Bound-preserving and mass-conserving projection BCFD methods for incompressible miscible displacement in porous media
Abstract
In this paper, we propose and analyze two novel structure-preserving projection schemes for the incompressible miscible displacement problem in porous media. These schemes are specifically designed to preserve two essential physical properties--concentration bounds and the mass balance identity, and are accompanied by a rigorous error analysis for the projection approach. Both schemes are second-order accurate in space but differ in temporal accuracy: one is first-order and the other is second-order. Methodologically, both schemes comprise a prediction step for the concentration variable, based on a second-order modified upwind discretization together with an implicit-explicit finite difference approximation, followed by a projection (or correction) step that enforces the desired physical constraints, namely bounds and mass conservation. For the first-order temporal scheme, the velocity and pressure are solved at the first step on staggered spatial grids with explicit treatment of concentration. In contrast, for the second-order scheme, to ensure well-posedness and to suppress non-physical oscillations in the simulation, the velocity and pressure are solved at the final step with the concentration treated implicitly. A rigorous numerical theoretical framework is established for the first-order scheme, encompassing optimal-order error estimates, stability, and proofs of the existence and uniqueness of the numerical solutions. The numerical analysis framework is further extended to the more involved second-order scheme, and the corresponding theoretical results are briefly outlined. Finally, numerical examples, including some 'real' simulations, are presented to verify the theoretical findings and demonstrate the strong performance of the proposed schemes. In particular, the occurrence of viscous fingering is successfully captured and confirmed by the numerical results.
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BibTeX
@misc{wang2026boundpreserving,
title = {{Bound-preserving and mass-conserving projection BCFD methods for incompressible miscible displacement in porous media}},
author = {Xiaoying Wang and Shusen Xie and Hongfei Fu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38219},
eprint = {2610.38219},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38219},
}
RIS
TY - UNPB
TI - Bound-preserving and mass-conserving projection BCFD methods for incompressible miscible displacement in porous media
AU - Xiaoying Wang
AU - Shusen Xie
AU - Hongfei Fu
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38219
UR - https://arxiv.org/abs/2610.38219
AB - In this paper, we propose and analyze two novel structure-preserving projection schemes for the incompressible miscible displacement problem in porous media. These schemes are specifically designed to preserve two essential physical properties--concentration bounds and the mass balance identity, and are accompanied by a rigorous error analysis for the projection approach. Both schemes are second-order accurate in space but differ in temporal accuracy: one is first-order and the other is second-order. Methodologically, both schemes comprise a prediction step for the concentration variable, based on a second-order modified upwind discretization together with an implicit-explicit finite difference approximation, followed by a projection (or correction) step that enforces the desired physical constraints, namely bounds and mass conservation. For the first-order temporal scheme, the velocity and pressure are solved at the first step on staggered spatial grids with explicit treatment of concentration. In contrast, for the second-order scheme, to ensure well-posedness and to suppress non-physical oscillations in the simulation, the velocity and pressure are solved at the final step with the concentration treated implicitly. A rigorous numerical theoretical framework is established for the first-order scheme, encompassing optimal-order error estimates, stability, and proofs of the existence and uniqueness of the numerical solutions. The numerical analysis framework is further extended to the more involved second-order scheme, and the corresponding theoretical results are briefly outlined. Finally, numerical examples, including some 'real' simulations, are presented to verify the theoretical findings and demonstrate the strong performance of the proposed schemes. In particular, the occurrence of viscous fingering is successfully captured and confirmed by the numerical results.
ER -