_{1}

^{*}

The formation conditions and time sequences for various types of wrench-related fractures are not clear. Based on a parabola-type failure criterion, this paper has gotten new insights on those questions. In a simple shear, the occurrence of either tensional fractures or Riedel shears is controlled by the ratio (
*R*
_{tc}) of tensile strength to cohesion. In a pure shear, the occurrence of either second order tensional fractures or second order Riedel shears is controlled by the ratio (
*R*
_{tci}) of tensile strength to cohesion, given a constant inner frictional coefficient. Where the
*R*
_{tc} or the
*R*
_{tci} is less than a certain value, the en echelon tensional fractures will occur first. Where the
*R*
_{tc} or the
*R*
_{tci} is bigger than the certain value, the Riedel shears will occur first. Where the
*R*
_{tc} or the
* R*
_{tci} is equal to the certain value, the en echelon tensional fractures and the Riedel shears will occur simultaneously. The understandings will enhance the research on wrench related fractures and will be of significance in petroleum exploration and development, because fractures are both important accumulation spaces and key migration paths for oil and gas.

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

The earliest physical modeling of a wrench zone was conducted in a mud model [

There are debates on the time sequences of various structures all the time. Bartlett et al. [

All kinds of structures may occur in an outcrop wrench belt or in a subsurface oil-bearing area [

Based on parabola-type fracture or failure criterion, this paper discusses the stress status and rock mechanics for the occurrence of T-tensional fractures and Riedel shears. Furthermore, the time sequence of their occurrence is addressed as well.

There are two end members of rock deformation patterns, the simple shear and the pure shear (

A parabola-type failure criterion is [

τ 2 = τ 0 2 σ I ( σ I + σ ) (1)

σ 2 + τ 2 = τ c 2 (2)

where τ is shear stress with positive sign for counter-clock shear and negative sign for clockwise shear. σ is normal stress with positive sign for compression

and negative sign for extension. τ_{0} is cohesion. σ_{I} is tensile strength under each isotension. τ_{c} is radius for an extreme circle (c-circle) (

Given σ I = R t c τ 0 and τ c = b τ 0 , the solution of Equation (1) and (2) is

σ = − τ 0 ± τ 0 1 − 4 R t c 2 ( 1 − b 2 ) 2 R t c (3)

Suppose

4 R t c 2 ( 1 − b 2 ) = 1 (4)

Equation (3) is now simply

σ = − τ 0 2 R t c (5)

where σ = −σ_{I} = −R_{tc}τ_{0}, there is one intersection point (

R t c = 2 2 (6)

or

σ I = 2 2 τ 0 (7)

In this case of rock mechanics like Equation (7), the tensional fractures and Riedel shears will occur instantaneously. Where the tensile strength is less than

2 2 τ 0 , tensional fractures will be dominant (

intersection angles between the tensional fractures and their en echelon axis, the wrench zone are 45˚ (

2 2 τ 0 , the Riedel shears will be dominant instead (

The angles between the shears and the wrench zone will vary with inner frictional coefficient.

In a pure shear, the first order fractures are two conjugate shears (_{fs} and τ_{fs} in a shear fracture (

τ f s = μ σ f s (8)

where μ is inner frictional coefficient. σ_{fs} is the normal stress of the intersection (P) between the σ_{1} - σ_{3} circle (FC) and the parabola-type failure criterion (PF) (_{fs}, τ_{fs}) and (0, −τ_{fs}). The second order extreme Mohr circle in formula form can be expressed as

( σ − σ f s 2 ) 2 + τ 2 = σ f s 2 4 ( 1 + 4 μ 2 ) (9)

Given σ_{I} = R_{tci}τ_{0} and only one value of σ, connecting Equation (9) and (1), we have

( σ f s − τ 0 R t c i ) 2 + 4 μ 2 σ f s 2 − 4 τ 0 2 = 0 (10)

and

σ = R t c i σ f s − τ 0 2 R t c i (11)

The σ_{1} - σ_{3} circle is

( σ f s − σ 1 + σ 3 2 ) 2 + τ f s 2 = ( σ 1 − σ 3 2 ) 2 (12)

The Equation (1) is now

τ f s 2 = τ 0 2 σ I ( σ I + σ f s ) = τ 0 R t c i ( R t c i τ 0 + σ f s ) (13)

Connecting (12) and (13), we have

( τ 0 R t c i ) 2 − 2 ( σ 1 + σ 3 ) τ 0 R t c i + ( σ 1 − σ 3 ) 2 − 4 τ 0 2 = 0 (14)

and

σ f s = R t c i ( σ 1 + σ 3 ) − τ 0 2 R t c i (15)

Connecting (11) and (15), we have

σ = R t c i ( σ 1 + σ 3 ) − 3 τ 0 4 R t c i (16)

where σ = −R_{tci}τ_{0}, a tensional fracture will occur and we have

R t c i = ( σ 1 + σ 3 ) 2 + 48 τ 0 2 − ( σ 1 + σ 3 ) 8 τ 0 (17)

or

σ 1 + σ 3 = ( 3 − 4 R t c i 2 ) τ 0 R t c i (18)

Substitute (18) into (14), we have

σ 1 = τ 0 2 R t c i ( 3 − 4 R t c i 2 + 5 − 4 R t c i 2 ) (19)

and

σ 3 = τ 0 2 R t c i ( 3 − 4 R t c i 2 − 5 − 4 R t c i 2 ) (20)

Substitute (18) into (15), we have

σ f s = ( 1 − 2 R t c i 2 ) τ 0 R t c i (21)

Substitute (21) into (10), we have

R t c i = 1 2 ± 1 2 1 1 + 4 μ 2 (22)

or

σ I = τ 0 1 2 ± 1 2 1 1 + 4 μ 2 (23)

In the case of rock mechanics like Equation (23), second order tensional fractures and Riedel shears will occur instantaneously. Whether a positive sign or negative sign in Equation (23) will be determined by the maximum (σ_{1}) and minimum (σ_{3}) principal stresses.

where the tensile strength (σ_{I}) is less than τ 0 1 2 ± 1 2 1 1 + 4 μ 2 , tensional fractures will be dominant. Where the tensile strength (σ_{I}) is bigger than τ 0 1 2 ± 1 2 1 1 + 4 μ 2 , Riedel shears will be dominant instead.

Study of the relationship between the rock mechanics and the time sequences and types of fractures in a wrench zone can help us explain some natural fractures or physical modeling fractures.

In simple shear, the tensional fractures and R-shears array to be en echelon belts, and penetrative principal displacement zones are absent. Because of that, the rock veins filled in en echelon T-fractures in

In pure shear, first order conjugate shear fractures should be formed first. Then, second order T-fractures or Riedel shears would be formed and delimited

by the first order fractures. In Keping uplift, Tarim basin, rock veins filled echelon T-fractures which were delimited by a penetrative left handed principal displacement zone, the first order shear fracture (

In physical modeling, the mechanic properties of the materials should be considered while the results are discussed. The Riedel shears are common in physical models for the tensile strength is small less than the cohesion [

Whether the first order fractures in a simple shear are tensional fractures or Riedel shears depends on tensile strength (σ_{I}) and the cohesion (τ_{0}). In a given

parabola-type failure criterion, if the tensile strength is less than 2 2 τ 0 or the ratio (R_{tc}) of tensile strength to cohesion is less than 2 2 , the tensional fractures will occur first. If the tensile strength is bigger than 2 2 τ 0 or the ratio (R_{tc}) of tensile strength to cohesion is bigger than 2 2 , the Riedel shears will occur first. If the tensile strength is equal to 2 2 τ 0 , the tensional fractures and

the Riedel shears will occur instantaneously. The first order fractures in a pure shear should be two conjugate shears. Subsequently, if tensile strength is less than τ 0 1 2 ± 1 2 1 1 + 4 μ 2 or the ratio (R_{tci}) of tensile strength to cohesion is less than 1 2 ± 1 2 1 1 + 4 μ 2 , the second order T fractures will occur first. If the tensile strength is bigger than τ 0 1 2 ± 1 2 1 1 + 4 μ 2 or the ratio (R_{tci}) of tensile strength to cohesion is bigger than 1 2 ± 1 2 1 1 + 4 μ 2 , the Riedel shears will occur first. If the tensile strength is equal to τ 0 1 2 ± 1 2 1 1 + 4 μ 2 , tensional fractures and Riedel shears will occur instantaneously. The positive sign or negative

sign will be determined by the maximum (σ_{1}) and minimum (σ_{3}) principal stress. The most important thing is the occurrence of tensional fractures or Riedel shears being determined by the relative magnitude between the tensional strength and the cohesion with the inner frictional coefficient.

The formats and figures (

The occurrence of tensional fractures or Riedel shears in a simple shear depends on tensile strength (σ_{I}) and the cohesion (τ_{0}). Where the ratio (R_{tc}) of tensile

strength to cohesion is less than 2 2 , the tensional fractures will occur first. Where the R_{tc} is bigger than 2 2 , the Riedel shears will occur first. Where the R_{tc} is equal to 2 2 , the tensional fractures and the Riedel shears will occur instantaneously.

The occurrence of second order tensional fractures or Riedel shears in a pure shear after the formation of first order conjugate shears depends on tensile strength (σ_{I}), the cohesion (τ_{0}) and the inner frictional coefficient (μ). Where the

ratio (R_{tci}) of tensile strength to cohesion is less than 1 2 ± 1 2 1 1 + 4 μ 2 , the second order T-fractures will occur first. Where the R_{tci} is bigger than 1 2 ± 1 2 1 1 + 4 μ 2 , the Riedel shears will occur first. Where the R_{tci} is equal to 1 2 ± 1 2 1 1 + 4 μ 2 , tensional fractures and Riedel shears will occur instantaneously.

This study was jointly funded by the “national key research and development plan-ultra-deep layer, Mesoproterozoic and Neoproterozoic cap sealing property and oil-gas preservation mechanism” (no.2017YFC0603105), the “Mechanism of deep hydrocarbon migration and enrichment in key areas of Sichuan basin” (no.XDA14010306), 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 Occurrences of Fractures in Wrench Zones. World Journal of Mechanics, 10, 27-38. https://doi.org/10.4236/wjm.2020.103003

FC: first order extreme stress circle

FS: first order shear fracture

P: intersection between the first order extreme stress circle (FC) and the parabola-type failure criterion (PF)

PF: Parabola-type failure criterion

R_{tc}: ratio of tensile strength to cohesion in simple shear

R_{tci}: ratio of tensile strength to cohesion in pure shear

R: synthetic Riedel shear (R-shear)

R': antithetic Riedel shear (R'-shear)

SC: second order extreme stress circle

T: tensile fracture (T-fracture)

α: inner frictional angle

σ: normal force

σ_{1}: maximum principal stress

σ_{3}: minimum principal stress

σ_{fs}: normal stress in a pre-existing shear fracture

σ_{I}: tensile strength

τ: shear stress

τ_{0}: cohesion

τ_{c}: radius for an extreme circle (c-circle)

τ_{fs}: shear stress in a pre-existing shear fracture