Small perturbations solution for steady but nonuniform infiltration

The small perturbations method is applied to a two‐dimensional flow problem due to a steady but nonuniform infiltration into the soil and a given head at the lower boundary. The perturbation solution is compared to an analytic solution for periodic distributions of the infiltration. Good accuracy is obtained by only two terms of the expansion. The first term is simply the vertical flow with no lateral terms. In the second term the horizontal flow divergence depends parametrically only on the horizontal distribution and the zero‐order approximation, i.e., the vertical flow. For deep lower boundaries the naive perturbation expansion leads to some secular terms that made the convergence poorer. The uniform asymptotic expansion is then used and shown to give an improved accuracy in one term only. The perturbation method is extended to nonperiodic distribution, in fact, any distribution. It may be used for two‐ and three‐dimensional problems that evade any analytic solutions. One has to solve numerically or analytically only one dimensional and ordinary linear equations for any number of approximations. The problem is singular in that it accepts at this stage only impermeable vertical boundaries or symmetrical lines. However, this covers a very important class of practical problems.