Solute transport along a single channel with radial diffusion in the porous rock matrix: A simple analytical solution and the implementation of time domain random walk algorithm

Abstract

Numerous field observations show that the channels in one fracture are narrow and the solute penetration depth might be larger than the width. For this case, the diffusion from a channel into the matrix is more realistic to be modeled as radial diffusion than one-dimensional. In present work, the single channel model with radial diffusion is revisited and a simple and robust analytical solution is developed. This solution takes a convolution form of two functions, in which different transport mechanisms are accounted for. The statistical interpretations of the two functions and the analytical solution aid to develop a simple Time Domain Random Walk (TDRW) algorithm and an extension is made to improve its accuracy, efficiency and applicability. To demonstrate the accuracy and efficacy of the extended algorithm, three groups of simulations are performed and it is found that the results of all approaches are identical. The TDRW algorithm, having the same performance as that of inverse Laplace transform solution, is superior to Gaussian quadrature method in computational time. However, due to Monte Carlo nature of the algorithm, the computational burden of TDRW algorithm is dependent on the number of particles applied, which also influences the calculation accuracy. Therefore, a trade-off between computational burden and calculation accuracy should always be made, once the TDRW algorithm is used.