3D Mohr diagram to explain reactivation of pre-existing planes due to changes in applied stresses

Rock Stress and Earthquakes Xie (ed.) 2010 Taylor & Francis Group, London, ISBN 978-0-415-60165-8 3D Mohr diagram to explain reactivation of pre-existing planes due to changes in applied stresses S.-S. Xu, A.F. Nieto-Samaniego & S.A. Alaniz-Álvarez Universidad Nacional Autónoma de México, Centro de Geociencias, Querétaro, Qro., México ABSTRACT: In this work, we analyze the characteristics of three-dimensional Mohr diagram. Based on this analysis, the conditions of reactivation of pre-existing planes on a Mohr diagram due to changes in applied stress state are investigated. Our results indicate that: (1) On a three-dimensional Mohr diagram, one point, which is an intersection of three cycles (arcs) with direction angles θ 1, θ 2 and θ 3, indicates a stress state in terms of shear and normal stresses, which represents four non-parallel planes due to the orthorhombic symmetry of the stress tensor. This implies that four planes may be reactivated, as long as a point on the diagram is located above the critical slip line; (2) The reactivated planes that originally had the identical normal and shear stresses can have two different angles of pitch; (3) If the planes represented by a point on the diagram rotate a magnitude about a certain axis, some of them could be reactivated, whereas the others could not be reactivated; (4) Reactivation of a pre-existing plane is dependent on not only change in the maximum differential stress (σ 1 σ 3 ), but also the value of intermediate stress (σ 2 ). No matter what the maximum differential stress increases or decreases or maintains constant, a pre-existing plane may be reactivated due to changes in any principal stresses. (1) The range of the dips of the reactivated planes is larger for the smaller values of coefficient of friction µ and cohesion C. Also, the range of dip of the reactivated planes increases or decreases as the magnitudes of the principal stresses change. 1 INTRODUCTION Two-dimensional Mohr diagram is widely used in structural geology, seismology, soil mechanics, engineering geology etc (e.g. Sibson 1985, Streit & Hillis 2002). Three-dimensional Mohr diagram is also used to explain mechanism of faulting and reactivation of pre-existing fault (e.g. Yin & Ranalli 1992, Jolly and Sanderson 1997, McKeagney et al. 2004). Triaxial stress state has two general conditions: (a) σ 1, σ 2, and σ 3 have non-zero values; (b) σ 1 >σ 2 >σ 3, and can be tensile or compressive. The measurements of in-situ stress indicate that the crustal stress is generally in three-dimensional stress state (e.g. Hast 1969, Tsukahara et al. 1996). In this way, mechanical behavior of crustal rocks should be explained by threedimensional Mohr diagram (e.g. Jaeger & Cook 1979). Crustal stress state could be considered as the result of superimposition from some sub-stress tensors. The main regional sub-stress tensors are lithostatic, pore fluid, and tectonic stress tensors (e.g. Fleitout 1991, Tobin & Saffer 2009). Local sub-stress tensors can be thermal stress tensor, stress tensor due to chemical changes, etc. The changes of any sub-stress tensors will alter the stress state. In this way, the pre-existing planes could be reactivated. In this paper, we will explain this mechanism of reactivation by using 3D Mohr diagram. 2 CONSTRUCTION OF A 3D MOHR DIAGRAM According to Ramsay (1967) and Moeck et al. (2009), the normal stress (σ) on a plane is expressed by where n i is direction cosine related to principal stress σ ii. Also, the total stress on the plane is calculated by where τ is maximum shear stress. On the other hand, since n i is a unit vector, we can have By resolving these three equations, the following three results can be obtained 739

Figure 1. Construction of a 3D Mohr diagram. Three families of concentric circles are shown at center O 12,O 31,O 23 in the σ 1 σ 2, σ 3 σ 1, and σ 2 σ 3 planes, respectively. The common region (grey area) of three families of concentric circles represents the stress state on all planes in three dimensions. Equations (4), (5), (6) can also be written as following forms at (σ 1 + σ 3 )/2 by giving the values of n 2 equal to from 1 to 1, or direction angle θ 2 equal to from 0 to 360. For these circles, the maximum diameter is (σ 1 σ 3 )/2 when n 2 = 0, and the minimum diameter is ((σ1 σ 3 )/2) 2 (σ 2 σ 3 )(σ 1 σ 2 ) when n 2 =±1. In the same way, from equation (9) other concentric circles can be drawn on the diagram at a center (0, (σ 1 + σ 2 )/2), given the values of n 3 equal to from 1 to 1. The minimum diameter of these circles is (σ 1 + σ 2 )/2, and the maximum diameter is ((σ 1 σ 2 )/2) 2 + (σ 3 σ 1 )(σ 3 σ 2 ). In this way, six typical circles are drawn as shown in Figure 1. The common area for all circles is shown as grey. This area is enclosed by three circles: ( σ σ 2 + σ 3 ) 2 2 + τ 2 = ( σ 2 σ 3 ) 2, ( 2 σ σ 1 + σ 3 ) 2 2 + τ 2 = ( σ1 σ 3 ) 2, ( 2 and σ σ 1 + σ 2 ) 2 2 + τ 2 = ( σ 1 σ 2 ) 2. 2 These three circles present the Mohr circles on three principal planes, respectively. 3 CHARACTERISTICS OF THE REACTIVATED PLANES ON 3D MOHR DIAGRAM As shown above, the applied stress on an arbitrary plane under a stress state is dependent on the orientation of the plane. According to Mohr-Coulomb theory, for a pre-existing plane, the critical condition to slip is These three equations have the form (x a) 2 + y 2 = r 2, which is the formula for a circle centered at x = a, y = 0. Therefore, equation (7) represents a series of circles that are centered at (σ 2 + σ 3 )/2 with the values of n 1 varying from 1 to 1 or direction angel θ 1 equal to from 0 to 360. The minimum diameter is (σ 2 σ 3 )/2, for which n 1 = 0. The maximum diameter is ((σ 2 σ 3 )/2) 2 + (σ 1 σ 2 )(σ 1 σ 3 ), for which n 1 =±1. Similarly, from equation (8) we can obtain a series of concentric circles with a center where τ is the magnitude of shear stress and σ is normal stress on the pre-existing plane; C is the shear strength on the pre-existing plane when σ is zero, and µ the coefficient of friction on the pre-existing plane. For this equation, only above half of the Mohr diagram is used for common analysis. Equation (10) indicates that on a reactivated plane there is not only shear stress but also normal stress. On the Mohr diagram, above the slip envelope, there is stress difference enough to initiate slip for a range of pre-existing plane orientations. In this region, the states of stress are unstable for slip. On the other hand, below the slip envelope, slip will not occur. On the 3D Mohr diagram, four types of reactivated planes can be distinguished (Fig. 2). On the first type of planes, the normal stress is compressional as shown by the area with vertical black lines in Figure 2. On second type of planes, no normal stress Figure 2. There are four types of reactivated planes according to the normal and stress on the planes. When all principal stresses are larger than zero, the normal stress on the planes is positive (Fig. 2a). If minimum principal stress is less than zero, other types of reactivated planes may appear (Fig. 2b, 2c). On one type of planes, only shear stress exists (points on line AB). On another type of planes, the normal stress is negative (grey area in figures 2b, 2c). Specially, for point G, there is only extensional stress. 740

exists and there is only shear stress. These planes are expressed by the points on line AB in Figure 2b and Figure 2c. For the third type of planes, the normal stress is tensional on them. The grey area represents this type of planes in Figures 2b, 2c. The forth type of planes is vertical, whose strike is parallel to the maximum principal stress (σ 1 ) and perpendicular to the minimum principal stress (σ 3 ). On these planes, there is only tensional stress and there is no shear stress. For example, the planes on G in Figure 2c are this type of planes. As commonly known, the function of cosine is periodic, whose value is from 1 to 1. Therefore, for a certain value of direction cosine (n i ), two direction angles can be obtained in a period. For example, for n 1 = 0.5, two direction angels are θ 1 = 60 and θ 1 = 300 or 60. According to equations (3), only two values of n 1, n 2, and n 3 are independent. In terms of the theory of permutation and combination, the planes represented by a point on the diagram are then equal to 2 2 = 4. The four planes are with direction angles (θ 1, θ 2, θ 3 ), (θ 1, θ 2, θ 3 ), (θ 1, θ 2, θ 3 ), and (θ 1, θ 2, θ 3 ), where 0 θ i 90. Specially, in solid mechanics, the stress on a right octahedron is always proposed (Pitarresi & Shames 1999). Among eight planes, there are pairwise symmetric planes related to origin of coordinates, their direction cosines have opposite signs. For example, for plane (θ 1, θ 2, θ 3 ), its symmetric plane is ( θ 1, θ 2, θ 3 ). The pairwise symmetric planes are parallel to each other with only different facings or normal directions. One is downward or face-down normal, and the other is outward or face-up normal. Based on this feature, if the shear stress is expressed by the absolute value, a point on the 3D Mohr diagram with the same resolved stresses represent 4 independent planes. The normal stress on octahedral planes is (σ 1 + σ 2 + σ 3 )/3 and the shear stress is (σ 1 σ 2 ) 2 + (σ 2 σ 3 ) 2 + (σ 3 σ 1 ) 2 /3.The direction angels are θ 1 = θ 2 = θ 3 = 54 45, and the four planes can be (54 45,54 45,54 45 ), (54 45, 54 45, 54 45 ), (54 45, 54 45, 54 45 ), and (54 45, 54 45,54 45 ). According to Sibson (1985), for the twodimensional case, the stress condition for reactivation of a plane with a dip of θ k to σ 1 is where p is pore pressure and µ is friction coefficient. This indicates that only two planes represented by a point on 2D Mohr diagram could be reactivated under the 2D stress state. This is different from those for the 3D stress state. The maximum shear stress vector is parallel to the slickenlines on the fault plane (e.g. Etchecopar et al. 1981). According to Bott (1959), the pitch (R) ofaset of slickenlines can be calculated Figure 3. Morh diagram explaining the effects of block rotation. In (a), (b), (c), effects of block rotation are shown. The method ofallmendinger (2002) is used to calculated rotation. (a) Planes 1 and 2 are two crosscutting planes. The attitude of plane 1 is 135 /60 SW, and plane 2, 45 /60 SE. Points 1 and 2 are the projections of planes 1 and 2 after rotation. The axis of rotation is 360 /0 N, and rotation angle is clockwise 30. After rotation, plane 1 is moved to the point 1 that is located in the slip area, and plane 2 is moved to point 2, that is farther from the criterion line τ = C + µσ than before rotation. (b) The axis of rotation is 90 /0 E, and rotation angle is 30 clockwise. After rotation, two planes are still below and farther from the critical line of slip. (c) The axis of rotation is 90 /0 E, and rotation angel is 30 anticlockwise. After rotation, two planes are located above the critical slip line. Because the values of n 1, n 2 and n 3 for a point on the diagram can be either positive or negative, the value of tanr may be positive and negative depending on the signs of n 1, n 2 and n 3. Here, the pitch of slip defied as an angle ranging from 0 to 180 measured from the strike to the slickenline on the plane. Therefore, for a given value of tanr, two values of angle can be obtained. Similarly, by using the minus sign for value of tanr, we can obtain other two values of angle. This indicates that the four planes may have two pitch 741

Figure 4. Cases of reactivation of pre-existing planes due to changes in the principal stresses during which the maximum differen-tial stress is not changed. σ 01 - Original maximum principal stress; σ 02 - Original intermediate principal stress; σ 03 Original minimum principal stress; σ 1 - Original maximum principal stress after change; σ 2 Original intermediate principal stress after change; σ 3 Original minimum principal stress after change. For all cases, P 1 > 0, P 2 > 0, and P 3 > 0. angles of slikenlines. For example, for tanr = 2, the pitch can be 63, 117. The senses of the slickenlines can determine that the faults are normal-oblique or inverse-oblique. 4 EFFECT OF BLOCK ROTATION The axes of the Mohr circle have no geographic significance. Therefore, in order to study the effect of block rotation, the geographic axes are assumed parallel to the principal stress direction as shown in Figure 3. It worth pointing out that in practice, the principal axes are rarely parallel to the geographic north. Both the strike and dip of a fault could be changed during block rotation. As a result, the applied stress on the fault plane will be changed. Here, for simplicity, only two planes are shown in Figure 3. There are three results of rotation if two pre-existing planes below the slip envelope rotate. First, after rotation, one plane is located above the slip envelope, whereas another is still located below the slip envelope and farther to it than before rotation (Fig. 3a). For this scenario, the two planes do not induce interaction. Second, after rotation, two planes are still located in the stable region of slip and farther to the slip envelope than before rotation (Fig. 3b). In this case, two planes cannot be initiate slip. Third, after rotation, two planes are located in the unstable region of slip (Fig. 3c). In this case, the two planes become to slip and there may be a kinematic interaction between two planes. These results imply that if the planes represented by a point on the Mohr diagram rotate a certain degree, not all of them can be reactivated. 5 EFFECT OF CHANGES IN THE APPLIED PRINCIPAL STRESSES Crustal stress state could be considered as a combination of sub-stress tensors. The common known sub-stress tensor is lithostatic stress tensor. If lithostatic stress tensor is superimposed by pore fluid or tectonic stress tensor or any other local sub-stress 742

Figure 5. Cases in which the pre-existing planes are reactivated due to changes in the principal stresses with decrease in the maximum differential stress. The signs of σ 01, σ 02, σ 03, σ 1, σ 2, and σ 03 have the same meaning as in Fig. 4. For all cases, P 1 > 0, P 2 > 0, and P 3 > 0. tensors such as thermal stress tensor, stress tensor due to chemical changes etc., the magnitudes of the principal stress could be changed. As a result, according to equations (7), (8) and (9) the positions of 3D Mohr circles can be translated along the axis σ on the 3D Mohr diagram. In this way, positions of the pre-existing planes could be changed and would be reactivated when they are located above the critical slip line. Crustal stresses are quite inhomogeneous. For example, Tsukahara et al. (1996) obtained that the fracture zone has small differential stress (σ 1 σ 3 )in Ashio, Japan. They show that the differential stress is large in the earthquake swarm region. But, it is extremely small at narrow zones adjoining fracture zones. If one of the three principal stresses is changed, the differential stress may be altered. On the 3D Mohr diagram, three trends of maximum differential stress are studied. The first case is that the maximum differential stress maintains constant when a plane is moved to the location above the critical slip line (Fig. 4). Five sub-cases can be distinguished. The important for these sub-cases is that change in only the intermediate principal stress can produce reactivation of a plane (Figs. 4b, 4c), and high pore fluid pressure always cause some plane to be reactivated (Fig. 4d). The second case is that the maximum differential stress decreases when a pre-existing plane initiates slip (Fig. 5). Four sub-cases are distinguished. These cases could be the results of combinations of high pore fluid pressure and tectonic stress. For example, the sub-case in Figure 5c may represent the following combination: (a) The pore fluid stress is P 1 ; (b) The tectonic stress is tensional and is applied in the plane of σ 02 σ 03, whose components in the σ 03 and in σ 02 is less than P 1. The third case is that the maximum differential stress increases when the stress state of a pre-existing plane reaches critical slip condition. Only five but not all the sub-cases are presented in Figure 6. For example, the condition in Figure 6f is σ 1 = σ 01 + p 1, σ 2 = σ 02 + p 2, and σ 3 = σ 03 + p 3. This condition can be further divided into some sub-cases such as p 1 < p 2 < p 3, p 3 < p 2 < p 1, p 1 < p 3 < p 2, etc., where p 1, p 2, and p 3 are large than zero. The superimposed sub-stress tensors in these cases could be more complicated than those in Figure 5. For example, the static stress can be changed due to co-seismic dislocations. 743

Figure 6. Cases of reactivation of pre-existing planes due to changes in the principal stresses during which the maximum differential stress increases. The signs of σ 01, σ 02, σ 03, σ 1, σ 2, and σ 3 have the same meaning as in Fig. 4. For all cases, P 1 > 0, P 2 > 0, and P 3 > 0. These induced changes in static stress on neighboring faults that may delay, advance, or trigger impending earthquakes (e.g. King et al. 1994, Muller et al. 2006). The above cases indicates that reactivation of a plane is dependent on not only the maximum differential stress, but also the intermediate stress. No matter how the maximum differential stress changes (increases or decreases or maintains constant), a pre-existing plane could be reactivated after certain changes of magnitudes in principal stresses. 6 RANGE OF THE DIPS OF REACTIVATED PLANES From the diagram, a range of dips of the reactivated planes can be evaluated. For simplicity, only the case in the normal fault regime is analyzed. In the normal fault regime, the direction angle related to maximum principal stress is the dip of a reactivated plane. The dips of reactivated planes are strong affected by the values of µ and C. Smaller the values of µ and C, lager the range of the dips of reactivated planes (Fig. 7). Then, if the values of µ and Care small enough, the dips can less than 45, which is consistent with equation (11). On the other hand, the change of applied stress can also influence the range of dips of the reactivated plane. This effect is shown in Figures 4, 5 and 6. (1) Case where the maximum differential stress is constant (Figs. 4b and 4c). The change of intermediate stress causes a little increase in the dips of the reactivated planes. High pore fluid pressure generally increases the range of the dips of the reactivated planes (Figs 4d, 4e and 4f). (2) Case in which the maximum differential stress decreases. In the sub-case in Figure 5b, the range of dips of the reactivated planes increases a little. Whereas in sub-cases in Figures 5c, 5d, 5e and 5f, the range of dips increases evidently. (3) Case where the differential stress increases. In any sub-cases, the range of the dips of the reactivated planes also increases (Fig. 6). In general, the range of the dips of the reactivated planes is dependent on the values of µ and C. Also, the range of dips for the reactivated planes changes with the principal stresses. 7 CONCLUSIONS In this paper, we analyze the characteristics of the reactivated planes on three-dimensional Mohr diagram. We 744

REFERENCES Figure 7. The range of dips (θ 1 ) of the reactivated planes changes in the normal fault regime due to the changes in the value of µ in (a) and the value of C in (b). obtain following results. (1) On a three-dimensional Mohr diagram, a point is determined by three Mohr circles. This point has unique combined values of shear and normal stresses. In real space, there are four planes with the same shear and normal stresses if the signs of stress are ignored. This implies that four planes may be reactivated, if a point on the diagram is located above the critical slip line. (2) The reactivated planes, on which there are the identical normal and shear stresses, can have two different pitches of the slickenlines. In this work, we also analyze changes in stress state on pre-existing planes on a Mohr diagram due to change of the applied stress. First, the effect of block rotation is analyzed. Our results indicate that if the magnitude of rotation about a certain axis is the identical for the planes represented by a point on the diagram, which one will be reactivated depends on magnitude and direction of the block rotation. On the other hand, reactivation of a pre-existing plane is not only dependent on change in the maximum differential stress. Under the constant differential stress, a pre-existing plane may also be reactivated due to appropriate changes in the intermediate principal stresses. Finally, three parameters such as the values of τ and C, the magnitudes of the principal stresses influence the range of the dips of the reactivated planes. High pore fluid pressure commonly increases the range of dips of the reactivated planes. Allmendinger, R.W. 2002. StereoWin for Windows: ftp://www.geo.cornell.edu/pub/rwa. Bott, M.H.P. 1959. The mechanics of oblique slip faulting. Geological Magazine 96: 109 117. Etchecopar, A., Vasseur, G., & Daigniéres, M. 1981. An inverse problem in microtectonics for the determination of stress tensors from fault striation analysis. Journal of Structural Geology 3: 51 65. Fleitout, L. 1991. What are the sources of the tectonic stresses? Philosophical Transactions: Physical Sciences and Engineering 337: 73 81. Hansen, D.L. & Nielsen, S.B. 2003. Why rifts invert in compression. Tectonophysics 373: 5 24. Hast, H. 1969. The state of stress in the upper part of the earth crust. Tectonophysics 8: 169 211. Jaeger, J.C. & Cook, N.W.G. 1979. Fundamentals of Rock Mechanics. New York: Chapman and Hall. Jolly, R.J.H. & Sanderson, D.J. 1997. A Mohr circle reconstruction for the opening of a pre-existing fracture. Journal of Structural Geology 19: 887 892. King, G.C.P., Stein, R.S., & Lin, J. 1994. Static Stress changes and the triggering of earthquakes. Bulletin of the Seismological Society of America 84: 935 953. McKeagney, C.J., Boulter, C.A., Jolly, R.J.H. & Foster R.P. 2004. 3-D Mohr circle analysis of vein opening, Indarama lode-gold deposit, Zimbabwe: implications for exploration. Journal of Structural Geology 26: 1275 1291. Moeck, I., Kwiatek, G. & Zimmermann, G. 2009. Slip tendency analysis, fault reactivation potential and induced seismicity in a deep geothermal reservoir. Journal of Structural Geology 31: 1174 1182. Muller, J.R., Aydin,A. & Wright, T.J. 2006. Using an elastic dislocation model to investigate static Coulomb stress change scenarios for earthquake ruptures in the eastern Marmara Sea region, Turkey. Geological Society, London, Special Publications 253: 397 414. Pitarresi, M.J. & Shames, I.H. 1999. Introduction to solid mechanics (3rd Edition). Prentice Hall. Ramsay, J.G. 1967. Folding and Fracturing of Rocks. New York: McGraw-Hill. Sibson, R.H. 1985. A note on fault reactivation. Journal of Structural Geology 7: 751 754. Streit, J.E. & Hillis, R.R. 2002. Estimating fluid pressures that can induce reservoir failure during hydrocarbon depletion. In: Rock mechanics conference. Texas: Irving, Paper SPE 78226. Tsukahara, H., Ikeda, R. & Omura, K. 1996. In-situ stress measurement in an earthquake focal area. Tectonophysics 262: 281 290. Tobin, H.J. & Saffer, D.M. 2009. Elevated fluid pressure and extreme mechanical weakness of a plate boundary thrust, Nankai Trough subduction zone. Geology 37: 679 682. Yin, Z.M. & Ranalli, G. 1992. Critical stress difference, fault orientation and slip direction in anisotropic rocks under non-andersonian stress systems. Journal of Structural Geology 14: 237 244. ACKNOWLEDGEMENT This work was supported by the 049049 and 089867 Conacyt projects of Mexico. 745