7.1 Outline of the Structure of a Novel Integrated Modeling System
7.1.1 Wadi Flow Routing Models
Two flow routing models are developed aiming at a comprehensive representation of the specific flow processes in the upstream and downstream reaches of a recharge dam. For the pronounced flow in the upstream reaches, a KW model is implemented, following the outcomes of the discussion in Section 3.7.1. To accommodate for the more complex character of the flow downstream of the dam, flow advance and recession are treated separately. It is important to match the dynamics and nonlinearity of the advancing flow in an initially dry channel under considerable infiltration. To circumvent numerical instabilities and to pay attention to the influence of infiltration, a tailor-made analytical ZI model, as derived in Section 5.1, is set up for modeling the advancing flow domain. In turn, flow recession in the downstream reaches—which is not covered by the analytical ZI solution—is modeled with the KW equations again. The same applies for dam outflow influenced by spillway operation (cf. Section 3.7.1). Evaporation from wadi flow is neglected since evaporation rates are usually two orders of magnitude below infiltration rates (cf. Section 1.1).
7.1.1.1 Numerical Kinematic Wave Model (Upstream Model)
Making use of the continuity equation (3.7), the KW momentum equation (3.4)—which assumes a parallelism of bottom and friction slope—and the uniform flow formula (3.5) yield an explicit expression of the KW model. According to the derivation outlined in Appendix A.1 and with β set to 2
3 for the Manning-Strickler formula, the KW model reads
∂Q ∂x = KStS 1 2 0 2 3R 1 3∂R ∂xA + KStS 1 2 0R 2 3∂A ∂x (7.1)
where KSt is the respective Strickler roughness coefficient [L
1 3T−1].
Equation (7.1) is inserted into the continuity Eq. (3.7) and the resulting equation is numerically solved using a finite differencing scheme on the basis of the argumentation given in Section 4.4. A second-order Runge-Kutta method is applied for approximating the partial differential quotients of
7.1 Outline of the Structure of a Novel Integrated Modeling System
Eqs. (7.1) and (3.7).1 Further following Section 4.4, an explicit formulation of the difference equations
is chosen. Appendix A.2 exemplarily shows the derivation of the corresponding second-order scheme as is used in this work.
The numerical solution requires the specification of boundary and initial conditions. The upstream boundary condition at x = 0 is the inflow hydrograph
Q0= Q0(t) = Q(x = 0, t) (7.2)
The downstream boundary can be characterized either by an advancing wave tip or a rating curve, e.g., coming from the assumption of outflow at normal depth at the end of the model domain. For the first case, the boundary conditions would read
A(xtip, t) = 0 (7.3)
u(xtip, t) = utip(t) =
dxtip
dt (7.4)
where xtip(t) is the location of the advancing wave tip.
The ideal dry-channel initial condition would be
xtip(t = 0) = 0 (7.5)
For this study, this ideal dry-channel initial condition is alleviated to prevent numerical issues. A constant minimum flow is introduced and the other dependent hydraulic variables are calculated prior to the numerical integration, assuming uniform flow conditions. This practice is quite common, albeit introducing some errors (Cunge et al., 1980). Nevertheless, for greater flood magnitudes, such as those present in the upstream wadi sections, the incorporation of a nonzero minimum flow is feasible, as discussed in Section 4.4. A normal-depth lower boundary condition is placed at the
lowermost cross section, i.e., the location of the recharge dam.2
Since cross-sectional infiltration is dependent on the wetted perimeter, and vice versa, the flow equation is coupled with the time-dependent Kostiakov-Lewis infiltration function (Eqs. (6.1) to (6.3)) by employing alternating iterative coupling, based on a fixed-point iteration scheme (cf. Section 5.1.7). Algorithm 7.1 illustrates a pseudocode implementation of the incorporated iteration scheme. In contrast to the iterative procedures given by Eqs. (5.47);(5.48) and (5.63);(5.64) which employ an adaptive temporal discretization (cf. Section 4.4), the coupling is carried out at the equidistant spatiotemporal nodes ∆(x, t) of the underlying finite difference scheme. Furthermore, it is important to account for the transient spatial extents of the flow domain in order to obtain an exact assessment of infiltration. To encounter the nonzero minimum flow assumption, only flow above the initial flow rate is taken into account for the calculation of infiltration.
1 The class of Runge-Kutta methods essentially comprises single-step methods of various order (which might be
evaluated for intermediate steps as discussed in Section 4.2.2) and, therefore, also covers the first-order Euler method (cf. Appendix A.1). The classification is made regardless of the applied formulation of the solution scheme (i.e., explicit or implicit). The fourth-order Runge-Kutta method is the “original” method, which is usually addressed by referring to “the Runge-Kutta method”. Concisely, the Runge-Kutta method applied herein is a second-order scheme with intermediate nodes, located at 12∆t (the so-called midpoint method).
2 Since the employed flow model is based upon the (steady) kinematic wave assumptions and perturbations cannot
travel in upstream direction, this boundary condition type poses no serious restriction for the validity of the KW results calculated for the interior points.
Algorithm 7.1 Algorithm for the alternating iterative procedure for coupling the numerical kinematic wave model with the Kostiakov-Lewis infiltration model.
1: EPS = 10−4 . Define iteration precision criterion
2: for j = 1 : ∆t : tend do . Loop over time
3: for i = 1 : ∆x : xend do . Loop over space
4: k = 1 . Set iteration counter
5: q(kφ−1)= 0 . Initially estimate cross-sectional infiltration qφ
6: repeat
7: Solve the KW model with respect to q(kφ−1) (Eqs. (7.1) and (3.7), according to
Appendices A.1 and A.2) to obtain infiltration opportunity times and wetted perimeter
8: qφ(k)from Eqs. (6.1) and (6.2) . Calculate new estimate of qφ
9: ICC = qφ(k)− q(kφ−1) . Calculate iteration convergence criterion
10: k = k + 1 . Update iteration counter
11: until ICC < EPS . Compare ICC with EPS
12: end for
13: end for
The alternating iterative coupling of the surface flow and the infiltration model yields a proper convergence behavior, which allows for a straightforward coupled computation of flow and infiltration with, for instance, less than 20 iteration loops under a quite strict iteration precision criterion of
10−4. The nonprismatic cross-sectional geometry is included via an analytical power law fit of the
empirical profile functions ˜h(x, A) and ˜R(x, A), derived from topographic data. The procedure
required because of this is the same as used for the processing of the cross-sectional data for the analytical ZI model, shown in Section 5.1.2. Thus, the corresponding values of water depth, wetted cross-sectional area, and hydraulic radius can be mapped onto each other on an analytical basis.
A prerequisite model validation was carried out by Six (2011) for a prismatic rectangular test
channel (width of 100 m; slope of 0.008; Strickler roughness coefficient of KSt = 30.30 m
1 3 · s−1;
zero infiltration) in order to compare the results of the KW model with those obtained from a
full hydrodynamic model, which was implemented in HEC-RAS.1 It can be seen exemplarily from
Fig. 7.2 that the KW model slightly overestimates values around the peak and underestimates their timing in comparison to the HD model. These effects are more pronounced for higher flow rates, i.e., the more unsteady portions of the hydrograph, and are attributable to the neglecting of secondary terms in the KW model (cf. Section 3.7.1). In contrast, the HD model preserves those terms, leading to the observed wave deceleration and dispersion. However, the results of the two models are in good agreement, especially for the portions of the hydrographs which are associated with moderate flow rates.
7.1.1.2 Coupled Analytical ZI Advance Model–Numerical Kinematic Wave Recession
Model (Downstream Model)
Dam release leads to an outflow which is advancing in the downstream direction. Therefore, the flow processes can be modeled with the analytical zero-inertia approach, presented in Section 5.1. Generally, the advance rate of the wave tip decreases with increasing time and increasing extent 1 The comparison was carried out disregarding the influence of the applied numerical solution schemes, namely an
explicit Runge-Kutta scheme for the KW model (cf. Appendix A.2) and an implicit Preissmann scheme for the HD model (cf. Appendix A.3). However, some of the difference in the model outputs may emerge from the differing solution schemes.
7.1 Outline of the Structure of a Novel Integrated Modeling System 0 4 8 12 16 20 0 100 200 300 400 500 600 700 800 900 Time t (h) Flow Q (m 3⋅s –1) Inflow at +0,000 m KW output at +10,000 m HD output at +10,000 m
Figure 7.2: Comparison of simulation results obtained by the kinematic wave model with those of a full hydrodynamic model for station +10,000 m of a rectangular test channel under zero infiltration. Inflow event recorded on 06/06/07 at Afi gauging station, according to Section 7.2.1. Values in the output were aggregated to 10 min intervals; spatial discretization for both models was 100 m; temporal discretization was 10 s for the HD model and 1 s for the KW model, respectively.
of the infiltrating domain. Assuming an infinitely long permeable channel bed and a quasisteady inflow, the advance would cease if infiltration rates equal inflow rates. Approaching such a state of zero advance leads to a rapidly growing number of iterations for solving the system (5.47);(5.48). Furthermore, if dam outflow rates are lower than infiltration rates, the flow domain would start receding in the upstream direction. Such conditions are not covered by the iterative solution procedure of the ZI model presented herein.
Therefore, from the point in time when the flow approaches such a zero-advance condition, hydrodynamics are modeled with a KW approach, following the concepts discussed in Section 7.1.1.1. This approach is reasonable since (a) the flow momentum is negligible when the flow advance velocity converges to zero, and (b) inflow rates have already become comparably low, which justifies the KW
assumption that the change of the water depth along the channel is very small (∂h∂x ≈ 0). Practically,
specific criteria are required in order to evaluate the zero-advance condition; the analytical ZI model is, therefore, applied until one of the following relations is harmed
(tn− tn−1)
(tn−1− tn−2) !
≤ σ (7.6)
k≤ ς! (7.7)
with σ: an upper limit for the allowed increase rate of advance times and ς: maximum tolerable number of iterations for solving the procedure (5.47);(5.48).
For the first case, the rate of increase of the advance times between the equidistant channel
locations xn,...,n−2 is evaluated, as illustrated by Fig. 7.3. If the condition (7.6) is not fulfilled
Location x Arrival time tn at location n t n–2 t n–1 t n x n–2 xn–1 xn (t n–1 – tn–2) (tn – tn –1)
Figure 7.3: Illustration of the employed zero-advance criterion, given by Eq. (7.6).
KW model, which is set up following the concepts already discussed.1 The KW model is employed
to simulate the further development of the flow variables, related to the initial condition for t = tn
and to the upper boundary condition Q0(t) from t = tn to t = tend. The end time of the simulation,
tend, might be defined a priori the simulation or dynamically, e.g., related to the condition if all
inflow volume had already infiltrated or left the modeling domain through the channel. In case condition (7.7) is harmed first, differing from the aforesaid, the solution procedure (5.47);(5.48) had
not yet converged and, therefore, yielded no result for tn. Therefore, switching between the ZI and
the KW model is performed for the time slice t = tn−1. Practically, adjusting σ to values between
3 and 5 and ς to 200 to 500 leads to a balancing of computation time and model accuracy. Besides culvert release, a comprehensive modeling approach has to account for spillway release as well. However, spillway release is a fairly rare condition for the operation of the recharge dam investigated in this thesis, located in Wadi Ma’awil (cf. Section 7.2.1). The dam construction report
(MAF, 1989; cf. Table 7.2) assesses the design storage to 10· 106m3, which is related to a return
period of roughly 30 years.
Generally, total outflow rates under spillway release are high compared to culvert outflow alone. This implies a negligible impact of infiltration on flow momentum during spillway operation. Furthermore, the spillway outflow features strongly falling hydrographs, which renders the analytical ZI approach not applicable. Dam outflow during spillway operation is, therefore, simulated again with a KW model, set up as outlined previously. When the spillway is activated, ZI results are passed to the initial condition of the KW model, and vice versa when the spillway outflow ceases. The downstream hydrodynamic model combined in this way, consisting of a ZI model and a KW model for routing the advancing/receding culvert outflow and an additional KW model for flow routing during spillway operation, is referred to as a coupled ZI/KW model in the following.