Research Paper Analytical solutions for contaminant transport in a semi-infinite porous medium using the source function method Bing Bai ⇑ , Huawei Li, Tao Xu, Xingxin Chen School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, PR China article info Article history: Received 5 December 2014 Received in revised form 5 April 2015 Accepted 6 May 2015 Keywords: Hydraulic permeability Contaminant transport Release effect Dispersive coefficient Elementary solution abstract The general solutions for contaminant transport in a saturated semi-infinite porous media are derived by using the Laplace transform and Fourier transform, along with their transform inversions, under the conditions of one-dimensional seepage flow and the three-dimensional dispersive effect. The analytical expression of contaminant concentration in a porous medium, subjected to a local contaminant source with arbitrary geometry, and intensity that varies with time and coordinates is derived by the source function method based on the elementary solution of an instantaneous point contaminant source. The results show that an exponentially degenerated contaminant source injected into the porous medium migrates gradually toward the depth and width of the porous medium due to the convective water flow and diffusion induced by molecular and mechanical movement, along with deposition of the contaminant on the solid matrix surface. The contaminant concentration in a porous medium subjected to a cyclic contaminant source exhibits cyclic fluctuations due to the fluctuation of the contaminant source applied on the porous surface; concentrations reach a quasi-steady state, with the same fluctuation phase as the contaminant source. The hydrodynamic dispersion effect accelerates the migration processes of the contaminant in the vertical direction as well as the diffusion in the horizontal direction, resulting in a dramatic rise in the contaminant concentration in a short period of time. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction Studies of contaminant transport in porous media based on hydraulic permeability are of great importance for a wide range of engineering, biophysical, and biomedical applications, such as contaminant treatment, oil extraction, and disposal of high-level nuclear waste [1,2]. Over the past few decades, intense research has been conducted to better understand the migration process, transport mechanism, and absorption and release effects of con- taminants, such as lead, arsenic, mercury, cadmium, colloid parti- cles, phosphorus, and plant nitrogen nutrients [3–5]. Several mathematical models have been developed to account for contaminant transport in porous media under different condi- tions of groundwater movement, with particular attention paid to the migration mechanism of pollutants. Chang et al. [4] simu- lated copper and cadmium transport in a lateritic silty-clay soil col- umn by using the Freundlich nonlinear equilibrium-controlled sorption parameters to determine the retardation factor used in column leaching experiments. Jungnickel et al. [5] described the solution of a fully coupled set of transport equations describing the simultaneous diffusion of several ion species through a clayey soil. Seetharam et al. [6] derived a multicomponent reactive trans- port model, coupled with an existing thermal, hydraulic, and mechanical model for porous media, based on conservation of mass/energy principles for flow and stress–strain equilibrium. Considerable effort has also been devoted to the modeling of consolidation-induced contaminant transport. Peters and Smith [7] derived flow and transport equations for a deforming porous medium based on the mass balance law and illustrated the differ- ences between the theory for the rigid porous medium and deforming porous medium using solute transport through an engi- neered landfill liner as an example. Fox et al. [8] experimentally and computationally investigated the importance of the consolida- tion process to solute transport in compressible porous media. Analytical solutions for more complex problems, such as vari- able coefficients, unsteady flow, and multi-layered media, have gained attention in recent years. Kumar et al. [9] used the Laplace transform to derive the analytical solutions for the one-dimensional advection–diffusion equation, with variable http://dx.doi.org/10.1016/j.compgeo.2015.05.002 0266-352X/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +86 010 51684815. E-mail address: Baibing66@263.net (B. Bai). Computers and Geotechnics 69 (2015) 114–123 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo