The Traditional Model of UShape Flow Paths

Figure 2.18 depicts a cross-section of a model that has been much quoted in the hydrological literature from Hubbert (1940) to most recent textbooks and articles. The modeling is restricted to a valley flank in a small drainage basin. The cross-section is through half a valley, from a local divide to a nearby low-order valley. The three marked planes are assumed in the model to be impermeable (no water is assumed to flow across the vertical planes

Fig. 2.18 A cross-section of a much-quoted model (following Freeze and Cherry, 1979, who cited Hubbert, 1940). The surface is described as undulating in a mode that can be expressed by a simple mathematical equation, and the water table is assumed to follow topography in a fixed mode. The stippled section describes a water system from a low-order divide to a nearby low-order valley; the thick lines mark there impermeable planes that are an intrinsic part of the U-shape flow paths model, enlarged in Fig. 2.19. The cross-section emphasizes topographic undulations and disregards the location of the terminal base of drainage and the location of the main water divide.

Fig. 2.18 A cross-section of a much-quoted model (following Freeze and Cherry, 1979, who cited Hubbert, 1940). The surface is described as undulating in a mode that can be expressed by a simple mathematical equation, and the water table is assumed to follow topography in a fixed mode. The stippled section describes a water system from a low-order divide to a nearby low-order valley; the thick lines mark there impermeable planes that are an intrinsic part of the U-shape flow paths model, enlarged in Fig. 2.19. The cross-section emphasizes topographic undulations and disregards the location of the terminal base of drainage and the location of the main water divide.

that are marked beneath the local divide and beneath the valley, and an impermeable horizontal rock bed is assumed at some depth). The water table is modeled as a moderated replica of the topography, which in turn is assumed to be expressible by a simple mathematical equation. Figure 2.19 shows a more detailed cross-section of the modeled area. The system is assumed to be composed of homogeneous permeable rocks. Equipotential lines, calculated by the model, were computed, and lines of force, or flow lines, were deduced perpendicular to the former. These model calculations, incorporating Darcy's law, deduced U-shape flow paths that are composed of zones of recharge (points C, E, G, and H in Fig. 2.19) and zones of discharge (points A, B, D, and F in Fig. 2.19) closely coinciding with topographic highs and lows, respectively. The resulting flow lines were divided (Toth, 1963) into three flow regions: local, intermediate, and regional. Realistic dimensions for the model were suggested to be a width on the order of 7 km and a depth on the order of 3 km, and in a 1995 paper, Toth suggested the model to represent entire sedimentary basins.

The early U-shape flow paths models assumed homogeneous permeable rocks for the entire system, but this restriction was later removed by the argument that "all rocks are permeable to some degree.'' Thus the road was paved to apply the U-shape model to entire basins.

A basic outcome of the U-shape flow paths model and its many derivations was that all groundwater systems were discussed in terms of

kilometers

Fig. 2.19 A more detailed cross-section of the U-shape model area shown in Fig. 2.18 (condensing several figures of Toth, 1963, 1995). Groundwater flow paths deduced by the U-shape flow paths model are marked with arrows denoting flow directions. Three flow zones have been concluded: local, intermediate, and regional, with alternating points of discharge (e.g., points A, B, D, F) and points of recharge (points C, E, G, H). The symmetry of the suggested flow lines, centered in the modeled box, reveals that they are a direct outcome of the assumption of the three impermeable flow planes.

kilometers

Fig. 2.19 A more detailed cross-section of the U-shape model area shown in Fig. 2.18 (condensing several figures of Toth, 1963, 1995). Groundwater flow paths deduced by the U-shape flow paths model are marked with arrows denoting flow directions. Three flow zones have been concluded: local, intermediate, and regional, with alternating points of discharge (e.g., points A, B, D, F) and points of recharge (points C, E, G, H). The symmetry of the suggested flow lines, centered in the modeled box, reveals that they are a direct outcome of the assumption of the three impermeable flow planes.

through-flow, at various flow velocities, and stagnant (static) groundwater was disregarded.

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