MIKE 21 & MIKE 3 Flow Model FM. Mud Transport Module. Scientific Documentation

Transcription

1 MIKE 21 & MIKE 3 Flow Model FM Mud Tranport Module Sientifi Doumentation MIKE 2017

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3 PLEASE NOTE COPYRIGHT Thi doument refer to proprietary omputer oftware, whih i proteted by opyright. All right are reerved. Copying or other reprodution of thi manual or the related programme i prohibited without prior written onent of DHI. For detail pleae refer to your DHI Software Liene Agreement. LIMITED LIABILITY The liability of DHI i limited a peified in Setion III of your DHI Software Liene Agreement : IN NO EVENT SHALL DHI OR ITS REPRESENTATIVES (AGENTS AND SUPPLIERS) BE LIABLE FOR ANY DAMAGES WHATSOEVER INCLUDING, WITHOUT LIMITATION, SPECIAL, INDIRECT, INCIDENTAL OR CONSEQUENTIAL DAMAGES OR DAMAGES FOR LOSS OF BUSINESS PROFITS OR SAVINGS, BUSINESS INTERRUPTION, LOSS OF BUSINESS INFORMATION OR OTHER PECUNIARY LOSS ARISING OUT OF THE USE OF OR THE INABILITY TO USE THIS DHI SOFTWARE PRODUCT, EVEN IF DHI HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. THIS LIMITATION SHALL APPLY TO CLAIMS OF PERSONAL INJURY TO THE EXTENT PERMITTED BY LAW. SOME COUNTRIES OR STATES DO NOT ALLOW THE EXCLUSION OR LIMITATION OF LIABILITY FOR CONSEQUENTIAL, SPECIAL, INDIRECT, INCIDENTAL DAMAGES AND, ACCORDINGLY, SOME PORTIONS OF THESE LIMITATIONS MAY NOT APPLY TO YOU. BY YOUR OPENING OF THIS SEALED PACKAGE OR INSTALLING OR USING THE SOFTWARE, YOU HAVE ACCEPTED THAT THE ABOVE LIMITATIONS OR THE MAXIMUM LEGALLY APPLICABLE SUBSET OF THESE LIMITATIONS APPLY TO YOUR PURCHASE OF THIS SOFTWARE. MIKE 2017

4 CONTENTS MIKE 21 & MIKE 3 Flow Model FM Mud Tranport Module Sientifi Doumentation 1 Introdution What i Mud General Model Deription Introdution Governing Equation Numerial Sheme Coheive Sediment Introdution Depoition Settling Veloity and Floulation Sediment Conentration Profile Teeter 1986 profile Roue profile Eroion Bed Deription Fine Sand Sediment Tranport Introdution Supended Load Tranport Requirement for ediment in upenion Settling Veloity Sediment Conentration Profile Diffuion fator Damping fator Conentration profile Depoition Eroion Hydrodynami Variable Bed Shear Stre Pure urrent Pure wave motion Combined urrent and wave Vioity and Denity Mud impat on denity Mud influene on vioity Multi-Layer and Multi-Fration Appliation Introdution i

5 7.2 Dene Conolidated Bed Soft, Partly Conolidated Bed Morphologial Feature Morphologial Simulation Bed Update Referene ii

6 Introdution 1 Introdution The preent Sientifi Doumentation aim at giving an in-depth deription of the theory behind and equation ued in the Mud Tranport Module of the MIKE 21 & MIKE 3 Flow Model FM. Firt a general deription of the term "mud" i given. Thi i followed by a number of etion giving the phyial, mathematial and numerial bakground for eah of the term in the oheive ediment and fine and tranport equation. Speial etion deribe the ae of multi-layer and multi-fration appliation a well a the option of morphologial imulation. DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 1

7 What i Mud 2 What i Mud Mud i a term generally ued for fine-grained and oheive ediment with grain-ize le than 63 miron. Mud i typially found in heltered area proteted from trong wave and urrent ativity. Example are the upper and mid reahe of etuarie, lagoon and oatal bay. The oure of the fine-grained ediment may be both fluvial and marine. Fine-grained upended ediment play an important role in the etuarine environment. Fine ediment i brought in upenion and tranported by urrent and wave ation. In etuarie, the tranport mehanim (ettling and our lag) ating on the fine-grained material tend to onentrate and depoit the fine-grained material in the inner heltered part of the area (Van Straaten & Kuenen, 1958; Potma, 1967; and Pejrup, 1988). A zone of high onentration upenion i alled a turbidity maximum and will hange it poition within the etuary depending on the tidal yle and the input of freh-water from river, et. (e.g. Dyer, 1986). Fine ediment are harateried by low ettling veloitie. Therefore, the ediment may be tranported over long ditane by the water flow before ettling. The oheive propertie of fine ediment allow them to tik together and form larger aggregate or flo with ettling veloitie muh higher than the individual partile within the flo (Krone, 1986; Burt, 1986). In thi way they are able to depoit in area the individual fine partile would never ettle. The formation and detrution of flo are depending on the amount of ediment in upenion a well a the turbulene propertie of the flow. Thi i in ontrat to non-oheive ediment, the partile are tranported a ingle grain. Figure 2.1 Muddy (left) and andy (right) ediment Fine ediment i laified aording to grain-ize a hown in Table 2.1. Table 2.1 Claifiation of fine ediment Sediment type Grain ize Floulation ability Clay < 4 m high Silt 4-63 m medium Fine and m very low/no floulation DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 2

8 General Model Deription 3 General Model Deription 3.1 Introdution In order to inlude the tranport and depoition proee of fine-grained material in the modelling ytem, it i neeary to integrate the deription with the advetion-diffuion equation aued by the water flow. MIKE 21 & MIKE 3 Flow Model FM i baed on a flexible meh approah and ha been developed for appliation within oeanographi, oatal and etuarine environment. In the ae of 2D, the model i depth-integrated. Thi mean that the imulation of the tranport of fine-grained material mut be averaged over depth and appropriate parameteriation of the ediment proee mut be applied. In the MIKE 21 & MIKE 3 Flow Model FM model omplex, the tranport of fine-grained material (mud) ha been inluded in the Mud Tranport module (MT), linked to the Hydrodynami module (HD), a indiated in Figure 3.1. Hydrodynami (HD) Current and turbulent diffuion Advetion-Diperion (HD) Advetion/diperion proee Mud Tranport (MT) Eroion, depoition and bed proee Figure 3.1 Data flow and phyial proee for MIKE 21 & MIKE 3 Flow Model FM, Mud Tranport alulation DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 3

9 General Model Deription The proee inluded in the Mud Tranport module are kept a general a poible. The Mud Tranport module inlude the following proee: Multiple mud fration Multiple bed layer Wave-urrent interation Floulation Hindered ettling Inluion of a and fration Tranition of ediment between layer Simple morphologial alulation The above poibilitie over mot ae appropriate for 2D modelling. In ae peial appliation are required uh a imulating the influene of high ediment onentration on the water flow through formation of tratifiation and damping of turbulene, the modeller i referred to 3D modelling. 3.2 Governing Equation The ediment tranport formulation are baed on the advetion-diperion alulation in the Hydrodynami module. The Mud Tranport module olve the o-alled advetion-diperion equation: + u + v = t x y 1 h hd x + x x 1 h hd y y + Q y L C L 1 - S h (3.1) - depth averaged onentration (g/m 3 ) u,v depth averaged flow veloitie (m/) D x,d y diperion oeffiient (m 2 /) h water depth (m) S depoition/eroion term (g/m 3 /) Q L oure diharge per unit horizontal area (m 3 //m 2 ) C L onentration of the oure diharge (g/m 3 ) In ae of multiple ediment fration, the equation i extended to inlude everal fration while the depoition and eroion proee are onneted to the number of fration. DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 4

10 General Model Deription 3.3 Numerial Sheme The advetion-diperion equation i olved uing an expliit, third-order finite differene heme, known a the ULTIMATE heme (Leonard, 1991). Thi heme i baed on the well-known QUICKEST heme (Leonard, 1979; Ekebjærg & Juteen, 1991). Thi heme ha been deribed in variou paper dealing with turbulene modelling, environmental modelling and other problem involving the advetion-diperion equation. It ha everal advantage over other heme, epeially that it avoid the "wiggle" intability problem aoiated with entral differentiation of the advetion term. At the ame time it greatly redue the numerial damping, whih i harateriti of firt-order up-winding method. The heme itelf i a Lax-Wendroff or Leith-like heme in the ene that it anel out the trunation error term due to time differentiation up to a ertain order by uing the bai equation itelf. In the ae of QUICKEST, trunation error term up to third-order are anelled for both pae and time derivative. The olution of the eroion and the depoition equation are traightforward and do not require peial numerial method. DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 5

11 Coheive Sediment 4 Coheive Sediment 4.1 Introdution The Mud Tranport module of MIKE 21 & MIKE 3 Flow Model FM deribe the eroion, tranport and depoition of fine-grained material < 63 m (ilt and lay) under the ation of urrent and wave. For a orret olution of the eroion proee, the onolidation of ediment depoited on the bed i alo inluded. The model i eentially baed on the priniple in Mehta et al. (1989) with the innovation of inluding the bed hear tree due to wave. Clay partile have a plate-like truture and an overall negative ioni harge due to broken mineral bond on their fae. In aline water, the negative harge on the partile attrat poitively harged ation and a diffue loud of ation i formed around the partile. In thi way the partile tend to repel eah other (Van Olphen, 1963). Still, partile in aline water floulate and form large aggregate or flo in pite of the repulive fore. Thi i beaue in aline water, the eletrial double layer i ompreed and the attrative van der Waal fore ating upon the atom pair in the partile beome ative. Floulation i governed by inreaing onentration, beaue more partile in the water enhane meeting between individual partile. Turbulene alo play an important role for floulation both for the forming and breaking up of flo depending on the turbulent hear (Dyer, 1986). A determiniti phyially baed deription of the behaviour of oheive ediment ha not yet been developed, beaue the numerou fore inluded in their behaviour tend to ompliate matter. Conequently, the mathematial deription of eroion and depoition are eentially empirial, although they are baed on ound phyial priniple. The lak of a univerally appliable, phyially baed formulation for oheive ediment behaviour mean that any model of thi phenomenon i heavily dependent on field data (Anderen & Pejrup, 2001; Anderen, 2001; Edelvang & Auten, 1997; Pejrup et al., 1997). Extenive data over the entire area to be modelled i required uh a: bed edimentology bed erodibility biology ettling veloitie upended ediment onentration urrent veloitie vertial veloity and upended ediment onentration profile ompation of bed layer effet of wave ation ritial hear tree for depoition and eroion Of oure, the dynami variation of water depth and flow veloitie mut alo be known along with boundary value of upended ediment onentration. The Mud Tranport module onit of a 'water-olumn' and an 'in-the-bed' module. The link between thee two module i oure/ink term in an advetion-diperion model. The tranport and depoition of fine-grained material i governed by the fat that ettling veloitie are generally low ompared to and. Hene, the onentration of upended DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 6

12 Coheive Sediment material doe not adjut immediately to hange in the hydrauli ondition. In other word, the ediment onentration at a given time and loation i dependent on the ondition uptream of thi loation at an earlier time. Potma (1967) firt deribed thi proe, alled ettling- and our-lag. Thi i the main fator for the onentration of fine material in etuarie often reulting in a turbidity maximum. In order to deribe thi proe, the ediment omputation ha been built into the advetion-diperion formulation in the Hydrodynami module. For 3D alulation the vioity and denity may be influened by large onentration of mud in the water olumn. The oure and ink term S in the advetion-diperion equation depend on whether the loal hydrodynami ondition aue the bed to beome eroded or allow depoition to our. Empirial relation are ued, and poible formulation for evaluating S are given below. The mobile upended ediment i tranported by long-period wave only, whih are tidal urrent, a the wind-wave are onidered a "haker". Combined they are able to re-entrain or re-upend the depoited or onolidated ediment. The proee in the bed are deribed in a multi-layer bed; eah layer i deribed by a ritial hear tre for eroion, eroion oeffiient, power of eroion, denity of dry ediment and eroion funtion. The bed layer an be oft and partly onolidated or dene and onolidated. Conolidation i inluded a a tranition rate of ediment between the layer and liquefation by wave i inluded a a weakening of the bed due to breakdown of the bed truture. A oneptual illutration of the phyial proee modelled by a "multi bed layer approah" i hown in Figure 4.1. Figure 4.1 Multi-layer model and phyial proee for example with three bed layer DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 7

13 Coheive Sediment 4.2 Depoition In the MT model, a tohati model for flow and ediment interation i applied. Thi approah wa firt developed by Krone (1962). Krone ugget that the depoition rate an be expreed by: Depoition : SD wb pd (4.1) w ettling veloity (m/) b near bed onentration (kg/m 3 ) p d probability of depoition The probability of depoition p d i alulated a: pd b 1 b d (4.2) d b the bed hear tre (N/m 2 ) d the ritial bed hear tre for depoition (N/m 2 ) 4.3 Settling Veloity and Floulation The ettling veloity of the fine ediment depend on the partile/flo ize, temperature, onentration of upended matter and ontent of organi material. Uually one ditinguihe between a regime the ettling veloity inreae with inreaing onentration (floulation) and a regime the ettling veloity dereae with inreaing onentration. The latter i referred too a hindered ettling. The firt i the mot ommon of the two in the etuary. Following Burt (1986), the ettling veloity in aline water (>5 ppt) an be expreed by: w y 3 k for 10 kg / m (4.3) w ettling veloity of flo (m/) volume onentration k,γ oeffiient γ 1 to 2 The relation for 10 kg/m 3 deribe the floulation of partile baed on partile olliion. The higher onentration the higher poibility for the partile to floulate. > 10 kg/m 3 orrepond to 'hindered' ettling, partile are in ontat with eah other and do not fall freely through the water. Alternative ettling formulation are alo available: The formulation of Rihardon and Zaki (1954) i the laial equation for hindered ettling: DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 8

14 Coheive Sediment w, r gel w, n w 1 (4.4) Where w,r i a referene value, w,n a oeffiient and gel the onentration at whih the flo tart to form a real elf-upported matrix (referred to a the gel point). Winterwerp (1999) propoed the following for hindered ettling: 1 1 * p w w (4.5), r p (4.6) and i the denity of ediment grain. Floulation i enhaned by high organi matter ontent inluding organi oating, et. (Van Leuen, 1988; Eima, 1993). In freh water, floulation i dependent on organi matter ontent, a in aline water alt floulation alo our. The influene of alt on floulation i primarily important in area freh water meet alt water uh a etuarie. The following expreion i ued to expre the variation of ettling veloity with alinity. Pleae note that the referene value w i the value repreentative for aline water: w C2 w 1 C1e (4.7) C 1 and C 2 are alibration parameter. Figure 4.2 how an example of C 1 ={0, 0.5, 1} and C 2 = -1/3. Figure 4.2 Settling veloity and alinity dependeny DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 9

15 Coheive Sediment The deription of alt floulation i baed on Krone' experimental reearh (Krone, 1962) Whitehoue et al., (1960) tudied the effet of varying alinitie on floulation of different lay mineral in the laboratory. Gibb (1985) howed that in the natural environment, floulation i more dependent on organi oating. Therefore, the effet of mineral ontitution of the ediment i not taken into aount in the model. 4.4 Sediment Conentration Profile Two expreion for the ediment onentration profile an be applied. Either an expreion that i baed on an approximate olution to the vertial ediment fluxe during depoition (Teeter) or an expreion that aume equilibrium between upward and downward ediment fluxe (Roue). The differene between the two expreion i depited in Figure 4.3. Figure 4.3 Definition of Roue profile v. Teeter profile The near bed onentration b i proportional to the depth averaged ma onentration and i related to the vertial tranport, i.e. a ratio of the vertial onvetive and diffuive tranport repreented by the Pelet number P e: C r Pe (4.8) C rd C r onvetive Courant number = w t / h C rd diffuive Courant number = w 2 D z t / h mean ettling veloity of the ediment D z depth mean eddy diffuivity DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 10

16 Coheive Sediment Teeter 1986 profile In thi expreion the near bed onentration b i related to the depth averaged onentration (Teeter, 1986): b (4.9) Pe 1 (4.10) p 2.5 d P e w h (4.11) D z 6w U f Von Karman' univeral ontant ( = 0.4) U f Frition veloity b / Roue profile The upended ediment i affeted by turbulent diffuion, whih reult in an upward motion. Thi i balaned by ettling of the grain. The balane between diffuion and ettling an be expreed: dc wc (4.12) dz C z diffuion oeffiient onentration a funtion of z vertial Carteian oordinate By auming that i equal to the turbulent eddy vioity, and applying the paraboli eddy vioity ditribution z U f z1 (4.13) h h height of water olumn DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 11

17 Coheive Sediment the following vertial onentration profile will be given: R a h z C Ca a z h h a z, (4.14) C a a R referene onentration at z = a referene level Roue number R w (4.15) U f It i poible to hooe a vertial variation of the onentration of upended ediment in order to determine the ettling ditane. The average depth, h* through whih the partile ettle at depoition i given by: h* = h R R d d (4.16) = h/z and the term h h* i named the relative height of entroid. The near bed upended ediment onentration b i related to the depth-averaged onentration uing the Roue profile b (4.17) RC in whih RC i the relative height of entroid. 4.5 Eroion Eroion an be deribed in two way depending upon whether the bed i dene and onolidated (Partheniade, 1965) or oft and partly onolidated (Parhure and Mehta, 1985). For dene, onolidated bed the eroion (S E) i defined by: S E e b E, n 1 b (4.18) e E erodibility of bed (kg/m 2 /) b bed hear tre (N/m 2 ) DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 12

18 Coheive Sediment e ritial bed hear tre for eroion (N/m 2 ) n power of eroion For oft, partly onolidated bed the eroion i defined by: S E ½ b e b e E exp, (4.19) oeffiient ( m/ N ) 4.6 Bed Deription It i poible to deribe the bed a having more than one layer. Eah layer i deribed by the ritial hear tre for eroion, e,j, power of eroion, n j, denity of dry bed material, i, eroion oeffiient, E j, and j-oeffiient. The depoited ediment i firt inluded in the top layer. The layer repreent weak fluid mud, fluid mud and underonolidated bed (Mehta et al., 1989) and are aoiated with different time ale. The model require an initial thikne of eah layer to be defined. The onolidation proe i deribed a the tranition of ediment between the layer (Teion, 1992; Sanford and Maa, 2001). The influene of wave i taken into aount a liquefation reulting in a weakening of the bed due to breakdown of bed truture. Thi may aue inreaed urfae eroion, beaue of the redued trength of the bed top layer (Delo and Okenden, 1992). DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 13

19 Fine Sand Sediment Tranport 5 Fine Sand Sediment Tranport 5.1 Introdution The major differene between the deription of oheive ediment tranport and and tranport i the ditintion of the upended ediment onentration profile. In the Mud Tranport module, a imple deription of the vertial onentration profile i applied. The time-ale needed to deform the profile by the flow-ondition i long in omparion to a onentration profile of and, whih i primarily tranported a bed-load. However, it i poible to ativate and tranport formulation in the Mud Tranport module in ae a ertain perentage of the bed material i in the fine and fration between 63 and 125 m that may be tranported both in upenion and a bed load. Thee formulation are built into the advetion-diperion formulation. 5.2 Supended Load Tranport The equilibrium onentration e i defined a: q e (5.1) uh u q i the depth averaged flow veloity upended load tranport (kg/m/) The upended load tranport i found a the integral of the urrent veloity profile, u, and the onentration profile of upended ediment, : q dy (5.2) h a u h a onentration of ediment (kg/m 3 ) at ditane y from bed flow veloity (m/) at ditane y from bed water depth (m) thikne of the bed layer (m) Uually little i known about the bed layer, uh a the height of the bed form. Thi reult in the approximation: a k (5.3) 2d 50 k d 50 equivalent roughne height (m) 50 perent fratile of grain-ize of ediment (m) DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 14

20 Fine Sand Sediment Tranport In the advetion-diperion model, the upended tranport i alulated baed on depthaveraged flow veloitie, u, v, a ompound onentration,, and the water depth, h, (approx. h h a ). The and tranport i deribed through a depth-averaged equilibrium onentration, e and an aretion (depoition), eroion term, S, whih mean that no bed tranport take plae. The tranport i highly dependent upon two parameter, namely the ettling veloity, w, and the turbulent ediment diffuion oeffiient,, beaue thee parameter have an effet on both the flow veloity and onentration profile. For a normal ediment load the effet on the veloity profile i negligible Requirement for ediment in upenion The following deription i mainly baed on Van Rijn (1982, 1984), Yalin (1972) and Engelund and Fredøe (1976). The preene of ediment upenion demand that the atual frition veloity, U f, i larger than a o-alled ritial frition veloity, U f,r, and that the vertial turbulene i uffiient to reate vertial veloity omponent higher than the ettling veloity. The firt aumption i expreed through the tranport tage parameter, T: T 2 U f 1, U f U U f, r 0, U f U f, r f, r (5.4) U f,r i found from Shield urve, ee Rijn (1982), uing the input parameter, d 50, relative denity of ediment,, and the dimenionle grain ize, d*, defined a the ube root of the ratio of immered weight to viou fore: 1 g * d 50 2 v 1/ 3 d (5.5) n i the kinemati vioity of water (m 2 /). The frition veloity, U f, read: U f g ghi V (5.6) C z I C z d 90 energy gradient (lope) Chezy Number (m ½ /) (= 18 ln (4h/d90)) 90 perent fratile of grain ize of ediment (m) DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 15

21 Fine Sand Sediment Tranport V flow peed (m/) The eond aumption i expreed through ome relation between the ritial frition veloity, U f,r for initiation of upenion, the ettling veloity, w and d*: U f, r w 4, 1 d * 10 d * 0.4 d* 10 (5.7) 5.3 Settling Veloity The ettling veloity for fine and depend on the partile ize, vioity and ediment denity. The ettling veloity i expreed by: w 18v 10v gd 1 2 d v 1 gd gd ,,, d 100m 100 d 1000m d 1000m (5.8) d v g grain ize ediment denity vioity aeleration due to gravity DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 16

22 Fine Sand Sediment Tranport 5.4 Sediment Conentration Profile The onentration profile i dependent upon the turbulent ediment diffuion oeffiient,, and the ettling veloity, w. When alulating mud tranport = f i aumed, f i the turbulent flow diffuion oeffiient, a for fine and tranport the aumption i: (5.9) f fator, whih deribe the differene in the diffuion of a direte ediment partile and the diffuion of a fluid "partile". fator, whih expree the damping of the fluid turbulene by the ediment partile. Dependent upon loal ediment onentration Diffuion fator The interpretation of i not quite lear. Some think that < 1, beaue partile annot repond fully to turbulent veloity flutuation. However, ome think that > 1, beaue in the turbulent flow the entrifugal fore on the ediment partile would be greater than thoe on the fluid partile, thereby auing the ediment partile to be thrown to the outide of the eddie with a onequent inreae in the effetive mixing length and diffuion rate. In thi model the following i ued: 2 w w 1, 0.5 U f U f w 1, (5.10) U f w no upenion, 2.5 U f Damping fator The fator expree the influene of the ediment partile on turbulene truture (damping effet) of the fluid. In order to deribe the onentration profile the following equation hall be olved: d dz w 1 5 (5.11) The determination of and the olution of thi equation are very time-onuming, whih lead to a more implified method, whih i hoen in thi model. DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 17

23 Fine Sand Sediment Tranport Conentration profile The ditribution of the onentration profile i deribed by the Pelet number, P e: C r Pe (5.12) C rd C r onvetive Courant number ( = w t/h) C rd diffuive Courant number ( = f t/h 2 ) f depth averaged fluid diffuion oeffiient Thi Pelet number i alo alled a upenion parameter, Z: Z w (5.13) U f Z upenion parameter Von Karman' univeral ontant ( = 0.4) fator (a deribed in Eq. (5.10) above) To take into aount effet other than thoe aued by the fator, a modified upenion parameter, Z' i defined a: Z ' Z (5.14) i an overall orretion fator, whih read: w a (5.15) U f 0 a onentration at referene level, z = a 0 onentration at bed, z = 0 The a/ 0 onentration ratio i found through the following profile: C C a a a a h a a h z z h a Z Z exp 4Z z h 0.5,, z h z h (5.16) a i baed on meaured and omputed onentration profile, and read: d T a (5.17) 0.3 a d* DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 18

24 Fine Sand Sediment Tranport The equilibrium onentration, e in the advetion-diperion equation read: e 6 10 F C (5.18) a F i a relation between the bottom onentration and the mean onentration baed on numerial integration of the upended onentration profile expreed by the ratio / a previouly mentioned, and the relative denity equal to If you ue a ale fator, e i multiplied with thi fator. 5.5 Depoition Depoition i deribed by: e Sd, e t (5.19) t i a time-ale given by h t (5.20) w h i equal to h * deribed in the previou etion. 5.6 Eroion Eroion i deribed by: e Se, e t (5.21) DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 19

25 Hydrodynami Variable 6 Hydrodynami Variable 6.1 Bed Shear Stre The ediment tranport formula deribed above apply hydrodynami variable for deribing the bed hear tre. Thi mut be determined for pure urrent or a ombined wave-urrent motion Pure urrent In the ae of a pure urrent motion the flow reitane i aued by the roughne of the bed. The bed hear tre under a urrent i alulated uing the tandard logarithmi reitane law: 2 ½ fv (6.1) V f bed hear tre (N/m2) denity of fluid (kg/m3) mean urrent veloity (m/) urrent frition fator f 30h 22.5In 1 k 2 (6.2) h k water depth (m) bed roughne (m) Pure wave motion In the ae of pure wave motion, the mean bed hear tre read: ½ fu (6.3) 2 w w b bed hear tre f w w U b wave frition fator horizontal mean wave orbital veloity at the bed (m/) U b 2H 1 T 2 z inh h L (6.4) H ignifiant wave height (m) DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 20

26 Hydrodynami Variable T z zero-roing wave period () An expliit approximation given by Swart (1974) for the wave frition fator i ued: f w f w a 0.47, 1 k a a exp , k k a fw , 3000 k (6.5) a a H horizontal mean wave orbital motion at bed (m) 1 2 inh h L (6.6) An expliit expreion of the wave length i given by Fenton and MKee (1990): 2 gt z 2 h L tanh 2 Tz g 3/ 2 2 / 3 (6.7) Combined urrent and wave Soulby et al Three wave-urrent hear tre formulation are offered. Two of the formulation ue a parameteried verion of Fredøe (1984) derived by Soulby et al The default option for the parameteried model i to alulate and ue the mean hear tre. Another option i to alulate and ue the maximum hear tre. The mean hear tre and maximum hear tre are given below (Soulby et al., 1993): p mean 1 b 1 w w w w max 1 a 1 w w w m n q (6.8) urrent alone hear tre wave alone hear tre amplitude w DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 21

27 Hydrodynami Variable b, p, q, a, m, n ontant, whih vary for different wave-urrent theorie parameteried For the Fredøe (1984) model, thee ontant are: j j j j i i i b b b o b b o log r j p p p o p p o log r j q q q o q q o log r i a a a o a a o log r i m m m o m m o log r i n n n o n n o log r a 1, a 2, et. are given in the table below, γ i the angle between wave and urrent, i = 0.8, j = 3.0 and r = 2 f w / f. Table 6.1 Contant for wave-urrent hear tre formulation a m n b p q Fredøe A third option i to alulate and ue the bed hear tre found from Fredøe (1981). For ombined wave-urrent motion the eddy vioity i trongly inreaed in the wave boundary layer loe to the bed, and the near bed urrent profile i retarded. The effet on the outer urrent veloity profile i deribed by introduing a wave roughne, k w, whih i larger than the atual bed roughne. It i aumed that the wave motion i dominant loe to the bed ompared to the urrent, whih mean that the wave boundary layer thikne, δ w, and wave frition, f w, an be determined by onidering the wave parameter only. The wave boundary thikne, δ w, i found by (Johnon and Carlen, 1976): a w 0.072k k 0.75 (6.9) The veloity profile outide the wave boundary layer, whih i influened by the wave boundary layer i given by: DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 22

28 Hydrodynami Variable 30 U z U f z 2.5ln kw (6.10) Where U (z) U f z k w Veloity at vertial oordinate z (m/) Frition veloity (m/) vertial oordinate (m) wave roughne In ae of weak wave motion, the wave roughne k w i le than the bed roughne k for pure urrent motion, the latter will be ued. The max bed hear tre for ombined wave-urrent motion i given by: b fw U b U 2U bu o (6.11) 2 Where Uδ α Current veloity at top (z = δ w ) of wave boundary layer Angle between mean urrent and diretion of wave propagation The reulting bed hear tre i found by the larget value of the bed hear tre for pure urrent derived by Equation (6.1) and the value derived by Equation (6.11). 6.2 Vioity and Denity Mud impat on denity Thee two proee impat the HD-module by impating the denity and vioity. The influene from the mud on the water denity i by definition given by: m w w 1 i i (6.12) Mud influene on vioity The influene on the kinemati vioity from the mud an be parameteried by: M a/ k v 2 k v 1 (6.13) Where kv1 and kv2 are alibration parameter and a i. i Thi expreion i aumed to be valid for applying a lower limit for the eddy vioity, hene: DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 23

29 Hydrodynami Variable max, (6.14) T T M Utiliing M a/ k v 2 kv1 (6.15) T DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 24

30 Multi-Layer and Multi-Fration Appliation 7 Multi-Layer and Multi-Fration Appliation 7.1 Introdution The MT model i a multi-layer and multi-fration model. In the water olumn the ma onentration 1, 2, et. to i are defined. In the bed 1,1, 2,1, et. to i,1 are defined for the firt layer and 1,2, 2,2, et. to i,2 for the eond layer and oneutively. See alo Figure 7.1. Water olumn Bed layer 1 Bed layer 2 Bed layer n Ma onentration 1, 2,... i Dry denity 1,1, 2,1,... i,1 Dry denity 1,2, 2,2,... i,2 Dry denity 1,n, 2,n,... i,n Figure 7.1 Definition keth for multi fration-layer The fration are defined by their ediment harateriti. For the oheive ediment fration thi give the following extenion to the formulae above for depoition and eroion. The depoition for the i mud fration i: i i i i D w b p (7.1) D i p D i p D i a probability ramp funtion of depoition: b max 0,min 1, 1 (7.2) i d Eroion of the top layer of the bed i onidered one inident alulated for one time tep updating the ediment fration ratio of the bed. In the Mud Tranport module, the and tranport deription i baed on the aumption that eroion take plae imultaneouly for both and and oheive ediment. Therefore, the eroion of eah layer i alulated uing the normal mud tranport eroion equation. Afterward, the fration of the ediment that may be kept in upenion under the preent hydrodynami ondition i alulated. DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 25

31 Multi-Layer and Multi-Fration Appliation 7.2 Dene Conolidated Bed For a dene onolidated bed the eroion rate from the top layer j, an be alulated in the following way: j E j total E0 j E m, p (7.3) E j p E i a probability ramp funtion of eroion and E 0 i the erodibility. j p E b max 0, 1 j e (7.4) The eroion rate for the fration i i then alulated a: E i j M M i, j total, j E j total, (7.5) in whih M i the ma of ediment in the layer j. 7.3 Soft, Partly Conolidated Bed Similarly for a oft, partly onolidated bed: E j total 0.5 j exp j j E0 b (7.6) e The eroion rate for the fration i i then alulated a: E i j M M i, j total, j E j total, (7.7) Eah layer in the bed ontain a ertain onentration of ediment defined by a dry denity exluding water ontent. Thi denity i aigned the et of fration applied. For example 60 perent partile < 63 m and 40 perent fine and. Thi ratio i not fixed but an vary throughout the imulation, dependent on the advetion-diperion proee. An aount i kept of the ediment ratio for the bed. Thi allow for a grain orting proe to take plae. DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 26

32 Morphologial Feature 8 Morphologial Feature 8.1 Morphologial Simulation The morphologial evolution i ought to be inluded by updating the bathymetry for every time tep with the net edimentation rate. Thi enure a table evolution in the model that will not detabilie the hydrodynami imulation. n 1 n n bat bat neted (8.1) bat n bat n+1 n Bathymetry at preent time tep Bathymetry at next time tep Time tep The Mud Tranport module alo allow the morphologial evolution to be peeded up in the following way. n 1 n n bat bat neted Speedup (8.2) Speedup i a dimenionle fator. Thi fator i relevant for ae the edimentation proee are governed by yli event, uh a tide or eaonal variation. Thi doe NOT apply to tohati event, uh a torm. The thikne of the individual bed layer i updated at the ame time a the bathymetry. Thi i not the ae for the upended onentration. 8.2 Bed Update The bed layer i updated uing the following logiti (only 1 layer and 1 fration i onidered) I i i 1. The net depoition i alulated a ND D E 2. The bed ma M i alulated. 3. If net eroion our (ND > 0) and M + ND < 0, the depoition and eroion rate are adjuted uh that M + ND = 0. i0 t DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 27

33 Morphologial Feature 4. The bed layer thikne, H, and denity,, are updated a: H new bed H H old bed old bed ND t i ND t old bed,, ND 0 ND 0 (8.3) old H bed ND t (8.4) new bed old bed new H bed DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 28

34 Referene 9 Referene /1/ Anderen, T.J., Seaonal variation in erodibility of two temperate mirotidal mudflat. Etuarine, Coatal and Shelf Siene 56, pp /2/ Anderen, T.J. & Pejrup, M., Supended ediment tranport on a temperate, mirotidal mudflat, the Danih Wadden Sea. Marine Geology 173, pp /3/ Burt, T.N., Field ettling veloitie of etuary mud. In Etuarine Coheive Sediment Dynami. Leture Note on Coatal and Etuarine Studie. (Mehta, A.J., ed.). Springer Verlag, Berlin, pp /4/ Delo, E.A., Okenden, M.C., "Etuarine Mud Manual" HR Wallingford, Report SR309, May /5/ Dyer, K.R., Coatal and Etuarine Sediment Dynami. John Wile & Son, pp 342. /6/ Edelvang, K. & Auten, I., The temporal variation of flo and feal pellet in a tidal hannel. Etuarine, Coatal and Shelf Siene 44, /7/ Eima, D., Supended matter in the aquati environment. Springer Verlag, pp 315. /8/ Ekebjærg, L. and Juteen, P., "An Expliit Sheme for Advetion-Diffuion Modelling in Two Dimenion". Comp. Meth. App. Meh. Eng., pp /9/ Engelund, F., Fredøe, J., "A ediment tranport model for traight alluvial hannel". Nordi Hydrology 7, pp /10/ Fenton and MKee, "On Calulating the Length of Water Wave". Coatal Engineering, Vol. 14, Elevier Siene Publiher BV, Amterdam, pp /11/ Fredøe, J., "Mean Current Veloity Ditribution in Combined Wave and Current". Progre Report No. 53, ISVA, Tehnial Univerity of Denmark. /12/ Fredøe, J., "Turbulent boundary layer in wave-urrent motion". J Hydr Eng, A S C E, Vol 110, HY8, /13/ Gibb, R.J Etuarine flo: Their ize, ettling veloity and denity. Journal of Geophyial Reearh 90, pp /14/ Grihanin, K.V, and Lavygin, A.M., "Sedimentation of dredging ut in and bottom river". PIANC, Bulletin, No. 59, pp DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 29

35 Referene /15/ Jonon, I.G. and Carlen, N.A., "Experimental and Theoretial Invetigation in an Oillatory Rough Turbulent Boundary Layer". J. Hydr. Re. Vol. 14, No. 1, pp /16/ Krone, R.B., "Flume Studie of the Tranport of Sediment in Etuarial Proee". Hydrauli Engineering Laboratory and Sanitary Engineering Reearh Laboratory, Univ. of California, Berkely, California, Final Report. /17/ Krone, B The ignifiane of aggregate propertie to tranport proee. In: Mehta A.J. (Ed) Etuarine Coheive Sediment Dynami, Springer Verlag, pp /18/ Leonard, B.P., "A table and aurate onvetive modelling proedure baed on quadrati uptream interpolation". Comput. Meth. Appl. Meh. Eng. 19, pp /19/ Leonard, B.P., "The ULTIMATE Conervative Differential Sheme Applied to Unteady One- Dimenional Advetion". Comput. Meth. Appl. Meh. Eng. 88, /20/ Mehta, A.J., Hayter, E.J., Parker, W.R., Krone, R.B., Teeter, A.M., "Coheive Sediment Tranport. I: Proe Deription". Journal of Hydrauli Eng., Vol. 115, No. 8, Aug. 1989, pp /21/ Parhure, T. M. and Mehta, A.J., Eroion of oft oheive ediment depoit. Journal of Hydrauli Engineering - ASCE 111 (10), /22/ Partheniade, E., Eroion and depoition of oheive oil. Journal of the Hydrauli Diviion Proeeding of the ASCE 91 (HY1), /23/ Pejrup, M., 1988a. Floulated upended ediment in a miro-tidal environment. Sedimentary Geology 57, /24/ Pejrup, M., Laren, M. and Edelvang, K., A fine-grained ediment budget for the Sylt-Rømø tidal bain. Helgoländer Meereunteruhungen 51, /25/ Potma, H., "Sediment Tranport and Sedimentation in the Etuarine Environment". In: Lauff G.H: (Ed) Etuarie AAAS Publ. 83, p /26/ Rijn, L.C., "Sediment Tranport, Part I Bed Load Tranport". Journal of Hydrauli Engineering, Vol. 110, No. 10, Otober, /27/ Rijn, L.C., "Sediment Tranport, Part II Supended Load Tranport". Journal of Hydrauli Engineering, Vol. 110, No. 10, Otober, /28/ Rijn, Van L.C., "Handbook on Sediment Tranport by Current and Wave". Delft Hydrauli, Report H461, June 1989, pp DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 30

36 Referene /29/ Sanford, L.P. and Maa, J.P., A unified eroion formulation for fine ediment. Marine Geology 179 (1-2), /30/ Soulby R.L., Hamm L., Klopman G., Myrhaug D., Simon R.R., and Thoma G.P., "Wave-urrent interation within and outide the bottom boundary layer". Coatal Engineering, 21, /31/ Swart, D.H., "Offhore ediment tranport and equilibrium beah profile". Delft Hydr. Lab. Publ., 1312, Delft Univ. tehnology Di., Delft. /32/ Teeter, A.M., "Vertial Tranport in Fine-Grained Supenion and Nearly-Depoited Sediment". Etuarine Coheive Sediment Dynami, Leture Note on Coatal and Etuarine Studie, 14, Springer Verlag, pp /33/ Teion, C., "Coheive upended ediment tranport: feaibility and limitation of numerial modelling". Journal of Hydrauli Reearh, Vol. 29, No. 6. /34/ Van Leuen, W Aggregation of partile, ettling veloity of mud flo. A review: In: Dronker & Van Leuen (Ed.): Phyial proee in etuarie. Springer Verlag, pp /35/ Van Olphen, H "An introdution to lay olloid hemitry. ". Interiene Publiher (John Wiley & Son) pp 301. /36/ Van Straaten, L.M.J.U. and Kuenen, Ph.H., Tidal ation a a aue of lay aumulation. Journal of Sedimentary Petrology 28 (4), /37/ Whitehoue, U.G., Jeffrey, L.M., Debbreht, J.D Differential ettling tendenie of lay mineral in aline water. In: Swineford (Ed) Clay and lay mineral. Proeeding 7th National Conferene, Pergamon Pre, p /38/ Yalin, M.S, "Mehani of Sediment Tranport". Pergamon Pre Ltd. Hendington Hill Hall, Oxford. DHI - MIKE 21 & MIKE 3 Flow Model FM - Mud Tranport Module 31