Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients

Discontinuities in effective subsurface transport properties commonly arise (1) at abrupt contacts between geologic materials (i.e., in composite porous media) and (2) in discrete velocity fields of numerical groundwater-flow solutions. However, standard random-walk methods for simulating transport and the theory on which they are based (diffusion theory and the theory of stochastic differential equations (SDEs)) only apply when effective transport properties are sufficiently smooth. Limitations of standard theory have precluded development of random-walk methods (diffusion processes) that obey advection dispersion equations in composite porous media. In this paper we (1) generalize SDEs to the case of discontinuous coefficients (i.e., step functions) and (2) develop random-walk methods to numerically integrate these equations. The new random-walk methods obey advection-dispersion equations, even in composite media. The techniques retain many of the computational advantages of standard random-walk methods, including the ability to efficiently simulate solute-mass distributions and arrival times while suppressing errors such as numerical dispersion. Examples relevant to the simulation of subsurface transport demonstrate the new theory and methods. The results apply to problems found in many scientific disciplines and offer a unique contribution to diffusion theory and the theory of SDEs.