Effective Media for Transversely Isotropic Models Based on Homogenization Theory: With Applications to Borehole Sonic Logs

Abstract

Sonic log records, including measurements of wave speeds in boreholes, provide critical input to the geological, geophysical, and petrophysical studies of a region under exploration. 1D background models are routinely built based on sonic log records for applications such as seismic imaging of hydrocarbon reservoirs and microseismic source inversions. Smoothing or ‘upscaling’ techniques are required to produce models in coarser scales than the very fine layers in the raw log data. In this paper, we follow the recently popular homogenization theory, derive its application to the special case of 1D TI models for both P-SV and SH waves, and show that it is consistent with the Backus averaging technique commonly used to upscale 1D fine-layered models. We examine a study case of sonic log data from a well in the Horn River Basin in northeastern British Columbia, a region known for its tight shale-gas deposit. We demonstrate the computational accuracy and efficiency gained by proper upscaling procedures for spectral-element simulations of seismic wave propagation, and discuss the effect of control parameters on wavefield recovery.

Appendices

Appendix A: Kelvin notation for second-order and fourth-order tensors

Helbig (1994) demonstrated one of the canonical ways of transforming a second-order symmetric 3D tensor (e.g., stress and strain) to a 6-D vector, and a fourth-order symmetric 3D tensor (e.g., elastic tensor) to a $6\times 6$ matrix is the Kelvin notation (Kelvin and Thomson 1878; Tarantola et al. 2000), where indices (ij) ($i=1,2,3$, $j=1,2,3$) are mapped into an index $\alpha$, ($\alpha =1,\ldots ,6$), as

$$\begin{aligned} \begin{array}{cccccccc} ij &{} = &{} 11 &{} 22 &{} 33 &{} 23,32 &{} 13,31 &{}12,21. \\ \Downarrow &{} {} &{} \Downarrow &{} \Downarrow &{} \Downarrow &{} \Downarrow &{} \Downarrow &{} \Downarrow \\ \alpha &{} = &{} 1 &{} 2 &{} 3 &{} 4 &{} 5 &{} 6. \\ \end{array} \end{aligned}$$

(40)

For example, the stress and strain tensors are transformed to vectors

$$\begin{aligned} \begin{bmatrix} S_1 \\ S_2 \\ S_3 \\ S_4 \\ S_5 \\ S_6 \end{bmatrix}=\begin{bmatrix} \sigma _{11} \\ \sigma _{22} \\ \sigma _{33} \\ \sqrt{2}\,\sigma _{23} \\ \sqrt{2}\,\sigma _{13} \\ \sqrt{2}\,\sigma _{12} \end{bmatrix} \quad \text { and }\quad \begin{bmatrix} E_1 \\ E_2 \\ E_3 \\ E_4 \\ E_5 \\ E_6 \end{bmatrix}=\begin{bmatrix} e_{11} \\ e_{22} \\ e_{33} \\ \sqrt{2}\,e_{23} \\ \sqrt{2}\,e_{13} \\ \sqrt{2}\,e_{12} \end{bmatrix}, \end{aligned}$$

(41)

while the symmetric fourth-order tensor B becomes ($B_{ijkl}=B_{jikl}=B_{ijlk}$)

$$\begin{aligned} B_{\alpha \beta }={\left\{ \begin{array}{ll} B_{iikk} \quad{ \text {when} }\quad i=j,\, k=l\\ \sqrt{2} B_{iikl} \quad {\text {when }}\quad i=j,\,k\ne l\\ \sqrt{2} B_{ijkk} \quad {\text {when }}\quad i\ne j, \,k=l\\ 2B_{ijkl} \quad {\text {when} }\quad i\ne j,\, k\ne l. \end{array}\right. } \end{aligned}$$

(42)

Note here, we do not use different symbols for the tensor $B_{ijkl}$ and the matrix representation $B_{\alpha \beta }$, as the distinction should be clear based on the number of indices. Under this notation, the norm of the fourth-order tensor is preserved in the matrix representation:

$$\begin{aligned} B_{ijkl}B_{ijkl}=B_{\alpha \beta }B_{\alpha \beta }, \end{aligned}$$

(43)

and double-dot product of two fourth-order tensors B and D can be computed by matrix multiplications, and converted back based on Eq. (42):

$$\begin{aligned} E_{ijmn}=B_{ijkl}D_{klmn} \quad \Leftarrow \quad E_{\alpha \gamma }=B_{\alpha \beta }D_{\beta \gamma }. \end{aligned}$$

(44)

For example, elastic tensor can be represented by the matrix

$$\begin{aligned} \mathbf {C}=\begin{bmatrix} C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16} \\ C_{21}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26} \\ C_{31}&C_{32}&C_{33}&C_{34}&C_{35}&C_{36} \\ C_{41}&C_{42}&C_{43}&C_{44}&C_{45}&C_{46} \\ C_{51}&C_{52}&C_{53}&C_{54}&C_{55}&C_{56} \\ C_{61}&C_{62}&C_{63}&C_{64}&C_{65}&C_{66} \end{bmatrix}= \begin{bmatrix} C_{1111}&C_{1122}&C_{1133}&\sqrt{2}\, C_{1123}&\sqrt{2}\,C_{1113}&\sqrt{2}\,C_{1112} \\ C_{2211}&C_{2222}&C_{2233}&\sqrt{2}\,C_{2223}&\sqrt{2}\,C_{2213}&\sqrt{2}\,C_{2212} \\ C_{3311}&C_{3322}&C_{3333}&\sqrt{2}\,C_{3323}&\sqrt{2}\,C_{3313}&\sqrt{2}\,C_{3312} \\ \sqrt{2}\,C_{2311}&\sqrt{2}\, C_{2322}&\sqrt{2}\,C_{2333}&2C_{2323}&2C_{2313}&2C_{2312} \\ \sqrt{2}\,C_{1311}&\sqrt{2}\,C_{1322}&\sqrt{2}\,C_{1333}&2C_{1323}&2C_{1313}&2C_{1312} \\ \sqrt{2}\,C_{1211}&\sqrt{2}\,C_{1222}&\sqrt{2}\,C_{1233}&2C_{1223}&2C_{1213}&2C_{1212}. \end{bmatrix}. \end{aligned}$$

(45)

Since the elastic tensor is symmetric $C_{ijkl}=C_{klij}$, the corresponding $\mathbf {C}$ matrix is also symmetric. The Kelvin matrix notation of the elastic tensor for a TI medium becomes

$$\begin{aligned} C_{ijkl} \Rightarrow {C}_{\alpha \beta } = \left[ \begin{array}{cccccc} A &{} B &{} F &{} 0 &{} 0 &{} 0 \\ B &{} A &{} F &{} 0 &{} 0 &{} 0 \\ F &{} F &{} C &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2L &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 2L &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2N \\ \end{array}\right] , \end{aligned}$$

(46)

where A, C, L, N, and F are the Love parameters discussed in Sect. 2.2, and $B=A-2N$.

Appendix B: Detailed expressions for H and G matrices

With the expressions for $\partial _m{\chi _n^{kl}}$ computed in Sect. 2.2, if we define auxiliary variables

$$\begin{aligned}&p=\left\langle \frac{C_{13}}{C_{33}} \right\rangle , \quad q=\left\langle C^{-1}_{33} \right\rangle ,\quad r=\left\langle \frac{C_{23}}{C_{33}} \right\rangle , \quad s=\left\langle C^{-1}_{44} \right\rangle ,\quad t=\left\langle C^{-1}_{55} \right\rangle \nonumber \\&\eta =\frac{C_{13}}{C_{33}},\quad \theta =C^{-1}_{33},\quad \xi =\frac{C_{23}}{C_{33}},\quad \phi =C^{-1}_{44},\quad \psi =C^{-1}_{55},\quad \nu =C_{66} \nonumber \\&\alpha =C_{11}-\frac{C_{13}^2}{C_{33}},\quad \beta =C_{12}-\frac{C_{13}C_{23}}{C_{33}},\quad \gamma =C_{22}-\frac{C_{23}^2}{C_{33}} \end{aligned}$$

(47)

then the Kelvin notation of the fourth-order tensor $\mathbf {G}$ in Sect. 2.2 becomes

$$\begin{aligned} {\mathbf G} = \begin{bmatrix} 1&0&0&0&0&0 \\ 0&1&0&0&0&0 \\ \frac{p}{q} \theta - \eta&\frac{r}{q} \theta - \xi&\frac{\theta }{q}&0&0&0 \\ 0&0&0&\frac{\phi }{s}&0&0 \\ 0&0&0&0&\frac{\psi }{t}&0 \\ 0&0&0&0&0&1 \end{bmatrix}. \end{aligned}$$

(48)

When the velocity model does not have any fine-scale variations, clearly, $G_{31}$ and $G_{32}$ vanish and ${\mathbf G}$ becomes an identity matrix.

The product of ${\mathbf C}$ matrix (Eq. 46) and ${\mathbf G}$ matrix (Eq. 34) gives

$$\begin{aligned} {\mathbf H} = \begin{bmatrix} \alpha +\frac{p}{q}\eta&\beta +\frac{r}{q}\eta&\frac{\eta }{q}&0&0&0 \\ \beta +\frac{p}{q}\xi&\gamma +\frac{r}{q}\xi&\frac{\xi }{q}&0&0&0 \\ \frac{p}{q}&\frac{r}{q}&\frac{1}{q}&0&0&0 \\ 0&0&0&\frac{1}{s}&0&0 \\ 0&0&0&0&\frac{1}{t}&0 \\ 0&0&0&0&0&\nu \\ \end{bmatrix}, \end{aligned}$$

(49)

and it can be shown that in the absence of fine-scale structures, ${\mathbf H}$ matrix becomes the ${\mathbf C}$ matrix.

With the expressions for ${\mathbf H}$ and ${\mathbf G}$, we can compute the effective moduli based on the matrix representation of Eq. (8):

$$\begin{aligned} {\mathbf C}^* =\mathscr {F}\big( {\mathbf H}\big) : \mathscr {F}\big( {\mathbf G}\big) ^{-1}. \end{aligned}$$

(50)

Clearly, the filter operator $\mathscr {F}\left( {\cdot }\right)$ only needs to be applied to $\alpha$, $\beta$, $\gamma$, $\eta$, $\theta$, $\xi$, $\phi$, $\psi$, and $\nu$ (i.e., the Greek letters), as p, q, r, s, and t already have fine-scale structures averaged out. To reduce symbol clutter, we write the filtering of the Greek letters $\mathscr {F}\left( {\cdot }\right)$ as $(\cdot )^*$, and the effective elastic tensor becomes

$$\begin{aligned} {\mathbf C}^* = \begin{bmatrix} \alpha ^*+\frac{(\eta ^*)^2}{\theta ^*}&\beta ^*+\frac{\eta ^*\xi ^*}{\theta ^*}&\frac{\eta ^*}{\theta ^*}&0&0&0 \\ \beta ^*+\frac{\eta ^*\xi ^*}{\theta ^*}&\gamma ^*+\frac{(\xi ^*)^2}{\theta ^*}&\frac{\xi ^*}{\theta ^*}&0&0&0 \\ \frac{\eta ^*}{\theta ^*}&\frac{\xi ^*}{\theta ^*}&\frac{1}{\theta ^*}&0&0&0 \\ 0&0&0&\frac{1}{\phi ^*}&0&0 \\ 0&0&0&0&\frac{1}{\psi ^*}&0 \\ 0&0&0&0&0&\gamma ^* \end{bmatrix}. \end{aligned}$$

(51)

Note interestingly that all the cell averaged variables p, q, r, s, and t naturally cancel out from this matrix product, which is a result of the special structure of ${\mathbf G}$ as in Eq. (34).

If we plug in the expressions for TI media, Eq. (46), to the auxiliary variables (47), we have

$$\begin{aligned}&\alpha =\gamma =A-F^2/C,\quad \beta =B-F^2/C, \nonumber \\&\eta =\xi =F/C,\quad \theta =1/C, \quad \phi =\psi =1/(2L), \quad \nu =2N, \end{aligned}$$

(52)

then we have the effective elastic tensor

$$\begin{aligned} \mathbf {C}^*= \begin{bmatrix} A^*&A^*-2N^*&F^*&0&0&0 \\ A^*-2N^*&A^*&F^*&0&0&0 \\ F^*&F^*&C^*&0&0&0 \\ 0&0&0&2L^*&0&0 \\ 0&0&0&0&2L^*&0 \\ 0&0&0&0&0&2N^* \\ \end{bmatrix} \end{aligned}$$

(53)

where

$$\begin{aligned}&A^* = \mathscr {F}\left( {A-F^2/C}\right) + \left[ \mathscr {F}\left( {1/C}\right) \right] ^{-1} \left[ \mathscr {F}\left( {F/C}\right) \right] ^{2}\nonumber \\&F^*=\left[ \mathscr {F}\left( {1/C}\right) \right] ^{-1}\mathscr {F}\left( {F/C}\right) \nonumber \\&C^*=\left[ \mathscr {F}\left( {1/C}\right) \right] ^{-1}\nonumber \\&L^*=\left[ \mathscr {F}\left( {1/L}\right) \right] ^{-1}\nonumber \\&N^*=\mathscr {F}\left( {N}\right) . \end{aligned}$$

(54)

Therefore, the homogenization of a TI model also produces a TI media with the effective moduli given by the expressions (54).