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A Fault-Cored Anticline Boundary Element Model Incorporating the Combined Fault Slip and Buckling Mechanisms

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We develop a folding boundary element model in a medium containing a fault and elastic layers to show that anticlines growing over slipping reverse faults can be significantly amplified by mechanical layering buckling under horizontal shortening. Previous studies suggested that folds over blind reverse faults grow primarily during deformation increments associated with slips on the fault during and immediately after earthquakes. Under this assumption, the potential for earthquakes on blind faults can be determined directly from fold geometry because the amount of slip on the fault can be estimated directly from the fold geometry using the solution for a dislocation in an elastic half-space. Studies that assume folds grown solely by slip on a fault may therefore significantly overestimate fault slip. Our boundary element technique demonstrates that the fold amplitude produced in a medium containing a fault and elastic layers with free slip and subjected to layer-parallel shortening can grow to more than twice the fold amplitude produced in homogeneous media without mechanical layering under the same amount of shortening. In addition, the fold wavelengths produced by the combined fault slip and buckling mechanisms may be narrower than folds produced by fault slip in an elastic half space by a factor of two. We also show that subsurface fold geometry of the Kettleman Hills Anticline in Central California inferred from seismic reflection image is consistent with a model that incorporates layer buckling over a dipping, blind reverse fault and the coseismic uplift pattern produced during a 1985 earthquake centered over the anticline forelimb is predicted by the model.
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doi: 10.3319/TAO.2015.06.18.01(TT)
* Corresponding author
E-mail: huang22@ncu.edu.tw
Terr. Atmos. Ocean. Sci., Vol. 27, No. 1, 73-85, February 2016
A Fault-Cored Anticline Boundary Element Model Incorporating the
Combined Fault Slip and Buckling Mechanisms
Wen-Jeng Huang1, * and Kaj M. Johnson 2
1 Graduate Institute of Applied Geology, National Central University, Taoyuan, Taiwan, R.O.C.
2 Department of Geological Sciences, Indiana University, Indiana, U.S.A.
Received 30 September 2014, revised 12 April 2015, accepted 18 June 2015
ABSTRACT
We develop a folding boundary element model in a medium containing a fault and elastic layers to show that anticlines
growing over slipping reverse faults can be significantly amplified by mechanical layering buckling under horizontal shorten-
ing. Previous studies suggested that folds over blind reverse faults grow primarily during deformation increments associated
with slips on the fault during and immediately after earthquakes. Under this assumption, the potential for earthquakes on
blind faults can be determined directly from fold geometry because the amount of slip on the fault can be estimated directly
from the fold geometry using the solution for a dislocation in an elastic half-space. Studies that assume folds grown solely by
slip on a fault may therefore significantly overestimate fault slip. Our boundary element technique demonstrates that the fold
amplitude produced in a medium containing a fault and elastic layers with free slip and subjected to layer-parallel shortening
can grow to more than twice the fold amplitude produced in homogeneous media without mechanical layering under the same
amount of shortening. In addition, the fold wavelengths produced by the combined fault slip and buckling mechanisms may
be narrower than folds produced by fault slip in an elastic half space by a factor of two. We also show that subsurface fold
geometry of the Kettleman Hills Anticline in Central California inferred from seismic reflection image is consistent with a
model that incorporates layer buckling over a dipping, blind reverse fault and the coseismic uplift pattern produced during a
1985 earthquake centered over the anticline forelimb is predicted by the model.
Key words: Folding, Earthquake hazard, Anticline, Fault-related fold, Blind fault, Fault slip rate
Citation: Huang, W. J. and K. M. Johnson, 2016: A fault-cored anticline boundary element model incorporating the combined fault slip and buckling
mechanisms. Terr. Atmos. Ocean. Sci., 27, 73-85, doi: 10.3319/TAO.2015.06.18.01(TT)
1. INTRODUCTION
Geodetic observations have been interpreted as evi-
dence that anticlines over blind faults grow as a consequence
of slip on faults during earthquakes. King and Stein (1983),
Stein and King (1984), and Stein and Ekström (1992) pro-
posed that a string of Quaternary folds in Central California
at Coalinga and Kettleman Hills are produced by sudden,
incremental growth during repeated earthquakes on underly-
ing blind reverse faults (Fig. 1). This idea is leveraged by
observations at Coalinga Anticline that show a similarity in
pattern between uplifted river terraces, current topography
and coseismic vertical displacements determined from level-
ing measurements before and after the 1983 Coalinga earth-
quake. Stein and Ekström (1992) inferred a fault slip rate for
the blind fault underlying Coalinga assuming that the uplift
rate of the fold directly reflects the slip rate on the fault.
This view that anticlines grow primarily by slip on the
underlying fault is reflected in other fault-related folding
studies (e.g., Taboada et al. 1993; Myers et al. 2003; Mynatt
et al. 2007). In addition, a growing number of studies at-
tempt to link active anticline growth with slip on an under-
lying fault. The approach in these studies is to determine
the geometry and uplift rate of active fault-related folds and
then infer the fault slip rate from an assumed kinematic rela-
tionship between the fault slip and fold shape (e.g., Grant et
al. 1999; Shaw and Shearer 1999; Allmendinger and Shaw
2000; Shaw et al. 2002; Ishiyama et al. 2004; Le Béon et
al. 2014).
The main message of this paper is that the geometry
of anticlines produced solely by slip on underlying reverse
Wen-Jeng Huang & Kaj M. Johnson
74
faults is significantly different from the geometry of anti-
clines produced by the combined fault slip and buckling
mechanisms of mechanical layers under horizontal shorten-
ing. Layer buckling has the effect of amplifying and nar-
rowing the fold produced by slip on the underlying fault.
To date, mechanical analyses of fault-related folding have
largely ignored the buckling process because either the
mechanical layers required for buckle folding are absent
in such analyses or passive layer folding due to slip on a
fault is considered without horizontal shortening. Perhaps
the influence of buckling in fault-related folds has been
largely ignored because previous studies on the mechanics
of buckle folding have focused on the formation of repeti-
tive fold forms in layered media (e.g., Johnson and Fletcher
1994), which are rarely observed in sedimentary rocks. Yet,
field observations (e.g., Erslev and Mayborn 1997) show
that layer-parallel slip appears ubiquitously in fault-cored
anticlines, and therefore the role of buckling accommodated
by slip at layer contacts should not be neglected in fault-
related folding analyses.
We develop a boundary element model of the growth
of an anticline over a fault embedded in a medium with
elastic layers that slip at the contacts. We demonstrate that
fault-cored folds in a mechanically layered medium can be
significantly amplified and localized by buckling under hor-
izontal compression. To demonstrate this point, we examine
the subsurface geometry and surface deformation measure-
ments from the active Kettleman Hills and Coalinga Anti-
clines in Central California.
2. KINEMATICS AND MECHANICS OF
FAULT-CORED ANTICLINES
In this paper, we focus our attention on anticlines that
form above buried reverse faults. Three classes of fault-
related folds that are most relevant to this particular geom-
etry are fault-bend folds, fault-tip/fault-propagation folds
and forced-folds. Fault-bend folds form when rock moves
through a flat-ramp-flat fault geometry and generates repeti-
tion of the section and a ramp anticline (Rich 1934; Suppe
1983). Fault propagation and fault-tip folds form by short-
ening and shear generated at the terminations of propagat-
ing or non-propagating reverse faults. Forced folds form in a
medium overlying displaced rigid basement blocks. A num-
ber of kinematic models, similar to the Suppe (1983) fault-
bend-fold model with straight limbs and sharp hinges, have
been constructed to capture the basic geometry of fault-tip
and fault-propagation folds (e.g., Chester and Chester 1990;
Suppe and Medwedeff 1994). The trishear kinematic model
(e.g., Erslev 1991; Cardozo 2008) is particularly popular in
the recent literature because it produces rounded fold forms
that look more like natural folds and has been used to model
forced-folds and fault-tip folds.
Theoretical models investigating the passive folding
of markers in various rheology materials in response to slip
on a underlying fault have been developed by a number
of researchers [models for ramp folding (e.g., Elliot 1976;
Wiltschko 1979; Berger and Johnson 1980, 1982; Johnson
and Berger 1989; Erickson and Jamison 1995; Strayer and
Suppe 2002); models for forced folds (e.g., Sanford 1959;
Reches and Johnson 1978; Patton and Fletcher 1995; John-
son and Johnson 2002; Cardozo et al. 2003, 2005; Finch et
al. 2003); models for fault propagation folding (e.g., Car-
dozo et al. 2005)]. To our knowledge few studies have incor-
porated buckling, mainly interlayer slip. Boundary element
models were developed by Cooke and Pollard (1997) and
Shackleton and Cooke (2007) to analyze the contribution of
frictional slip along bedding planes to fault-related folding
of layered rocks. The former mainly investigated the defor-
mation of frictional bedding planes near dipping faults under
layer-parallel contraction and extension. The latter focused
on evaluating the validity of the plane strain assumption in
non-cylindrical folds. Finite element models were developed
by Niño et al. (1998) to analyze the propagation of a blind
thrust in a deformable basement in terms of the evolution
of strain localization in the overlying elastoplastic layers;
studying the layer thickness, bedding-parallel slip and fault
dip roles. In addition, Smart et al. (2009) used finite-element-
based geomechanical models of fault related folds to show
the impact of interlayer slip on fracture prediction.
While little attention has been afforded to the mechan-
ics of fault-related fold buckling, the theory of folding of
initial perturbations in isolated layers or multilayers with-
out faulting is quite mature (e.g., Biot 1963, 1964; Chapple
1969; Fletcher 1977; Johnson 1977; Kilsdonk and Fletcher
1989; Johnson and Fletcher 1994; Mancktelow 1999). Of
particular relevance to this paper are theoretical studies on
the physical conditions of multilayer folding that lead to
significant amplification of initially small perturbations. In
linear, homogenous materials, the rate at which an initial
perturbation is amplified is a function of the number of lay-
ers in the multilayer, N, the thickness of the individual lay-
ers, h, and the wavelength, L, of the initial perturbation. The
rate at which initial sinusoidal perturbations are amplified
by buckling under horizontal compression was quantified
by Biot (1961) and Fletcher (1977) as the “amplification
factor”. The amplification factor is a scalar quantity that
determines the rate at which the amplitude of an initially
small perturbation grows with increased shortening of the
medium (e.g., Johnson and Fletcher 1994).
Figure 2, produced from the folding theory developed
by Johnson and Pfaff (1989), shows the amplification fac-
tor as a function of the perturbation wavelength normalized
by the thickness of a single layer. The layers have viscosity
equal to the surrounding media and free slip at layer contacts.
The amplification factor is shown for multilayers with two,
four, or ten layers. Figure 2 illustrates that the amplification
factor (i.e., the rate at which the amplitude grows) increases
Fault-Cored Anticline Boundary Element Model 75
with the number of layers in the multilayer. For a given
layer thickness and number of layers, a so-called dominant
wavelength exists at which the amplification factor is largest
and the fold grows the fastest (the peak of the curves). The
amplification factor concept is relevant in a general way to
fault-cored folding in a multilayer under layer-parallel short-
ening. In this case, we expect the length of the perturbation
produced by slip on the fault would be controlled by the fault
geometry. We would expect the rate of growth of the fault-
cored fold to be a function of the shortening rate, the rate
of slip on the fault, and the layer thickness and number of
layers. A fault-cored fold in a medium with no mechanical
layering is expected to grow more slowly than a fold overly-
ing a fault in a medium with many mechanical layers.
3. BOUNDARY ELEMENT MODEL OF
FAULT-RELATED FOLDING
We develop a boundary element model to examine the
fault-related fold amplification by buckling. The boundary
element method (BEM) is different from the finite element
method (FEM) in that the medium is discretized only at
boundaries in the BEM whereas the entire medium is dis-
cretized in the FEM.
3.1 Basic Formulation
In layered sedimentary rocks mechanical interfaces be-
tween sedimentary layers may form because of differences
(a) (b)
Fig. 2. (a) Plots of amplification factor for periodic folds in viscous layers as a function of wavelength, L, normalized by the thickness of a single
layer, h. N indicates number of layers in multilayer. Layers slip freely at contacts. Layers and surrounding medium have same viscosity. The plots
are produced using the folding theory developed by Johnson and Pfaff (1989). (b) Illustration of a multilayer bounded above and below by semi-
infinite media. The number of layers in the multilayer, N, is 6.
Fig. 1. Geological map and cross sections of Coalinga and Kettleman Hills (modified after Stein and Ekström 1992). Geometry is based on seismic
reflection images. (a) Coalinga profile with moment tensor for 1983 earthquake. Hypocenters of small earthquakes are shown with small circles. (b)
Kettleman Hills North Dome with moment tensor for 1985 earthquake. Hypocenters of small earthquakes are shown with small circles. (c) Kettle-
man Hills South Dome.
(a)
(b)
(c)
Wen-Jeng Huang & Kaj M. Johnson
76
in physical properties at the interfaces such as grain size and
cementation. Soft layers interbedded with stiff layers may
localize shear, allowing the stiff layers to slide past each
other. These conditions are important in folding because
the bedding-plane slip can allow the strata to mechanically
buckle with flexural slip. We model these conditions with
multiple elastic layers with frictional contacts (Fig. 3).
The basic geometry and boundary conditions of two
classes of models are illustrated in Figs. 3a and b. We model
mechanical layers with initially horizontal slip surfaces of
finite length within an otherwise homogeneous elastic half-
space. The fault may be embedded in the layers or below the
layers in the half-space. In general the layers and the fault
are assumed to slip according to a Coulomb friction law,
C
sn
#xnv+
, where
s
x
is shear stress, C is cohesion,
n
is
the coefficient of friction and
n
v
is normal stress (compres-
sion is positive). The entire medium is subjected to incre-
ments of either uniform strain (Fig. 3a) or uniform displace-
ment above the detachment (Fig. 3b). If the shear stress on
the fault or layers exceeds the strength as defined by the Cou-
lomb friction law during each increment of applied displace-
ment or far-field strain the interfaces slip in order to reduce
the shear stress to the strength. In this paper we restrict our
attention to the frictionless, cohesionless case, C =
n
= 0, be-
cause this special case simplifies the illustration of the effect
of mechanical layering on fold growth.
The BEM numerical technique has been clearly de-
scribed by Crouch and Starfield (1983). Our boundary ele-
ment algorithm is largely similar to their two-dimensional
displacement discontinuity method (TWODD) which was
succinctly summarized by Martel and Muller (2000). We for-
mulate the elastic boundary element models using the solu-
tion for an edge dislocation in an isotropic, homogeneous,
elastic half-space assuming infinitely long faults and bedding
contacts in the strike direction (2D plane-strain conditions).
We give a brief outline of our formulation of the bound-
ary element model. Assume we have a N × 1 vector of incre-
mental values of the dip component of slip, s, on N patches.
From the solution for a 2D edge dislocation, we can relate
the vector of shear stresses,
s
v
, at the center of each patch to
slip on all the patches through the N × N matrix,
Gv
,
Gs
s
v=v
(1)
We assume a coordinate system with x in the horizon-
tal direction and y in the vertical direction. We apply incre-
ments of far-field uniform strain,
(2)
with corresponding uniform far-field stress,
0, 2
ff
yy
ff
xy
ff
xx
ff
xx
ff
xx
ff
yy
vfffvv nm== =++
^h
(3)
where
m
and
n
are Lame’s elastic constants,
2( )12omn o=-
, and
o
is Poisson’s ratio. We normalize
all stresses using
n
and assume
o
= 0.25. From the far-field
stress we compute the shear component of stress resolved
onto each patch,
ff
s
v
. Assuming cohesionless, frictionless
contacts, we satisfy the condition that the shear stress is zero
on each patch after each increment of deformation,
0Gs
ff
s
v+=
v
(4)
The distribution of incremental slip on the patches that gives
this condition is
sG ff
s
1v=-
v
-
^h
(5)
N × N matrices xGd and yGd relating the x and y components
(a) (b)
Fig. 3. Geometry and boundary conditions of two models of a fault embedded in an elastic medium with mechanical layering. Notation is
v
: stress,
n
v
: normal traction,
s
v
: shear traction, ffε: remote strain, and S: uniform slip. Wiggly edges indicate that the medium extends to infinity. (a) Embed-
ded fault case. The loading condition is horizontal shortening, i.e., ffεxx. (b) Ramp fault case. The loading condition is a uniform slip applied to the
detachment on the far right side of the model domain. (Color online only)
Fault-Cored Anticline Boundary Element Model 77
of displacements of the endpoints of the patches to the slip
on each patch is constructed using the solution for the edge
dislocation. Note that we only specify one boundary condi-
tion and solve for only one slip component on each patch
because we assume that the normal component of displace-
ment discontinuity across patches is zero. The incremental
displacements,
ux
D
and
uy
D
, of the patch endpoints during
the small deformation increment are then calculated as
,uGsuuGsu
xxd
ff
xd
ff
yy y
DDDD=+ =+
(6)
with the contribution to the displacements from the far-field
strain being
,ux uy
ff
x
ff
xx
ff
y
ff
yy
ffDD==
(7)
New patch endpoint positions are calculated from the
previous endpoints and the incremental displacements. A
new far-field strain increment is then applied and the cal-
culations in Eqs. (1) - (7) are repeated. For each far-field
strain increment we fix the y-coordinate of the coordinate
system origin at the ground surface (free surface). Note that
because we assume zero resistance to sliding on the faults
and layer interfaces, we do not need to consider confining
pressure due to the lithostatic load. Also for simplicity we
ignore the topography build-up and assume that erosion and
deposition maintain a flat and horizontal ground surface.
Any points on the interfaces that move above the ground
surface are discarded.
It is important to recognize that we have adopted the
linear (infinitesimal strain) elastic solution for an edge dis-
location, yet we do not restrict our analysis to small strains.
We assume that each deformation increment can be mod-
eled with the small strain theory, ignoring nonlinear effects
due to the initial stress condition at the beginning of each
increment. This is equivalent to assuming that the elastic
stresses in the medium surrounding the faults and layer in-
terfaces are somehow relaxed before the beginning of the
next deformation increment. Inelastic processes for relaxing
stresses include: micro-cracking (e.g., Meglis et al. 1995),
grain boundary sliding (e.g., Langdon 1970), twinning (e.g.,
Yamashita and Ojima 1968), pressure solution (e.g., Mc-
Clay 1977), recrystallization, and so on (Sibson 1986). Be-
cause we do not account for these processes in our model,
results from this analysis must be viewed with mindfulness
of the assumptions. Furthermore, we assume an incremen-
tal far-field strain of ffεxx = -0.02 in all applications in this
paper which is about an order of magnitude larger strain
than permissible using linear elasticity theory. However, we
examine the incremental far-field strain effect with a range
of magnitudes between 0.005 and 0.2 on the final fold form.
We find that deformation increments equal to or smaller
than 0.02 do not produce an appreciable difference in the
final fold form indicating that our choice of incremental far-
field strain is not a severe limitation.
3.2 Simulations
We now show fold forms produced with different fault
geometries. For each simulation, we compare folds pro-
duced in mechanical layering with folds produced in pas-
sive markers with no mechanical layering. We refer to the
mechanical layering folds as fault- or ramp-cored buckle
folds, and we refer to passive marker folds as fault- or ramp-
cored passive folds.
3.2.1 Fault in Basement Underlying Sedimentary
Layering
The setting for this case is a fault in a massive rock
unit (basement) underlying layers of sedimentary rocks. We
model this scenario with a fault embedded in an elastic half-
space underlying a stack of elastic layers. The fault initially
dips 25°. Figure 4 shows the results for two extreme condi-
tions for the layer interfaces: bonded (interfaces are passive
markers) or freely sliding. We show three stages of folding
with maximum fault slip of 0W, 0.14W, and 0.30W, where
W is the initial fault width (down-dip distance).
The distinct differences between the bonded and freely
sliding layers are as follows: (1) the amplitudes of folded
interfaces of freely sliding layers grow faster than the am-
plitudes of the folded passive markers. (2) After maximum
fault slip of 0.30W, the fold in the mechanical layer model
is nearly symmetric with tight hinges and curved limbs
while the fold in the passive markers is broader. (3) The
fold wavelength, measured as the distance between syncli-
nal hinges on the flanks of the anticline, is shorter in the
folded mechanical layers than in the folded passive mark-
ers. (4) The ratio of average amplitude of folded interfaces
to maximum slip on the fault in the fault-cored buckle fold
is 1.1 while the ratio for the fault-cored passive fold is 0.23.
Therefore, given the same amount of slip on the fault, the
fold amplitude in the fault-cored buckle fold is about 5 times
the amplitude of the fault-cored passive fold.
3.2.2 Fault Embedded in Layers
Figure 5 shows a model in which an originally straight
fault is embedded in the layering. The fault initially dips
25°. We show three stages with maximum fault slip of 0W,
0.18W, and 0.30W. The fault-cored buckle fold form is dif-
ferent from the previous case with the fault below the layers.
The crest of the anticline of the fault-cored buckle fold in
Fig. 5 forms over the midpoint of the underlying fault, rath-
er than above the fault tip as in the fault-cored buckle fold
in the previous model with the fault below the layers and in
the fault-cored passive folds in Figs. 4 and 5. The distinct
Wen-Jeng Huang & Kaj M. Johnson
78
(a) (b)
Fig. 4. Models of fault in an elastic half space underlying a stack of elastic layers. Entire medium is subjected to far-field horizontal shortening. W
is initial fault width. Fault dips 25° at onset. (a) Fault-cored passive fold (layers are passive markers.) (b) Fault-cored buckle fold (layers slip freely
at contacts). (Color online only)
(a) (b)
Fig. 5. Models of a fault embedded in layers. W is initial fault width. Fault dips 25° at onset. (a) Fault-cored passive fold (layer interfaces are
bonded). (b) Fault-cored buckle fold (Layers slip freely at contacts). (Color online only)
Fault-Cored Anticline Boundary Element Model 79
differences between the passive and buckle folds in Fig. 5
are as follows: (1) the amplitudes of the fault-cored buckle
fold grow faster than the amplitudes of the fault-cored pas-
sive fold. (2) After the maximum fault slip of 0.30W, the fold
in the fault-cored buckle fold is more highly localized with
steeper limb dips than the fault-cored passive fold. (3) The
buckle fold is nearly concentric while the passive is some-
what asymmetric with a short forelimb dipping to the left.
(4) The fold wavelength, measured as the distance between
synclinal hinges on the flanks of the anticline, is shorter in
the fault-cored buckle fold than in the fault-cored passive
fold. (5) The ratio of the average folded interface amplitude
to the maximum slip on the fault in the fault-cored buckle
fold is 0.6 while the ratio for the fault-cored passive fold is
0.3. Thus, with the same amount of fault slip, the buckle
fold amplitude is about 2 times the passive fold amplitude.
3.2.3 Ramp Anticline
The setting for this case is fault-bend folding over a
flat-ramp-flat fault embedded in the layering. We show
three stages in which the total displacement on the semi-
infinite dislocation is 0W, 0.38W, and 0.71W, where W is the
width of the ramp in Fig. 6. The ramp initially dips 25°. An
anticline forms above the ramp in both the bonded and free-
ly sliding cases. However, the geometry of the anticlines is
distinctly different, namely: (1) like the previous two mod-
els, the amplitudes of ramp-cored buckle fold grow faster
than the amplitudes of the ramp-cored passive fold. (2) Af-
ter total hanging wall displacement of 0.71W, the folded in-
terfaces in the ramp-cored buckle fold are box-like with two
localized shear bands with opposite-facing limbs above the
upper and lower ends of the ramp. The localized folding of
the forelimb and backlimb with nearly uniform limb dips
closely resembles the geometry produced in passive lay-
ers by the ramp fold model for anisotropic materials (e.g.,
Erickson et al. 2001). The folding somewhat resembles the
angular fault-bend kinematic model fold (e.g., Suppe 1983),
however, the relatively flat anticline crest is tilted in this
model in contrast with the horizontal crest assumed in the
fault-bend fold kinematic model. In comparison, the ramp-
cored passive fold is broad and gentle. (3) The fold wave-
length, measured as the distance between synclinal hinges
on the anticline flanks, is similar in the two models. (4) The
ratio of the average amplitude of folded interfaces to the
maximum slip on the fault in the ramp-cored buckle fold is
0.27 while the ratio in the ramp-cored passive fold is 0.15.
Thus, with the same amount of fault slip, the fold amplitude
in the ramp-cored buckle fold is nearly twice the amplitude
of the fold in the ramp-cored passive fold.
3.3 Influence of Buckling on Fold Form
We demonstrated that fault-cored fold forms can be sig-
nificantly influenced by fault geometry, properties of layer
contacts, and loading conditions. Hereafter in this section, we
will use the setting of a fault in basement underlying sedi-
mentary layering to examine the effects of two factors on fold
forms: the free ground surface and the thickness of layers.
3.3.1 Ground Surface Effect
We plot the fold amplitudes of the folded interfaces in
Fig. 4 with a maximum fault slip of 0.3W in Fig. 7 in addi-
tion to the folded interface amplitudes of a buckle fold and a
fault-cored passive fold in a full-space medium with the same
setting. The amplitude at a layer horizon is measured as the
vertical difference between the anticline crest and the lower
synclinal hinge on one of the anticline flanks. Figure 7 shows
the amplitudes of the fault-cored buckle folds are larger than
the fault-cored passive folds. Furthermore, the amplitude of
the two fault-cored passive folds decreases upwards whereas
the amplitude of the fault-cored buckle fold near the ground
surface increases upward away from the fault. This is clearly
an effect of the free surface. Figure 7 shows that the ampli-
tude of a fault-cored buckle fold far below the free surface
does not steadily increase upwards from the fault.
3.3.2 Layer Thickness Effect
A series of fault-cored folds with different layer thick-
nesses is shown in Fig. 8. The thicknesses of mechanical
layers vary but the total thickness of the entire stack of lay-
ers is the same in each fold. All folds in Fig. 8 are formed
by shortening until fault slip reaches a maximum of 0.18W.
The ratio of their amplitudes to the amplitudes of fault-cored
passive fold with the same fault slip is plotted as a function
of the number of layer interfaces in Fig. 9a. Fold amplitude
increases as the number of layers increases. The amplitude
of the top layer interface of the fault-cored buckle fold with
25 layer interfaces is nearly 6 times the amplitude of the
fault-cored passive fold under the same amount of shorten-
ing. The wavelength of the fault-cored buckle folds is plot-
ted as a function of the number of layer interfaces in Fig. 9b.
The wavelength decreases as the number of layer interfaces
increases. The wavelength of the fault-cored buckle fold
with 10 or more layers is less than half the wavelength of
the fault-cored passive fold (i.e., n = 0).
4. COALINGA AND KETTLEMAN HILLS
ANTICLINES, CALIFORNIA
We now examine actively growing anticlines at Coal-
inga and Kettleman Hills in Central California for which
we have data relating slip on the fault to the growth of the
folds. Surface displacements were recorded from moderate
earthquakes in 1983 and 1985 on the faults underlying the
Coalinga and Kettleman Hills anticlines.
Wen-Jeng Huang & Kaj M. Johnson
80
4.1 Setting
A 110 km-long chain of Quaternary fault-cored en ech-
elon anticlines is located in Central California approximately
30 km east of the San Andreas fault and on the western edge
of the San Joaquin Valley (Fig. 1). The anticlinal axes of the
folds trend nearly parallel to the San Andreas fault. During
1982 - 1985, three moderate earthquakes (5.4 ≤ Mw 6.5)
occurred along this chain of anticlines on reverse faults un-
derlying the folds (e.g., Stein and Ekström 1992). Figure 1
shows profiles across the chain of folds constructed from
well and seismic reflection data (Meltzer 1989; Wentworth
and Zoback 1989; Stein and Ekström 1992).
As discussed in the introduction a series of papers on
the earthquakes (King and Stein 1983; Stein and King 1984;
Ekström et al. 1992) suggested that the Kettlemen Hills
Anticlines grow primarily by slip on the underlying fault
during repeated large earthquakes like the 1982 - 1985 se-
quence. In contrast, the peak of the vertical displacement
pattern for the 1985 Kettleman Hills earthquake is offset
(a) (b)
Fig. 6. Models of a ramp anticline. W is the width of ramp. Ramp initially dips 25°. The maximum fault slip is the amount of slip applied to the
detachment at the far right side of the model domain. (a) Ramp-cored passive fold (layer interfaces are bonded). (b) Ramp-cored buckle fold (layers
slip freely at contacts). (Color online only)
Fig. 7. Comparison of fold amplitudes in Fig. 5 and fault-cored passive and buckle fold amplitudes in a full space (no free surface). (Color online
only)
Fault-Cored Anticline Boundary Element Model 81
about 3 km NE of the fold axis of the Kettleman Hills North
Dome. Figure 10 shows the vertical displacement contours
from an elastic dislocation model with uniform slip that best
reproduces the vertical displacement measurements during
the 1985 earthquake (Stein and Ekström 1992). Stein and
Ekström (1992) suggested that the North Dome probably
grew as a result of repeated earthquakes on a dipping fault
under the fold, similar to the fault that slipped in the 1985
earthquake. However, that fault has perhaps begun migrat-
ing to the northeast, generating uplift to the northeast of the
anticline during the 1985 earthquake.
We will demonstrate that an alternative explanation is
that the 1985 earthquake may very well be typical of earth-
quakes on the major fault underlying the anticline, but the
coseismic deformation pattern does not match the fold geom-
etry because the fold did not grow solely as a consequence of
Fig. 8. Models of a fault embedded in stacks of mechanical layers with different layer thicknesses. The faults in all models have the same maximum
fault slip of 0.18W. W is initial fault width. n is number of layer interfaces. (Color online only)
(a) (b)
Fig. 9. Comparisons of fold features with different numbers of layer interfaces. (a) Ratio of fault-cored buckle fold amplitudes in Fig. 8 to fold ampli-
tude in passive markers under the same amount of shortening plotted with number of layer interfaces. (b) Fold wavelengths from Fig. 8 normalized
by the initial fault width plotted with number of layer interfaces. (Color online only)
Wen-Jeng Huang & Kaj M. Johnson
82
slip on the underlying fault.
4.2 Mechanical Analysis
Results from our analyses of fault-cored buckle folds
with an embedded shallowly dipping fault (Fig. 5) show that
the anticline crest is well behind the fault tip, whereas mod-
els of the same type but with passive layering show the fold
crest above the fault tip. Figure 10b shows the fold in a me-
chanically layered medium along with vertical displacement
pattern at the ground surface due to slip on the buried reverse
fault. The relationship between the location of the peak co-
seismic uplift and the axial trace of the anticline is similar
to that observed from the 1985 Kettleman Hills earthquake.
The peak coseismic uplift is shifted to the front limb of the
anticline, not centered on the anticline. We therefore sug-
gest that this result indicates that the Kettleman Hills North
Dome likely formed as a buckle fold overlying a reverse
fault, similar to those produced in our model. The shape of
the vertical displacement pattern due to earthquakes on the
underlying reverse fault does not directly reflect the shape
of the anticline because the anticline grows by combined
fault slip and layer buckling mechanisms.
The subsurface shape of the Kettleman Hills South
Dome (Fig. 1c) is further evidence that buckling contributes
significantly to the growth of the anticline. Figure 11 com-
pares models of folds produced by shortening of a medium
with a 45° dipping reverse fault with either passive markers
or mechanical layers. The geometry of the fold produced by
the model with mechanical layers is similar to the Kettle-
man Hills South Dome. The seismic profile and the model
both show a slightly asymmetric fold localized above the
dipping reverse fault. The fold produced in a homogeneous
medium with passive markers clearly does not resemble the
actual fold and the modeled amplitude is too small.
This analysis of the Kettleman Hills Anticlines is
revealing, but is far from complete. To determine the ex-
tent to which the fold has recently grown between large
(a)
(b)
Fig. 10. Predicted elevation change of 1985 Kettleman Hills earthquake. (a) Vertical deformation field predicted from a rectangular dislocation
with reverse-slip by Ekström et al. (1992). (b) Model of a fault embedded in mechanical layers like in Fig. 6. W is initial width of the fault and Uy
is vertical displacement. The plot of coseismic uplift attributed to only fault slip in the upper part of (b) is centered above the forelimb of Kettleman
Hills Anticline, much like the observed pattern in (a).
(a) (b) (c)
Fig. 11. Comparison of South Dome anticline and simulations. (a) Profile of Kettleman Hills South Dome (profile in Fig. 11c). (b) Result from
mechanical layer model. (c) Result from passive marker model. The anticline in (b) resembles the South Dome anticline better than the anticline in
(c). (Color online only)
Fault-Cored Anticline Boundary Element Model 83
earthquakes, either by slip on the underlying fault or buck-
ling, one would want to examine interseismic data showing
the deformation pattern. One would also want to examine
geomorphic evidence for Holocene deformation of the fold
in order to determine the longer-term deformation history.
Acquisition and analysis of such data in future work would
likely yield valuable insight about the fold growth processes
at Kettleman Hills.
5. CONCLUSIONS
We have constructed boundary element models with
a medium containing a fault and elastic layers subjected to
layer-parallel shortening to demonstrate the influence of
buckling on fault-cored fold growth. Free slip is assumed
on the fault and layer contacts. We compare folding sim-
ulations in the mechanically layered elastic medium with
passive marker folding in a non-layered medium. Given the
same amount of far-field shortening for both conditions, the
mechanically layered medium produces more highly local-
ized folds with higher amplitude and shorter wavelength.
The horizontal shortening that causes a fault in an an-
ticline core to slip can cause significant amplification of the
fold by buckling of the strata. Under the conditions consid-
ered in this paper, the contribution to fold growth by slip on
the underlying fault alone is only about 20 - 50% of the total
growth. Therefore, studies that seek to estimate fault slip
from fold geometry by assuming the fold is built by slip on
the fault alone could significantly overestimate the amount
of fault slip.
At Kettleman Hills Anticline in Central California,
published seismic profiles show the subsurface fault and
fold geometry and slip on the fault can be inferred from
coseismic uplift data from the 1985 Kettleman Hills earth-
quake. We show that the general features of the fold form
are consistent with the BEM model that incorporates buck-
ling of layers over a dipping, blind reverse fault. Further-
more, the general coseismic uplift pattern centered over the
anticline forelimb is predicted by the model.
Acknowledgements We thank Arvid M. Johnson for pro-
viding his analytical viscous folding theory results to verify
our BEM method and many useful suggestions that helped
improve this article.
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APPENDIX A1: COMPARISON OF ANALYTICAL
AND BEM SOLUTIONS FOR AMPLIFICATION
FACTOR
To verify that our BEM program is reliable for solving
folding problems, we compare amplification factors com-
puted with the analytical solution (Johnson and Pfaff 1989)
and with the boundary-element solution. It was not intuitive-
ly obvious that a folding model (BEM) consisting of layers
modeled with small boundary elements at their interfaces
would be equivalent (or nearly so) to analytical folding the-
ory (Johnson and Fletcher 1994). The amplification factor
is related exponentially to the amplitude growth rate of a
fold. The larger the amplification factor, the faster the fold
amplitude grows. The amplification factor for a given set of
conditions is a function of the wavelength to thickness ratio
of a single layer within a multilayer. Thus, Fig. A1 shows
plots of amplification factors calculated using the two meth-
ods for two, four and ten identical layers. The peak in each
curve has coordinates of the maximum amplification factor
and the dominant wavelength. The dominant wavelength is
the wavelength that will grow the fastest.
By comparing amplification factors calculated with the
BEM and the analytical model, we see that the results are
very similar (Fig. A1). The similarity shows that the two
models, derived in quite different ways, are almost certainly
addressing the same mechanical problem. Thus we can de-
pend on the BEM model to solve some folding problems
such as those addressed in this paper that would be very
difficult with the analytical folding theory.
Fig. A1. Plots of amplification factor of periodic folds using analyti-
cal viscous folding theory and boundary element theory. (Color online
only)
... To understand how brittle faulting and folding can occur simultaneously, we develop a framework to incorporate bedding plane slip at earthquake timescales, by estimating the downdip width (length-scale) of the actively deforming hinge zone, and incorporating hinge-related OFD into earthquake sequence simulations and related inverse problems. We first use a Boundary Element Method (BEM) framework to model a ramp-décollement system in an elasto-plastic medium, incorporating the anisotropy of sedimentary layers (Huang & Johnson, 2016;K. M. Johnson, 2018). ...
... Previous work on multi-layer folding suggests that folding in the brittle crust varies from sinusoidal (large hinge width) to kink bands (narrow hinge width), depending on a combination of material properties-the rigidity of the medium, frictional strength of the layer interfaces, and bed thickness (Biot, 1961(Biot, , 1964Honea & Johnson, 1976;Huang & Johnson, 2016 (Figures 7a and 7b) is instead exerted by the number of bedding planes and the angularity of the primary fault-bend (inset Figure 4a), rather than the shear modulus of the medium, friction coefficient of the bedding plane contacts or the number of slip increments ( Figure 6). We highlight the insensitivity of the fold morphology and bedding plane slip to the friction coefficient by running an additional experiment with 0.6   ( Figure A1). ...
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Observations of fold growth in fold-thrust belt settings show that brittle deformation can be localized or distributed. Localized shear is associated with frictional slip on primary faults, while distributed brittle deformation is recognized in the folding of the bulk medium. The interplay of these processes is clearly seen in fault-bend folds, which are folds cored by a fault with an abrupt change in dip (e.g., a ramp-décollement system). While the kinematics of fault-bend folding were described decades ago, the dynamics of these structures remain poorly understood, especially the evolution of fault slip and off-fault deformation over different periods of the earthquake cycle. In order to investigate the dynamics of fault-bend folding, we develop a numerical modeling framework that combines a long-term elasto-plastic model of folding in a layered medium with a rate-state frictional model of fault strength evolution in order to simulate geologically and mechanically consistent earthquake sequences. In our simulations, slip on the ramp-décollement fault and inelastic fold deformation are mechanically coupled processes that build geologic structure. As a result, we observe that folding of the crust (like fault slip) does not occur steadily in time but is modulated by earthquake cycle stresses. We suggest combining seismological and geodetic observations with geological fault models to uncover how elastic and inelastic crustal deformation generate fault-bend folds. We find that distinguishing between the elastic and inelastic response of the crust to fault slip is possible only in the postseismic period following large earthquakes, indicating that for most fault systems this information currently remains inaccessible.
... However, the importance of fold buckling and distributed strain to the component of shortening and fold growth is not clear (e.g., Ainscoe et al., 2017;Gonzalez-Mieres & Suppe, 2006;Veloza et al., 2015;Yonkee & Weil, 2010). In other words, it is also recognized in the literature that there is not a simple relationship between slip and fold growth (e.g., Bonanno et al., 2017;Huang & Johnson, 2016;Johnson, 2018). ...
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Questions regarding the development of folds and their interactions with the seismic faults within thrust systems remain unanswered. However, estimating fault slip and earthquake hazards using surface observations and kinematic models of folding requires an understanding of how the shortening is accommodated during the different phases of the earthquake cycle. Here, we construct 16-years of InSAR time series across the North Qaidam thrust system (NE Tibet), where three Mw 6.3 earthquakes occurred along basement faults underlying shortened folded sediments. The analysis reveals spatio-temporal changes of post-earthquake surface displacement rates and patterns, which continue more than ten years after the seismic events. The decomposition of the Sentinel-1 ascending and descending LOS velocities into vertical and shortening post-earthquake components indicates that long-term transient uplift and shortening is in agreement with the deformation that might be expected from kinematic models of folding. Long-term uplift coincides spatially with young anticlines observed in the geomorphology, with steep gradients in the forelimbs, gentle gradients in the back-limbs, an absence of subsidence in the footwalls, and higher gradients along interpreted bedding planes. Long-term shortening is also different from the surface displacements expected for typical time-varying creep on a narrow dislocation interface and shows rates higher than the average convergence across the whole region. These findings provide evidence for anelastic fold buckling during the post-earthquake phase and highlight the contribution of distributed aseismic deformation to the growth of topography.
... However, the importance of fold buckling and distributed strain to the component of shortening and fold growth is not clear (e.g., Ainscoe et al., 2017;Gonzalez-Mieres & Suppe, 2006;Veloza et al., 2015;Yonkee & Weil, 2010). In other words, it is also recognized in the literature that there is not a simple relationship between slip and fold growth (e.g., Bonanno et al., 2017;Huang & Johnson, 2016;Johnson, 2018). ...
... This direction is nearly compatible with southward tilting that has been observed over the past 10 years. Different studies (Davis et al. 1983;Yeats 1986;Huang and Johnson 2016) have proposed the model in which an anticline on an active thrust fault shows growth and increase of the anticline topographic slopes. A southward tilting, such as the one documented by the tiltmeters, could be interpreted by this model as being the expression of the southward movement of the massif along the two thrust faults that mark the margin of the southern border of the massif (Fig. 1) and are responsible of the southward tilting of the slope facing the thrust fault. ...
Chapter
Ten years’ geodetic observations (2006–2016) in a natural cave of the Cansiglio Plateau (Bus de la Genziana), a limestone karstic area in northeastern Italy, are discussed. The area is of medium–high seismic risk: a strong earthquake in 1936 below the plateau (Mm = 6.2) and the 1976 disastrous Friuli earthquake (Mm = 6.5) are recent events. At the foothills of the karstic massif, three springs emerge, with average flow from 5 to 10 m³/s, and which are the sources of a river. The tiltmeter station is set in a natural cavity that is part of a karstic system. From March 2013, a multiparametric logger (temperature, stage, electrical conductivity) was installed in the siphon at the bottom of the cave to discover the underground hydrodynamics. The tilt records include signals induced by hydrologic and tectonic effects. The tiltmeter signals have a clear correlation to the rainfall, the discharge series of the river and the data recorded by multiparametric loggers. Additionally, the data of a permanent GPS station located on the southern slopes of the Cansiglio Massif (CANV) show also a clear correspondence with the river level. The fast water infiltration into the epikarst, closely related to daily rainfall, is distinguished in the tilt records from the characteristic time evolution of the karstic springs, which have an impulsive level increase with successive exponential decay. It demonstrates the usefulness of geodetic measurements to reveal the hydrological response of the karst. One outcome of the work is that the tiltmeters can be used as proxies for the presence of flow channels and the pressure that builds up due to the water flow. With 10 years of data, a new multidisciplinary frontier was opened between the geodetic studies and the karstic hydrogeology to obtain a more complete geologic description of the karst plateau.
... This direction is nearly compatible with southward tilting that has been observed over the past 10 years. Different studies (Davis et al. 1983;Yeats 1986;Huang and Johnson 2016) have proposed the model in which an anticline on an active thrust fault shows growth and increase of the anticline topographic slopes. A southward tilting, such as the one documented by the tiltmeters, could be interpreted by this model as being the expression of the southward movement of the massif along the two thrust faults that mark the margin of the southern border of the massif (Fig. 1) and are responsible of the southward tilting of the slope facing the thrust fault. ...
Article
Full-text available
Ten years’ geodetic observations (2006–2016) in a natural cave of the Cansiglio Plateau (Bus de la Genziana), a limestone karstic area in northeastern Italy, are discussed. The area is of medium–high seismic risk: a strong earthquake in 1936 below the plateau (Mm = 6.2) and the 1976 disastrous Friuli earthquake (Mm = 6.5) are recent events. At the foothills of the karstic massif, three springs emerge, with average flow from 5 to 10 m3/s, and which are the sources of a river. The tiltmeter station is set in a natural cavity that is part of a karstic system. From March 2013, a multiparametric logger (temperature, stage, electrical conductivity) was installed in the siphon at the bottom of the cave to discover the underground hydrodynamics. The tilt records include signals induced by hydrologic and tectonic effects. The tiltmeter signals have a clear correlation to the rainfall, the discharge series of the river and the data recorded by multiparametric loggers. Additionally, the data of a permanent GPS station located on the southern slopes of the Cansiglio Massif (CANV) show also a clear correspondence with the river level. The fast water infiltration into the epikarst, closely related to daily rainfall, is distinguished in the tilt records from the characteristic time evolution of the karstic springs, which have an impulsive level increase with successive exponential decay. It demonstrates the usefulness of geodetic measurements to reveal the hydrological response of the karst. One outcome of the work is that the tiltmeters can be used as proxies for the presence of flow channels and the pressure that builds up due to the water flow. With 10 years of data, a new multidisciplinary frontier was opened between the geodetic studies and the karstic hydrogeology to obtain a more complete geologic description of the karst plateau. SharedIt link to article: https://rdcu.be/bbZvk
... Off-fault folding should fold the entire hanging wall of the MFT. Moreover, we did not see evidence of significant bedding plane slip or buckling (Huang and Johnson, 2016) in the field or reported in the literature that could explain the large scale topography of the Mohand Range. However, we acknowledge that the slip along the MFT could be accommodated by flexural-slip folding within the Mohand Range and the secondorder folding of Middle Siwalik rocks in the MFT fault zone is most likely off-fault. ...
Article
The Main Frontal thrust (MFT) uplifts the Himalayan topographic front. Deciphering MFT deformation kinematics is crucial for understanding how the orogen accommodates continuing continental collision and assessing associated hazards. Here, we (a) detail newly discovered fault-zone exposures along the MFT at the Mohand Range front in northwestern India and (b) apply contemporary fault zone theory to show that the MFT is an emergent fault with a well-developed, fault zone overlain by uplifted Quaternary gravels over a horizontal length of ∼700 m. Northward from the front, the fault zone grades from a central, gouge-dominated core to a hanging-wall, rock-dominated damage zone. We observed incohesive, non-foliated breccia, fault gouge, and brittle deformation microstructures within the fractured country rocks (Middle Siwaliks) and outcrop scale, non-plunging folds in the proximal hanging wall. We interpret these observations to suggest that (1) elastico-frictional (brittle) deformation processes operated in the fault zone at near surface (∼1–5 km depth) conditions and (2) the folds formed first at the propagating MFT fault tip, then were subsequently dismembered by the fault itself. Thus, we interpret the Mohand Range as a fault-propagation fold driven by an emergent MFT in contrast to the consensus view that it is a fault-bend fold. A fault-propagation fold model is more consistent with these new observations, the modern range-scale topography, and existing erosion estimates. To further evaluate our proposed structural model, we used a Boundary Element Method-based dislocation model to simulate topographic growth from excess slip at a propagating fault tip. Results show that the frontal topography could have evolved by slip along a (a) near-surface fault plane consistent with the present-day MFT location, or (b) blind MFT at ∼3 km depth farther north near the drainage divide. Comparing modelled vs. measured high resolution (∼16 cm) topographic profiles for each case provides permissible end-member scenarios of a either a dynamically-evolving, high erosion, northward-migrating fold or a static, low, and symmetric, MHT-related fold, respectively. Our integrated approach is expected to deliver a novel approach for improved understanding of coupled fault-generated deformation and topographic growth that may be applied more broadly across the entire Himalayan front.
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Observations of fold growth in fold-thrust belt settings show that brittle deformation can be localized or distributed. Localized shear is associated with frictional slip on primary faults, while distributed brittle deformation is recognized in the folding of the bulk medium. The interplay of these processes is clearly seen in fault-bend folds, which are folds cored by a fault with an abrupt change in dip (e.g., a ramp-décollement system). While the kinematics of fault-bend folding were described decades ago, the dynamics of these structures remain poorly understood, especially the evolution of fault slip and off-fault deformation over different periods of the earthquake cycle. In order to investigate the dynamics of fault-bend folding, we develop a numerical modeling framework that combines a long-term elasto-plastic model of folding in a layered medium with a rate-state frictional model of fault strength evolution in order to simulate geologically and mechanically consistent earthquake sequences. In our simulations, slip on the ramp-décollement fault and inelastic fold deformation are mechanically coupled processes that build geologic structure. As a result, we observe that folding of the crust does not occur steadily in time but is modulated by earthquake cycle stresses. We suggest combining seismological and geodetic observations with geological fault models to uncover how elastic and inelastic crustal deformation generate fault-bend folds. We find that distinguishing between the elastic and inelastic response of the crust to fault slip is possible only in the postseismic period following large earthquakes, indicating that for most fault systems this information currently remains inaccessible.
Article
Fault-related folds develop due to a combination of slip on the associated fault and distributed deformation off the fault. Under conditions that are sufficient for sedimentary layering to act as a stack of mechanical layers with contact slip, buckling can dramatically amplify the folding process. We develop boundary element models of fault-related folding of viscoelastic layers embedded with a reverse fault to examine the influence of such layering on fold growth. The strength of bedding contacts, the thickness and stiffness of layering, and fault geometry all contribute significantly to the resulting fold form. Frictional contact strength between layers controls the degree of localization of slip within fold limbs; high contact friction in relatively thin bedding tends to localize bedding slip within narrow kink bands on fold limbs and low contact friction tends to produce widespread bedding slip and concentric fold form. Straight ramp faults tend to produce symmetric folds whereas listric faults tend to produce asymmetric folds with short forelimbs and longer backlimbs. Fault-related buckle folds grow exponentially with time under steady loading rates. At early stages of folding, fold growth is largely attributed to slip on the fault, but as the fold increases amplitude, a larger portion of the fold growth is attributed to distributed slip across bedding contacts on the limbs of the fold. An important implication for geologic and earthquake studies is that not all surface deformation associated with blind reverse faults may be attributed to slip on the fault during earthquakes.
Chapter
Monoclinal flexures, which are isolated asymmetric flexures, range in scale from a few millimetres in kink bands to hundreds of metres in monoclines on the Colorado Plateau. A general model of monoclinal flexuring of multilayers is proposed here; the multilayers include layers with various rheologies, densities, thicknesses, and strengths of contacts between layers. The multilayers are subjected to displacements at their base, stresses at their edges, and a free surface at their tops. We study in detail three modes of this general model, assuming linear, incompressible elastic or viscous multilayers: Drape folding, in which a monoclinal flexure develops over a vertical fault; buckling, in which an initial monoclinal flexure is amplified by layer-parallel compression; and kinking, in which monoclinal kink bands develop unstably by compression inclined to the layering. Selected solutions are presented for the first two modes, and previous research is summarized for the kinking mode. According to analyses of the three special cases of the general model, the profile of the monoclinal flexure, the displacement field, and the strain distribution within the flexure are useful criteria for distinguishing among the three modes of monoclinal flexuring. The Palisades monocline, described in detail in Part I (this volume), is interpreted to be a result of a combination of drape folding over a fault in Precambrian basement rocks and buckling, which together appear to account for most of the field observations. The Yampa monocline in Dinosaur National Park changes form along its length, but each form can be compared with characteristics of a combination of modes, including faulting at depth and layer-parallel compression. In some places it closely resembles a large kink band.
Article
Deformation twins in pure iron crystals containing many subgrain boundaries have been investigated by transmission electron microscopy. Some of the deformation twins have the serrated shapes on the subgrain boundaries, and in other cases cause the change in thickness on passing through the subgrain boundaries. There are the twin tips which are within the subgrains, or in contact with or near the subgrain boundaries. The shapes of the tips are irregularly curved or fiat. The crystallographic orientation of the flat surface of the twin tip varies as the tip is different. Dislocations, ahead of the twin tip, which are restricted within almost the same thickness as the twin are seen. They are considered to be introduced as a result of the relaxation of the stress due to the twinning dislocations piled-up at the twin tip. The amount of the relaxation of the stress due to the twinning dislocations on the flat surface of the twin tip is discussed on the basis of dislocation theory.
Article
The folding of a single viscous layer embedded in an uniformly shortening medium is analyzed mathematically by superposing the mean flow corresponding to uniform shortening and a perturbing flow corresponding to folding. The mean flow is coupled to the perturbing flow in the boundary conditions applying at the layer—medium interfaces. These are expanded according to the surface-wave approximation in hydrodynamics, and the solution applies when the amplitude of folding is small. Folding with interfacial adherence and interfacial slip are both treated. The dominant wavelenghts obtained do not agree with those presented in Biot (1959a) and Remberg (1963a, 1970b).
Article
We describe the three-dimensional geometry and Quaternary slip history of the Puente Hills blind-thrust system (PHT) using seismic reflection profiles, petroleum well data, and precisely located seismicity. The PHT generated the 1987 Whittier Narrows (moment magnitude [Mw] 6.0) earthquake and extends for more than 40 km along strike beneath the northern Los Angeles basin. The PHT comprises three, north-dipping ramp segments that are overlain by contractional fault-related folds. Based on an analysis of these folds, we produce Quaternary slip profiles along each ramp segment. The fault geometry and slip patterns indicate that segments of the PHT are related by soft-linkage boundaries, where the fault ramps are en echelon and displacements are gradually transferred from one segment to the next. Average Quaternary slip rates on the ramp segments range from 0.44 to 1.7 mm/yr, with preferred rates between 0.62 and 1.28 mm/yr. Using empirical relations among rupture area, magnitude, and coseismic displacement, we estimate the magnitude and frequency of single (Mw 6.5-6.6) and multisegment (Mw 7.1) rupture scenarios for the PHT.
Article
Understanding and interpreting the timing, location, orientation, and intensity of natural fractures within a geologic structure are commonly important to both exploration and production planning activities. Here we explore the application of finite-element-based geomechanical models to fracture prediction. Our approach is based on the idea that natural fractures can be interpreted or inferred from the geomechanical-model-derived permanent strains. For this analysis, we model an extensional fault-tip monocline developed in a mechanically stratified limestone and shale sequence because field data exist that can be directly compared with model results. The approach and our conclusions, however, are independent of the specific structural geometry. The presence or absence of interlayer slip is shown to strongly control the distribution and evolution of strain, and this control has important implications for interpreting fractures from geomechanical models.