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

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*World Journal of Mechanics*,

**10**, 199-209. doi: 10.4236/wjm.2020.1011014.

1. 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 synthetic shears (R-shears) and antithetic shears (R’-shears) were defined to be Riedel shears [19] [20]. Other secondary structures in a wrench zone include P-shears, Y-shears, and convergent structures like folds and reverse faults [21] [22] [23].

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.

2. 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
${\sigma}_{1}^{L}$ and
${\sigma}_{3}^{L}$. 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

Figure 1. Types of wrench. (a) parallel, (b) divergent (transtensional), (c) convergent (transpressional). The fault planes are vertical.

Figure 2. Two types of pre-existing faults. (a) hidden pre-existing fault (F_{1}) with its propagation part of fault F_{2} after the sedimentation of layer L_{2}, (b) surface pre-existing fault (F_{3}). L_{1}-pre-growth layer, L_{2}-growth layer with identical rock mechanic properties with layer L_{1}.

the layer L_{2} occurs.

The rock deformation in the upper lithosphere is governed by Coulomb behavior, and the brittle fracture or frictional sliding applies for most the deformation in the upper lithosphere [28] [29]. So, this paper will focus on the discussions on brittle fractures. Typical rock failure criterions include Mohr-Coulomb failure criterion and Byerlee’s law [27].

In Mohr-Coulomb failure criterion (Figure 3),

$\tau ={\tau}_{0}+\mu {\sigma}_{n}={\tau}_{0}+{\sigma}_{n}\mathrm{tan}\psi $ (1)

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).

$\tau ={\tau}_{f}+\mu {\sigma}_{n}={\tau}_{f}+{\sigma}_{n}\mathrm{tan}\psi $ (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

${\tau}_{f}=\frac{{h}_{2}}{{h}_{1}+{h}_{2}}{\tau}_{0}=K{\tau}_{0}$ (3)

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 the cohesion of a weak fabric to the cohesion of the intact homogenous rock. The Equation (3) is shown in Figure 3. For the surface pre-existing fault (Figure 2(b)), the cohesion is considered to be zero because the fault F_{3} is a syn-sedimentary fault and the layer L_{2} is faulted. The Equation (2) becomes to be the Byerlee’s law which is

$\tau =\mu {\sigma}_{n}={\sigma}_{n}\mathrm{tan}\psi $ (4)

which equation is shown in Figure 3 as well.

Figure 3. Failure criteria. τ_{0} is the cohesion of intact rock. τ_{f} is cohesion of hidden pre-existing fault. Ψ is the inner frictional angle of rock. The triangle Δjo_{1}h is the range for surface pre-existing faults with zero cohesion to rejuvenate before new Coulomb fractures occur. The triangle Δgo_{1}f is the range for hidden pre-existing faults with certain cohesion to rejuvenate before new Coulomb fractures occur.

For a pre-existing fault, the stresses on the fault plane are σ_{n} and τ. The stress Mohr circle for the local induced
${\sigma}_{1}^{L}$ and
${\sigma}_{3}^{L}$ stresses crosses through the points (σ_{n}, τ) and (0, −τ). Where the normal stress (σ_{n}) is positive, a transpressional wrench will occur, in which situation the center of the
${\sigma}_{1}^{L}\text{-}{\sigma}_{3}^{L}$ circle is on the positive part of the σ axis. Oppositely, where the normal stress (σ_{n}) is negative, a transtensional wrench will occur. Then the center of the
${\sigma}_{1}^{L}\text{-}{\sigma}_{3}^{L}$ 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.

3. 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(a) is a pre-existing fault (PF) and its applied stress state while a left-handed transpressional or transtensional wrench occurs. Figure 4(b) shows local induced normal stress and shear stress. Figure 4(c) display Mohr stress circles for applied stress state (σ_{1} - σ_{3} circle) and local induced stress state (
${\sigma}_{1}^{L}\text{-}{\sigma}_{3}^{L}$ circle) in transpressional wrenches. Figure 4(d) shows the geometric relationships between local induced maximum principal stress (
${\sigma}_{1}^{L}$ ) and the pre-existing fault (PF) in transpressional wrenches. Figure 4(e) shows Mohr stress circles for applied stress state (σ_{1} - σ_{3} circle) and local induced stress state (
${\sigma}_{1}^{L}\text{-}{\sigma}_{3}^{L}$ circle) in transtensional wrenches. Figure 4(f) shows the geometric relationships between local induced minimum principal stress (
${\sigma}_{3}^{L}$ ) and the pre-existing fault (PF) in transtensional wrenches. The n is the normal of the pre-existing fault, the PF the pre-existing fault, the ψ the inner frictional angle, the β the axis-fault angle either between the

Figure 4. Direction of local induced principal stress axis around a rejuvenating pre-existing fault. See the text for the meanings of the letters.

local induced maximum principal stress axis (
${\sigma}_{1}^{L}$ ) and the normal of the PF where transpressional wrench occurs or between the local induced minimum principal stress axis (
${\sigma}_{3}^{L}$ ) and the normal of the PF where transtensional wrench occurs, the α the angle between the maximum principal stress axis (σ_{1}) and the normal of the PF. The σ_{1} - σ_{3} circle with the circle center of o_{1} is Mohr stress circle for applied maximum and minimum principle stresses σ_{1} and σ_{3}. The
${\sigma}_{1}^{L}$ &
${\sigma}_{3}^{L}$ circle is Mohr stress circle with the circle center of o_{2} for local induced maximum and minimum principle stresses
${\sigma}_{1}^{L}$ and
${\sigma}_{3}^{L}$. 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(b)). A Mohr cicle can be gotten crossing the point of p (σ_{n}, τ) and the point (0, −τ) (the bigger cicle, Figure 4(c)). The local induced principal stresses of
${\sigma}_{1}^{L}$ and
${\sigma}_{3}^{L}$ are created. The angle between the
${\sigma}_{1}^{L}$ and the normal of the pre-existing fault is β (Figure 4(d)). In terms of the Figure 4(c), we have

$\mathrm{tan}2\beta =\left(\frac{\stackrel{\xaf}{ro}}{\stackrel{\xaf}{{o}_{2}q}}+2\right)\mathrm{tan}\psi $ (5)

where β is called the axis-fault angle, being the angle between the local induced maximum principal stress axis
${\sigma}_{1}^{L}$ and the normal of the pre-existing fault where a transpressional wrench occurs(Figure 4(d)). The axis-fault β is measured clockwise from the normal of the pre-existing fault to the local induced maximum principal stress axis
${\sigma}_{1}^{L}$ (Figure 4(d)). The
$\stackrel{\xaf}{ro}$ is the length of the line section ro and the
$\stackrel{\xaf}{{o}_{2}q}$ is the length of the line section o_{2}q.

If a negative normal stress acts on the pre-existing fault, this fault is a left-handed transtensional one (Figure 4(e) and Figure 4(f)). We have

$\mathrm{tan}2\beta =\left(\frac{\stackrel{\xaf}{ro}}{\stackrel{\xaf}{{o}_{2}q}}-2\right)\mathrm{tan}\psi $ (6)

where β is the angle between the local induced minimum principal stress axis
${\sigma}_{3}^{L}$ and the normal of the pre-existing fault where a transtensional wrench occurs (Figure 4(e), Figure 4(f)). The axis-fault β is measured anticlockwise from the normal of the pre-existing fault to the local induced minimum principal stress axis
${\sigma}_{3}^{L}$ (Figure 4(f)). The
$\stackrel{\xaf}{ro}$ is the length of the line section ro and the
$\stackrel{\xaf}{{o}_{2}q}$ is the length of the line section o_{2}q.

Integrating Equations (5) and (6), we have

$\mathrm{tan}2\beta =\left(\frac{\stackrel{\xaf}{ro}}{\stackrel{\xaf}{{o}_{2}q}}\pm 2\right)\mathrm{tan}\psi $ (7)

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 ${\sigma}_{1}^{L}$ (Figure 4(d)) which in the meantime is measured anticlockwise from the normal of the pre-existing fault to the local induced minimum principal stress axis ${\sigma}_{3}^{L}$ (Figure 4(f)).

Because

$\stackrel{\xaf}{ro}=\frac{{\tau}_{f}}{\mu}$ (8)

Then we can substitute Equations (3) and (8) into Equation (7), and we get

$\mathrm{tan}2\beta =\left(\frac{K{\tau}_{0}}{\mu \cdot \stackrel{\xaf}{{o}_{2}q}}\pm 2\right)\mathrm{tan}\psi =\frac{K{\tau}_{0}}{\stackrel{\xaf}{{o}_{2}q}}\pm 2\mathrm{tan}\psi $ (9)

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.

4. A Surface Pre-Existing Fault

Given a thin growth layer L_{2} and a large inner frictional coefficient, the cohesion coefficient approximately equals to zero. The Coulomb failure criterion line moves to cross through the coordinate origin o (Figure 6(a)). On a transpressional wrench,

$\mathrm{tan}2\beta =2\mathrm{tan}\psi $ (10)

or

$\beta =\frac{\mathrm{arctan}\left(2\mathrm{tan}\psi \right)}{2}$ (11)

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.

Figure 5. Stress state in a transpressional wrench fault with a zero cohesion. n-normal of the pre-existing fault, PF-pre-existing fault, ψ-inner frictional angle, β-axis-fault angle between the local induced maximum principal stress axis (
${\sigma}_{1}^{L}$ ) and the normal of the PF, α-angle between the maximum principal stress axis (σ_{1}) and the normal of the PF.

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 $\stackrel{\xaf}{{o}_{2}q}$ 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.

5. Conclusions

There are two kinds of pre-existing faults in the crust, one of which is called

Figure 6. Comparison among the fractures in various wrench faults with zero cohesion. (a) left-handed transpressional wrench, (b) left-handed parallel wrench, (c) left-handed transtensional wrench with a cohesion of 0. R’-antithetic shear, R-synthetic shear, T-tensional fracture, ω-angle between the applied minimum principal stress axis and the pre-existing fault.

hidden pre-existing faults composed by a buried fault and its overlying sedimentary layer. The other is called surface pre-existing faults with zero cohesion. Quantitative geometric relationships between the pre-existing faults and the fractures related to their transpressional and transtensional wrenches are established based on the Coulomb fracture criterion and Byerlee’s frictional law for brittle deformation regimes. These geometric relationships are controlled by both the cohesion coefficients and the applied stresses for a given rock, where the cohesion coefficient refers to the ratio of the thickness of the unfaulted covering layer to the sum thickness of the whole wrench body. This cohesion coefficient can also be understood to be the ratio of a pre-existing weak fabric to an intact homogeneous rock.

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. For the transtensional wrenches of the pre-existing faults with some cohesion, the geometric relationships of the local induced stresses to the pre-existing fault are determined by the applied stresses. In a surface pre-existing fault without cohesion, there would be no wrench related fracture and the directions of the fractures are determined by the applied stresses. For parallel wrenches, there are certain relationships between the wrench related fractures and the principal displacements.

Although the theoretical geometric relationships between the pre-existing faults and their wrench related fractures are derived based on left handed wrenches, they are of referring significance for right handed wrenches. These theoretical geometric relationships can help to understand and analyze wrench zones both in outcrops and in subsurface like oil bearing areas. They are of important guidance in petroleum exploration. The pre-existing faults can also be considered to pre-existing fabrics, and the cohesion coefficient is the ratio between the cohesion of the pre-existing weak fabric and that of the intact homogenous rock.

Acknowledgements

This study was jointly funded by the “Mechanism of deep hydrocarbon migration and enrichment in key areas of Sichuan basin” (no.XDA14010306), the “National key research and development plan-“ultra-deep layer, Mesoproterozoic and Neoproterozoic cap sealing property and oil-gas preservation mechanism” (no.2017YFC0603105), the “Development in Large-scale oil-gas field and coalbed methane project”—“Reservoir formation conditions and controlling factors in deep-ultra-deep Cambrian in Tarim basin” (no.2017ZX05005-002-005) and the “Quantitative characterization on various types of strike-slip faults in Jiyang depression” (no.30200018-19-ZC0613-0118). The author will thank Zongpeng Chen for his revision of English draft.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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