General solution of the infiltration-advance problem in irrigation hydraulics

Abstract

The Lewis-Milne equation, describes the advance of water down a border check when constant flow q per unit width is introduced at its head, c is the mean water depth, x is the distance of advance at time t, and the cumulative infiltration function is y(t). The paper deals primarily with the solution of (0) and the interdependence of x(t) and y(t). The results are of relevance to other methods of surface irrigation. (0) is valid only if x is a monotonic increasing function of t. Sufficient conditions are y ≥ 0, dy/dt ≥ 0, d2y/dt2 ≤ 0. Normally, the physical restrictions on y are such that these conditions are realized and (0) is valid. The general solution of (0), ‘the infiltration-advance problem’ is found by means of the Faltung theorem of the Laplace transformation. The solutions of (0) for the various common forms of y follow readily. The ‘Horton’ solutions had been found previously by other methods. The new particular solutions are for the Kostiakov-Lewis equation and for the equation y = St1/2 + At, which was developed from a detailed physical study of infiltration. The solutions for this form of y are treated at some length. The power series and asymptotic expansion forms give insight into behavior at small and large times. The relative importance of the parameters is shown to be c, S, and A at small times and A, S, and c at large times. Appropriate dimensionless forms simplify the graphical presentation of the solutions. The forms of x for various y(t) in the limit c → 0 are presented. The inverse problem of deducing y from observed x and c is considered. The formal solution is given, and a simple method is developed in which known solutions of the infiltration-advance problem are used. With additional assumptions it applies also to the case where no data on c are available.