Analysis of 3D Ellipsoids and 2D Ellipses: Geophysics Study 2015
Investigate aspect ratio histograms of 3D ellipsoids and 2D ellipses with implications for representing pore crack grain structures. Explore the relationship between 2D and 3D aspect ratios, transformations between dimensions, and comparisons with simulated and actual data. The study delves into uniform and non-uniform distributions, standard deviations, and validation against existing research by Rudge et al., 2008.
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Aspect ratio histograms of 3D Ellipsoids and 2D Ellipses* * Paper submitted to Geophysics M. Landr ROSE meeting 2015
3D ellipsoids might represent Pore Crack Grain
Cutting an ellipsoid with a plane 2 3 2 1 2 2 x x x = T ( , , ) n n n n + + = 1 n 1 2 3 2 2 2 a b c Klein (2012) shows that the cut is an ellipse with half axes 1 d 1 d = = A 2 = B 2D aspect ratio 1 2 1 2 3 b 2 1 c 2 2 c 1 b 1 c 1 a 1 c 1 a 1 b n n n + + + + + + + + = 2 2 1 2 2 2 3 0 n n n 2 2 2 2 2 2 2 2 2 2 2 2 b a a = c/ a Special case 1: two shortest semiaxes, a =b < c => 3D aspect ratio = .. Simple relation between 2D and 3D aspect ratio + ) 1 2 3 2 1 ( n
The histogram of the 2D aspect ratio = + ) 1 2 3 2 1 ( n Histogram given by the derivative of n3with respect to the 2D aspect ratio: 2 dn const = 3 ( ) h d 2 2 2 2 ( 1 )( ) 2 N = ( ) h N = number of realizations 2 2 2 2 ( 1 )( ) Note: h is independent of n3-distribution
The 3D to 2D transform: ( = ( ( , d ) ) ) h h K 2 3 D D + 2 = ( , ) K + 2 2 2 2 ( 1 )( ) The 2D to 3D transform: ( = ( ( , d 1 ) ) ) h h K 3 2 D D ( ) + 2 1 = 1 ( , ) K ( ) 3 + 2 2 2 1 ( ) 2
Uniform distribution - 10 000 realizations 3 = D ' = = 4 2 N = ( ) h Simulated 2 2 2 2 ( 1 )( ) 2 D
Non-uniform example Normal distribution: = = n n = . 0 47 n 1 2 3 = . 0 33 Decreasing the standard deviation yields a worse fit Simulated 2 N = ( ) h 2 2 2 2 ( 1 )( ) Note: This formula is general, not dependent on the assumed distribution independent of n3
Rudge et al., 2008: Rudge et al., 2008 use spherocylinders and use a random close packing followed by one single 2D cut. I computed the 2D aspect ratio from this figure and compared to my results =>
Comparison with Rudge et al., 2008 Rudge (2008) Simulated 2 N = ( ) h 2 2 2 2 ( 1 )( ) Rudge et al., 2008 use spherocylinders and use a random close packing followed by one single 2D cut
A normal distribution of 3D aspect ratio forward modeling of 2D aspect ratio histogram 3D distribution ( = ( ( , d ) ) ) h h K 2 3 D D 10 = 7 10 = 8 + 2 = ( , ) K + 2 2 2 2 ( 1 )( )
Testing the 2D to 3D transform 10 = 7 ( = ( ( , d 1 ) ) ) h h K 3 2 D D ( ) + 2 1 = 1 ( , ) K ( ) 3 + 2 2 2 1 ( ) 2
So far, a=b; what if a , b and c are different? Stochastic modeling of 2D histograms c = 10 a b c alfa 1 1 10 1 10 1 2 10 2 5 10 1 5 10 2 5 10 Shorter axes apsect ratios easier to detect
An attempt to find an equation for the general case (a=1, b and c different) dn const = ( ) 1 1 b 1 c h 1 = d + 2 2 4 1 bc b b b 2 2 3 bc Note: Approximate equation far from exact! a=1; b=8 ; c=10
The importance of 3D selection of cuts to reveal the aspect ratio = + = + = 2 3 2 2 2 2 . 0 09 . 0 57 . 0 33 n = + ) 1 2 3 2 1 ( n = . 0 09 = = = . 0 57 n n n 1 2 3 = + . 1 = / 4 1 . 0 33 15 63 The true value of 4 is not observed on the 2D histogram, but the wrong value of 1.6
Coupling aspect ratio (3D) to rock physics K 2 2 1 ( 4 / )( 1 ) 4 1 ( ) c a Penny-shaped pore: c >> a=b = = 3 1 ( 0 2 ) 3 1 ( 0 2 ) K K Mavko et al., 2009 Poisson ratio (solid mineral) K Bulk modulus (solid mineral) 0 K 1 1 = + 3D aspect ratio Kdry K 0 K is the pore stiffness Another application: Anisotropy
Discussion and conclusions Simple equation to derive 2D aspect ratio histograms from 3D ellipsoids is derived 2D to 3D transform of aspect ratio is derived and tested by stochastic simulations Using thin sections to estimate 3D aspect ratio might be misleading Useful for rock physics applications?
Acknowledgments Jon Marius Venstad, Kjetil Eik Haavik, Dave Hale, Mike Batzle, Kenneth Duffaut, Roger Sollie and C. Morency for discussions and comments Tor Arne Johansen, Bernardo Moyano for inviting me as an opponent for Bernardo s PhD defense. The Norwegian Research Council is acknowledged for financial support to the ROSE consortium at NTNU Geophysics Department at Colorado School of Mines for hosting me the autumn semester 2014.
