Evaporation-driven wicking of soil water through porous domains with contrasting hydraulic properties is studied analytically by conformal mappings and compared to numerical solutions. Initially, the connected rectangular domains are fully saturated. The first rectangle, G(p), is comprised of a coarse-textured porous medium with large permeability and low capillary forces. Evaporation-induced capillary flow pulls water horizontally across the domains to the surface of fine-textured rectangular domain, G(z), through an interfacial hydraulic exchange region that shrinks with time. The flow field in G(z) is 2-D and is analytically expressed by the Vedernikov-Bouwer model that assumes a constant hydraulic conductivity for pressure heads higher than the air-entry value. The rate of 1-D drop of the phreatic surface in G(p) is proportional to the evaporation rate (decreasing with time) from the G(z) surface. The complex potential domain G(w) shrinks with time, and at any time instance, it is conformally mapped onto G(z) via two auxiliary planes using the Schwarz-Christoffel and Mobius transformations. The resulting Cauchy problem for an integro-differential equation with respect to an affix of the conformal mapping is solved using numerical algebra routines. A similar capillary coupled flow problem was numerically simulated using HYDRUS2D considering 2-D flow in both G(p) and G(z). New insights into process dynamics are gained from a solution of an auxiliary optimization for a vertical imbibition in a column brought in contact with a water table where particle size (linking capillarity and permeability) is used as a control variable and counter-gravity front propagation dynamics as criteria.