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
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)
Conventional Triaxial Testing σ 1 = σ 3 Strength Criterion σ = f ( σ ) 1 3 1 1 q= g( p), q= ( σ1 σ3), p= ( σ1+ σ3) 2 2 1 τ = h( σ), σ = ( σ1+ σ2+ σ3) 3 τ = ss ij ij /2 1 = ( ) + ( ) + ( ) 6 Since, σ = σ { 2 2 2 σ } 1 σ2 σ2 σ3 σ3 σ1 2 3 2 1 τ = q, σ = p+ q 3 3
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)
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
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
Mineral Feldspar Quartz Clay Mica Three rocks tested Westerly granite (igneous) 66 28 3 Volume percentage (%) KTB amphibolite (metamorphic) 25 Amphibole 58 Property Westerly granite 2 KTB amphibolite TCDP siltstone (sedimentary) 10 68 20 3 TCDP siltstone Density, kg/m 3 2630 2920 2594 Porosity, % 0.9 0.7 6.9 UCS, MPa 201 164 80 Elastic Modulus, GPa 59 95 14
True triaxial strengths (peak σ 1 ) of three tested rocks 1200 Westerly granite = σ 3 100 10 KTB amphibolite = σ 3 150 MPa 0 TCDP siltstone σ 1 (MPa) 800 38 20 77 1200 800 100 MPa MPa 400 = σ 3 100 MPa MPa 40 MPa 400 2 σ 3 = 0 MPa 400 σ 3 = 0 MPa 30 MPa = σ 1 200 25 MPa σ 3 = 10 MPa 0 = σ 1 0 100 200 300 400 (MPa) KTB amphibolite 0 0 200 400 0 σ (MPa) 2 0 = σ 1 0 200 400 (MPa)
TCDP Siltstone 250 200 200 τ 150 100 150 100 τ = 3.036 * p 0.739 50 50 R 2 = 0.990 0 100 200 300 100 200 300 200 200 150 150 q 100 100 50 50 0 50 100 150 200 250 300 350 σ = (σ 1 + +σ 3 )/3 100 200 300 p = (σ 1 +σ 3 )/2
True triaxial strength criteria (All stresses MPa) τ 750 0 450 300 150 Amphibolite τ = 2.216*p 0.852 R 2 = 0.989 0 0 200 400 0 p = (σ 1 + σ 3 )/2 200 TCDP Slitstone 0 Westerly Granite 150 450 τ 100 50 τ = 3.036 * p 0.739 R 2 = 0.990 τ 300 150 τ oct = 1.975*p 0.881 R 2 = 0.995 100 200 300 p = (σ 1 + σ 3 )/2 150 300 450 0 p = (σ 1 + σ 3 )/2
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
Tresca s 2 axisymmetric compression σ = σ > σ (+ in tension) 1 2 3 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
TCDP Siltstone τ, MPa 240 180 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 0 50 100 150 200 250 300 350 σ = (σ 1 + + σ 3 )/3, MPa
Typical fault planes under true triaxial stress σ 1 σ 1 σ 1 σ 3 Westerly granite KTB amphibolite TCDP siltstone
Fracture dip angle increases with (for given σ 3 ) σ 1 θ o 80 75 70 65 σ 3 = 10 MPa 25 40 100 85 80 75 θ o 70 σ 3 = 0 MPa 77 20 100 θ σ 1 σ 3 90 80 θ o 70 TCDP siltstone 55 0 100 200 300 400 (MPa) σ 3 = 0 MPa 100 150 65 Westerly granite 55 0 50 100 150 200 250 - σ 3 (MPa) KTB amphibolite 50 0 100 200 300 400 500 0 700 (MPa)
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
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=0 90 80 70 τ = A*p B RR β + µ = 0.75 β + µ = 1.50 β + µ = 2.25 θ b 50 Pure shear Axisymmetric Extension 40-1.0-0.5 0.0 0.5 1.0 2*sin(θ) = 3 *N Axisymmetric Compression
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 90 80 70 TCDP Siltstone Data σ = 50 MPa σ = 150 MPa σ = 300 MPa Band (dip) angle 90 80 70 TCDP Siltstone Data Axisym Extension Pure Shear Axisym Compression -1.0-0.5 0.0 0.5 1.0 3 N 0 75 150 225 300 σ = σ 1 + + σ 3, MPa Predictions from 1 τ = 3.036 σ1+ σ3, ν = 0.35 2 ( ) 0.739
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 ) 72 68 64 Band Angle (θ b ) 72 68 64 0 50 100 150 200 250 300 350 σ = (σ 1 + + σ 3 )/3, MPa -1.0-0.5 0.0 0.5 1.0 3N Predictions from 1 τ = 3.036 σ1+ σ3 2 ν = 0.35 ( ) 0.739
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.