Drainage of groundwater resting on a sloping bed with uniform rainfall

Abstract

The differential equation derived by Childs for groundwater flow over a sloping bed, the streamlines being assumed to be parallel to the slope, is integrated for the case of ditches and uniform rainfall. Expressions are obtained for the maximum vertical water table height, the location of this maximum and of the watershed, and the water table shape. Calculated water table heights are in much better agreement with previously published experimental data than those calculated from an earlier theory based on the assumption that streamlines are horizontal. Thus Childs' revised assumption is confirmed. An approximate solution for the form of the water table under small rainfall rates is also derived.Childs [1971] pointed out that in applying the Dupuit-Forchheimer approximations to groundwater flow over sloping impermeable beds to transverse ditch drains in the presence of uniform rainfall, Werner [1957] and Schmid and Luthin [1964] had incorrectly assumed that the equipotentials were vertical rather than perpendicular to the sloping bed, Childs presented a revised analysis leading to a differential equation for the groundwater flow incorporating the necessary modifications but indicated that there were no analytical solutions available except in a few special cases considered by Wooding and Chapman [1966], and he therefore proceeded by numerical methods. However, it is possible to obtain analytical solutions, which are moreover very similar in form to those given by Schmid and Luthin, and these are presented in this paper. THEORYThe particular problem considered is that of the drainage of groundwater in equilibrium with rainfall flowing down a sloping bed by equidistant parallel ditches traversing the slope and penetrating to the impermeable bed. It is assumed that the water levels in the drains are maintained at bed level. The

Steady Groundwater Flow to Drains on a Sloping Bed: Comparison of Solutions Based on Boussinesq Equation and Richards Equation

“…The physically inappropriate solution of Schmid and Luthin (1964) best fits the numerical data for the maximum pressure head ( m ) expressed as the ratio L/ m but not phreatic height. The Towner (1975) solution gave the best fit to the numerical results for both the height of the phreatic surface and the location of the maximum on the slope. This suggests that in design of drains on slopes that Schmid and Luthin (1964) would be the best analytical solution if the pressure head on the impermeable boundary is required, but the Towner solution will be best if the water-table height is the required value.…”

“…However, Fig. 7b shows that the phreatic surface height, from the simulations, is very closely modelled by the Towner (1975) and Chapman (1980) solutions. The Wooding and Chapman (1966) solution underestimates L/ h m when λ > 1.…”

“…The position of the maximum phreatic height was best fitted overall by the Towner (1975) solution (sum of the squares of the difference = 0.002). The Wooding and Chapman (1966) and Chapman (2003) solutions also gave very good fits compared to the numerical results when a > 0.3, even though their fits to the numerical data were not good over the whole range of values tested (sum of the squares of the difference = 0.37 and 0.77 for Wooding and Chapman (1966) and Chapman (2003), respectively).…”

“…This extends the original work Wooding and Chapman (1966) to be fully analytical, and the requirement of numerical calculations to derive the scaling factor is removed. Towner (1975), using the approach of Childs (1971), was able to derive a solution without the scaling factor. This solution is identical to that of Wooding and Chapman for the situation where no flow occurs to the upslope drain but is different when flow occurs to both drains.…”

Rapid water-level fluctuations in a thin colluvium landslide west of Cincinnati, Ohio

“…They found that flow through an inclined soil slab that has a length-to-thickness ratio greater than 10:1 could be accurately represented by a one-dimensional, slopeparallel flow model. Childs (1971) and Towner (1975) give analytic solutions for steady-state, unconfined, slopeparallel flow both with and without recharge. Chauan and others (1968) and Ram and Chauan (1987) present analytic solutions for the problem of transient slope-parallel flow between two trenches on a slope with constant recharge.…”