Three-Dimensional DC Anisotropic Resistivity Modeling Using a Method in the Mixed Space–Wavenumber Domain

Abstract

A newly developed method in the mixed space–wavenumber domain is proposed for 3D DC resistivity modeling in anisotropic structures. Unlike the traditional method for numerical simulation of DC resistivity, this method performs a 2D Fourier transform directly on the partial differential equation of DC abnormal potential along the horizontal direction, so that the horizontal direction is converted into the wavenumber domain while the vertical direction remains in the spatial domain. In this way, the 3D partial differential equation in the spatial domain is transformed into a 1D ordinary differential equation in different wave numbers. Therefore, a large-scale 3D numerical simulation problem is decomposed into many 1D numerical simulation problems. The 1D finite element method is used to solve the equations, and a contraction operator is used for iterative calculation to obtain a highly accurate solution. The accuracy of the algorithm is verified by comparison with the goal-oriented adaptive finite element method. Compared with the COMSOL Multiphysics commercial finite element simulation software, the algorithm in this paper has high computational efficiency. To identify the electrical anisotropy, we propose a new measurement and then use it to study the response characteristics of a half-space model with two prisms. The test results show that this measurement can accurately reflect the underground electrical anisotropy. Finally, the code is applied to the numerical simulation of a vertical-borehole survey and shows that the significant information on the hydraulically induced fracturing system can be reflected well by the electric anisotropy.

Appendices

Appendix 1: Electric Field in Homogeneous Anisotropic Half-Space

According to Li and Uren (1997), in homogeneous anisotropic media, the coordinates of the source are ${\mathbf{r}}_{0} = \left( {x_{0},y_{0},z_{0} } \right)$, and the expression of the coordinate of its mirror source ${\mathbf{r^{\prime}}}_{0} = \left( {x^{\prime}_{0},y^{\prime}_{0},z^{\prime}_{0} } \right)$ is

$$x^{\prime}_{0} = 2z_{0} s_{1} + x_{0},$$

(34)

$$y^{\prime}_{0} = 2z_{0} s_{2} + y_{0},$$

(35)

$$z^{\prime}_{0} = - z_{0},$$

(36)

where

$$s_{1} = \frac{{\rho_{yy} \rho_{xz} - \rho_{xy} \rho_{yz} }}{{\rho_{yy} \rho_{xx} - \rho_{xy} \rho_{xy} }},$$

(37)

$$s_{2} = \frac{{\rho_{xx} \rho_{yz} - \rho_{xy} \rho_{xz} }}{{\rho_{yy} \rho_{xx} - \rho_{xy} \rho_{xy} }},$$

(38)

Then, the electric field at the transceiver distance r is

$$\begin{gathered} E_{x} = \frac{I}{{8\pi \sqrt {\left| {{\varvec{\upsigma}}} \right|} {\mathbf{B}}^{\frac{3}{2}} }}\left[ {2\rho_{xx} \left( {x - x_{0} } \right) + 2\rho_{xy} \left( {y - y_{0} } \right) + 2\rho_{xz} \left( {z - z_{0} } \right)} \right] \\ + \frac{I}{{8\pi \sqrt {\left| {{\varvec{\upsigma}}} \right|} {\mathbf{B^{\prime}}}^{\frac{3}{2}} }}\left[ {2\rho_{xx} \left( {x - x^{\prime}_{0} } \right) + 2\rho_{xy} \left( {y - y^{\prime}_{0} } \right) + 2\rho_{xz} \left( {z - z^{\prime}_{0} } \right)} \right], \\ \end{gathered}$$

(39)

$$\begin{gathered} E_{y} = \frac{I}{{8\pi \sqrt {\left| {{\varvec{\upsigma}}} \right|} {\mathbf{B}}^{\frac{3}{2}} }}\left[ {2\rho_{yx} \left( {x - x_{0} } \right) + 2\rho_{yy} \left( {y - y_{0} } \right) + 2\rho_{yz} \left( {z - z_{0} } \right)} \right] \\ + \frac{I}{{8\pi \sqrt {\left| {{\varvec{\upsigma}}} \right|} {\mathbf{B^{\prime}}}^{\frac{3}{2}} }}\left[ {2\rho_{yx} \left( {x - x^{\prime}_{0} } \right) + 2\rho_{yy} \left( {y - y^{\prime}_{0} } \right) + 2\rho_{yz} \left( {z - z^{\prime}_{0} } \right)} \right], \\ \end{gathered}$$

(40)

$$\begin{gathered} E_{z} = \frac{I}{{8\pi \sqrt {\left| {{\varvec{\upsigma}}} \right|} {\mathbf{B}}^{\frac{3}{2}} }}\left[ {2\rho_{zx} \left( {x - x_{0} } \right) + 2\rho_{zy} \left( {y - y_{0} } \right) + 2\rho_{zz} \left( {z - z_{0} } \right)} \right] \\ + \frac{I}{{8\pi \sqrt {\left| {{\varvec{\upsigma}}} \right|} {\mathbf{B^{\prime}}}^{\frac{3}{2}} }}\left[ {2\rho_{zx} \left( {x - x^{\prime}_{0} } \right) + 2\rho_{zy} \left( {y - y^{\prime}_{0} } \right) + 2\rho_{zz} \left( {z - z^{\prime}_{0} } \right)} \right], \\ \end{gathered}$$

(41)

where ${\mathbf{B}} = {\mathbf{r}}_{0}^{T} \cdot {{\varvec{\uprho}}} \cdot {\mathbf{r}}_{0}$, ${\mathbf{B^{\prime}}} = {\mathbf{r^{\prime}}}_{0}^{T} \cdot {{\varvec{\uprho}}} \cdot {\mathbf{r^{\prime}}}_{0}$,${{\varvec{\upsigma}}} = {{\varvec{\uprho}}}^{ - 1}$.

Appendix 2: The Matrix Coefficient of the Equation

$${\mathbf{K}}_{1e} = \frac{{\sigma^{p} }}{3l}\left( {\begin{array}{*{20}l} 7 &\quad { - 8} &\quad 1 \\ { - 8} &\quad {16} &\quad { - 8} \\ 1 &\quad { - 8} &\quad 7 \\ \end{array} } \right),$$

(42)

$${\mathbf{K}}_{{{\text{2e}}}} = \frac{{l\sigma^{p} \left( {k_{x}^{2} + k_{y}^{2} } \right)}}{30}\left( {\begin{array}{*{20}l} 4 &\quad 2 &\quad { - 1} \\ 2 &\quad {16} &\quad 2 \\ { - 1} &\quad 2 &\quad 4 \\ \end{array} } \right),$$

(43)

$${\mathbf{P}}_{1e} = \frac{{ik_{x} l}}{30}\left( {\begin{array}{*{20}l} 4 &\quad 2 &\quad { - 1} \\ 2 &\quad {16} &\quad 2 \\ { - 1} &\quad 2 &\quad 4 \\ \end{array} } \right),$$

(44)

$${\mathbf{P}}_{2e} = \frac{{ik_{y} l}}{30}\left( {\begin{array}{*{20}l} 4 &\quad 2 &\quad { - 1} \\ 2 &\quad {16} &\quad 2 \\ { - 1} &\quad 2 &\quad 4 \\ \end{array} } \right),$$

(45)

$${\mathbf{P}}_{3e} = \frac{1}{6}\left( {\begin{array}{*{20}l} { - 3} &\quad 4 &\quad { - 1} \\ { - 4} &\quad 0 &\quad 4 \\ 1 & \quad { - 4} &\quad 3 \\ \end{array} } \right),$$

(46)

$${\mathbf{B}} = \left( {\begin{array}{*{20}l} 0 & \quad {\ldots} &\quad 0 \\ {\ldots} &\quad {\ldots} &\quad {\ldots} \\ 0 &\quad {\ldots} &\quad { - k_{2} \sigma_{zz}^{b} } \\ \end{array} } \right),$$

(47)

where l is the unit length.