1、83A SPATIALLY DISTRIBUTED HYDROLOGICAL MODEL FOR THE OKAVANGO DELTAPeter Bauer, Thomas Gumbricht, Wolfgang Kinzelbach1 and George Thabeng2Abstract The Okavango Delta in Botswana is one of the worlds most fascinating wetland systems. A hydrological model of the Delta is presented, which is based on a
2、 finite difference formulation of the relevant flow processes (surface water and groundwater). Spatially distributed input data include rainfall, evapotranspiration and microtopography. The model results are compared to flooding patterns derived from remote sensing. The water budget can be assessed
3、for the whole model area as well as for arbitrary sub-regions down to the scale of the individual model cell. Some scenarios are discussed, where the model may help to analyze long-term impacts of man-made changes in the Delta and its catchment.INTRODUCTIONIn northern Botswana, southern Africa, the
4、Okavango River forms a huge wetland system called the Okavango Delta (Figure 1). The waters of the Okavango River originate in the humid tropical highlands of Angola, flow southward into the Kalahari basin, spill into the Okavango Delta and are consumed by evapotranspiration. A variety of hydrologic
5、al, geochemical, sedimentological and biological processes are shaping the Delta over different spatial and temporal scales (Ellery et al., 1993, Gumbricht et al., 2001, McCarthy and Ellery, 1994, Modisi et al., 2000). The natural system has to satisfy the water demands of various users all over the
6、 river basin. Locally, domestic water supply, mining industry and tourism compete with the ecosystem and its spectacular wildlife for the scarce water resource. Interna-tionally, the classic conflict between countries located upstream (Angola, Namibia) and downstream (Namibia, Botswana) is being obs
7、erved.Sound management of the system calls for a tool for a priori analysis of different management options. This was recognized a long time ago and several modelling efforts have been carried out (Dincer et al., 1987, Gieske, 1997, SMEC, 1987). However, the previous models were designed as box mode
8、ls and could not reproduce the spatially distributed flooding patterns in the Delta. Recent progress in remote sensing technology provides time series of flooding patterns, which can be used to calibrate a spatially1 Institute of Hydromechanics and Water Resources Management, ETH Hoenggerberg CH-809
9、3 Zrich.Corresponding author: bauerihw.baug.ethz.ch 2Department of Water Affairs, Gaborone, Botswana84Figure 1. The Okavango Delta. Coordinates are UTM Zone 34 S, Cape Datumdistributed hydrological model (McCarthy et al., 2002). To take advantage of these new developments and to generate a more reli
10、able and flexible tool, a spatially distributed hydrological model of the Okavango Delta is being developed together with Botswanas Department of Water Affairs.DESCRIPTION OF THE HYDROLOGICAL MODELModeling ApproachThe hydrological model of the Okavango Delta is a finite difference surface and ground
11、water flow model, based on the groundwater modelling software MODFLOW (McDonald and Harbaugh, 1988). In the Delta, surface and groundwater are in close contact and in continuous exchange. They are therefore represented in the model as two interacting horizontal layers. The lower layer represents the
12、 underlying sand aquifer. In this layer, water flows according to Darcys law. In the upper layer, which represents the wetland, the model provides two optional flow laws, which can be assigned individually to every cell in the layer: Darcy flow and normal discharge (Darcy-Weisbach equation). The int
13、erface between the two horizontal layers is given by the topographic surface. The single most important driving force for the flooding dynamics is the inflow into the Delta at Mohembo (Figure 2), which is modeled as 85a series of temporally variable recharge cells. Rainfall (recharge) and evapotrans
14、piration are added to or taken from the highest active model cell at each horizontal location.Equations governing the flowIn the major channels, the water is assumed to flow according to the Darcy-Weisbach equation (1).(1)2/13/5)()(hbothyKQcel Q is the cell-by-cell flow (m3/s), K is the Strickler co
15、efficient (m1/3/s), h is the hydraulic head (m); bot is the elevation of the channel bottom (m) and is the size of the cell perpendicular to the flow direction. yThis equation can be derived from the more fundamental Navier-Stokes equation under the assumptions of one-dimensional, steady and uniform
16、 flow. The geometry of the major channels is explicitly put into the model and is assumed to be stable.In the underlying sand aquifer and in the swamp, the water is assumed to flow according to the Darcy equation (2).(2)hbotyKQcel )(In this case, K is the saturated hydraulic conductivity (m/s). This
17、 parameter can vary widely and is much greater in the swamp than in the sand aquifer.Upscaling the topographic variabilityThe spatial resolution of the hydrological model is much coarser than the typical size of terrain features (channels, islands, floodplains etc.) in the Okavango Delta. The repres
18、entation of the flow processes on such a coarse grid is only possible, if the local parameters are upscaled to the grid scale to yield effective parameters, taking into account the topographic variability at the sub-grid scale. The idea86Figure 2. Inflow of the Okavango River at Mohemboof this upsca
19、ling procedure is to skip the exact form of the topographic surface and to consider the statistical properties of successively higher order instead. As an “order 0” approach, one could just take the mean value of the topographic variation in every cell and neglect all higher statistical moments. If
20、this approach is used, the parameters dont change at all and the mathematical description of this system is equivalent to the one for an unconfined, leaky aquifer.The “order 1” approach is to not only consider the mean value of the topographic variation in each cell, but also its variance. For the m
21、athematical treatment, some statistical model of the terrain has to be postulated. In the following sections, the effective parameters for the order 1 approach are presented, given the terrain is uncorrelated and normally distributed. Topographic transects measured in the field (Figure 3) suggest th
22、at the correlation length of the terrain is about 300-500 m, i.e. the grid spacing of 2 km is about 5 times the correlation length. The assumption of uncorrelated terrain elevation is therefore certainly critical, but is useful as a first order approach. 87Figure 3. Variograms of the topographic sur
23、face at different locations in the Delta.Effective kf for swamp cellsIf the terrain is uncorrelated (i.e. if the cell size is large compared to the intrinsic correlation length of the topographic surface), a combination of percolation theory and homogenisation theory can be applied. Homogenisation t
24、heory gives the effective parameter as the problem approaches the limiting case of an unconfined aquifer with a rough bottom, where the size of the roughness elements is small compared to the overall thickness of the aquifer. Percolation theory describes the effective parameter as the thickness of t
25、he water layer tends to zero and the connectivity of the flooded areas within one model cell becomes important (Stauffer and Aharony, 1994):(3)wetgocef bhpPKT)(1)(4)hdbH)()(5)hwetgo dbpbh)(lnx88where K is the hydraulic conductivity, b is the topographic elevation, pc is the percolation threshold and
26、 h is the water table elevation. The characteristics of the terrain are now summarized in the probability density function p(b).Water level dependent specific yieldConsider a two-layered aquifer system with an undulating interface between the two layers. The specific yield of the two layers the amou
27、nt of water released from storage per unit water level change has to be corrected for the portion of the aquifer area which is actually flooded at a certain water level. In the first layer, flooding starts with some isolated ponds and since the flooded area is so small, even a small quantity of wate
28、r can significantly increase the water level. Once the entire cell is flooded, much more water is needed per unit water level increase. This effect is mathematically expressed in the two equations for the specific yields in layer 1 and 2:(6)2)(,1bherfcSloa(7)2)(,2bherfcSloaEffective Evapotranspirati
29、onThe local evapotranspiration is parameterized as follows: It takes its maximum value, once the spot is flooded and decreases exponentially, if the water table falls below the topographic surface (8).(8)bot hfr expma bothETHowever, due to topographic variability, one model cell will contain a mixtu
30、re of different depths to the water level. The effective evapotranspiration from both layers is then calculated as89(9)2)(2exp2)( 2maax1 bherfcbhETrfclayerlayerSpatially distributed input parameters: Rain and EvapotranspirationThe exchange of water between land and atmosphere, i.e. the balance of ra
31、in and evapotranspi-ration is parameterized as a function of the depth to the water table only. This concept is covered in more detail in Bauer et al., 2002. The net exchange of water (rain-ET) can be quantified for the wetland situation (depth to water table equal zero), using remote sensing techni
32、ques. For the deep groundwater situation, the chloride method can be used to derive an estimate for the net exchange. In between these two points, the net exchange is assumed to decrease exponentially with a given ET-extinction-depth.MODELLING RESULTS AND IMPLICATIONS FOR SUSTAINABLE MANAGEMENTCalib
33、ration strategy: comparison with satellite-derived flooding patternsMcCarthy and Gumbricht (2002) have derived a time series of the flooded area of the Okavango Delta starting as early as 1973. The flooding patterns are used to calibrate the hydrological model. Of all the pixels, which are potential
34、ly subject to flooding, 70-80% are correctly predicted over the calibration period of 15 years (1985-2000). The global statistics of the flooded area (average extent and standard deviation in time) are quite accurately reproduced. Calibration will eventually be carried out systematically in a Bayesi
35、an framework; the probability of correctly predicting a pixels flooding status will be used to update initial probability density functions for the parameters, which are based on informed guesses. However, due to the high amount of computer time needed for an individual model run (ca 24 h for 30 yea
36、rs on a modern PC), no systematic calibration has been finalized yet.90Apart from the flooding patterns, the model outputs consist of water levels for all the model cells as well as flux and budget terms, also on the level of the individual cell (Figure 5, Figure 6). These outputs can be retrieved f
37、or every time step of the simulation. Figure 4. Global statistics of observed (remote sensing) and modeled flooding patterns91Figure 5: Model Results for one time step (I)92Figure 6. Model Results for one time step (II)The hydraulic head shows the regional gradient in layer 1 and the groundwater mou
38、nd below the Delta in layer 2. Cell-by-cell flow is high in layer 1 within the channel cells; in layer 2 it is concentrated along the swamps fringe. Evapotranspiration is high on the flooded surfaces, vertical exchange is again concentrated along the fringes. Storage is plotted in absolute values. I
39、t can be seen that the panhandle, during this particular time step is storing big quantities of water, whereas the peripheral parts are destoring.If the net exchange of water with the atmosphere and the topography are assumed to be known, the model output is most sensitive to the following parameter
40、s: Hydraulic conductivities of channels and swamp in the wetland layer. Together with the kf value of the swamp, the Strickler coefficient of the channels (kst) determines the temporal response of the Delta. If the conductivities are either too big or too small, the annual fluctuation in the flooded area is completely suppressed, or the amplitude of that fluctuation is too small. In addition to the overall effect of the conductivity, the ratio between the channel and the swamp conductivity determines the shape of the Delta. In the case of high channel conductivity but relatively low
