Numerical investigations of hillslopes with variably saturated subsurface and overland flows

Transcription

1 Numerical investigations of hillslopes with variably saturated subsurface and overland flows ARC DYNAS H. Beaugendre, T. Esclaffer, A. Ern and E. Gaume DYNAS Workshop 06-08/12/04 DYNAS Workshop 06-08/12/04 p.1

2 Outline Introduction: hillslope models Governing equations and boundary conditions Computational schemes Results Conclusions DYNAS Workshop 06-08/12/04 p.2

3 Introduction infiltration runoff exfiltration DYNAS Workshop 06-08/12/04 p.3

4 Introduction mechanisms leading to surface runoff in hillslopes Hortonian models (1933): flood solely caused by rainfall above infiltration capacity of soil partial area contribution concept (Dunne & Black 70): segmentation into infiltration, runoff, and exfiltr. zones field studies: site-dependent; laboratory tests: difficult need for computer simulations DYNAS Workshop 06-08/12/04 p.4

5 Introduction questions to address: impact of soil hydraulic parameters partition of storm hydrographs: relative importance of surface and subsurface water do we need to model surface runoff? DYNAS Workshop 06-08/12/04 p.5

6 Governing equations Richards equation { t θ(h) + x v(h) = 0 v(h) = k(h) x (h + z) θ: water volume fraction; h: soil water pressure head; k: unsaturated hydraulic conductivity; v: Darcy flow velocity soil hydrodynamic functions h θ(h) and h k(h) DYNAS Workshop 06-08/12/04 p.6

7 Governing equations van Genuchten Mualem (VGM) model θ(h) = θ(h) θ r θ s θ r = k(h) = k s k r ( θ(h)). { (1 + ( αh) n ) m, h < 0, 1, h 0, n, θ s, θ r, α: VGM parameters Mualem model for k r DYNAS Workshop 06-08/12/04 p.7

8 Comparison of two strategies Obstacle Type Model (OTM): assumption: height of the overland flow and re-infiltration processes are neglected Diffusive Wave model (DWM): modeling the overland flow: simplified form of the Saint-Venant equations DYNAS Workshop 06-08/12/04 p.8

9 OTM code: boundary conditions Rainfall intensity: v r = ie z PSfrag replacements lacements lacements n = v r n lacements n = h v r = n0 Ω n = h v = Ω r n0 t h = Ω 0 Ω Ω + t Ω Ω r Ω Ω + t Ω r Ω + t Ω r non saturated zone water table saturated zone v(h) n = v r n h = 0 Ω t Ω Ω + t Ω r Ω t : non-saturated region h 0 and v n = v r n Ω + t : saturated region h = 0 and v n v r n Ω r : no-flow BC (BC1) or constant total hydraulic head (BC2) DYNAS Workshop 06-08/12/04 p.9

10 OTM code: computational schemes unstructured, non-uniform triangulations Richards equation discretized by P 1 conforming finite elements nonlinear soil hydrodynamic behavior handled with Newton s method unsteady obstacle-type problem handled with fixed-point iteration DYNAS Workshop 06-08/12/04 p.10

11 OTM code: computational schemes steady-state solution exists whenever i k s assuming Ω + t known, set V Ω + t = { } φ H 1 (Ω); φ = 0 on Ω + t, a Ω + t (h, φ) = k(h) ( x h + e z ) x φ Ω + (v r n) φ. Ω t \ Ω + t DYNAS Workshop 06-08/12/04 p.11

12 OTM code: computational schemes Fixed-point scheme: choose Ω + t solve h V Ω + t s.t. a Ω + t (h, φ) = 0, φ V Ω + t check conditions: (ii) h 0 on Ω t and (iii) v(h) n v r n on Ω + t both satisfied: done only one satisfied: move Ω + t one mesh cell (or more) up or down depending on the condition DYNAS Workshop 06-08/12/04 p.12

13 OTM code: computational schemes unsteady problem: implicit Euler scheme for a time step k 1: set ( Ω + t )k+1 = ( Ω + t )k solve Richards equation in V ( Ω + t ) k+1 1 δt Ω ( θ(h k+1 ) θ(h k ) ) φ + a ( Ω + t )k+1 (h k+1, φ) = 0 update ( Ω + t )k+1 according to head and flux conditions at Ω t DYNAS Workshop 06-08/12/04 p.13

14 Modeling the runoff wave does the runoff flow influence the water table dynamics? non saturated zone runoff wave water table saturated zone DYNAS Workshop 06-08/12/04 p.14

15 DWM code: modeling the runoff wave 1D St-Venant equation posed on domain Ω + t { t y + x (uy) = w t u + u x u = g x h + g(i J) y: water height, u: speed, h: head, I: slope, J: friction, w: source term due to rainfall and exfiltration Maning Strickler correlation u 2 = k 2 ms Jy 4 3 DYNAS Workshop 06-08/12/04 p.15

16 DWM code: modeling the runoff wave diffusive wave approximation: neglect inertia terms in momentum balance assume hydrostatic pressure in runoff flow: y = h ) t h + x (k ms h 3 5 ( x (h) + S o ) = w this PDE plays the role of BC for Richards equation on Ω + t DYNAS Workshop 06-08/12/04 p.16

17 Numerical Results Metric-scale problem influence of soil hydrodynamic parameters influence of BC influence of runoff wave Hectometric-scale problem effects of geometrical properties influence of runoff wave DYNAS Workshop 06-08/12/04 p.17

18 Metric-scale problem Final water table Ls 1 m 0.8 m initial water table 0.7 m 1.4 m a constant rainfall intensity is imposed: i/k s = 10% output: saturated length: Ls, exfiltration flow rate: Q exf DYNAS Workshop 06-08/12/04 p.18

19 Impact of hydrodynamic parameters θ r (-) θ s (-) α (1/m) n (-) k s (m/h) SAND YLC SCL SCL YLC SAND SCL soil YLC soil SAND soil kr Ls/L (%) h (cm) t/te DYNAS Workshop 06-08/12/04 p.19

20 Impact of Ω r and runoff wave non saturated zone non saturated zone water table water table saturated zone initial water table saturated zone BC1 BC Ls/L (%) Qrunoff/Qrain (-) OTM code: BC1 BC2 DWM code: BC1 BC OTM code: BC1 BC2 DWM code: BC1 BC Time t (h) Time t (h) DYNAS Workshop 06-08/12/04 p.20

21 Hectometric-scale problem y i water table L Ls S o g Initial water table D x constant rainfall intensity: i = 30mm/h, BC2 BC IC: horizontal water table located at the toe of the slope θ r (-) θ s (-) α (1/m) n (-) k s (m/h) Sand OW DYNAS Workshop 06-08/12/04 p.21

22 Impact of geometrical properties L = 30, D = 1 L = 50, D = 1 L = 50, D = 2 T e (h) Dk ss o il 44.4% 66.7% 33.3% L s /L 45.6% 67.3% 34.7% L=50m, D=1m, So=10% L=30m, D=1m, So=10% L=50m, D=2m, So=10% Ls/L (%) Qexf (kg/h) t/te 0.2 L=50m, D=1m, So=10% L=30m, D=1m, So=10% L=50m, D=2m, So=10% t/te DYNAS Workshop 06-08/12/04 p.22

23 Impact of runoff wave L = 50m, D = 1m and k ms = 10m 1/3 s 1 similar results for L s /L slight difference for the exfiltration flux (BC) Ls/L (%) Qexf/Qrain (-) DWM code: BC2 OTM code: BC Time t (h) DWM code: BC2 OTM code: BC Time t (h) DYNAS Workshop 06-08/12/04 p.23

24 Dynas test case A H L 1 L = L + L 1 2 B L 2 A (0;15) B (10;14.8) C (39.85;11) D (54.85;10.8) E (54.85;7.8) F (39.85;8) G (10;11.8) H (0;12) G Ls C water table D F E Runoff 56 flux/qrain (%) zoom time (h) 20 Soil Exfiltration DWM code OTM code time (day) DYNAS Workshop 06-08/12/04 p.24

25 Conclusions for a fixed ratio i/k s, the VGM parameters strongly influence the dynamic response of the water table metric scale: Ls/L depends on BC hectometric scale: good agreement with Ogden and Watts analytical considerations in both cases: the mechanisms leading to saturation are controlled by subsurface flows DYNAS Workshop 06-08/12/04 p.25

26 Perspectives re-infiltration processes spatial and temporal heterogeneity of rainfall influence of the initial condition discontinuous Galerkin, box schemes DYNAS Workshop 06-08/12/04 p.26