrix}{\mathbf{R }}(\theta '). \end{aligned}$$
(4)
For a 1D Earth, the distortion matrix will be the identity matrix. The presence of galvanic distortion will be manifest as an amplitude shift, and a twist and shear operation (Lilley 2015). For 2D structures, the distortion matrix can be rotated to reflect the changes in E along and across strike.
Targets for hydraulic stimulation are generally in sedimentary basins that are laterally extensive, and for most of the bandwidth the responses are 1D. The injection of conductive fluids into the subsurface will alter the telluric distortion matrix which can be mapped to show the lateral constraints of fluid migration. Lilley (2015) proposed the application of both eigenvalue analysis (EA) and singular value decomposition (SVD) on the telluric distortion matrix, with Mohr diagrams introduced as a versatile way of visualising properties of the matrix. These diagrams can be used to determine the extent to which the matrix is diagnostic of 1D, 2D or 3D geological structure as well as determining a strike direction (with \(90^\circ\) ambiguity) and a relative amplitude change. Grids of Mohr circles can show where the greater amplitude changes occur at depth and may be used to determine the extent to which fluid has migrated from the injection point.
Following from Lilley (2015), EA of the telluric distortion matrix involves finding a direction of E\(_M\) for which the change in E\(_B\) is in the same direction, with the eigenvalue of the direction giving the gain of the process. When real eigenvectors exist, the characteristic equation for D,
$$\begin{aligned} \zeta ^2 - (D_{xx}+D_{yy})\zeta + D_{xx}D_{yy} - D_{xy}D_{yx} = 0, \end{aligned}$$
(5)
can be solved for the two eigenvalues \(\zeta _{1}\) and \(\zeta _{2}\) with solutions
$$\zeta _{1},\zeta _{2} = \frac{1}{2}(D_{xx}+D_{yy}) \pm \frac{1}{2}\sqrt{(D_{xx}+D_{yy})^2 + 4(D_{xy}D_{yx}-D_{xx}D_{yy})}.$$
(6)
Eigenvectors can then be found corresponding to these eigenvalues. The solutions to EA can be plotted on a Mohr circle with centre \([(D_{xx} + D_{yy})/2, (D_{xy} - D_{yx})/2]\) and radius \(r = \sqrt{(D_{xy} + D_{yx})^2 + (D_{xx} - D_{yy})^2}\). Axes for \(D^{\prime }_{yx}\) and \(D^{\prime }_{yy}\) can also be plotted to display the variation with axis rotation of all components of D\(^\prime\). Figure 1a shows an example of such a Mohr diagram, where P represents the observed point, H and J mark the EA positions and C represents the centre of the circle. The eigenvalues are the \(D^{\prime }_{xx}\) axis values of H and J.
The SVD of D produces a rotation for the axes at the regional site B and a different rotation for the axes at local site M. Therefore, a change in E at rotated site B produces a change in E along the corresponding but differently rotated axis at site M, with the change generally amplified or attenuated (Lilley 2015). This rotation of the local and regional axes reduces the telluric distortion tensor to an ideal 2D form. The results of SVD can also be displayed on a Mohr diagram as shown in Fig. 1b. Here, OG and OF represent the greater and lesser singular values, P is the observed point and C is the centre. Grids of SVD Mohr circles can show the magnitudes and directions of the injected near surface anomaly. For a detailed explanation on the theory of EA, SVD and Mohr diagrams related to the telluric distortion matrix, the reader is referred to Lilley (1993, 2012, 2015).
The goal of this study is to test the viability of utilising the TS method for monitoring hydraulic stimulation at depth. Ideally we would like to be able to constrain the spatial and temporal dimensions of resistivity changes. Spatial changes can be constrained laterally from using multiple sites, and depths can be estimated given resistivity data to map periods to depths. One advantage of the TS method is that it is relatively easy to measure E with many dipoles and multi-channel systems and therefore E arrays could be deployed for continuous monitoring. Additionally, hydraulic stimulation targets are generally laterally extensive sedimentary basins where E transfer functions are essentially the identity matrix. Therefore, monitoring would involve plotting deviations relative to the identity matrix, with static shift appearing as a galvanic multiplier at all periods. The impedance on the other hand has real and imaginary components that vary with frequency. Finally, the TS method is relatively low cost when compared with traditional magnetotelluric measurements and may prove a simple and favourable method for monitoring fluid movement.