DBHM have fifteen prognostic physical state variables: twelve variables of land surface sub-model SiB2 and three water depth variables (groundwater depth ha, surface overland flow depth hs and river water depth hr). The water exchange between groundwater and soil water, overland flow and flow in river channel will be described in this subsection. The transfer of water between groundwater and soil water is given by:
where Q3g is vertical water exchange between soil recharge zone (the third soil layer) and groundwater layer (m/s), K3g is estimated effective hydraulic conductivity between soil layer and groundwater layer (m/s), 15#15 soil moisture potential (m), and z3g is vertical distance. The soil moisture potential 16#16 of soil recharge zone is taken from the empirical relationship of Clapp and Hornberger (1978): where W3 is soil moisture wetness fraction in the soil recharge zone, equals to volumetric soil moisture to saturation soil moisture, B is empirical parameter. The soil hydraulic conductivity of soil recharge zone K3 is obtained from the saturation hydraulic conductivity Ks (Clapp and Hornberger, 1978): The effective hydraulic conductivity K3g then is estimated as: where the soil freeze factor fice is a function of temperature defined in SiB2 (Sellers et al., 1996).The surface overland flow is simply described by the one dimensional kinematic wave model including the continuity equation (Hager, 1984; Lighthill and Whitham, 1955):
and momentum equation:where qs is the overland discharge per unit width (m2/s), t is time (s), x is the distance along the overland flow (m), i is surface runoff in water depth (m), S0 is the friction slope gradient, and n is Manning's roughness parameter.
The flow between the river network and the groundwater is considered as groundwater flow to a ditch over a sloping impermeable bed (Towner, 1975; Childs, 1971). This conceptual representation of river-groundwater exchange is shown in Figure 2. Assuming flow lines are approximately parallel to the bed, according to Dupuit-Forchheimer approximation, the flow of water per unit width to the river can be written in terms of the hydraulic conductive and the absolute slope of the water table:
where qg is the flow between groundwater and river water, 35#35 is bed slope, s is the distance along the bed, and hg is the aquifer thickness. If qg is positive, it is base flow for gaining streams. Whereas if qg is negative, it is river recharge for losing streams. The equation 13 could be written as: where hg40#40 is the averaged aquifer thickness, and hg41#41 = ha + hg40#40 - hrcos35#35 is the aquifer thickness at the river. The river flow is governed by continuity equation (Lighthill and Whitham, 1955; Chow, 1959): and momentum equation: where Q is river discharge (m3/s), B is river width (m), h is flow depth (m), and Sr is river bed slope. Estimation of the Manning's roughness parameter n for natural streams is provided by field observations guided by (Acrement and Schneider, 1989; Chow, 1959). A value of n equal to 0.035 is used in this study. The river width B is specified along the river network, using the geomorphological relationship between river width and mean annual discharge (Arora and Boer, 1999): where Qm is the mean annual discharge (m3/s) passing through a given river section, and Qm, mouth is the mean annual discharge at the mouth of the river, 48#48 is empirical parameter. A value of 48#48 equal to 0.5 is used following Arora and Boer (1999). The river bed depth D (m) is assigned as the below implicit equation: where Qmax is the maximum discharge (m3/s) passing through the given river section.