4.5 Summary and Conclusions
5.1.7 Coupling Surface Flow and Infiltration
As discussed in Section 3.5, surface flow and infiltration are strongly interdependent phenomena. Such intricate and mutual processes can be modeled using a coupling approach, which includes 1 In order to deliver the comprehensive derivation of Appendix A.5, Eq. (5.47) differs slightly from the corresponding
expression in Philipp et al. (2010).
2 Since the values of k are spread equidistantly, the iterative procedure belongs to the class of fixed-point iteration
schemes. Fixed-point schemes applied to differential equations (as applies for the present case) are also referred to as Picard iteration schemes (Lindel¨of, 1894).
5.1 Novel Analytical Solution Approach for Zero-Inertia Open Channel Flow
the flow model and a loss model. Morita and Yen (2002) and W¨ohling (2005) give an extensive overview of literature on coupled physically-based surface–subsurface modeling. If infiltration can be quantified with a functional relationship which is analytically determinable and differentiable, this relationship and its derivatives can be directly included in the hydrodynamic equations by means of mathematical analysis.
As previously shown, the iterative procedure (5.47);(5.48) establishes a fixed-point iteration scheme for coupling the surface flow and the infiltration model in order to obtain the wetted cross-
sectional area at the inflow boundary A0,nand the wave tip’s arrival time tn at a specific location n,
which allows for a space-discrete solution of the problem. The advantage of the fixed-point scheme is that no derivatives of the surface flow and the infiltration function are involved. According to the Banach fixed-point theorem (Banach, 1922), the method converges linearly under the given problem-specific conditions. The consecutively presented iterative solution procedure for the coupled surface flow–infiltration model for advancing wadi flow was implemented in MATLAB.
By defining N observation points along the channel, the iteration (5.47);(5.48) is carried out
under an equidistant space step ∆x.1 As already discussed in Section 4.4, this leads to an adaption
of the time step to wave dynamics. The spatial interval (i.e., the number of observation points N ) is chosen according to a desired accuracy of the results and is typically in the range of some ten to some hundred meters. W¨ohling (2005) and W¨ohling et al. (2006) comprehensively investigated the numerical behavior of a comparable iterative procedure for coupled surface–subsurface flow, based on the analytical solution of the ZI equations for flow in irrigation furrows by Schmitz and Seus (1992). Both time and space discrete formulations were investigated. They showed that a space discretization of the problem—as applies for the iterative scheme presented in this thesis—leads to improved stability and convergence of the iteration procedure and is, therefore, economical with respect to computational effort.
Herein, the alternating iterative coupling strategy is applied, which is the method of choice if a more complex functional description of infiltration (e.g., via Richards’ equation) is desired and/or surface flow and infiltration are strongly interconnected, which is, for example, the case in irrigated furrows or ephemeral channels with flow under transmission losses. Alternating iterative coupling means that the flow equations and the loss relationship are solved separately but for the same discretization step. Surface flow and losses are interlinked via infiltration as an internal boundary condition. Variables related to the momentum and volume balance are used to check for convergence with respect to a specific tolerance criterion. Figure 5.2 shows a simplified sketch of an alternating iterative coupling procedure for one observation location n. When convergence is obtained, the calculation proceeds to the next observation point, i.e., the next cross section. It is further assumed that the infiltration rate at one specific point in space and time is only dependent on the infiltration opportunity time, as also applies for the herein incorporated Kostiakov-Lewis
infiltration model (cf. Section 6.1.1.1).2 Following the concepts outlined by W¨ohling et al. (2004b)
1 Consistent with Section 5.1.8, Schmitz and Seus (1992) state: “This, however, must not be confused with the
discretization of differential equations used to gain a numerical solution by replacing infinitely small differentials by finite differences. A procedure like this would include the well-known numerical effects of attenuation, as well as phase and discretization errors. Avoiding those undesirable issues was, aside from saving computer time, one of the main reasons for developing an analytical solution.”
2 It has to be emphasized that choosing the Kostiakov-Lewis model is not a concession made in order to cope with a
potentially inadequate coupling approach. In fact, alternating iterative coupling allows for including any arbitrary infiltration relationship in the flow model, which is, furthermore, not constricted by the analytical solution strategy presented in this thesis. In fact, the relatively simple Kostiakov-Lewis infiltration model was selected in order to account for the specific transmission loss conditions present in ephemeral rivers, as already discussed in Section 1.3.
Momentum and continuity equation
A0,nand tn
Volumetric infiltration rate and total infiltration
Kostiakov-Lewis infiltration model
qΦ Infiltration opportunity time
Spatial extent of the flow domain Wetted cross-sectional perimeter
Regarding one specific observation point n
Surface flow model Infiltration model
Figure 5.2: Principles of alternating iterative coupling.
Algorithm 5.1 General algorithm for the alternating iterative coupling procedure of the analytical ZI surface flow model and an infiltration model.
1: EPS = 10−4 . Define iteration precision criterion
2: for n = 1 : N do . Loop over space
3: k = 1 . Set iteration counter
4: A(k0,n−1)from Eq. (5.47) . Calculate A0,n according to initial conditions
5: t(k−1)n from Eq. (5.48) . Calculate tn according to initial conditions
6: repeat
7: τ from tn . Calculate infiltration opportunity time τ
8: P from Eqs. (5.42) and (5.6) . Calculate wetted perimeter P acc. to A0,n and P = AR
9: qφ from Eqs. (6.1) and (6.2) . Calculate qφ according to P and τ
10: A(k)0,nfrom Eq. (5.47) . Calculate A0,n considering qφ
11: t(k)n from Eq. (5.48) . Calculate tn considering qφ
12: ICC from Eq. (5.49) . Calculate iteration convergence criterion
13: k = k + 1 . Update iteration counter
14: until ICC < EPS . Compare ICC with EPS
15: end for
and W¨ohling (2005), the incorporated fixed-point iteration scheme comprises four consecutive steps (Algorithm 5.1 delivers a pseudocode implementation):
Initialization: Within the first iteration (iteration count k = 1), the variables A(k0,n−1)and t
(k−1)
n are
evaluated according to the initial condition and values of the inflow hydrograph Q0
t(kn−1)
. For the following iteration cycles, the values of the variables are updated by employing the result of the respective preceding iteration cycle.
Infiltration calculation: The cross-sectional infiltration rate qφis a term in both the momentum
equation (5.47) and the continuity equation (5.48), and is at the same time expressed by the incorporated infiltration model. Hence, for the current iteration count k, the cross-
sectional infiltration rate qφ = f (τ (x, t)), given by the Kostiakov-Lewis model in form of
Eq. (6.2), is calculated dependent on the infiltration opportunity time τ (x, t) at a predefined spatial location n, and taking into account the transiently wetted perimeter at each cross
5.1 Novel Analytical Solution Approach for Zero-Inertia Open Channel Flow
section, which can be calculated by using Eqs. (5.42), (5.6), and the hydraulic principle
P = A
R. During the advance of the infiltrating flow domain, the infiltration opportunity
time at the upstream boundary equals the total simulation time, thus τ (x0) = t. In the
downstream direction, the opportunity time decreases nonlinearly towards the moving wave tip, where, finally, the opportunity time equals zero. The total infiltration volume, given by the term´t(k−1)n
0
´xn
0 q
φ(ξ, τ )dξdτ in Eq. (5.48)1, is calculated by integrating qφover the wetted
channel reach, i.e., the interval [0, ..., xtip]. Furthermore, for being inserted into Eq. (5.47),
the infiltration volume at the upper boundary qφ0
t(kn−1)
is calculated by employing the
transiently wetted perimeter P0at x = 0.
Evaluation of A(k)0,n and t (k) n : A(k0,n−1) and q φ 0 t(kn−1)
are inserted into Eq. (5.47) and A(k)0,n is
calculated. Consecutively, A(k0,n−1)and the results of the integration´t(k−1)n
0
´xn
0 q
φ(ξ, τ )dξdτ
are inserted into Eq. (5.48) in order to calculate t(k)n . As commonly indicated for the iterative
solution of nonlinear equations, a relaxation is included in order to prevent an overshooting of
the solution or an alternating of subsequent solutions for consecutive observation points n.2
Check for convergence: The first three steps of the procedure yield the values of A(k)0,n, A(k−1)0,n ,
t(k)n , and t(kn−1). Steps two and three are executed until the iteration convergence criterion
(5.49) is not yet fulfilled. After convergence is reached, the scheme turns to the next observation point, i.e., n + 1.
After the convergence of the iteration procedure (5.47);(5.48) is achieved, the wetted cross-sectional
area A(x, tn) can be straightforwardly computed from Eq. (5.42). To compute the discharge Q(x, t),
Eq. (5.42) is inserted into Eq. (5.30). This step again includes some numerical integration as well as
again requires a functional relationship for quantifying the infiltration rate qφ(ξ, τ ), i.e., established
by the Kostiakov-Lewis model with regard to this work.