GEOGRAPHICAL REPORTS

OF TOKYO METROPOLITAN UNIVERSITY 47 (2012) 11-25

SIMULATION OF FAULT FLEXURE ASSOCIATED WITH THE

1999 CHICHI, TAIWAN EARTHQUAKE

USING THE CIP METHOD

Koichi ANDO

Abstract An active fault which develops in a plain is often covered with a weak surface

stratum. The large-scale fault slip in the bed rock may spread to the surface stratum and will

often generate a fault rupture on the surface. Depending on the case, the fault slip sometimes

appears on the surface as a fault flexure. When the fault flexure is generated, estimation of a

fault angle of a bed rock is impossible. The September 21, 1999 Chichi earthquake in central

Taiwan produced a 95 km long surface rupture. A trench was excavated 7 m deep and 27 m long,

across the Chichi earthquake fold scarp at the Shijia site. However, since a fault flexure was

present, the fault angle of the bed rock could not be determined. This study presents an attempt

to determine the fault angle of the bed rock using simulation of the deformation. The simulation

technique uses the CIP (Constrained Interpolation Profile) method, which is a new technique that

overcomes the weak point of numerical diffusion in the finite difference method. The fault angle

of the bed rock and the maximum slip rate in the simulation ware 49° and 1.25-1.5 m/s,

respectively, for the Shijia trench site.

Key words: active fault simulation, Chichi earthquake, surface rupture, fault flexure,

Constrained Interpolation Profile method

1. Introduction

An active fault which develops in a plain is often covered with a weak surface stratum. A

large-scale fault slip in the bed rock may propagate to a surface stratum, and will often generate the

fault rupture on the surface, which is indicated by the generation of a shear belt in the stratum.

Depending on the case, a fault slip will sometimes appear on the ground surface not as a shear belt but

as a fault flexure. The stratum keeps its continuity without being disrupted by the underground shear

belt. As such an example, the Tachikawa fault in the western suburbs of Tokyo Japan (Yamazaki

1978) form flexure. It is impossible to evaluate the fault angle of the bedrock in the fault

produced by the fault flexure because no shear belt was generated in the stratum. Nevertheless,

it is essential to determine the fault angle of the bed rock in order to predict the character of future

earthquakes. For example, the displacement of a fault is H/sin (θ), where H is the height of the

fault for one event scarp, and θ is the fault angle of the bed-rock. The magnitude of an

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earthquake is proportional to the amount of displacement for one event (Matsuda 1975);

therefore, the smaller the θ, the larger the earthquake. Moreover, the determination of the fault

angle of the bed rock is important for the estimation of the earthquake occurrence probability of

active faults using ΔCFF (Coulomb Failure Function).

The September 21, 1999 Chichi earthquake in central Taiwan produced a 95 km long surface

rupture (Chen et al. 2007) (Fig. 1). Based on seismic reflection profiles and focal mechanisms of the

main shock, the earthquake occurred on a shallow-dipping (20-30°E) thrust ramp of the Chelungpu

fault (Chen et al. 2007).

Chen et al. (2007) excavated a trench in the earthquake surface rupture area (the Shijia site) (Fig.

1), and confirmed that the stratum forms a flexure. They also drilled boreholes near the earthquake

surface rupture and found shear zones at two places (Fig. 2). The shear zones assumed to be on the

fault plane of the bed rock. Accordingly, two fault angles of the bed rock 25° and 49° estimated from

the depth of the shear zones. However, it was not possible to specify which of these angles is the bed

rock fault angle.

In the Shijia site, the stratum was silty sand. When simulating the deformation of sandy soil,

it is necessary to take dilatancy into consideration (Johansson and Konagai 2007). Because, it is

Fig. 1 The location of the earthquake surface fault generated by the Chichi earthquake, and the

location of the Shijia trench site.

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known that the material that forms the stratum sand or silt changes the appearance configuration

of the fault scarp (Kawai and Tani 2003). Therefore, in this research, a simulation using the CIP

(Constrained Interpolation Profile) method was performed considering the dilatancy of the stratum,

and the fault angle of the bed rock was estimated by calculating the shape of the flexure.

The simulation program used in this research is SDSSC (The Stratum Deformation

Simulation System using the CIP method). SDSSC is a program for calculating a deformation

of the stratum.

2. Surface Rupture and Trench Site

The characteristics of the surface rupture suggest that the Chelungpu fault system consists of

three segments with different slip directions and vertical displacements; N30°-40°W, 3-8 m for

the Shihkang fault, N70°-90°W, 0.2-4 m for the Chelungpu fault, and N50°E (right-lateral strike

slip fault), 0.2-1 m for the Tajianshan fault (Chen et al. 2007) (Fig. 1).

Fig. 2 Spatial relationship of the Shijia trench and the boreholes.

Fig. 3 The trace of the shape of cw1, cw2 and cw3 in the Shijia trench.

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The Shijia site is located on Chelungpu fault, and sits is on an alluvial fan, which exhibits a

gentle westward-dipping slope (Chen et al. 2007) (Fig. 1). Chen et al. (2007) excavated a 7 m

deep, 27 m long trench across the Chichi earthquake fault scarp (Fig. 2). They revealed shallow

subsurface deposit consisting predominantly of well-sorted fine sand inter-bedded with mud and

humic paleosoil that represent an overbank deposits. Trench-wall exposures show three

depositional units, cw1, cw2, and cw3 (Fig. 3). In the hanging wall side, cw1 and cw2 are

lacking, as a result of erosion. Only cw3 can observe with the whole trench. Therefore, cw3 was

set as the comparison target of the simulation. Three continuous core drilled boreholes with

depths of 13 and 43 m on the hanging wall and 17 m on the footwall provided further

subsurface constraints (Fig. 2). Boreholes 1 and 2 data show that the depth to the bed rock at the

side of the hanging wall is 10-11 m. Shear zones were found at depths of 13 m in borehole 2 and

20.7 m and 30 m in borehole 1. Linking the shear zones from the two boreholes suggests two

faults, one dipping to be 25°E and another one to be 49°E, respectively (Fig. 2).

3. Simulation Method

About SDSSC

SDSSC is the program for simulation of the deformation of the unconsolidated stratum. The

characteristics of SDSSC are described below.

1. The unconsolidated stratum is treated as Bingham fluid. 2. High viscosity non-Newtonian

fluid (Bingham fluid) is stably calculable. 3. The governing equation is calculated by having

separated into the advection term and the non-advection term. 4. The CIP method (Yabe et al.

1991; Yabe et al. 2003) is used for calculation of an advection term. 5. Poisson's equation of a

non-advection term is calculated using BiCGSTAB (Bi-Conjugate Gradient Stabilized) method

(van der Vorst 1992). 6. The interaction of the fluid (stratum) and a solid (bed rock) is calculable.

7. In the incompressible-fluid cord, dilatancy is introduced in approximately. 8. Parallel

computation is supported.

Compared with the traditional finite difference method, the highly accurate calculation with

little numerical diffusion is achieved by using the CIP method.

The CIP method

To simulate the deformation of a stratum, methods such as the finite difference method, finite

element method (e.g., Gregory et al. 2000; Lin et al. 2007; Loukidis et al. 2009), and discrete element

method (e.g., Onizuka 2000; Finch et al. 2003; Benesh et al. 2007) are used. In the finite element

method, since the mesh changes with the migration of the matter, numerical diffusion does not occur;

it is beneficial to have a clear matter boundary. This presents a drawback in cases with large

deformation of the matter, where fission and fusion are incalculable. The stratum deformation caused

by a fault slip is a problem, because, fracture and a large deformation are generated in the fault plane.

The discrete element method can calculate large deformation and fission of the matter, but has higher

computation costs compared to the finite difference method. In the finite difference method, the

mesh is fixed to space. Hence, this method can calculate large deformation of the matter, fission, and

fusion. However, a problem may arise whereby the calculation becomes inaccurate because of the

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numerical diffusion, resulting in the disadvantage mentioned previously with regards to the finite

difference method. Therefore, the finite difference method is not used in the field of stratum

deformation. Although the CIP method is a finite difference method, it includes a technique which

solves the problem of numerical diffusion (Fig. 4).

Approximation to non-Newtonian fluid in the stratum

The study area consists of strata that approximate a Bingham fluid (e.g., Sawada et al. 2000;

Moriguchi et al. 2005), which is a type of non-Newtonian fluid. In a Newtonian fluid, the

viscosity does not change with shear strain rate, whereas in a non-Newtonian fluid it does, and is

given as follows:

��

� �������

�

��

��

�1�

where �� is the shear strain rate (deformation rate), ��

is the apparent viscosity coefficient,

η0 is the viscosity coefficient of the Newtonian fluid, τy is the shear stress, and n is a material

parameter. If n = 1 and τy ≠ 0, the equation expresses the behavior of a Bingham fluid. The shear

strain rate (deformation rate) is the derivative of strain (deformation). In actual strata, because

the shear strain acts on particles consisting of gravel or sand, the strata are deformed. In a

Bingham fluid, the higher the deformation rate, the lower the viscosity. We apply the

Mohr-Coulomb failure criterion

��� � sin� � ��2�

Fig. 4 The time variation of the function when carrying out advection of the pulse form function at a

fixed speed. Continuous lines show the initial functions. Dotted lines are the results after 100

step calculations. Dashed lines are after 200 step calculations. Left: The result of the previous

finite difference method. Right: The CIP method, where the numerical diffusion is hardly

noticeable.

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to the stratum to represent the soil mechanics (Ishihara 1997), where p is the hydrostatic

pressure, c is the cohesion, and � is the internal friction angle. The characteristics of the

stratum are determined by ��, � and c. For details on a Bingham fluid and the Mohr-Coulomb

failure criterion, refer to Sawada et al. (2000) and Moriguchi et al. (2005).

Evaluation of the dilatancy

In dilatant sandy soil, the volume changes are accompanied by soil deformation. The

volume change is generated by staggering of sand particles changes with distortions (Fig. 5).

The deformation causes an increase in the volume of the compacted sandy soil (Nakai 1989). In

this research, since the simulation is performed using an incompressible-fluid code, simulation

of the dilatancy could not be carried out directly. Therefore, the simulation of the dilatancy was

carried out using the following assumption.

The energy which results in an increase in volume is expressed as an increase in viscosity (if

the viscosity is high, the amount of energy required to distort the material will increase). The

Fig. 5 Conceptual diagram of the volume increase of the stratum by dilatancy.

Fig. 6 Change in the internal friction angle of the stratum when taking dilatancy into

consideration. The abscissa shows the deformation of the stratum. The ordinate shows

the internal friction angle of a stratum.

Fig. 7 Conceptual diagram of the stratum deformation simulation by the fault slip.

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energy required to increase the volume is proportional to the hydrostatic pressure p. In a

Bingham fluid, the viscosity of a medium is calculated as � tan��� � �, where p, � and c are

static pressure, internal friction angle and cohesion, respectively. Therefore, increases in

viscosity and volume can be calculated by increasing � in proportion to the amount of

distortion in the stratum. The increase in the internal frictional angle is defined as ψ (Fig. 6).

4. Implementation of Simulation

In this study, we carried out two-dimensional computer simulations assuming that the

stratum is cut perpendicular to the fault line (Fig. 7). Most of the upper region is air. Below the

air is the stratum, which is deformed by the fault slip. The stratum lies over the bed rock, which

is split into a hanging wall and a foot wall by the fault plane (Fig. 8). The bed rock is treated as a

solid body. In Fig. 8 we represent the stratum by different shades to emphasize the layer

deformation; however, the parameters are the same throughout the layer. The time evolution of

the slip velocities of the hanging wall is set to an absolute value of a sine curve (Aharonov

2004). The maximum slip rate of the fault was determined from the seismograph records near

the earthquake fault of the Taiwan Chichi earthquake in 1999, and was estimated to be < 2 m/s

(e.g., Wang et al. 2002; Bray and Rodriguez-Marek 2004; Strasser et al. 2009). From the result

of the boring at the Shijia trench site, the thickness of the stratum is estimated to be 11 m (Figs.

2 and 8). The internal friction angle and cohesion of the stratum were set to 45° and 25.0 kPa,

respectively, from the experiment on the suction of silty sand (Nakayama et al, 2008). The

internal friction angle increased by dilatancy ψ (Fig. 6) was 25° (Sakakibara et al. 2008). The

density of sand particle was assumed to be 2.6 g/cm3

(Sakakibara et al. 2008), and the void ratio

Fig. 8 The initial condition of the simulation. The cross section cut is perpendicular by the fault

plane.

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used was 0.7 (dense sand) (Nakayama et al. 2008). The density of the stratum was set at 1.8

g/cm3

from this assumption. The resolution of the simulation was set to 0.2 m.

The fault angle in the simulation was performed at 25° and 49° by borehole data (Fig. 2).

Since the maximum slip rate in case of the Chichi earthquake was unknown, the simulation was

performed for 0.5, 1.0, 1.25 and 1.5 m/s. The value for the sum of the total vertical displacement

of the fault slip was set to 3.4 m from the cw3 layer as seen at the Shijia trench site (Fig. 3) and

the co-seismic vertical displacement by the Chichi earthquake was 0.8 m (Chen et al. 2007).

Therefore, the simulation was performed for the amount of unit vertical displacement as 0.85 m

and 0.68 m. For the amount of unit vertical displacement of 0.85 m, the displacement after

deposition of the layer cw3 was 4 times. For a unit vertical displacement of 0.68 m, the

displacement after deposition of layer cw3 was 5 times. In order to judge the influence by

dilatancy at the deformation of cw3, the simulation is performed by changing the existence of

dilatancy. Table 1 lists the parameters varied in these simulations.

Fig. 9 Simulation results (models 23 and 24). The shape of the deformation of the stratum is shown.

The white shaded area denotes the deformation area of stratum.

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5. The Result of Simulations

The sample of the calculation results of the models are shown in Fig. 9. Some parameters

ware established in order to evaluate the shape of cw3. 1. θmax is the maximum slope angle of

cw3 (Fig. 9). 2. L is the maximum slope angle point from fault tip. 3. W is the width of the fault

scarp. The place where the angle of layer cw3 is over 5.0° is defined as the width of the fault

scarp. 4. Δwl is the distance from the maximum slope angle point to the terminal point of layer

cw3. The terminal point is defined as the point at which the angle of layer cw3 becomes < 10°. 5.

Δwr is the distance from the maximum slope angle point to the starting point of the cw3. The

starting point was defined as the point that the angle of the layer cw3 becomes < 10°. 6. AD is

the value, which shows the symmetry of the angle distribution of the cw3 layer. The point of

symmetry is the point where the angle of cw3 reaches the maximum value. AD is calculated

from Δwl and Δwr, and cw3 is symmetrical when AD is close to 0. The values of these

parameter are shown in the Table 2.

In order to evaluate the difference of the trench data of cw3 and the simulation results of

cw3, the difference of the parameter of the Table 2 is taken. 1. Δθ is the difference between the

θmax from the trenching data for cw3, and that of the simulation result. 2. ΔL is the difference

between the L of the trenching data, and that of the simulation result. 3. Δw is the difference

Table 1 The values of the parameters varied in the simulations

Model Fault angle

Maximum slip rate

[m/s]

Unit vertical

displacement [m]

Dilatancy

Cw3 (trench data) - - - -

1 49 0.5 0.85 no

2 49 1 0.85 no

3 49 1.5 0.85 no

4 25 0.5 0.85 no

5 25 1 0.85 no

6 25 1.5 0.85 no

7 49 0.5 0.68 no

8 49 1 0.68 no

9 49 1.5 0.68 no

10 25 0.5 0.68 no

11 25 1 0.68 no

12 49 0.5 0.68 yes

13 49 1 0.68 yes

14 49 1.5 0.68 yes

15 49 0.5 0.85 yes

16 49 1 0.85 yes

17 49 1.5 0.85 yes

18 25 0.5 0.68 yes

19 25 1 0.68 yes

20 25 1.5 0.68 yes

21 25 0.5 0.85 yes

22 25 1 0.85 yes

23 25 1.5 0.85 yes

24 49 1.25 0.85 yes

25 49 1.125 0.85 yes

26 49 1.25 0.85 no

27 49 1.125 0.85 no

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Model θmax

[degree] L [m] W [m] Δwl [m] Δwr [m] AD

cw3 (trench data) 33 5.0 10.0 4.0 2.7 0.19

1 64 5.6 6.0 3.7 1.3 0.48

2 53 7.7 5.7 3.6 1.0 0.57

3 60 7.9 5.0 3.5 0.2 0.89

4 60 8.0 8.0 4.5 1.2 0.58

5 over hang - 5.0 - - -

6 over hang - 4.5 - - -

7 44 4.9 7.0 3.9 2.2 0.28

8 55 6.4 5.7 3.7 1.2 0.51

9 69 7.7 7.0 4.1 0.7 0.71

10 53 7.7 9.0 4.6 1.2 0.59

11 over hang - 9.0 - - -

12 46 3.2 7.5 2.6 1.9 0.16

13 69 5.5 7.5 2.8 2.1 0.14

14 55 6.0 7.3 3.6 2.2 0.24

15 70 4.7 6.5 3.2 0.8 0.60

16 64 4.7 5.0 2.6 0.9 0.49

17 37 5.4 7.8 2.7 3.6 0.14

18 34 4.2 13.1 3.1 8.8 0.48

19 55 5.7 10.2 2.7 6.8 0.43

20 48 5.0 11.5 2.0 9.0 0.64

21 46 8.6 9.0 6.9 0.9 0.77

22 35 5.6 11.9 2.3 8.4 0.57

23 57 5.1 13.8 2.3 10.6 0.64

24 42 6.0 7.2 3.2 2.1 0.21

25 57 5.2 6.3 2.7 2.5 0.04

26 47 6.2 5.4 3.0 1.4 0.36

27 45 6.2 5.8 3.4 1.3 0.45

Table 2 The values of parameter which evaluated the shape of cw3

Model Δθ [degree] ΔL [m] ΔW [m] Δa residual error

1 31 0.6 4.0 0.29 0.29

2 20 2.7 4.3 0.37 0.39

3 27 2.9 5.0 0.70 0.52

4 7 3.0 2.3 0.38 0.32

5 - - 5.0 - -

6 - - 0.0 - -

7 11 0.1 3.0 0.08 0.13*

8 22 22.0 22.0 0.32 1.79

9 36 2.7 3.0 0.51 0.44

10 2 1.3 3.3 0.39 0.25

11 - - 1.0 - -

12 13 1.8 2.5 0.04 0.20

13 36 0.5 2.5 0.05 0.20

14 22 1.0 2.7 0.05 0.19

15 37 0.3 3.5 0.41 0.31

16 31 0.3 5.0 0.29 0.30

17 4 0.4 2.2 0.05 0.10*

18 1 0.8 3.1 0.28 0.19

19 22 0.7 0.2 0.24 0.16

20 15 0.0 1.5 0.44 0.19

21 13 3.6 1.0 0.58 0.38

22 2 0.6 1.9 0.38 0.18

23 24 0.1 3.8 0.45 0.28

24 9 1.0 2.8 0.01 0.15*

25 24 0.2 3.7 0.16 0.21

26 14 1.2 4.6 0.12 0.24

27 12 1.2 4.2 0.12 0.23

Table 3 The values of parameter which evaluated the difference between the configurations of the

calculated layer cw3 and the trench data

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between the W of the trenching data that of the simulation result. 4. Δa is the difference between

the AD of the trenching data, and that of the simulation result. The residual errors are

normalized and totaled Δθ, ΔL, Δw, and Δa. The values of these parameter are shown in the

Table 3. The with an asterisk cells in Table 3 indicate models residual error of 0.15.

The slope angle distribution of layer cw3 in the Shijia trench and the simulation results of

cw3 are shown in Fig. 10.

Fig. 10 Slope angle distribution of layer cw3 in a Shijia trench and simulation results. Dotted lines

are the trench data. Black lines are the models in consideration of dilatancy. Gray lines are

the models which does not take dilatancy into consideration. The vertical continuous line

extending down into the negative angle values indicates overhang of the stratum, for

example, model 5, model 10 and model 11.

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L for the 25° model is further away from the fault tip compared to the model of 49 ° (Fig.

11). When dilatancy is taken into consideration, W becomes large (Fig. 12). When dilatancy is

taken into consideration, the θmax becomes small (Fig. 12).

6. Conclusions

The trench data of layer cw3 and the simulation results are compared using the residual

errors.

The fault angle of the bed rock and the maximum slip rate in the Shijia trenching site are

determined from these results. Thus, the most suitable selected where models 17 and 24 (Table

Fig. 11 Influence which the fault angle of bed rock has to the maximum slope angle point of the

fault scarp. Distribution of maximum slope angle distance from the result of simulation.

Fig. 12 Influence which the dilatancy has to maximum slope angle point of the fault scarp and the

fault scarp width.

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3), hence, the fault angle of the bed rock and the maximum slip rate obtained for the Shijia

trench were 49° and 1.25-1.5 m/s, respectively.

The fault slip value computed for the Shijia trench of the simulation 1.25-1.5 m/s is fast

compared to a typical fault slip rate of an earthquake source fault 1.0 m/s (e.g., Erdik and

Durukal 2001; Bray and Rodriguez-Marek 2004). The reason why the slip rate is slow in the

Shijia trench site is a future work. In the model in consideration of dilatancy, residual error is

small. For example, models 3 and 17; models 24 and 26 (Table 3). Therefore, it is confirmed

that consideration of dilatancy for sandy soil is important.

The width of the fault scarp became narrower as the fault angle increased. When dilatancy is

taken into consideration, the width of the fault scarp becomes large and the maximum angle of

the fault scarp becomes small.

Acknowledgments

I would like to thank LIS (the Library of Iterative Solvers for linear systems) of the SSI

(Scalable Software Infrastructure for Scientific Computing) project for providing facilities for

the parallel matrix computation, Professor Haruo Yamazaki (Tokyo Metropolitan University) for

the helpful research guidance. Professor Yasushi Nakamura (Tokyo Institute of Technology) for

offering the base program of the simulation, and Mr. Shuji Moriguchi (Gifu University) for

providing technical guidance for the simulation.

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