Strain Associated with the Fault-Parallel Flow Algorithm During Kinematic Fault Displacement

Abstract

Deformation studies require that geological bodies are kinematically moved along faults. Fault-parallel flow is one of a small number of kinematic restoration algorithms developed for this purpose. This scale-independent method describes how material nodes are displaced parallel to the fault plane, in the direction of fault movement. The one-dimensional strain of linear objects and two-dimensional strain of bodies within the hanging-wall during the restoration is shown for all cutoff angles and all angles of fault bends. A line moving over a fault bend is either shortened or extended depending on its initial orientation. However, the elongation of the line is significantly different under shortening and extension, with respect to the fault bend angle. The geometries of compressional fault systems, in which faults change angle by about 20 to 40°, generate low values of elongation. Modeling of extensional faults, which typically have steeper dips (60 to 80°) and therefore have tighter fault bends, causes high, unnatural values of elongation. The calculated strain ellipse ratios are directly proportional to the fault bend angle, corroborating the one-dimensional results. The fault-parallel flow method should be used primarily to kinematically restore and forward-model compressional faults, and other faults where the fault bend angles do not exceed 40°.

Appendices

Appendix A: Calculation of Elongation and Change in Angles During Fault-Parallel Flow

The following equations were modified after Tanner et al. (2003), and are related to Fig. 10. They show shortening and extension of material lines for concave (case 1) and convex (case 2) fault bends. The material lines move along flow paths (dashed line) that are parallel to the fault surface, in the direction of movement. The two fault segments are defined by a flow deflector (fd) that bisects the angle of the fault bend apex. If two fault segments are defined by the angles θ ₁ and θ ₂, then η is equal to the bisecting angle. The equations are related to triangles in Fig. 10. Every point in Fig. 10 is defined by its x and y coordinates.

Fig. 10

The fault-parallel flow method for concave fault bends (case 1, η<90°) (a) shortening, (b) extension and convex fault bends (case 2, η>90°), (c) shortening, (d) extension (fd—flow deflector)

The calculation of the half fault bend angle η for all variations is defined as

(1)

where the angles θ ₁ and θ ₂ define the dip of the fault segments.

Appendix B: Concave Fault Bends, Case 1: 0°<η<90° for α ₁ (Fig. 10(a))

The original length of the line A–B (l ₀) is given by the sine rule on the triangle ABD,

(2)

The length h is the distance between the fault surface and the flow path line. Using the sine rule, h is given by

(3)

After deformation the length of line C–D (l ₁) is given by the triangle ACD,

(4)

and the new cutoff angle ϕ is, using the arc sine rule, given as

(5)

The elongation e of a line is defined as

(6)

Using the sine rule, the shear strain g is

(7)

in which the angular shear γ describes the change in angle between the original line (l ₀) and the deformed line (l ₁). γ is defined as

(8)

The strain ratio R _f is equal to the ellipticity of a body that was originally a circle and is given as

(9)

After displacement z along the flow path lines, the coordinates of the points C and D that define a line can be determined by the following equations

(10)

and

(11)

To restore initial line length and cutoff angle, the displaced hanging-wall line can be transformed by simple shear, parallel to the path flow lines of the second dip domain, to the new line C–E. The coordinates of E are defined as

(12)

Appendix C: Concave Fault Bends, Case 1: 0°<η<90° for α ₂ (Fig. 10(b))

For e and g see Eqs. (6) and (7). Other equations change accordingly. Thus

(13)

The equation for h is equivalent to Eq. (3), except angle α ₁ replaces α ₂. Thus

(14)

(15)

(16)

(17)

x _C , y _C and x _D , y _D are as in Eqs. (10) and (11). Thus

(18)

Appendix D: Convex Fault Bends, Case 2: 90°<η<180° for α ₂ (Fig. 10(c))

For e, g, l ₀, h, l ₁ and ϕ, see Eqs. (6) and (7), and (13) to (16),

(19)

(20)

(21)

(22)

Appendix E: Convex Fault Bends, Case 2: 90°<η<180° for α ₁ (Fig. 10(d))

(23)

(24)

Appendix F: Calculation of Two-Dimensional Area Change

The calculation of the original thickness of the hanging-wall block is defined as follows (see Fig. 7)

(25)

and

(26)

where t ₀ and t ₁ are the thicknesses before and after deformation. The areas A ₀ and A ₁ are given by

(27)

The length b and h _b are the sides of the parallelogram formed by the bed between flow lines (e.g. Fig. 7).