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L’acqua nei suoli e nel sottosuolo
Richards 1-D
Riccardo Rigon
G.VDuNoyer,Oldredsandstones,Doneen
Rock,Mallow,Co,Cork
R. Rigon
Iverson,2000;D’Odoricoetal.,2003,
CordanoeRigon,2008
!2
L’equazione di Richards su un versante piano
s
L’equazion...
R. Rigon
Il termine transiente della pressione si può calcolare se si
assume K ~ costante e trascurando i termini sorgente...
R. Rigon
!4
Dove:
L’equazione Richards 1-D
Capacità idraulica
dei suoli
Pressione dell’acqua
Conducibilità idraulica
verti...
R. Rigon
L’equazione diventa LINEARE e, trovata una soluzione con
un impulso unitario istantaneo al contorno, la soluzione...
R. Rigon
!6
Assunta che f(t,z) sia una soluzione dell’equazione
di Richard - 1D sottoposta ad un impulso di pressione
Ques...
R. Rigon
!7
L’equazione Richards 1-D
Una soluzione dell’equazione di Richards
R. Rigon
!8
Per un impulso di precipitazione di intensità costante, la soluzione
può scriversi:
D’Odoricoetal.,2003
L’equa...
R. Rigon
In quel caso l’equazione ammette una soluzione analitica
D’Odoricoetal.,2003
!9
L’equazione Richards 1-D
TD :=
z2...
R. Rigon
!10
In quel caso l’equazione ammette una soluzione analitica
L’equazione Richards 1-D
R. Rigon
L’EQUAZIONEDIRICHARDS1D
D’Odoricoetal.,2003
!11
TD
TD
TD
TD
Una soluzione dell’equazione di Richards
R. Rigon
I metodi di soluzione analitica dell’equazione di avvezione-dispersione
(anche non lineare) che risulta dall’equa...
R. Rigon
L’EQUAZIONEDIRICHARDS1D
!13
Soluzioni dell’equazione di Richards
R. Rigon
!14
R. Rigon
!15
R. Rigon
!16
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12.10 acqua nei suoli-richards-1d

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Solving the 1-D Richards equation

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12.10 acqua nei suoli-richards-1d

  1. 1. L’acqua nei suoli e nel sottosuolo Richards 1-D Riccardo Rigon G.VDuNoyer,Oldredsandstones,Doneen Rock,Mallow,Co,Cork
  2. 2. R. Rigon Iverson,2000;D’Odoricoetal.,2003, CordanoeRigon,2008 !2 L’equazione di Richards su un versante piano s L’equazione di Richards semplificata
  3. 3. R. Rigon Il termine transiente della pressione si può calcolare se si assume K ~ costante e trascurando i termini sorgente ⇤⇥ ⇤t = D0 cos2 ⇤2 ⇥ ⇤t2 !3 L’equazione Richards 1-D: C( ) @ @t = Kz 0 @2 @z2 D0 := Kz 0 C( ) Diffusività idraulica L’equazione di Richards semplificata
  4. 4. R. Rigon !4 Dove: L’equazione Richards 1-D Capacità idraulica dei suoli Pressione dell’acqua Conducibilità idraulica verticale di riferimento Una soluzione dell’equazione di Richards
  5. 5. R. Rigon L’equazione diventa LINEARE e, trovata una soluzione con un impulso unitario istantaneo al contorno, la soluzione dipendente da una precipitazione variabile viene a dipendere dalla convoluzione di questa soluzione e la precipitazione. ⇤⇥ ⇤t = D0 cos2 ⇤2 ⇥ ⇤t2 !5 L’equazione Richards 1-D Una soluzione dell’equazione di Richards
  6. 6. R. Rigon !6 Assunta che f(t,z) sia una soluzione dell’equazione di Richard - 1D sottoposta ad un impulso di pressione Questo significa che: Condizione iniziale Soluzione Impulsiva Variazione di pressione in superficie dovuto alla precipitazione Una soluzione dell’equazione di Richards
  7. 7. R. Rigon !7 L’equazione Richards 1-D Una soluzione dell’equazione di Richards
  8. 8. R. Rigon !8 Per un impulso di precipitazione di intensità costante, la soluzione può scriversi: D’Odoricoetal.,2003 L’equazione Richards 1-D Una soluzione dell’equazione di Richards
  9. 9. R. Rigon In quel caso l’equazione ammette una soluzione analitica D’Odoricoetal.,2003 !9 L’equazione Richards 1-D TD := z2 D0 Tempo scala infiltrazione Una soluzione dell’equazione di Richards
  10. 10. R. Rigon !10 In quel caso l’equazione ammette una soluzione analitica L’equazione Richards 1-D
  11. 11. R. Rigon L’EQUAZIONEDIRICHARDS1D D’Odoricoetal.,2003 !11 TD TD TD TD Una soluzione dell’equazione di Richards
  12. 12. R. Rigon I metodi di soluzione analitica dell’equazione di avvezione-dispersione (anche non lineare) che risulta dall’equazione di Richards, si possono trovare nei libri che trattano la diffusione del calore (l’equazione linearizzata è la stessa), per esempio in Carslaw e Jager, 1959, pg 357. ! In genere, le strategie di soluzione sono 4 e basate: ! - Sul metodo di separazione delle variabili - L’uso delle trasformate di Fourier - L’uso delle trasformate di Laplace - Metodi geometrici basati sulla simmetria delle equazione (e.g. Kevorkian, 1993) ! Tutti i metodi mirano a ridurre l’equazione differenziale alle derivate parziali ad un sistema di equazioni differenziali ordinarie L’EQUAZIONEDIRICHARDS1D !12 Una soluzione dell’equazione di Richards
  13. 13. R. Rigon L’EQUAZIONEDIRICHARDS1D !13 Soluzioni dell’equazione di Richards
  14. 14. R. Rigon !14
  15. 15. R. Rigon !15
  16. 16. R. Rigon !16

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