Effect of the intermediate principal stress on fault strike and dip - theoretical analysis and experimental verification

Transcription

1 Effect of the intermediate principal stress on fault strike and dip - theoretical analysis and experimental verification B. Haimson University of Wisconsin, USA J. Rudnicki Northwestern University, USA

2 We experimentalists are not like theorists: the originality of an idea is not for being presented in a paper but for being shown in implementation of an original experiment. Patrick M. S. Blackett, London, 1962 (from plaque outside lecture hall)

3 Conventional Triaxial Testing σ 1 = σ 3 Strength Criterion σ = f ( σ ) q= g( p), q= ( σ1 σ3), p= ( σ1+ σ3) τ = h( σ), σ = ( σ1+ σ2+ σ3) 3 τ = ss ij ij /2 1 = ( ) + ( ) + ( ) 6 Since, σ = σ { σ } 1 σ2 σ2 σ3 σ3 σ τ = q, σ = p+ q 3 3

4 σ 1 hydrostat σ 1 = σ 2 =σ3 Axisymmetric compression σ 3 σ = 3 τ = f ( σ ) τ ( q) 3 3 Axisymmetric extension σ = σ 2 3 σ ( p)

5 What if σ σ? 2 3 Ans.: Make assumptions, e.g. Mohr Coulomb: 1 1 ( σ1 σ 3) + µ c ( σ1+ σ3) = c 2 2 No dependence on σ! Drucker-Prager (Rudnicki-Rice): τ = h( σ) No dependence on 3rd stress invariant! 1 J3 = det( s ), s = σ δ σ 3 ij ij ij ij kk 2

6 True- Triaxial (polyaxial) testing σ 1 Load cell σ 1 σ 3 Biaxial cell Polyaxial apparatus Rock specimen σ 3 Hydraulic fluid Strength criterion: σ 1 = f(,σ 3 ) (Mises, Nadai, Mogi) True-triaxial testing apparatus

7 Mineral Feldspar Quartz Clay Mica Three rocks tested Westerly granite (igneous) Volume percentage (%) KTB amphibolite (metamorphic) 25 Amphibole 58 Property Westerly granite 2 KTB amphibolite TCDP siltstone (sedimentary) TCDP siltstone Density, kg/m Porosity, % UCS, MPa Elastic Modulus, GPa

8 True triaxial strengths (peak σ 1 ) of three tested rocks 1200 Westerly granite = σ KTB amphibolite = σ MPa 0 TCDP siltstone σ 1 (MPa) MPa MPa 400 = σ MPa MPa 40 MPa σ 3 = 0 MPa 400 σ 3 = 0 MPa 30 MPa = σ MPa σ 3 = 10 MPa 0 = σ (MPa) KTB amphibolite σ (MPa) 2 0 = σ (MPa)

9 TCDP Siltstone τ τ = * p R 2 = q σ = (σ 1 + +σ 3 )/ p = (σ 1 +σ 3 )/2

10 True triaxial strength criteria (All stresses MPa) τ Amphibolite τ = 2.216*p R 2 = p = (σ 1 + σ 3 )/2 200 TCDP Slitstone 0 Westerly Granite τ τ = * p R 2 = τ τ oct = 1.975*p R 2 = p = (σ 1 + σ 3 )/ p = (σ 1 + σ 3 )/2

11 2 axisy met rico mpre sion T resca σ=σ 1 >3 in (+ tensio n ) Mis es 30ο pur eshe 30ο s= 2 0, s= 1 -s 3 s 3 axi sym etric exten sion s1 σ>σ= 1 2 σ(+ 3 inten sion ) σ 1 hydrostat σ 1 = σ 2 =σ3 σ 3 σ = 3

12 Tresca s 2 axisymmetric compression σ = σ > σ (+ in tension) Mises 30 ο 2τ 1 27J θ = arcsin 3 3 2τ 3 30 ο pure shear s 2 = 0, s = -s 1 3 s 3 s 1 axisymmetric extension σ > σ = σ (+ in tension) 1 2 3

13 TCDP Siltstone τ, MPa TCDP Data Fit, Axisym Ext Fit, Pure Shear Fit, Axisym Comp θ = 30 o θ = 0 o θ = 30 o 120 Fit for 1 τ = A σ1+ σ3 2 ( ) 1 1 τ = A σ + τ sinθ 3 3 B B σ = (σ σ 3 )/3, MPa

14 Typical fault planes under true triaxial stress σ 1 σ 1 σ 1 σ 3 Westerly granite KTB amphibolite TCDP siltstone

15 Fracture dip angle increases with (for given σ 3 ) σ 1 θ o σ 3 = 10 MPa θ o 70 σ 3 = 0 MPa θ σ 1 σ θ o 70 TCDP siltstone (MPa) σ 3 = 0 MPa Westerly granite σ 3 (MPa) KTB amphibolite (MPa)

16 Band Angle Predictions π 1 Mohr Coulomb: θmc = + arctan µ MC 4 2 π 1 Rudnicki-Rice: θrr = + arcsin α 4 2 where dilatancy factor friction coefficient (2 / 3)( β + µ ) N(1 2 ν) = 2 4 3N Poisson s ratio s θ 2 2 N = = sin( θ ) τ 3 b Generalize, for yield condition f ( τσθ,, ) = 0 and plastic potential g( τσθ,, ) = 0 with g = f, g = f but f α 1 3 ( ) τ τ θ θ 2 N sin( θ + φ ), tan φ = 3 ( g + f ) g σ σ β + µ cosφ g τ σ gθ / τ g τ

17 Comparison of band angle predictions vs. deviatoric stress state for Rudnicki-Rice (Drucker Prager) with constitutive relation derived from Haimson strength criterion. Normalized to agree at deviatoric pure shear, N= τ = A*p B RR β + µ = 0.75 β + µ = 1.50 β + µ = 2.25 θ b 50 Pure shear Axisymmetric Extension *sin(θ) = 3 *N Axisymmetric Compression

18 Band angle data against deviatoric stress state with predictions for fixed mean normal stress. Band angle data against mean normal Stress with predictions for axisym ext, axisym comp and pure shear. band (dip) angle TCDP Siltstone Data σ = 50 MPa σ = 150 MPa σ = 300 MPa Band (dip) angle TCDP Siltstone Data Axisym Extension Pure Shear Axisym Compression N σ = σ σ 3, MPa Predictions from 1 τ = σ1+ σ3, ν = ( ) 0.739

19 Band angle vs. mean normal stress (for different deviatoric stress states, i. e. N) Band angle vs. deviatoric stress state for different mean normal stresses. 76 Data Prediction 76 Data Prediction Band angle (θ b ) Band Angle (θ b ) σ = (σ σ 3 )/3, MPa N Predictions from 1 τ = σ1+ σ3 2 ν = 0.35 ( ) 0.739

20 Conclusions The intermediate principal stress affects all aspects of mechanical behavior of rock under compressive stresses. The strength is well-described by a relation τ = Αp B (neither Mohr- Coulomb nor Drucker Prager (RR)). Fault dip angle increases steadily as is raised for a given σ 3 (prediction based on τ = Αp B models trends with mean stress and deviatoric stress state adequately but, in general, angles are less than observed). True triaxial testing is essential for constraining constitutive relations for applications and numerical calculations. True triaxial testing provides the opportunity to interrogate the role of constitutive behavior in predicting failure strength and fault orientation.