The article is a continuation of the author's study, in the framework of filtration theory [see \textit{V. N. Monakhov}, Sib. Math. J. 41, 907-920 (2000; Zbl 0958.76073)], of the mathematical questions of modeling fluid filtration in a polygonal channel with free boundary. In the above-mentioned article, the author proposed the cyclic iteration method for perturbation operator in the parameter problem: choose some initial simple polygon for which a solution to the four parameters is known. It was proven that, for all polygons of a finite family, the perturbation vector of the corresponding nonlinear equations can be found by a simple iteration. The main aim of the article under review is to present a new convergent algorithm for numerical solution of the nonlinear problem of finding the parameters of conformal mappings describing free contact boundaries of fluid filtration flows in a porous medium. For calculating the parameters of filtration flows with a horizontal drainage, an equivalent equation is suggested whose right-hand side does not involve two-dimensional integrals with moving singularity. As a concrete application of the results obtained, the author studies the classical filtration problem for a ground dam on an impermeable base with a horizontal drainage.