A bounded-interval multiwavelet formulation with conservative finite-volume transport for one-dimensional Buckley--Leverett waterflooding

TL;DR

This paper introduces a hybrid finite-volume and multiwavelet method for solving the Buckley--Leverett equation, effectively capturing shocks and providing multiresolution insights, validated against benchmark profiles.

Contribution

It develops a novel hybrid approach combining finite-volume and bounded-interval multiwavelet methods for accurate, hierarchical simulation of one-dimensional Buckley--Leverett waterflooding.

Findings

Excellent agreement with reference profiles in saturation histories and spatial profiles.

The multiwavelet reconstruction accurately tracks the finite-volume state.

The method effectively captures shocks and provides multiresolution analysis.

Abstract

We develop a hybrid conservative finite-volume / bounded-interval multiwavelet formulation for the deterministic one-dimensional Buckley--Leverett equation. Because Buckley--Leverett transport is a nonlinear hyperbolic conservation law with entropy-admissible shocks, the saturation update is performed by a conservative finite-volume scheme with monotone numerical fluxes, while the evolving state is represented and reconstructed in a bounded-interval multiwavelet basis. This strategy preserves the correct shock-compatible transport mechanism and simultaneously provides a hierarchical multiresolution description of the solution. Validation against reference Buckley--Leverett profiles for a Berea benchmark shows excellent agreement in probe saturation histories, spatial profiles, front-location diagnostics, and global error measures. The multiwavelet reconstruction also tracks the internal finite-volume state with essentially exact fidelity. The resulting formulation provides a reliable first step toward more native multiwavelet transport solvers for porous-media flow.