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

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.

Figures (6)

• Figure 1: Comparison of the saturation history at the probe location $x=7.61\ \mathrm{cm}$ between the reference Buckley--Leverett solution and the present FV/MW solver. The agreement is essentially pointwise over the full simulated time interval. In particular, the numerical solution reproduces both the sharp breakthrough jump and the subsequent post-front increase in saturation, confirming that the conservative transport backbone captures the physically correct one-dimensional front propagation while the multiwavelet representation does not degrade the local saturation history.
• Figure 2: Spatial saturation profiles $S_w(x,t)$ at selected pore volumes injected (PVI). Solid lines denote the reference Buckley--Leverett profiles and dashed lines denote the bounded-interval MW-reconstructed numerical solution.
• Figure 3: Snapshot-based error measures between the MW-reconstructed numerical state and the reference Buckley--Leverett profile as functions of pore volumes injected. Shown are the RMSE, the mean absolute error $L^1$, the maximum pointwise error $L^\infty$, and the internal consistency error between the finite-volume state and the MW reconstruction.
• Figure 4: Additional transport diagnostics as functions of pore volumes injected. Left: absolute front-position error between the numerical and reference profiles, computed from a fixed saturation-threshold criterion. Right: mass-balance defect obtained by comparing the observed change in total water content with the time-integrated net boundary flux.
• Figure 5: Temporal evolution of the most active dyadic detail energies $E_\ell(t)$ extracted from the reconstructed saturation state. Here, level $\ell$ denotes a dyadic multiresolution level obtained from repeated coarse--fine splitting of the state, and the plotted quantity is the corresponding detail energy.