Simulation of solute transport in heterogeneous aquifers using stochastic partial differential equations

Abstract

Flow in porous media has been a subject of active research for the last four to five decades. In this paper, a simulation model developed using stochastic partial differential equations to describe hydrodynamic dispersion of a tracer in a confined aquifer is presented with appropriate visualisations. Since the velocity of a tracer particle depends very much on the pore structure of the medium, it can be described by the average Darcian velocity and a Wiener process in space and in time accounting for uncertainty in the pore structure. We can simulate random paths of tracer particles using this stochastic process for the velocity and correlation functions can be used to control the behaviour. In this way, we can model the hydrodynamic dispersion of a tracer without resorting to perturbation solutions in a porous medium. In addition, pore-scale diffusion is modelled as a stochastic differential equation relating the instantaneous concentration gradient of the solute at a specific location to the stochastic diffusive flux through a coefficient containing noise. This noise term is also characterised in terms of a Wiener process in space and in time with its correlation functions. The total stochastic flux is the addition of the flux due to the velocity and the flux due to the pore-scale diffusion. Based on these concepts, a stochastic solute transport model is developed incorporating the properties such as porosity and hydraulic conductivity either as deterministic functions or as random quantities from appropriate distributions if sufficient amount of data is available from the porous formation. The stochastic transport model is solved numerically for 1-dimensional, 2-dimensional and 3-dimensional cases.