**Simplified Multi-Block Constitutive Model Predicting the Seismic Displacement of Saturated Sands along Slip Surfaces with Strain Softening**

Constantine A. Stamatopoulos

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59657

## **1. Introduction**

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and Earthquake Engineering, 16: 385-394.

Slopes consisting of saturated sands have recently moved down-slope tens or hundreds of meters under the action of earthquakes [1-3]. This large slide movement is usually associated with the generation of large excess pore pressures, as a result of grain crushing [3]. As these landslides have caused much destruction and fatalities, there is a need to propose easy-to-use and cost-effective methods predicting the triggering and movement of such slides.

The sliding-block model [4] is frequently used to simulate movement of slides triggered by earthquakes [5]. When ground displacement is large, this model may be inaccurate because of (a) reduction of shear resistance along the slip surface and (b) rotation of the sliding mass towards a more stable configuration [6]. It has been modeled in a cost-effective manner by an iterative procedure using the Jambu stability method by Deng et al [7], as well as by the multiblock model by Stamatopoulos et al. [8]. The multi-block model, described below, has the advantage of ensuring displacement compatibility during motion and will be applied in the present work.

Regarding effect (a) above, recently ring shear devices where sandy samples can be sheared under undained conditions have been developed and applied to study the response of saturated sands along slip surfaces [2], [3], [9-15]. Constitutive equations modeling this soil response coupled with the multi-block sliding system model are needed in order to simulate the triggering of the slides and predict accurately the slide displacement. In the general case, constitutive models must be formulated in terms of effective stress in order to predict not only the shear stress, but also the generation of excess pore pressure along slip surfaces. Gerolymos and Gazetas [16] proposed an effective stress model predicting the displacement of saturated sands along slip surfaces based on grain crushing theory which requires 9 model parameters.

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

In addition, Stamatopoulos and Korai [17] proposed and validated with a number of ring shear tests a constitutive model in terms of effective stresses simulating the change of resistance along slip surfaces with shear displacement for sands either under undrained or drained conditions. The model requires 12 model parameters evaluation, which require the availability or performance of ring shear tests at different densities and drainage conditions.

On the other hand, in sliding-block models only the shear resistance versus shear displacement soil response affects the solution. This response depends on the drainage conditions and may alter as a result of dissipation of excess pore pressure in saturated sand. Yet, under earthquakeinduced slides, triggering and slide movement are so rapid that dissipation of excess pore pressure does not occur and saturated sands behave in an undrained manner [1-3], [16]. It is inferred that for predicting the triggering and the slide movement of such slides, only the shear resistance versus shear displacement soil response under undrained conditions may be predicted.

The purpose of the present chapter is to present a cost-effective, but accurate, method pre‐ dicting the triggering and displacement of slides consisting of saturated sands during earth‐ quakes. For this purpose, first the shear stress -displacement response of saturated sands along slip surfaces is described and a model predicting this response with the minimum number of parameters is proposed, calibrated and validated. Then, this constitutive model is coupled with the multi-block sliding system model and the steps needed to predict earthquake-induced slide triggering and large displacement along slip surfaces of saturated sand using this improved model are specified in detail. Finally, the new improved method is validated by application at the well-documented Higashi Takezawa earthquake-induced slide.

## **2. Soil response**

For the simulation of the response of sands along slip surfaces the ring shear device has the advantage that it is the only device where, similarly to landslides, shear displacement can be very large, larger than centimeters, or even meters. The sample in the ring-shear box is doughnut shaped. Undrained response is maintained by pressing a necessary contact pressure in order to keep the specimen volume constant. Additional details on the ring shear apparatus are given by Sassa et al. [18]. Ring-shear tests under both monotonic and cyclic loading have been performed and illustrated that the virgin shear stress-displacement response of saturated sands is almost identical under monotonic and cyclic loading [13]. This investigation also illustrated that the unloading-reloading stiffness is much larger than that due to virgin loading, especially when shear displacement that has already accumulated is considerable. Typical results are given in Fig. 1.

Table 1 gives tests found in the literature where saturated sands are sheared under undrained conditions in the ring shear device, the soils used in these tests, the initial density and stress conditions and the relevant references. Table 2 gives their particle diameter D50 and D10. Fig 2 gives the measured shear stress-displacement response of these tests. In addition, table 1 and Fig. 3 gives two tests performed in a simple shear device that represent adequately the undrained response of the saturated sand along the slip surface of the Higashi Takezawa landslide [1]. It can be observed that in all tests the shear stress first increases at a decreasing rate with shear displacement and reaches its peak value in a few millimeters or centimeters. Then, it decreases at a decreasing with shear displacement rate and reaches its residual value in many centimeters or meters.

In addition, Stamatopoulos and Korai [17] proposed and validated with a number of ring shear tests a constitutive model in terms of effective stresses simulating the change of resistance along slip surfaces with shear displacement for sands either under undrained or drained conditions. The model requires 12 model parameters evaluation, which require the availability

On the other hand, in sliding-block models only the shear resistance versus shear displacement soil response affects the solution. This response depends on the drainage conditions and may alter as a result of dissipation of excess pore pressure in saturated sand. Yet, under earthquakeinduced slides, triggering and slide movement are so rapid that dissipation of excess pore pressure does not occur and saturated sands behave in an undrained manner [1-3], [16]. It is inferred that for predicting the triggering and the slide movement of such slides, only the shear resistance versus shear displacement soil response under undrained conditions may be

The purpose of the present chapter is to present a cost-effective, but accurate, method pre‐ dicting the triggering and displacement of slides consisting of saturated sands during earth‐ quakes. For this purpose, first the shear stress -displacement response of saturated sands along slip surfaces is described and a model predicting this response with the minimum number of parameters is proposed, calibrated and validated. Then, this constitutive model is coupled with the multi-block sliding system model and the steps needed to predict earthquake-induced slide triggering and large displacement along slip surfaces of saturated sand using this improved model are specified in detail. Finally, the new improved method is validated by

For the simulation of the response of sands along slip surfaces the ring shear device has the advantage that it is the only device where, similarly to landslides, shear displacement can be very large, larger than centimeters, or even meters. The sample in the ring-shear box is doughnut shaped. Undrained response is maintained by pressing a necessary contact pressure in order to keep the specimen volume constant. Additional details on the ring shear apparatus are given by Sassa et al. [18]. Ring-shear tests under both monotonic and cyclic loading have been performed and illustrated that the virgin shear stress-displacement response of saturated sands is almost identical under monotonic and cyclic loading [13]. This investigation also illustrated that the unloading-reloading stiffness is much larger than that due to virgin loading, especially when shear displacement that has already accumulated is considerable. Typical

Table 1 gives tests found in the literature where saturated sands are sheared under undrained conditions in the ring shear device, the soils used in these tests, the initial density and stress conditions and the relevant references. Table 2 gives their particle diameter D50 and D10. Fig 2 gives the measured shear stress-displacement response of these tests. In addition, table 1 and Fig. 3 gives two tests performed in a simple shear device that represent adequately the undrained response of the saturated sand along the slip surface of the Higashi Takezawa

application at the well-documented Higashi Takezawa earthquake-induced slide.

or performance of ring shear tests at different densities and drainage conditions.

236 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

predicted.

**2. Soil response**

results are given in Fig. 1.

**Figure 1.** Typical measured effect of cyclic loading on shear stress-displacement relationship. (Trandafir and Sassa [13]).


**Table 1.** Constant-volume ring shear tests in saturated sands found in the literature. Relevant reference (Ref.), sand name and initial conditions are given.


**Table 2.** D50 and D10 values of the sands reported in table 1.

**Fig. 2.** Shear stress-displacement response of the constant-volume ring shear tests of table 1. Tests on sands (a) S8, (b) ING, (c) WG **Figure 2.** Shear stress-displacement response of the constant-volume ring shear tests of table 1. Tests on sands (a) S8, (b) ING, (c) WG. Model predictions are also given. **1. FIGURES**

Higashi, σ'o=90, τo=60 kpa Higashi, σ'ο=50, το=40kPa

 *R* 0 ' where (1a) **In Fig. 3 the labels of the y axes are incorrect. Replace Fig. 4 with Figure 3.** Shear stress-displacement response of the constant-volume ring shear tests of table 1. Tests on the Higashi Takezawa landslide. Model predictions are also given.

res r u

r 1 ( r

res

1 a 1

<sup>1</sup> <sup>R</sup> { <sup>2</sup>

<sup>u</sup> <sup>R</sup> <sup>R</sup> (1b)

2 a

a 1

(u <sup>u</sup> ) 1)

Rres

1

a

r u

r

res

1 (

res

r

**For u>u2** R Rres **(d)** 

<sup>1</sup> <sup>R</sup> {

a 1

<sup>u</sup> <sup>R</sup> <sup>R</sup> **(b)** 

2

2

a 1

a

(u <sup>u</sup> ) 1)

2 1

 **(c)**

2 1

(1c)

a

For 0<u<u2 <sup>1</sup>

R /r res

 **WG ING S8** 

**(mm) 0.23 0.11 0.06** 

**(mm) 0.04 0.03 0.015**

**Equations (1b-d) are incorrect. Replace with:** 

**(Items needing correction is given in yellow).** 

**For u1>u>u2** } (u <sup>u</sup> )

**For 0<u<u1** <sup>1</sup>

α <sup>1</sup>

<sup>0</sup> <sup>u</sup>

**2. TABLES**

**D50**

**D10**

**3. EQUATIONS**

R=τ/σ΄

o

For u2>u>u3 } (u <sup>u</sup> )

α <sup>2</sup>

u (log scale) <sup>1</sup> <sup>u</sup> 2 3 <sup>u</sup>

Table 2**. is incorrect. Delete last column. Thus, the table becomes** 

Fig. 4. **The proposed model and its model parameters.** 

4

## **3. Proposed model**

**WG ING S8**

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 **u (m)**

Test 18, Dr=29%, σ'o=196, τo=0kPa Test 19, Dr=29%, σ'o=250, τo=0kPa Test 20, Dr=29%, σ'o=290, τo=0kPa Test 21, Dr=44%, σ'o=203, τo=0kPa Test 22, Dr=44%, σ'o=235, τo=0kPa Test 23, Dr=44%, σ'o=290, τo=0kPa

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 **u (m)**

 Test 6, Dr=68%, σ'ο=196, τo=56kPa Test 7, Dr=68.2%, σ'o=196, τo=0kPa Test 8, Dr=69%, σ'o=196, τo=40kPa Test 9, Dr=69%, σ'o=196, τo=75kPa Test 10, Dr=69%, σ'o=196, τo=110kPa Test 11, Dr=95%, σ'o=196, τo=0kPa

4

D50 (mm) 0.23 0.11 0.06 D10 (mm) 0.04 0.03 0.015

238 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

Test 24, Dr=44%, σ'o=366, τo=0kPa

**τ (Kpa)**

 **WG** 

**τ (Kpa)**

**Fig. 2.** Shear stress-displacement response of the constant-volume ring shear tests of table

**Figure 2.** Shear stress-displacement response of the constant-volume ring shear tests of table 1. Tests on sands (a) S8,

(b) (c)

Higashi, σ'o=90, τo=60 kpa Higashi, σ'ο=50, το=40kPa

**In Fig. 3 the labels of the legends are incorrect. Replace Fig. 3 with** 

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 u (m)

Higashi, σ'o=90, τo=60 kpa Higashi, σ'ο=50, το=40kPa

**Fig. 3.** Shear stress-displacement response of the constant-volume ring shear tests of table

The following simplified equations are used to simulate shear stress-displacement response

**Figure 3.** Shear stress-displacement response of the constant-volume ring shear tests of table 1. Tests on the Higashi

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 u (m)

' where (1a)

**In Fig. 3 the labels of the y axes are incorrect. Replace Fig. 4 with** 

Table 2**. is incorrect. Delete last column. Thus, the table becomes** 

res r u

r 1 ( r

res

1 a 1

<sup>1</sup> <sup>R</sup> { <sup>2</sup>

<sup>u</sup> <sup>R</sup> <sup>R</sup> (1b)

2 a

a 1

(u <sup>u</sup> ) 1)

Rres

1

a

r u

r

res

1 (

res

r

**For u>u2** R Rres **(d)** 

<sup>1</sup> <sup>R</sup> {

a 1

<sup>u</sup> <sup>R</sup> <sup>R</sup> **(b)** 

2

2

a 1

a

(u <sup>u</sup> ) 1)

2 1

 **(c)**

2 1

(1c)

a

**Table 2.** D50 and D10 values of the sands reported in table 1.

(a)

 Test 1, Dr=61%, σ'o=196, τo=56kPa Test 2, Dr=63%, σ'o=196, τo=0 kPa Test 3, Dr=63%, σ'o=196, τo=4 kPa Test 4, Dr=63%, σ'o=200, τo=0 kPa

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 **u (m)**

Test 12, Dr=29%, σ'o=202, τo=0kPa Test 13, Dr=29%, σ'o=262, τo=0kPa Test 14, Dr=31%, σ'ο=290, το=0kPa Test 15, Dr=44%, σ'o=196, το=0kPa Test 16, Dr=44%, σ'ο=280, το=9kPa Test 17, Dr=44%, σ'o=374, τo=9kPa

**S8 1/2**

Test 5, Dr=65%, σ'o=196, το=56 kPa

 **ING**

**1. FIGURES**

**τ (Kpa)**

**τ (Kpa)**

1. Tests on sands (a) S8, (b) ING, (c) WG

(b) ING, (c) WG. Model predictions are also given.

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 **u (m)**

1. Tests on the Higashi Takezawa landslide

τ (Kpa)

0

50

100

of sands under undrained conditions:

**3. Proposed Model** 

*R* 0

α <sup>1</sup>

 

<sup>0</sup> <sup>u</sup>

**2. TABLES**

**D50**

**D10**

**3. EQUATIONS**

R=τ/σ΄

o

0

50

τ (Kpa)

For 0<u<u2 <sup>1</sup>

R /r res

Takezawa landslide. Model predictions are also given.

 **WG ING S8** 

**(mm) 0.23 0.11 0.06** 

**(mm) 0.04 0.03 0.015**

**Equations (1b-d) are incorrect. Replace with:** 

**(Items needing correction is given in yellow).** 

**For u1>u>u2** } (u <sup>u</sup> )

**For 0<u<u1** <sup>1</sup>

For u2>u>u3 } (u <sup>u</sup> )

α <sup>2</sup>

u (log scale) <sup>1</sup> <sup>u</sup> 2 3 <sup>u</sup>

Fig. 4. **The proposed model and its model parameters.** 

100

The following simplified equations are used to simulate shear stress-displacement response of sands under undrained conditions:

$$
\sigma = \sigma \, ^\prime \prescript{}{0}{R} \text{ where} \tag{a}
$$

$$\begin{aligned} \text{For } 0 \le \mathbf{u} \le \mathbf{u}\_1 \quad \mathbf{R} = \mathbf{R}\_{\text{res}} \frac{\mathbf{u}^{\mathbf{a}\_1}}{\mathbf{r} \cdot \mathbf{u}\_1^{\mathbf{a}\_1}} \\ \text{For } \mathbf{u}\_1 \ge \mathbf{u} \ge \mathbf{u}\_2 \quad \mathbf{R}\_{\text{res}} \left\{ \frac{1}{\mathbf{r}} - (\frac{1}{\mathbf{r}} - 1) \frac{(\mathbf{u} \cdot \mathbf{u}\_1)^{\mathbf{a}\_1}}{(\mathbf{u}\_2 \cdot \mathbf{u}\_1)^{\mathbf{a}\_2}} \right\} \quad \text{(c)} \\ \text{For } \mathbf{u} \ge \mathbf{u}\_2 \quad \mathbf{R} = \mathbf{R}\_{\text{res}} \end{aligned} \tag{1}$$

where σ'ο is the applied effective normal stress and u is the shear displacement. The model has 6 parameters: Rres, r, u1, u2, a1, a2. As illustrated in Fig. 4, Rres is the stress ratio (τ/ σ'ο) of the material at the residual state. The parameter r equals to (Rres/Rmax). Thus, it varies from 0 to 1. The parameters (a) u1 and (b) u2 give the shear displacement corresponding to (a) the peak shear stress and (b) the minimum shear displacement corresponding to the residual shear stress. The parameters (i) a1 and (ii) a2 determine the rate of change of the shear stress with shear displacement for (i) u1<u and (ii) u1<u<u2 respectively.

In the proposed constitutive model plastic shear displacements are assumed to accumulate when the stress ratio (τ/σ') increases, while the elastic shear displacement is ignored. This is consistent with the response depicted in Fig. 1.

An excel worksheet was programmed to simulate the undrained response of sands, as described by equations (1). A modification is needed for tests with initial shear stress (τo). In this case the shear displacement is u' and equals:

$$\mathbf{u}" = \mathbf{u} - \boldsymbol{\mu}\_0 = \boldsymbol{\mu} - \boldsymbol{\mu}\_1 \left[ \frac{\boldsymbol{\sigma}\_0}{\sigma \prime \prime\_0 R\_m} \right]^{\frac{1}{a\_l}} \mathbf{u} \text{\textquotedblleft } \mathbf{0} \tag{2}$$

where u is the displacement predicted by eq. (1).

The model parameters of the tests of Table 1 were estimated using the procedure described above. They are given in Table 3. In addition, Fig. 2 gives the computed shear stress-displace‐ ment curves. Table 3 gives the standard deviation of the ratio of predicted by measured values of all points defining the shear stress-displacement curves of all tests. It can be observed that in all tests the standard deviation is less than 0.3, while the average value is 0.12. In addition, from Fig 2 it can be observed that adequate prediction of undrained response is achieved in all tests. Furthermore, from table 3 it can be observed that the values of the model parameters used do not vary considerably from case to case and are in general in a rational range: the parameters a1 and a2 do not take values greater than 1 and less than 0 and the parameter Rm generally increases with Dr. All the above verify the proposed model.

Advantage of the proposed model is simplicity. However, it has the disadvantage of generality. Inspection of table 3, illustrates that the model parameters depend on the relative density, confining stress and initial shear stress. Thus, when applying this simplified model, tests with similar relative density, confining stress and initial shear stress as that existing in-situ should be used.


**Table 3.** The parameters of equation (1) that fit the tests of table 1. Statistical analysis of the accuracy of the predictions is also given, where N represents the number of points defining the shear stress-displacement curve of each test.

0 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 u (m) Simplified Multi-Block Constitutive Model Predicting the Seismic Displacement of Saturated Sands along Slip… http://dx.doi.org/10.5772/59657 241

50

τ (Kpa)

100

**In Fig. 3 the labels of the legends are incorrect. Replace Fig. 3 with** 

**In Fig. 3 the labels of the y axes are incorrect. Replace Fig. 4 with** 

Table 2**. is incorrect. Delete last column. Thus, the table becomes** 

Higashi, σ'o=90, τo=60 kpa Higashi, σ'ο=50, το=40kPa

Fig. 4. **The proposed model and its model parameters.** 

**Figure 4.** The proposed model and its model parameters.

**2. TABLES**

**1. FIGURES**

### **4. The Multi-Block Sliding System Model**

**D50**

#### **4.1. Introduction**

Advantage of the proposed model is simplicity. However, it has the disadvantage of generality. Inspection of table 3, illustrates that the model parameters depend on the relative density, confining stress and initial shear stress. Thus, when applying this simplified model, tests with similar relative density, confining stress and initial shear stress as that existing in-situ should

240 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

**u2 (m)**

 0.42 0.20 0.001 7.00 0.08 0.16 12 0.07 0.27 0.35 0.001 0.50 1.00 0.15 11 0.17 0.27 0.52 0.002 0.02 0.57 0.48 12 0.15 0.27 0.39 0.001 0.70 1.00 0.15 10 0.12 0.57 0.31 0.001 7.00 0.30 0.40 12 0.18 0.75 0.30 0.005 12.50 0.20 0.40 12 0.13 0.25 0.42 0.001 1.00 1.00 0.38 10 0.12 0.35 0.32 0.001 10.00 0.10 0.14 12 0.07 0.45 0.26 0.001 10.00 0.08 0.15 12 0.10 0.60 0.20 0.005 10.00 0.02 0.24 12 0.07 0.99 0.20 0.023 100.00 0.40 0.20 9 0.15 0.24 0.10 0.002 0.10 1.00 0.30 5 0.01 0.25 0.17 0.003 0.01 0.50 0.40 11 0.17 0.36 0.23 0.002 0.50 1.00 0.15 10 0.12 0.37 0.42 0.010 10.00 0.10 0.25 1 0.14 0.50 0.30 0.010 2.00 0.30 0.20 10 0.18 0.40 0.26 0.010 2.00 0.10 0.30 11 0.14 0.30 0.17 0.001 0.03 0.35 0.20 6 0.14 0.30 0.30 0.001 4.00 0.50 0.15 4 0.16 0.34 0.12 0.004 0.10 0.20 0.10 9 0.18 0.50 0.13 0.010 2.00 0.20 0.20 10 0.18 0.52 0.12 0.010 1.00 0.40 0.20 11 0.17 0.65 0.17 0.009 1.00 0.50 0.25 10 0.14 0.60 0.15 0.009 1.00 0.40 0.20 9 0.15 1.10 0.28 0.020 10.00 0.40 0.25 5 0.05 0.60 0.28 0.010 5.00 0.05 0.20 6 0.04

**Table 3.** The parameters of equation (1) that fit the tests of table 1. Statistical analysis of the accuracy of the predictions is also given, where N represents the number of points defining the shear stress-displacement curve of each test.

**a1 a2 N SDev**

All Mean 0.12 All SDev 0.05

**(Pred/Meas)**

be used.

**Test no**

**Rm r u1**

**(m)**

**(mm) 0.23 0.11 0.06 D10 (mm) 0.04 0.03 0.015 3. EQUATIONS Equations (1b-d) are incorrect. Replace with:**  A slip surface which consists of linear segments is considered (Fig. 5a). In order for the mass above the slip surface to move, interfaces inside the sliding mass must be formed at the nodes between the linear segments [20]. In this manner the mass is divided into n blocks. Soil is assumed to behave as a rigid-perfectly plastic material with a Mohr-Coulomb failure criterion at both the slip surface and the interfaces. The forces that are exerted in block "i" are given in Fig. 5a.

 **WG ING S8** 

**For 0<u<u1** <sup>1</sup> 1 a 1 a res r u <sup>u</sup> <sup>R</sup> <sup>R</sup> **(b)**  When the slide moves, two options exist regarding the relative movement of blocks: (a) no separation and (b) separation. When blocks are not separated, the velocity must be continuous at the interface. This rule predicts that the relative displacement of the n blocks is related to each other as:

$$\frac{du\_i}{du\_{i+1}} = \frac{\cos\left(\mathcal{S}\_l + \mathcal{J}\_{l+1}\right)}{\cos\left(\mathcal{S}\_l + \mathcal{J}\_l\right)}\tag{3}$$

1

2

a 1

**For u>u2** R Rres **(d) (Items needing correction is given in yellow).**  where u is the displacement moved by the sliding mass along the linear segment "i" of the trajectory, d refers to increment, the subscripts i and i+1 refer to trajectory segments i and i+1 counting uphill and β<sup>i</sup> and (90-δ<sup>i</sup> ) are the inclinations of the trajectory segment and interface i respectively.

Fig. 5a illustrates the forces exerted in each block i. In the case that separation of blocks does not occur and a horizontal acceleration is applied, as indicated by Stamatopoulos et al [8], the governing equation of motion is

$$\begin{split} \widecheck{i}\left(t\right) \cdot \sum\_{i=1}^{n} \Big( \boldsymbol{m}\_{i} \boldsymbol{q}\_{i} \cos\left(\phi\_{i}\right) \cdot \prod\_{j=i}^{n-1} \frac{d\boldsymbol{d}\_{j+1}}{\boldsymbol{\varvarprojlim}} \Big) &= \sum\_{i=1}^{n-1} \Big[ \boldsymbol{-b}\boldsymbol{c}\_{i} \cos\left(\phi \boldsymbol{b}\_{i}\right) + \boldsymbol{U} \boldsymbol{b}\_{i} \sin\left(\phi \boldsymbol{b}\_{i}\right) \Big] \cdot \frac{\boldsymbol{S} \boldsymbol{S}\_{i}}{\boldsymbol{\varvarlim}} \cdot \prod\_{j=i}^{n-1} \frac{d\boldsymbol{d}\_{j+1}}{\boldsymbol{\varvarlim}} + \\ + \sum\_{i=1}^{n} \Big[ \Big( \boldsymbol{x}\boldsymbol{x}\_{i} \Big( \boldsymbol{m}\_{i} \boldsymbol{g} \boldsymbol{k}\left(t\right) + \boldsymbol{H}\_{i}\big) - \boldsymbol{v}\_{i} \Big[ \boldsymbol{m}\_{i} \boldsymbol{g} + \boldsymbol{Q}\_{i} \big] - \boldsymbol{c}\_{i} \boldsymbol{l}\_{i} \cos\left(\phi\_{i}\right) + \boldsymbol{U}\_{i} \boldsymbol{\upalpha}\left(\phi\_{i}\right) \Big\} \cdot \prod\_{j=i}^{n-1} \frac{d\boldsymbol{d}\_{j+1}}{\boldsymbol{\varvarlim}} \right) \end{split}$$

wher e

$$\mathbf{r}.\mathbf{s}\_{i} = \sin\left(\beta\_{i+1} - \beta\_{i} + \eta b\_{i} + \eta b\_{i+1}\right)\tag{b}$$

$$\mathbf{r}.\mathbf{r}.\mathbf{r}.\mathbf{r}.\mathbf{r}.\mathbf{r}.\mathbf{r}.\mathbf{a}.\mathbf{r}.\mathbf{a}$$

$$\text{ff}\_i \equiv \cos(\varphi\_i + \varphi \mathbf{b}\_{i+1} - \beta\_i - \delta\_i)$$

$$\text{xxx}\_{l} = \cos\left(\varphi\_{l} - \beta\right) \tag{6}$$

$$\mathbf{v}\_i = \sin \left( \wp\_i - \wp\_i \right) \tag{d}$$

$$\mathbf{d} \mathbf{d}\_i \mathbf{d} = \cos \left( \rho\_i + \rho \mathbf{b}\_i - \beta\_i - \delta\_{i\perp} \right) \tag{5}$$

$$\mathbf{q}\_{i} = \stackrel{\text{n-1}}{\stackrel{\text{G}}{\bullet}} \frac{\mathbf{\hat{\mathbf{G}} \text{os}} \left( \boldsymbol{\beta}\_{i+1} + \boldsymbol{\epsilon}\_{i} \right)}{\mathbf{\hat{\mathbf{G}} \text{os}} \left( \boldsymbol{\beta}\_{i} + \boldsymbol{\epsilon}\_{i} \right)} \tag{f} \tag{f}$$

where u is the relative shear displacement of the uppermost block (block n), down-hill displacements being considered positive, g is the acceleration of gravity, Qi is the vertical external load on block i, Hi is the horizontal external load on block i, mi is the mass of block i, Ui is the pore water force acting on block i and Ubi is the pore water force along the interface between blocks i and i+1, φ<sup>i</sup> and ci is the angle of friction and cohesion on the slip surface at block i along the slip surface and φb<sup>i</sup> and cbi is the angle of friction and cohesion on the interslice surface between blocks i and i+1. It should be noted that all the above masses and forces are taken per unit length, normal to the paper in Fig. 5a.

Equation (4) can be written in the form

$$\begin{aligned} \dot{\boldsymbol{u}}\_{1} &= \; \mathbf{Z}\_{1} \; \mathbf{g} \left( \mathbf{k}(t) \cdot \mathbf{k}\_{c} \right) \\ \dot{\boldsymbol{Z}}\_{1} &= \sum\_{i=1}^{n} \bigg( -\mathbf{x} \mathbf{x}\_{i} \mathbf{m}\_{i} \cdot \prod\_{j=1}^{n-1} \frac{d \boldsymbol{d}\_{j+1}}{\iint\_{\mathcal{I}\_{j}} \boldsymbol{d} \mathbf{f}\_{j} \, \bigg) \\ &= \sum\_{i=1}^{n} \bigg( \boldsymbol{m}\_{i} \boldsymbol{q}\_{i} \cos(\phi\_{i}) \cdot \prod\_{j=i}^{n-1} \frac{d \boldsymbol{d}\_{j+1}}{\iint\_{\mathcal{I}\_{j}} \boldsymbol{f} \mathbf{f}\_{j} \, \bigg) \end{aligned} \tag{6}$$

$$k\_c = \frac{\sum\_{i=1}^{n-1} \left[ \left\{ cch\_l b\_l \cos\left(\phi b\_l\right) - Ub\_l \sin\left(b\_l \phi\_l\right) \right\} \right] \cdot \frac{SS\_l}{\iint\_{\hat{I}\_l} \cdot \prod\_{j=1}^{n-1} \frac{dd\_{j+1}}{\iint\_{\hat{I}\_j}}}{\int\_{\hat{I}\_l} \left( -\text{xx}\_j m\_l \cdot \prod\_{j=1}^{n-1} \frac{dd\_{j+1}}{\iint\_{\hat{I}\_j}} \right)} + \tag{5}$$

$$+ \frac{\sum\_{i=1}^{n} \left( \left\{ -\text{x}v\_i\left(H\_i\right) + v\_i\left[W + Q\_l\right] + c\_l l\_i \cos\left(\phi\_l\right) - U\_i \sin\left(\phi\_l\right) \right\} \cdot \prod\_{j=1}^{n-1} \frac{dd\_{j+1}}{\iint\_{\hat{I}\_j}} \right)}{\text{g-} \sum\_{i=1}^{n} \left( -\text{x}v\_i m\_i \cdot \prod\_{j=1}^{n-1} \frac{dd\_{j+1}}{\iint\_{\hat{I}\_j}} \right)} \tag{6}$$

where kc is the critical acceleration factor, defined as the limit horizontal acceleration needed to initiate motion (ac) normalized by the acceleration of gravity (g).

For large displacement, the masses and lengths of each block in equation (5) must be updated in terms of the distance moved. Assuming that the displacement u is less than the initial length of ln, the change of lengths li in each displacement increment Δu equals

$$\begin{aligned} \Delta l\_i &= \Delta \iota \cdot q\_1 & \quad & \text{(a)}\\ \Delta l\_n &= -\Delta \iota & \quad & \text{(b)}\\ \Delta l\_2 = \Delta l\_3 &= \dots = \Delta l\_{n-1} = 0 & \quad & \text{(c)} \end{aligned} \tag{6}$$

The incremental change in the interslice lengths bi is

23 1

*n*

( ) ( ) ( ) ( )

*u t mq bc b Ub b*

cos cos sin

*<sup>n</sup> <sup>n</sup> <sup>j</sup> i i i i i i ii i i i i j i j*

*ii i i ii i*

*xx m gk t H v m g Q c l U ff*

æ ö + ç ÷ + - +- + × è ø

*n n <sup>n</sup> <sup>n</sup> <sup>j</sup> <sup>j</sup> <sup>i</sup>*

242 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

<sup>æ</sup> ö æ <sup>ö</sup> <sup>×</sup> <sup>ç</sup> × = é- + ù × × + ÷ ç <sup>÷</sup> ë û <sup>ç</sup> ÷ ç <sup>÷</sup> <sup>è</sup> ø è <sup>ø</sup>

xx cos (c) v =sin (d) dd =cos + b (e)

cos +δ q =Õ (f ) cos +δ

and cbi

where u is the relative shear displacement of the uppermost block (block n), down-hill displacements being considered positive, g is the acceleration of gravity, Qi is the vertical

Ui is the pore water force acting on block i and Ubi is the pore water force along the interface

surface between blocks i and i+1. It should be noted that all the above masses and forces are

Z g ( k(t) - k ) where (a)

is the horizontal external load on block i, mi

and ci is the angle of friction and cohesion on the slip surface at

<sup>1</sup> <sup>1</sup>

*dd*


is the angle of friction and cohesion on the interslice

*i i j i j i j j i*

= = = =

å å Õ Õ

1 1

f

1

s n e i

ff =cos + b

i ii

j b

= - -

wher

*i*

*ss*

&&

1

( ) ( )

i i i+1 i i

( ) ( )

external load on block i, Hi

between blocks i and i+1, φ<sup>i</sup>

11 c

1

1

*c*

1

*u*

&&

=

block i along the slip surface and φb<sup>i</sup>

Equation (4) can be written in the form

( )

f

*m q ff*

<sup>1</sup> <sup>1</sup> <sup>1</sup>

*<sup>n</sup> <sup>n</sup> <sup>j</sup> i i i j i j <sup>n</sup> <sup>n</sup> <sup>j</sup> ii i i j i j*

æ ö ç ÷ - × è ø <sup>=</sup> æ ö ç ÷ × è ø

cos

å Õ

= =

= =

å Õ

*dd xx m ff <sup>Z</sup>*

( ) ( )

å Õ

*<sup>n</sup> <sup>n</sup> <sup>j</sup> <sup>i</sup> ii i i i i i j j i*

cos sin

1

=

*i*

n

*cb b b Ub b ff ff <sup>k</sup>*

f

<sup>1</sup> <sup>1</sup> <sup>1</sup> 1 1

æ ö ç ÷ é - ù× × + ë û è ø <sup>=</sup> ×- ×


 f

*i i*

{ ( ) [ ] ( ) ( )}

å Õ

= =

1

*g xx m ff*

å Õ

*n n*

1

<sup>1</sup> <sup>1</sup>

*dd*


cos sin

ff

+

*ss dd*

*j*

*j i j*

=

*xx H W Q c l U ff*

è ø

*<sup>n</sup> <sup>n</sup> <sup>j</sup> i i i i ii i i i i j i j <sup>n</sup> <sup>n</sup> <sup>j</sup> i i i j i j*

*g xx m ff*

å Õ

è ø <sup>+</sup> æ ö ×- × ç ÷

= =

1


æ ö ç ÷ è ø æ ö ç ÷ - + ++ - ×

*dd*

taken per unit length, normal to the paper in Fig. 5a.

<sup>1</sup> <sup>1</sup>


*dd*


i i i i i-1 n-1 i+1 i <sup>i</sup> j=i i i

 bj

( ) ( )

 b d


( )

 b d


*i i ii*

+

= -+

jj

j b

i

jj

b

b

b

<sup>1</sup> <sup>1</sup> <sup>1</sup> <sup>1</sup> <sup>1</sup>

*ff ff ff*

*dd ss dd*


f

cos sin

f

<sup>1</sup> <sup>1</sup>

*dd*


 f

 f (a)

(b)

is the mass of block i,

(b)

(5)

(c)

(4)

{ ( ( ) ) [ ] ( ) ( )}

å Õ

= =

1

*i*

j*b*

+ <sup>+</sup>

$$
\Delta b\_{i} = \frac{\sin \theta\_{i}}{\cos(\theta\_{i} + \beta\_{i} + \delta\_{i})} \cdot q\_{i} \Delta u \tag{7}
$$

where the angle θ<sup>i</sup> is given in Fig. 5b. Fig. 5c illustrates the deformation that these rules predict. In addition, the incremental change in the mass is

$$\begin{split} \Delta m\_{i} &= \rho\_{i+1} \cdot (b\_{(i+1)-c} \cdot \cos(\beta\_{i+1} + \delta\_{i+1}) \cdot q\_{i} \Delta u + \frac{0.5 \cdot \cos(\beta\_{i+1} + \delta\_{i+1}) \cdot \sin \theta\_{i+1}}{\cos(\theta\_{i+1} + \beta\_{i+1} + \delta\_{i+1})} \cdot \left(q\_{i} \Delta u\right)^{2} \\ &- b\_{i-c} \cdot \cos(\beta\_{i} + \delta\_{i}) \cdot q\_{i} \Delta u - \frac{0.5 \cdot \cos(\beta\_{i} + \delta\_{i}) \cdot \sin \theta\_{i}}{\cos(\theta\_{i} + \beta\_{i} + \delta\_{i})} \cdot \left(q\_{i} \Delta u\right)^{2} \end{split} \tag{8}$$

where bi-c and b(i+1)-c correspond to the values of bi and b(i+1) at the previous increment and ρ is the total unit weight of the soil.

Separation of blocks occurs when an interslice force, Ni , is negative [21]. Fig. 6c illustrates a typical case where this occurs: when the angle β<sup>m</sup> at the trajectory is larger than the angle β<sup>m</sup> +1. In this case, the soil mass of the block along the "m+1" segment of the trajectory cannot maintain contact with the rest of the sliding material and is detached from the system. The detached mass is no longer considered in the solution.

#### **4.2. Multi-block model with constitutive equations**

The multi-block sliding system can be coupled with the constitutive model described above by assuming zero cohesion and varying only the friction angles, φ<sup>i</sup> , of equation (5) as

$$\varphi\_l = \arctan\left(\frac{\tau\_l}{\sigma\_{o-l}^{\prime}}\right) \tag{9}$$

In equation (9), τ<sup>i</sup> and σ'o-i are the shear stress and the initial (prior to slide movement) effective normal stress at the base of block i.

Application of eq. (9) with the multi-block numerical code first requires to estimate the shear initial stresses τ<sup>i</sup> , or equivalently, the initial friction angles φ<sup>i</sup> . This is performed by iterations, by assuming all φ<sup>i</sup> are equal and by increasing them incrementally until critical equilibrium is achieved in the initial slide configuration. Then, at each increment, for each block, τ<sup>i</sup> is updated from equations (1) in terms of the incremental shear displacement. It should be noted that the values of τ, u and σ'o of equations (1) correspond to τ<sup>i</sup> , ui and σ'o-i in equation (9).

#### **4.3. Computer program**

A computer program which solves the equations described above has been developed by the author. The input geometry is specified as the nodes of the linear segments defining the trajectory, ground and water table surfaces. Different soil properties may be specified at each segment of the slip surface and at the interfaces. Along the slip surface the Mohr-Coulomb Model or the constitutive model may be applied. At the interfaces the Mohr-Coulomb Model is applied. Output of the program includes the final slide geometry and acceleration velocity and displacement of nodes of the sliding mass versus time.

Pore pressures at the mid-points of the linear segments of the linear segments of the slip and the interfaces prior to the application of earthquake loading are estimated from the water table surface according to the general equation:

$$P\_i = h\_{W-i} \cos^2 \left(\theta w\_i\right) \tag{10}$$

where θw<sup>i</sup> is the inclination and hi is the height of the water table surface above the mid-point of slip segment "i". The program includes graphical representations of the input and final geometries of the slope. More details of the multi-block method and the associated numerical code are given by Stamatopoulos et al [8].

#### **4.4. Application of the model along (pre-defined) slip surfaces and very large displacement under earthquake loading**

In the case that the slip surface is not pre-existing, generally, application of the sliding-block model first requires the prediction of the location of the slip surface by stability analysis. However, this determination, and the ability of stability methods to estimate the location of this slip surface is beyond the purpose of the present work.

In order to apply the improved multi-block model along pre-defined slip surfaces under earthquake loading first the slip surface, ground surface and water table surface are simulated as a series of linear segments. If the inclinations of the interfaces is not predifined according to existing faults, as proposed by Sarma [20] they are obtained based on the condition of the minimum critical acceleration value, by iterations. Along the slip surface the Mohr-Coulomb model the value is used with soil strength corresponding to large displacement, as it is the most representative of the soil strength during occurrence of the landslide. At the interfaces, peak values of strength are used. The reason is that as the internal interfaces are fixed in space, they are continuously reforming with new material and thus the strength cannot be at residual [23]. In addition, at large deformations the method described above to estimate the interslice angles of the sliding mass according to the condition of minimum critical acceleration at the initial slide configuration may not be adequate. The reason is that some segments of the trajectory do not have mass at the initial configuration and thus their interface angles cannot be defined. This can be resolved by applying the criterion of minimum acceleration not only at the initial, but also at the final slide configuration and taking the average values at the common interfaces of the two interfaces. The final slide configuration can be obtained by applying the multi-block model assuming that the interface angles not defined at the initial slide configuration equal zero.

The slide triggering is investigated and the potential slide deformation is estimated using the multi-block model. For the representative seismic motion, the proposed constitutive model is used along the slip surface, while at the interfaces the Mohr-Coulomb model with peak values of strength is used.

## **5. Application at the Higashi Takezawa landslide**

### **5.1. The slide**

5.2 Model predictions

applied normal stress, predictions are adequate.

results of the triaxial tests, equals φ'=36o

achieved in the initial slide configuration. Then, at each increment, for each block, τ<sup>i</sup> is updated from equations (1) in terms of the incremental shear displacement. It should be noted that the

244 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

A computer program which solves the equations described above has been developed by the author. The input geometry is specified as the nodes of the linear segments defining the trajectory, ground and water table surfaces. Different soil properties may be specified at each segment of the slip surface and at the interfaces. Along the slip surface the Mohr-Coulomb Model or the constitutive model may be applied. At the interfaces the Mohr-Coulomb Model is applied. Output of the program includes the final slide geometry and acceleration velocity

Pore pressures at the mid-points of the linear segments of the linear segments of the slip and the interfaces prior to the application of earthquake loading are estimated from the water table

q

where θw<sup>i</sup> is the inclination and hi is the height of the water table surface above the mid-point of slip segment "i". The program includes graphical representations of the input and final geometries of the slope. More details of the multi-block method and the associated numerical

**4.4. Application of the model along (pre-defined) slip surfaces and very large displacement**

In the case that the slip surface is not pre-existing, generally, application of the sliding-block model first requires the prediction of the location of the slip surface by stability analysis. However, this determination, and the ability of stability methods to estimate the location of

In order to apply the improved multi-block model along pre-defined slip surfaces under earthquake loading first the slip surface, ground surface and water table surface are simulated as a series of linear segments. If the inclinations of the interfaces is not predifined according to existing faults, as proposed by Sarma [20] they are obtained based on the condition of the minimum critical acceleration value, by iterations. Along the slip surface the Mohr-Coulomb model the value is used with soil strength corresponding to large displacement, as it is the most representative of the soil strength during occurrence of the landslide. At the interfaces, peak values of strength are used. The reason is that as the internal interfaces are fixed in space, they are continuously reforming with new material and thus the strength cannot be at residual [23]. In addition, at large deformations the method described above to estimate the interslice angles of the sliding mass according to the condition of minimum critical acceleration at the initial slide configuration may not be adequate. The reason is that some segments of the trajectory do not have mass at the initial configuration and thus their interface angles cannot

( ) <sup>2</sup> cos *Ph w i Wi* = -

, ui

and σ'o-i in equation (9).

*<sup>i</sup>* (10)

values of τ, u and σ'o of equations (1) correspond to τ<sup>i</sup>

and displacement of nodes of the sliding mass versus time.

this slip surface is beyond the purpose of the present work.

surface according to the general equation:

code are given by Stamatopoulos et al [8].

**under earthquake loading**

**4.3. Computer program**

Fig. 6 gives the cross-section of the Higashi Takezawa landslide [1]. According to Deng et al. [1] a representative acceleration record for the slide is that reported by the Japan Meteorological Agency (JMA) in 2004. It is given in Fig. 6. Triaxial and shear laboratory tests of material along the slip surface of the slide have been performed by Deng et al., [1] and are given in Fig. 6.

**Fig. 5 (**a) The multi-block stability method proposed by Sarma (1979). (b) Definition of the angle θi of the ground surface of block i. The angle θi changes when ui>uθi (c) Deformation assumed in the multi-block model for a case of two blocks. The x-axis gives the horizontal distance, while the y-axis gives the elevation. **Figure 5.** (a) The multi-block stability method proposed by Sarma (1979). (b) Definition of the angle θ<sup>i</sup> of the ground surface of block i. The angle θ<sup>i</sup> changes when ui >uθ<sup>i</sup> (c) Deformation assumed in the multi-block model for a case of two blocks. The x-axis gives the horizontal distance, while the y-axis gives the elevation.

First the model parameters of the constitutive model are obtained by the prediction of the shear test results. Table 4 gives the model parameters and Fig. 7 compares model predictions of the constitutive model with the measured response. Very good agreement is observed. It can be observed that even though all model parameters do not depend on the

From Fig. 6 it can be observed that all of the slip surface is below the water table line. In addition, according to Deng et al. [1] the material at the slip surface is approximately uniform. Thus, the model parameters of table 4 are used to simulate the soil response at the slip surface. At the interfaces, the peak soil strength must be used, which, according to the

Typically, the criterion of minimum critical acceleration factor at the initial and final configurations is applied to estimate the interface angles of the sliding mass. The Mohr Coulomb model was applied both along the slip surface and at the interfaces. Along the slip surface the residual soil strength was applied, while at the interface the peak soil strength. Fig. 8 gives the critical acceleration in terms of the interslice angles at the (a) initial and (b)

.

12

### **5.2. Model predictions**

First the model parameters of the constitutive model are obtained by the prediction of the shear test results. Table 4 gives the model parameters and Fig. 7 compares model predictions of the constitutive model with the measured response. Very good agreement is observed. It can be observed that even though all model parameters do not depend on the applied normal stress, predictions are adequate.

From Fig. 6 it can be observed that all of the slip surface is below the water table line. In addition, according to Deng et al. [1] the material at the slip surface is approximately uniform. Thus, the model parameters of Table 4 are used to simulate the soil response at the slip surface. At the interfaces, the peak soil strength must be used, which, according to the results of the triaxial tests, equals φ'=36<sup>o</sup> .

Typically, the criterion of minimum critical acceleration factor at the initial and final configu‐ rations is applied to estimate the interface angles of the sliding mass. The Mohr Coulomb model was applied both along the slip surface and at the interfaces. Along the slip surface the residual soil strength was applied, while at the interface the peak soil strength. Fig. 8 gives the critical acceleration in terms of the interslice angles at the (a) initial and (b) final slide configurations. Table 5 gives the obtained interslice angles according to the criterion of minimum acceleration factor and the results of Fig. 8.

Once the interface angles are obtained, the multi-block model with the constitutive model along the slip surface is applied. Fig. 9a gives the input geometry used to simulate the slide with the multi-block model. As the seismic motion of Fig. 6d is applied, Fig. 10 gives the computed equivalent friction angle of block 2 (equation (9) and the slide acceleration (a), velocity (V) and distance moved (u). In addition, Fig. 9b gives the computed versus measured final slide configuration. It can be observed that the proposed method predicted both the triggereing of the slide and with good accuracy the final slide configuration. From Fig. 9 it can be observed, as the earthquake is applied, some shear displacement accumulates. This causes the friction angle at the base of the blocks to increase. Once the peak friction angle is reached, due to material softening, the friction angle decreases, to its residual value. At this point, the critical acceleration of the sliding system is negative (this means that the slide is unstable) and the slide velocity starts to increase and displacement to accumulate rapidly. As the slide moves, the mass slides at a progressively smaller average inclination. The critical acceleration of the sliding mass gradually increases and it becomes positive. Then, the slide velocity starts to decrease, and becomes zero at t=51s.

Finally, parametric analyses were performed. In some of these the Mohr-Coulomb with residual or maximum strength was applied, while in other no seismic motion was applied. Their results are given in Table 6. They illustrate that even though for the prediction of the triggering of the slide the applied motion and the constitutive model are important, for the prediction of the final slide configuration, use of the Mohr-coulomb model with the residual soil strength produces relatively accurate results, even when the seismic motion is not applied.


**Table 4.** The Higashi Takezawa landslide. Model parameters

**5.2. Model predictions**

predictions are adequate.

tests, equals φ'=36<sup>o</sup>

.

factor and the results of Fig. 8.

decrease, and becomes zero at t=51s.

First the model parameters of the constitutive model are obtained by the prediction of the shear test results. Table 4 gives the model parameters and Fig. 7 compares model predictions of the constitutive model with the measured response. Very good agreement is observed. It can be observed that even though all model parameters do not depend on the applied normal stress,

246 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

From Fig. 6 it can be observed that all of the slip surface is below the water table line. In addition, according to Deng et al. [1] the material at the slip surface is approximately uniform. Thus, the model parameters of Table 4 are used to simulate the soil response at the slip surface. At the interfaces, the peak soil strength must be used, which, according to the results of the triaxial

Typically, the criterion of minimum critical acceleration factor at the initial and final configu‐ rations is applied to estimate the interface angles of the sliding mass. The Mohr Coulomb model was applied both along the slip surface and at the interfaces. Along the slip surface the residual soil strength was applied, while at the interface the peak soil strength. Fig. 8 gives the critical acceleration in terms of the interslice angles at the (a) initial and (b) final slide configurations. Table 5 gives the obtained interslice angles according to the criterion of minimum acceleration

Once the interface angles are obtained, the multi-block model with the constitutive model along the slip surface is applied. Fig. 9a gives the input geometry used to simulate the slide with the multi-block model. As the seismic motion of Fig. 6d is applied, Fig. 10 gives the computed equivalent friction angle of block 2 (equation (9) and the slide acceleration (a), velocity (V) and distance moved (u). In addition, Fig. 9b gives the computed versus measured final slide configuration. It can be observed that the proposed method predicted both the triggereing of the slide and with good accuracy the final slide configuration. From Fig. 9 it can be observed, as the earthquake is applied, some shear displacement accumulates. This causes the friction angle at the base of the blocks to increase. Once the peak friction angle is reached, due to material softening, the friction angle decreases, to its residual value. At this point, the critical acceleration of the sliding system is negative (this means that the slide is unstable) and the slide velocity starts to increase and displacement to accumulate rapidly. As the slide moves, the mass slides at a progressively smaller average inclination. The critical acceleration of the sliding mass gradually increases and it becomes positive. Then, the slide velocity starts to

Finally, parametric analyses were performed. In some of these the Mohr-Coulomb with residual or maximum strength was applied, while in other no seismic motion was applied. Their results are given in Table 6. They illustrate that even though for the prediction of the triggering of the slide the applied motion and the constitutive model are important, for the prediction of the final slide configuration, use of the Mohr-coulomb model with the residual soil strength produces relatively accurate results, even when the seismic motion is not applied.


**Table 5.** The Higashi Takezawa landslide. Obtained interslice angles (δi) according to the criterion of minimum acceleration and the result of Fig. 8.


**Table 6.** Parametric analyses. Slide displacement in terms of the soil model and seismic motion applied.

**Fig. 6.** The Higashi Takezawa landslide. (a).Cross-section of slide (Deng et al. [1]), (b). Simple shear test results (Deng et al. [1]), (c) Triaxial test results (Deng et al. [1]) (d) representative applied acceleration in terms of time (Deng et al. [1]) **Figure 6.** The Higashi Takezawa landslide. (a).Cross-section of slide (Deng et al. [1], (b). Simple shear test results (Deng et al. [1], (c) Triaxial test results (Deng et al. [1], (d) representative applied acceleration in terms of time (Deng et al. [1]

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 **u (m)**

(a) (b)

0.2


δ1 δ2

0.25

0.3

**kc**

0.35

0.4

Test σ'o=99, τo=60kPa Test σ'o=50, τo=40kPa **Fig. 7.** The Higashi Takezawa landslide. Computed versus measured response in terms of

**Fig. 8.** The Higashi Takezawa landslide.Critical acceleration in terms of the interslice

initial consolidation stress for the unique model parameters of table 4.

δ3 δ2

angles. (a) Initial slide configuration, (b) Final slide configuration.



**kc**

**τ (kPa)**

14


a-applied (m/s2)

(a) (b)

(c) (d)

**Figure 7.** The Higashi Takezawa landslide. Computed versus measured response in terms of initial consolidation stress for the unique model parameters of table 4. Test σ'o=99, τo=60kPa Test σ'o=50, τo=40kPa

**Fig. 7.** The Higashi Takezawa landslide. Computed versus measured response in terms of

initial consolidation stress for the unique model parameters of table 4.

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 **u (m)**

angles. (a) Initial slide configurati **Figure 8.** The Higashi Takezawa landslide. Critical acceleration in terms of the interslice angles. (a) Initial slide config‐ on, (b) Final slide configuration. uration, (b) Final slide configuration.

**Fig. 8.** The Higashi Takezawa landslide.Critical acceleration in terms of the interslice

**Fig. 9.** The Higashi Takezawa landslide. (a). Input used to simulate the slide with the multiblock model, (b). Computed versus measured final slide configuration. (a) (b) **Figure 9.** The Higashi Takezawa landslide. (a). Input used to simulate the slide with the multi-block model, (b). Com‐ puted versus measured final slide configuration.

**Fig. 10.** (a). Computed equivalent friction angle of block 3 and critical acceleration of the slide (ac) and (b) slide acceleration (a), velocity (v), distance moved (u), all in terms of time

Slopes consisting of saturated sands have moved down-slope tens or hundreds of meters under the action of earthquakes recently. The chapter presents a simplified method predicting the triggering and displacement of such earthquake-induced slides of saturated

For this purpose, a simplified constitutive model simulating soil response of saturated sands along slip surfaces is proposed. Comparison of the model predictions with results of ring shear tests illustrated that the model predicts with good accuracy the shear stressdisplacement response of saturated sands. Advantage of the proposed model is simplicity. However, it has the disadvantage of generality. Thus, when applying this simplified model, tests with similar relative density, confining stress and initial shear stress as that existing in-

Then, this constitutive model was coupled with the multi-block sliding system model to predict the triggering and displacement of earthquake-induced slides of saturated sands.

0 10 20 30 40 50 60 **Time (seconds)**

a (m/s2) V (m/s) u (m/10)

ac (m/s2)

φ3 (ο)

sands.

**6. Conclusions** 

situ should be used.

0 10 20 30 40 50 60 **Time (seconds)**

15


Slip surface and trajectory Initial ground surface Final measured ground surface Final predicted ground surface

(a) (b)

Slip surface, Trajectory and Initial ground surface

Water table

**Fig. 9.** The Higashi Takezawa landslide. (a). Input used to simulate the slide with the multi-

block model, (b). Computed versus measured final slide configuration.

**6. Conclusions Figure 10.** (a). Computed equivalent friction angle of block 3 and critical acceleration of the slide (ac) and (b) slide ac‐ celeration (a), velocity (v), distance moved (u), all in terms of time

Slopes consisting of saturated sands have moved down-slope tens or hundreds of meters

**Fig. 10.** (a). Computed equivalent friction angle of block 3 and critical acceleration of the slide (ac) and (b) slide acceleration (a), velocity (v), distance moved (u), all in terms of time

#### under the action of earthquakes recently. The chapter presents a simplified method predicting the triggering and displacement of such earthquake-induced slides of saturated **6. Conclusions**

sands.

14

15

**τ (kPa)**

initial consolidation stress for the unique model parameters of table 4.

δ3 δ2

(d) representative applied acceleration in terms of time (Deng et al. [1])

 **(kPa)**

for the unique model parameters of table 4.



puted versus measured final slide configuration.

0 10 20 30 40 50 60 **Time (seconds)**

Slip surface, Trajectory and Initial ground surface

angles. (a) Initial slide configurati

Water table

uration, (b) Final slide configuration.


**kc**

sands.

**6. Conclusions** 

situ should be used.

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 **u (m)**

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 **u (m)**


0 10 20 30 40 50 60

a-applied (m/s2)

**Time (seconds)**


> Slip surface and trajectory Initial ground surface Final measured ground surface Final predicted ground surface

‐50 0 50 100 150 200 250 300 350 400 450

a (m/s2) V (m/s) u (m/10)

0 10 20 30 40 50 60 **Time (seconds)**

δ1 δ2

(a) (b)

(c) (d)

Test 'o=99, o=60kPa Test 'o=50, o=40kPa

**Fig. 6.** The Higashi Takezawa landslide. (a).Cross-section of slide (Deng et al. [1]), (b). Simple shear test results (Deng et al. [1]), (c) Triaxial test results (Deng et al. [1])

248 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

**Figure 7.** The Higashi Takezawa landslide. Computed versus measured response in terms of initial consolidation stress

(a) (b)

**Fig. 8.** The Higashi Takezawa landslide.Critical acceleration in terms of the interslice

**Figure 8.** The Higashi Takezawa landslide. Critical acceleration in terms of the interslice angles. (a) Initial slide config‐ on, (b) Final slide configuration.

(a) (b)

**Fig. 9.** The Higashi Takezawa landslide. (a). Input used to simulate the slide with the multi-

**Figure 9.** The Higashi Takezawa landslide. (a). Input used to simulate the slide with the multi-block model, (b). Com‐

(a) (b)

**Fig. 10.** (a). Computed equivalent friction angle of block 3 and critical acceleration of the slide (ac) and (b) slide acceleration (a), velocity (v), distance moved (u), all in terms of time

Slopes consisting of saturated sands have moved down-slope tens or hundreds of meters under the action of earthquakes recently. The chapter presents a simplified method predicting the triggering and displacement of such earthquake-induced slides of saturated

For this purpose, a simplified constitutive model simulating soil response of saturated sands along slip surfaces is proposed. Comparison of the model predictions with results of ring shear tests illustrated that the model predicts with good accuracy the shear stressdisplacement response of saturated sands. Advantage of the proposed model is simplicity. However, it has the disadvantage of generality. Thus, when applying this simplified model, tests with similar relative density, confining stress and initial shear stress as that existing in-

Then, this constitutive model was coupled with the multi-block sliding system model to predict the triggering and displacement of earthquake-induced slides of saturated sands.

block model, (b). Computed versus measured final slide configuration.

ac (m/s2)

φ3 (ο)

0.2

0.25

0.3

**kc**

0.35

0.4

Test σ'o=99, τo=60kPa Test σ'o=50, τo=40kPa **Fig. 7.** The Higashi Takezawa landslide. Computed versus measured response in terms of

> For this purpose, a simplified constitutive model simulating soil response of saturated sands along slip surfaces is proposed. Comparison of the model predictions with results of ring shear tests illustrated that the model predicts with good accuracy the shear stress-Slopes consisting of saturated sands have moved down-slope tens or hundreds of meters under the action of earthquakes recently. The chapter presents a simplified method predicting the triggering and displacement of such earthquake-induced slides of saturated sands.

> displacement response of saturated sands. Advantage of the proposed model is simplicity. However, it has the disadvantage of generality. Thus, when applying this simplified model, tests with similar relative density, confining stress and initial shear stress as that existing insitu should be used. Then, this constitutive model was coupled with the multi-block sliding system model to predict the triggering and displacement of earthquake-induced slides of saturated sands. For this purpose, a simplified constitutive model simulating soil response of saturated sands along slip surfaces is proposed. Comparison of the model predictions with results of ring shear tests illustrated that the model predicts with good accuracy the shear stress-displacement response of saturated sands. Advantage of the proposed model is simplicity. However, it has the disadvantage of generality. Thus, when applying this simplified model, tests with similar relative density, confining stress and initial shear stress as that existing in-situ should be used.

> 15 Then, this constitutive model was coupled with the multi-block sliding system model to predict the triggering and displacement of earthquake-induced slides of saturated sands. The multiblock model considers a general mass sliding on a slip surface which consists of n linear segments. In order for the mass to move, at the nodes between the linear segments, interfaces inside the sliding mass must be formed. Steps needed to apply the method are: (a) define the trajectory, ground and water table surfaces, (b) obtain the model parameters of the soil resistance by the results shear tests, (c) obtain the interface angles by applying the principle of minimum critical acceleration factor and (d) to simulate the triggering and displacement of the slide apply the multi-block model for a representative seismic motion is applied.

The method was applied successfully to predict the triggering, the motion and the final configuration of the well-documented Higashi Takezawa earthquake-induced slide.

## **Acknowledgements**

The work was funded partly by the project "Novel methodologies for the assessment of risk of ground displacement" under ESPA 2007-2013 of Greece, under action: Bilateral S & T Cooperation between China and Greece. Graduate student of the Hellenic Open University Kelly Gouma assisted in the analysis of the case study. Graduate students of the Hellenic Open University P. Sidiropoulos and J. Bakratzas assisted in the collection of the laboratory tests.

## **Author details**

Constantine A. Stamatopoulos1,2\*


## **References**


[7] Deng, J., Tsutsumi, Y, Kameya, H., Koseki, J. A Modified Procedure to Evaluate Earthquake-induced Displacement of Slopes Containing a Weak Layer. Soils and Foundations. 2010, 50, 3; 413-420

**Acknowledgements**

**Author details**

**References**

Constantine A. Stamatopoulos1,2\*

Address all correspondence to: k.stam@saa-geotech.gr

Foundations Vol. 51. 2011. No. 5, 929-943. Oct.

tions,Japan Geotechnical Society. 1996. 53-64.

Vol. 15, No. 2, London, England. 1965. June, 139-160.

engineering and structural dynamics. 1988. 16, 7, 985-1006.

1 Hellenic Open University, Athens, Greece

2 Stamatopoulos and Associates Co, Greece

ology. 2010, Vol. 32., No. 1, Mar.

10, 1075-1093.

The work was funded partly by the project "Novel methodologies for the assessment of risk of ground displacement" under ESPA 2007-2013 of Greece, under action: Bilateral S & T Cooperation between China and Greece. Graduate student of the Hellenic Open University Kelly Gouma assisted in the analysis of the case study. Graduate students of the Hellenic Open University P. Sidiropoulos and J. Bakratzas assisted in the collection of the laboratory tests.

250 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

[1] Deng J., Kameya H., Miyashita Y., J., Kuwano R., Koseki J. Study on dip slope failure at Higashi Takezawa induced by 2004 Niigata-Ken Chuetsu earthquake. Soils and.

[2] Sun P., Wang F., Yin Y., Wu S.. An experimental study of the mechanism of rapid and long run-out landslides triggered by Wenchuan earthquake. Seismology and ge‐

[3] Sassa, K., Fukuoka, H., Scarascia-Mugnozza, G., Evans, S. Earthquake-induced land‐ slides: Distribution, motion and mechanisms. Special Issue of Soils and Founda‐

[4] Newmark, N. M. Effect of earthquakes on dams and embankments, Geotechnique,

[5] Ambaseys N. and Menu J. Earthquake induced ground displacements, Earthquake

[6] Stamatopoulos, C. A. Sliding System Predicting Large Permanent Co-Seismic Move‐ ments of Slopes. Earthquake Engineering and Structural Dynamics. 1996. Vol. 25, No,


## **Detection of Accelerating Transient of Aseismic Rock Strain using Precursory Decline in Groundwater Radon**

Ming-Ching T. Kuo

[20] Sarma S.K. Stability analysis of embankments and slopes. Journal of Geotechnical En‐

252 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

[21] Sarma S.K. and Chlimitzas G. Co-seismic & post-seismic displacements of slopes, 15th ICSMGE TC4 Satellite Conference on "Lessons Learned from Recent Strong

[22] Sarma S. K., Tan D. Determination of critical slip surface in slope analysis, *Geotechni‐*

[23] Ambraseys N., Srbulov M. Earthquake induced displacements of slopes, Soil Dynam‐

gineering ASCE. 1979, Vol.105, No. 12, 1511-1524.

Earthquakes". 2001, 25 August, Istanbul, Turkey

ics and Earthquake Engineering. 1995, 14, 59-71.

*que*, 2006, Vol. 56, 539-550

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/59530

## **1. Introduction**

A seismic slip can be preceded by accelerating aseismic slips near the hypocenter of an impending earthquake. Compared with the strain step recorded at the time of the earthquake, the precursory strain from aseismic fault slips are transient and small. It is of practical interest to be able to detect the accelerating aseismic rock strain for the warning of local disastrous earthquakes. From a series of laboratory experiments in un-drained conditions, I discovered a significant drop in groundwater radon, greater than 50 %, by a mechanism of radon volati‐ lization into the vapor phase. Both radon volatilization and rock dilatancy offer attractive insitu physical mechanisms for a premonitory decrease in groundwater radon. In addition, we have been monitoring groundwater radon at well (D1) in the Antung hot spring in eastern Taiwan since July 2003 (Fig. 1). The Antung hot spring is located near the Chihshang fault that ruptured during two 1951 earthquakes of magnitudes M=6.2 and M=7.0 (Hsu, 1972). Based on the long-term observation at the Antung hot spring, I discovered that an un-drained brittle aquifer near an active fault can be used as a natural strain meter for detecting recurrent precursory radon declines. The observed radon minimum decreases as the earthquake magnitude increases. The un-drained condition at well (D1) is essential for the development of two phases (vapor and water) in the rock cracks, which is attributed to the happening of recurrent precursory radon declines. The concurrent concentration declines in groundwaterdissolved gases (radon, methane, and ethane) support the un-drained condition at well (D1) and the mechanism of in-situ volatilization of groundwater-dissolved gases. Radon precursory declines in groundwater can only be detected in certain locations with favorable geological conditions. An un-drained brittle aquifer near an active fault is recommended to monitor recurrent precursory radon declines. The objective of this chapter is to provide a practical means to detect the accelerating aseismic rock strain using the precursory decline in ground‐

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

water radon. A case study is provided in this chapter to illustrate the application of an undrained brittle aquifer as a natural strain meter to detect the accelerating transient of aseismic rock strain by monitoring precursory decline in groundwater radon.

**Figure 1.** Map of the epicenters of the earthquakes that occurred on December 10, 2003, April 1 and 15, 2006, February 17, 2008, July 12, 2011 near the Antung hot spring. (a) Geographical location of Taiwan. (b) Study area near the Antung hot spring (filled stars: mainshocks, filled triangle: radon-monitoring station).

## **2. Monitoring methods**

We have been monitoring groundwater radon at a well (D1) located at the Antung hot spring in eastern Taiwan since July 2003. Figure 1 shows that the radon-monitoring well (D1) is about 3 km southeast of the Chihshang fault. Discrete samples of groundwater were pumped and collected from the radon-monitoring well (D1) twice per week for analysis of radon content. We also started to analyze methane and ethane from November 2007 and November 2010, respectively.

The production interval of the radon-monitoring well (D1) ranges from 167 m to 187 m. Water samples were collected in a 40 mL glass vial with a TEFLON-lined cap. To obtain representative water samples for radon, methane, and ethane measurements, a minimum purge of three wellbored volumes were required before taking samples. A minimum of 50 min purging-time was required with a pumping rate at around 200 L/min.

It is important to ensure the radon not to escape during the sampling procedure and the sample transportation and preparation. For every sampling, the sample vial was inverted to check if any bubbles were present in the vial. The sample water with gas bubbles present was discarded and sampling was repeated. The sampling time of sample collection were recorded and radon measurement was done within 4 days. The samples were stored and transported in a cooler to minimize biological degradation of methane and ethane.

The liquid scintillation method was applied to determine the concentration of radon in groundwater [12]. Radon was first extracted into a mineral-oil based scintillation cocktail from the water samples, and then measured with a liquid scintillation counter (LSC). The measure‐ ment results were corrected for the radon decay between sampling and counting.

Calibration factor for the LSC measurements should be at least 6 cpm/pCi with the background not exceeding 6 cpm. For a count time of 50 min and background less than 6 cpm, we achieved a detection limit below 18 pCi/L using the sample volume of 15 ml.

The head space method was used to determine the concentrations of methane and ethane in groundwater with a gas chromatograph (Shimadzu GC-14A), a HP-Plot/Q, 30 m, 0.53 mm i.d. capillary column, and a flame ionization detector (FID). The concentrations of methane and ethane measured in the head space were converted to groundwater concentrations using Henry's constants of 31.5 and 23.9, respectively, for methane and ethane at a room temperature of 27 ℃.

## **3. In-situ radon-volatilization and rock-dilatancy**

water radon. A case study is provided in this chapter to illustrate the application of an undrained brittle aquifer as a natural strain meter to detect the accelerating transient of aseismic

254 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

**Figure 1.** Map of the epicenters of the earthquakes that occurred on December 10, 2003, April 1 and 15, 2006, February 17, 2008, July 12, 2011 near the Antung hot spring. (a) Geographical location of Taiwan. (b) Study area near the Antung

We have been monitoring groundwater radon at a well (D1) located at the Antung hot spring in eastern Taiwan since July 2003. Figure 1 shows that the radon-monitoring well (D1) is about 3 km southeast of the Chihshang fault. Discrete samples of groundwater were pumped and collected from the radon-monitoring well (D1) twice per week for analysis of radon content. We also started to analyze methane and ethane from November 2007 and November 2010,

The production interval of the radon-monitoring well (D1) ranges from 167 m to 187 m. Water samples were collected in a 40 mL glass vial with a TEFLON-lined cap. To obtain representative water samples for radon, methane, and ethane measurements, a minimum purge of three wellbored volumes were required before taking samples. A minimum of 50 min purging-time was

hot spring (filled stars: mainshocks, filled triangle: radon-monitoring station).

required with a pumping rate at around 200 L/min.

**2. Monitoring methods**

respectively.

rock strain by monitoring precursory decline in groundwater radon.

Radon partitioning into the gas phase can explain the anomalous decreases of radon concen‐ tration in groundwater precursory to the earthquakes [7]. Based on radon phase-behavior and rock-dilatancy process [1], [8] developed a mechanistic model to correlate the observed decline in radon with the rock strain precursory to an earthquake. We will present the model with two parts, i.e., the radon-volatilization model and the rock-dilatancy model. The radon-volatiliza‐ tion model which correlates the radon decline to the gas saturation can be expressed as follows.

$$C\_0 = C\_w \left( \ H \ S\_g + 1 \right) \tag{1}$$

where *C*0 is the stabilized radon concentration in groundwater before a radon anomaly, pCi/L; *Cw* is the observed radon decline in groundwater during a radon anomaly, pCi/L; *Sg* is the gas saturation in aquifer during a radon anomaly, fraction; *H* is Henry's constant for radon, dimensionless. The Henry's constant (*H* ) at formation temperature (60 ℃) is 7.91 for radon [5].

The rock-dilatancy model which correlates the rock strain to the gas saturation can be ex‐ pressed as follows.

$$d\mathcal{E} \triangleq \phi \; S\_{\mathcal{S}} \tag{2}$$

where *dε* is the rock strain precursory to an earthquake, fraction; *ϕ* is the fracture porosity of aquifer, fraction; *Sg* is the gas saturation in aquifer during an radon anomaly, fraction.

Based on the radon volatilization and rock dilatancy models, we can correlate the radon decline in groundwater radon to the rock strain precursory to an earthquake. Combining equations (1) and (2), we obtain equation (3) as follows.

$$d\mathcal{E} \triangleq \frac{\phi}{H} \left( \frac{C\_0}{C\_\ast} - 1 \right) \tag{3}$$

where ( *<sup>C</sup>*<sup>0</sup> *Cw* −1) is normalized radon decline precursory to an earthquake, dimensionless. Given the precursory decline in groundwater radon such as, Figure 2, equation (3) can be used to calculate the precursory crustal-strain transient from aseismic fault slips. Case studies are provided to illustrate the application of an un-drained brittle aquifer as a natural strain meter to detect the accelerating transient of aseismic rock strain by observing premonitory decline in groundwater radon.

**Radon concentration (pCi/L) Figure 2.** Observed radon decline and calculated crustal-strain transient prior to 2003 *MW* 6.8 Chengkung earthquake at the monitoring well (D1) in the Antung hot spring (solid circles: observed radon concentration; open triangles: calcu‐ lated crustal-strain). Stage 1 is buildup of elastic strain. Stage 2 is development of cracks and gas saturation. Stage 3 is influx of groundwater.

## **4. A case study**

*<sup>g</sup> d S* e

256 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

*<sup>C</sup> <sup>d</sup> H C* f

e

 **786 ± 43 pCi/L**

(1) and (2), we obtain equation (3) as follows.

where ( *<sup>C</sup>*<sup>0</sup> *Cw*

in groundwater radon.

**Radon concentration (pCi/L)**

influx of groundwater.

**0**

**200**

**400**

**600**

**800**

**1000**

 f

where *dε* is the rock strain precursory to an earthquake, fraction; *ϕ* is the fracture porosity of aquifer, fraction; *Sg* is the gas saturation in aquifer during an radon anomaly, fraction.

Based on the radon volatilization and rock dilatancy models, we can correlate the radon decline in groundwater radon to the rock strain precursory to an earthquake. Combining equations

> <sup>0</sup> 1 *w*

−1) is normalized radon decline precursory to an earthquake, dimensionless. Given

æ ö @ - ç ÷

the precursory decline in groundwater radon such as, Figure 2, equation (3) can be used to calculate the precursory crustal-strain transient from aseismic fault slips. Case studies are provided to illustrate the application of an un-drained brittle aquifer as a natural strain meter to detect the accelerating transient of aseismic rock strain by observing premonitory decline

**2003/8/1 2003/9/1 2003/10/1 2003/11/1 2003/12/1** 

**Figure 2.** Observed radon decline and calculated crustal-strain transient prior to 2003 *MW* 6.8 Chengkung earthquake at the monitoring well (D1) in the Antung hot spring (solid circles: observed radon concentration; open triangles: calcu‐ lated crustal-strain). Stage 1 is buildup of elastic strain. Stage 2 is development of cracks and gas saturation. Stage 3 is

**Stage 1 2 3**

**326 ± 9 pCi/L**

@ (2)

è ø (3)

**Strain change (ppm)-1**

**0**

**1**

**2**

**3**

**4**

**5**

**6**

▼

**2003** *M***w = 6.8 Chengkung Mainshock**

**65 days 45 days**

[7] discovered that a significant drop in groundwater radon, greater than 50 %, can be generated in the laboratory by a process of radon volatilization into the gas phase in un-drained conditions. We conducted a series of radon-partitioning experiments to determine the variation of the radon concentration remaining in groundwater at various levels of gas saturation. Figure 3 shows the results of vapor-liquid, two-phase radon-partitioning experi‐ ments conducted at formation temperature (60 ℃) using formation brine from the Antung hot spring. The processes of rock dilatancy, under-saturation and radon volatilization offer an attractive mechanism to monitor anomalous radon declines in groundwater radon precursory to an earthquake. Given an observed radon decline precursory to an earthquake, we can apply Figure 3 to estimate the amount of gas saturation in micro-cracks developed in aquifer during an radon anomaly. For example, a gas saturation of 10 % in cracks developed in the aquifer rock when the radon concentration in groundwater decreased from 780 pCi/L to 330 pCi/L precursory to the 2003 *Mw* 6.8 Chengkung earthquake. [16] reported that the fracture porosity for naturally fractured rocks ranges from 0.00008 to 0.0003. To generate an in-situ gas saturation of 10% in a fractured aquifer, a crustal strain of 8.0 ppm and 30.0 ppm is required for a fracture porosity of 0.00008 and 0.0003, respectively. Both low-porosity and un-drained conditions are favorable for applying a fractured aquifer as a natural strain meter by monitoring precursory decline in groundwater radon. It is of practical interest to be able to detect the accelerating aseismic rock strains by monitoring the radon concentration in groundwater.

**Figure 3.** Variation of radon concentration remaining in groundwater with gas saturation at 60℃ using formation brine from the Antung hot spring.

Since July 2003, we have observed recurrent recurrent anomalous declines in groundwater radon at the Antung well (D1) in eastern Taiwan precursory to the 2003 *Mw* = 6.8 Chengkung, 2006 *Mw* = 6.1 Taitung, 2008 *Mw* = 5.4 Antung, and 2011 *Mw* = 5.0 Chimei earthquakes that occurred on December 10, 2003, April 1, 2006, February 17, 2008, and July 12, 2011, respectively. The epicenters of the 2003 *Mw* = 6.8, 2006 *Mw* = 6.1, 2008 *Mw* = 5.4, and 2011 *Mw* = 5.0 earthquakes were located only 24 km, 52 km, 13 km, and 32 km, respectively, from the observation well (D1).

The observed radon anomalies can be correlated with the magnitude and precursor time of upcoming earthquakes. We define the precursor time for radon as the time interval between the moment when the trend of the radon concentration starts to decline and the time of occurrence of the earthquake. Based on the radon anomalies observed prior to (1) 2003 *Mw* 6.8 Chengkung, (2) 2006 *Mw* 6.1 Taitung, (3) 2008 *Mw* 5.4 Antung, and (4) 2011 *Mw* 5.0 Chimei earthquakes, Table 1 summarizes the precursor time and radon minima. [11] shows that as the magnitude of earthquakes increases, the precursor time for radon anomalies increases and the observed radon minima decrease. Monitoring precursory decline in groundwater radon at a suitable geological site can be a useful means of forecasting the magnitude and precursor time of local disastrous earthquakes.


**Table 1.** Observed precursory time and radon minimum at well (D1) prior to (1) 2003 *MW* = 6.8 Chengkung, (2) 2006 *MW* = 6.1 Taitung, (3) 2008 *MW* = 5.4 Antung, and (4) 2011 *MW* = 5.0 Chimei earthquakes.

[4] show that the Antung hot spring situated in an andesitic block and surrounded by a ductile mudstone of the Lichi mélange. Figure 4 shows the geological map and cross section near well (D1) in the area of Antung hot spring which is a low-porosity fractured confined aquifer. The groundwater is in un-drained conditions at well (D1). [13] and [15] suggested that the development of new cracks in aquifer rock could occur at a rate faster than the recharge of pore water in un-drained conditions. Gas saturation developed in the rock cracks and groundwater-dissolved radon then volatilized into the gas phase. We also observed simulta‐ neous anomalous declines in groundwater-dissolved radon and methane precursory to the 2008 *Mw* = 5.4 Antung earthquake [9]. The mechanism of in-situ radon volatilization was substantiated.

The composition of groundwater-dissolved gases taken from a separator flow test at well (D1) on December 26, 2006 consists of 62.8 % of nitrogen, 36.7 % of methane, and 0.5 % of ethane

Detection of Accelerating Transient of Aseismic Rock Strain using Precursory Decline in Groundwater Radon http://dx.doi.org/10.5772/59530 259

Since July 2003, we have observed recurrent recurrent anomalous declines in groundwater radon at the Antung well (D1) in eastern Taiwan precursory to the 2003 *Mw* = 6.8 Chengkung, 2006 *Mw* = 6.1 Taitung, 2008 *Mw* = 5.4 Antung, and 2011 *Mw* = 5.0 Chimei earthquakes that occurred on December 10, 2003, April 1, 2006, February 17, 2008, and July 12, 2011, respectively. The epicenters of the 2003 *Mw* = 6.8, 2006 *Mw* = 6.1, 2008 *Mw* = 5.4, and 2011 *Mw* = 5.0 earthquakes were located only 24 km, 52 km, 13 km, and 32 km, respectively, from the observation well

258 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

The observed radon anomalies can be correlated with the magnitude and precursor time of upcoming earthquakes. We define the precursor time for radon as the time interval between the moment when the trend of the radon concentration starts to decline and the time of occurrence of the earthquake. Based on the radon anomalies observed prior to (1) 2003 *Mw* 6.8 Chengkung, (2) 2006 *Mw* 6.1 Taitung, (3) 2008 *Mw* 5.4 Antung, and (4) 2011 *Mw* 5.0 Chimei earthquakes, Table 1 summarizes the precursor time and radon minima. [11] shows that as the magnitude of earthquakes increases, the precursor time for radon anomalies increases and the observed radon minima decrease. Monitoring precursory decline in groundwater radon at a suitable geological site can be a useful means of forecasting the magnitude and precursor time

> **Precursory time (day)**

**Radon minimum (pCi/L)**

(D1).

of local disastrous earthquakes.

substantiated.

**Earthquake Moment magnitude,**

*MW, (dimensionless)*

*MW* = 6.1 Taitung, (3) 2008 *MW* = 5.4 Antung, and (4) 2011 *MW* = 5.0 Chimei earthquakes.

2003 Chengkung 6.8 65 326 ± 9 2006 Taitung 6.1 61 371 ± 9 2008 Antung 5.4 56 480 ± 43 2011 Chimei 5.0 54 447 ± 18

**Table 1.** Observed precursory time and radon minimum at well (D1) prior to (1) 2003 *MW* = 6.8 Chengkung, (2) 2006

[4] show that the Antung hot spring situated in an andesitic block and surrounded by a ductile mudstone of the Lichi mélange. Figure 4 shows the geological map and cross section near well (D1) in the area of Antung hot spring which is a low-porosity fractured confined aquifer. The groundwater is in un-drained conditions at well (D1). [13] and [15] suggested that the development of new cracks in aquifer rock could occur at a rate faster than the recharge of pore water in un-drained conditions. Gas saturation developed in the rock cracks and groundwater-dissolved radon then volatilized into the gas phase. We also observed simulta‐ neous anomalous declines in groundwater-dissolved radon and methane precursory to the 2008 *Mw* = 5.4 Antung earthquake [9]. The mechanism of in-situ radon volatilization was

The composition of groundwater-dissolved gases taken from a separator flow test at well (D1) on December 26, 2006 consists of 62.8 % of nitrogen, 36.7 % of methane, and 0.5 % of ethane

**Figure 4.** Geological map and cross section near the radon-monitoring well (D1) in the area of Antung hot spring. (B: tuffaceous andesitic blocks; 1): Chihshang, or, Longitudinal Valley Fault, 2): Yongfeng Fault)

by volume. In addition to radon and methane, we initiated the monitoring of groundwaterdissolved ethane at well (D1) in the Antung hot spring since November 30, 2010 to corroborate the in-situ volatilization mechanism. The in-situ radon-volatilization model for groundwaterdissolved radon, methane, and ethane can be expressed as follows.

$$C\_{0, \text{Re}} = C\_{\text{w, \text{Re}}} (H\_{\text{Re}} S\_{\text{g}} + 1) \tag{4}$$

$$C\_{0, \text{Mc}} = C\_{\text{w,Mc}} (H\_{\text{Mc}} S\_{\text{g}} + 1) \tag{5}$$

$$C\_{0,Et} = C\_{w,Et}(H\_{Et}S\_g + 1)\tag{6}$$

where *C*0,*Rn* is the stabilized radon concentration in groundwater before a radon anomaly, pCi/ L; *Cw*,*Rn* is the observed radon decline in groundwater during a radon anomaly, pCi/L; *Sg* is the gas saturation in aquifer during a radon anomaly, fraction; *HRn* is Henry's constant for radon, dimensionless; *C*0,*Me* is the stabilized methane concentration in groundwater before a methane anomaly, mg/L; *Cw*,*Me* is the observed methane decline in groundwater during a methane anomaly, mg/L; *HMe* is Henry's constant for methane, dimensionless; *C*0,*Et* is the stabilized ethane concentration in groundwater before a ethane anomaly, mg/L; *Cw*,*Et* is the observed ethane decline in groundwater during a ethane anomaly, mg/L; *HEt* is Henry's constant for ethane, dimensionless. The Henry's coefficients at 60 °C are 7.91, 37.6, and 38.2 for radon, methane, and ethane, respectively. According to equations (4), (5), and (6), the mechanism of in-situ volatilization predicts the concurrent concentration declines in ground‐ water-dissolved radon, methane, and ethane.

Prior to the 2011 Chimei earthquake, the concurrent concentration declines in groundwaterdissolved radon, methane, and ethane were observed [10]. Figure 5 shows the observed concentration anomalies for radon, methane, and ethane, respectively, precursory to the 2011 Chimei earthquake. The concentration errors are ±1 standard deviation after simple averaging of triplicates. Radon, methane, and ethane decreased from background levels of 752 ± 24 pCi/ L, 8.24 ± 0.48 mg/L, and 0.217 ± 0.010 mg/L to minima of 447 ± 18 pCi/L, 5.81 ± 0.30 mg/L, and 0.161 ± 0.008 mg/L, respectively (Figure 5). The mechanism of in-situ radon volatilization was confirmed again by the simultaneous anomalous declines in groundwater-dissolved radon, methane, and ethane.

The anomalous decline of radon concentration in groundwater was observed prior to the 2003 Chengkung earthquake. Figure 2 shows that the sequence of events can be divided into three stages. During Stage 1 (from July 2003 to September 2003), radon concentration in groundwater was fairly stable (around 780 pCi/L). During Stage 1, there was a slow, steady increase of effective stress and an accumulation of tectonic strain. Sixty-five days before the 2003 *Mw* = 6.8 Chengkung earthquake which occurred on December 10, 2003, the concentration of radon started to decrease and reached a minimum value of 330 pCi/L twenty days before the earthquake. We define this 45-day period as Stage 2. Dilation of the rock mass occurred during Stage 2. When the aquifer is in un-drained conditions, the development of new cracks in aquifer rock could occur at a rate faster than the recharge of pore water. Gas saturation then developed in the rock cracks and groundwater-dissolved radon volatilized into the gas phase. After the minimum point of radon concentration (Stage 3), groundwater continued to encroach into the rock cracks and the water saturation in the aquifer began to increase. During Stage 3, ground‐ water-dissolved radon increased and recovered to the background level. The main shock produced a sharp coseismic anomalous decrease (~300 pCi/L).

The 2003 Chengkung earthquake's dislocation fault model was analyzed by Wu et al using a computer code by [14]. [17] determined the fault geometry utilizing aftershock distribution and geology (CGS 2000a, b) and assumed a thrust fault parallel to Coastal Range with strike N20˚E, with a bend at a depth of 18 km. The fault-plane dips 60˚SE and 45˚SE above and below 18 km respectively. Assume both the lower and upper fault-planes extend a maximum of 33 km from north to south. Rupturing of the lower and upper fault-planes occurred within depths of 18-36 km and 5-18 km, respectively. For an optimal fit with the coseismic ground deforma‐ tion, the lower fault slipped 61.6 cm with a rake of 81.7˚ and the upper fault slipped 26 cm with Detection of Accelerating Transient of Aseismic Rock Strain using Precursory Decline in Groundwater Radon http://dx.doi.org/10.5772/59530 261

radon, dimensionless; *C*0,*Me* is the stabilized methane concentration in groundwater before a methane anomaly, mg/L; *Cw*,*Me* is the observed methane decline in groundwater during a methane anomaly, mg/L; *HMe* is Henry's constant for methane, dimensionless; *C*0,*Et* is the stabilized ethane concentration in groundwater before a ethane anomaly, mg/L; *Cw*,*Et* is the observed ethane decline in groundwater during a ethane anomaly, mg/L; *HEt* is Henry's constant for ethane, dimensionless. The Henry's coefficients at 60 °C are 7.91, 37.6, and 38.2 for radon, methane, and ethane, respectively. According to equations (4), (5), and (6), the mechanism of in-situ volatilization predicts the concurrent concentration declines in ground‐

260 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

Prior to the 2011 Chimei earthquake, the concurrent concentration declines in groundwaterdissolved radon, methane, and ethane were observed [10]. Figure 5 shows the observed concentration anomalies for radon, methane, and ethane, respectively, precursory to the 2011 Chimei earthquake. The concentration errors are ±1 standard deviation after simple averaging of triplicates. Radon, methane, and ethane decreased from background levels of 752 ± 24 pCi/ L, 8.24 ± 0.48 mg/L, and 0.217 ± 0.010 mg/L to minima of 447 ± 18 pCi/L, 5.81 ± 0.30 mg/L, and 0.161 ± 0.008 mg/L, respectively (Figure 5). The mechanism of in-situ radon volatilization was confirmed again by the simultaneous anomalous declines in groundwater-dissolved radon,

The anomalous decline of radon concentration in groundwater was observed prior to the 2003 Chengkung earthquake. Figure 2 shows that the sequence of events can be divided into three stages. During Stage 1 (from July 2003 to September 2003), radon concentration in groundwater was fairly stable (around 780 pCi/L). During Stage 1, there was a slow, steady increase of effective stress and an accumulation of tectonic strain. Sixty-five days before the 2003 *Mw* = 6.8 Chengkung earthquake which occurred on December 10, 2003, the concentration of radon started to decrease and reached a minimum value of 330 pCi/L twenty days before the earthquake. We define this 45-day period as Stage 2. Dilation of the rock mass occurred during Stage 2. When the aquifer is in un-drained conditions, the development of new cracks in aquifer rock could occur at a rate faster than the recharge of pore water. Gas saturation then developed in the rock cracks and groundwater-dissolved radon volatilized into the gas phase. After the minimum point of radon concentration (Stage 3), groundwater continued to encroach into the rock cracks and the water saturation in the aquifer began to increase. During Stage 3, ground‐ water-dissolved radon increased and recovered to the background level. The main shock

The 2003 Chengkung earthquake's dislocation fault model was analyzed by Wu et al using a computer code by [14]. [17] determined the fault geometry utilizing aftershock distribution and geology (CGS 2000a, b) and assumed a thrust fault parallel to Coastal Range with strike N20˚E, with a bend at a depth of 18 km. The fault-plane dips 60˚SE and 45˚SE above and below 18 km respectively. Assume both the lower and upper fault-planes extend a maximum of 33 km from north to south. Rupturing of the lower and upper fault-planes occurred within depths of 18-36 km and 5-18 km, respectively. For an optimal fit with the coseismic ground deforma‐ tion, the lower fault slipped 61.6 cm with a rake of 81.7˚ and the upper fault slipped 26 cm with

produced a sharp coseismic anomalous decrease (~300 pCi/L).

water-dissolved radon, methane, and ethane.

methane, and ethane.

**Figure 5.** Observed concentration anomalies (a) radon, (b) methane, and (c) ethane prior to 2011 *MW* 5.0 Chimei earth‐ quake. Stage 1 is buildup of elastic strain. Stage 2 is development of cracks. Stage 3 is influx of groundwater.

a rake of 47.3˚. The area and slip on the ruptured surface of the lower and upper fault-planes were used to calculate *Mw* (moment magnitude scale), with respective values of 6.7 and 6.3. The total *Mw* was about 6.8 and agreed with the result of the moment tensor inversion solution from the Harvard CMT database (http://www.seismology.harvard.edu/), indicating that the coseismic energy were mainly released by the lower fault. Coseismic strain distribution due to the 2003 Chengkung earthquake was calculated using the dislocation fault model [17] and a computer code by [14]. Contraction surface strain near the Antung hot spring area was approximately 20 ppm (Fig. 6).

**Figure 6.** Distribution of coseismic surface strain (ppm) calculated based on the computer code for dislocation models by [14]. Positive and negative values mean dilatation and contraction, respectively. The open star denotes the 2003 mainshock. The filled triangle denotes the radon-monitoring well (D1). EXT and COMP denote dilatation and contrac‐ tion, respectively.

A seismic slip can be preceded by accelerating aseismic slips near the hypocenter of an impending earthquake. Compared with the strain step about 20 ppm at the time of the 2003 Chengkung earthquake, the precursory strains from aseismic fault slips are small and accel‐ erating. It is of practical importance to detect the accelerating transient of aseismic rock strain for the warning of local disastrous earthquakes. With the help of a case study, we show the capability to monitor the precursory decline in groundwater radon and to detect the acceler‐ ating transient of aseismic rock strain prior to the 2003 *Mw* = 6.8 Chengkung earthquake. Based on the precursory decline in groundwater radon observed at the Antung hot spring (Fig. 2), equation (3) can be used to calculate the accelerating transient of crustal-strain from aseismic

fault slips prior to the 2003 *Mw* = 6.8 Chengkung earthquake. The open triangles in Fig. 2 show the calculated crustal-strain transient prior to 2003 *MW* 6.8 Chengkung earthquake at the monitoring well (D1) in the Antung hot spring with an average fracture porosity of 0.00003.

## **5. Conclusions**

The total *Mw* was about 6.8 and agreed with the result of the moment tensor inversion solution from the Harvard CMT database (http://www.seismology.harvard.edu/), indicating that the coseismic energy were mainly released by the lower fault. Coseismic strain distribution due to the 2003 Chengkung earthquake was calculated using the dislocation fault model [17] and a computer code by [14]. Contraction surface strain near the Antung hot spring area was

262 Earthquake Engineering - From Engineering Seismology to Optimal Seismic Design of Engineering Structures

**Figure 6.** Distribution of coseismic surface strain (ppm) calculated based on the computer code for dislocation models by [14]. Positive and negative values mean dilatation and contraction, respectively. The open star denotes the 2003 mainshock. The filled triangle denotes the radon-monitoring well (D1). EXT and COMP denote dilatation and contrac‐

A seismic slip can be preceded by accelerating aseismic slips near the hypocenter of an impending earthquake. Compared with the strain step about 20 ppm at the time of the 2003 Chengkung earthquake, the precursory strains from aseismic fault slips are small and accel‐ erating. It is of practical importance to detect the accelerating transient of aseismic rock strain for the warning of local disastrous earthquakes. With the help of a case study, we show the capability to monitor the precursory decline in groundwater radon and to detect the acceler‐ ating transient of aseismic rock strain prior to the 2003 *Mw* = 6.8 Chengkung earthquake. Based on the precursory decline in groundwater radon observed at the Antung hot spring (Fig. 2), equation (3) can be used to calculate the accelerating transient of crustal-strain from aseismic

approximately 20 ppm (Fig. 6).

tion, respectively.

In a series of laboratory experiments in un-drained conditions, I discovered a significant drop in groundwater radon, greater than 50 %, by a mechanism of radon volatilization into the gas phase. In-situ radon volatilization offers an attractive mechanism for a premonitory decrease in groundwater radon. An un-drained brittle aquifer near an active fault can be employed as a natural strain meter for detecting recurrent precursory radon declines. Anomalous declines in groundwater radon consistently recorded at a well (D1) prior to large earthquakes on the Chihshang fault in eastern Taiwan provide the reproducible evidence. Compared with the coseismic strain, the precursory strain from aseismic fault slips are transient and small. In this chapter, a quantitative method using the precursory radon decline as a tracer to detect the accelerating aseismic rock strain is presented with the help of a case study.

## **Acknowledgements**

Supports by the National Science Council (NSC-103-2116-M-006-013), Central Geological Surveys, Industrial Technology Research Institute (L550001060), Radiation Monitoring Center, and Institute Earth Sciences of Academia Sinica of Taiwan are appreciated.

## **Author details**

Ming-Ching T. Kuo

Address all correspondence to: mctkuobe@mail.ncku.edu.tw

Department of Mineral and Petroleum Engineering, National Cheng Kung University, Tainan, Taiwan

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