Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media (extended version)

Abstract

In this article, we present new random walk methods to solve flow and transport problems in unsaturated/saturated porous media, including coupled flow and transport processes in soils, heterogeneous systems modeled through random hydraulic conductivity and recharge fields, processes at the field and regional scales. The numerical schemes are based on global random walk algorithms (GRW) which approximate the solution by moving large numbers of computational particles on regular lattices according to specific random walk rules. To cope with the nonlinearity and the degeneracy of the Richards equation and of the coupled system, we implemented the GRW algorithms by employing linearization techniques similar to the L-scheme developed in finite element/volume approaches. The resulting GRW L-schemes converge with the number of iterations and provide numerical solutions that are first-order accurate in time and second-order in space. A remarkable property of the flow and transport GRW solutions is that they are practically free of numerical diffusion. The GRW solutions are validated by comparisons with mixed finite element and finite volume solutions in one-and two-dimensional benchmark problems. They include Richards' equation fully coupled with the advection-diffusion-reaction equation and capture the transition from unsaturated to saturated flow regimes. For completeness, we also consider decoupled flow and transport model problems for saturated aquifers.

Global random walk solvers for reactive transport and biodegradation processes in heterogeneous porous media

“…The binomial random variables δn used in the BGRW algorithm are approximated by using the unaveraged relations (11), summing up the reminders of multiplication by r and of the floor function ⌊N f /θ⌋, and allocating one particle to the lattice site where the sum reaches the unity (see also [36,38]). By giving up the particle's indivisibility and using the un-averaged relations (11) one obtains a deterministic BGRW algorithm.…”

“…From the definition (8) of the BGRW parameters and the constraints (12) it follows that the allowable value of the local Péclet number is limited by Pé ≤ 2 (see [38,Remark 4]).…”

“…[6], where one uses solutions for linear degradation and saturated flows with constant velocity. Using manufactured analytical solutions, instead, allows code verification tests for nontrivial situations such as flows governed by degenerate Richards equation and one-way coupling with nonlinear reactions [29] or fully coupled one-dimensional flow and transport with transition from unsaturated to saturated conditions [38]. Here, we adopt this approach to test the new schemes for problems of increased complexity: constant flow and nonlinear reactions, saturated flows coupled with Monod model, and space-time variable flows with transition from unsaturated to saturated regime coupled with Monod biodegradation reactions.…”

“…1.5 • 10 5 for each molecular species in [24]) and a scaling factor was necessary to convert numbers of particles into moles. The GRW solvers for reactive transport introduced below in this article rely on the "reduced fluctuations" algorithm which can manage arbitrarily large numbers of particles [41], also used in our recent papers [37,38]. In this way, one achieves fast and highly accurate simulations, which cannot be done with the sequential PT and biased random walk CA approaches described above.…”

“…In case of saturated flows, the nonlinear reaction terms are evaluated with an explicit linearization by computing them with concentration values from the previous steps of the numerical scheme. Since numerical tests indicate that the explicit linearization is no longer accurate for variable water content, we propose an iterative approach for reactive transport in variably saturated flows similar to the GRW L-schemes introduced in [38,39].…”