The Arnoldi reduction technique for efficient direct solution of radionuclide decay chain transport in dual-porosity media

An efficient technique is presented for the numerical solution of multi-species radionuclide decay chain transport problems in dual-porosity media. The method is based on the Arnoldi modal reduction technique that uses orthogonal matrix transformations to reduce the discretized transport equations. The reduced equation system is much smaller than the original one. This new system can be solved by a standard Crank–Nicolson scheme with very little computational effort and then the original solutions at desired time steps are obtained by using a matrix–vector multiplication. In this paper, we also develop two new alternative methods for choosing a common starting vector for all the transport species. The new methods can be used for most popular cases of first or second type boundary conditions and ensure the convergence of Arnoldi method. In addition, a new method for calculating mass exchange between the matrix block and fracture is presented. This method calculates the leakage terms directly using reduced space data for both slab and spherical matrix block and is highly efficient compared to the traditional iterative methods. The technique is verified through the comparison with analytical solutions. The efficiency and accuracy of the Arnoldi method are demonstrated by applying to the case study of three-species decay chain transport in a heterogeneous dual-porosity aquifer system. The proposed technique shows an impressive 98% saving in computing time and 75% saving in storage space for the case study problem. The Arnoldi reduction method (ARM) affords an efficient means of solving large problems particularly when time durations are long or many species are involved.