Van Hieu Bui 1,2,* , Minh Duc Bui 1 and Peter Rutschmann 1
1 Institute of Hydraulic and Water Resources Engineering, Technische Universität München, Arcisstrasse 21, D-80333 München, Germany; [email protected] (M.D.B.); [email protected] (P.R.) 2 Faculty of Mechanical Engineering, Thuyloi University, 175 Tay Son, Dong Da, Hanoi 100000, Vietnam * Correspondence: [email protected]; Tel.: +49-152-134-15987
Received: 15 February 2019; Accepted: 13 March 2019; Published: 17 March 2019
Abstract: Understanding the alterations of gravel bed structures, sediment transport, and the effects on aquatic habitat play an essential role in eco-hydraulic and sediment transport management. In recent years, the evaluation of changes of void in bed materials has attracted more concern. However, analyzing the morphological changes and grain size distribution that are associated with the porosity variations in gravel-bed rivers are still challenging. This study develops a new model using a multi-layer’s concept to simulate morphological changes and grain size distribution, taking into account the porosity variabilities in a gravel-bed river based on the mass conservation for each size fraction and the exchange of ﬁne sediments between the surface and subsurface layers. The Discrete Element Method (DEM) is applied to model inﬁltration processes and to conﬁrm the effects of the relative size of ﬁne sediment to gravel on the inﬁltration depth. Further, the exchange rate and the bed porosity are estimated while using empirical formulae. The new model was tested on three straight channels. Analyzing the calculated results and comparing with the observed data show that the new model can successfully simulate sediment transport, grain sorting processes, and bed change in gravel-bed rivers.
Keywords: numerical modelling; bed porosity; grain sorting; gravel-bed rivers
1. Introduction Sediment transport processes that are due to ﬂowing water in gravel-bed rivers can be formed from bed load, suspended load, and movement inside the bed layer. More importantly, in the same hydraulics condition, the transport rate of coarse size fraction may be different from the rate of ﬁne size fraction. These reasons lead to the distinct forms of gravel-bed rivers. Due to high-ﬂow velocity, all of the ﬁner particles may be eroded, leaving a layer of coarse particles. When the ﬂow is unable to carry the coarse particles, then no more erosion occurs, which is known as an armoring process [1,2]. Inversely, under low-ﬂow conditions, sediment transport can cause the extensive inﬁltration of ﬁne sediment into void spaces in coarse bed material [3,4], which is also known as clogging or colmation [5,6] (see Figure1).
Water 2019, 11, 550; doi:10.3390/w11030550 www.mdpi.com/journal/water Water 2019, 11, 550 2 of 21 Water 2019, 10, x FOR PEER REVIEW 2 of 21
FigureFigure 1. 1. PorosityPorosity variatio variationn of of grain grain system system due due to to fine ﬁne grain grain exchange exchange..
TheThe change change in in grain size distribution ofof coarsecoarse andand ﬁnefine sediment sediment not not only only leads leads to to the the variation variation in inbed bed proﬁle, profile but, but it also it also mainly mainly results results in the in changethe change in bed in porosity bed porosity deﬁned defined as a fraction as a fraction of the volume of the volumeof voids of over voids the over total the volume total volume of the gravel-bedof the gravel river.‐bed Accordingriver. According to [7,8 ],to the [7,8 porosity], the porosity of sand-gravel of sand‐ gravelmixtures mixtures can vary can from vary aboutfrom about 0.10 to 0.10 0.50. to For 0.50. example, For example, the measured the measured porosity porosity values values in the in Rhine the RhineRiver River range range from 0.06from to 0.06 0.48 to [9 0.4]. 8 . TheThe study study of variationvariation inin porosity porosity is is vital vital for for ﬂuvial fluvial geomorphology geomorphology assessment assessment as well as well as in as river in riverecosystem ecosystem management. management. From From a river a river management management point point of view, of view, the amount the amount of ﬁne of sediment fine sediment stored storedin the void in the space void of space gravel-bed of gravel up‐ tobed 22% up [ 10 to] may 22% be[10 neglected] may be during neglected calculation during bycalculation considering by consideringconstant porosity. constant Furthermore, porosity. Furthermore the porosity, the may porosity exert may strong exert inﬂuence strong oninfluence the rate on of the bed rate level of bedchanges level [changes11,12]. The [11, impact12]. The of impact void space of void of gravel-bedspace of gravel on habitats‐bed on forhabitats ﬁsh and for aquaticfish and insects aquatic is insectsessential, is essential, and particularly and particularly the importance the importance of assessing of assessing the change the inchange the void in the structure void structure of the bed of thematerial bed material has been has pointed been pointed out strongly out strongly . . ToTo model model sediment sediment transport transport processes processes as as well well as as bed bed deformation deformation,, multi multi-layer‐layer models models were were appliedapplied for for graded graded sediment sediment transport. transport. Despite Despite the the considerable considerable variations variations of of porosity, porosity, most most of of the the conventionalconventional models models assumed assumed that that the the porosity porosity of of bed material is constant. Therefore, Therefore, sediment sediment transporttransport in in the the form form of infiltration of inﬁltration into intothe void the spaces void spaces of the co ofarse the bed, coarse or fine bed, particles or ﬁne in particles sublayers in andsublayers the entrainment and the entrainment of fine sediment of ﬁne sedimentinto flow into from ﬂow the from substrates the substrates,, is not taken is not into taken account. into account. This assumptionThis assumption can be can inappropriate be inappropriate for simulating for simulating the sediment the sediment transport transport and bed and variation bed variation in gravel in‐ bedgravel-bed rivers. Toro rivers.‐Escobar Toro-Escobar et al.  et al. accounted  accounted for this for process this process in terms in terms of an of appropriate an appropriate transfer transfer or exchangeor exchange function function at the at theinterface interface between between the the surface surface layer layer and and substrate. substrate. Based Based on onthe the data data from from a largea large-scale‐scale experiment experiment on onthe the aggradation aggradation and and the theselective selective depositio depositionn of gravel of gravel,, they they proposed proposed an empiricalan empirical form form for the for thetransfer transfer function, function, where where material material that thatis transferred is transferred to the to thesubstrate substrate can canbe representedbe represented as a as weighted a weighted average average of of bedload bedload and and surface surface material, material, with with a abias bias toward toward bedload. bedload. ApplyingApplying this this function function with with th thee assumption assumption of of a a constant constant porosity, porosity, Cui Cui [ ] developed developed a a numerical numerical modelmodel for for sand sand entrainment/infiltration entrainment/inﬁltration from/into from/into the the subsurface. subsurface. The The model model was was applied applied to to study study thethe dynamics dynamics of of grain grain size size distributions, distributions, including including the the fractions fractions of sand of sand in sediment in sediment depos depositsits and and on theon channel the channel bed surface. bed surface. Further, Further, Sulaiman Sulaiman et al. [ et16 al.] proposed  proposed an approximated an approximated bed variation bed variation model whilemodel considering while considering the change the change in bed inporosity, bed porosity, where where the thickness the thickness of each of eachlayer layer is assumed is assumed to be to constantbe constant and and equal. equal. Furthermore, Furthermore, the the exchange exchange be betweentween bed bed material material and and the the transported transported sediment sediment onlyonly takes takes place place on on the surface layer, andand sedimentsediment transporttransport fromfrom an an upper upper layer layer to to the the lower lower layer layer is isneglected. neglected These. These assumptions assumptions are are improper improper ingravel-bed in gravel‐bed rivers, rivers where, where ﬁner finer sediment sediment possibly possibly drops dropsinto the into subsurface the subsurface layer. layer. To bridge above knowledge gap, in this paper we develop a numerical hydromorphological model that considers the bed porosity change and the exchange flux of fine sediment between two
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To bridge above knowledge gap, in this paper we develop a numerical hydromorphological model that considers the bed porosity change and the exchange ﬂux of ﬁne sediment between two different bed layers. Further, the Discrete Element Method (DEM) is employed for modeling the inﬁltration processes. We test the new numerical model for three straight gravel-bed open channels. The calculated results are then compared with the observed data to verify the improved model.
2. Methodology As usual, the structure of our proposed model consists of two modules: (1)—hydrodynamic module; and, (2)—sediment transport and bed variation module. In the hydrodynamic module, the backwater equation is employed for low Froude number conditions and the quasi normal ﬂow assumption is applied for the high Froude number ﬂow conditions. The friction slope is calculated with the Keulegan formulation, which is used for calculating the transport rate in the sediment transport and bed variation module. More details regarding the water ﬂow equations can be found in the studies that were conducted by Cui and Parker [15,17–20]. The ﬂow module interacts with sediment transport and bed variation module that is based on the mechanism of quasi-steady morphodynamic time-stepping, in which the interaction between the ﬂow and bed proﬁle is considered in two different times. More speciﬁcally, at each time step of the ﬂow calculation, the bed level is considered to be constant, and the sediment transport is assumed not to effect the ﬂow, whereas in the process of bed calculation, the ﬂow is assumed to be invariant. Subsequently, the water depth, velocity, as well as the bed roughness are adjusted based on the results of the sediment transport and bed variation module. The calibrated roughness is used in the hydraulic module in the next time step. Since previous models contained some limitations, we developed a new sediment transport and bed variation module, which is presented in detail in the following sections. In principle, the bed topography in a channel is related to the transport of both bed load and suspended load. In the case of a coarse-material bed, the inﬂuence of suspended load is negligible, whereas the bed load is predominant. Further, suspended sediment load is the ﬁne material that moves through the channel in the water column. These materials are kept in suspension by the upward ﬂux of turbulence that is generated at the bed of the channel. The upward currents must equal or exceed the particle fall-velocity for suspended sediment load to be sustained. In this paper, we assume the turbulent ﬂow to be small, therefore we only consider bed load in sediment transport modelling. The governing equations for ﬂow, sediment transport, and bed variation are numerically solved while applying the ﬁnite volume method with the required initial and boundary conditions.
2.1. Sediment Transport and Bed Variation Module The size-fraction method is applied to divide bed material into some size-fractions. Each fraction is characterized by an average grain size and volume percentage of occurrence in the bed material and presented these components as the probability of different group. Applying the so-called multi-layer concept, we divide the vertical bed structure into different layers. Figure2 shows a diagram of this structure consisting of one active layer and several substrate layers. Sediment particles are continuously exchanged between the ﬂow and the active layer. Generally, the sediments in this layer can be transported as bed load, suspended load, or inﬁltration. Furthermore, they can move between the active layer and sublayers, depending mostly on the bed porosity and grain size distribution, as it will be discussed below in Section 2.3. For example, sediments are exchanged between the active layer and substrates when the bed scours or ﬁlls, as well as when ﬁltration occurs. As erosion occurs, the entrainment of sediment particles from the active layer and its ensuing downward displacement causes particles from substrate layers to be mixed with those in the active layer. On the contrary, the deposition of sediment particles on the bed leads to an upward displacement of the active layer and the initiation of new substrate layers. At high ﬂow discharges, ﬁne sediments in the top substrate (2) may also be attracted to the surface layer and entrained ﬂow. Water 2019, 11, 550 4 of 21 Water 2019, 10, x FOR PEER REVIEW 4 of 21
FigureFigure 2.2. TheThe structure structure diagram diagram of of the the vertical vertical cross cross-section‐section is based is based on the on multi the multi-layers‐layers model. model. (1) = active layer; (2) and (3) = substrate layers; Ea and Em = Active layer thickness and active‐stratum layer (1) = active layer; (2) and (3) = substrate layers; Ea and Em = Active layer thickness and active-stratum thickness; zb = bed elevation; zc and zd = substrate elevations. layer thickness; zb = bed elevation; zc and zd = substrate elevations.
Furthermore,Furthermore, thethe riverbedriverbed morphologymorphology inin naturenature cancan bebe relatedrelated toto allall threethree typestypes ofof sedimentsediment transporttransport thatthat are mentioned above above (bed (bed load, load, suspended suspended load load,, and and infiltration inﬁltration of offine ﬁne sediment). sediment). As Asmentioned mentioned above,above, when when turbulent turbulent flow ﬂow isis week week or or the the settling settling velocity velocity of of sediment particle isis moremore signiﬁcantsignificantthan than critical critical shear shear stress stress velocity, velocity, sediment sediment particle particle only only moves moves in thein the forms forms of bedof bed load load or inﬁltration.or infiltration. For For the the sake sake of simplicityof simplicity of this of this paper, paper, we assumewe assume that that
• TheThe horizontalhorizontal surface surface is is unchanged unchanged zd z=d = 0, 0, so so sediment sediment sorting sorting occurs occurs only only in thein the active active layer layer (1) and(1) and the upmostthe upmost sublayer sublayer (2); i.e.,(2); i.e. the, bedthe bed consists consists of only of only two two layers. layers. • TheThe bedbed sedimentsediment isismoving moving underunder twotwo forms:forms: inﬁltrationinfiltration or or bed bed load. load. • TheThe ﬂowflow andand sedimentsediment transporttransport are are one-dimensional. one‐dimensional.
TheThe changechange inin thethe bedbed elevationelevation isis calculatedcalculated usingusing thethe lawlaw ofof continuitycontinuity ofof bedbed sedimentsediment inin thethe activeactive layer:layer: Z z ∂ b ∂qb z(b1 − p)dz = −∂q (1) ∂t ∂xb 0 1 − p dz = − (1) ∂t ∂x Where x = The coordinate axis along the ﬂow0 direction; z = The coordinate axis along the vertical direction;Where t = x time; = The p coordinate = p(x,z,t) = bedaxis porosity;along the q flowb = q direction;b(x,t) = total z = speciﬁc The coordinate volumetric axis bed along load the discharge. vertical Itsdirection; fraction t q=b,j time;is determined p = p(x,z,t) by= bed experiment porosity; orqb semi-experiment= qb(x,t) = total specific when volumetric considering bed non-equilibrium load discharge. bedIts fraction load [21 ].qb,j In is this determined study, the by fractional experiment bed loador semi rate‐experiment is calculated when while considering using the bed-load non‐equilibrium equation proposedbed load by[21 Wilcock]. In this and study, Crowe the [22 fractional]. Further, bed Equation load rate (1) can is calculatedalso be rewritten while as:using the bed‐load equation proposed by Wilcock and Crowe . Further, Equation (1) can also be rewritten as: Z z −Ea Z z ∂ b b ∂qb ∂ ( 1 − p)dz + (1 − p)dz = −∂q (2) ∂ − ∂ t 0 (1 − p)dz + zb Ea (1 − p)dz = − x (2) ∂t ∂x Applying the theorem of mean values and Leibniz integral rule for two ﬁnite integrals in the left-handApplying side of the Equation theorem (2) of yields: mean values and Leibniz integral rule for two finite integrals in the left‐hand side of Equation (2) yields: ∂z ∂q ∂p ∂p ∂Ea − b = − b + = (∂z − )∂q s + a + − 1 ps S(p1S−p p ) zb =E−a + SE a pa ps (3) ∂t ∂x ∂t ∂∂xt ∂t ∂t (3) ∂p ∂p ∂E In which pa = pa(x,t) = theS = average (z − E porosity) + ofE active + layer;(p − p ps)= p s (x,t) = the average porosity ∂t ∂t ∂t of sublayer layer (Figure3). If the porosity of layers is constant, the porosity source term S p = 0 and EquationIn which (3) becomes pa = pa(x,t) the Exner= the average equation. porosity Therefore, of active Equation layer; (3) p cans = p bes(x,t) considered = the average as the porosity expansion of ofsublayer the Exner layer equation, (Figure when 3). If consideringthe porosity the of variationlayers is ofconstant, porosity the in spaceporosity and source time. term Sp = 0 and Equation (3) becomes the Exner equation. Therefore, Equation (3) can be considered as the expansion of the Exner equation, when considering the variation of porosity in space and time.
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FigureFigure 3. 3.Bed Bed level level change change due due to to porosity porosity variation variation in in different different time time steps. steps.
AssumingAssuming that that the the grain grain sorting sorting in in an an active active layer layer only only occurs occurs under under the the ﬂow flow interaction interaction and and exchangeexchange process process between between the the active active layer layer and and the the active active stratum stratum layer layer and and while while applying applying the the mass mass conservationconservation law law in in the the active active layer, layer, we we obtain obtain the the equation equation for for the the change change of of size size fraction, fraction as, as below: below: