Formation and Extension of Compaction Bands in Porous Sandstone

Transcription

1 Formation and Extension of Compaction Bands in Porous Sandstone J. W. Rudnicki Dept. of Civil and Env. Eng. and Dept. of Mechanical Eng. Northwestern University September 8, 2011 Statoil, Bergen, Norway

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3 Collaborators Bill Olsson, Dave Holcomb (Sandia, Albuquerque) Eric Grueschow (Exxon-Mobil) Kathleen Issen (Clarkson University) Sheryl Tembe, Teng-Fong Wong (SUNY Stony Brook) Kurt Sternlof, Dave Pollard (Stanford) Jose Andrade (NU, now Caltech), Steve Sun (NU, now Sandia Livermore), Nicola Lenoir (NU, now Ecole des Ponts ParisTech) Peter Eichhubl (Texas Bureau of Economic Geology) Support: U.S. Dept of Energy, Office of Basic Energy Sciences

4 Outline What are compaction bands? Why are they important? What do they look like? Why do they form? Models for propagation

5 What are compaction bands? o shear band (fault) o shear-enhanced compaction band compaction band

6 Why are compaction bands important?

7 From Vajdova, Baud and Wong, JGR, 2004 Permeability evolution during localized deformation in Bentheim Sandstone Aydin and Ahmadov, Bed-parallel compaction bands in aeolian sandstone: their identification, characterization and implications, Tectonophysics, (2009)

8 Uniform compaction Localized compaction

9 high permeability low permeability

10 Chicago Tribune Front Page, January 27, 2003

11 What do compaction bands look like in the field?

12 From Sternlof, Rudnicki and Pollard, JGR, 2005; Sternlof, 2006.

13 Compaction Bands: Earliest Structural Fabric of the Aztec From Kurt Sternlof, Stanford

14 From Sternlof, Rudnicki and Pollard, JGR, 2005

15 From Sternlof, Rudnicki and Pollard, JGR, 2005

16 Eichhubl, Hooker, Laubach, J. Struc. Geology, 2010

17 pocket knife Eichhubl, Hooker, Laubach,J. Struc. Geology, 2010

18 Compactive Shear Band with 2.5 cm reverse slip Eichhubl, Hooker, Laubach, J. Struc. Geology, 2010 Detail of Shear band with 1 cm slip Macroscopic band is composed of multiple parallel bands of more intense grain-size reduction

19 shear enhanced compaction bands shear enhanced compaction bands pure compaction bands Fossen, Schultz &Torabi J. Structural Geology in press

20 In situ permeability measurements inside compaction bands using X-ray CT and lattice Boltzmann calculations Lenoir, Andrade, Sun and Rudnicki, Proceedings of GEOX 10, 2009 /Geox2010: 3^rd International Workshop on X-Ray CT for Geomaterials/* and publication in the conference proceedings book entitled: *_GeoX2010: Advances in Computed Tomography for Geomaterials_*

21 Multiscale methods for characterization of porous microstructions and their effect on macroscopic permeability,int. J. Numer. Methods Engng (2011) Sun, Andrade, Rudnicki

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23 2.25mm CB1.75mm.75mm 9.0mm CB2 CB3 OUTSIDE1.75mm.75mm 6.3mm OUTSIDE2 OUTSIDE OCCLUDED POROSITY CONNECTED POROSITY

24 Shortest Flow Path Inside and Outside Compaction Bands INSIDE CB = 0.14 = 2.79 = 2.15 = 2.56 K= 3.4e-13 m 2 K= 5.3e-13 m 2 K= 4.4e-13 m 2 OUTSIDE CB = 0.21 = 1.77 = 1.76 = 1.81 K= 1.3e-12 m 2 K= 1.2e-12 m 2 K= 1.3e-12 m 2 9/9/

25 Homogenization of Effective Permeability Across scale LBM FEM 9/9/

26 Effective Permeabilities Inside and Outside Compaction Bands FEM/LBM Scheme Inside CB K zz = 3.5e-13 m 2 K zz = 14.0e-13 m 2 Outside CB 9/9/

27 What do compaction bands look like in the laboratory?

28 Bentheim sandstone: Pc = 300 MPa 1 cm 1 cm 1 cm 1 cm ax = 1.4 % ax = 3.1 % ax = 4 % ax = 6 % Baud, Klein, and Wong, Compaction localization in porous sandstones:, JSG, 2004 GRL, 2001

29 Ben#12 (300 MPa) A compaction band in the Bentheim sandstone: typically the lateral width extends over 2 grains or so (~ 600 m) 0,5 mm 0.5 mm

30 Baud, Klein and Wong,, Compaction localization in porous sandstones: spatial evolution of damage and acoustic emission activity, J. Struct. Geology, 2004 Papka and Kyriakides, In-Plane Crushing of a Polycarbonate Honeycomb, Int. J. Solid Struct., 1998.

31 Bastawros, Bart-Smith and Evans Experimental analysis of deformation mechanisms in a closed-cell aluminum alloy foam, JMPS (2000)

32 Tembe, Baud and Wong, JGR, 2008 Shear bands Mixed failure modes Development of an array of discrete compaction bands

33 Photograph of sample showing ultrasonic transducers (pins protruding from sides of sample), copper and urethane coating and steel endcaps. Olsson and Holcomb, GRL, 2000; Holcomb and Olsson, JGR, 2003

34 Olsson and Holcomb, GRL, 2000; Holcomb and Olsson, JGR, 2003

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36 From Anthony DiGiovanni

37 From Haimson and Lee

38 From Haimson and Lee (Mansfield sandstone)

39 Why do compaction bands form?

40 Homogeneous and homogeneous deformation Continued Homogeneous deformation Stress Shear band or compaction band Strain

41 Kinematic Condition 0 du g n d band 0 u du g n x ( ) band 0 1 dε dε ( ng gn ) 2 For compaction band, g an, 90 o

42 Equilibrium Condition band dt n dσ n dσ n o

43 Constitutive relation d L d ij ijkl kl with equilibrium n d n d band 0 k ki k ki and compatibility Contrast with a fracture mechanics approach: material behavior is elastic (which does not allow localization, but acute geometry or a flaw elevates local stress sufficiently to cause propagation. band 0 1 d ij d ij ( ni g j gin j ) 2 Localization condition: n L n g i ijkl l k 0, det( n L n ) 0, for g 0 i ijkl l k For compaction band: g k n n L an k i j ijkl l k E uniaxial n n 0 0 Constrains constitutive parameters at localization and n gives normal to the band. Tangent modulus for uniaxial deformation relative to the plane of the band vanishes.

44 Elastic-plastic rate-independent d p d p 0 (dilation) constitutive model depending on first and second invariants (Drucker Prager type). ( a c) / 3 for standard test d p 1 0 d p yield surface d p 1 elastic response, <0 k 1 (compression) k < k crit for localization ( a 2 c) / 3 for standard test p

45 1 ( ) extension bands 1 (1 2 ) N 4 3N ( ) 3 2(1 ) 2 0 shear bands -1 compaction bands 1 (1 2 ) N 4 3N ( ) 3 2(1 ) axisymmetric extension 3N axisymmetric compression

46 Axisymmetric Compression 100 Dilation Band crit Compaction Band crit = angle between fault normal and most compressive principal stress.

47 , dilatancy angle tan / degrees Range of angles for shear enhanced compaction bands by Eichhubl et al. (2010) 53 degrees , fault angle crit

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49 Evaluate for an Elliptic Yield Cap Rudnicki, JGR, 2004 More elaborate yield surface model Grueschow and Rudnicki, Int. J. Solids Struc., 2005

50 Q (MPa) COMPACTIVE YIELD STRESSES OF 4 SANDSTONES: The critical stress levels C*at the onset of shear-enhanced compaction were fitted with elliptical caps (yield envelopes) Darley Dale 250 Bentheim 200 Berea Rothbach P (MPa)

51 ( c) a 2 2 b ( a/ b) 1.53 to 3.0 from Wong, David & Zhu (JGR, 1997) for Bentheim Sandstone from Klein et al. (Phys. Chem. Earth A, 2001) b 3 axisymmetric compression path c c a

52 increasing k crit k crit 0 (for normality) Shear bands compaction bands Elliptical cap surface increasing k crit increasing confining stress

53 k crit / G Bandangle (degrees) kcritcom1 Mar. 9, :50:46 PM shear band -0.4 compaction band bandangle S ( c)/ a c

54 From Baud, Klein and Wong, J. Struc. Geol., Wong, David and Zhu, J. Geophys. Res Klein, Baud, Reuschlé, and Wong, Phys. Chem Earth, Rock Type % P* (MPa) a/ b trans c (MPa) Confining pressure Deformation mode Darley Dale , cataclastic flow 80, 90, 95 shear bands Berea , discrete & diffuse CB 150, diffuse CB 90, 75, 60, 50 shear bands Rothbach , diffuse CB 40,55, 90, shear bands Bentheim , 350, 300, 250, 180, discrete CB Dimelstadt , diffuse CB 90 discrete CB and shear bands 160 discrete CB

55 Open circles, peak stress at brittle failure Solid circles, onset of shear enhanced compaction distributed cataclastic flow, no localization compaction localization, discrete compaction bands

56 More elaborate yield surface model Grueschow and Rudnicki, Int. J. Solids Struc., Essentially same model as used by Deshpande and Fleck, Isotropic constitutive models for metallic foams, JMPS, n a c 2 2 b 1 b b( p ) c a a( ) c p

57 yield surface F (,, ) 0 shear cap elastic response (compression) Two yield surface models: Issen, Engineering Fracture Mechanics, Issen and Challa, J. Geophys. Res., 2008.

58 shear cap Axisymmetric compression test path (constant lateral stress) 3 (mean compressive stress)

59 Conclusions Compaction bands are predicted to occur on portions of the cap surface where they are (roughly) observed to occur. Predictions are roughly consistent with lab data but do not agree very well quantitatively. Need for better constitutive data and modeling and a better understanding of the micromechanical factors affecting macroscopic response. More complex load paths (now being used) should be more useful for constraining constitutive models.

60 How do the bands propagate (extend)?

61 Possible Propagation Mechanisms Compaction band (anti) crack-like model (Fletcher Pollard, Sternlof): compressive stress ahead of the band is elevated sufficiently to cause compaction Lateral spreading as in Castlegate SS at stresses much less than those at the tip. Localization model: stress state ahead of the band is driven to localization

62 Normalized band thickness vs. distance from center normalized by half-length. From Sternlof, Rudnicki and Pollard (JGR, 2005) and Sternlof (PhD Thesis, 2006)

63 Ellipsoidal band with uniform inelastic compactive strain p compactive displacment w 1 p p w p 2 w band width compressive stress ahead of the band is elevated sufficiently to cause compaction

64 Anti-crack model (Pollard, Fletcher, Sternlof) K/ 2 x

65 Conclusions For aspect ratios representative of field data (Sternlof) and probably lab data, 10-3 to 10-4 the stress state in the band is essentially the same as for zero aspect ratio. The stress elevation at the tip is roughly times for field stress for representative field parameters. The stress elevation is dominated by the small aspect ratio and the inelastic compaction, relatively independent of the stiffness of the compaction band material. Consequently, we can use results from fracture mechanics

66 Energy Release Due to Compaction Band Propagation Rudnicki and Sternlof, GRL, 2005 (after J. R. Rice, 1968) energy release rate G W W 1 Mh M p ( M / M ) (1 ) h M h b 2 b p 2

67 p 2 energy release rate G h M ( / h) O( ) net compaction stress= From Sternlof and Rudnicki, GRL, 2005: h p G 40 MPa 0.01 (1 cm thick CB, spaced 1 meter apart) 0.1 (corresponding to 10% porosity reduction) kj/m (10 to 60 kj/m for range of stress estimates) Tembe et al., JGR, 2006 estimate compaction energies (equivalent to energy release rate) of kj/m -2 for Berea and Bentheim sandstones.

68 Midpoint Thickness (mm) Figure 1:Compaction Band Data 10 Sternlof (2006) Hill (1989) M & A (1996) Baud et al. (2004) Tembe et al. (2008) Fortin et al. (2005) Field 1 Lab Linear Fit: Field Data Linear fit: y A Bx A , B Band Half-Length (m)

69 Combined anti-crack and anti-dislocation model uniform compactive displacement, 2 w uniform resistive nomal traction a a L L

70 Dislplac Apr. 20, :58:37 PM 1.0 vx (,0) h a 0 (no dislocation) a 0.25L a 0.5 L 0.4 a 0.75L a 0.9L x / L

71 K w k L a 2 (1 v) L a E( k) (1 k ) K( k) where k 1 ( a / L) 2 2 F( k) (1 ) 2 G G K K 2 (1 ) 2 crit crit crit crit 8 (1 ) G w L F k L a crit 2 ( ), For =0.2, GPa, G 40 kj/m G crit 3 / mm 1/2 1 crit (Sternlof, Rudnicki & Pollard, 2005) 2w L m 2 F( k) for L 3a F( k) for L 5a

72 Midpoint Thickness (mm) Compaction Band Data 10 Sternlof (2006) Hill (1989) M & A (1996) Baud et al. (2004) Tembe et al. (in prep) Fortin et al. (2005) Field 1 Lab Linear Fit: Sternlof (2006) Linear Fit: Field Data Linear Fit: All Data Linear Fit: Model Band Half-Length (m)

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74 Initial compactive event 2a compactive displacement K >> K crit Final band length 2a K = K crit L >> a

75 Conclusions A simple fracture and dislocation model indicates a (surprising) consistency between lab and field data. Model is, however, speculative since little information is available on the initiation and propagation of bands in the field. Field and laboratory observations of the breakdown process near the tip of bands would be helpful in constructing more elaborate models.

76 Questions How do lab, field and theoretical results fit together? What properties (or factors) other than porosity affect compaction band formation? What is a good physical microstructural microstructural model? How do microscale properties and lithology translate to macroscopic material (constitutive) behavior? What controls band thickness (and changes in thickness) and spacing? If bands formed under saturated conditions how does interaction of fluid flow with deformation affect formation and extension? How do we model the transition from initial band formation to extension of a fully formed band?

77 Thanks! Questions for me?