On anelliptic approximations for qP velocities in TI and orthorhombic media

Examples

To investigate the accuracy of the proposed approximations, we make the relative error comparison with both plots and tables using several anisotropy models based on values from laboratory rock samples. The plots in Figure 2 are generated using the stiffness tensor measurements of Greenhorn shales (Jones and Wang, 1981), which have been applied for various approximation comparisons in the past (e.g. Farra and Pšencík, 2013; Fomel, 2004; Stovas, 2010; Dellinger, 1991). Additionally, Tables 4 and 5 show the RMS relative error results of the new approximations, in comparison with results from some of the previously suggested approximations using the normalized stiffness tensor measurements given in Table 6. The RMS error computation is based on

$$= \sqrt{\sum_{\psi=0}^{90} (v_{exact}(\psi)-v_{approx}(\psi))^2}~,$$ (39)

where $\psi$ denotes phase or group angle as appropriate. In all comparisons, we apply the relationships shown in Figure 1 to reduce the number of parameters from four to three. For each model, the best-performing approximation is denoted in red and bold. The proposed approximations appear to be the most accurate in nearly all of the cases.

Sample	Thomsen (1986)	Alkhalifah (1998)	Proposed
1	0.6789	0.1422	0.0978
2	0.6482	0.2254	0.0503
3	0.4564	0.1399	0.0273
4	0.2978	0.0485	0.0506
5	0.1244	0.0541	0.0201
6	0.5710	0.1631	0.0149

Table 4. RMS relative error (%) from 0 - $90\,^{\circ}$ of phase-velocity approximations by Thomsen (1986), Alkhalifah (1998) (similar to Fomel (2004)), and of the proposed three-parameter approximation for transversally-isotropic elastic models from Table 6. Bold red highlight indicates the best-performing approximation. In all the cases, except sample 4, the proposed approximation appears to be the most accurate.

Sample	Alkhalifah and Tsvankin (1995)	Fomel (2004)	Farra and Pšencík (2013)	Proposed
1	1.0149	0.1210	0.2530	0.0801
2	0.3306	0.2179	0.1351	0.0564
3	0.4602	0.1311	0.0977	0.0194
4	0.1369	0.0467	0.0983	0.0492
5	0.0188	0.0540	0.0194	0.0202
6	0.4258	0.1541	0.1412	0.0084

Table 5. RMS relative error (%) from 0 - $90\,^{\circ}$ of group-velocity approximations by Alkhalifah and Tsvankin (1995), Fomel (2004), Farra and Pšencík (2013) (second-order) and of the proposed three-parameter approximation for transversally-isotropic elastic models from Table 6. Bold red highlight indicates the best-performing approximation. In all the cases, except samples 4 and 5, the proposed approximation appears to be the most accurate.

Shales sample	$c_{11}$	$c_{33}$	$c_{13}$	$c_{55}$	$V_{P0}$	$V_{S0}$	$\epsilon$	$\delta$
1. Greenhorn	14.47	9.57	4.51	2.28	3.094	1.510	0.256	-0.0505
2. Hard (brine)	20.89	13.89	3.048	5.655	3.727	2.378	0.252	0.0347
3. North Sea (brine)	7.292	5.248	1.578	1.798	2.291	1.341	0.195	-0.0139
4. Dog Creek	5.098	3.5163	2.4832	0.6823	1.875	0.826	0.225	0.0998
5. Mesaverde	17.653	14.055	1.3391	6.87	3.749	2.621	0.128	0.0781
6. North Sea (dry)	22.051	14.90	5.336	4.928	3.860	2.220	0.240	0.0199

Table 6. Normalized stiffness tensor coefficients (in $km^2$ /$s^2$ ) from different TI samples: 1 is from Jones and Wang (1981), 2 and 3 are from Wang (2002), 4 and 5 are from Thomsen (1986), and 6 is from Vernik and Liu (1997).

vtiphaseplotlegnew,vtigroupplotleg1new,vtigroupplotleg2new
Figure 2. Relative error plots using Greenhorn Shale measurements. a) Phase velocity. b) Group velocity. c) Group velocity (finer scale).

On anelliptic approximations for qP velocities in TI and orthorhombic media