A Krylov finite element approach for multi-species contaminant transport in discretely fractured porous media

Simulation of non-ideal transport of multi-species solutes in fractured porous media can easily introduce hundreds of thousand to millions of unknowns. In this paper, a Krylov finite element method, the Arnoldi reduction method (ARM), for solving these type problems has been introduced. The Arnoldi reduction technique uses orthogonal matrix transformations to reduce each of the aforementioned coupled systems to much smaller size. In order to speed convergence of the Arnoldi process, an eigenvalue shift in each finite element system is introduced. This approach greatly improves the diagonal dominant properties of the matrices to be solved. This property leads to great enhancement of the iterative solution and the convergence rate for Arnoldi reduction process. In addition, the use of the eigenvalue shift technique greatly relaxes the grid Peclet restrictions. Courant number criteria restrictions are also effectively removed. We utilize an ORTHOMIN procedure to carry out the equation system reductions for discrete fractured media. The proposed numerical method has been verified by comparison against analytical solutions. The developed model is highly efficient in computing time and storage space. Simulations of radioactive decay chain and trichloroethylene transport are made and compared to the Laplace transform Galerkin (LTG) method where appropriate. Examples with about one million unknowns are solved on personal computers and shown that the ARM is even more efficient than the LTG method, by allowing for similar speed increases with multi-components. Therefore, the Arnoldi approach will allow for a variety of complex, high-resolution problems to be solved on small computer platforms.