Conjunctive modelling of free/porous flows provides a powerful and cost-effective tool for designing industrial filters used in the process industry and also for quantifying surface–subsurface flow interactions, which play a significant role in urban flooding mechanisms resulting from sea-level rise and climate changes. A number of well-established schemes are available in the literature for simulation of such regimes; however, three-dimensional (3D) modelling of such flow systems still presents numerical and practical challenges. This paper presents the development of a fully 3D, transient finite element model for the prediction and quantitative analyses of the hydrodynamic behaviour encountered in industrial filtrations and environmental flows represented by coupled flows. The weak-variational formulation in this model is based on the use of C0 continuous equal-order Lagrange polynomial functions for velocity and pressure fields represented by 3D hexahedral finite elements. A mixed UVWP finite element scheme based on the standard Galerkin technique satisfying the Ladyzhenskaya–Babuska–Brezzi stability criterion through incorporation of an artificial compressibility term in the continuity equation has been employed for the solution of coupled partial differential equations. We prove that the discretization generates unified stabilization for both the Navier–Stokes and Darcy equations and preserves the geometrical flexibility of the computational grids. A direct node-linking procedure involving the rearrangement of the global stiffness matrix for the interface elements has been developed by the authors, which is utilized to couple the governing equations in a single model. A variety of numerical tests are conducted, indicating that the model is capable of yielding theoretically expected and accurate results for free, porous and coupled free/porous problems encountered in industrial and environmental engineering problems representing complex filtration (dead-end and cross-flow) and interacting surface–subsurface flows. The model is computationally cost-effective, robust, reliable and easily implementable for practical design of filtration equipments, investigation of land use for water resource availability and assessment of the impacts of climatic variations on environmental catastrophes (i.e. coastal and urban floods). The model developed in this work results from the extension of a multi-disciplinary project (AEROFIL) primarily sponsored by the European aerospace industries for development of a computer simulation package (Aircraft Cartridge Filter Analysis Modelling Program), which was successfully utilized and deployed for designing hydraulic dead-end filters used in Airbus A380.Copyright © 2012 John Wiley & Sons, Ltd.