Fuzzy Analytical Solution to Vertical Infiltration

Abstract

In this article, we examine the solution of the fuzzy linear vertical infiltration equation, which represents the water movement in porous media in that part which is called the vadose zone. This zone is very important for semi-arid areas, due to complex phenomena related to the moisture content in it. These phenomena concern the interchange of moisture content between the vadose zone and the atmosphere, groundwater and vegetation, transfer of moisture and vapor and retention of moisture. The equation describing the problem is a partial differential parabolic equation of second order. The calculation of water flow in the unsaturated zone requires the knowledge of the initial and boundary conditions as well as the various soil parameters. But these parameters are subject to different kinds of uncertainty due to human and machine imprecision. For that reason, fuzzy set theory was used here for facing imprecision or vagueness. As the problem concerns fuzzy differential equations, the generalized Hukuhara (gH) derivative was used for total derivatives, as well as the extension of this theory for partial derivatives. The results are the fuzzy moisture content, the fuzzy cumulative infiltration and the fuzzy infiltration rate versus time. These results allow researchers and engineers involved in Irrigation and Drainage Engineering to take into account the uncertainties involved in infiltration.