Microseismic Full Waveform Modeling in Anisotropic Media with Moment Tensor Implementation

Abstract

Seismic anisotropy which is common in shale and fractured rocks will cause travel-time and amplitude discrepancy in different propagation directions. For microseismic monitoring which is often implemented in shale or fractured rocks, seismic anisotropy needs to be carefully accounted for in source location and mechanism determination. We have developed an efficient finite-difference full waveform modeling tool with an arbitrary moment tensor source. The modeling tool is suitable for simulating wave propagation in anisotropic media for microseismic monitoring. As both dislocation and non-double-couple source are often observed in microseismic monitoring, an arbitrary moment tensor source is implemented in our forward modeling tool. The increments of shear stress are equally distributed on the staggered grid to implement an accurate and symmetric moment tensor source. Our modeling tool provides an efficient way to obtain the Green’s function in anisotropic media, which is the key of anisotropic moment tensor inversion and source mechanism characterization in microseismic monitoring. In our research, wavefields in anisotropic media have been carefully simulated and analyzed in both surface array and downhole array. The variation characteristics of travel-time and amplitude of direct P- and S-wave in vertical transverse isotropic media and horizontal transverse isotropic media are distinct, thus providing a feasible way to distinguish and identify the anisotropic type of the subsurface. Analyzing the travel-times and amplitudes of the microseismic data is a feasible way to estimate the orientation and density of the induced cracks in hydraulic fracturing. Our anisotropic modeling tool can be used to generate and analyze microseismic full wavefield with full moment tensor source in anisotropic media, which can help promote the anisotropic interpretation and inversion of field data.

Appendices

Appendix 1: Moment Tensor Source Radiation Pattern

A seismic moment tensor is the combination of nine generalized couple forces which have three possible directions and act on three possible arms. It can be used to simulate seismic sources which have body-force equivalent given by pairs of forces. The seismic moment tensor source equivalent has been verified by the radiation patterns of teleseismic data and also seismic data obtained very close to the source region (Aki and Richards 2002). A common seismic moment tensor can be expressed as

$$\begin{aligned} \mathbf {m}= \begin{pmatrix} m_{xx} & m_{xy} & m_{xz}\\ m_{yx} & m_{yy} & m_{yz}\\ m_{zx} & m_{zy} & m_{zz} \end{pmatrix} . \end{aligned}$$

(9)

 The source radiation pattern of P- and S-waves can be derived from the Green’s function in an isotropic elastic medium (Aki and Richards 2002). For far-field P-waves, the radiation pattern is given by

$$\begin{aligned} R_n^{\rm p}=\gamma _n\gamma _{p}\gamma _{q}m_{pq} . \end{aligned}$$

(10)

 For far-field S-waves, the radiation pattern is given by

$$\begin{aligned} R_n^{\rm s}=-\,(\gamma _n\gamma _{p}-\delta _{np})\gamma _{q}m_{pq}. \end{aligned}$$

(11)

 In these equations, $R_n$ represents the nth component of the radiation pattern vector for P- or S-wave, $\gamma _p$ is the direction cosine of the source-receiver unit direction vector, $m_{pq}$ is the moment tensor component. Implicit summation over the repeated index is applied in these equations.

If using the unit basis vectors in spherical coordinates, then we can further obtain the radiation pattern for P-waves (Chapman 2004)

$$\begin{aligned} R^{\rm p}&= \left( m_{xx}\cos ^2\phi +m_{yy}\sin ^2\phi +m_{xy}\sin 2\phi \right) \sin ^2\theta \\&\quad +\,m_{zz}\cos ^2\theta +\left( m_{zx}\cos \phi +m_{yz}\sin \phi \right) \sin 2\theta , \end{aligned}$$

(12)

for SV-waves

$$\begin{aligned} R^{\rm sv}&= \frac{1}{2}\left( m_{xx}\cos ^2\phi +m_{yy}\sin ^2\phi -m_{zz}+m_{xy}\sin 2\phi \right) \sin 2\theta \\&\quad +\,\left( m_{zx}\cos \phi +m_{yz}\sin \phi \right) \cos 2\theta , \end{aligned}$$

(13)

for SH-waves

$$\begin{aligned} R^{\rm sh}= \left( \frac{1}{2}\left( m_{yy}-m_{xx}\right) \sin 2\phi +m_{xy}\cos 2\phi \right) \sin \theta +\left( m_{yz}\cos \phi -m_{zx}\sin \phi \right) \cos \theta , \end{aligned}$$

(14)

in which $\theta$ and $\phi$ represent the coordinate components (polar angle and azimuth angle) in the spherical coordinates, respectively.

Appendix 2: Analytical Solutions in Homogeneous Isotropic Medium

The displacement field in a homogeneous isotropic medium can be obtained by convolving the Green’s function with the seismic moment tensor (Aki and Richards 2002, Eq. 4.29)

$$\begin{aligned} u_n&=M_{pq}*G_{np,q} = R_{n}^{ne}\frac{M_0}{4\pi \rho r^4}\int _{r/v_{\rm p}}^{r/v_{\rm s}}\tau S(t-\tau )d\tau +R_n^{ip}\frac{M_0}{4\pi \rho v_{\rm p}^2r^2}S\left( t-r/v_{\rm p}\right) \\&\quad +R_n^{is}\frac{M_0}{4\pi \rho v_{\rm s}^2r^2}S\left( t-r/v_{\rm s}\right) +R_n^{fp}\frac{M_0}{4\pi \rho v_{\rm p}^3r}\dot{S}\left( t-r/v_{\rm p}\right) +R_n^{fs}\frac{M_0}{4\pi \rho v_{\rm s}^3r}\dot{S}\left( t-r/v_{\rm s}\right) , \end{aligned}$$

(15)

where $u_n$ is the nth component of displacement field, r is the distance between source point and receiver point, $G_{np,q}$ is the Green’s function describing the wave propagation between source and receiver, $R_{n}^{ne}$, $R_n^{ip}$, $R_n^{is}$, $R_n^{fp}$, $R_n^{fs}$ are near-field, intermediate-field P-wave, intermediate-field S-wave, far-field P-wave, far-field S-wave radiation pattern, respectively. The comma indicates the spatial derivative with respect to the coordinate after the comma (e.g., $G_{np,q}=\partial G_{np}/\partial q$), and the dot above the source time function S(t) indicates the time derivative. Thus, the displacement field in the far-field is proportional to particle velocities at the source. The elastic properties of the medium are described by density $\rho$, P-wave velocity $v_{\rm p}$ and S-wave velocity $v_{\rm s}$.

The first term in Eq. 15 is called the near-field term, which is proportional to $r^{-4}\int _{r/v_{\rm p}}^{r/v_{\rm s}}\tau S(t-\tau )d\tau$ (hereafter referred to as the proportional part of near-field term). The two middle terms are called the intermediate-field terms, which are proportional to $(vr)^{-2}S(t-r/v)$. The last two terms are called the far-field terms, which are proportional to $v^{-3}r^{-1}\dot{S}(t-r/v)$. Since there is no intermediate-field region where only the intermediate-field terms dominate, it is common to combine the intermediate-field and near-field terms. If a Ricker wavelet $S(t)=(1-2\pi ^2f_m^2t^2)e^{-\pi ^2f_m^2t^2}$ ($f_m$ is the peak frequency of the wavelet) is used as the source time function, the integration in the near-field term is very small and its peak amplitude is approximately proportional to $r/f_m$ with ratio often smaller than $10^{-6}$ in SI units. The derivative term of the source time function in the far-field terms is much larger than the Ricker wavelet and its integration, and its peak amplitude is approximately proportional to $f_m$ with an approximate ratio of 6.135 for Ricker wavelet.

Appendix 3: Distortion of Near- and Far-Field Due to Source Radiation Pattern

Normally, the near- and far-field are just defined using source-receiver distance and seismic wavelength. However, through examining Eq. 15 and numerical experiments, we find that the near- and far-field are also influenced by source radiation patters. Figure 29a shows the relative magnitude of peak amplitude of the proportional part of the near-field term, intermediate-field terms and far-field terms at different source-receiver distances. The elastic parameters of the medium used are $v_{\rm p}=3500$ m/s, $v_{\rm s}=2000$ m/s and $\rho =2400$ kg/m$^3$. The source time function is a Ricker wavelet with a peak frequency of 40 Hz and a time delay of $1.1/f_m$. The X-axis of Fig. 29a is the ratio of the source-receiver distance to the dominant S-wave wavelength. It is obvious that at a distance larger than three or four dominant S-wave wavelengths, the far-field term dominates the wavefield (with a proportion higher than 95%). This far-field approximation is quite pervasive in microseismic monitoring because of the widely used ray-based methods and relatively high dominant frequencies of the recorded data. Furthermore, most focal mechanism inversion methods are also based on the far-field approximation. However, at a distance less than two dominant S-wave wavelengths, the near-field terms and intermediate-field terms will have a non-negligible effect on the whole wavefield and may even dominate the wavefield, especially when very close to the source region (less than one half the dominant S-wave wavelength). For microseismic downhole monitoring arrays, which are deployed close to the microseismic source area, larger errors may occur due to the significant contribution of the near-field and intermediate-field terms.

Fig. 29

a Relative magnitude of peak amplitude of the proportional part for near-field term, intermediate-field terms and far-field terms under certain parameters. b 3D map which shows the far-field distance in terms of S-wave wavelength in different directions for a $45^{\circ }$ dip-slip double-couple source. Beyond this far-field distance, the far-field terms will occupy more than $80\%$ energy in the whole wavefield. c Relative magnitude of wavefields for near-field term, intermediate-field S-wave term and far-field S-wave term for a double-couple source in different directions. The solid lines show the scenario in direction which has a zenith angle of $45^\circ$ and azimuth angle of $0^\circ$. The dashed lines show the scenario in direction which has a zenith angle of $5^\circ$ and azimuth angle of $0^\circ$

The far-field approximation is not only related to the source-receiver distance but also the radiation patterns of the near-field terms (including intermediate-terms hereafter) and far-field terms. In directions where the strength of the far-field radiation pattern is weaker than the strength of the near-field radiation pattern, the contribution of near-field terms may bias the far-field approximation in the “far” field. Figure 29b is a 3D map which shows the far-field distance of a $45^{\circ }$ dip-slip double-couple source ($m_{xx}=-m_{zz}$ and other components are 0) in different directions. The elastic property of the medium is the same as before. The far-field distance is expressed in terms of S-wave wavelength. The color and shape in the figure shows the distance where the far-field terms will occupy $80\%$ energy in the whole wavefield. Beyond this distance, we can consider that the far-field terms dominate the wavefield. Figure 29b reveals an obvious directional feature. If there were no difference in radiation pattern between the far-field and near-field terms, Fig. 29b would show an uniform spherical distribution in different directions. However, the difference in radiation patterns has distorted the scope where the near-field could exert influence on the wavefield. In directions where the near-field radiation pattern is strong and the far-field radiation is weak, the distance in which the near-field terms have a non-negligible influence on the whole wavefield has been extended. The far-field distance in different directions in Fig. 29b ranges from about 2 times the dominant S-wave wavelength to 12 times the dominant S-wave wavelength. Thus, great care must be taken when receivers have been deployed in these directions. Figure 29c shows the variation of relative magnitude in two specific directions for the same double-couple source. The radiation patterns of the near-, intermediate- and far-field terms have been taken into consideration. When considering the source radiation pattern, the far-field distance shows strong dependence on directions. The far-field distance has been extended to 12 times the dominant S-wave wavelength in direction of $5^\circ$ zenith angle and $0^\circ$ azimuth angle (shown as the dashed lines). The far-field terms need a farther distance to dominate in the whole wavefield. This example demonstrates that the far-field distance is changed and is also affected by source radiation patterns. For microseismic monitoring, the receivers are normally deployed near microseismic events, especially for the downhole array. Therefore, the influence of source radiation patterns to far-field distance must be taken into consideration. When source–receiver geometry, source moment tensor and media elastic parameters are defined, the far-field distance in different directions where the far-field approximation is acceptable can be quantitatively evaluated. This will be very helpful for array deployment and data interpretation in microseismic monitoring.