SEDIMENT TRANSPORT CALCULATION CONSIDERING COHESIVE EFFECTS AND ITS APPLICATION TO WAVE-INDUCED TOPOGRAPHIC CHANGE

Transcription

1 Proeedings of the 7 th International Conferene on Asian and Paifi Coasts (APAC 03) Bali, Indonesia, September 4-6, 03 SEDIMENT TRANSPORT CALCULATION CONSIDERING COHESIVE EFFECTS AND ITS APPLICATION TO WAVE-INDUCED TOPOGRAPHIC CHANGE Y.H. Cho, T. Nakamura, N. Mizutani 3 and K.H. Lee 4 ABSTRACT: A sediment transport alulation onsidering ohesive fore is proposed to deal with the transport phenomena of ohesive sediment. In the proposed alulation, eah sand partile is assumed to be surrounded by a thin layer of lay. The ritial Shields parameter and bed-load sediment transport rate are modified to inlude the ohesive fore ating on the sand partile. The proposed alulation is inorporated into a two-way oupled fluid-struturesediment interation model, and applied to wave-indued topographi hange of an artifiial shallow. Numerial results show that an inrease in the ontent ratio of the lay, ohesive resistane fore per unit surfae area and water ontent ause inreases in the ritial Shields parameter and dereases in the bed-load sediment transport rate, reduing the topographi hange of a shallow without hanging its trend. This suggests that mixing lay in the pores of the sand partiles an redue the topographi hange of shallows. Keywords: Cohesive fore, fine sediment, sediment transport, shallow, numerial analysis INTRODUCTION Dredging is a periodially required ativity in oastal and fluvial regions with the purpose of maintaining sea routes, environment protetion and reservoir apaity. River hannel and harbor dredging produes millions of tons of soil annually. Managing these enormous amounts of dredged soil remains an unsolved problem. One of the proposed methods of handling dredged soil is to reate an artifiial shallow in a nearby bay. From an environment perspetive, it was onfirmed that there are benefiial effets suh as inreasing bottom life and enhaning water purifiation potential (Kazama et al. 006). In ontrast, the stability of an artifiial shallow omposed of dredged soil against inident waves is of onern beause of the low weight of fine sediments in the dredged soil. Nakamura et al. (0a, 0b) onduted experimental and numerial researh on the harateristis of the topographi hange of a shallow omprised of fine sand. However, these studies had limitations beause they only onsidered non-ohesive fine sand, whereas atual dredged soil ontains fine sediments with ohesive fores. Thus, there is little understanding of the harateristi of ohesive fine sediment transport and the topographi hange of an artifiial shallow made up of dredged soil. Most researh on sediment transport phenomena has been based on non-ohesive sediments and there is little researh that aounts for the effets of ohesive fores. Ashida et al. (98) proposed a formula onerning the non-dimensional ritial shear stress and the nonequilibrium sediment transport of the sand partiles. This formula onsidered the effets of the fine sediments (hereinafter alled lay) with ohesive fore existing in the pore spae of the sand partiles and onfirmed the validity of the formula through omparison with experimental tests. However, there was no attempt to apply the formula to wave fields, whih is a more energeti environment, to analyze the harateristis of sediment transport overing the ohesive effets in mixed soil oexisting with sand and lay suh as dredged soil or the real sea bottom. Consequently, the harateristis of sediment transport in mixed soil are not fully understood, partiularly in a oastal field of study. In this study, a sediment transport alulation onsidering the ohesive effets indued by lay in the pores of sand partiles is proposed and the basi harateristis of the alulation are investigated taking into aount the orrelation between ohesive intensity and the bed-load sediment transport rate. In addition, the alulation is inorporated into a three-dimensional oupled fluid-struture-sediment interation model (FSSM) developed by Nakamura et al (0), whih an analyze the interation between waves and topographi hange, and applied to the ondition of hydrauli experiments on the topographi hange of a shallow omposed of fine sand (Nakamura et al. 0a). Furthermore, the preditive apability of the modified model is verified with experimental results and a Department of Civil Engineering, Nagoya University, Furo-ho, Chikusa-ku, Nagoya , JAPAN Institute for Advaned Researh, Nagoya University, Furo-ho, Chikusa-ku, Nagoya , JAPAN 3 Department of Civil Engineering, Nagoya University, Furo-ho, Chikusa-ku, Nagoya , JAPAN 4 Maritime & Oean Engineering Researh Institute, KIOST, Yuseong-daero, Yuseong-gu, Daejeon , KOREA 338

2 Sediment Transport Calulation Considering Cohesive Effets and Its Appliation to Wave-Indued Topographi Change sensitivity analysis is arried out in terms of the effets of ohesive fore on the topographi hange of the shallow, whih omprises a soil mixture based on the assumption that lay exists in the pores of the sand partiles. SEDIMENT TRANSPORT CALCULATION CONSIDERING COHESIVE EFFECTS Sediment transport alulation onsidering the effets indued by lay ontained in the pores of the sand partile is proposed following Ashida et al. (98). Speifially, the ritial Shields parameter * and bedload sediment transport rate q i are modified to onsider the ohesive effets of the lay. Ashida et al. (98) onsidered the soil mixture ombined with spherial sand partiles with uniform partile diameter d 50 and lay filling in the pores of the sand partiles as bottom sediment. It was assumed that the frition between the sand partiles did not hange even for the ondition without lay. That is, even though the lay is eliminated, sand partiles an maintain a stati stable state. Thus, we now have the lay-indued ohesive fore F C expressed by: F k d f, () C ' 50 in whih k ' d 50 is the ohesive area of the lay ontributing to the ohesive resistane and f is the ohesive resistane fore of the lay per unit surfae area. Here, assuming the ohesive area per ontat point between sand partiles is a, it an be obtained approximately from a simple geometri relationship as in Ashida et al. (98). When one sand partile is supported by the number of ontat points, n, this an be given by k ' d 50 = an, and finally, Eq. () an be expressed as: /3 d ( ) 50 pf s FC nf, () 4 m p f s FD CD wd50vr CFDv, (4) r 8 FL CLwd50vr klcfdv, (5) r 8 in whih g is the gravitational aeleration; is the shading oeffiient; C D and C L are the turbulent drag and lift oeffiients; k L is the ratio between C D and C L (= C L /C D ); v r is the relative flow veloity ating on the partile for the mean transport veloity v b of a sediment partile in bed-load motion; and C FD = C D w d 50 /8. In the following, * and q i are derived by inorporating the effets of F C. Critial Shields Parameter In the ritial state, the fore ating on a sand partile exists as shown in Fig.. Here, v f is the frition veloity; C vf is the oeffiient of the frition veloity; is the angle between the flow veloity C vf v f and the diretion of the steepest bed slope; is the angle between the flow veloity C vf v f and the sediment transport veloity v b ; and is the angle between the relative flow veloity v r and the sediment transport veloity v b. The values F DC, F LC, v f, v b, v r,, and represent those of F D, F L, v f, v b, v r,, and in the ritial state. In addition, is the angle of the steepest bed slope and s is the stati frition oeffiient of the sand partile. In this ase, v b = 0, whih means that v r = C vf v f and =, beause it is under the ritial state. The formulations of and v r are given as follows from the fore balane in the diretions the same as and perpendiular to v b : { s os sin os( )} W F vr, (6) (os k ) C s L FD W sin sin artan. (7) CFDvr W sin os in whih m is the porosity; p f is the ontent ratio of the lay; is the water ontent; and s is the speifi weight of the sand partile (= s / w, where s is the sand partile density and w is the water density). Additionally, there are the submerged weight of the partile, W, and the drag and lift fores, F D and F L, whih are triggered by the bottom veloity. The fores ating on the sand partiles, apart from the ohesive fore F C expressed in Eq. (), are expressed by: 3 W ( s w ) gd, (3) 50 6 Fig. Fore balane on a sediment partile on a sloping bed in the ritial state 339

3 Y.H. Cho, et al. In Eq. (7), and have the same sign and. Here, the unknown parameters, v r and, are determined by iterative alulations. Defining *0 as a ritial Shields parameter in the horizontal plane ( = 0) without onsidering the ohesive fore (F C = 0), * an be written by the following equation: v /{( s ) gd } v /{ C ( s ) gd } v * f 50 r vf 50 r * 0 vf0 /{( s ) gd50} vr0 /{ Cvf ( s ) gd50} vr0, (8) in whih v r0 and v f0 are v r and v f in the horizontal plane ( = 0) without the ohesive fore (F C = 0), and v r0 is obtained as follows from substituting F C = = = = 0 into Eq. (6): v r0 sw. (9) ( k ) C s L FD Furthermore, for the ross-setional two-dimensional phenomenon, the following equation is obtained beause of = = = 0: * ( s os sin ) W F. (0) W * 0 s In addition, the influene of the ohesive fore is onsidered to be a pikup funtion of suspended sediment through the value of the ritial Shields parameter *. Bed-Load Sediment Transport Rate The bed-load sediment transport rate q i per unit width and unit time is defined as follows (Engelund and Fredsøe 976): qi d50 pefv, () bi 6 where p EF is the perentage of sediment partiles in bedload motion in the surfae layer of the bed, whih is expressed as: p EF 0 if * * 6, () ( * * ) if * * d Fig. Fore balane on a moving sediment partile on a sloping bed Figure shows the agitating and stabilizing fores ating on a sand partile in the bed-load motion. The equation of motion in the same diretion as the sand partile movement (i.e., the diretion of v b ) is given as: FD os Wsin os( ). (4) ( W os F ) F d L In the same way, the equation of motion in the diretion perpendiular to the sand partile movement is: F sin Wsin sin( ). (5) D In addition, the simple geometri relationship between v b, C vf v f and v r gives: v sin C v sin, (6) r vf f v os v C v os. (7) r b vf f Here, there are four unknown variables (i.e., v b, v r, and ) and four equations, and v bi is omputed through iterative alulations. Thus, the bed-load sediment transport rate q i is given by Eqs. () and () together with *, whih is obtained in the previous setion. Furthermore, for the ross-setional two-dimensional phenomenon, the equation is summarized as follows from the relationship of = = = 0: vb Cvf v f v, (8) r ( d os sin ) W F vr CFD ( drl ) /. (9) in whih d is the dynami frition oeffiient ( d s ) and * is the Shields parameter, whih is defined as: v f * ( s ) gd 50. (3) 340

4 Sediment Transport Calulation Considering Cohesive Effets and Its Appliation to Wave-Indued Topographi Change (a) (b) () Fig. 3 Non-dimensional bed-load sediment transport rate q* versus the Shields parameter *: (a) the harateristi of q* aording to ; (b) the harateristi of q* aording to pf ; and () the harateristi of q* aording to f CHARACTERISTICS OF SEDIMENT TRANSPORT CALCULATION CONSIDERING COHESIVE EFFECTS In this setion, the fundamental harateristis of the proposed sediment transport alulation are investigated based on Eqs. (0), (8) and (9) for a ross-setional two-dimensional ondition, for whih there is no need to perform any iterative alulation. Speifially, sand partiles with d 50 = 0. mm, s = kg/m 3 and *0 = 0.03 were assumed to be on the horizontal plane ( = 0. The three ases for the water ontent (0., 0.5 and 0.8), two ases for the ontent ratio of the lay p f (0.05 and 0.0) and three ases for the ohesive fore per unit surfae area f (0., 0.5 and.0 N/m ) were then used as the main variables. The alulation without the ohesive fore effet (f = 0.0 N/m ) was also arried out for omparative analysis. Other parameters were g = 9.80 m/s, w = kg/m 3, C D = 0.45, C vf = 0.0, k L = 0.85, = 0.4, s = 0.63 and d = 0.5, whih are the same as used in Nakamura et al. (0). Figure 3 shows the relationship of the nondimensional bed-load sediment transport rate q * versus the Shields parameter *. As shown in Fig. 3(a), an inrease in auses an inrease in the ritial Shields parameter *. In addition, it is observed that q * tends to derease with an inrease in when * is the same. However, the effet of on q * varies aording to the value of f beause the deline in q * inreases with an inrease in f. A similar phenomenon is also observed in Fig. 3(b). Although a derease in q * aording to an inrease in p f is small for f = 0. N/m, it is found that a deline in q * for f =.0 N/m is larger than that for f = 0. N/m (see Fig. 3(b)). In Fig. 3(), the value of * inreases with an inrease in f, and it is also found that q * dereases with an inrease in f for the same *. Furthermore, the ase of *0 = 0.046, whih is the solid green line without ohesive effets (f = 0.0 N/m ), and the ase of f =.0 N/m and *0 = 0.03, whih is the blue two-dot hain line, are ompared based on the same ritial Shields number * beause * is about when f =.0 N/m and *0 = A different trend in q * is observed in spite of the same ritial Shields parameter, that is, q * for f =.0 N/m is smaller than that for non-ohesive effet onsideration. This result emphasizes the importane of appropriate modeling taking into aount the ohesive effets aused by the lay in sediment transport alulation beause onsidering only the inreasing effets of *0 aused by the ohesive fore is insuffiient. THREE-DIMENSIONAL COUPLED FLUID- STRUCTURE-SEDIMENT INTERACTION MODEL The three-dimensional oupled fluid-struturesediment interation model developed by Nakamura et al. (0) is briefly desribed in this setion. The model is omposed of a main solver and three modules. The main solver is a large-eddy simulation (LES) based on a ontinuity equation and a Navier-Stokes equation for analyzing the motion of an inompressible visous airwater two-phase fluid inluding the pore fluid inside the porous media taking into aount the movement of the movable struture and topographi hange. The three modules are a volume-of-fluid (VOF) module based on the multi-interfae advetion and reonstrution solver (MARS) to trak the gas-liquid interfae, an immersedboundary (IB) module based on the body-fore type of IB method dealing with the movable struture, and the sediment transport module to analyze the onentration of suspended sediment transport and the topographi hange assoiated with bed-load and suspended sediment 34

5 Y.H. Cho, et al. transport. The three modules are inorporated into the main solver with a two-way oupling tehnique to take into aount the fluid-struture-sediment interation. Here, the proposed sediment transport alulation was applied into the sediment transport module to take into aount the ohesive effets aused by lay filling in the pores of the sand partiles. It should be noted that, in this study, the IB module was not inluded beause motion of the struture was not dealt with. APPLICATION OF THE MODIFIED MODEL CONSIDERING COHESIVE EFFECTS The modified model was applied to hydrauli experiments on the topographi hange of an artifiial shallow onsisting of fine sand (Nakamura et al. 0a) and its preditive apability was verified by omparison with experimental results. Subsequently, a numerial simulation was arried out on the supposed ondition that the pores of the sand partiles ontained the lay with the ohesive fore, and the sensitivity analysis of the modified model was onduted fousing on the relationship between the variation of the parameters (f, p f, ) and topographi hange of the shallow. Verifiation of The Modified Model Numerial ondition Figure 4 shows a shemati of a omputational domain. As shown in Fig. 4, a shallow (height: 0.0 m; rown width: 00.0 m; slope gradient: /0) omposed of fine sand with a median grain size, d 50 = 0. mm, was installed based on the initial topography measured in the hydrauli experiments. In addition, a wave generating soure/sink was plaed 40.0 m away from the beginning of the shallow, and damping zones were installed in the offshore side of the wave generating soure and the onshore side of the vertial wall. Here, the length of the damping zones was about twie as long as the inident wavelength to redue the wave refletion from the boundaries. Regular waves were generated with a wave height H i = 6.5 m, a wave period T =.0 s, and a still water depth h =.50 m. The entire domain exept for the damping zones was disretized using orthogonal staggered ells with a uniform size of Δx =.0 m and Δz = 0.5 m, whereas non-uniform ells widening gradually were used in the damping zones for effiient numerial simulation. The boundary onditions were the same as in Nakamura et al. (0b). The parameters were the porosity of fine sand m = 0.4, the density s = kg/m 3, the ritial Shields parameter on the horizontal plane *0 = 0.03, the submerged angle of repose r = 30.0 and the ontent ratio of the lay p f = 0.0 beause the shallow was omposed of only fine sand without lay. Other parameters were referred to in the previous setion and Nakamura et al. (0). Preditive apability of the modified model Although the topographi hanges have been measured in the hydrauli experiments at, 0, 30, 60 and 300 min after the waves were generated, only the result of the topographi hange with t = min was adopted in this study to avoid the heavy omputational load and impratial work from long simulation times of over 0 min. The omparison between experimental and numerial results for free surfae elevations observed from W to W6 (see Fig. 4) is presented in Fig. 5, in whih t denotes time. As shown in Fig. 5, although a slight phase shift was observed for W loated at the offshore side of the shallow and W3 at the middle of the seaward slope of the shallow, reasonable results were obtained. A derease in the wave height, whih ourred subsequent to wave breaking, is shown from the upper slope (W4) to the rown of a shallow (W6), and the amplitude has Fig. 4 Computational domain for the topographi hange of the shallow Fig. 5 Comparison between experimental (irle) and numerial (dotted line) results for free surfae elevation 34

6 Sediment Transport Calulation Considering Cohesive Effets and Its Appliation to Wave-Indued Topographi Change Fig. 6 Comparison of the amount of topographi hange z between experimental (irle) and numerial results (solid line) at t = 60 s (a) (b) Fig. 7 Topographi hange and the suspended sediment onentration at t = 40 s for the different still water depths: (a) h =.50 m; (b) h =.75 m reasonably good agreement. However, espeially at W5, there is the tendeny for the mean water depth to vary as time passes. This is beause, in the hydrauli experiments, the 0.40 m wide wave flume was designed by dividing the. m wide wave tank with a partition board, and the shallow mentioned above was installed on the 0.40 m wide wave flume. In this ase, irulation flow an be generated by the mean water depth variation in the wave tank, whih affeted the experimental results beause the partition board did not extend adequately to the wave absorbing beah at the end of the wave tank. Thus, it seemed that a disrepany between the numerial and experimental results for W4 to W6, partiularly in a low water depth area, was evident beause a ross-setional two-dimensional simulation used in this study has the restrition of reproduing ompletely the phenomena that emerged in the three dimensional experiments. This dimensional limitation is also refleted in the numerial results for the topographi hange. Figure 6 shows the experimental and numerial results of the topographi hange Δz of the shallow at t = 60 s, where a positive value of Δz indiated deposition. It shows that when the water depth of.50 m, the same as the experimental ondition, was adopted for the numerial simulation (blak solid line), substantial erosion took plae on the upper slope to the middle of the rown, whih was different from the experimental results. Note that similar results were also obtained in a previous study (Nakamura et al. 0b). From this result, it an be onsidered that the trend for the topographi hange altered beause of the mean water variation, espeially the phenomenon of an inrease in mean water depth in the experiment at W5. Consequently, the numerial simulation was onduted by raising the water depth slightly to enhane the preditive apability of the model in the hydrauli experiments for the ross-setional two-dimensional alulation. As a result, when the water depth was raised to.75 m, 0.5 m deeper than the experiments, the preditive auray of at W4 and W5 was improved as shown in Fig. 5 and overall the trend for the free surfae elevations orresponded to that for the experimental results. Moreover, the orrelation with the experimental results was improved in terms of the topographi hanges beause there was no signifiant topographi hange, espeially for the erosion that ourred in the slope to the rown (see Fig. 6) unlike the ase for h =.50 m. These results for the different topographi hange patterns were also obvious in the suspended sediment onentration results (see Fig. 7). The red ontour indiates a high onentration of suspended sediment and the blue ontour means the reverse. From Fig. 7(b), some suspended sediment was observed in the upper slope and at the end of the rown, the same as shown in Fig. 7(a). However, when h =.50 m (Fig. 7(a)), high suspended sediment onentration was observed on the rown of the shallow as shown in Fig. 6, whereas there was little suspended sediment on the rown with h =.75 m (Fig. 7(b)). Consequently, it was established that this model was appliable to the topographi hange phenomena of a shallow, sine the preditive apability of the model was verified by raising the water depth to.75 m. Therefore, in the following, the sensitivity analysis for the main parameters (f, and p f ) ontributing to ohesive fore was arried out with h =.75 m to investigate the effets of the ohesive fore on the topographi hanges of the shallow. Effets of The Cohesive Fore on The Topographi Changes of a Shallow It is assumed that lay filled the pores of the sand partiles forming the shallow. The ohesive resistane fore per unit surfae area f, the water ontent and the ontent ratio of the lay p f are noted as the main parameters whih lead to ohesive fore, and the effets of these parameters on the topographi hange of the shallow were investigated. The effets of f, and p f on the topographi hange z at t = 60 s are shown in Fig. 8. In the figure, only z 343

7 Y.H. Cho, et al. the pores of the sand partiles with lay, the waveindued topographi hange of the shallow an be redued without hanging the topographi hange pattern. (a) (b) () Fig. 8 Differene in the amount of topographi hange z aused by the ohesive fore at t = 60 s: (a) the effets of the ohesive fore per unit surfae area ( = 0.5, pf = 0.5); (b) the effets of the water ontent (pf = 0.5, f = 0. N/m ); () the effets of the ontent ratio of the lay ( = 0.5, f = 0. N/m ) in the range of x = m of the whole domain was expressed beause the topographi hange mainly ourred on the slope of the shallow as shown in Fig. 6. It was onfirmed from Fig. 8(a) that the amount of z tends to derease with an inrease in f. Speifially, when f was inreased from 0. N/m to.0 N/m, it was observed that little deposition emerged on the loation from the upper slope to the rown and little erosion appeared on the middle of the slope. This was beause, as shown in Fig. 3(), the ritial Shields parameter inreased and the bed-load sediment transport rate dereased with an inrease in f, and finally this sequene had the effet of dereasing z. In addition, although z dereased with an inrease in the distintion was very small as shown in Fig. 8(b). This was beause the small value of f, 0. N/m, was adopted on aount of the little topographi hange that ourred for f =.0 N/m, and the effets of on the ritial Shields parameter and the bed-load sediment transport rate were very small when f = 0. N/m as shown in Fig. 3(a). A similar trend to that shown in Fig. 8(b) is also observed in Fig. 8(). Furthermore, the variation of the parameters f, and p f seems to have little effet on the trend for the topographi hange of the shallow. Therefore, by filling CONCLUSIONS In this study, the sediment transport alulation onsidering the ohesive fore generated by lay filled sand partiles was proposed. The numerial simulation model inorporating the proposed alulation was verified by applying it to the hydrauli experiments on the topographi hange of a shallow omposed of fine sand, and the basi harateristis of the model were examined. In addition, the validation of the model and the sensitivity analysis of the ohesive effets on the topographi hange of the shallow were onduted. As a result, the onlusion an be summarized as follows:. Through the relationship between the nondimensional bed-load sediment transport rate and Shields parameter, an inrease in the ohesive resistane fore per unit surfae area f, the water ontent and the ontent ratio of the lay p f result in an inrease in the ritial Shields parameter and a derease in the bed-load sediment transport rate. Furthermore, the existene of lay in the pores of the sand partiles indiates the importane of proper modeling taking into aount the ohesive effets. This is beause the results presented different patterns for the bed-load sediment transport rate for ases onsidering ohesive and non-ohesive effets, in spite of having the same ritial Shields number.. By arrying out a sensitivity analysis on the effets of the three parameters, f, and p f, on the topographi hange of the shallow, it was onfirmed that the topographi hange was dereased with an inrease in f and p f,. In addition, the results show that taking into aount the ohesive effets had little influene on the topographi hange pattern. Consequently, the possibility of reduing topographi hange by ontaining the lay in the sand partiles without any hange in its trend is suggested. From these results, the omputational apability of this proposed model for analyzing sediment transport problems involving various onditions of mixed sediments oexisting with sand and lay was onfirmed. However, it is desirable to validate the proposed alulations with hydrauli experiments. 344

8 Sediment Transport Calulation Considering Cohesive Effets and Its Appliation to Wave-Indued Topographi Change ACKNOWLEDGEMENT The authors are grateful for the finanial support of the Steel Foundation for Environmental Protetion Tehnology, Japan (SEPT). REFERENCES Ashida K., Egashira, S. and Kamoto, M. (98). Study on the erosion and variation of mountain streams - on the erosion and transportation of sand-lay mixtures. Disaster Prevention Researh Institute Annuals B, 5(B-): (in Japanese). Engelund, G. and Fredsøe, J. (976). A sediment transport model for straight alluvial hannels. Nordi Hydrology, 7: Kazama, T., Nakata, K., Tanabe, Y., Hasegawa, M., Oshima, I. and Nagakura, T. (006). Tidal flats and shallows reated through the utilization of dredged sand and its effetiveness on oastal sea environment and its problem. Pro., JSCE in the Oean, : (in Japanese). Nakamura T., Yim, S.C. and Mizutani, N. (0). Numerial simulation on loal souring around bottom-mounted movable short ylinder. Pro., Coastal Strutures 0, ASCE, C4-087, p. Nakamura, T., Nezasa, Y. and Mizutani, N. (0a). Experimental study on evolution of surfae profile of artifiial shallow and temporal hange in pore-water pressure inside surfae layer of shallow. J. JSCE Series B (Coastal Eng.), 68: I_54-I_545 (in Japanese). Nakamura, T., Ishihara, R. and Mizutani, N. (0b). Numerial analysis of topographi hange of artifiial shallow onsidering pore-water pressure in surfae layer of shallow. J. JSCE Series B (Coastal Eng.), 68: I_6-I_65 (in Japanese). 345