_{1}

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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 determine 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.

Wrench zones and their related structures were common both in outcrops and in oil-bearing areas [

There are three types of strike-slip faults: pure strike-slip, transtensional and transpressional wrenches (

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 [

Based on rock failure criterions like the Byerlee’s law [

The pre-existing faults are classified in two types, a hidden one and the other surface one (_{1}) and a minimum stress (σ_{3}). The local induced principal stresses caused by the wrenches of pre-existing faults are σ 1 L and σ 3 L . The fault F_{1} in _{2} forms the fault F_{2}. In _{3} keeps alive when the sedimentation of

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 [

In Mohr-Coulomb failure criterion (

τ = τ 0 + μ σ n = τ 0 + σ n tan ψ (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 (

τ = τ f + μ σ n = τ f + σ n tan 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 (

τ f = h 2 h 1 + h 2 τ 0 = K τ 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 _{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

τ = μ σ n = σ n tan ψ (4)

which equation is shown in

For a pre-existing fault, the stresses on the fault plane are σ_{n} and τ. The stress Mohr circle for the local induced σ 1 L and σ 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 σ 1 L - σ 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 σ 1 L - σ 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.

For a given hidden pre-existing fault with a cohesion of τ_{f} (_{1} and σ_{3}. _{1} - σ_{3} circle) and local induced stress state ( σ 1 L - σ 3 L circle) in transpressional wrenches. _{1} - σ_{3} circle) and local induced stress state ( σ 1 L - σ 3 L circle) in transtensional wrenches.

local induced maximum principal stress axis ( σ 1 L ) and the normal of the PF where transpressional wrench occurs or between the local induced minimum principal stress axis ( σ 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 σ 1 L & σ 3 L circle is Mohr stress circle with the circle center of o_{2} for local induced maximum and minimum principle stresses σ 1 L and σ 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 (_{n}, τ) and the point (0, −τ) (the bigger cicle,

tan 2 β = ( r o ¯ o 2 q ¯ + 2 ) tan ψ (5)

where β is called the axis-fault angle, being the angle between the local induced maximum principal stress axis σ 1 L and the normal of the pre-existing fault where a transpressional wrench occurs(_{2}q.

If a negative normal stress acts on the pre-existing fault, this fault is a left-handed transtensional one (

tan 2 β = ( r o ¯ o 2 q ¯ − 2 ) tan ψ (6)

where β is the angle between the local induced minimum principal stress axis σ 3 L and the normal of the pre-existing fault where a transtensional wrench occurs (_{2}q.

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

tan 2 β = ( r o ¯ o 2 q ¯ ± 2 ) tan ψ (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 σ 1 L (

Because

r o ¯ = τ f μ (8)

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

tan 2 β = ( K τ 0 μ ⋅ o 2 q ¯ ± 2 ) tan ψ = K τ 0 o 2 q ¯ ± 2 tan ψ (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.

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 (

tan 2 β = 2 tan ψ (10)

or

β = arctan ( 2 tan ψ ) 2 (11)

The angle between the local induced maximum principal stress axis and the pre-existing fault is 90-β (

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

In a parallel wrench, the 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

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 (

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

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.

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.

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

Chen, S.P. (2020) On the Orientation of Fractures with Transpressional and Transtensional Wrenches in Pre-Existing Faults. World Journal of Mechanics, 10, 199-209. https://doi.org/10.4236/wjm.2020.1011014