Richards equation is a robust mathematical model for predicting spontaneous imbibition in unsaturated porous materials, whether granular or fibrous. The main restriction of this imbibition model is that it is valid only for Newtonian fluids. In the present work, we extend the classic Richards equation to viscoelastic fluids using a modified version of the single-phase Darcy's law reported in the literature for several polymer solutions. In two-dimensional flows, the viscoelastic Richards model obtained this way turns out to be in the form of a system of three highly nonlinear coupled partial differential equations for the moisture content and the velocity field. The model is numerically validated against published experimental data for a shear-thinning polymer solution imbibed in a tight sandstone core sample extracted from a typical oil reservoir. The viscoelastic Richards model is then used to investigate the effect of a fluid's elasticity on the quasi-stationary regime in a two-dimensional porous membrane of complex shape typically used in diagnostic kits. The obtained numerical results suggest that elasticity has a retarding effect on spontaneous imbibition in capillary-driven, paper-based kits. Based on this new imbibition model, it is predicted that viscoelasticity of the displacing liquid can extend the duration of the quasi-stationary regime on the test line of diagnostic kits. The conclusion is that, for extending the quasi-steady regime in capillary-driven kits, it might prove useful to enhance the degree of elasticity of the test fluids using appropriate rheology modifiers.