Numerical modeling, Finite difference method; Finite element method method; Finite element method, Computer models

M odu le 4 – ( L1 2 - L1 8 ) : “ W a t e r sh e d M ode lin g”

  

St a n da r d m ode lin g a ppr oa ch e s a n d cla ssifica t ion s, syst e m con ce pt St a n da r d m ode lin g a ppr oa ch e s a n d cla ssifica t ion s, syst e m con ce pt

for w a t e r sh e d m ode lin g, ove r a ll de scr ipt ion of diffe r e n t h ydr ologic

pr oce sse s, m ode lin g of r a in fa ll, r u n off pr oce ss, su bsu r fa ce flow s a n d gr ou n dw a t e r flow g

  N u m e r ica l W a t e r sh e d N u m e r ica l W a t e r sh e d 1 7 1 7 M ode lin g

L1 7 – L1 7 N u m e r ica l W a t e r sh e d M ode lin g M d li

   

Topics Cove r e d Topics Cove r e d

    Physically based wat ershed m odeling, Physically based wat ershed m odeling, Num erical m odeling, Finit e difference Num erical m odeling, Finit e difference m et hod; Finit e elem ent m et hod, m et hod; Finit e elem ent m et hod, m et hod; Finit e elem ent m et hod m et hod; Finit e elem ent m et hod Com put er m odels Com put er m odels

    Physically based wat ershed Physically based wat ershed

Keywords: Keywords:

  m odeling, Num erical m odeling, FDM, FEM. m odeling, Num erical m odeling, FDM, FEM. ode ode

  g, u u e ca e ca ode ode

  g,

  g, , ,

  g,

W a t e r sh e d M ode lin g

• Transform at ion of rainfall int o runoff over a wat er
• Generat ion of flow hydrograph for t he out let • Generat ion of flow hydrograph for t he out let
• Use of t he hydrograph at t he upst ream end t o rout e t o t he downst ream end

  H d l i i l t i d l t h t i l t i • Hydrologic sim ulat ion m odels use m at hem at ical equat ions t o calculat e result s like runoff volum e or peak flow

• Com put er m odels allows param et er variat ion in space and p p

  p t im e – w it h use of num erical m et hods

• Ease in sim ulat ion of com plex rainfall pat t erns and het erogeneous wat ersheds

  12.00 ^15.00 _{e c)} ^0.0 _{5.0 ty}

  het erogeneous wat ersheds

  3.00 ^{6.00 9.00} _is ^{c har ge (m 3 /s e 5.0 10.0 15.0} _R ^{a in fa ll i n te rn si (mm/ h r)}

H ydr ologic M ode ls y g Model Type Example of Model Lumped Parameter Synder Unit Hydrograph p

  y y g p Distributed Kinematic wave Event HEC-1, SWMM Event

  

Continuous Stanford Watershed Model,

SWMM, HSPF, , , Physically based HEC-1, SWMM, HSPF Stochastic Synthetic stream flows Stochastic Synthetic stream flows Numerical

  Explicit kinematic wave Analytical Nash IUH Analytical Nash IUH

N e ce ssit y of D ist r ibu t e d m ode ls

   

  Flow of wat er in a wat ershed is a dist ribut ed process Flow of wat er in a wat ershed is a dist ribut ed process

  

  Models should be physically based

  

  Governing equat ions – St . Venant equat ions g q q

  

  Com put er m odels- based on t he St . Venant equat ions

  

  Allows com put at ion of flow rat e and wat er level as funct ions of space and t im e f d t i

  

  Model m ore closely approxim at es t he act ual unst eady non- uniform nat ure of flow propagat ion in channels p p g

  Rainf Rainf Channel flow al l Infiltration I filt ti

  Overland flow Interflow

H ydr ologic/ H ydr a u lic M ode lin g

   Hydrological / Hydraulic m odel- concept ual or

  physically based procedure- num erically solving hydrological processes diagnose or forecast hydrological processes - diagnose or forecast processes.

  

Physical based: descript ion of nat ural syst em using

basic m at hem at ical represent at ion of flows of m ass, basic m at hem at ical represent at ion of flows of m ass

m om ent um and various form s of energy.

  

Dist ribut ed: consider spat ial variat ion of variables &

param et ers .

   Applicat ions: Rainfall t o runoff , Surface wat er/

  groundwat er assessm ent , Flood/ drought predict ions, groundwat er assessm ent Flood/ drought predict ions Evaluat ion of wat ershed / cat chm ent m anagem ent st rat egies, River basin / Agricult ural wat er m anagem ent et c. m anagem ent et c Hydrologic/ Hydraulic Modeling

Ra in fa ll  Ru n off - > w a t e r sh e d A

  Fie ld da t a Con st r u ct ion M ode l Con ce pt u a liz a t ion M ode l Con ce pt u a liz a t ion Pe r for m a n ce Pe r for m a n ce cr it e r ia - M ode l M a t h e m a t ica l

  Ca libr a t ion M ode l M ode l Fie ld da t a Fie ld da t a & com pa r ison N u m e r ica l M ode l

  V a lida t ion Com pu t e r M ode l M ode l ve r ifica t ion M ode l ve r ifica t ion Se n sit ivit y An a lysis Re su lt s & D iscu ssion M ode l Ade qu a cy

  Fie ld da t a Fi ld d t P Post a u dit t dit B An a lysis

Ph ysica lly ba se d dist r ibu t e d m ode ls:

  Rainfall _{Channel phase flow} Infiltration Infiltration _{Overland flow} Rainfall

  Fig. Flow in a watershed – Typical flow pattern Flow at the Outlet _{Channel flow of watershed Overland flow} Infiltration Infiltration

  Fig. General concept of flow modeling

  Ph ysica lly Ba se d Ph ysica lly Ba se d M ode l M ode l – – Ove r la n d Ove r la n d Flow Flow Equ a t ion s Equ a t ion s Equ a t ion s Equ a t ion s Con t in u it y e qu a t ion Con t in u it y e qu a t ion M om e n t u m Equ a t ion M om e n t u m Equ a t ion I n it ia l a n d Bou n da r y con dit ion s y

  I C for overland is usually of dry bed condit ion. At t im e t = 0, h = 0 and q = 0 at all nodal point s Upst ream boundary condit ion is assum ed as zero inflow s; h = 0 and q = 0 at all t im es

  Gov. Equat ion for Channel Flow    

  Q Q A A   q 

  Equat ion of cont inuit y

   x  t _{2 2}       Q Q   Q Q

    h h 

  Mom ent um equat ion

    gA S  S  gA   _f

   

 t  x A

 x  

  

  Diffusion Diffusion

  h h  

    Q Q   A A

  1 _{2 / 3 1 / 2} S S     q  _{o f}

  1

  Q  R S A _{h f} x  x  t  n

  

  Q Q   A A



  Kinem at ic:

    q 

 x  t

  S = S

  f 

  q- lat eral inflow; Q- discharge in t he channel;

  A- area of flow in t he channel, S - bed slope; S S - frict ion slope of channel frict ion slope of channel.

  f f Solut ion Met hodologies 

  An a lyt ica l m e t h od: For t he given m at hem at ical form ulat ion, an analyt ical expression involving t he param et ers and t he independent variables are obt ained using various m at hem at ical procedures.

   Main lim it at ion- only for a sm all class of m at hem at ical

form ulat ions w it h sim plified governing equat ions, boundary

condit ions & geom et ry, analyt ical solut ions can be obt ained.

   Ph ysica l m e t h od: As t he m at hem at ical m odel represent s a real physical syst em , alt hough on cert ain idealized p y y , g assum pt ions, variables and param et ers of t he m odel can be considered as having physical dim ensions and can be analyzed som et im es in t he laborat ory or in t he field it self.

   The physical m odels are used less frequent ly since it is expensive, cum bersom e and difficult in pract ice.

   Com pu t a t ion a l m e t h od Com pu t a t ion a l m e t h od Com put at ional Met hod 

  I n t he com put at ional m et hod, t he solut ion is obt ained w it h t he help of som e approxim at e m et hods using a com put er. Com m only,

num erical m et hods are used t o obt ain solut ion

i t h in t he com put at ional m et hod. t t i l t h d

   Wider class of m at hem at ical form ulat ions & Wid l f t h t i l f l t i & advent of fast com put ers, com put at ional m odels have becom e t he m ost widely used m odels have becom e t he m ost widely used valuable t ool for solving t he engineering p problem s.

  Num erical Modeling 

Variet y of num erical m et hods such as

• - M e t h od of ch a r a ct e r ist ics

• - Fin it e D iffe r e n ce M e t h od ( FD M )

• - Fin it e V olu m e M e t h od ( FV M ) i i l h d ( ) - Fin it e Ele m e n t M e t h od ( FEM ) - Fin it e Ele m e n t M e t h od ( FEM ) - Bou n da r y Ele m e n t M e t h od ( BEM ) .

Finit e Difference Met hod 

  Cont inuous variat ion of t he funct ion concerned by a set C i i i f h f i d b of values at point s on a grid of int ersect ing lines.

   The gradient of t he funct ion are t hen represent ed by e g ad e o e u c o a e e ep ese ed by differences in t he values at neighboring point s and a finit e difference version of t he equat ion is form ed.

   

  At point s in t he int erior of t he grid, t his equat ion is used At point s in t he int erior of t he grid t his equat ion is used t o form a set of sim ult aneous equat ions giving t he value of t he funct ion at a point in t erm s of values at nearby point s. i t

   At t he edges of t he grid, t he value of t he funct ion is fixed, or a special form of finit e difference equat ion is , p q used t o give t he required gradient of t he funct ion. Met hod of charact erist ics ( MOC) 

  MOC MOC - reduce a part ial different ial equat ion t o a fam ily d i l diff i l i f il of ordinary different ial equat ions along which t he solut ion can be int egrat ed from som e init ial dat a given on a suit able hyper surface

  

  For a first - order PDE, MOC discovers curves ( called charact erist ic curves or charact erist ics) along which PDE charact erist ic curves or charact erist ics) along which PDE becom es an ODE. I t is solved along t he charact erist ic curves & t ransform ed int o a solut ion for original PDE.

  

  V Variant of FDM – suit able for solving hyperbolic i t f FDM it bl f l i h b li equat ions

  

  MOC t o sim ulat e advect ion dom inat ed t ransport p

  

  Track idealized part icles t hrough flow field

  

  Efficient & m inim ize num erical inst abilit ies Finit e Elem ent Met hod 

  The region of int erest is divided in a m uch m ore flexible way flexible way

   The nodes at which t he value of t he funct ion is found have t o lie on a grid syst em or on a flexible m esh

  

  The boundary condit ions are handled in a m ore h b d d h dl d convenient m anner.

   Direct approach, variat ional principle or weight ed Direct approach, variat ional principle or weight ed residual m et hod is used t o approxim at e t he governing different ial equat ion

  Boundary Elem ent Met hod 

  The part ial different ial equat ions describing t he dom ain, is t ransform ed in t o an int egral equat ion dom ain, is t ransform ed in t o an int egral equat ion relat ing only t o boundary values.

   The m et hod is based on Green’s int egral t heorem . 

  The boundary is discret ized inst ead of t he dom ain. h b d d d d f h d

  

  A 3- Dim ensional problem reduces t o a 2- Dim ensional problem and 2- Dim ensional problem in 2 Dim ensional problem and 2 Dim ensional problem in t o 1- Dim ensional problem .

  

  BEM is ideally suit ed t o t he solut ion of m any t wo and t hree- dim ensional problem s in elast icit y and pot ent ial h di i l bl i l i i d i l t heory Analyt ical Solut ion–Kinem at ic wave S S _{f f}

    n _o

  5    

  3

• Analyt ical solut ion for one- dim ensional kinem at ic wave equat ions is given by above equat ions ( Jaber and Moht ar, 2003) ; t is t im e of concent rat ion ( sec) ; t is

  c r rainfall durat ion ( sec) ; t is t he sim ulat ion t im e ( sec) ; f

  Lw is t he lengt h of w at ershed ( m ) in t he direct ion of m ain slope. ( Jaber, F.H., and Moht ar, R.H. ( 2003) . “ St abilit y and accuracy of t wo dim ensional kinem at ic wave overland flow m odeling.” Advances in Wat er l d fl d li ” Ad i W t Resources, 26( 11) , 1189- 1198) .

  

Fin it e D iffe r e n ce M e t h od ( FD M )

• FD M : Calculat ions are perform ed on a grid placed over t he ( x, t ) plane
• Flow and wat er surface elevat ion are obt ained for  Flow and wat er surface elevat ion are obt ained for increm ent al t im e and dist ances along t he channel

   Ex plicit m e t h ods: calculat es values of velocit y &

  d dept h over a grid syst em based on a previously known h d b d l k dat a for t he river reach

   I m plicit m e t h ods: I m plicit m e t h ods:

  set up a series of sim ult aneous set up a series of sim ult aneous num erical equat ions over a grid syst em for t he ent ire river & equat ions are solved at each t im e st ep.

  Fig: x - t pla n e for fin it e diffe r e n ce sch e m e

Typical St eps for FDM m odel

• – Governing Part ial

  Different ial Equat ions wit h Subsidiary condit ions Subsidiary condit ions

• – Divide dom ain int o Grids – Transform at ion by Finit e y

  Difference Met hod

• – Syst em of difference equat ions equat ions
• – Applicat ion of Boundary Condit ions

  I,J+1 Δy I-1 J

  I J I+1 J

  Δy I 1,J I,J I+1,J Δx I,J-1

• – Solve by direct or it erat ive m et hod
• – Solut ion
• – Solut ion

Finit e Difference Schem e

  There are t hree com m only used finit e difference approxim at ions finit e difference approxim at ions for t he solut ion of PDE

        

      

      h h c) Cent ral difference schem e. _{1 1 2 2 I I} I h h h x x

     

  I I x x

  1 I

  c) Cent ral difference schem e

  g sought b) Forward difference schem e

  I h h h

  a) Backward difference

  1 I

   

  I  x x 

  schem e: We consider t he node in t he backward direct ion of t he node at which gradient is

        

      

  I h h h

  1 I

         

  _{i j i 1 j i+1 j} Finit e Difference Schem e

  ∆x ∆x ^{i, j i-1, j i+1, j} Cross-sectional view in x-t plane h , Q , t ₁ ^h ₁ ^{, Q} ₁ ^{, t} ₁ ^h ₂ ^{, Q} ₂ ^{, t} ₂ x-t plane p , , _{1 1} ^, ₁ ^, _{1 2} ^, ₂ ^, ₂

  ∆t _{h , Q , t h 1 , Q 1 , t h 2 , Q 2 , t} ∆x ∆x

Fin it e D iffe r e n ce Appr ox im a t ion s

    Ex plicit I m plicit j  ^{1 j  1 j} _{j  1 j  1 j j} u u u

    _{i i i} u u u u u

      _{i i 1 i i 1}  

  

  Tem poral derivat ive

    t  t  t _{j  1 j } 2  t _{1 j j}

  Spat ial derivat ive j j j

  u u u u u    _{i 1 i i 1 i}

    u u u

    _{i i 1 i}

₁

 

     ( 1   )   x  x  x  x

  2  x Spat ial and t em poral Spat ial derivat ive is writ t en using derivat ives use unknow n t im e t erm s on know n t im e line t erm s on know n t im e line lines for com put at ion lines for com put at ion

Fin it e Ele m e n t M e t h od

  

  1D- Kinem at ic & Diffusion Wave Models for Overland Flow

   

  One- dim ensional m odel wit h linear line elem ent s One- dim ensional m odel wit h linear line elem ent s

  

  Apply Galerkin FEM for 1D cont inuit y equat ion

  

• ( 1)

  

• ( 2) ( 2)

   

  Expansion of Eq considering it for one elem ent is given as Expansion of Eq considering it for one elem ent is given as

  

• ( 3)

Fin it e Ele m e n t M e t h od Fin it e Ele m e n t M e t h od

  

Shape funct ion N for a linear elem ent can be expressed as [ N] =

[ N1 N2] Where N = 1- ( x/ L) and N = x/ L i j

   Equat ion can be w rit t en in m at rix form as follows: E t i b it t i t i f f ll

  

• ( 4) ( 4)

  

Assem bling t he overland flow line elem ent s and applying im plicit

finit e difference schem e for t im e dom ain

  

• ( 5)

Fin it e Ele m e n t M e t h od

  

  Aft er rearranging t erm s, t he final form of equat ion as:

  

  Syst em of equat ions will be solved aft er applying t he y q pp y g boundary condit ions

  Typical Finite element Grid map

  Ca se st u dy: Harsul Wat ershed

  ( Venkat a Reddy, 2007) ( V k R dd 2007)

  Locat ion- Nashik dist rict , Maharasht ra, I ndia

  Maj or Soil class – Gravelly loam Rem ot ely Sensed Dat a- I RS 1D LI SS I I I im agery of January, 1998 J 1998 Them at ic Maps- Drainage, DEM, Slope and LU/ LC

• 0.25 km

  

  Channel flow

  

  Overland flow

  

  Manning’s roughness

  

  Channel flow M i ’ h

  

  Overland flow

  

  Slope

  Slope

  

  

  Average bed widt h - 18 m

  

  Channel flow Elem ent lengt h

  

  Channel flow elem ent s - 22

  

  Overland flow nodes - 188

  

  Overland flow nodes 188

  

  Overland flow elem ent s - 144

  Finite element grid map

  Ca se st u dy: Harsul Wat ershed

  ( Venkat a Reddy, 2007) ( Venkat a Reddy 2007)

  

  Diffusion wave- GAML m odel

   

  Calibrat ion - 3 Rainfall event s Calibrat ion - 3 Rainfall event s

  

  Validat ion - 2 Rainfall event s

  Calibrated parameters for rainfall events (Harsul)

  10 ^{20 (m 30 a rg e 3 /s ) 5 10 15 ll in te n s ity m m/h r) 6 8 10 g e( m 3 /s ) 2 4 6 8 Rainfall 10 n te ns ity /hr ) Observed 6 8 10 e( m 3 /s ) 0.0 10.0 20.0 n ten s ity /hr ) Rainfall Simulated Observed 10} _{200 400 600} Time(min) ^{D isch a 15 20} _Rainfall ^{25 Ra in fa (m 2 D 4 500 1000 1500 2000 2500 3000 is c har g 10 12 14 16 18 Simulated 20 R a in fa ll i n (m m / 2 Di 4} _{250 500 750 1000} ^{s c h a rg 30.0 40.0 50.0 R a in fa ll i n (m m /} _{Rainfall Observed Simulated Time(min) Time(min)} August 22, 1997 September 23, 1997 September 26, 1997

  2 ^{2.5 3 3.5 4 g e( m 3 /s ) 5 Rainfall 10 n tens ity /h r) Observed Simulated 40 50 60 70 g e( m 3 /s ) 0.0 10.0 20.0 s ity (mm/ hr ) Rainfall} _{0.5 1 1.5} ² 500 1000 1500 ^{D is c har g 10 15 20 R a in fa ll i n (m m 10 20 D 30 500 1000 1500 is c har g 30.0 40.0 50.0 R a in fa ll In te n s Rainfall Observed Simulated 500 1000 1500} _Time(min) ^{500 1000 1500} _Time(min) August 21, 1997 August 23, 1997

  

Observed & simulated hydrographs of calibration & validation rainfall events

g , g ,

Re fe r e n ce s Re fe r e n ce s

  Raj Vir Singh ( 2000) , Wat ershed Planning and Managem ent , Yash • Publishing House J.V.S Murt hy ( 1991) , Wat ershed Managem ent , New Age int ernat ional • P bli Publicat ions t i

   Venkat a Reddy K., Eldho T. I ., Rao E.P. and Hengade N. ( 2007) “ A kinem at ic wave based dist ribut ed wat ershed m odel using FEM, GI S and

rem ot ely sensed dat a. Journal of Hydrological Processes, 21, 2765 rem ot ely sensed dat a ” Journal of Hydrological Processes 21 2765-

2777

   Chow , V.T., Maidm ent , D.R., and Mays, L.W. ( 1988) . Applied Hydrology, McGraw- Hill, I nc., New York. y gy, , ,

   Bedient , P.B. and Huber W.C.( 1988) . Hydrology and flood plain analysis, Addison- Wesley Publishing Com pany., London

   Cunderlik, J. M. ( 2003) . “ Hydrologic m odel select ion for t he CFCAS proj ect : Assessm ent of Wat er Resources Risk and Vulnerabilit y t o

changing Clim at ic Condit ions – Proj ect Report 1” , Depart m ent of Civil

and Environm ent al engineering, Universit y of West ern Ont ario

Tu t or ia ls - Qu e st ion !.?

   I llust rat e t he necessit y of physically based w at ershed m odeling. w at ershed m odeling

    Develop a concept ual m odel for a t ypical Develop a concept ual m odel for a t ypical w at ershed, for physically based m odeling. w at ershed, for physically based m odeling.

  Describe t he m erit s & dem erit s of physical Describe t he m erit s & dem erit s of physical m odeling. m odeling.

  Prof. T I Eldho, Department of Civil Engineering, IIT Bombay

Se lf Eva lu a t ion - Qu e st ion s!. Q

   Why dist ribut ed m odeling required for w at ershed m odeling?. w at ershed m odeling?

   I llust rat e various solut ion m et hodologies for problem solut ion. problem solut ion

   Different iat e bet w een explicit & im plicit FDM schem es. schem es.

  

Describe FEM solut ion m et hodology w it h salient feat ures

  Prof. T I Eldho, Department of Civil Engineering, IIT Bombay P f T I Eldh D t t f Ci il E i i

  IIT B b

Assign m e n t - Qu e st ion s?. g Q

   Wit h t he help of a flow chart , illust rat e hydrologic/ hydraulic m odeling. hydrologic/ hydraulic m odeling

   Describe FDM solut ion m et hodology w it h salient feat ures salient feat ures.

   Different iat e bet w een FDM & MOC.  

  Describe BEM solut ion m et hodology w it h Describe BEM solut ion m et hodology w it h salient feat ures.

  

Prof. T I Eldho, Department of Civil Engineering, IIT Bombay

Un solve d Pr oble m !. Un solve d Pr oble m !

    St udy t he salient feat ures & problem s of St udy t he salient feat ures & problem s of your wat ershed area. I dent ify how various your wat ershed area. I dent ify how various t t h d h d I d I d t if h t if h i i physically based m odels can be used for physically based m odels can be used for various problem solut ions such as: rainfall various problem solut ions such as: rainfall-- various problem solut ions such as: rainfall various problem solut ions such as: rainfall runoff, flooding, drought m anagem ent , runoff, flooding, drought m anagem ent , rainw at er harvest ing, soil erosion et c. rainw at er harvest ing, soil erosion et c.

  g, Prof. T I Eldho, Department of Civil Engineering, IIT Bombay

  g,

  Dr. T. I. Eldho Dr. T. I. Eldho Professor, Professor, Department of Civil Engineering, Department of Civil Engineering, p p g g g g Indian Institute of Technology Bombay, Indian Institute of Technology Bombay, Mumbai, India, 400 076. Mumbai, India, 400 076. Email: Email: Email: Email: eldho@iitb.ac.in eldho@iitb.ac.in eldho@iitb.ac.in eldho@iitb.ac.in Phone: (022) – Phone: (022) – 25767339; Fax: 25767302 25767339; Fax: 25767302