Development of new equation to estimate the maximum soil depth by using the safety factor

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1 Lndslides nd Engineered Slopes. Experiene, Theory nd Prtie Avers et l. (Eds) 2016 Assoiione Geoteni Itlin, Rome, Itly, ISBN Development of ne eqution to estimte the mximum soil depth by using the sfety ftor G.P. Mihel Snt Ctrin Stte University, Ibirm, Bril M. Kobiym Federl University of Rio Grnde do Sul, Porto Alegre, Bril ABSTRACT: Soil depth is n importnt prmeter hih is hrd to be mesured nd estimted in hydrogeomorphi nd pedologil studies. Furthermore, the soil depth is one of the most unknon vribles in the hillslope system. Therefore, the objetive of the present study s to develop ne eqution to estimte the mximum soil depth by using the ftor of sfety. For this development, uniform hydrologil model s ombined to n infinite slope model. The ne eqution is lled MEMPS (Modelo de Estimtiv d Máxim Profundidde do Solo). Differently from other equtions to lulte mximum soil depth, MEMPS does not require the ssumptions of omplete sturtion or totl bsene of ter in the soil. Thus, MEMPS represents ondition loser to the relity hen ompred to other equtions. Moreover, due to the inorportion of hydrogeomorphi prmeters in its formultion, MEMPS quires fitness to reflet the influene of hydrogeomorphi proesses on the soil depth estimtion. 1 INTRODUCTION Soil depth, defined here s the vertil distne beteen the soil surfe nd the bedrok, is n importnt prmeter hih is hrd to be mesured nd estimted in hydrogeomorphi nd pedologil studies. Furthermore, the soil depth is one of the most unknon vribles in the hillslope system (Ctni et l., 2010). Tht is hy there re vrious ttempts to estimte the soil depth. These ttempts n be lssified into to: field methods (Piegri et l., 2009, Bery, 2013) nd mthemtil methods (Sulnier et l., 1997, Segoni et l., 2012). Eh ttempt hs its on good nd bd points. Borg et l. (2002) performed sensitivity nlysis of slope stbility model nd demonstrted tht depth vrition n signifintly hnge the slope stbility results. It mens tht the suessful mpping of lndslide hrd res depends upon the orret determintion of the sptil distribution of soil depth in bsin. As the soil depth mesurement in hole bsin is quite diffiult nd time-onsuming (Dietrih et l., 1995), it is usully required to use some mthemtil formultions hih estimte the soil depth distribution. The proess-bsed methods re those tht use physilly-bsed equtions to desribe vrious mehnisms ffeting the soil formtion nd evolution. Though generting stisftory results, they require tht the physil proesses ting in the study re re ell knon nd tht input prmeters dt re urte. In this ontext, the objetive of the present study s to develop ne eqution to estimte the mximum soil depth by using the Ftor of Sfety (FS). This ne eqution resulted from the ombintion of n infinite slope stbility model ith uniform hydrologil model. The present study shos the eqution development nd omments its limittion use. 2 THEORETICAL DEVELOPMENT OF MEMPS Aording to Selby (1993), the vlue of FS n be formulted: FS s g g + ( ρ os θ ρ h os θ) tnφ ρ g osθ s 2 2 (1) here FS ftor of sfety; soil ohesion; ρ ter density; soil depth; h height of the ter olumn inside the soil lyer; ρ s soil density; g grvittionl elertion; θ slope ngle; nd f soil internl frition ngle (Figure 1). Iid (1999) exerised limit equilibrium nlysis for n infinite slope model, nd defined the ritil 1417

2 of ter in the soil olumn; nd (ii) its omplete sturtion. In ft, both onditions hrdly our. The present study, therefore, extended his theory in order to tret the more rel sitution, by ombining n infinite slope stbility model to uniform hydrologil model. Then, the present study modifies the Eqution 1 nd obtins: Figure 1. Representtion of n infinite slope model. P eight of the soil; τ sher stress; σ norml stress; u pore-pressure. (From Mihel et l., 2014). soil depth s the mximum vlue of the depth tht n settle on hillslope, by onsidering tht the FS vlue t prtiulr lotion does not reh loer vlues thn 1. In the se of ter bsene, here soil lyer is perfetly dry, the ritil depth ( 0 ) s desribed by: ρ g os θ tnθ tnφ 0 2 s ( ) (2) here 0 ritil depth for dry soil. The 0 vlue n be lulted, but it is never rehed, sine the soil depth sloly inreses over time nd periodi storms produe ertin hillslope sturtion (i.e. the ssumption of bsene of ter is not stisfied), using lndslides. Therefore, Iid (1999) proposed, for the se tht the soil is totlly sturted, nother eqution to determine the ritil depth ( 1 ): os θ g ( tnθ tnφ)+ ρ g tnφ 1 2 (3) here 1 ritil depth for the sturted soil. If the tul soil depth is smller thn 1, lndslide ill never our even ith very intensive rinfll. In this se, this lotion quires immunity sitution for ertin period. If the soil depth is greter thn 1, the lndslide n our hen the soil sturtion rehes threshold vlue. Iid (1999) demonstrted tht his equtions (Equtions 2 nd 3) hd good performne through omprison ith the vlues mesured in the field. As bove-mentioned, Iid (1999) determined the mximum soil depth, by onsidering to extreme onditions of soil moisture: (i) the totl bsene h s g ρ osθ 1 ρ θ tnφ os FS (4) The term h/ in Eqution 4 represents the sturtion degree of the hillslope soil. In stedy stte hydrologil model, hih is ommonly lled TOPOG, O Loughlin (1986) defined the etness index () ith the folloing eqution: h bt (5) here q uniform stedy stte rehrge; upstrem bsin re t one point; b ontour length; nd T soil trnsmissivity. Obviously, the vlue of is limited to 1, nd this ondition mens tht the height of the ter olumn inside the soil is equl to the soil depth. When the pplition of Eqution 5 results in vlue lrger thn one, is there overlnd flo. In Eqution 5 sturted hydruli ondutivity (Ks) is onsidered onstnt throughout the soil profile, nd then T n be lulted: T Ks osθ (6) By substituting the Eqution 6 in the Eqution 5, it is obtined: h or b Ks osθ h b Ks osθ (7) (8) As the ter blne of the soil lyer is onsidered permnent, the Eqution 7 is ble to estimte n verge behvior of the soil sturtion level over time. The mximum vlue of h is equl to. When the pplition of Eqution 8 results in h vlue lrger thn, the soil is ompletely sturted nd there is overlnd flo. Then, substituting the Eqution 7 in the Eqution 4, the folloing expression is obtined: 1418

3 q s g ρ os θ ρ osθ 1. b Ks osθ tn φ FS (9) The ssumption, tht the stbility threshold in hillslope ours hen FS is equl to 1, permits to estimte the mximum depth tht the soil n hieve ithout using instbility, only bsed on soil strength hrteristis, geomorphi hrteristis of the bsin nd hydrologil situtions tht tke ple. Hene, the Eqution 9 n be modified s follos: The present study lls the Eqution 13 Modelo de Estimtiv d Máxim Profundidde do Solo (MEMPS) hih mens model to estimte the mximum soil depth, in Portuguese. 1 ρ os θ. ρ g b Ks tnφ s osθ osθ 1 < (10) Isolting the term, the Eqution 10 beomes: ρ tnφ g osθ b Ks < osθ tnφ (11) The Eqution 11 performs n estimte of the mximum depth rehed by the soil under ertin hydrologil onditions. Then, by dividing the Eqution 11 by osθ nd treting the mximum soil depth, the vlue of the mximum soil depth ( ) n be obtined: tnφ ρ 2 g os θ b Ks osθ tnθ tnφ (12) The term in the brkets in the Eqution 12 represents the ter height of the soil lyer (h) of the Eqution 8. The mximum vlue of h must be equl or less thn the vlue of. Thus, it n be sid tht, under threshold blne ondition hen h beomes equl to, the ritil depth beomes 1 of the Eqution 3. Then, the Eqution 12 n be modified: tnφ ρ min s g s, 2 1 ρ os θ ρ b Ks osθ tnθ tnφ (13) 3 COMPARISON BETWEEN MEMPS AND IIDA (1999) The MEMPS (Eqution 13) hs intermedite hrteristis beteen to equtions (Equtions 2 nd 3) of Iid (1999) hih used n inversion of the FS eqution to limit the mximum soil depth physilly fesible on slope. The differene beteen the equtions of Iid (1999) nd the MEMPS is grphilly illustrted in Figure 2. For elborting these grphs, the lultion s done under the folloing onditions: 11.9 kp, q m/dy, 300 m², b 5 m, f 30.5, Ks 0.38 m/dy, ρ s 1800 kg/m 3, nd ρ 1000 kg/m 3. Considering tht the im of the present study is just to demonstrte the performne of Eqution 13, nd not Figure 2. Vrition of mximum soil depth vlues estimted by Iid s equtions nd MEMPS. 1419

4 to modelling the tul mximum soil depth in slope, some of these prmeters ere defined from Mihel et l. (2014) nd other ones ere rbitrrily defined. The only re in the definition of these prmeters s to selet vlues of q,, b nd Ks tht ere not ble to generte omplete sturtion in the slope. It is lerly observed tht the Eqution 13 hs its pplition limit. When the stedy stte rehrge (q) used to define the soil depth distribution ith MEMPS results in omplete sturtion of the soil, the results of MEMPS re equl to those ith Eqution 3. When the vlue of q is set to ero (i.e. there is no ter in the bsin) the results of MEMPS re equl to those ith Eqution 2. Thus, MEMPS n desribe three different situtions (bsene of ter, omplete sturtion nd n intermedite ondition) ith only one eqution. Beuse it, the performne of the model depends on the orret estimton of q. Furthermore, this model is only pplible on slopes tht re steeper thn f, i.e., θ > f. In Figure 2, it is observed tht the etness ondition used by MEMPS is intermedite beteen omplete sturtion nd totl bsene of ter. For lrge vlues of soil depth, the etness ondition in MEMPS s more similr to bsene of ter. Insted, for smll vlues of soil depth, the etness ondition in MEMPS s more similr to sturtion. The prtilly sturtion ondition in the one bove the ter tble s not took into ount in this study. When nlying the mximum stble soil depth, only the onditions developed in possible rupture surfe re relevnt. In other ords, only the onditions in the interfe beteen soil nd bedrok re tking into ount, beuse this one is the one more suseptible to filure. In the pplition of MEMPS (Eqution 13), it is neessry to set uniform stedy stte rehrge rte, in order to mimi period of preipittion bove the regulr onditions, but ithout the pbility to trigger lndslides. The ondition ssumed genertes pttern of soil depth distribution in hih the hillslope tends to its stbility during norml or slightly more thn norml rinfll sesons. When hevy rinfll tht is more thn norml tkes ple, the soil grdully redues its FS vlue nd n reh unstble onditions. Thus, for exmple, n initil ondition of relted to the nnul men rinfll ould generte et ondition insuffiient to use the instbility. When hevy rinfll similr to tht used to define the soil depth distribution ours in the bsin, the sitution turns grdully from stble ondition to the threshold. If the rinfll persists nd the etness exeeds the ondition used to define the soil depth, the sitution evolves from limit equilibrium to unstble onditions. The hrteristi of MEMPS, in hih initilly there re stble onditions tht grdully turn in to unstble onditions ith the rinfll event, gives to this model the qulity to be quite dequte to define initil onditions to slope stbility modelling. Most of times, in the slope stbility modelling, the initil onditions re of instbility over lrge prt of the terrin (minly in steep onve res), even before the ourrene of the triggering event. In regions here the lndslides re triggered by hydrologil ftors, the initil onditions (before the rinfll or triggering event ourrene) of the terrin must be stble, ht is hieved ith pplition of MEMPS. Thus, ith this more relisti senrio, the slope stbility modelling ould be ble to desribe the speifi proess involved in the lndslide triggering. Furthermore, in lndslide rning systems, it is neessry to define the moment in hih the terrin ill turn from stble to unstble onditions. Thus, in these ses, n initil ondition of stbility is required. Therefore, MEMPS n be useful tool for soil depth nd lndslide hrd mpping, nd lndslide rning systems. 4 FINAL CONSIDERATIONS The present study developed ne eqution, lled MEMPS, to estimte the mximum soil depth through ombintion of stedy stte hydrologil model ith n infinite slope stbility model. This eqution does not need to hve ny of ssumptions of Iid (1999), representing ondition loser to the relity. In future, it should be verified ith field dt. Even if the MEMPS n be only used for res here the slope ngle is lrger thn the soil internl frition ngle, i.e., mountinous bsins, it is very useful. One of the gols of the development of this eqution is to rete more relisti initil senrios for slope stbility modelling. In these ses, MEMPS n be very useful both in to optimie lndslide modelling nd to improve lndslide rning systems. REFERENCES Bery, A.A High resolution in seismi refrtion tomogrphy for environmentl study. Interntionl Journl of Geosienes 4: Borg, M., Dll Fontn, G., Gregoretti, C., Mrhi, L Assessment of shllo lndsliding by using physilly bsed model of hillslope stbility. Hydrologil Proesses 16: Ctni, F., Segoni, S., Florni, G An empiril geomorphology-bsed pproh to the sptil predition of soil thikness t thment sle. Wter Resoures Reserh 46:

5 Dietrih, W.E., Reis, R., Hsu, M.L., Montgomery, D.R A proess-bsed model for olluvil soil depth nd shllo lndsliding using digitl elevtion dt. Hydrologil Proesses 9: Iid, T A stohsti hydro-geomorphologil model for shllo lndsliding due to rinstorm. Cten 34: Mihel, G.P., Kobiym, M., Goerl, R.F Comprtive nlysis of SHALSTAB nd SINMAP for lndslide suseptibility mpping in the Cunh River bsin, southern Bril. Journl of Soils nd Sediments 14: O Loughlin, E.M Predition of surfe sturtion ones in nturl thments by topogrphi nlysis. Wter Resoures Reserh 22: Piegri, E., Ctudell, V., Di Mrio, R., Milno. L., Niodemi, M., Soldovieri, M.G Eletril resistivity tomogrphy nd sttistil nlysis in lndslide modelling: A oneptul pproh. Journl of Applied Geophysis 68(2): Sulnier, G.M., Beven, K., Obled, C Inluding sptilly vrible effetive soil depths in TOPMODEL. Journl of Hydrology 202: Segoni, S., Rossi, G., Ctni, F Improving bsin sle shllo lndslide modelling using relible soil thikness mps. Nturl Hrds 61: Selby, M Hillslope mterils nd proesses. 2nd ed. Oxford University Press, Oxford. 1421