A hybrid fuzzy evaluation method for curtain grouting efficiency assessment based on an AHP method extended by D numbers

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A hybrid fuzzy evaluation method for curtain grouting efficiency assessment based on an AHP method extended by D numbers Q1

Guichao Fan a, Denghua Zhong a,∗, Fugen Yan a,b, Pan Yue a

Q2

a b

State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China Hydraulic Complex Design Department, Changjiang Institute of Survey, Planning, Design and Research, Wuhan 430010, China

a r t i c l e

i n f o

Keywords: Curtain grouting efficiency assessment Hybrid fuzzy evaluation method D-AHP method Water pressure tests (WPT) Rock quality designation (RQD) Fracture filled rate (FFR)

Curtain grouting is designed to create a hydraulic barrier that reduces water seepage under a dam foundation. However, due to the inability of observing grout penetration in the fractures and voids under a dam foundation, it is difficult to assess grouting effectiveness. Existing single-factor evaluation methods, including in situ water pressure tests (WPT), numerical modeling and experiments, cannot estimate the grouting efficiency from the perspectives of both permeability and tightness. This paper proposes a hybrid fuzzy comprehensive evaluation method to assess curtain grouting efficiency by considering both the permeability and tightness of a grout curtain. Permeability (LU), Rock Quality Designation (RQD), and the Fracture Filled Rate (i.e., the ratio of the fractures filled by grout, FFR), as derived from water pressure tests, core specimen drilling and borehole television imaging, are the key assessment parameters of grouting efficiency. A three-level hierarchical model of curtain grouting efficiency assessment is established that considers permeability and tightness as criteria and LU, RQD and FFR as alternatives. Because determining the weights of the alternatives is a fuzzy and uncertain problem, the D-AHP method, which extends traditional AHP with D numbers and can handle uncertain problems more effectively, is introduced to determine the weights of the alternatives. The detailed procedures of the proposed hybrid evaluation method are demonstrated step by step. A case study is presented to evaluate the curtain grouting efficiency of a hydropower project in China. The results indicate that low permeability does not result in high grouting efficiency because although the RQD and FFR may be very low, only low permeability, high RQD and high FFR indicate high grouting efficiency. The proposed method provides a detailed understanding of grouting efficiency evaluation; with the help of an efficiency index (EI), the grouting efficiency of grouted units or areas can be easily marked and classified. © 2015 Published by Elsevier Ltd.

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1. Introduction

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The geologic condition of a dam foundation is typically complex because of the numerous fractures, voids, and weak intercalated layers underneath the dam foundation. Because these fractures or voids provide a water leakage path that threatens a dam’s foundation, it is necessary to improve the rock conditions under the dam foundation. Curtain grouting is a common approach for improving a dam foundation (Houlsby, 1990; JSIDRE, 1994); grouting not only reduces hydraulic conductivity but also improves the strength of the ground under the dam foundation. The grout penetrates into the fractures and voids and thus spreads into the rock mass (Saeidi, Azadmehr, & Torabi, 2014; Stille, Gustafson, & Hassler, 2012). Case studies (Bruce & Dugnani, 1996; Bryson, Ortiz, & Leandre, 2014; Simpson, Phillips, & Ressi, 2006; USACE, 2011) have shown the efficacy of curtain grout-

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Q3

a b s t r a c t

∗

Corresponding author. Tel.: +86 22 2789 0911; fax: +86 22 2789 0910. E-mail addresses: [email protected] (G. Fan), [email protected] (D. Zhong), [email protected] (F. Yan), [email protected] (P. Yue).

ing in minimizing seepage. However, because it is impossible to observe the dam foundation and the penetration of grout into the voids and fractures, this technology is frequently described as more of an art than a science (Dreese, Wilson, Heenan, & Cockburn, 2003), thus indicating that the success of seepage prevention and permeability reduction is unknown. Therefore, it is important and necessary to effectively evaluate the efficiency of curtain grouting. To evaluate cement grouting efficiency in the improvement of rock masses, many in situ tests, such as water pressure tests (WPT), core specimen drilling, borehole television imaging, acoustic velocity tests and seismic tests have been commonly applied. Among these tests, WPT is the most popular method and is extensively used to determine rock mass permeability in curtain grouting effect assessment (Ewert, 1994, 1997; Hao & Li, 2003; Magoto, 2014; Xia, Yue, Zhang, & Zheng, 2013; Zadhesh, Rastegar, Sharifi, Amini, & Nasirabad, 2014). In situ water pressure testing over discrete zones in the foundation provides an excellent method for verification of the effectiveness of the grouting (Roman, Hockenberry, Berezniak, Wilson, & Knight, 2013). However, WTP cannot determine how many

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fractures and voids have been filled by grout or whether potential leakage paths still exist. Measuring the sonic or acoustic velocity before grouting and its subsequent increasing rate can be used to assess grouting effectiveness (Chen, Lu, Zhang, Yan, & Zhou, 2015; Yang, & Li, 2008). In addition, some in situ geotechnical tests, such as borehole expansion tests (Kikuchi, Igari, Mito, & Utsuki, 1997; Utsuki, 2013; Zolfaghari, Bidar, Javan, Haftani, & Mehinrad, 2015) and plat load tests (Zolfaghari et al., 2015), and geophysical tests such as seismic tests/seismic tomography (Haramy, Henwood, & Szynakiewicz, 2012; Lynch, Dodson, & McCartney, 2012; Turk & Dearman, 1987), electrical conductivity measurements (Bryson et al., 2014) and electromagnetic wave and elastic wave tests (Kikuchi et al., 1997), can be used to evaluate rock mass improvement from cement grouting. However, these tests are more commonly applied in consolidation grouting and compaction grouting than in curtain grouting. In addition to in situ tests, numerical modeling and experimental methods are also used to evaluate the effectiveness of grout curtains. Rahmani (2009) estimated grout effectiveness by simulating the distribution of grout inside fractures with numerical models. Bryson et al. (2014) analyzed the relationship between electrical conductivity and hydraulic conductivity based on a laboratory-scale physical model, and recommended that this relationship would help evaluate the effectiveness of grout curtains. However, these methods are single-factor evaluation methods that can determine the permeability but not the tightness of a rock mass under a dam foundation. Thus, these existing methods are insufficient at comprehensively describing curtain grouting efficiency from the perspectives of both permeability and tightness. To overcome the shortcomings of the existing evaluation methods, this study proposes a comprehensive evaluation of curtain grouting efficiency based on the results of water pressure testing, core specimen drilling, and borehole television imaging. These results include the Permeability (Lugeon, LU), the Rock Quality Designation (RQD), and the Fracture Filled Rate (i.e., the ratio of the fractures filled by grout, FFR). The Lugeon values from water pressure testing verification holes can provide a direct indication of residual permeability (Roman et al., 2013). The RQD is an easy, quick, and inexpensive measurement of the degree of jointing along the core drill hole (Palmstrom, 2005; Sadeghiyeh, Hashemi, & Ajalloeian, 2013). The FFR can describe the ratio of the fractures filled by grout. The parameters RQD and FFR remedy the defect of the LU value being unable to describe the fracture distribution and the absence or presence rate of grout within the rock discontinuities. This comprehensive evaluation overcomes the shortcomings of existing grouting efficiency evaluation methods that primarily focus on the effects of permeability on the curtain grouting efficiency while disregarding the effect of tightness. Because the comprehensive evaluation of curtain grouting efficiency is a fuzzy concept with multiple indicators and classes, this study applies a fuzzy comprehensive evaluation method based on the theory of fuzzy mathematics (Bˇehounek & Cintula, 2006; Regression, 2002) and precise mathematical language to describe and judge the fuzziness of a given problem. This method was first developed by Zadeh (1965); because it is an efficient evaluation method for evaluating objects that are affected by various factors, it has been widely applied in many fields such as water quality safety assessment (Zhou, Zhang, & Dong, 2013), feed product safety assessment (Chen, Jin, Qiu, & Chen, 2014), petrochemical industry safety management (Li, Liang, Zhang, & Tang, 2015; Li, Wang, Liu, & Zhou, 2015; Zuo, Zhang, Deng, & Wang, 2012), chemical risk assessment (Han, Song, Duan, & Yuan, 2015), flood risk evaluation (Lai et al., 2015), forest fire risk evaluation (Wang, Ma, Song, & Xu, 2013), performance assessment (Bai, Dhavale, & Sarkis, 2014; Chen, Hsieh, & Do, 2015; Jiang, Wu, Ren, & Yang, 2015), urban development (Feng, & Xu, 1999), equipment criticality evaluation (Guo, Gao, Yang, & Kang, 2009), supplier selection (Igoulalene, Benyoucef, & Tiwari, 2015) and engineering machinery selection (Yan,

Zhang, Zhang, & Wu, 2008). However, it has rarely been applied in the field of grouting efficiency evaluation. As mentioned earlier, LU, RQD and FFR are proposed as the indicators of the comprehensive evaluation for curtain grouting efficiency. However, qualitative and quantitative data always simultaneously exist in actual multiple criteria assessment situations (Chen et al., 2014). After identifying these indicators, a methodology is needed to determine the significance order of the indicators. The analytic hierarchy process (AHP), first developed by Saaty (1977, 1980), is a structured technique for handling complex decision making by combining qualitative and quantitative analyses. Because of its simplicity in concept and convenience in mathematical calculations, AHP has been extensively used in complex decision making (Chen et al., 2014; Dehe, & Bamford, 2015; Kar, 2014; Kilincci & Onal, 2011; Lu, Lian, & Lien, 2015; Ngai & Chan, 2005; Racioppi, Marcarelli, & Squillante, 2014; Saaty, 2008; Vaidya & Kumar, 2006; Wang et al., 2014). However, there are still some deficiencies and limitations when applying this methodology. First, the comparative judgments are subjective because they rely heavily on expert opinion, which may sometimes cause inconsistency. Furthermore, AHP lacks the ability to adequately cope with any inherent uncertainty and imprecision in the data. Finally, in a real situation, an expert may have limited knowledge of and experience with alternatives; the preferred information may contain fuzziness and incompleteness, and AHP is unable to handle this incomplete information (Dehe, & Bamford, 2015; Deng, Hu, Deng, & Mahadevan, 2014a; Dong, Li, & Zhang, 2015; Su, Mahadevan, Xu, & Deng, 2015). Fortunately, there are several theories that can handle the uncertainty of certain information, including fuzzy set (Boran, Genc, Kurt, & Akay, 2009; Igoulalene, Benyoucef, & Tiwari, 2015; Naili, Boubetra, Tari, Bouguezza, & Achroufene, 2015; Ozkoka & Cebi, 2014), intuitionistic fuzzy set (Atanassov, 1986; Melo-Pinto et al., 2013; Nguyen, 2015), rough set (Pawlak, 1982; Pawlak & Skowron 2007a; 2007b; Wang, Peng, & Liu, 2015), grey system (Deng, 1989; Memon, Lee, & Mari, 2015) and Dempster–Shafer theories of evidence (Compare & Zio, 2015; Dempster, 1967; Shafer, 1976). Among these theories, the Dempster–Shafer theory is popular and extensively used to address uncertain problems because it has the advantage of directly expressing uncertain information. However, there are still several inherent shortcomings caused by the theory itself. First, the mathematical framework of this theory is based on the strong hypotheses that the elements in the frame of discernment are required to be mutually exclusive. However, in many situations, these assessments inevitably contain intersections because they are prone to subjective judgment (Deng et al., 2014a; Li, Hu, Zhang, & Deng, 2015). In addition, a normal basic probability assignment (BPA) must be subjected to the completeness constraint; i.e., the sum of all focal elements in a BPA must equal 1. However, in the real world, expert assessments may be incomplete due to a lack of knowledge or cognition. More importantly, the Dempster’s rule of combination cannot address the incomplete BPAs (Deng et al., 2014a; Liu, You, Fan, & Lin, 2014). To effectively solve these problems, a decision-making method using uncertain information called the D-AHP method (Deng et al., 2014a) is introduced in this paper. The D-AHP method is developed from the AHP method by extension with D numbers-based preferences. D numbers were developed by Deng (2012) as an effective representation of uncertain information, and they are widely used in many applications such as supplier selection (Deng et al., 2014a), failure analysis (Liu et al., 2014), environmental impact assessment (Deng, Hu, Deng, & Mahadevan, 2014b; Rikhtegar et al., 2014) and product quality evaluation (Li et al., 2015). The D-AHP method can represent uncertain information more effectively because it overcomes the shortcomings and deficiencies of the traditional AHP and Dempster–Shafer theories. First, the D-AHP method uses a D numbers preference relation instead of a pair wise comparison; the D numbers preference relation is the classical fuzzy preference relation (Tanino, 1984) extended by D numbers. Although the preference relations of

Please cite this article as: G. Fan et al., A hybrid fuzzy evaluation method for curtain grouting efficiency assessment based on an AHP method extended by D numbers, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.09.006

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3

Fig. 1. Research framework.

First level: Objective

Second level: Criteria

Third level: Alternatives

Curtain Grouting Efficiency Assessment

Permeability

Permeability Rate (LU)

Tightness

Rock Quality Designation (RQD)

Fracture Filled Rate (FFR)

Fig. 2. Hierarchical model for curtain grouting efficiency evaluation.

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the alternatives or criteria given by the experts are imprecise, fuzzy and incomplete, the D numbers preference relation can effectively express this uncertain information without causing inconsistency. Furthermore, the sum of all focal elements in a D numbers preference relation need not equal 1; i.e., if the assessment information given by experts is incomplete, this value may be less than 1. The remainder of this paper is organized as follows: in Section 2, we introduce the research framework of this paper. In Section 3, we construct the hierarchical model of curtain grouting efficiency evaluation and explain the derivation and calculation of the evaluation indicators. In Section 4, we briefly review the AHP method and D numbers and present the proposed hybrid fuzzy evaluation method based on D-AHP step by step. In Section 5, we demonstrate an application in China to show the effectiveness of the proposed method. In Section 6, we present a discussion. Finally, we provide concluding remarks and recommendations for further research in Section 7.

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2. Research framework

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As shown in Fig. 1, the research framework of this paper includes three major parts: obtaining the evaluation indicators, establishing the hybrid fuzzy evaluation method, and addressing a case study. First, a hierarchical model for curtain grouting efficiency evaluation is established; the LU, RQD and FFR, which are derived from the analysis results of water pressure testing, core specimen drilling and borehole television imaging, are proposed as the indicators of curtain grouting efficiency assessment. Second, a hybrid fuzzy evaluation method is proposed to assess curtain grouting efficiency by combining the D-AHP method with the fuzzy comprehensive evaluation method. Fi-

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nally, the proposed method is applied as a case study to a curtain grouting efficiency assessment for a hydropower project in China.

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3. Curtain grouting efficiency evaluation system and its indicators

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3.1. Hierarchical evaluation system of curtain grouting efficiency

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As mentioned by Roman et al. (2013), the aim of foundation seepage prevention is to ensure that the curtain is tight and that the hydraulic barrier has low permeability; we thus set permeability and tightness as the criteria of the evaluation system. Based on these criteria, we establish the following three-level hierarchical model for curtain grouting evaluation: the first level represents the overall objective, i.e., curtain grouting efficiency assessment; the second level represents the criteria, including permeability and tightness; and the third level represents the evaluation indicators, consisting of LU, RQD, and FFR, as shown in Fig. 2.

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3.2. Derivation and calculation of the evaluation indicators

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LU, RQD, and FFR represent the water pressure test, core specimen drilling and borehole television imaging, respectively. Grouting of each borehole, including verification holes, was monitored by an automated grout monitoring system; grouting parameters such as pressure and flow rate were measured and recorded in real time. According to Lugeon (1933), WPT results are expressed in Lugeon units (LU); a Lugeon is defined as the water loss of 1 L/min per meter length

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tion of the evaluation system f as the relationship function from the indicators (i.e., LU, RQD, and FFR) to the efficiency index EI. Eq. (3) defines the solution method set for the objective function M as the solution method set, which includes the fuzzy comprehensive evaluation method (FCEM, Mf ) and the D-AHP method (Md ), where y is the membership function, k is the function from the indicators (i.e., LU, RQD and FFR) to the weight vector W, g is the fuzzy comprehensive function used to calculate the fuzzy comprehensive evaluation set D, and ⊗ is the fuzzy operator. Eq. (4) defines the parameter set of the solution method P as the parameter set composed of the evaluation indicator set U, the evaluation criteria set V, the fuzzy membership matrix R, the weight set W, and the fuzzy comprehensive evaluation set D; Sm represents the establishment method of these parameter sets. The following equations result from these assumptions:

EI = f (LU, RQD, F F R)

⎧ ⎪ ⎨M = M f ∪ Md R = y( ∪ γ (ui )(v j )) W = k(LU, RQD, F F R) ⎪ ⎩ D = g(W ⊗ R) 

P = U ∪V ∪R∪W ∪D S = Sm

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of a test section at an effective pressure of 1 MPa. If the grout pressure, flow rate and length of the drill hole section are known, then the Lugeon value can be calculated by Eq. (1):

LU = Q/P · L

(1)

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where LU is the Lugeon value in Lu; Q is the water take (i.e., discharge) in L/min; P is the effective pressure in MPa; and L is length of the tested interval in m. Based on Eq. (1), the Lugeon value can be automatically calculated and recorded by the automated grout monitoring system in real time. According to Deere (1968) and Palmstrom (2005), RQD is a quantitative estimate index for the rock mass quality of drill core logs. It is defined as the percentage of intact core pieces longer than 10 cm in the total length of a core. Fig. 3 shows the procedure for measuring and calculating RQD. However, one tested interval contains more than one RQD, as shown in Fig. 4; to address this problem, we propose an arithmetic average value of all RQD in one tested interval to be the overall RQD value of each single tested interval; this value is denoted by RQDavg . FFR refers to the ratio of the fractures filled by grout and can be calculated as the number of fractures filled by grout divided by the total number of fractures in one tested interval, as shown in Fig. 5. If the number of fractures filled by grout is six and the total number of fractures in one tested interval is eight, then the FFR value is 75%.

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4. Methodology

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4.1. Mathematical model of the evaluation system

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This paper focuses on describing the relationship between the indicators (i.e., LU, RQD, and FFR) and the curtain grouting efficiency to obtain an efficiency index for curtain grouting. The mathematical model (as shown in Eqs. (2)–(4)) of the hybrid fuzzy evaluation system is composed of three parts: an objective function, the solution method set and the parameter set. Eq. (2) defines the objective func-

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(4)

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(5)

with

D(B) ≤ 1andD(∅) = 0

250

(3)

Definition 1. Let  be a finite nonempty set, a D number is a mapping formulated by



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4.2.1. D numbers D numbers, first developed by Deng (2012), are a good representation of uncertain information. Through extension with a fuzzy preference relation and the AHP method, the D-AHP method can handle uncertain information more effectively, thus overcoming the existing deficiencies in the traditional AHP and Dempster–Shafer theories. If we take the assumed situations presented in Deng et al. (2014a) as an example, then 10 experts assess two alternatives. In one situation, 8 experts assess that the first alternative is preferred to the second alternative and that the preference is 0.8; the remaining 2 experts also assess that the first alternative is preferred to the second alternative but the preference degree is 0.7. In another situation, 8 experts assess that the first alternative is preferred to the second alternative and that the preference degree is 0.8, and the remaining 2 experts do not give any information about the preference relation of these two alternatives because of the lack of background knowledge. Neither the pair wise comparison relation nor the fuzzy preference relation can express both of these two situations. However, the D numbers extended fuzzy preference relation can represent these two situations, as shown later. Deng (2012) and Deng et al. (2014a, 2014b) define D numbers as follows:

D :  → [0, 1]

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(2)

4.2. Determining the weight of each indicator using the D-AHP method

Fig. 3. Procedure for measuring and calculating RQD (Palmstrom, 2005).

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(6)

B⊆

where Ø is an empty set and B is a subset of . From this definition, we notice that the completeness constraint  is released if D numbers are used. If B⊆ D(B) = 1, then the infor mation is complete; and if B⊆ D(B) < 1, the information is incomplete. Suppose that the set  = {b1 , b2 ,…, bi ,…, bn }, where bi ∈R and bi an bj if i if j. Then, a special form of D numbers can be expressed as:

D({bi }) = vi (i = 1, 2, · · · , n)

(7)

Please cite this article as: G. Fan et al., A hybrid fuzzy evaluation method for curtain grouting efficiency assessment based on an AHP method extended by D numbers, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.09.006

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Fig. 4. The overall RQD calculation for one single tested interval. Left is a sketch of the Lugeon and RQD values of each tested interval; right is the overall RQD value for each tested interval. RQDavg represents the overall RQD value of each tested interval.

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or simply denoted as D = {(b1 , v1 ), (b2 , v2 ),..,(bi , vi ),…,(bn , vn )}, where  vi > 0 and ni=1 vi ≤ 1. Definition 2. Let D = {(b1 , v1 ), (b2 , v2 ),…,(bi , vi ),…,(bn , vn )} be a D numbers, the integration representation of D is defined as

I(D) =

n 

bi vi n

Definition 3. A D numbers preference relation RD on a set of alternatives A is represented by a D matrix on the product set A×A, whose elements are formulated by

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where bi ∈R, vi > 0 and

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4.2.2. D numbers extended fuzzy preference relation The fuzzy preference relation (Tanino, 1984) is provided to construct pair wise comparison matrices based on expert judgment and is described by a fuzzy pair wise comparison with an additive reciprocal (i.e., rij + rji = 1) that is different from the multiplicative preference relation. rij denotes the preference degree of an alternative Ai over another alternative Aj and can be expressed as follows:

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ri j =

⎧ 0 ⎪ ⎪ ⎨∈ (0, 0.5) 0.5

⎪ ⎪ ⎩∈ (0.5, 1) 1

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(8)

i=1

301

fuzzy preference relation. To overcome these shortcomings, Deng et al. (2014a) extended the classical fuzzy preference relation by using D numbers. The derived relation is called a D numbers preference relation, and the corresponding matrix is called a D numbers preference matrix, which can be abbreviated as a D matrix. The D matrix is defined as follows.

i=1

vi ≤ 1.

A j is absolutely preferred to Ai A j is preferred to Ai to some degree; indifference between Ai and A j ; Ai is preferred to A j to some degree; Ai is absolutely preferred to A j .

(9)

There are some shortcomings when using the fuzzy preference relation to represent certain situations. For example, if the expert assessments are uncertain or incomplete, it is difficult to construct the

RD : A × A → D

··· ··· ··· .. . ···

A2 D12 D22 .. . Dn2

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An ⎤ D1n Dn ⎥ .. ⎥ ⎦ . Dnn

(11)

where Di j = {(b1 , v1 ), (b2 , v2 ), · · · , (bm , vm )}, D ji = {(1 − b1 , v1 ), ij

(1 − bi2j , vi2j ), · · ·

∀k∈{1, 2, …, m}.

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(10)

the D numbers preference relation in matrix form is

⎡ A1 A1 D11 D21 RD = A2 ⎢ .. ⎢ .. ⎣ . . An Dn1

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ij

ij

ij

ij ij , (1 − bm , vm )},

ij

ij

∀i, j∈{1, 2, …, n}, and

ij

ij bk

ij

∈[0, 1],

Consequently, with the help of the D numbers preference relation, the preference relations of the two situations presented above as an

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where rik refers to the membership degree of the ith indicator for the kth class, Fik is the frequency of the ith indicator on the interval [α ik , α ik+1 ], i refers to a particular evaluation indicator, k refers to a particular class, l is the number of a class, [α ik , α ik+1 ] is the interval corresponding to the kth class, and xi refers the test value of the ith indicator. Step 4: Calculate the weight vector. As mentioned earlier, the weights can be calculated using the D-AHP method introduced by Deng et al. (2014a). Step 5: Calculate the fuzzy comprehensive evaluation set. The weighted average operator M(+, •) can be used as the fuzzy operator; this operator ensures that the evaluation coefficient maintains every indicator by using each indicator’s weight, as shown in Eq. (15):

dk = g(w, r) =

m 

wi · rik

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(15)

i=1

Fig. 5. Calculation of FFR.

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example are shown in Eqs. (12) and (13), respectively.

RD1 = A1 A2

A1  {(0.5, 1.0)} {(0.2, 0.8), (0.3, 0.2)}

A2  {(0.8, 0.8), (0.7, 0.2)} {(0.5, 1.0)} (12)

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RD2 = A1 A2

A1  {(0.5, 1.0)} {(0.2, 0.8)}

A2  {(0.8, 0.8)} {(0.5, 1.0)}

(13)

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4.2.3. AHP method extended by the D numbers preference relation (D-AHP) Detailed procedures of the D-AHP method have been provided by Deng et al. (2014a). The process of applying the D-AHP method to calculate the alternative priority weights in the D matrix can be divided into four steps. First, convert the D matrix into a crisp matrix by using the integration representation of the D numbers. Second, construct a probability matrix based on the crisp matrix to represent the preference probability between the pair wise alternatives. Third, rank the alternatives using the triangularization method (Bartnick, 1991). Fourth, calculate the priority weights of the alternatives.

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4.3. Procedures for fuzzy comprehensive evaluation

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The detailed procedures for the hybrid fuzzy comprehensive evaluation are summarized in Fig. 6. These procedures are divided into the following six steps:

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Step 1: Determine the evaluation indicators. The evaluation indicators should be representative, rational, and accurate to reflect the efficiency. The indicator set can be expressed as U = (ui ), (i = 1, 2,…, m). Step 2: Determine the evaluation criteria. In general, evaluation criteria are often divided into several classifications from which the criteria set is established as V = (vk ), (k = 1, 2,.., l). Step 3: Establish a fuzzy membership matrix. The elements of this matrix represent the degree to which the specified concentration belongs to the fuzzy set and can be determined by the membership function (Eq. (14)), which is described using

rik = Fik /

l  k=1

Fik ,

αik ≤ xi ≤ αik+1

(14)

where dk refers to the evaluation coefficient for the kth class, g is the fuzzy comprehensive function, and wi refers to the weight of the ith indicator. Step 6: Calculate the efficiency index. According to the evaluation coefficient, an object’s class can be determined by the membership class to which the maximum evaluation coefficient corresponds by using the maximum membership principle. However, there are still some defects when applying the maximum membership principle in a fuzzy comprehensive evaluation; for example, when this principle determines an object’s class, the other class of an evaluation coefficient is abandoned. To solve this problem, we apply the comprehensive index method to calculate the efficiency index EI, as shown in Eq. (16). According to the efficiency index, not only can the grouting efficiency of each class be judged, but also, the grouting efficiency of two different objects with the same class can be distinguished. The efficiency index is described by

EI =

m 

dk · z k

367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384

(16)

k=1

where EI is the efficiency index and zk is the grouting efficiency standard grade.

385 386

5. Case application

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5.1. Project overview

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The hybrid fuzzy evaluation method is applied to a case study to determine the curtain grouting efficiency evaluation for a hydropower project X located on the Jinsha River between the Yunnan province and Sichuan province of China. Hydropower project X includes a dam, a spillway, a powerhouse, an underground powerhouse, and a ship lift, as shown in Fig. 7. There are numerous dam monoliths in this project, which are considered as grouting units for simplicity. We select three monoliths (i.e., units) from the overall curtain grouting lines for study, as shown in Fig. 8. These three units are selected from the powerhouse dam monolith to determine the curtain grouting efficiency. Verification boreholes (Fig. 8, right) are designed and located in each grouting unit. The total number of verification boreholes in the three grouting units is 14, which include 303 grouting intervals. The longitudinal layout of the 14 boreholes is shown in Fig. 9.

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5.2. Determining evaluation indicators and criteria

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As mentioned earlier, LU, RQD and FFR are proposed as the indicators. The indicator set can thus be expressed as U = {I1 , I2 , I3 } = {LU,

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Fuzzy Comprehensive Evaluation Method (FCEM)

AHP method extended by D numbers (D-AHP method)

Construct the hierarchy structure of the evaluation system

Establish the D numbers preference matrix (D matrix)

Determine evaluation indicators

Convert the D matrix to a crisp matrix

Construct a probability matrix based on the crisp matrix

Determine evaluation criteria

Rank the alternatives using the triangularization method Calculate the relative weights of alternatives

Establish fuzzy membership matrix

Calculate the fuzzy comprehensive evaluation set

Calculate the efficiency index (EI)

Judge the assessment using EI Fig. 6. Detailed procedures for hybrid fuzzy comprehensive evaluation.

Table 1 Efficiency-level classification standard for the evaluation indicators. Efficiency level

I (good)

II (fine)

III (ordinary)

IV (poor)

V (bad)

LU (Lu) RQD FFR (%) Dimensionless value

< 0.1 90–100 90–100 5

0.1–1 75–90 75–90 4

1–1.5 50–75 50–75 3

1.5–2 25–50 25–50 2

>2 < 25 < 25 1

Table 2 Calculated data of the evaluation indicators using the method introduced in Section 3.2.

Notes: The dimensionless value can help identify which level a grout unit should be located according to the value of its efficiency index (EI). For example, if the EI value of a grout unit is 3, this indicates that the grouting efficiency of this unit is of an ordinary level.

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RQD, FFR}. In general, the efficiency of curtain grouting is divided into five classifications: good, fine, ordinary, poor, and bad. This classification scheme establishes the criteria set as V = {good, fine, ordinary, poor, bad} = {I, II, III, IV, V}.

411

5.3. Establishing the fuzzy membership matrix

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412 413 414 415 416 417 418

As mentioned in Section 4.3, the fuzzy membership matrix can be established by the membership function (Eq. (14)). Before constructing the membership function, we must determine the standard values for efficiency-level classifications of the evaluation indicators, as shown in Table 1. However, we calculate the evaluation indicators from original test data using the methods introduced in Section 3.2, as shown in Table 2. The frequency of each indicator in each classifica-

Borehole name

Interval sequence

LU (Lu)

RQD

FFR (%)

ZC5-SM-J-1 ZC5-SM-J-1 ZC5-SM-J-1 … ZC7-SM-JSJC-1 ZC7-SM-JSJC-1 ZC7-SM-JSJC-1

1 2 3 … 25 26 27

0.42 0.80 0.35 … 0.78 4.60 1.83

0 70 27 … 81 74 40

33 0 0 … 100 0 0

tion can be calculated based on the efficiency-level classification and calculated data of the indicators; this is illustrated in the histogram shown in Fig. 10. Thus, the membership matrix of the three selected grouting units is established, as shown in Eqs. (17)–(19), where R1 , R2 , and R3 represent the fuzzy membership matrices of unit 1, unit 2, and unit 3, respectively.



R1 =

0.021 0.032 0.307

 R2 =

0 0.094 0.339



0.968 0.126 0

0.011 0.442 0.068

0 0.316 0.136

0 0.084 0.489

0.974 0.137 0

0.026 0.368 0.037

0 0.325 0.165

0 0.077 0.459

421 422 423 424

(17)



425

(18)

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Fig. 7. Details and location of hydropower project X, China.

Fig. 8. Selected study regions from the overall curtain grouting lines. Left is the location and layout of the curtain grouting lines; right is the magnified selected study region from the powerhouse dam monolith. These circles represent the locations of the verification boreholes for curtain grouting efficiency checks. Each borehole has a name beside its circle; the little circles in the center are the real size of the verification boreholes. To improve the readability of this figure, the authors have magnified the circle into the bigger circle on the periphery.

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Fig. 9. The longitudinal layout of the verification boreholes.



426

R3 =

0.022 0.077 0.470

0.934 0.275 0

0.022 0.407 0.048

0.011 0.154 0.096



0.011 0.088 0.386

435

(19)

 I1 I1 0.5 Rc = I(RD ) = I2 0.1 I3 0.28

427

5.4. Calculating the weights of the indicators

428

The D numbers preference matrix (i.e., D matrix) must be constructed before calculating the weights of the indicators using the DAHP method. As an example, we present the process of determining the weights of the indicators with respect to the criteria permeability. First, we calculate the relative significance of LU, RQD, and FFR relative to the feed permeability to construct the D matrix based on the D numbers preference relation as follows:

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I RD = 1 I2 I3



I1

{(0.5, 1.0)} {(0.1, 1.0)} {(0.3, 0.8), (0.2, 0.2)}

I2

{(0.9, 1.0)} {(0.5, 1.0)} {(0.8, 0.8)}

(1) The D matrix is converted to a crisp matrix Rc using the integration representation of D numbers (Eq. (10)) as follows:

I2 0.9 0.5 0.64

I3  0.72 0.16 0.5

 I1

0 0 0

I2 1 0 1

I3 1 0 0



I3  {(0.7, 0.8), (0.8, 0.2)} {(0.2, 0.8)} {(0.5, 1.0)}

(22)

(20)

Please cite this article as: G. Fan et al., A hybrid fuzzy evaluation method for curtain grouting efficiency assessment based on an AHP method extended by D numbers, Expert Systems With Applications (2015), http://dx.doi.org/10.1016/j.eswa.2015.09.006

437

(21)

(2) According to the rules proposed to generate the probability matrix by Deng et al. (2014a), the probability matrix is constructed as below:

I Rp = 1 I2 I3

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Using the weight relation of the indicators represented in the matrix, the weight equations are constructed by incorporating necessary constraints:

⎧ λ(w1 − w3 ) = 0.72 − 0.5 ⎪ ⎪ ⎪ ⎪ λ( ⎨ w3 − w2 ) = 0.64 − 0.5 w1 + w 2 + w 3 = 1

⎪ ⎪ ⎪ λ>0 ⎪ ⎩

wi ≥ 0,

λ, λ = n,

n2 /2,

Fig. 10. Frequency distribution of the indicators in each efficiency-level classification.

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(3) Using the triangularization method, the ranking of the indicators is calculated as: I1 I3 I2 , where the symbol “” indicates preference. (4) Calculate the relative weights of the indicators. First, based on the ranking of the indicators, the matrix Rc is converted to a triangulated crisp matrix Rc T :

RTc =

I1 I3 I2

 I1 0.5 0.28 0.1

I3 0.72 0.5 0.16

I2  0.9 0.64 0.5

(23)

449

(24)

∀i ∈ {1, 2, 3}

where wi refers to the weight of the ith indicator, and λ indicates the granular information about the pair wise comparison and is associated with the cognitive ability of the experts. According to Deng et al. (2014a), a feasible scheme of λ is:



447 448

The information is with high credibility The information is with mudium credibility The information is with low credibility

450 451 452 453

(25)

where λ = [h] = 1. The weight of I1 , I2 , I3 is calculated as 0.527, 0.167 and 0.306, respectively. Similarly, the weights of the indicators are derived with respect to other criteria, and the weights of the criteria are derived with respect to the overall objective, as shown in Table 3. Eventually, by integrating the weights at each level, the weights of the indicators for the overall evaluation problem can be calculated, as shown in the right column of Table 3.

454 455 456 457 458 459 460

5.5. Evaluation results

461

Based on the weights of the indicators determined in Section 5.4 and the fuzzy membership matrix established in Section 5.3, the efficiency index (EI) of these three grouting units is calculated

462

Fig. 11. Comparison of grouting efficiency evaluated by the original and proposed method. (a) and (b) show contour maps and 3D surface maps of the grouting efficiency for the selected study region evaluated by the original method; (c) and (d) show contour maps and 3D surface maps of the grouting efficiency for the selected study region, as evaluated by the proposed method.

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Fig. 12. Comparison of the permeability (LU), RQD, FFR and grouting efficiency assessment evaluated by the original and proposed method for the selected typical boreholes. The left and right map represent the typical borehole I and II, respectively, which are selected from units 2 and 3 (Fig. 9), respectively. EI-O and EI-P represent the efficiency index evaluated by the original and proposed method, respectively. Table 3 Weights of the indicators with respect to the overall objective. Criteria

Indicators I1 (LU) I2 (RQD) I3 (FFR)

C1 (Permeability) 0.55

0.527 0.167 0.306

C2 (Tightness) 0.45

0.467 0.167 0.366

Weight

0.5 0.167 0.333

Table 4 Results obtained from the proposed method. Grout unit

Efficiency index (EI)

Efficiency level

Ranking

Unit 1 Unit 2 Unit 3

3.29 3.33 3.51

III (ordinary) III (ordinary) III (ordinary)

3 2 1

472

following the detailed procedures presented in Section 4.3; the results are shown in Table 4. The results indicate that the grouting efficiencies of these three units are not favorable because the efficiency level of these three units is only III (i.e., ordinary). Although their efficiency levels are the same, their EI values are different. It is significant that the grouting efficiency of unit 2 is better than that of unit 1, but worse than that of unit 3; thus, we can rank them according to their EI.

473

6. Discussion

474

To illustrate the superiority of the method presented in this paper, we compare it with a traditional single-factor evaluation method.

465 466 467 468 469 470 471

475

The water pressure test (WPT) is one of the most widely used singlefactor evaluation methods for curtain grouting efficiency evaluation. We consider WPT and the method presented in this paper to be the “original” method and the “proposed” method, respectively. The grouting efficiency of each grouting interval is thus evaluated by using these two methods. Consequently, the evaluation results are illustrated in the contour maps and 3D surface maps shown in Fig. 11. The evaluation results indicate that the EI of most parts of the selected study region obtained from the original method is equal to or more than 4 (Fig. 11a); actually, the absolute majority is equal to 4 (Fig. 11b). However, the ratio of the EI obtained from the proposed method that is equal to or greater than 4 is barely 30% (Fig. 11c). This indicates that the grouting efficiency evaluated by the original method seems much better than that by the proposed method, and the proposed method seems to decrease the efficiency level. This statement makes sense in some situations, such as zone Y (marked in pink in Fig. 11), where the EI of this zone obtained from the original method is greater than 3 (i.e., level III) but is less than 2.6 using the proposed method (i.e., level II). However, in another situation, the statement that the proposed method decreases the efficiency level is no longer accurate. For example, the EI of zone X (marked in green in Fig. 11) is between 3.8 and 4.2 when using the original method but increases to more than 4.6 when using the proposed method; the efficiency level increases. Therefore, this proposed method does not decrease or increase the efficiency level from the original method but does improve the grouting efficiency evaluation, thus creating a more scientific and rational curtain grouting efficiency assessment because the proposed method has combined RQD and FFR with LU. The superiority of the proposed method can also be illustrated by a comparative analysis of the permeability (LU), RQD, FFR and

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Fig. 13. Core specimen images of the selected typical boreholes. (a) shows the core specimen image of the typical borehole I from depths of 70–86 m; (b) shows the typical borehole I from depths of 104–117 m; (c) shows the typical borehole II from depths of 100–116 m.

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EI obtained from the original method and the proposed method for typical boreholes (Fig. 12). The left map of the figure shows the typical borehole I (named ZC6-SM-JSJC-1) selected from unit 2 (Fig. 9) This borehole was drilled from an elevation of 212 m to a depth of 127 m; this measurement does not include the concrete cover. The right map of the figure shows the typical borehole II (named ZC7SM-JSJC-1) selected from unit 3. This borehole was drilled from an elevation of 220 m to a depth of 128 m; this measurement does not include the concrete cover. The RQD and FFR values are presented in ascending order from left to right on the primary (lower) horizontal axis in Fig. 12. The LU, EI-O (the EI obtained from the original method) and EI-P (the EI obtained from the proposed method) are shown in ascending order from left to right on the secondary horizontal (upper) axis in the figure. According to the classification standard shown in Table 1, the permeability of each interval of the typical borehole I is less than 1 Lu, which results in “fine” efficiency level, i.e., the EI-O is equal to 4, as shown in the left of Fig. 12. However, the EI-P of these intervals varies greatly with RQD and FFR. Particularly at depths from 70 m to 127 m (i.e., the shadowed area in the left map of Fig. 12), the efficiency levels of these intervals from depths of 70 m to 90 m are upper level (EI-P > 4). Because the FFR of these intervals is 100% and the RQD is greater

than 80, it is evident that the rock of this area is relatively integrated and the grout has spread into the fractures and voids. This can be proven by examining the core specimen (Fig. 13a). Thus, a comparatively tight and low permeability hydraulic barrier is formed. Conversely, the efficiency levels of these intervals from depths of 100 m to 127 m are lower level (EI-P < 3). This can be explained by the low FFR and RQD values, as shown in Fig. 13b. It is observed that the integrity of the rocks is not sufficient, and many fractures can be observed without grout; this indicates that the grout has not spread into these fractures. Therefore, these unfilled fractures will likely become leakage paths that will endanger the dam foundation. While the LU is very high, and the FFR and RQD are relatively low, the EI obtained from both the original and proposed methods will be very low. For example, at depths from 90 m to 128 m for the typical borehole II, as shown in the shadowed area of the right map of Fig. 12, the core specimen shows a poor rock condition and inefficient grouting (Fig. 13c). Vagueness and incompleteness exist in the preference information when calculating the weights of indicators using the D-AHP method in the proposed method, whereas the original method is based on complete and certain assessment information. To compare the original method and proposed method and rank the grouting efficiency of the three units of the selected study region, the

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G. Fan et al. / Expert Systems With Applications xxx (2015) xxx–xxx Table 5 Comparison of results obtained from the original method and proposed method. Grout unit

Unit 1

Unit 2

Unit 3

Original method

Proposed method

EI-O

EL

Ranking

4.01

II (fine)

1

3.97

3.95

III (ordinary)

III (ordinary)

EI-P

EL

Ranking

2

3

Notes: The EL represents efficiency level.

581

7. Conclusion

586

Curtain grouting can effectively reduce hydraulic conductivity and improve the tightness of the ground under a dam foundation. However, grouting technology is currently described as more of an art than a science, even though evaluating grouting efficiency is important and necessary. Existing methods cannot simultaneously evaluate the permeability and tightness of the curtain; it is also difficult to assess the condition of fractures filled by grout, leading to evaluation results that are not rational. To evaluate curtain grouting efficiency from the perspectives of both permeability and tightness, LU, RQD, and FFR are proposed as key indicators; the derivations of these parameters are presented in this paper. Considering that the curtain grouting efficiency assessment is a fuzzy problem, this paper proposes a hybrid fuzzy evaluation method for curtain grouting efficiency assessment based on a fuzzy comprehensive evaluation method and a D-AHP method. The weights of the indicators are determined by the D-AHP method. This proposed method is successfully applied in the curtain grouting efficiency assessment of a hydropower project in China by comparing its results with those of an existing typical single-factor evaluation method. The findings of this study can be summarized in the following conclusions:

587

Fig. 14. Comparison of evaluation results for the selected grout units.

•

551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576

efficiency indexes (EI) of unit 1, unit 2 and unit 3 are calculated using the original method as 4.01, 3.97 and 3.95, respectively. The comparison of results obtained from these two methods is illustrated in Table 5 and Fig. 14. It is obvious that the efficiency levels of the three units have been maintained in the proposed method except for that of unit 1. In other words, the same efficiency level can be obtained from both the original and proposed methods. This means that even though the expert assessment information may be incomplete and fuzzy when determining the weights of the evaluation indicators, the evaluation results based on the D-AHP method are credible and acceptable. More importantly, with the help of the D-AHP method, uncertain information containing subjective judgments can be expressed effectively. In addition, the proposed method provides more detailed information than the original method. For example, even though the efficiency levels obtained from both the original method and the proposed method are almost the same, the EI-P of the three units is lower than the EI-O; this is because the original method only considers one indicator (i.e., permeability), which indicates that only if the permeability is low will the grouting efficiency be regarded as high. However, in the proposed method, permeability is not the only factor that affects grouting efficiency; RQD and FFR are also proposed as evaluation indicators. Thus, low permeability does not guarantee high grouting efficiency. When rock integrity and FFR are low, unfilled fractures can still become leakage paths even though permeability is low.

13

In the original method, the efficiency assessment of unit 2 is worse than that of unit 1 but better than that of unit 3. However, the ranking of the efficiency assessment is reversed in the proposed method because the permeability of unit 1 is lower than that of the other two units; the integrity of the rocks and the fracture filled ratio by grout is much lower than in the other two units. After the fuzzy comprehensive evaluation, the efficiency ranking of the three units is reversed, which indicates that RQD and FFR can also influence the final evaluation results.

•

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•

•

Because it considers perspectives of both permeability and tightness, the proposed hybrid fuzzy evaluation method overcomes the deficiencies of the existing method, which primarily focuses on the effect of permeability on curtain grouting efficiency but ignores the effects of tightness. In considering various types of uncertainties, including fuzziness, incompleteness, and imprecision, the D-AHP method is shown to more effectively represent and address uncertain information and make weighting more scientific. This study reveals that low permeability does not guarantee high grouting efficiency. When rock integrity and FFR are low, unfilled fractures can still become leakage paths even though permeability is low. By proposing LU, RQD, and FFR as evaluation indicators, the proposed method integrates several types of grouting efficiency evaluation technologies and ensures a more comprehensive evaluation result. Thus, the proposed method can effectively expose inefficient grouting regions, which can help grouting engineers more scientifically judge grouting efficiency and design and remedy those regions.

In future research, more in situ tests or other available methods, such as acoustic velocity tests, can be integrated into the proposed hybrid fuzzy evaluation method for more efficient grouting assessment. In addition, because the proposed method can handle uncertain problems more effectively, it should be applied to other fuzzy and uncertain problems such as dam safety assessments. Moreover, the proposed method could be integrated into a real-time grouting monitoring and analysis system to more efficiently control and evaluate grouting quality. Finally, the theory of cement-grout penetration needs more insightful research because a deeper understanding of the mechanisms of grout spreading will provide a better method for evaluating grouting efficiency in real time.

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Acknowledgments

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This research is supported by Innovative Research Groups of the National Natural Science Foundation of China (grant no. 51321065), the National Basic Research Program of China (973 Program) (grant no. 2013CB035904) and the Natural Science Foundation of China (grant nos. 51439005 and 51339003).

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