Zhuo-fu WANG*, Ji-yong DING, Gao-sheng YANG

Institute of Engineering Management, Hohai University, Nanjing 211100, P. R. China

Risk analysis of slope instability of levees under river sand mining conditions

Zhuo-fu WANG*, Ji-yong DING, Gao-sheng YANG

Institute of Engineering Management, Hohai University, Nanjing 211100, P. R. China

Levees are affected by over-exploitation of river sand and river adjustments after the formation of sand pits. The slope stability is seriously threatened, drawing wide concern among experts and scholars in the area of water conservancy. This study analyzed the uncertainties of slope stability of levees under river sand mining conditions, including uncertainty caused by interestdriven over-exploitation by sand mining contractors, and uncertainty of the distance from the slope or sand pit to the bottom of the levee under the action of cross-flow force after the sand pit forms. Based on the results of uncertainty analysis, the distribution and related parameters of these uncertainties were estimated according to the Yangtze River sand mining practice. A risk model of the slope instability of a levee under river sand mining conditions was built, and the possibility of slope instability under different slope gradients in a certain reach of the Yangtze River was calculated with the Monte Carlo method and probability combination method. The results indicated that the probability of instability risk rose from 2.38% to 4.74% as the pits came into being.

sand mining; levee; risk analysis; slope instability; Monte Carlo method; probability combination method

1 Introduction

Sand resources are abundant in some rivers, such as the Yangtze River and Pearl River in China. Sand is one of the components of the riverbed and also a construction material with high economic value. Since the beginning of the last century, illegal sand mining activities have been rampant in many rivers in China, which have caused a serious threat to the safety of levees. Take the Chiding reach of the Xijiang River in Guangdong Province of China as an example: its levee is an earth embankment with a total width of 6 m and a 5-m-wide concrete pavement. Illegal sand mining activities taking place before 2000 seriously corroded the lower part of this levee, and its bottom was heavily incised, which threatened the safety of the levee. Under these circumstances, a landslide finally took place in February of 2001, causing a 100-m-long levee to slip into the river (Wang et al. 2004). The largest river in South India, the Pamba River, also encountered similar incidents (Padmalal et al. 2008). In recent years, localgovernments in different places, on the one hand, have attacked the illegal activities of sand mining, and, on the other hand, have utilized river sand resources scientifically and rationally with sand exploitation planning as guidance. But there still exist two problems: first, sand mining contractors always try their best to over-exploit beyond the scope or depth, driven by their interests, and even under strict supervision; and second, the state of river flow changes after sand mining pits come into being, corroding the river bank or the slope of the levee. Experts and scholars are paying close attention to the heavy risk of slope instability caused by these two factors.

River sand mining affects not only the levee of the reach containing the pit, but also the upper reaches of the pit in both vertical and horizontal directions, which causes incision deformation of riverbed. In the Chiding reach of the Xijiang River levee in Guangdong Province, after sand mining caused a sand pit in 1998, the riverbed was cut down by almost 2 m (Wang et al. 2004). Macdonald (1988) conducted systematic analyses of riverbed recovery in the sand mining reach of the American Naugatuck River and the floodplain near it. He predicted that channel recovery to pre-mining morphology was expected to require up to several hundred years for instream sites and longer for riparian pits. Mao (2003) simulated and studied the secondary flow problem of sand mining pits in natural rivers with the anisotropic three-dimensional algebraic stress turbulent model. The result demonstrated that sand mining changed the original steady state of the river, caused a vertical vortex along the mainstream direction, and scoured the upper edge of the sand pit, while the transverse secondary flow in the sand pit caused transversal erosion. Research on turbulent characteristics has indicated that sand mining have some negative influences on riverbed stability. Li (2008) indicated that sand mining lowered the upstream water level, and the height of the pit played a decisive role. Transverse circulation took place in some parts of the pits and scoured the lateral riverbed. The deeper the pit was and the larger the area was, the stronger the circulation flow would be. The speed of backward erosion was very fast, but the influencing distance was limited, and the stream-wise erosion downstream did not scour deeply, but the influencing distance was relatively large and deposition phenomena occurred after scouring. Consequently, in this study on the instability risk of levees under river sand mining conditions, we mostly focus on levees in reaches containing pits and in reaches upstream of the pits, as well as the instability risk caused by the uncertainty of slopes or levees.

To study the instability risk of levee and dam projects, Wang et al. (1998) calculated the risk taking into account the physical and mechanical indices of the soil body under uncertain flood-preventing water levels, and calculated probability using the Monte Carlo (MC) method. By combining the reality of safe operation and management of the levee, Wu and Zhao (2003) proposed a risk estimation model and solution method based on the reliability theory with consideration of slope stability and seepage stability. The model was applied to the risk estimation system of parts of the Yangtze River levee. Cao (2006) and Srivastava andSivakumar (2010) built corresponding single-risk calculation models and comprehensive risk calculation models quantitatively, according to the main invalidation types of levees, including overtopping, seepage, and instability. Zhou et al. (2010) conducted an analysis of the safety risk of dams using the Bayesian network technology. Bi et al. (2010) introduced the radial basis function (RBF) neural network technology to improve the efficiency and accuracy in assessing the safety and reliability of slope stability.

The literature review shows that sand mining increases the uncertainty of slope stability, which should not be ignored. However, all the existing research focuses on the stability risk problem of levees under certainty circumstances of slope and levees. Therefore, this study intended to examine the stability risk of levees under sand mining conditions in rivers, for the purpose of offer support to sand mining planning design and the establishment of sand mining control policy.

2 Uncertainty analyses

In the Yangtze River sand mining practice, out-of-scope exploitation by sand mining contractors is one of the reasons for slope uncertainty in the reaches containing sand mining pits. Research has also revealed that, due to the effect of water flow, there exists uncertainty of slope stability in the reaches containing sand pits and the reaches upstream of the pits (Mao 2003; Li 2008). Therefore, it is necessary to discuss the uncertainties of slope stability in the two different kinds of reaches.

2.1 Uncertainty of physical and mechanical indices of soil

Because of the inhomogeneity of soil as well as the interference of different factors, there exists uncertainty in both physical and mechanical indices of soil in the slope of levees (Francisco et al. 2008). However, Wang (1998) has pointed out, on basis of a great deal of analysis and research, that the cohesion and friction angle have the greatest uncertainty and effects on slope instability risk of all of the indices of physical and mechanical properties. In contrast, the influence of other parameters is relatively small, and can be ignored during calculation. Besides, a lot of scholars consider that the uncertain distributions of cohesion and friction angle obey normal distribution (Kuo et al. 2007).

2.2 Uncertainty of slope stability in reach containing sand pits

There are two factors that contribute to the uncertainty of slope stability in reaches containing sand pits. The first is that sand mining contractors always try to over-exploit in scope and depth, even though supervision exists, and this over-exploitation is affected by many uncertain factors. The second is that after the formation of sand mining pits, transverse circulation occurs in some parts of the pits and erodes the riverbed and levee slope, and this kind of erosion becomes uncertain under the influence of many uncertain factors.

Transverse circulation occurs in some parts of the sand pits, influencing the erosion of theriverbed and levee slope. The pit and slope evolution in the reach containing the pit are simplified and shown in Fig. 1, in whichBstands for the erosion extent of the riverbed or slope caused by local transverse circulations after the sand pit forms. The value ofBis influenced by many factors, such as river characteristics, flow, water level, and the position and depth of the sand pit. Quantitative research on this aspect has not yet been reported except that Wang et al. (2004) revealed that the riverbed was cut down by nearly 2 m after the formation of sand mining pits in the Xijiang River levee in Guangdong Province of China in 1998. Observation of the Anhui sand mining district of the Yangtze River shows an uncertainty of bottom heights of the sand mining pits, even under the circumstances of strict supervision. This kind of uncertainty is not very significant, and, after the formation of the sand pits, their bottom heights are not lowering, but experiencing deposition and in fact rising. Therefore, the uncertainty of the bottom heights of sand pits is not taken into consideration in this paper.

Fig. 1 Sketch of deformation line of river bank and sand pit

Over-exploitation by the sand mining contractors in scope and depth will influence the uncertainty of the slope and levee. Even though the owner of sand resources (the owner is always the government or its agency) generally sends representatives to supervise the work of the contractors, the supervision is quite minimal because of underwater operation. For example, in the horizontal direction, we can control the sand mining activities only by observing the relative position of the sand mining ship and control boundary, but this kind of observation will be made on the ship, which causes the uncertainty to be relatively great. In Fig. 1,Astands for the erosion extent of the sand pit along the direction of the levee line. Practice shows that the greatest width ofAcan be as wide as a sand mining ship, about 10 m. In the vertical direction, the sand mining suction head is fixed on the ship. The ship fluctuation in the water, as well as the fuzziness of the suction head position and power control, cause the uncertainty of sand mining depth. But observation reveals that the maximum of over-exploitation in the depth direction is limited to one to two meters under strict supervision circumstances. Also, after the formation of sand pits, their bottoms are not developing downward, but experiencing deposition and rising slowly. Consequently, the uncertainty of the pit bottom in the depth direction is not taken into consideration for the time being.

Slope uncertainty in the reach containing sand mining pits needs to be estimated. The uncertainty ofAis influenced by both the over-exploitation by sand mining contractors and the water flow after the pits form. The over-exploitation degree is related to many factors, such as the operation behavior of workers and the supervision behavior of the supervisor. After the pitstake shape, the influence of the flow onAandBis related to factors such as river characteristics, the size of water flow, and the depth of sand pits. Therefore,AandBare influenced by many factors, maybe none of which is a dominating factor. Accordingly, we can suppose that bothAandBaccord with normal distribution based on the probability and statistical theory. To simplify calculation, we suppose that bothAandBobey triangular distribution.

2.3 Uncertainty of slope stability in reaches upstream of sand pits

The uncertainty of slope stability of levees in reaches upstream of sand mining pits results from the following: the sand mining pits change the original stable state of water flows after the pits form, bring about vertical scroll that scours the upstream edge of the sand pits, and cause transverse erosion. According to research by Wang et al. (2004) and Li (2008), as well as practical experience, the cross-section of the upper reach of the sand mining pits can be simplified as shown in Fig. 2 so as to simplify the calculation. Similarly, the uncertainty ofBin Fig. 2 is affected by many factors, of which it is difficult to find one that plays the leading role. Thus, we also suppose it obeys triangular distribution according to the probability and statistical theory.

Fig. 2 Sketch of slope line deformation of reaches upstream of sand pit

3 Model of slope stability risk under sand mining conditions

3.1 Slope stability risk

Under river sand mining conditions, the stability risk of the river bank slope or levee slope (both referred to as the slope) can be defined as the failure probability that the load suffered by the levee exceeds its resistance when the slope gradient exceeds a certain value under river sand mining conditions. As to the suffered load of the levee, it can be the retaining water level of the levee or the sliding moment of force of the slope, and it can also represent the seepage gradient of the levee under the effect of hydraulic action. Similarly, the resistance that the levee holds can be the height of the levee (or the design water level), the resisting moment of the slope, or the impermeability of the soil in the levee.

In this paper, we mainly discuss the slope instability risk of levees under sand mining conditions. That is to say, we only consider the risk probability when slope deformation occursunder sand mining conditions, namely, when the sliding moment of force denoted asMSand the resisting moment denoted asMRchange. Suppose the risk probability isR; according to the general principle of soil mechanics,Rcan be expressedas

In Eq. (1), bothMSandMRare random variables, which are related to different river sand mining conditions.

3.2 Calculation model of slope instability risk

Suppose the probability density distribution ofMSisf(MS). Then, the calculation model ofRcan be expressed as

In Eq. (2), it is difficult to calculateRdirectly. On the one hand, bothMSandMRare relevant to the slope gradient. On the other hand,MSandMRare also related to a lot of uncertain factors such as indices of physical and mechanical characteristics of the soil. However, we can utilize the probability combination method to estimateRindirectly.

Suppose the joint probability density distribution ofMSand the slope gradient denoted asiisf(MS,i); by using Bayes’ conditional probability formula, we can obtain

wheref(MS|i) is the conditional probability density function ofMSat a given slope gradienti, andf0(i) is the probability density function ofi. According to the total probability theorem, we can obtain

Therefore, Eq. (2) can be expressed as follows:

wherei1represents the slope gradient at the upstream face before sand mining, andi2represents the slope gradient at the upstream face after sand mining.

In practical calculatio n, after the discretization of Eq. (5) and through the probability combination method, we obtain

whereRjis risk probability of thejth subinterval;, which is the probability of the situation in whichMSis greater thanMRwhen the slope gradien t of the levee is given asis the probability of thejth subinterval in the frequency curve ofi; andnstands for the number of subintervals in the frequency curve ofi.

3.3 Calculation of

To calculatewe need to determine the minimum stability safety coefficient first, then adopt a performance functiong(x) expressed as Eq. (7) (Ayyub et al. 2009), as well as the MC method, to obtain the value of

where the subscriptkdenotes the soil strip serial number,Ckand?kare the cohesion and friction angle of soil body, respectively,Ukis the pore water pressure,Wkis the self-gravity that the soil strip suffers,bkis the width of the soil strip, andθkis the angle formed by the tangent line at the middle point of the soil strip bottom and the horizontal line. Wheng(x)=0, we get the famous Bishop’s terminal state equation. Wheng(x)＜0, it means thatMS＞MR.

The procedure of calculatingcan be described as follows, using the MC method:

(1) First of all, we calculate the minimum safety coefficient of slope stability, and determine the slip arc.

(2) We generate pseudo-random numbers, and samplesCkand?kwith an indefinite quantity.

(3) With the given slip arc, we calculateg(x).

(4) WithNbeing the times wheng(x)＜ 0, we countN, that is, ifg(x)＜0,N=N+1.

(5) We repeat the procedures from (2) to (4)Mtimes (M≥100000).

(6) We calculate, and

4 Case study

A case study was carried out in a sand mining area in a reach called Reach W of the Yangtze River when a sand mining planning design is conducted. It is determined to be 2 000 m long, 150 m wide, and 8 m deep. Both sides of the sand mining area are levees, and the location of one side of the slopes and sand pits in the designed mining area are shown in Fig. 3. The physical and mechanical indices of the soil in the levee and its foundation are shown in Table 1.

Fig. 3 Location of sand pit and slope of levee

Table 1 Average physical and mechanical indices of soil in levee and its foundation

According to practical experience, we suppose bothAandBto obey the triangular distribution, as shown in Figs. 4(a) and (b), respectively. We then get the probabilities of thejth subinterval in frequency curves ofAandB, which are shown in Table 2.

Fig. 4 Density distribution function ofAandB

The preliminary calculation results indicate that in this example the slip arc will not go down below the original riverbed surface because the mechanical indices of the soil at the levee bottom are fine. Therefore, we mainly discuss the slope instability risk under hydraulicaction when the state of water flow changes due to the formation of sand pits.

Table 2 Calculation o fRj

Using the MC method, we determine that the risk probability of slope instability before sand mining in Reach W is 2.38%. Similarly,can be calculated with the MC method. The results shown in Table 2 indicate that the risk probability of slope instability increases very quickly when different slope gradients are given. Of the five different slope gradients, the maximum value of the risk probability is 2.28 times the minimum value. Withwe obtainRjby further calculation using Eq. (6), and the results are shown in Table 2. Finally, we can obtain the risk probability in Reach W after sand mining by calculating the sum ofRj. The result turns out to beR= 4.74%. Apparently, the risk has nearly doubled at this point, which means that sand mining has a large influence on the slope stability of the levee in this reach of the Yangtze River.

5 Conclusions

There are two problems caused by river sand mining: one is that it changes the state of river flows when the sand mining pits come into being, corroding the slope and levee; another is the over-exploitation by mining contractors in scope and depth, even though supervision exists. These two factors not only make the slope steeper, but also reduce the distance from the bottom line of the levee to the sand pits. A case study indicated that the probability of instability risk nearly doubled after sand mining. It is obviously necessary to pay close attention to sand mining planning and design.

After the sand pits form, the erosion extent of the slope and the incision extent of riverbed are supposed to obey a triangular distribution. In fact, in this study, the distribution law of the erosion extent appeared to be affected by many factors, including river depth, flow velocity, river shape, and size of sand mining pits. This is worth further investigation.

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(Edited by Yun-li YU)

This work was supported by the Special Fund for Public Welfare Industry of the Ministry of Water Resources of China (Grant No. 201001007).

*Corresponding author (e-mail:zfwang@hhu.edu.cn)

Received Jun. 13, 2011; accepted Feb. 22, 2012