Numerical Exploration of Nonlinear Dispersion Effects via a Strongly Coupled Two-scale System

TL;DR

This work addresses the numerical solution of a strongly coupled two-scale dispersion model for porous media, where the macroscopic transport $oxed{u}$ depends nonlinearly on microscopic drift via a dispersion tensor $D^*(W)$ determined by cell problems for $W=(w_1,w_2)$. It develops two decoupled finite element schemes—the Picard-type iterative scheme and a time-stepping scheme—and introduces an offline precomputing strategy to approximate $D^*(W)$ with an interpolated tensor $D^{int}$, significantly speeding up online computations while controlling interpolation error. Theoretical convergence for the Picard-type scheme is established, and the precomputing framework is shown to yield provable error bounds under suitable assumptions. Numerical experiments demonstrate that the time-stepping scheme is computationally more efficient, especially when combined with precomputing, and reveal the influence of microstructure on dispersion, including penetration depth in structured materials. Overall, the results provide a practical, scalable approach for simulating nonlinear two-scale dispersion in heterogeneous media with potential applications in geoscience and materials engineering.

Abstract

The effective, fast transport of matter through porous media is often characterized by complex dispersion effects. To describe in mathematical terms such situations, instead of a simple macroscopic equation (as in the classical Darcy's law), one may need to consider two-scale boundary-value problems with full coupling between the scales where the macroscopic transport depends non-linearly on local (i.e. microscopic) drift interactions, which are again influenced by local concentrations. Such two-scale problems are computationally very expensive as numerous elliptic partial differential equations (cell problems) have to constantly be recomputed. In this work, we investigate such an effective two-scale model involving a suitable nonlinear dispersion term and explore numerically the behavior of its weak solutions. We introduce two distinct numerical schemes dealing with the same non-linear scale-coupling: (i) a Picard-type iteration and (ii) a time discretization decoupling. In addition, we propose a precomputing strategy where the calculations of cell problems are pushed into an offline phase. Our approach works for both schemes and significantly reduces computation times. We prove that the proposed precomputing strategy converges to the exact solution. Finally, we test our schemes via several numerical experiments that illustrate dispersion effects introduced by specific choices of microstructure and model ingredients.

Figures (10)

• Figure 1: Schematic illustration of a typical two-scale geometry: the macroscopic domain $\Omega$ and the microscopic domain $Y$ with the internal boundary $\Gamma_N$.
• Figure 2: Solution to the Stokes problem, $B(y) = (B_1(y),B_2(y))$, over domain \ref{['micro_domain1']}. Left: $B_1(y)$. Right: $B_2(y)$.
• Figure 3: Computed values for the entries of the dispersion tensor $D^*$ for different values of $p$ and its interpolated values: The main-diagonal entries (left) and off-diagonal entries (right).
• Figure 4: Comparison of the time evolution of the dispersion tensor $D^*$ and $D^{int}$ at the point $(0.5714, 0.5714)$.
• Figure 5: Concentration profile approximated via scheme $2$ (left) and via scheme $2$ with precomputing (middle). The pointwise difference between these two approximations at $T = 2$ and $M = 50$ (right).

Theorems & Definitions (7)

• Definition 2.1
• Theorem 1: Solvability of Problem $(P)$, cf. raveendran2023strongly
• Definition 3.1
• Theorem 2
• Definition 3.2
• Definition 3.3
• Theorem 3