Estimation of anisotropy parameters using the P‐wave velocities on a cylindrical shale sample

Abstract

In this paper we present a new approach to the estimation of the Thomsen anisotropy parameters and symmetry axis coordinates from the P-wave traveltime measurements on cylindrical shale samples. Using the tomography-style array of transducers, we measure the ultrasonic P-wave ray velocities to estimate the Thomsen anisotropy parameters for a transversely isotropic shale sample. This approach can be used for core samples cut in any direction with regard to the bedding plane, since we make no assumption about the symmetry axis directions and will estimate it simultaneously with the anisotropy parameters. We use the very fast simulated re-annealing to search for the best possible estimate of the model parameters. The methodology was applied to a synthetic model and an anisotropic shale sample.

An upscaling based three parameter elastic anisotropy model

“…O T H g Z 3 A K q q F A j s L + 8 X k K z m E m g g m j d d Z 0 U / J w o 4 F S w S d n L N E s J H Z I + 6 x q U J G b a z 2 e n T / C J c X o 4 S p R 5 E v D M / T 2 R k 1 j r c R y a z p j A Q C / W p u Z / t W 4 G 0 a W f c 5 l m w C S d L 4 o y g S H B 0 x x w j y t G Q Y w N E K q 4 u R X T A V G E g k m r b E J w F 7 + 8 D K 2 z m m v 4 9 r x a d 4 o 4 S u g I H a N T 5 K I L V E c 3 q I G a i K J 7 9 I x e 0 Z v 1 a L 1 Y 7 9 b H v H X F K m Y O 0 R 9 Z n z / / j Z R o < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 O T H g Z 3 A K q q F A j s L + 8 X k K z m E m g g m j d d Z 0 U / J w o 4 F S w S d n L N E s J H Z I + 6 x q U J G b a z 2 e n T / C J c X o 4 S p R 5 E v D M / T 2 R k 1 j r c R y a z p j A Q C / W p u Z / t W 4 G 0 a W f c 5 l m w C S d L 4 o y g S H B 0 x x w j y t G Q Y w N E K q 4 u R X T A V G E g k m r b E J w F 7 + 8 D K 2 z m m v 4 9 r x a d 4 o 4 S u g I H a N T 5 K I L V E c 3 q I G a i K J 7 9 I x e 0 Z v 1 a L 1 Y 7 9 b H v H X F K m Y O 0 R 9 Z n z / / j Z R o < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 O T H g Z 3 A K q q F A j s L + 8 X k K z m E m g g m j d d Z 0 U / J w o 4 F S w S d n L N E s J H Z I + 6 x q U J G b a z 2 e n T / C J c X o 4 S p R 5 E v D M / T 2 R k 1 j r c R y a z p j A Q C / W p u Z / t W 4 G 0 a W f c 5 l m w C S d L 4 o y g S H B 0 x x w j y t G Q Y w N E K q 4 u R X T A V G E g k m r b E J w F 7 + 8 D K 2 z m m v 4 9 r x a d 4 o 4 S u g I H a N T 5 K I L V E c 3 q I G a i K J 7 9 I x e 0 Z v 1 a L 1 Y 7 9 b H v H X F K m Y O 0 R 9 Z n z / / j Z R o < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 O T H g Z 3 A K q q F A j s L + 8 X k K z m E m g g m j d d Z 0 U / J w o 4 F S w S d n L N E s J H Z I + 6 x q U J G b a z 2 e n T / C J c X o 4 S p R 5 E v D M / T 2 R k 1 j r c R y a z p j A Q C / W p u Z / t W 4 G 0 a W f c 5 l m w C S d L 4 o y g S H B 0 x x w j y t G Q Y w N E K q 4 u R X T A V G E g k m r b E J w F 7 + 8 D K 2 z m m v 4 9 r x a d 4 o 4 S u g I H a N T 5 K I L V E c 3 q I G a i K J 7 9 I x e 0 Z v 1 a L 1 Y 7 9 b H v H X F K m Y O 0 R 9 Z n z / / j Z R o < / l a t e x i t > Data G vh [GPa] < l a t e x i t s h a 1 _ b a s e 6 4 = " z + G 3 + O g E c Q m M W c k w Y q O W k u J C f U g = " > A A A C C H i c b Z D J S g N B E I Z 7 4 h b j F v X o w c E g e A o z I u g x o B C P E c w C k 2 G o 6 X S S J j 0 L 3 T X B M E x u X n w V L x 4 U 8 e o j e P N t 7 C w H T f y h 4 e O v K q r r 9 2 P B F V r W t 5…”

“…There are many more other authors who investigated anisotropic elastic properties of shales, see e.g. [4,5,6,7,8,9,10,11]. Some use dynamic velocity measurements and some focus on the static experiments.…”

“…0 T N y 4 U c e u T u P N t n L Z Z a O s P A x / / O Y d z 5 v c T w R X a 9 r e x t L y y u r Z e 2 C h u b m 3 v 7 J q l v Y a K U 8 m g z m I R y 5 Z P F Q g e Q R 0 5 C m g l E m j o C 2 j 6 g 8 t J v T k E q X g c 3 e E o A T e k v Y h 3 O a O o L c 8 s d R A e M L u J A x D j p y t v e O y Z Z b t i T 2 U t g p N D m e S q e e Z X J 4 h Z G k K E T F C l 2 o 6 d o J t R i Z w J G B c 7 q Y K E s g H t Q V t j R E N Q b j Y 9 f W w d a S e w u r H U L 0 J r 6 v 6 e y G i o 1 C j 0 d W d I s a / m a x P z v 1 o 7 x e 6 F m / E o S R E i N l v U T Y W F s T X J w Q q 4 B I Z i p I E y y f W t F u t T S R n q t I o 6 B G f + y 4 v Q O K 0 4 m m / P y l U 7 j 6 N A D s g h O S E O O S d V c k 1 q p E 4 Y u S f P 5 J W 8 G Y / G i / F u f M x a l 4 x 8 Z p / 8 k f H 5 A 3 Q e l A w = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e6 4 = " G M Z / Q A 7 s 2 A T Z w i f e g j d 8 W 1 s K W a s = " > A A A B + n i c b Z D L S s N A F I Y n X m u 9 p b p 0 E y y i q 5 K I o M u C C G 6 E C v Y C b Q i T y W k 7 d H J h 5 q R a Y n 0 T N y 4 U c e u T u P N t n L Z Z a O s P A x / / O Y d z 5 v c T w R X a 9 r e x t L y y u r Z e 2 C h u b m 3 v 7 J q l v Y a K U 8 m g z m I R y 5 Z P F Q g e Q R 0 5 C m g l E m j o C 2 j 6 g 8 t J v T k E q X g c 3 e E o A T e k v Y h 3 O a O o L c 8 s d R A e M L u J A x D j p y t v e O y Z Z b t i T 2 U t g p N D m e S q e e Z X J 4 h Z G k K E T F C l 2 o 6 d o J t R i Z w J G B c 7 q Y K E s g H t Q V t j R E N Q b j Y 9 f W w d a S e w u r H U L 0 J r 6 v 6 e y G i o 1 C j 0 d W d I s a / m a x P z v 1 o 7 x e 6 F m / E o S R E i N l v U T Y W F s T X J w Q q 4 B I Z i p I E y y f W t F u t T S R n q t I o 6 B G f + y 4 v Q O K 0 4 m m / P y l U 7 j 6 N A D s g h O S E O O S d V c k 1 q p E 4 Y u S f P 5 J W 8 G Y / G i / F u f M x a l 4 x 8 Z p / 8 k f H 5 A 3 Q e l A w = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " 5 K H 7 n y u a A c p 8 h v H V C G t j U Z x c 2 f E = " > A A A C C X i c b Z C 7 S g N B F I Z n v c Z 4 W 7 W 0 G Q y C V d g V Q c u A R W y E C O Y C y R J m J y f J k N k L M 2 e D Y d m U N r 6 K j Y U i t r 6 B n W / j 5 F J o 4 g 8 D H / 8 5 h z P n 9 2 M p N D r O t 7 W y u r a + s Z n b y m / v 7 O 7 t 2 w e H N R 0 l i k O V R z J S D Z 9 p k C K E K g q U 0 I g V s M C X U P c H 1 5 N 6 f Q h K i y i 8 x 1 E M X s B 6 o e g K z t B Y b Z u 2 E B 4 w v Y 0 6 I L N x u Z 0 O + 9 m 4 O T P L F Z Z 5 b b v g F J 2 p 6 D K 4 c y i Q u S p t + 6 v V i X g S Q I h c M q 2 b r h O j l z K F g k v I 8 q 1 E Q 8 z 4 g P W g a T B k A W g v n V 6 S 0 V P j d G g 3 U u a F S K f u 7 4 m U B V q P A t 9 0 B g z 7 e r E 2 M f + r N R P s X n m p C O M E I e S z R d 1 E U o z o J B b a E Q o 4 y p E B x p U w f 6 W 8 z x T j a M L L m x D c x Z O X o X Z e d A 3 f X R R K z j y O H D k m J + S M u O S S l M g N q Z A q 4 e S R P J N X 8 m Y 9 W S / W u / U x a 1 2 x 5 j N H 5 I + s z x 8 X B J s k < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 K H 7 n y u a A c p 8 h v H V C G t j U Z x c 2 f E = " > A A A C C X i c b Z C 7 S g N B F I Z n v c Z 4 W 7 W 0 G Q y C V d g V Q c u A R W y E C O Y C y R J m J y f J k N k L M 2 e D Y d m U N r 6 K j Y U i t r 6 B n W / j 5 F J o 4 g 8 D H / 8 5 h z P n 9 2 M p N D r O t 7 W y u r a + s Z n b y m / v 7 O 7 t 2 w e H N R 0 l i k O V R z J S D Z 9 p k C K E K g q U 0 I g V s M C X U P c H 1 5 N 6 f Q h K i y i 8 x 1 E M X s B 6 o e g K z t B Y b Z u 2 E B 4 w v Y 0 6 I L N x u Z 0 O + 9 m 4 O T P L F Z Z 5 b b v g F J 2 p 6 D K 4 c y i Q u S p t + 6 v V i X g S Q I h c M q 2 b r h O j l z K F g k v I 8 q 1 E Q 8 z 4 g P W g a T B k A W g v n V 6 S 0 V P j d G g 3 U u a F S K f u 7 4 m U B V q P A t 9 0 B g z 7 e r E 2 M f + r N R P s X n m p C O M E I e S z R d 1 E U o z o J B b a E Q o 4 y p E B x p U w f 6 W 8 z x T j a M L L m x D c x Z O X o X Z e d A 3 f X R R K z j y O H D k m J + S M u O S S l M g N q ZA q 4 e S R P J N X 8 m Y 9 W S / W u / U x a 1 2 x 5 j N H 5 I + s z x 8 X B...…”

Estimation of the anisotropy parameters of transversely isotropic shales with a tilted symmetry axis

“…In the context of the inverse problem, this small sensitivity would not be enough to have a reliable estimate of β 0 from actual experimental data. However, in a noise-free numerical example as discussed by Nadri et al (2011b), it is possible to estimate β 0 using a gradient search method such as the conjugate gradient with a prior model close to solution. Within the limit of the weak anisotropy (|ε| 1, |δ| 1), linearized P-wave phase velocity equation (Thomsen 1986) does not depend on the β 0 .…”

“…Some of the 204 measurements are recorded from reciprocal shots on a given pair of transducers; these repetitive measurements are averaged. To check the accuracy of the parameter estimation, we have estimated the model parameters using the noise-free synthetic data for the above cylinder with the same acquisition configuration (Nadri et al 2011b). Although the VFSR recovered all the parameters except β 0 accurately, a conjugate gradient method applied afterward recovered all the parameters including β 0 .…”