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The reactivation of pre-existing faults is a common phenomenon in a basin. This paper discusses the relationship between the pre-existing faults and the newly formed Coulomb shear fractures regarding pore fluid pressures. Based on the Coulomb fracture criterion and Byerlee frictional sliding criterion, an equation relating pore pressure coefficient (
*λ*
_{e}), minimum dip angle (
*α*
_{e}) of the reactive pre-existing fault and the intersection point depth (
*z*) between the pre-existing fault and a newly formed Coulomb shear fault in an extensional basin, is established in this paper. This equation enhanced the understanding on the reactivation of pre-existing faults and can be used to calculate paleo-pore fluid pressures. The bigger the pore fluid pressure in a pre-existing fault is, the less the minimum dip angle for a reactive pre-existing fault will be. The minimum dip angle is less in shallow area than that in deep area. This will be of significance in petroleum exploration and development.

The reactivation of pre-existing faults is a common phenomenon in a basin [

Coulomb criterion or frictional sliding criterion is applicable in most of the deformation in the upper lithosphere which always is shown as:

τ = τ o + μ σ n = τ 0 + tan ϕ σ n (1)

where τ_{o} is cohesion, μ is coefficient of internal friction, ϕ is internal frictional angle and σ_{n} is effective normal stress [

In terms of the principal stresses, the Coulomb criterion for normal faults can be written to be [

σ 1 = ρ g z ( 1 − λ ) (2)

and

σ 1 − σ 3 = K − 1 K ρ g z ( 1 − λ ) + S K (3)

with

S = 2 τ o sin 2 θ f 1 + cos 2 θ f , K = 1 − cos 2 θ f 1 + cos 2 θ f (4)

and

2 θ f = 90 + ϕ , μ = tan ϕ (5)

where K is a parameter depending on the fracture angle; S is the fracture strength under uniaxial compression with zero confining pressure; θ_{f} is the fracture angle; ϕ is the internal friction angle and λ is pore fluid pressure coefficient, the ratio of pore pressure to overburden pressure. In a rift basin, the maximum stress is vertical and the pore fluid pressure coefficient is [

λ = P ρ g z (6)

where P is pore fluid pressure, ρ is density of overlying rocks, g is gravity acceleration and z is depth.

For a pre-existing fault, its cohesion is zero and the frictional sliding criterion, for the same rocks becomes to be

τ = μ σ n = tan ϕ σ n (7)

where τ is critical shear stress, μ is frictional sliding coefficient equal to the internal frictional coefficient for a specific rock [_{n} is normal stress and ϕ is frictional angle. The frictional coefficient is 0.85 or 0.6 where the confining pressure is less than or larger than 200 MPa in Byerlee’s law.

Under the stresses σ_{1} and σ_{3}, corresponding to total stresses σ 1 t and σ 3 t , the pre-existing faults with their normal lines within the ΔOLM (_{e}.

The normal stress on the fault AB is

σ n = σ 1 t cos α e + σ 3 t sin α e − σ 1 t λ e (8)

with

σ 1 t = ρ g z , σ 3 t = 1 K ρ g z ( 1 − λ + K λ ) − S K (9)

The shear stress on the pre-existing fault AB is

τ = σ 1 t sin α e − σ 3 t cos α e (10)

According to Equation (7), we have

( σ 1 t sin α e − σ 3 t cos α e ) = μ ( σ 1 t cos α e + σ 3 sin α e t − σ 1 t λ e ) (11)

Given ρ = 2.7 g/cm^{3}, ϕ = 30˚, τ = 23 Mpa, g = 10 m/s^{2} and λ = 0.413 (a salt water density of 1.073 g/cm^{3} is assumed), in terms of the Equations (4) and (5), we get

S = 79.67 , K = 3 (12)

In terms of the Equations (2) and (7), we get

σ 1 t = 27 z (13)

and

σ 3 t = 16.43 z − 26.56 (14)

where the unit of σ 1 t and σ 3 t is MPa and that of z is km. The depth z is defined to be the depth of the intersection point between a pre-existing fault and a newly formed Coulomb fracture (

Substituting Equations (13) and (14) into Equation (11) and considering μ = tan 30˚ = 0.577, we get

λ e = 2.03 cos α e − 1.14 sin α e − 1.71 cos α e + 1.03 sin α e z (15)

According to Equation (15) we know that there is a specific relationship between the pore fluid pressure (λ_{e}) and the minimum dip angle (α_{e}) in a reactive pre-existing fault and the intersection depth (z) between the pre-existing fault and a newly formed Coulomb shear fault. The pore pressure coefficient is rational in the range of 0 to 1 and the minimum dip angle is rational in the range of 0˚ - 60˚. For a typical rock with an inner frictional angle of 30˚, the dip angle of a normal fault, a Coulomb shear fracture with a maximum vertical stress and a minimum horizontal stress, is 60˚. As shown in _{e} and α_{e} is close to linear. We can get one of the three parameters like z, λ_{e} and α_{e} in terms of the Equation (15) or

For the cases with the same intersection depth of z, the pore fluid pressure coefficient in a reactive pre-existing fault will decrease with the increase of the minimum dip angle for the pre-existing fault (

Sharing the same dip angle, the bigger the intersection depth of z is, the bigger the pore fluid pressure coefficient is. Similarly, when pore pressure coefficient keeps the same, the minimum dip angle for reactivating a pre-existing fault will increase with the increase in the intersection depth z (_{e}_{1}) of pre-existing faults in less confining pressure is less than those (α_{e}_{2}) in higher confining pressure.

Rock deformation in the upper lithosphere is governed by Coulomb behavior, and the brittle fracture [

occurrence of new Coulomb fractures will be accompanied by reactivation of pre-existing faults to form fluid flowage paths [_{1}). The angle relationship between the fault dip and the maximum principal stress (σ_{1}) is not involved in the Byerlee frictional sliding criterion. Seldom work has been addressed on the effect of pore fluid pressures [

In terms of the Equation (15) and based on the analysis in section of implication of the equations, the effect of pore fluid pressures on the reactivations of pre-existing faults can be addressed. A high pore fluid pressure will decrease the minimum dip angles of reactive pre-existing faults. Paleo fluid pressure would be calculated and this will be helpful in determining fault sealing property. On the other hand, the minimum dip angles of reactive pre-existing faults will increase with the increasing depth in an extensional environment where the maximum principal stress is vertical (

Given certain rocks in a basin, a quantitative relationship between the pore fluid pressure (λ_{e}), the minimum dip angle (α_{e}) in a reactive pre-existing fault and the intersection depth (z) can be established. The intersection depth (z) refers to the depth of the intersection point between the pre-existing fault and a newly formed Coulomb shear fault. This relationship will help us understand both the reactivation of pre-existing faults and the pore fluid pressures in the pre-existing faults. Two improvements have been made on the reactivation of pre-existing normal faults. The first is that the pore fluid pressures affect the reactivations of pre-existing faults. A high pore fluid pressure will decrease the minimum dip angles of reactive pre-existing faults. This is of significance in petroleum exploration. The second is that the minimum dip angles of reactive pre-existing faults will increase with the increasing depth in an extensional environment. This is of significance in explaining some downward branching faults and some upward steepening faults.

This study is financially supported by the National Natural Science Foundation of China (No. 41572105, 41172124) and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA14010306).

Chen, S.P. and Chen, Z.P. (2018) On the Reactivation of the Pre-Existing Normal Fault. World Journal of Mechanics, 8, 210-217. https://doi.org/10.4236/wjm.2018.85016