1. Emmanuel Mouche, Marie Alice Harel (LSCE)
Modélisation des pics de crue comme phénomènes critiques: de la théorie des files d'attente à la physique statistique.
Emmanuel Mouche, Marie Alice Harel (LSCE)
Michel Bauer (IPhT)
17/05/2017

2. Outline Hydrologic context Connectivity issue 1D & 2D Models
Queueing theory
Statistical physics approaches

3. (same question for erosion and sediment yield)
Hydrologic Context
Rainfall Intensity (mm/h)
Flow rate
(l/s) R Q
Watershed
Hyetogram (R) – hydrogram (Q)
Q(t) = f(R(t), t) ?
(same question for erosion and sediment yield)

4. Infiltration and Runoff (ruissellement)
Infiltration model
R > I1 : Infiltration and runoff
R < I2 : Infiltration
! Ponding time
Diffuse runoff and runoff in rills

5. Runoff runon (ruissellement réinfiltration)
Diffuse runoff and runoff in rills t2
Runoff runon t3
Coalescence

6. Fill and Spill (remplissage et fuite)
Precipitation
Infiltration
! Sediment transport
Same concept applies to subsurface water stored in depressions above the bedrock
Different scales: plot to region

7. Connectivity (Allard, 1993)
Connectivity issue
Rainfall Intensity (mm/h)
Flow rate
(l/s)
Connectivity (Allard, 1993) x
x+h
X and X+h connected
X and X+h
disconnected
Coalescence of flow patterns and divergence of the flow connectivity.
→ Stochastic approach (≠ deterministic)

8. Issues Q(t) = f(R(t), t) ? Connectivity index (t) = ? (at the stream)
Rainfall Intensity (mm/h)
Flow rate
(l/s)
Q(t) = f(R(t), t) ?
Connectivity index (t) = ? (at the stream)
Functionnal connect. = F(Structural connect.) ?
Rainfall thresholds (intensity and duration) for a given watershed ?

9. 1D runoff runon model (steady state)
I infiltrability (random), R rainfall rate (may be random),
→ Q runoff flow rate (is random)
If R < I all the rainfall infiltrates
If R > I a fraction of the rainfall infiltrates and the runoff flow rate is Q = R – I
0 D mass balance
1 D steady state mass balance
𝑄 𝑖 = 𝑄 𝑖−1 +𝑅− 𝐼 𝑖−1 +
𝑋 + =𝑀𝑎𝑥(𝑋,0

10. 2D runoff runon model (steady state)
I infiltrability, R rainfall rate, Q runoff flow rate
2 D steady state mass balance
𝑄 𝑖,𝑗 = 1−2𝜖)𝑄 𝑖−1,𝑗 + 𝜖(𝑄 𝑖−1,𝑗−1 + 𝑄 𝑖−1,𝑗+1 )+𝑅− 𝐼 𝑖,𝑗 +
Where ε is a mixing (or dispersion) factor. May be random (εij) to simulate channels or rills.

11. 1D runoff runon model (transient)
I infiltrability, Q runoff flow rate, R rainfall rate (may be random in time)
1 D transient mass balance
If Courant number = 1 and transport is linearized and ponding time is neglected
𝑄 𝑖 𝑛+1 = 𝑄 𝑖−1 𝑛 + 𝑅 𝑛 − 𝐼 𝑖 +
n = time

12. 1D Fill and spill model ℎ 𝑖 𝑛+1 = ℎ 𝑖 𝑛 + 𝑄 𝑖−1 𝑛 − 𝑄 𝑖 𝑛 +𝑅− 𝐼 𝑖 +
I infiltrability, R rainfall rate, Q spilling flow rate, h pool volume
Spill
Fill
Precipitation
Infiltration
ℎ 𝑖 𝑛+1 = ℎ 𝑖 𝑛 + 𝑄 𝑖−1 𝑛 − 𝑄 𝑖 𝑛 +𝑅− 𝐼 𝑖 +
1 D transient mass balance Q
𝑄 𝑖 𝑛 = 𝛼[ ℎ 𝑖 𝑛 − ℎ 𝑖 ∗ + h h*
If I = 0 no steady state

13. 1D runoff runon model (steady)
I infiltrability, Q runoff flow rate, R rainfall rate
𝑄 𝑖 = 𝑄 𝑖−1 +𝑅− 𝐼 𝑖−1 +
Transport equation with positivity constraint (Max plus algebra, cellular automaton).
Random walk with a special boundary condition at x=0: X(t+dt) = [X(t) + f(t)]+
Queueing theory
Ruin problems in mechanics, …
(see Feller, « An intro. to probability theory »)

14. Link 1D runoff runon model – Queueing theory
Pixel i-1 i
i+1 Ii Qi
Qi+1
waiting – service - interarrival
(Lindley 1952, Harel 2014)

15. Physics (1) ρ < 1 - Q is correlated - Renewal process
Infiltrability
Rainfall
Slope
Runon
Effective rainfall
ρ < 1
- In an infinite domain submersion occurs for ρ=1
- Q is correlated
- Renewal process

16. Infiltrability distribution (uncorrelated)
Physics (2)
ρ < 1
Infiltrability distribution (uncorrelated)

17. A few results
Mean flowrate
Wet area fraction
Mean number of wet area zones
Rainfall rate
Connectivity function
Theoretical results based the work of Erlang, Lindley, Takacs, … (generating function and Wiener Hopf problem)
𝑒 𝑧𝑄 = 𝑒 𝑧 𝑄+𝑅−𝐼 + …𝐹 𝑄 = 0 ∞ 𝐹 𝑢 𝑑𝐺(𝑢−𝑄) …
Q and R-I are uncorrelated
Analytical representation of any physical variable as a function of ρ (Harel 2014, 2015)
Distance

18. The bimodal queue The only known queue with PdF discontinuities
No analytical representation of the physical variables as functions of ρ (Mouche 2016)
The discrete generating function of the problem leads to a characteristic polynomial which exponents are discontinuous functions of ρ.
α and β are the prob. of the two modes (α + β = 1)

19. 1D runoff runon model (transient state)
I infiltrability, Q runoff flow rate, R rainfall rate (may be random in time)
𝑄 𝑖 𝑛+1 = 𝑄 𝑖−1 𝑛 + 𝑅 𝑛 − 𝐼 𝑖 +
1 D transient mass balance
n = time
Solution (rising and recession) obtained with Spitzer identity (1957).
Thresholds analyzed and, in the exponential case, analytical expressions have been obtained. (Mouche 2016)
Wet area fraction
Prob(Q=0)

20. 2D runoff runon model (steady) (Work done in collab
2D runoff runon model (steady) (Work done in collab. with Michel Bauer of SPhT)
I infiltrability, R rainfall rate, Q runoff flow rate
2 D steady state mass balance
(Harel 2016)
𝑄 𝑖,𝑗 = 1−2𝜖)𝑄 𝑖−1,𝑗 + 𝜖(𝑄 𝑖−1,𝑗−1 + 𝑄 𝑖−1,𝑗+1 )+𝑅− 𝐼 𝑖,𝑗 +

21. ρ = 0.2 R > I In white Slope Q > 0 In white
Black: dry pixel P(Q=0), White: wet pixel P(Q>0), Red: largest connected wet pixel

22. ρ = 0.4 R > I In white Q > 0 In white
Black: dry pixel P(Q=0), White: wet pixel P(Q>0), Red: largest connected wet pixel

23. ρ = 0.6 R > I In white Q > 0 In white
Black: dry pixel P(Q=0), White: wet pixel P(Q>0), Red: largest connected wet pixel

24. ρ = 0.8 R > I In white Q > 0 In white
Black: dry pixel P(Q=0), White: wet pixel P(Q>0), Red: largest connected wet pixel

25. 2D runoff runon model (steady)
I infiltrability, R rainfall rate, Q runoff flow rate
2 D steady state mass balance
𝑄 𝑖,𝑗 = 1−2𝜖)𝑄 𝑖−1,𝑗 + 𝜖(𝑄 𝑖−1,𝑗−1 + 𝑄 𝑖−1,𝑗+1 )+𝑅− 𝐼 𝑖,𝑗 +
If i is considered as time axis we have the transient diffusion equation with positivity constraint
𝑄 𝑖 = 𝑄 𝑖−1 + 𝑄 𝑖+1 )+𝑅− 𝐼 𝑖 +
At steady state
Advection
𝑄 𝑖 = 𝑄 𝑖−1 +𝑅− 𝐼 𝑖−1 +
𝑄 𝑖 = 𝑄 𝑖−1 + 𝑄 𝑖+1 )+𝑅− 𝐼 𝑖 +
Diffusion

26. 𝑄 𝑖 = 𝑄 𝑖−1 + 𝑄 𝑖+1 )+𝑅− 𝐼 𝑖 +
It does not describe a renewal process. The equation must be solved at the whole scale of a pattern. Numerically the transient state must be solved to obtain the steady state. Submersion occurs for ρ = 1 (physics)
ρ = 0.1
ρ = 0.2
ρ = 0.5
ρ = 0.7
ρ = 0.9
Probably anomalous diffusion (extremely long relaxation times for high rho values).
The wet area fraction does not depend on the type of equation (diff. or adv.) and on the dimension (1D or 2D) !.
M. Bauer propose to solve « iteratively » the problem starting from the divergence ρ = 1

27. 𝑄 𝑖,𝑗 = 1−2𝜖)𝑄 𝑖−1,𝑗 + 𝜖(𝑄 𝑖−1,𝑗−1 + 𝑄 𝑖−1,𝑗+1 )+𝑅− 𝐼 𝑖,𝑗 +
Following the recent work of Boxma (2016) on
we solve
𝑄 𝑖 = 𝑎𝑄 𝑖−1 +𝑅− 𝐼 𝑖− , 𝑎≤1
𝑒 𝑧 𝑄 𝑠 = 𝑒 𝑧 𝑄 𝑒 +𝑅−𝐼 +
We show that the wet area fraction (Prob(Q=0))is independent of ε. We obtain a relationship between the mean flowrate Q and its covariance
Connectivity ?

28. Statistical physics approaches
Exact resolution of the different problems seem to be (very) difficult.
Runoff runon dynamics for a rainfall duration and intensity close to their threshold values may qualified of critical phenomenon (assumption).
For the steady state runoff runon problem we look for solutions
𝑄= (1−𝜌 ) −𝜇 𝑎𝑛𝑑 𝜆= ( 𝜌 𝑐 −𝜌 ) −𝜈
where 𝜌 𝑐 is the critical rainfall for an infinite connectivity length 𝜆.
We are looking for the critical exponents 𝜇, 𝜈, … and the rainfall threshold intensity 𝑅 𝑐 . For the transient problem there is a threshold duration 𝜏 𝑐 and a relationship 𝜏 𝑐 ( 𝑅 𝑐 ).

29. 𝑄= (1−𝜌 ) −𝜇 𝑎𝑛𝑑 𝜆= ( 𝜌 𝑐 −𝜌 ) −𝜈
Renormalization (1)
I2 ,R
I1 ,R
I3 ,R
I4, R
I’(I1 , I2 , …,4R)
The idea:
Upscale the transport equation (« à la Kadanoff »)
For the steady state runoff runon problem we look for solutions
𝑄= (1−𝜌 ) −𝜇 𝑎𝑛𝑑 𝜆= ( 𝜌 𝑐 −𝜌 ) −𝜈
where 𝜌 𝑐 is the critical rainfall for an infinite connectivity length 𝜆.
We are looking for the critical exponents 𝜇, 𝜈, … and the rainfall threshold intensity 𝑅 𝑐 . For the transient problem there is a threshold duration 𝜏 𝑐 and a relationship 𝜏 𝑐 ( 𝑅 𝑐 ).
Scaling laws, Fixed point, universality ?
« à la Kadanoff »
The PdF P(I’=nR) is known !
𝑄 𝑖,𝑗 = 1−2𝜖)𝑄 𝑖−1,𝑗 + 𝜖(𝑄 𝑖−1,𝑗−1 + 𝑄 𝑖−1,𝑗+1 )+𝑅− 𝐼 𝑖,𝑗 +

30. Renormalization (2): the sandpile problem
(Bak 1987)
One add a grain somewhere (pixel i) in the pile. If 𝑧 𝑖 < 𝑧 𝑐 ok, if not, toppling occurs and may create an avalanche. The statistics of the avalanche size s is 𝑃 𝑠 ≈ 𝑠 −𝜏 and the linear size of an avalanche scales with time t≈ 𝑙 𝑧 (Self Organized Criticality).
There are similarities and disimilarities between the sandpile problem and our problem. The grain is the rainfall in our problem and the sandpile is a max plus problem but grains are added one at a time and there is no threshold in the sandpile.
Nevertheless the renormalization approaches of Pietronero (1994), Ivashkevich (1999) should be useful tools.
The PdF 𝑃 𝐼 1 + 𝐼 2 +…+ 𝐼 𝑛 =𝑛𝑅 is known.

31. Correlated percolation
! Correlated percolation may be useful to compute the connectivity length 𝜆 , not the runoff flowrate Q
If we introduce the indicator 𝜔 𝑖 which tells if pixel i is wet (𝜔 𝑖 =1) or not (𝜔 𝑖 =0) we have a percolation problem.
Then comes a central question: is 𝜔 𝑖 spatially correlated ? The answer is yes
Then comes a second central question: is it long range or short range ?
According to (Harris, 1974 and Weinrib 1984)
If the correlations are short ranged (fall off faster than 𝑟 −𝑎 ) they do not change the critical behaviour of the system and consequently the behaviour does not depend on infiltrability distribution.
If they are long ranged : 𝜆= ( 𝜔 𝑐 −𝜔) −𝜈 where 𝜔 𝑐 =𝜔( 𝜌 𝑐 ) and 𝜈=2/𝑎 and 𝜔 𝑐 …
Correlation function of 𝜔 𝑖 is unknown and we need to know 𝜔(𝜌) !

32. 1D Fill and spill model ℎ 𝑖 𝑛+1 = ℎ 𝑖 𝑛 + 𝑄 𝑖−1 𝑛 − 𝑄 𝑖 𝑛 +𝑅− 𝐼 𝑖 +
I infiltrability, R rainfall rate, Q spilling flow rate, h pool volume
Spill
Fill
Precipitation
Infiltration
1 D transient mass balance
ℎ 𝑖 𝑛+1 = ℎ 𝑖 𝑛 + 𝑄 𝑖−1 𝑛 − 𝑄 𝑖 𝑛 +𝑅− 𝐼 𝑖 + Q
𝑄 𝑖 𝑛 = 𝛼[ ℎ 𝑖 𝑛 − ℎ 𝑖 ∗ + h
Highly complex problem: all the variables are correlated!