On the Orientation of Fractures with Transpressional and Transtensional Wrenches in Pre-Existing Faults

The orientation of fractures with transpressional and transtensional wrenches in pre-existing faults has not been quantitatively determined. Based on Coulomb failure criterion and Byerlee’s frictional sliding criterion, this paper has indicated quantitative geometric relationships between the pre-existing fault and the local induced principal stress axes caused by the rejuvenation of the pre-existing fault. For a hidden pre-existing fault with some cohesion, the angles between the local induced principal stress axes and the pre-existing fault quantitatively vary with the applied stress and the cohesion coefficient, the ratio of the thickness of the cover layer to the thickness of the whole wrench body, whether transpressional or transtensional wrenches occur. For a surface pre-existing fault with zero cohesion, the angles between the pre-existing fault and the local induced principal stress axes are related to the rock inner frictional angle regardless of both the applied stress and the cohesion coefficient where transpressional wrenches occur, and the local induced maximum principal stress axis is identical with the applied maximum principal stress axis where transtensional wrenches occur. Therefore, the geometric relationships between the pre-existing faults and their related fractures are defined, because the local induced principal stress axes deter-mine the directions of the related fractures. The results can be applied to pre-existing weak fabrics. They can help to understand and analyze wrench structures in outcrops or subsurface areas. They are of significance in petroleum exploration.


Introduction
Wrench zones and their related structures were common both in outcrops and in oil-bearing areas [1]- [11]. They are of significance in exploration of oil and gas [12] [13] [14] [15].
There are three types of strike-slip faults: pure strike-slip, transtensional and transpressional wrenches (Figure 1) [16]. The earliest physical modeling of a wrench zone was conducted in a mud model [17]. Based on that model, En echelon tensional fractures (T-fracture) and shear fractures were identified [18].
The rejuvenation of preexisting faults can be compared to be transtensional and transpressional wrench. Although there are certain geometric relationships between the fractures and the principal displacement zone in a pure strike-slip [21] [24], there is little analytical discussion on the geometric relationship between the fractures and the pre-existing faults with transtensional and transpressional wrenches [25].
Based on rock failure criterions like the Byerlee's law [26] and Mohr-Coulomb failure criterions [27], this paper discusses the geometric relationships between the transpressional or transtensional pre-existing faults and their related fractures in upper crust where brittle deformation occurs. The results will help to understand wrench related structures both in outcrops and in basins. They can help to the exploration of mineral resources, such as iron, oil and gas. Also, the results will help to analyze structures in a normal fault or a reverse fault.

Methodology
The pre-existing faults are classified in two types, a hidden one and the other surface one (Figure 2). A stress state is applied to cause their transtensional and transpressional wrenches with a maximum principal stress (σ 1 ) and a minimum stress (σ 3 ). The local induced principal stresses caused by the wrenches of pre-existing faults are 1 L σ and 3 L σ . The fault F 1 in Figure 2(a) is a hidden pre-existing fault whose rejuvenation after the sedimentation of the layer L 2 forms the fault F 2 . In Figure 2(b), the F 3 keeps alive when the sedimentation of τ is the shear stress with positive sign for counter-clock shear and negative sign for clockwise shear. σ n is the normal stress with positive sign for compression and negative sign for extension. τ 0 is cohesion. μ is inner frictional coefficient and ψ is inner frictional angle.
In a transtensional or transpressional pre-existing fault, the normal stress and shear stress on the fault plane are σ n and τ ( Figure 3). tan f n f n τ τ µσ τ σ ψ = + = + (2) can be easily derived with Equation (1), in which τ f is cohesion for the pre-existing fault. Given the layer L 1 and layer L 2 having identical rock mechanic properties, the cohesion of the hidden pre-existing fault (Figure 2(a)) can be represented as where the h 1 is the thickness of layer L 1 and the h 2 is the thickness of the layer L 2 and K is the cohesion coefficient, which also can be considered to be the ratio of  σ σ circle will be on the negative part of the σ axis and this would be meaningless in the surface pre-existing fault. Where the normal stress (σ n ) is zero, a pure wrench will occur.

A Hidden Pre-Existing Fault
For a given hidden pre-existing fault with a cohesion of τ f (Figure 4), it will rejuvenate under the applied stresses of σ 1 and σ 3 . Figure 4   σ . The r is the intersection of the failure criterion to the abscissa axis. The p is the intersection of the two Mohr stress circles. The pq is perpendicular to the abscissa axis.
If a positive normal stress (σ n ) acts on the pre-existing fault, this fault is a left-handed transpressional one. The plane parallel to the fault has a normal stress of σ n and a shear stress of τ (Figure 4(b)). The plane perpendicular to the fault has a normal stress of zaro and a shear stress of −τ (Figure 4 Integrating Equations (5) and (6) A plus sign is adopted for a transpressional wrench and a minus sign is adopted for a transtensional wrench accordingly. It will be noted that the axis-fault β is measured clockwise from the normal of the pre-existing fault to the local induced maximum principal stress axis 1 L σ (Figure 4(d)) which in the meantime is measured anticlockwise from the normal of the pre-existing  (Figure 4(f)).
Because f ro τ µ = (8) Then we can substitute Equations (3) and (8) into Equation (7), and we get which concludes that the axis-fault angle (β) is determined by both the cohesion coefficient (K) and the applied stress state for a given rock layer. The directions of fractures related to those wrenches could be determined based on the Coulomb failure criterion.

A Surface Pre-Existing Fault
The angle between the local induced maximum principal stress axis and the pre-existing fault is 90-β ( Figure 5(b)). The angle between the applied maximum principal stress axis and the pre-existing fault is 90-α. In terms of Coulomb failure criterion, the geometric relationships of the fractures accompanying the transpressional wrench of the pre-existing fault with zero cohesion to the pre-existing faults are shown in Figure 6(a). The angle between the pre-existing fault and the R shear would be 45˚ + ψ/2−β. The angle between the pre-existing fault and the T fracture would be 90˚ − β. The angle between the pre-existing fault and the R' shear would be 135˚ − β − ψ/2.
In a parallel wrench, the 2 o q is zero. In terms of the equation (9), the tan (2β) is infinitely great. So the 2β is equal to 90˚ and the β is equal to 45˚. This indicates the local induced maximum principal stress axis is 45˚ to the pre-existing fault, and the geometric relationships of the fractures to the pre-existing faults are shown in Figure 6(b). Considering all the possible R shears occurring in transpressional wrenches, the angles between the shears and the pre-existing faults will be within a range from ψ/2 ( Figure 6(b)) to 45˚ + ψ/2 − β ( Figure   6(a)). The angles between the possible R' shears and the pre-existing faults will be within a range from 90˚ − ψ/2 ( Figure 6(b)) to 135˚ − β − ψ/2 ( Figure 6(a)).
The angles between the possible T faults and the pre-existing faults will be within a range from 45˚ ( Figure 6(b)) to 90˚ − β ( Figure 6(a)).
In a transtensional wrench with a zero cohesion, the Equation (9) has no result and the local induced principal stress axes will be in the same direction as the applied principal stress axes for there is no friction along the fault surface.
The tensional fractures will be perpendicular to the applied minimum principal stress axis and have no relationships to the pre-existing fault (Figure 6(c)). In a transtensional wrench with some cohesion coefficient, the geometric relationships between the wrench related fractures and the pre-existing weak fabrics are determined by the applied stresses like the Equation (9), and they will not be discussed in detail in this paper.

Conclusions
There are two kinds of pre-existing faults in the crust, one of which is called For transpressional wrenches of the pre-existing faults, the angles between the R shears and the pre-existing faults will be within a range from ψ/2 to 45˚ + ψ/2−β, where the ψ is the rock inner frictional angle and the β is the angle between the normal of the pre-existing fault and the local induced principal stress axes. The angles between the possible R' shears and the pre-existing faults will be within a range from 90˚ − ψ/2 to 135˚ − β − ψ/2. The angles between the possible T faults and the pre-existing faults will be within a range from 45˚ to 90˚ − β.