Modeling overland flow and soil erosion on nonuniform hillslopes - Wiley

WATER RESOURCES RESEARCH, VOL. 45, W05423, doi:10.1029/2008WR007502, 2009

Modeling overland flow and soil erosion on nonuniform hillslopes: A finite volume scheme B. C. Peter Heng,1 Graham C. Sander,1 and Cecil F. Scott1 Received 3 October 2008; revised 3 March 2009; accepted 20 March 2009; published 23 May 2009.

[1] This paper presents a finite volume scheme for coupling the Saint-Venant equations

with the multiparticle size class Hairsine-Rose soil erosion model. A well-balanced monotone upstream-centered schemes for conservation laws-Hancock (MUSCL-Hancock) method is proposed to minimize spurious waves in the solution arising from an imbalance between the flux gradient and the source terms in the momentum equation. Additional criteria for numerical stability when dealing with very shallow flows and wet/dry fronts are highlighted. Numerical tests show that the scheme performs well in terms of accuracy and robustness for both the water and sediment transport equations. The proposed scheme facilitates the application of the Hairsine-Rose model to complex scenarios of soil erosion with concurrent interacting erosion processes over a nonuniform topography. Citation: Heng, B. C. P., G. C. Sander, and C. F. Scott (2009), Modeling overland flow and soil erosion on nonuniform hillslopes: A finite volume scheme, Water Resour. Res., 45, W05423, doi:10.1029/2008WR007502.

1. Introduction [2] Soil erosion due to rainfall and overland flow is a complex phenomenon. The erodibility of a soil is a function of its particle size distribution, cohesiveness, resistance to aggregate breakdown, antecedent moisture conditions, and soil texture. External factors include rainfall intensity, topography, vegetative cover, slope angle and length. These factors and their interactions can potentially be captured in a model founded on the physical laws governing erosion processes. Hairsine and Rose [1991, 1992a, 1992b] developed one such model that has since been applied successfully to various scenarios of soil erosion [Sander et al., 1996; Hairsine et al., 1999; Beuselinck et al., 2002; Rose et al., 2003; Hogarth et al., 2004a; Van Oost et al., 2004; Rose et al., 2007; Sander et al., 2007a]. The Hairsine-Rose (H-R) model considers erosion and deposition processes separately, accounts for size-selective sediment transport, and recognizes the development of a deposited layer with different properties from the underlying parent soil. These features together facilitate the modeling of complex evolving scenarios of soil erosion. [3] As shown in some of the above-cited papers [see also Parlange et al., 1999; Hairsine et al., 2002; Sander et al., 2002; Tromp-van Meerveld et al., 2008], the H-R model can be solved analytically for simple scenarios and with appropriate assumptions. Numerical methods would have to be used for more complex problems involving multiple concurrent erosion processes, nonuniform topography, etc. We

1

Department of Civil and Building Engineering, Loughborough University, Loughborough, UK. Copyright 2009 by the American Geophysical Union. 0043-1397/09/2008WR007502

present a finite volume scheme in this paper for the solution of the one-dimensional H-R model coupled with the full Saint-Venant equations. Particular attention is given to developing a well-balanced scheme for the solution of the Saint-Venant equations to minimize the errors in discharge (leading to errors in erosion rates) that can arise because of varying topography. Related works include papers by Nord and Esteves [2005], Simpson and Castelltort [2006], and Murillo et al. [2008] using different (notably single-class) soil erosion models. Both Nord and Esteves [2005] and Murillo et al. [2008] noted the need to use better models in future work. Our work represents a step in that direction. [4] The outline of the paper is as follows. A brief description of the governing equations is given section 2. This is followed by details of the numerical implementation in section 3. We then verify the implementation with respect to the hydraulics component as well as the full erosion model against benchmark tests in the literature.

2. Governing Equations [5] The one-dimensional H-R equations for sediment size class i are @ðhci Þ @ðqci Þ þ ¼ ei þ eri þ ri þ rri di @t @x @mi ¼ di eri rri ; @t

where x is horizontal distance (approximately equal to distance downslope for small gradients), h is flow depth, q is unit discharge, ci is sediment concentration in mass per unit volume, and mi is the deposited sediment mass per unit area. The source terms ei, eri, ri, rri, and di are detachment and redetachment rates due to rainfall,

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entrainment and reentrainment rates due to runoff, and the rate of deposition with units of mass per unit area per unit time, respectively. [6] The rainfall-related erosion rates are evaluated as

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where, with I sediment classes, 2

2 3 3 h q 6 6 2 7 7 6 6q 7 7 6 q 7 6 h þ 12 gh2 7 6 6 7 7 6 6 7 7 6 6 7 7 hc1 7; F ¼ 6 qc1 7; U¼6 6 6 7 7 6 6 7 7 6 . 7 6 7 . 6 .. 7 6 7 .. 6 6 7 7 4 4 5 5

ei ¼ ð1 HÞpi aP mi eri ¼ H ad P; mt

where P is rainfall intensity, a is the detachability of undisturbed soil, and ad the detachability of deposited sediment, in mass per unit area per unit rainfall, pi is the proportion of class i sediment in the original soil [Sander et al., 2007b], mt = Smi is the total deposited sediment mass per unit area, H = min(mt/m*t, 1) represents the degree of shielding provided by the deposited sediment, and m*t is the mass of deposited sediment required to shield the original soil completely. [7] Runoff entrainment and reentrainment rates are evaluated as

hcI

qcI

2

3

P

6 7 6 7 6 ghðS0 þ Sf Þ Pu 7 6 7 6 7 6 7 6 7; e þ e þ r þ r d 1 r1 1 r1 1 S¼6 7 6 7 6 7 .. 6 7 . 6 7 4 5 eI þ erI þ rI þ rrI dI

FðW Wcr Þ ri ¼ ð1 HÞpi J mi FðW Wcr Þ rri ¼ H mt rs rw gh rs

2

2

3

d1 er1 rr1

3

.. .

7 7 7: 7 5

6 7 6 6 . 7 6 .. 7; D ¼ 6 M¼6 6 7 6 4 5 4

respectively, where W is stream power and Wcr the critical stream power at incipient motion, F is the fraction of excess stream power effective in entrainment and reentrainment, J is the energy required per unit mass of sediment for entrainment, g is the acceleration due to gravity, rs is sediment density and rw the density of water. Flow stream power is determined using the equation W = rwgSf q, where Sf = n2q2h10/3 is friction slope and n is Manning’s roughness coefficient. The rate of deposition for sediment class i is di = vici, where vi is the settling velocity of particles in that class. [8] The Saint-Venant equations with lateral inflow due to rainfall are

dI erI rrI

mI

[10] The system of equations (1) can be written in quasilinear form @ @ U þ A U ¼ S; @t @x

where 2

0

6 6 6 gh u2 6 6 @F 6 uc1 A¼ ¼6 @U 6 6 6 .. 6 . 6 4

@h @q þ ¼P @t @x 2 @q @ q 1 þ þ gh2 ¼ ghðS0 þ Sf Þ Pu; @t @x h 2

ucI

where S0 = dz/dx is bed slope, z is bed elevation (positive upward), u is flow velocity, and the x component of raindrop velocity is assumed negligible. [9] The coupled system of governing equations is therefore

1

0

2u

0

c1

u

.. .

.. .

cI

0

..

.

0

3

7 7 07 7 7 7 0 7: 7 7 .. 7 .7 7 5 u

The eigenvalues of A (corresponding to the wave speeds of pffiffiffiffiffi pffiffiffiffiffi the system) are l1 = u gh, l2 = u + gh, and l3,. . .,I+2 = u (with an algebraic multiplicity equal to the number of sediment classes). The corresponding eigenvectors (waves) are

@ @ Uþ F¼S @t @x @ M ¼ D; @t

m1

ð1Þ

2 3 2 3 0 0 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 607 607 6 l1 7 6 l2 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 c1 7; r2 ¼ 6 c1 7; r3 ¼ 6 1 7; . . . ; rIþ2 ¼ 6 0 7: r1 ¼ 6 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6.7 6.7 6 . 7 6 . 7 6 .. 7 6 .. 7 6 .. 7 6 .. 7 6 7 6 7 6 7 6 7 4 5 4 5 4 5 4 5 2

1

cI 2 of 11

3

2

1

cI

3

0

1


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The quasi-linear form is the basis for linearized approximate Riemann solvers commonly used in numerical solutions of hyperbolic equations.

of steady state conditions), where qj = 0 and hj = hj + zj = h0, a constant, for all j. The gradient in h between xj1 and xj is

3. Numerical Solution [11] We adopt a cell-centered discretization of bed elevation as well as state variables. Defining z at cell centers rather than at cell interfaces is convenient for mobile-bed applications as erosion and deposition rates are computed at cell centers. [12] Given initial values and boundary conditions, we obtain the solution at the next time step explicitly using the monotone upstream-centered schemes for conservation laws-Hancock (MUSCL-Hancock) method [van Leer, 1984], which can be outlined as follows: (1) piecewiseslope linear state reconstruction at tn with an appropriate 1 limiter, (2) a predictor step, advancing to t nþ2 , (3) solution of the Riemann problem at each cell interface, and (4) a corrector step to obtain solution at tn+1. [13] The numerical problems associated with solving the Saint-Venant equations with varying topography are well known. Spurious waves can arise because of an imbalance between the flux gradient and the source terms in the momentum equation, and various solutions have been proposed [e.g., Bermudez and Vazquez, 1994; LeVeque, 1998; Hubbard and Garcia-Navarro, 2000; Zhou et al., 2001; Li and Chen, 2006; Noelle et al., 2007; George, 2008]. [14] Within the framework of the MUSCL-Hancock approach, we introduce two modifications to preserve the flux source balance in steady flow. First, we impose conditions on state reconstruction to ensure steady state is preserved in the predictor step. Second, following Hubbard and GarciaNavarro [2000], source term integrals at cell interfaces are split into left- and right-going components for updating state variables in adjacent cells. [15] The numerical scheme is discussed in greater detail in the following subsections. 3.1. State Reconstruction and Predictor Steps [16] The MUSCL approach reconstructs cell state in a piecewise-linear manner to achieve second-order accuracy in space [van Leer, 1979]. Slope limiters are used in this step to preserve monotonicity. We have found the van Leer limiter to be robust while not excessively diffusive. We follow Quirk [1994] in applying the slope limiter to wave strengths rather than elements of the state gradient vector. [17] The state reconstruction step gives, for cell j, Unjþ1;L 2 and Unj1;R , the states left of xjþ12 and right of xj12 respec2 tively. The predictor equation then advances the solution by half a time step, giving nþ12

Uj

¼ Unj

¼

Unjþ1;L 2

Unj1;R 2

2

and, between xj and xj+1, Dh Dz ¼ : Dx jþ1 Dx jþ1 2

2

[19] The slope limiter algorithm gives

Dz Dhj Dh ¼ fA ðqhj Þ ¼ fA qhj ; Dx jþ1 Dx jþ1 Dxj 2

2

where fA() is an arbitrary limiter function and

Dh Dz Dx Dx 1 j j1 qhj ¼ 2 ¼ 2 : Dh Dz Dx jþ1 Dx jþ1 2

2

[20] This gradient in h across cell j results in a momentum flux gradient 2 2 g hjþ12;L hj12;R 2 Dxj

!

Dz ; ¼ ghj fA qhj Dx jþ1 2

which has to be balanced by the source term ghjDzj/Dxj if quiescent conditions are to be preserved in the predictor step. That is, we require Dzj Dz ¼ fA ðqhj Þ ; Dx jþ1 Dxj 2

which implies that z has to be reconstructed with the same limiter function that is used in the reconstruction of h. This is the only condition required to preserve quiescent conditions in the predictor step. [21] Let us now consider steady flow conditions (qj = q0 for all j). Between cells j and j + 1, steady flow requires Z

xjþ1

Fjþ1 Fj ¼

ð2Þ

S dx; or xj

Dt Dt n Fjþ1;L Fnj1;R þ Sn ; 2 2 2Dxj 2 j

nþ1 Uj12;R 2

Dh hj hj1 zj þ zj1 Dz ¼ ¼ ¼ xj xj1 xj xj1 Dx j1 Dx j1 2

h i1 Z b j ; Ujþ1 Þ Ujþ1 Uj ¼ AðU

xjþ1

S dx;

ð3Þ

xj

with the state gradient unchanged over the half step, that is, nþ1 Ujþ12;L 2

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b (V, W) is Roe’s Jacobian evaluated between states where A V and W. Similarly, steady flow between cells j 1 and j requires

¼ DUj :

nþ1

[18] For steady state, we require Uj 2 = Unj or Fnjþ1;L Fnj1;R = 2 2 DxjSnj . We consider first quiescent conditions (a special case 3 of 11

h i1 Z b j1 ; Uj Þ Uj Uj1 ¼ AðU

xj

S dx: xj1

ð4Þ


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Assuming (3) and (4) are satisfied exactly, setting state gradient in cell j as some function of ðDU=DxÞj1 and 2 ðDU=DxÞjþ1 would not in general satisfy

tional Riemann problem that can be solved using any standard method. We use Roe’s solver [Roe, 1981], which gives an approximation to the flux across the interface as

DUj b 1 ¼ Aj Sj ; Dxj

1 1 b b b 1 F ¼ ðFL þ FR Þ R LR ðUR UL Þ; 2 2

2

ð5Þ

bj = A b (Uj1;R ; Ujþ1;L ). Thus the condition for where A 2 2 steady flow in cell j is violated. An imbalance between the flux gradient and the source term in the predictor step nþ1 means Uj 2 6¼ Unj , with the error propagating into the corrector step. This explains the errors observed by Hubbard and Garcia-Navarro [2000] with their slope-limited scheme. [22] Solving (5) for the required state gradient across cell j is nontrivial because DUj appears on both sides of the equation. Consider the frictionless Saint-Venant equations without rainfall, where 2 b ¼4 A

0

1

^ u2 þ g h 2û

2

3

5 and S ¼ 4

0 Dz gh Dx

ghj Dzj q2 h2j 0:25Dh2j

5:

:

Dt

n Uj

þ

FðUnjþ1 Þ

FðUnj1 Þ

xjþ1 xj1

¼

¼ 1 I R sgnðLÞR1 S; S 2

where the left-going (right-going) component is denoted by the negative (positive) superscript and 2 6 6 6 6 sgnðLÞ ¼ 6 6 6 6 4

SðUnj1 ; Unjþ1 Þ;

3

sgnðl1 Þ

7 7 7 7 7: 7 7 7 5

sgnðl2 Þ ..

. sgnðlIþ2 Þ

where Uj = 0.5(Uj+1 +Uj1). Near steady state, Un+1 Unj j n would be small relative to Uj . We therefore check that FðUnjþ1 Þ FðUnj1 Þ n n n Dt þ SðUj1 ; Ujþ1 Þ Uj ; x x jþ1

The subscript j + 0.5 is implicit in the above equations. The entropy correction function proposed by Harten and Hyman [1983] is used to eliminate entropy-violating discontinuities where there is a transonic rarefaction. [2000], the [25] Following Hubbard and Garcia-Navarro = R xR S dx is split into leftquasi-cell source term integral S xL and right-going components:

ghj

We can use a rough estimate for Dhj on the right to obtain a second approximation on the left. A good initial estimate for Dhj is 0.5(hj+1 hj1) (zjþ12;L zj12;R ). We have found that a single iteration is sufficient to get a close approximation. [23] The above adjustment to DUj is only valid at steady state so a check has to be performed to determine if the flow is steady. Consider a ‘control volume’ between xj1 and xj+1. Then nþ1 Uj

hL þ hR h¼ ; pffiffiffiffiffi2 pffiffiffiffiffi hL uL þ hR uR ^ u ¼ pffiffiffiffiffi pffiffiffiffiffi ; hL þ hR pffiffiffiffiffi pffiffiffiffiffi hL ci;L þ hR ci;R pffiffiffiffiffi pffiffiffiffiffi : ^ci ¼ hL þ hR

3

The steady state condition (in addition to the trivial Dqj = 0) in this case is Dhj ¼

bR b is the approximate Jacobian matrix with bL b 1 = A where R primitive variables

j1

where is much smaller than unity, say, 0.001. We adjust the state slope only if the above condition is met for all state variables. 3.2. Solving the Riemann Problem at Cell Interfaces [24] The reconstruction step results in discontinuities in z as well as in U across each cell interface. Suppose there is a quasi-cell at each interface with a linearly varying z linking the bed elevation on either side (Figure 1). Extrapolating the states Ujþ12;L and Ujþ12;R , from the left and right, respectively, to the center of the quasi-cell, we end up with a conven-

[26] With reference to our system, the integral of the source term due to bed slope in the quasi-cell is g h (zR h = 0.5(hL + hR). Physically, this may be zL), where interpreted as the rate of change of momentum across the interface due to the step in bed elevation. Note that this is the only source term that needs to be considered in the quasi-cell: all other source term integrals vanish as (xR xL) ! 0. 3.3. State Update [27] The corrector equation advances the solution by a full time step on the basis of the fluxes and source terms at the half step: Dt nþ12 nþ1 F jþ1 F j12 2 2 Dxj Dt hnþ12 iþ h nþ12 i nþ1 þ DtSj 2 : Sj1 þ Sjþ1 þ 2 2 Dxj

Ujnþ1 ¼ Unj

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3.4. Critical Time Step [30] The CFL condition for stability [Courant et al., 1967] requires, for the advection problem, Dt max jlp j 1: Dx p

In the present work, it is only necessary to consider l1 and l2: the other wave speed lies between these two. [31] Besides the CFL condition, there is another restriction on time step due to the H-R equations. Consider the suspended sediment mass balance for class i sediment. Clearly, a negative hci does not make sense. Hence we require, on the basis of the corrector equation, Dt

Figure 1. The interface quasi-cell. The justification for this formulation is given by Hubbard and Garcia-Navarro [2000], and it can be shown that quiescent conditions are exactly preserved with this equation. To give the physical sense, we reason as follows. Quiescent state reconstruction (section 3.1) results in a constant surface level across the whole domain. The predictor step preserves this. In the corrector step, the momentum flux gradient across each cell is balanced exactly by the bed slope source term and the flux difference across each interface by the quasi-cell source integral. Thus = Unj for all j and quiescent conditions are preserved. Un+1 j [28] With steady flow, the situation is less intuitive. Suppose (2) is satisfied at tn. If we then set state gradients in cells j andR j + 1 so that (5) is satisfied in both cells, we R have DFj = Sj dx and DFj+1 = Sj+1 dx. Since Z

Z

xjþ1

S dx ¼ xj

xjþ1;L

xj

2

1þ S dx þ S jþ2

Z

xjþ1

S dx xjþ1;R 2

¼

DFj DFjþ1 þ Sjþ12 þ ; 2 2

Fjþ1

DFjþ1 2

F

nþ12 jþ12

ðhci Þnj

1 nþ1 nþ F j12 Sv 2 ; 2

hci

hci

where Shci = ei + eri + ri + rri di. We cannot obtain the upper bound on Dt directly using the above because the denominator depends on the state at tn + 0.5Dt. A rough estimate can be obtained by ignoring the flux gradient and the nþ1 erosion source terms and using the approximation di 2 dni , which leaves us with Dt

n hnj hci ¼ : di j vi

ð6Þ

Physically, this criterion states that the amount of sediment deposited over a time increment cannot be greater than that in the flow. The upper bound on Dt is determined by the ratio of the minimum flow depth over the domain to the settling velocity of the largest particles. This could be updated at every step or determined a priori for simpler problems where an estimated flow depth is available. Since the problems we are interested in are characterized by very small flow depths (5 mm), the time step will often be governed by this criterion.

4. Verification

we can rewrite (2) as

1 Dxj

Z x 1 jþ ;R DFj 2 S dx: Fj þ ¼ 2 xjþ1;L 2

The left-hand side is only approximately equal to Fjþ12;R Fjþ12;L because, while U is piecewise linear within cells, F is a nonlinear function of U. Thus the steady flow condition across xjþ12 is only approximately satisfied. It can be shown that the error is O(DU2j ) and, as such, is negligible in many cases. [29] The deposited masses are simply updated via nþ12

Mnþ1 ¼ Mnj þ DtDj j

:

[32] The numerical scheme described in the previous section was verified against a number of benchmarks. The first two tests show that accurate solutions (relative to comparable schemes in the literature) to the Saint-Venant equations with varying topography can be obtained. The next two demonstrate the robustness of the scheme in handling wetting and drying scenarios with and without friction. We compare the solution of the Saint-Venant equations with the kinematic wave approximation in section 4.5. Finally, we verify numerical solutions of the full system (including the H-R equations) against solutions in the literature for two different scenarios: rainfall-driven soil erosion and net deposition in overland flow. We used a Courant number of 0.8 for all tests except the last two, where the critical time step was governed by (6). 4.1. Steady Flow Over a Bump [33] The benchmark tests associated with steady flow over a bump [Goutal and Maurel, 1997] have been widely used [e.g., Zhou et al., 2001; Li and Chen, 2006; Noelle et

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containing the jump, arguing that the averaged flow depth in that cell does not correctly reflect the presence of a discontinuity anyway. 4.2. Quasi-Steady Flow [36] We use this test case, first proposed by LeVeque [1998] and since used by others [e.g., Hubbard and GarciaNavarro, 2000; Zhou et al., 2001; Li and Chen, 2006], to show that the modeling of unsteady flow is not adversely affected by the adjustment of the state slope where steady flow prevails. [37] The bed topography is described by

zðxÞ ¼

8 < 0:25fcos½10pðx 0:5Þ þ 1g

for 0:4 < x < 0:6

:

otherwise

0

:

The initial conditions are q = 0 and

hðxÞ ¼

Figure 2. Numerical and analytical solutions of the SaintVenant equations for steady subcritical flow over a bump (q = 4.42 m2 s1 upstream and h = 2 m downstream). The discharge values obtained with slope adjustment in the predictor step (circles) are clearly more accurate than those obtained without slope adjustment (crosses).

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